Padrões num Contexto de Formação Inicial de Educadores

Ficha Técnica Título: Padrões: Múltiplas perspectivas e contextos em educação matemática | Patterns: Multiple perspectives and contexts in mathematics education Organização de: Isabel Vale e Ana Barbosa Designer: Nelson Dias Editor: Escola Superior de Educação do Instituto Politécnico de Viana do Castelo - Projecto Padrões Dezembro 2009 Depósito Legal nº 303665/09 ISBN: 978-989-95980-4-1 Impressão: Gráfica Visão

Índice / Contents Nota de abertura

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Preface

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Capítulo 1 - Introdução | Chapter 1 - Introduction Mathematics and patterns in elementary schools: perspectives and classroom experiences of students and teachers Isabel Vale Reflections on pattern in the mathematics curriculum Anthony Orton

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Capítulo 2 - Números e Álgebra | Chapter 2 - Numbers and Algebra Patterns and relationships in the elementary classroom Elizabeth Warren

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Improving mathematics learning in numbers and algebra (IMLNA) – a current project Manuel Joaquim Saraiva

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Exploração de Padrões e Pensamento Algébrico António Borralho, Elsa Barbosa

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Padrões num Contexto de Formação Inicial de Educadores Isabel Cabrita

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Capítulo 3 - Geometria | Chapter 3 - Geometry Pupils’ perception of shape, pattern and transformations Jean Orton

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Wallpaper Patterns in Origami Luís António Teixeira de Oliveira

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Looking for patterns in geometric transformations with pre-service teachers Lina Fonseca

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Capítulo 4 - Visualização | Chapter 4 - Visualization Visuoalphanumeric Mechanisms that support Pattern Generalization Ferdinand D. Rivera

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Exploring generalization with visual patterns: tasks developed with pre-algebra students Ana Barbosa, Isabel Vale, Pedro Palhares

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Visual Pattern Tasks with Elementary Teachers and Students: a Didactical Experience Isabel Vale, Teresa Pimentel

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Capítulo 5 - Painel e Síntese | Chapter 5 - Panel and Synthesis Discussion Panel: Patterns in Teacher Training Rosa Antónia Tomás Ferreira

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Uma agenda para investigação sobre padrões e regularidades no ensino-aprendizagem da Matemática e na formação de professores João Pedro da Ponte

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Nota de abertura

Quando um sistema de símbolos “sem sentido” incorpora padrões que revelam ou representam com precisão diversos fenómenos do mundo, essa revelação ou representação impregna o próprio símbolo com algum grau de significado, na verdade, tal revelação ou representação não é mais nem menos do que o próprio significado. Dependendo da complexidade, subtileza e fiabilidade da revelação, assim surgem diferentes graus de significado.

(Hofstadter, D. 1979)

Serão os seres humanos uma amálgama de matéria que, emaranhada numa determinada forma e num padrão misterioso, encontra um sentido, um significado para a sua existência? Será isso que torna a vida tão fascinantemente bela e atraente e, ao mesmo tempo, tão misteriosa? Será por isso que sentimos uma estranha sensação de prazer ao descobrir um padrão, seja na natureza ou em símbolos abstractos? Estas podem ser questões complexas para filósofos, matemáticos, cientistas, ou simples curiosos. Aqui, neste livro, vamos explorar e investigar o ensino e a aprendizagem da matemática na sala de aula através dos padrões. Os padrões fazem parte de nossa vida. Sempre que olhamos à nossa volta, encontramos padrões. Quando as crianças organizam blocos por cores, elas seguem um padrão. Quando uma criança aprende a contar, ela segue um padrão. Quando uma criança observa que múltiplos de cinco terminam em cinco ou zero está a seguir um padrão. Quando faz uma pavimentação, descobriu outro tipo de padrão. Quando um adolescente estuda funções, da álgebra ao cálculo, está a trabalhar o conceito de padrão, analisando como um número se transforma noutro. Quando um estudante universitário aprende como a simetria numa molécula afecta o seu espectro infravermelho, está a usar padrões. Quando designers criam belas composições, seja um padrão numa parede de uma sala ou num vestido ou quando um cientista segue a propagação de um vírus numa população, estão a identificar padrões. Estas ideias foram o ponto de partida para a realização de um projecto no campo da educação em matemática sobre padrões no ensino básico e na formação de professores. Este livro reúne um conjunto de artigos desenvolvidos no âmbito desse projecto e discutidos durante o encontro temático que se realizou em Maio de 2009. Este encontro foi uma actividade do projecto, mas também uma consequência. Pretendíamos mostrar algum do trabalho realizado, clarificar alguns aspectos, consolidar, discutir e aprofundar outros. O encontro organizou-se em torno de três grandes temas - Números e Álgebra, Geometria e Visualização. Começou-se por uma conferência plenária proferida por investigadores internacionais que têm trabalhado a temática dos padrões, em diferentes contextos, seguida de comunicações satélite relacionadas com o tema. Depois da apresentação do projecto, prosseguiu-se com uma conferência que abordou os padrões numa perspectiva abrangente e finalizou-se com um painel sobre a formação de professores. O encontro encerrou com uma síntese das ideias principais analisadas e discutidas pelo consultor do projecto. Foram também convidados outros investigadores portugueses que têm incluído e desenvolvido, no seu trabalho de investigação, o estudo de padrões, tendo contribuído para uma reflexão conjunta sobre esta temática permitindo clarificar e aprofundar muitos dos aspectos tratados. Finalmente, queremos expressar, publicamente, a todos os participantes, convidados e moderadores o nosso profundo agradecimento pelo entusiasmo e empenho manifestado, sem o qual o encontro não teria sido tão profícuo.

A equipa do projecto Ana Barbosa, António Borralho, Elsa Barbosa, Isabel Cabrita, Isabel Vale, José Portela, Lina Fonseca, Teresa Pimentel

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Preface

When a system of “meaningless” symbols has patterns in it that accurately track, or mirror, various phenomena in the world, then that tracking or mirroring imbues the symbol with some degree of meaning—indeed, such tracking or mirroring is no less and no more then what meaning is. Depending on how complex and subtle and reliable the tracking is, different degrees of meaningfulness arise.

(Hofstadter, D. 1979)

Will be humans an amalgam of matter, tangled in a certain way and in a mysterious pattern, finding a sense, a meaning for its existence? Is that what makes life so fascinating, beautiful and attractive and yet so mysterious? Is that why we feel a strange sense of pleasure to discover a pattern, either in nature or abstract symbols? These may be complex questions for philosophers, mathematicians, scientists, or simply curious. Here, in this book, we will explore and investigate the teaching and learning of mathematics in the classroom through patterns. We know that patterns are part of our daily lives. Whenever we look around us we can find patterns. When children organize blocks by colours, they are following a pattern. When a child learns to count, he or she follows a pattern. When a 10-year-old child observes that multiples of five end in five or zero or make tessellations filling a plane he or she found a pattern. When a teenager learn that mathematics, from algebra to calculus, is all about function, which is the pattern of how one number changes into another, or when a college student learns how symmetry in a molecule affects its infrared spectrum he or she is using patterns. When designers of all kinds create beautiful compositions, being a pattern in a room wall or in a dress or a scientist follows the spread of a virus among a population they are identifying patterns. This book brings together a series of articles developed under the project and discussed during the thematic meeting that was held in May 2009. This meeting was an activity of the project but also a consequence, where the intension was to exhibit some of the research done to clarify some aspects, consolidate, discuss and develop others. The meeting was organized by three major themes: Numbers and algebra, Geometry and Visualization, which began with a plenary lecture given by international researchers who have worked the theme of patterns in different contexts, then followed by concurrent communications on the theme. It started by a conference that addressed the patterns from a comprehensive perspective and a panel on teacher education. The meeting ended with a summary of the main ideas analyzed and discussed by the consultant of the project. We also invited other researchers who have been developing and including in its research, the study of patterns who have contributed to a joint discussion on this issue by allowing clarification and deepen many of the issues addressed. For this publication was decided to follow the structure of the meeting Finally we wish to state publicly to all participants, guests and moderators, our deep appreciation for their enthusiasm and commitment without which the meeting would not be as constructive as, in our view, it was.

The project team Ana Barbosa, António Borralho, Elsa Barbosa, Isabel Cabrita, Isabel Vale, José Portela, Lina Fonseca, Teresa Pimentel

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Mathematics and patterns in elementary schools: perspectives and classroom experiences of students and teachers

Mathematics and patterns in elementary schools: perspectives and classroom experiences of students and teachers Isabel Vale [email protected], School of Education of the Polytechnic Institute Viana do Castelo Resumo Nesta apresentação dá-se uma panorâmica geral das principais linhas de investigação do projecto em curso que estamos envolvidos e que estuda a importância dos padrões e a sua relevância no ensino e aprendizagem da matemática de alunos e professores do ensino básico, assim como no desenvolvimento curricular.1 Palavras-Chave: padrões, resolução de problemas, generalização, visualização, pensamento algébrico, ensino e aprendizagem, formação de professores. Abstract This presentation intends to introduce a general overview of the main lines of our work in this ongoing project1 about patterns and its relevance in elementary mathematics education, both teaching and learning of mathematics of students and teachers and school mathematics curricular development. Key words: patterns, problem solving, generalization, visualization, algebraic thinking, teaching and learning, teacher training, curriculum development.

Introduction Our interest in patterns began a long time ago when we were studying Polya and we were personally involved in a research project on problem solving, and more recently the work of Keith Devlin (1999) reinforces our interest. And when we decided to go ahead with this project, the book of Anthony Orton (1999) was very important to increase our motivation and knowledge, contributing to the design of our conceptual framework. It is our belief that mathematics as the science of patterns can contribute for a new view of mathematics and provide significant learning contexts where we can develop the students’ mathematical power. The study of patterns in elementary curricula development of school mathematics in Portugal is only now being considered (ME-DGIDC, 2007) so there is yet insufficient research on the theme.

The team This research and development project has a team that is mainly based at the School of Education of the Polytechnic Institute of Viana do Castelo in the north of Portugal (Isabel Vale, Ana Barbosa, José Portela, Lina Fonseca and Teresa Pimentel) and two more institutions, one in the center, University of Aveiro (Isabel Cabrita) and the other in the south, University of Évora (António Borralho, Elsa Barbosa). We have also João Pedro Ponte, of the University of Lisbon, as consultant of the project.

The project Several national and international studies revealed serious gaps of Portuguese students concerning reasoning, problem solving and communication skills. The goal that all the students learn mathematics by themselves can be reached through a curricular proposal where tasks are defined as a support for a significant learning. In a constructivist perspecCapítulo 1 - Introdução | Chapter 1 - Introduction

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Isabel Vale

tive, the investigative tasks, based on the exploration of patterns, in diversified contexts, increase the development of mathematical skills (Orton, 1999; Devlin, 2003). In this context, the teacher has an important role in the classroom, so we must provide changes in teacher training programs in order to give teachers new ways of organizing their teaching practices, giving them the necessary mathematical competence. In particular, we intend to analyze the impact of an intervention centered on the study of patterns by preschool and primary students (K-9) in the learning of basic concepts, both numerical and geometric, and in the development of higher order thinking skills. Also we intend to analyze the development of didactic knowledge directly connected with pattern tasks and its application in the classroom, teaching practice and curriculum management and to contribute for a positive attitude towards mathematics of teachers and students. This project contains four fundamental components: (1) theoretical component, involving bibliography research, analysis and discussion of the existing literature (both national and foreign) about teacher training and mathematics learning, especially that concerning the study of patterns, with the aim of developing a theoretical framework that stands for the clarification of concepts and for the development of curricular materials; (2) empirical component, mainly based on qualitative and interpretative approaches, organized in case studies using interviews to both students and teachers and classroom observations. We preview the development of partial research projects involving post graduation students; (3) production component, where materials will be produced for application in concrete situations in classrooms of the K-9 system - developing numerical, algebraic and geometric competences; where teacher training modules will be produced and applied in training of preschool educators and elementary teachers; and (4) the results will be communicated through national and international symposiums and/or conferences.

Theoretical framework Many researchers have recognized for many years the study of patterns as setting up the very essence of mathematics and, more recently, as a fundamental part of the curriculum and the teaching of school mathematics (Davis & Hersh, 1981; Devlin, 1998; Polya, 1945; NCTM, 2000, Orton, 1999; Sawyer, 1955; Steen, 1988). More accurate are FerriniMundy, Lappan and Phillips (1997) when they refer that mathematical knowledge can be developed through the study of problems involving patterns and algebra arises as a way of generalizing and representing that knowledge. The depth and variety of connections that patterns allow with all the topics of mathematics leads to the consideration of this theme as crossing all school mathematics curriculum, both to prepare students for further learning and in developing the skills of problem solving and communication. Numerical patterns, relations between variables and generalization are currently considered important components of mathematics curricula in many countries. According to these ideas many researchers argue that patterns can be used to develop and deepen the basic concepts in number theory, prealgebra, algebra, geometry, probability and functions (Arcavi, 2006). Another feature that we gave special attention was visual/figurative contexts. The role of visual problem solving in learning mathematics, although recognized as important and subject of study by many researchers (eg. Arcavi, 2003, 8

Mathematics and patterns in elementary schools: perspectives and classroom experiences of students and teachers

Eisenberg & Dreyfus, 1986; English, 2004; Presmeg, 1986; Stylianides & Silver, 2004) is still an open question (Dreyfus 1991; Presmeg 2006) and doesn’t have a crucial role in school mathematics curricula. Rivera and Becker (2005) state that children and young adults are known to have a strong visual understanding of mathematical ideas and concepts. “Seeing” is an important component of generalization that young students should explore. So education must offer challenging tasks that emphasize the widespread understanding through its numerical and figurative aspects, exploiting this innate ability of students to think visually. Seeing a pattern is necessarily a first step in pattern exploration (Lee & Freiman, 2006). The visual approaches can enhance the discovery of different ways to represent a pattern, depending on how the students see it. So we must foster this skill of students and teachers as well. Despite the significant role of patterns in mathematics, it hasn’t been given great prominence to this subject in the curricula of mathematics education. Just presently this aspect is been considered in the development of mathematics education curriculum (ME-DGIDC, 2007). It is this connection of patterns with mathematics that we want to see reflected in the work we do with students to learn and teachers to teach mathematics. Nowadays, widespread tendencies in mathematical education suggest that effective learning requires that students be active and reflexive when they are involved in significant and diversified activities. This idea follows a way of thinking that has been appearing in the last decades, that doesn’t consider knowledge as given, established and transmissible, but where higher order and the critical thinking skills are privileged, where lectures are substituted by dialogue and discovery methods. Within this perspective problem solving tasks are powerful tools for teachers to use in their classroom. In particular patterns challenge students to use higher order thinking skills and emphasize exploration, investigation, conjecture and generalization. In this context, the teacher has an important role in the classroom while mediator between students and knowledge. They should create, select or adapt appropriate materials for students mathematical activity that allow them to conjecture, to investigate and to communicate, in order to contribute for an effective and significant learning of mathematical concepts. But in order for this to became a reality in our schools we must propose teachers, preservice and inservice, learning contexts identical to those that we want them to develop with their own students, because teachers tend to teach in the same way they were taught. Otherwise we must develop professional skills that keep improving their special expertise to be able to reach all students (e.g. Biehler, 1994; Lester, 1997). We believe that by solving problem with patterns, students can experience the power and utility of mathematics and also develop their knowledge about new concepts; and teachers can find interesting and diversified contexts for developing the mathematical power of students. At the same time they can contribute for a more positive view of mathematics. So, instructional mathematics programs should enable teachers and students, from pre-kindergarten through grade 12, to engage in several tasks involving the understanding of problems, patterns, relations and functions (NCTM, 2000). The tasks that teachers select for their classes are crucial to characterize its work and the way to face and deal with the mathematics curriculum. The tasks to work with students may be quite diversified in nature, context, subject, connections, cognitive processes and so on, since readings and exercises to problems, constructions, games, applications, Capítulo 1 - Introdução | Chapter 1 - Introduction

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investigations and projects (Smith & Stein, 1998; NCTM, 1991). The careful choice of tasks is necessary but not sufficient. The questioning that the teacher has to use in the classroom to explore the tasks and especially the nature of the questioning and inquiring is crucial to student learning. This comes only properly if the teacher has a sound knowledge of the matter to teach, how to teach and when to teach. Thus it was given special importance to the role of tasks in supporting teaching and learning in this project because as Doyle (1988) says they are the foundation of all learning. Figure 1 summarizes the ideas behind, our perspective on patterns through this project. Patterns are reached, within a constructive perspective, through problem solving and with strong focus on visualization where generalization is one of the main processes leading to the growth of mathematics not only to make mathematics larger but it also helps to tie the subjects together: this can only be reached through adequate tasks in classroom environment.

Fig. 1 Main ideas behind this project

Our Work As teacher’s educators in the field of the didactics of mathematics we are concerned with its two main dimensions, referred by Niss (1999): the descriptive/explanatory and the normative. We intend to identify and understand the phenomena and processes that are involved in the teaching and learning of patterns, mainly in the elementary levels, as well as in teacher education. On the other hand, we intend to identify curricular opportunities, different teaching approaches, learning environments, and construct different teaching and learning materials and an instructional sequence. In the beginning, during our theoretical phase, we were concerned with the definition of pattern so we began by discussing this concept and found that it was a hard task. Literature shows us that there are different terms with same 10

Mathematics and patterns in elementary schools: perspectives and classroom experiences of students and teachers

sense of pattern and as a result various investigators report that has not yet managed to find a satisfactory definition for mathematical pattern (e.g. Orton, 1999; Smith, 2003). However, apparently, when we work with patterns, everyone seems to understand what we are talking about. The multifaceted nature of the concept of pattern, as well as its many uses, can be characterized and represented in different ways, which makes its description complex. The different terms used in literature associated to the idea, concept and construct of pattern are summarized in the diagram of Figure 2.

Fig. 2 Terms related with pattern

Indeed we use the term pattern in mathematics when we want to look for order or structure and therefore regularity, repetition and symmetry are often present (Frobisher et al., 2007). We believe as Sawyer (1955) that a pattern is to be understood in a very wide sense, to cover almost any kind of regularity that can be recognized by the mind. But Davis e Hersh (1981) can helps us clarify the meaning of pattern when they introduce besides the idea of regularity, invariance as a fundamental idea of a pattern. During our study, algebra and algebraic thinking had become a strong line of research of our work as well visualization. In particular great part of our work was to develop a didactical sequence that constituted a teaching and learning trajectory in the sense that its aim was to promote a meaningful pattern approach to algebraic thinking through figural pattern tasks and a more meaningful comprehension of elementary mathematical concepts, both teacher to teach and students to learn. We had worked with preservice and inservice teachers and students of K-9 grades. This experience is reported in a booklet (Vale et al., 2009). The framework we used to construct our didactical experience was a dynamic cycle adapted from Nathan and Koellner (2007). This framework helps us, in particular, to understand and cultivate the transition from arithmetic to algebraic reasoning. We began to explore various kinds of pattern tasks with different populations that we created and/or adapted from literature and then explored and tested with teachers (preservice and inservice) and students, having the data collected through observations/interviews. These tasks are developed in a coherent way to relate the different concepts develop by each tasks in the different steps in order to consolidate students learning (Simon & Tzur, 2004). We develCapítulo 1 - Introdução | Chapter 1 - Introduction

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oped, applied, refined and applied, reflected, and refined the times that were necessary to have an adequate approach to the didactical sequence through the dynamic cycle in the Figure 3.

Fig. 3 Dynamic cycle for the didactical sequence

But we also studied another type of patterns those that we call the geometric patterns. These patterns are those that are related with geometric transformations, involved some sort of regularities related with several mathematics ideas such as shape recognition, congruence, similarity and tessellations. As we said before algebra has strong connections with patterns through its generalization patterns (numerical and visual contexts), but we must develop in students the ability to generalize not only with numbers but also including shape and space (Frosbisher et al., 2007). This area is the actual focus of our project. However we have other interests spread over different subjects and areas as are summarized in the Figure 4. All the members of this project have their own interests that are expressed in this book some of them are academic research.

Fig. 4 Themes and areas of this project

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Mathematics and patterns in elementary schools: perspectives and classroom experiences of students and teachers

Conclusion Patterns are a fascinating field with limitless potential for the study of mathematics in elementary education: cross all curriculum subjects and develop different skills such as mathematical problem solving, reasoning and communication in different contexts and using different representations. This work will only be enhanced if teachers have appropriate didactical materials and are motivated and prepared for their profession, so that they can develop the students’ math skills that we expected all students acquired. We do not understand patterns as the panacea for all problems in mathematics but so far we have empirical results that are indicative of their potential on mathematics learning and teaching in elementary grades.

References Arcavi, A. (2006). El desarrolo y el uso del sentido de los números. Em Vale, I. et al. (org.), Números e álgebra (pp.29-48). Lisboa: SPCE. Arcavi, A. (2003). The role of visual representations in the learning of mathematics. Educational Studies in Mathematics, 52, 215-241. Biehler, R. (1994). Teacher education and research on teaching. Em R. Biehler, R. Scholz, R. Sträber e B. Winkelmann (Eds.), Didactics of mathematics as a scientific discipline (pp. 55-60). Dordrecht: Kluwer Academic Pubishers. Davis, P. & Hersh, R. (1995). A experiência matemática. Lisboa: Gradiva. Devlin, K. (1999). Mathematics: the science of patterns. New York: Henry Holt and Company. Doyle, W. (1988). Work in mathematics classes: The context of students’ thinking during instruction. Educational Psycologist, 23, 167-80. Dreyfus, T. (1991). On the status of visual reasoning reasoning in mathematics and mathematics education. In F. Furinghetti (Ed.), Proceedings of the 15th Conference of the International Group for the Psychology of Mathematics Education, 1, 33-48 Eisenberg, T. & Dreyfus, T. (1989). Spatial visualization in the Mathematics curriculum. Focus on learning problems in Mathematics, 11, 1. English, L. & Warren, E. (1998). Introducing the variable through pattern exploration, Mathematics Teacher, 91(2), 166-170. English, L. D. (2004). Promoting the development of young children’s mathematical and analogical reasoning. In L. English (Ed.), Mathematical and analogical reasoning of young learners (pp. 210–215). Mahwah, USA: Lawrence Erlbaum. Ferrini-Mundy, J., Lappan, G. & Phillips, E. (1997). Experiences with Patterning, Teaching Children Mathematics, 3, 282-289. Frobisher, L., Frobisher, A., Orton, A. & Orton, J. (2007). Learning to teach shape and space. Cheltenham: Nelson Thornes. Lee, L. & Freiman, V. (2006). Developing algebraic thinking through pattern exploration, Mathematics Teaching in the Middle School, 11(9),428-433. Lester, F. (1997). Mathematics teacher education at Indian University: twenty-five years of innovative practice. Em D. Fernandes, F. Lester, A. Borralho e I. Vale (Coords.), Resolução de problemas na formação inicial de professores de matemática: múltiplos contextos e perspectivas (pp. 189-216).Aveiro: GIRP. Mason, J. (1996), Expressing Generality and Roots of Algebra. In N. Bednarz, C. Kieran and L. Lee (Eds.), Approaches to Algebra, Perspectives for Research and Teaching (pp. 65-86). Dordrecht: Kluwer Academic Publishers. Mason, J., Burton, L. & Stacey, K. (1985). Thinking Mathematically. Bristol: Adiisson-Wesley. ME-DGIDC (2007). Programa de Matemática do Ensino Básico. Lisboa: Ministério da Educação, Departamento de Educação Básica. NCTM (2000). Principles and standards for school mathematics. Reston: NCTM.

Capítulo 1 - Introdução | Chapter 1 - Introduction

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Niss, M. (1999). Aspects of the nature and state of research in mathematics education. Educational Studies in Mathematics, 40, 1–24 Orton, A. (1999) (Ed.). Pattern in Teaching and Learning of Mathematics. London: Cassel Polya, G. (1945). How to solve it: a New Aspect of Mathematical Method. Princeton, NJ: Princeton University Press. Presmeg, N. (2006). Research on visualization in learning and teaching mathematics: Emergence from psychology. In A. Gutiérrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education (pp. 205–235). Dordrecht: Sense Publishers. Presmeg. N. (1986). Visualisation in high school mathematics. For the Learning of Mathematics, 6(3), 42-46. Rivera, F. & Becker, J.R. (2005). Figural and numerical modes of generalizing in algebra. Mathematics Teaching in the Middle School 11(4), 198-2-3. Sawyer, W. (1955). Prelude to mathematics. Baltimore:Penguim Books Nathan, M. & Koellner, K. (2007). A Framework for Understanding and Cultivating the Transition from Arithmetic to Algebraic Reasoning, Mathematical Thinking and learning, 9(3), 179-192. Smith, E. (2003). Stasis and change: Integrating patterns functions and algebra throughout the K-12 curriculum. Reston:NCTM Simon, M. & Tzur, R. (2004). Explicating the role of mathematical tasks in conceptual learning: an elaboration of the hypothetical learning trajectory. Mathematics Thinking and Learning, 682, 91-104. Smith, M. & Stein, M. (1998). Selecting and creating mathematical tasks: From research to practice. Mathematics Teaching in the Middle School, 3, 344–350. Steen, L. (1988). The Science of Patterns, Science, 240, 611-616. Stylianides, G. J. & Silver, A. (2004). Reasoning and proving in school mathematics curricula: An analytic framework for investigating the opportunities offered to students. In D. McDougall & J. Ross (Eds.), Proceedings of the 26th Annual Meeting of the North American Chapter of the International Group for the PME (Vol. 2, pp. 611-619). Toronto, Canada: OISE/UT. Vale, I., Barbosa, A., Borralho, A., Barbosa, E., Cabrita, I., Fonseca, L. & Pimentel, T. (2009). Padrões no ensino e aprendizagem da matemática: propostas curriculares para o ensino básico. Viana do Castelo: ESEVC.

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Reflections on pattern in the mathematics curriculum

Reflections on pattern in the mathematics curriculum Anthony Orton [email protected], University of Leeds (retired)

RESUMO A maior parte do trabalho desenvolvido pelo Leeds Pattern Group já foi publicado (Orton, 1999). Esta apresentação inclui o trabalho não publicado nesse livro, bem como uma primeira análise e classificação do conceito de padrão e uma contextualização do envolvimento das escolas em vários projectos. Contempla ainda uma referência a futuras publicações referentes aos padrões em outras áreas do currículo de matemática como a geometria e o cálculo. São consideradas questões relacionadas com o método da diferença e as relações entre figuras e padrões numéricos. Por fim, é ainda desenvolvido e ilustrado o potencial contributo dos padrões geométricos, tais como as pavimentações, na aprendizagem da geometria e da álgebra. Palavras-chave: matemática, educação, padrão, geometria, cálculo.

ABSTRACT The major part of the work of the Leeds Pattern Group has already been published (Orton, 1999). This presentation includes work which was omitted from that book, such as an early analysis and classification of pattern and an account of the involvement of schools in the various projects. It also includes reference to further publications relating to pattern in other parts of the mathematics curriculum such as geometry and calculus. Issues relating to differencing and to the relationships between pictures and number patterns are considered. The potential contribution of geometrical patterns such as tessellations in learning both geometry and algebra are developed and illustrated. Key words: mathematics, education, pattern, geometry, calculus.

The Pattern in Mathematics Research Group It is ten years since the publication of the outcomes from the Pattern in Mathematics Research Group at the University of Leeds (Orton, 1999), and my retirement from full-time employment took place around the same time. This research group was then disbanded, but there are nevertheless a number of aspects of pattern in mathematics which I can present to this meeting. One is some of the work of the Leeds Pattern Group (it saves space to refer to it in this abbreviated way) which was omitted from the book, including an early analysis and classification of pattern in mathematics learning and an account of the involvement of schools in the various projects. Another involves results and reflections from my own personal research into using pattern in teaching mathematics.

Pattern in mathematics It was inevitable in the early days of developing a group project in 1992, that we would start asking ourselves fundamental questions. Sawyer (1955) had claimed that ‘mathematics is the classification and study of all possible patterns’, and this view had subsequently been accepted and repeated by many other writers. The first question for us was, therefore, is this true? This was and still is a very relevant question, because if it is true then the whole of mathematics learning should arguably be based around studying patterns. The curriculum at all levels should be completely reorganised to acknowledge this fact. Since this is clearly not happening in classrooms in Britain it is relevant to enquire why not? Is it not true after all? Or is it that school mathematics teachers do not accept the claim? Or is it that the rest Capítulo 1 - Introdução | Chapter 1 - Introduction

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of society, perhaps led by political opinion, wouldn’t accept a curriculum totally based on this claim, or that they have other priorities for the curriculum? Many other questions also came to mind, for example what is pattern in mathematics, and what kinds of patterns can lead to mathematics? And then, in what parts of the curriculum is the use of pattern most desirable, or even essential. In what other parts of the curriculum is it an option? One of my perennial interests throughout my working life has been in trying to use pattern to assist students to understand and learn mathematics. This is a reasonable objective, and it is possible to pursue it with some success even within the constraints of a statutory curriculum. At this point, and partly because I wish to refer to the same example later, I would like to insert a simple illustration relating to what is pattern. Figure 1 shows a sequence of figures. Is each individual figure in this sequence a pattern? If not, is the potentially never-ending sequence of shapes a pattern? And if not, is the corresponding number sequence, 1, 3, 6, 10, 15, .. (the numbers of chords) a pattern? How do you justify your opinions?

Fig. 1 Where is the pattern?

Next, we formulated questions about learning. How do children perceive pattern, for example? Do all children see the same patterns in the same ways? Do children perceive the same patterns that their teachers perceive? Can children be taught to perceive pattern? Are there skills associated with pattern recognition which must be learned, and if so can all children learn them? And then, are patterns in the world or are they in the mind of the person? Not surprisingly, we never achieved anywhere near complete answers to these questions and to the many others which we asked. We decided it might be useful to draw all these questions together into a structured framework which we thought might guide us and others in pattern research activities. The problem which emerged over time was that we were never sure that this framework provided any useful guidance at all. Our research projects often seemed to proceed without any reference to the framework. Consequently, we did not publish it in the book. Yet at the same time, it is possible that others may think it has some value as a reference document, and I have come to believe it should not be hidden away any longer. I am therefore presenting it as an Appendix to this paper. You will see from this that our analysis led us to try to classify our questions into four different domains of knowledge: mathematical, philosophical, psychological and educational. It is important to accept that this was our list, and that yours might be different.

Working in and with schools Involvement with local schools was essential to all of the separate studies within the overall research project, for two obvious reasons. Firstly, we needed to be able to teach, test, and generally carry out research with children. Secondly, we thought we needed to carry a message to the schools, namely that pattern is a fundamental dimension or strand of 16

Reflections on pattern in the mathematics curriculum

the mathematics curriculum. In this second context we thought we should provide ideas for teachers to use with their classes. When we wrote to selected local schools, more than fifty local teachers expressed an interest in being involved in the Leeds Pattern Project in some way. The majority worked in primary schools (up to age 11 years), but we did have enough interest from teachers in secondary schools (from 11 years upwards). In retrospect, I don’t think we needed to carry any strong message to these teachers; they all wanted to be involved because they were already convinced of the importance of pattern in mathematics. That was why they had volunteered to become involved. We held meetings about twice a year with these associated teachers, eventually discussing and exchanging ideas in smaller subgroups, each such subgroup concerned with one particular project. Another aspect of our association with schools was that we produced a Newsletter about twice each year. We used the front page of this to keep the teachers informed about progress on all of the projects, to inform them of meeting dates, and occasionally to provide teaching ideas. Overleaf, there was generally a collection of informal pattern questions or activities for use in classrooms. The first of these collections, distributed in December 1992, was totally devoted to activities with a Christmas theme. Subsequent back pages each followed just one particular theme, one on patterns from Bhutan where one of our team of tutors had recently worked, another on Rangaveli patterns, and yet another included a selection of bead patterns. In hindsight, and if we had been able to find more time for this aspect of our link with local schools, I believe we should have addressed particular topics of our National Curriculum and provided ideas as to how a pattern approach might contribute to learning these topics. That would have been a much more demanding task, which is perhaps why we never got around to it. However, I am pleased to say that we have now tried to address this issue in two recent books for primary school teachers – more information on these later. After a few years this regular contact with local teachers and schools had run its course, and we had more than enough data to work on.

Teaching ideas for schools In our early work with the schools it occurred to us that, with our number curriculum including specific reference to number patterns, children needed to be discouraged from mistakenly deducing that the first few terms of a number pattern automatically determine the complete sequence. We therefore suggested an activity in an early Newsletter, which I refer to as TASK 1, though we called it ‘What Comes Next?’ for the children. For me, the critical question in this task was the final one, and all the other questions merely paved the way for it. Unfortunately, being fully occupied with so many different research studies, we did not find the time to build a research project around this task, and it always remained just a suggested school activity. We did collect some research data concerning children’s responses, but we did not collect it systematically and have not been able to use what we obtained. I therefore do not even have a record of the variety of children’s responses, never mind how the responses changed according to the age of the child. I do not know whether there are responses which children frequently select and others that they do not perceive. I do not know to what extent children might be dependent on differencing in such a context. In fact, there is even no way of knowing whether it was a worthwhile question to ask. There may therefore still be a potential research project here. What I do Capítulo 1 - Introdução | Chapter 1 - Introduction

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know, however, is that there are countless ways of continuing this number pattern, and that children are sometimes able to find a number of ways of continuing the sequence. Subsequent personal research on my part reveals the final question to be a totally fascinating one which can occupy considerable time and can expose many startling new patterns. I commend it for your attention. What Comes Next? All of these number patterns start with the same three numbers: 1, 2, 4, 7, 11, 16, …, 1, 2, 4, 7, 12, 20, …, 1, 2, 4, 8, 15, 26, …, 1, 2, 4, 8, 16, 31, …, 1, 2, 4, 8, 16, 32, …, 1, 2, 4, 9, 19, 36, …, For each number pattern: What number comes next? What would be the tenth term? What would be the twentieth term? Can you say how a number is related to its position in the list? (That is, how is the 5th number related to 5; how is the nth number related to n?) Can you find any more patterns starting 1, 2, 4 ?

Analysing number patterns Differencing is clearly a useful and simple technique in the study of some number patterns, but we know that children are not always able to use the technique wisely and appropriately. There is plenty of published evidence of misguided attempts to use it, for example APU (undated), Stacey (1989), and Orton & Orton (1999). It has been accepted for some years now that children’s responses in dealing with linear patterns may be classified into categories, (see for example page 106 of Orton,1999). It is clear that children need help in understanding what they are doing when they use differencing, otherwise they will continue to misuse it.

1

3 2

6 3

10 4

15 5

.. 6

Table 1 Difference table for Figure 1

Also, when we consider it appropriate, we ought also to assist the children towards an understanding of how the number pattern is linked to the mathematical structure inherent in the task. Here is where I return to Figure 1, and link it to Table 1. The algebraic rule here may be more comprehensively understood not from the number pattern 1, 3, 6, 10, 15, .., but from the number of chords from each point. From Figure 1, we can see that when the nth new point is inserted on the circumference it generates (n–1) new chords. The differences 2, 3, 4, 5, .. , in Table 1 illustrate this. Then, if each point generates (n–1) chords, there must be ½n(n–1) chords altogether, division by 2 being required because otherwise we are counting each chord twice. This formula ½n(n–1) encapsulates the structure. This raises the question of whether there is a distinction between ‘pattern’ and ‘structure’ in mathematics. Do these two words describe the same thing or are they distinct? Or are they different facets of the same thing? 18

Reflections on pattern in the mathematics curriculum

There was a further possible discussion point which arises from TASK 1, and this relates to the practice of using sequences of shapes to generate the corresponding number patterns. Such tasks hardly ever seemed to be set the other way round, namely to present a given number pattern and ask for a drawing of an illustrative sequence of shapes. On reflection, we decided this was potentially an extremely difficult task, and that this was presumably why we couldn’t find many references to it in the literature. Can you draw a simple sequence of shapes which would give rise to the number pattern 1, 2, 4, 10, 23, 46, ..? The fact that going from number pattern to geometrical representation is difficult is surely relevant to the debate about the value of using patterns of shapes to generate number patterns. We need to be clear in our minds why we want to use pictures or diagrams if our objective is the number sequence.

Subsequent developments We have recently been involved in writing mathematics handbooks for primary school teachers. This activity forced us to try to answer some of the many questions which we had asked when the Leeds Pattern Project was initiated and which are listed in the Appendix. In Frobisher et al. (1999 and 2007, respectively) we tried to explain or define selected terminology about number patterns (Table 2), and geometrical patterns (Table 3). How do you react to these definitions and explanations? We spent a very long time trying to perfect them, but can they be bettered? It has been suggested that pattern is all about ‘consistent change’, but is that a concept which is helpful to children and their teachers? What words do you use when you want to help children to understand what we mean by pattern?

Term

Explanation

Pattern

Order, regularity, repetition and symmetry in and between mathematical ‘objects’ such as symbols.

Sequence

A set of mathematical ‘objects’ arranged or ordered according to some rule.

Number pattern

A sequence in which the ‘objects’ are numbers. Number patterns may sometimes lead to spatial patterns.

Repeating pattern

A sequence that repeats itself every given number of terms; the given number is called the ‘period of the pattern’. Table 2 Definitions relating to number patterns

Term

Explanation

Spatial pattern

A regular arrangement of spatial parts such as lines or shapes; an attractive and predictable spatial arrangement. Spatial patterns may sometimes be used to introduce number patterns.

Symmetry

An object or configuration possessing symmetry consists of matching parts which may be interchanged without changing the overall appearance.

Repeating pattern

A pattern involving repetition of component parts; a repeated decorative design.

Tessellation

A repeating pattern of shapes which can be extended infinitely across a surface. Table 3 Definitions relating to geometry patterns

Capítulo 1 - Introdução | Chapter 1 - Introduction

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Anthony Orton

Pattern and geometry

a

b

c

d

e

f

g

h

Fig. 2 Which of these are patterns?

I now turn more specifically to pattern in geometry, and there is so much that I could write that I must be very selective. I’ll begin by putting our definitions of spatial pattern to the test (see Table 3) – and returning to the question: ‘What is pattern in mathematics?’ Are the designs in Figure 2 patterns or not? It seems to me that teachers should present their students with this question from time to time, using different collections of designs each time. The most important issue is surely not to be seeking a definitive yes or no; it is to encourage discussion. If Figure 2h is thought to be a pattern, why is that so? And does it matter if we disagree amongst ourselves? Is it the teacher’s job to persuade students to accept his or her opinion? And are there shapes in Figure 2 about which you just cannot decide? Personally, I find 2b particularly disturbing, and I’m still unsure whether I believe it is a pattern or not. The fact that the three subsections are all equal in area doesn’t help me to decide.

I have started my discussion of pattern in geometry with this collection because I wanted to focus solely on shapes, without any reference to or interference from numbers. Yet the reference to area in connection with 2b shows how difficult it is to avoid measures. Of course, the origins of geometry lie in measurement, so perhaps we should not be surprised to find numbers sneaking back in, even though the intention was purely geometrical. In 2a, the shape may be considered as a smaller sub-shape which is repeated 6 times by rotation, or a slightly larger sub-shape which is rotated into 3 different positions, or yet another sub-shape rotated through two right angles. Also, can we determine the area of the various sub-sections of the shape if, say, the radius is 1 unit? There may be questions of a numerical nature which could be asked about all of the shapes in Figure 2. Clearly, it does not matter that numbers might be involved here; what is important is that shape patterns may be used to promote valuable mathematical thinking.

20

Reflections on pattern in the mathematics curriculum

Fig. 3 Patterns in regular hexagons

Using geometrical patterns to initiate mathematical thinking, including numerical thinking, has always seemed to me to be important. I think I first used the idea in Figure 3 nearly fifty years ago. Here, starting with a tessellation of regular hexagons, lines have been drawn from the centres and vertices so as to create a different design within each hexagon, thus producing a wealth of opportunities for exploration and discovery. Such designs may be created by the children themselves from a lattice of regular hexagons. In my figure the designs become progressively more complex as we move from left to right. Then, the first question might be which of these designs are patterns? Secondly, we see that each hexagon contains other shapes, so what are these shapes? After that, many other questions can follow. What are the angles within all of these shapes? Which are the equal lengths? What are the ratios of the different lengths? What are the areas of the various shapes if the sides of the original hexagons are taken as one unit? This is a good example of how shape patterns may be used to learn about angles, lengths, ratios, areas and geometrical properties.

a d b c b

d

a

Fig. 4 Scalene quadrilaterals

Capítulo 1 - Introdução | Chapter 1 - Introduction

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Anthony Orton

Tessellations often generate opportunities to investigate angles and other measurements. Figure 4 shows a tessellation of scalene quadrilaterals. Many children do not think that scalene shapes will tessellate, until they work with the shapes and can explore the angles. It is interesting first, however, to reflect on why this tessellation might be regarded as a pattern. Do you regard it as a pattern? A single scalene quadrilateral is not a pattern to me, not in the same way as I might regard a single regular hexagon as being a pattern. Neither do two adjacent scalene quadrilaterals form a pattern for me. If we believe the tessellation in Figure 4 is a pattern, how does our opinion accord with our definition of what pattern is? Looking now at how we might use this tessellation, there are four different angles around any vertex, labelled a, b, c and d in Figure 4. Using obvious equal angles in the tessellation we may deduce that the four interior angles of any quadrilateral are also a, b, c and d. Hence the sum of the interior angles of a quadrilateral is equal to the sum of the angles at a point, or one complete rotation. A similar process with a scalene triangle may be used to illustrate the angle sum of a triangle.

(a)

(b)

Fig. 5 Two semi-regular tessellations

Semi-regular tessellations may also be used to explore properties of angle, length and area, for example the two in Figure 5. What are the geometrical properties which make it possible to fit polygons together in these ways? What mathematics can the children discover in these tessellations?

(a) Fig. 6 Converting a tessellation of squares and triangles

(b)

A more exciting prospect, for me anyway, emerges when we use a semi-regular tessellation to create new polygons by joining up the centres of all the polygons around each vertex. Figure 6a shows a semi-regular tessellation which you may have difficulty accepting as a pattern. Figure 6b reveals that joining the polygon centres produces a new tessellation of pentagons. Or is it a tessellation of hexagons subdivided into pentagons? What are the interior angles of these pentagons and hexagons? If the sides of the polygons in 6a have unit length, what are the lengths in 6b? What other questions may be asked? 22

Reflections on pattern in the mathematics curriculum

(a)

(b)

Fig. 7 Converting a tessellation of triangles and hexagons

Figure 7a is yet another semi-regular tessellation. The process of joining up the centres around each vertex produces the delightful tessellation of pentagons in Figure 7b. What mathematics might we extract from that?

(b)

(a) Fig. 8 Analysing a famous tessellation

A famous tessellation which I refer to as the ‘hammerhead’ is shown in Figure 8a. This is a very difficult tessellation to draw. In coming to analyse it, you may see swastikas, but an alternative way to understand it is shown in Figure 8b, where we can see that four ‘unit’ shapes fit together to form a square with abutments at each corner. Assuming the lengths of the sides of the ‘unit’ shape are in the ratio 1:2, we may easily calculate the area of the ‘square with abutments’ as 4×4 + 1×4 = 20, and therefore the area of the unit shape as one quarter of that, namely 5. We then realise that the ratio of lengths of sides does not have to be 1:2. Figure 9 shows variations in this tessellation in which the ratios are 1:2.5 and 1:3. The corresponding areas of the unit shapes are clearly ¼(52 + 4) = 7.25 and ¼(62 + 4) = 10, respectively. In this way we may expose a number pattern, shown in Table 4, and the formula for the area may be deduced as A = n2 + 1.

(a)

(b)

Fig. 9 Variations on the hammerhead tessellation

Capítulo 1 - Introdução | Chapter 1 - Introduction

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Anthony Orton

Side ratio (1:r)

Area

1:2

5

1:2.5

7.25

1:3

10

1:3.5

12.25

1:4

17

1:n

n2 + 1

Table 4 Areas of hammerhead tessellations

And yet, this number pattern should be superfluous to us, because there is another way to look at the structure of the situation. We already know that the shape is one quarter of the area of the ‘square with abutments’. And in the general case this is ¼((2n)2 + 4) = n2 + 1. The question is whether the number pattern is superfluous to children. Do they need to go through the procedure of calculating the areas and seeing the number pattern before they can appreciate the underlying structure as summarised in the formula n2 + 1? All I know, from my own experience, is that having devised the table of results, some children can see the formula and its relationship with the diagrams straight away. What I am more certain of is that the table is not the end product, and it is important that children do come to an understanding of how the formula relates to the geometry. Incidentally, there is much more exciting mathematics hidden within this family of tessellations than I have attempted to deal with here (see Orton and Flower, 1989).

(a)

(b)

(c)

(d)

Fig. 10 Tessellations and fractals

A completely different tessellation is based on equilateral triangles. Figure 10a shows an equilateral triangle which is about to have smaller equilateral triangles removed from its sides. Having removed them, we may place them back on the original triangle so as to form the new shape in Figure 10b. This new shape clearly still tessellates. We may repeat the process of cutting out triangles on every side of this new shape in Figure 10b. If we place them back, as before, we produce the new shape in Figure 10c. This new shape once again obviously tessellates, and part of the tessellation is shown in Figure 10d, which also illustrates the next shape in this sequence. I believe that older children will understand that the area of the basic shape at each stage of this endless procedure is as it was at the start. It should also be clear that we have a simple introduction to fractals here. If there is a numerical aspect to this it is in calculating the perimeter at each stage and studying the number pattern thus derived. This is not an easy task, but it is another illustration of how numbers can arise naturally from tessellation activities (see Orton, 1991b).

24

Reflections on pattern in the mathematics curriculum

Pattern in elementary calculus At the age of 16, having recently taken important public examinations, students in England make choices. Some leave school, others stay on to continue their studies, probably aiming for university. Of those who stay, some opt to continue to study mathematics, but many do not. Thus, in the 16 – 18 age range, mathematics is taught to a largely self-selected group of students in each school. This group will almost certainly include all the very best mathematics students in their year group, but it would be a mistake to think of them as sophisticated mathematicians. In my view, there is still a place for an approach to mathematical ideas through pattern. One area of study in which I have written a number of papers in the past is the introduction to calculus. Now I am well aware that there is computer software which will draw tangents to curves, and will provide the numerical value of their gradients. I am also aware of software which allows students to zoom in at a point on a curve, so that the section of curve comes to look more and more like a straight line. Some software will provide the equation of the tangent, some will provide derivatives. But all of this can be too non-participatory for my liking, rather like idly watching television. At some stage students must come to appreciate the kind of analysis here, which relates to the curve y = x2:

How may we prepare them for that, and how might patterns help? Figure 11 shows part of the graph of y = x2, with a fixed point P, and a variable point Q. This is the standard kind of diagram which is found in British text books. The objective is to use the (x, y) coordinates of Q to calculate the gradient of the secant PQ, and to use a variety of positions of Q which gradually get closer and closer to P. Using calculators or computers, or perhaps spreading the work around the class if electronic help is not available, it does not take long to complete the task. For example, using P = (1, 1), and with Q starting at (2, 4), we may obtain values such as those in Table 5. x 2.00 1.80 1.60 1.40 1.20 1.10 1.08 1.06 1.04 1.02 1.01

y 4.0000 3.2400 2.5600 1.9600 1.4400 1.2100 1.1664 1.1236 1.0816 1.0404 1.0201

δx 1.00 0.80 0.60 0.40 0.20 0.10 0.08 0.06 0.04 0.02 0.01

δy 3.0000 2.2400 1.5600 0.9600 0.4400 0.2100 0.1664 0.1236 0.0816 0.0404 0.0201

δy /δx 3.00 2.80 2.60 2.40 2.20 2.10 2.08 2.06 2.04 2.02 2.01

Table 5 Calculating gradients

Fig. 11 Gradients of secants

In this way, the student is expected to deduce that, as Q → P, the secant → tangent and the gradient of that tangent = 2. It is then wise to repeat the procedure from below, so that the students may appreciate that the gradient → 2 Capítulo 1 - Introdução | Chapter 1 - Introduction

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from below as well as from above. Thus, we assume that the gradient of the curve at P is 2. By moving P to (2, 4) and repeating the procedure we deduce that the corresponding gradient at (2, 4) is 4. Using different positions of P, the final outcome is the pattern in Table 6a. x 1 2 3 4

Gradient 2 4 6 8

(a)

x 1 2 3 4

Gradient 3 = 3x1 12 = 3 x 4 27 = 3 x 9 48 = 3 x 16

(b) Table 6 Patterns of gradients

Compared with many patterns which the students will have encountered in earlier years, it is not difficult for them to express this relation algebraically, and they are generally happy to assume that the formula for the gradient at a point on the curve y = x2 is 2x. Similar investigation using y = x3 produces Table 6b, and the algebraic summary of this pattern as 3x2 is still relatively easy. This self-selected group of mathematics students generally needs no further encouragement to deduce an overall formula nxn-1, though other curves such as y = x4 may be studied to provide more evidence if there is time. What is more, when the algebraic analysis is subsequently demonstrated it may be matched exactly to this practical activity. A similar investigation may be used for integration, again making use of calculators and/or computers if available. Using a diagram such as that in Figure 12, we may investigate the area under the curve y = x2 from x = 0 to x = 1. The average of the two final sums 0.285 and 0.385 in Table 7, which is of course equivalent to using the trapezium rule, gives an approximate area of 0.335. x-range

Area of rectangle below

Area of overlapping rectangle

0.0 – 0.1

0.000

0.001

0.1 – 0.2

0.001

0.004

0.2 – 0.3

0.004

0.009

0.3 – 0.4

0.009

0.016

0.4 – 0.5

0.016

0.025

0.5 – 0.6

0.025

0.036

0.6 – 0.7

0.036

0.049

0.7 – 0.8

0.049

0.064

0.8 – 0.9

0.064

0.081

0.9 – 1.0

0.081

0.100

Sum

0.285

0.385

Fig. 12: Approaching integration

Table 7 Calculating the area under a curve

Clearly, the accuracy depends on the number of rectangles into which we subdivide a range. Using a school computer and 5000 rectangles the values which I recently obtained are shown in Table 8. 26

Reflections on pattern in the mathematics curriculum

Range on x-axis

x

Estimated area

Fractional equivalent

0–1

1

0.333333

1/3

0–2

2

2.666666

8/3

0–3

3

9.000000

27/3

0–4

4

21.333333

64/3

Table 8 Calculating areas with a computer

The fractional equivalents in Table 8 allow us to assume that the pattern of areas is summed up by the formula ⅓x3. If we now repeat the whole procedure for y = x3, we find that the pattern of our area estimates for the same x-ranges as in Table 9 suggest the formula ¼x4. Other curves such as y = x4 may also be investigated. The pattern of area formulae then indicates that for y = xn the formula for the area would be xn+1/(n+1). Of course, the entire process may be based on trapeziums rather than rectangles. What of the equivalent algebraic approach to integration? At this point I have to admit that in many British schools integration appears to be introduced as reverse differentiation. It is not part of my brief to argue the case for how integration should be introduced to students, nor to debate the place of the Fundamental Theorem of Calculus in school mathematics. All I am setting out to do here is to show how a pattern approach to calculus is possible, should you think it a good idea for your students. Further details of this approach are in Orton (1986). It is also possible to use pattern as the first introduction to the function of a function rule, to the differentiation of products, and to the differentiation of quotients, but I do not have space for that here (see Orton, 1991a).

References APU (undated), Mathematical Development: A Review of Monitoring in Mathematics 1978 – 1982. Slough: NFER. Frobisher, L., Monaghan, J., Orton, A., Orton, J., Roper, T. and Threlfall, J. (1999) Learning to Teach Number. Cheltenham: Stanley Thornes. Frobisher, L., Frobisher, A., Orton, A. and Orton, J. (2007), Learning to Teach Shape and Space. Cheltenham: Nelson Thornes. Orton, A. (1986), Introducing calculus: an accumulation of teaching ideas? International Journal of Mathematical Education in Science and Technology, 17 (6), 659-68. Orton, A. and Flower, S. M. (1989), Analysis of an ancient tessellation. The Mathematical Gazette, 73 (466). Orton, A. (1991a), Exploring first year calculus. Mathematics in School, 20 (1), 32-3. Orton, A. (1991b), From tessellations to fractals. Mathematics in School, 20 (2), 30-31. Orton, A. (Ed.) (1999), Pattern in the Teaching and Learning of Mathematics. London: Cassell. Orton, A. & Orton , J. (1999), Pattern and the approach to algebra. In A. Orton (ed.) Pattern in the Teaching and Learning of Mathematics. London: Cassell, 104-20. Sawyer, W. W. (1955), Prelude to Mathematics. Harmondsworth: Penguin. Stacey, K. (1989), Finding and using patterns in linear generalizing problems. Educational Studies in Mathematics, 20, 147-64.

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Appendix: The framework produced by the Leeds Pattern

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Patterns and relationships in the elementary classroom

Patterns and relationships in the elementary classroom1 Elizabeth Warren [email protected], Australian Catholic University RESUMO Generalizar padrões é visto como a chave para o desenvolvimento do raciocínio matemático e da compreensão algébrica. Este artigo explora o papel de padrões e relações no desenvolvimento da compreensão matemática dos 3 aos 5 anos. Os resultados são baseados num estudo longitudinal, tendo por base uma amostra de 220 alunos que atravessa os 4 níveis de escolaridade. Os dados foram recolhidos usando múltiplas técnicas, incluindo entrevistas, gravação de aulas e pré e póstestes. Este artigo apresenta uma teoria para uma trajetória de ensino/aprendizagem, delineada com o objectivo de contribuir para a compreensão matemática nos níveis elementares. O principal foco assenta no papel de diferentes representações e discursos em ajudar os alunos a “ver” as relações dentro e através de padrões. ABSTRACT Generalising patterns is seen as a key to developing mathematical thinking and algebraic understanding. This paper explores the role of patterns and relationships in developing mathematical understanding in Years 3 to 5. The results are based on a longitudinal study, following a cohort of 220 students across 4 years of schooling. Data were gathered from multiple sources including interviews, videos of classroom teaching and pre and post tests. This paper presents a theory for a teaching/learning trajectory designed to build mathematical understanding in the elementary school. The particular focus is on the role of different representations and discourse in assisting young students to ‘see’ relationships within and across patterns.

Introduction Generalising is a key element of mathematics and a guiding goal in the mathematics classroom. In fact many researchers have gone as far as to state that something should only be considered mathematical when it is fully generalised and that generalisation is at the heart of mathematics (e.g., Gattegno, 1987; Kaput, 1999), and learning involves the process of generalising from multiple experiences (Lakoff, & Nunez, 2000). The process of mathematical generalisation has been delineated as seeing the general in the particular (Kruteskii, 1976; Mason 1996). It involves students looking across a number of cases and identifying the common structure that underpins each, perceiving commonalities such as patterns and structures, and exploring these relationships (Kaput, 1999). But, there is a duality in this process. Mason (1996) terms this as seeing the general in the particular and the particular in the general. Kruteskii (1976) sees this as the ability to see something that is general and still unknown in what is isolated and particular compared with the ability to see something general and known in what is particular and concrete. This second aspect allows the individual to classify this case as an example of a more general concept, seeing the particular in the general. There is no special program/ curriculum that evokes this thinking. Rather it involves ‘awaking and sharpening’ both students’ and teachers’ sensitivity to the structure of mathematics and ways of exploiting situations that have the potential for algebraic thinking to occur (Mason, 1996), a process that Kaput and Blanton (2001) refer to as algebrafying the curriculum. But generalisation goes beyond just noticing the general in the particular. Kieran (1989) argues that in addition one must be able to express generalisations algebraically. She contended that to think algebraically is more than thinking about the general. It also entails thinking about the general in a way that makes it distinctively algebraic in its form of reason1

This research is funded by a grant from the Australian Research Council (No. LP0348820) Capítulo 2 - Números e Álgebra | Chapter 2 - Numbers and Algebra

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ing as well as its expression. In patterning situations Radford (2006) refers to this as being able to grasp a commonality noticed in some the elements of the sequence and use this to provide an ‘algebraic’ expression for all elements in the sequence. This is similar to Harel’s (2001) contention of two different forms of generalisations (i) results generalisation, developing generalisation from a few examples, and (ii) process generalisation, developing a generality from a few examples and then justifying it in terms that show its applicability to all examples for any number. It is the justification of the expression of the generalisation that seems to be a key component of being able to truly generalise. The ability to generalise represents a critical element of algebraic thinking and in our study geometric patterning situations represented the medium through which young students’ ability to generalise was explored.

The phases The literature identifies various phases through which adolescent students appear to advance as they engage in reaching generalisations from mathematical tasks. In brief these are expressed as first noticing a commonality in some given particular terms, second forming a general concept, third generalising the noticed commonality to all the terms in the sequence, and finally providing a rule that is applicable to all of the terms of the sequence (Radford, 2006). Research has indicated that most adolescent students experience difficulties in generalising patterns with few going beyond making conjectures from a few cases and testing their generalisation against particular examples (Steele & Johanning, 2004). Thus to generalise, students are required to identify the underlying structure from a number of particular cases and express this structure in symbols.

The categories Radford (2001) in his work with adolescent students identified three levels of generalisation related to geometric patterns. The first is factual generalisations, which is generalising the numerical aspects of the pattern thus enabling students to tackle particular cases. It is bound in the numerical schema and students when looking along the pattern, notice common features such as how the next term is related to the previous term.

For the dot pattern representing the odd numbers an example of a factual generalisation is each time you add 2 more dots. Second, contextual generalisations are performed on the conceptual spatio-temporal object. Students while still looking along the pattern have now abandoned the use of numbers in their discussion. For example, they are now looking at the figures position to help them see the pattern rather than simply looking at the pattern itself. For example, in the dot pattern representing odd numbers saying something like If you add the previous position number to the new position number you get the number of dots. Third, symbolic generalisations deal with the algebraic object. Using the dot pattern for odd numbers the symbolic 30

Patterns and relationships in the elementary classroom

generalisation entails expressing the generality as n+n+1. Radford (2001) suggested that symbolising terms in a pattern according to their position constitutes a key conceptual difference from symbolising an unknown in an equation. He termed this as the positioning problem. It entails the identification of the pattern and its link to its position, the suppression of spatial characteristics such as next, on the top, on the bottom, a de-subjectifying of the situation. To understand why students perceive (n+1)+n as different from (n+n)+1 we need to understand the difference in students’ objectification of the expression (see Figure above). It is conjectured that both involve different actions with the first requiring students to focus on the two rows in the odd number pattern and the second requiring them to separate one tile out so that the n+n is the centre of attention. The conjecture is that the symbols (n+1) and n cannot be added as long as the student sees them as pointing to particular objects. The reformulation of the object allows for the addition to meaningfully occur, reiterating the notion that there is a link between the types of generalisation that students construct and the scheme they are forming when engaging with the context. We contest that the type of visual used in these problems is crucial to reaching different generalisations but also the difficulty commonly associated with this process.

The strategies As many research projects have tended to use different contexts and types of problems with students, in order to analyse what each project is reporting we have chosen to take one context to exemplify research findings found in the literature. This approach allows us to examine the commonalities and differences between strategies identified in past research. The particular context we have chosen is the table seating problem.

How many chairs would be needed if three square tables were pushed together? What about four tables? …five? How would you determine the number of chairs needed for any number of tables?”

Using this context Ainley, Bills and Wilson (2003) identified two broad strategies, namely generalising the context (If it is a corner table you need three chairs, if it is a middle table you need two chairs) and generalising the calculation (You times how many tables there are two plus two). By contrast Lannin, Barker and Townsend (2006) first distinguished the strategy according to whether it includes the context or is contextually free. Within each of these there are a further four categories, namely (a) explicit (basing the rule on the information provided and incorporating a counting technique – there are 2 chairs at each table so you times the number of tables by four and add 2 for the ends), (b) whole object (using a proportion of the problem as a unit to build a generalisation of a larger proportion and making adjustments – 6 tables require 14 chairs so 12 tables require 28 chairs but we take 2 off as we have counted the end ones twice), (c) chunking (using the established rule for a smaller number and adding on extra units to reach the required number – For 6 tables we need 14 chairs so for 15 tables we need 14 + 3x2), and (d) recursive (describing the relationship that exists between consecutive terms –You keep adding a new table. You add two extra chairs). These closely mirror the four strategies delineated by Stacey (1989): linear, whole object, difference and counting. Capítulo 2 - Números e Álgebra | Chapter 2 - Numbers and Algebra

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In their work with young adolescents (ranging from 11 years to 15 year olds) Orton and Orton (1999) found that (i) finding the general term becomes progressively more difficult the further the terms are from those given in the question, (ii) more pupils can continue a pattern than can explain it, (iii) number pattern rules are described by a large proportion of pupils in relation to the difference between terms, (iv) generally oral explanations of rules are given by more pupils than can write an explanation, (v) students were more successful on linear patterns than quadratic patterns. Some conclusions were that as students mature so do their performance on aspects of tasks given in the test. Four obstacles to generalisation were recognised. These were students’ arithmetic incompetence, fixation of a recursive approach, temptation to use inappropriate methods such as difference (product or shortcuts), and idiosyncratic methods. There is some contention in the literature concerning formalising mathematics and using symbols. Pririe and Kieran (1994) suggested that formalising mathematics is characterised by a sense that one’s method works for all examples rather than expressing this understanding in algebraic symbols. They suggest that formalising must be based on and grow from personal mathematical structures, unfolding from one’s less formal images, connected between exploring mathematic thinking in informal settings then bridging this thinking to formal generalisations. There has been contention with regard to the types of problems utilised in past research. Researchers have explored young adolescents’ capabilities by using either geometric patterns (e.g., Radford, 2001, 2006,) or problems that are considered to have some ‘real world’ context, for example, people sitting at tables that are continually growing (e.g., Ainley, Bills, & Wilson, 2005), putting lights on an increasing number of Christmas trees (Stacey, 1989) or the number of eyes of an ever increasing number of dogs (Blanton & Kaput, 2004). Some researchers have made the distinction between these problems as being purposeful and non purposeful, with geometric patterns falling in the latter category. For the purpose of this paper we are taking the stance that all of these contexts, while engaging to students are not necessarily purposeful or even related to reality. The debate should not be about the context rather what the problem offers in terms of mathematical conversations that can lead to discussion about the mathematical structure which the problem contains. The literature presents two differing perspectives on the ontology of student learning: the learning trajectory and the learning-teaching trajectory. While both perspectives have many commonalities, the main differences lie in their emphasis on the act of teaching in the learning process, and the prescriptiveness of the resultant curriculum. From the first perspective, learning consists of a series of “natural” developmental progressions identified in empirically-based models of students’ thinking and learning (Clements, 2007). In conjunction with viewing learning as a progression through development hierarchical levels, the learning trajectory sees teaching as the implementation of “a set of instructional tasks designed to engender these mental processes” (Clements, 2007). From this perspective, the act of teaching is secondary to the act of learning. The resultant curriculum consists of diagnostic tests, learning hierarchies and purposely-selected instructional tasks. Fundamental to this perspective is (a) the existence of a large repertoire of empirically-based research evidencing the development of particular concepts, such as number, number sense, and counting, and (b) conducting extensive field tests trialing various instructional tasks. 32

Patterns and relationships in the elementary classroom

In contrast, the learning-teaching trajectory has three interwoven meanings, each of equal importance. These are: a learning trajectory that provides an overview of the learning process of students; a teaching trajectory that describes how teaching can most effectively link up with and stimulate the learning process; and finally, a subject matter outline, indicating which core elements of the mathematical curriculum should be taught (Van den Heuvel-Panhuizen, 2008). It is believed that the learning-teaching trajectory provides a “mental education map” which can help teachers make didactical decisions as they interact with students’ learning and instructional tasks. It serves as a guide at the metalevel. The resultant curriculum tends to be more open and flexible with teachers choosing and adapting activities in order to enhance student learning. Learning is not necessarily seen as a progression through hierarchical steps. It is this second perspective that has greatest resonance with the research presented in this paper. In our research we were particularly interested in the interaction between teaching and learning and young students’ ability to pattern. We focussed on studying the generalisation unfolding in classroom discourse as young students explore growing patterns represented by tiles and matchsticks, and number patterns represented by concrete materials. The focus was on both student learning and teacher actions, including the introduction of specific discourse, resources and representations that assisted students to go beyond the surface and engage in meaningful conversations exemplifying mathematical thinking, and generalisation. It was situated at the juncture between activity of the individuals and the culture of the classroom, the production of text co-jointly created by all participants. The specific questions were: -- How do students and teachers construct the general term in a pattern or sequence? -- What ‘signs’ and representations assist students to ‘notice’ the underlying structure of the pattern? -- What do students attend to as they reach generalisations? and -- What teacher actions and sequence of activities assist students to perceive the commonalities, and expressing this in language and symbolic notation? Underpinning the theoretical stance we have taken is the reason why adolescent students have difficulty with algebra. It is not related to their inability to engage with conversations about algebraic ideas, rather the teaching or curriculum to which the students have been exposed is preventing them from developing mathematical ideas and representations that assist them in meaningful participation in formal algebraic discussions.

How did we go about answering these questions. The methodology adopted for our Early Algebra Thinking Project (EATP) was the conjecture driven approach of Confrey and Lachance (2000). The conjecture consists of two dimensions, mathematical content and pedagogy linked to the content. The design aimed to produce both theoretical analyses and instructional innovations (Cobb, 2000). For example, many of the instructional tasks were generated prior to the teaching phase, but during lessons tasks were modified according to classroom discourse and interactions. New representations were introduced in order to challenge students’ thinking and encourage them to justify their responses. EATP was based on a re-conceptualisation of Capítulo 2 - Números e Álgebra | Chapter 2 - Numbers and Algebra

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content and pedagogy for algebra in the elementary school. It sought to identify the fundamental cognitive steps crucial for an understanding of equivalence and expressions. The participants across the years were a cohort of students, and their teachers, from 5 inner city middle class Queensland schools. During the study the cohort of students progressed from Year 3 to Year 6, with a small pilot study in Year 2. In total 220 -270 students and 40 teachers participated in the study. All schools were following the new “Patterns and Algebra” strand from the new Queensland Years 1-10 Mathematics Syllabus (Queensland Study Authority, 2004). All lessons were taught by one of the researchers (teacher – researcher). During and in between each lesson hypotheses were conceived ‘on the fly’ and were responsive to the teacher-researcher (TR) and the students. During the teaching phases, the second researcher and classroom teacher acted as participant observers, recording field notes of significant events. All lessons were videotaped using two video cameras, one on the teacher and the other on the students, focusing on the students that actively participated in the discussion. The video-tapes were transcribed and worksheets collected. Investigation with regard to the patterning dimension occurred over a three year period, beginning with students in Year 3 (average age 7 years and 6 months) and ending with students in Year 5 (average age 9 years and 6 months). We were particularly interested whtether they could recognise generality, talk about generality, and represent generality in a written format, whether it was a sentence, equation or simply using uncountable numbers. We began our exploration in the Year 3 class by focussing on arithmetic patterns such as the compensation principle and continued in the Year 4 and Year 5 classes using simple geometric growing patterns.

Year 3 participants This initial discussion relied on developing an understanding of addition compensation using unmeasured quantities. The underlying belief behind this decision was that it is sometimes easier to ‘see’ arithmetic structure in the absence of number, as numbers seem to evoke a propensity to compute.

Fig. 1 Strips of paper used to model the compensation principle.

First students were encouraged to name the three strips of paper and express the relationship between the three strips as; (Length) Red Strip + (Length) Green Strip = Yellow Strip, referring to the length of the strips. Then a piece was cut off the red strip and the class was asked: TR: What would I have to do to the green strip so that the length of the green strip plus the length of the red strip is still the same length as the yellow strip?

How much would I have to add to the green strip? Students were then asked to write their own ‘pattern rule’ using their own language. Writing sentences to explain mathematical processes was not a common classroom experience. While all of the students attempted to write their ‘pattern 34

Patterns and relationships in the elementary classroom

rule’, their responses varied in sophistication and understanding. Some of the students’ responses were: -- The yellow strip stays and the other two move. -- One pece is chopt of and you chop another pece of that will fit in the gap. -- A bit gets taken but it comes back in the other one. -- Taking some of but adding some back to the other one -- You Take some away then replace it so they are equal and the same. -- Cut a bit off and find a piece the same size and add it to one of the pieces. -- When you cut of Red and put the same amount on Green it still equals Yellow. Talking about the ’pattern rule’ in a number world appeared more difficult than talking about the rule using the length model. Some simply could not formulate a sentence to describe what was happening. The sophistication of responses varied. Some pattern rules were: -- Patterns going up and down in any direction. -- Make one number lower and make the other bigger. -- You add 22 and 37 – goes up 3 and down 3. -- You decrease and increase and make things the same. -- It is decreased and increased by the same number so it stays the same number. -- You increase one number then decrease the other number by how much you increased the other number. -- Decrease one number and add the same number that you took away and add it to the other number. Given that these students had had limited opportunities and encouragement to communicate their strategies and thinking, their responses were surprising. The sharing of responses did scaffold students into more sophisticated levels of responses. Previous arithmetic experiences impacted on the success of this lesson. Students, who had limited understanding of addition as change, struggled. In one instance this was the whole class. This class exhibited little understanding of how numbers could increase or decrease or in ascertaining by how much numbers had increased or decreased. The difficulties that these students experienced in interpreting concrete representations, linking these representations to mathematical structures, and expressing generalisations in everyday language underpinned many of the pedagogical decisions we made in the Year 4 and Year 5 classrooms. Conjecture 1: Effective models show the underlying pattern structure. Their translation to number patterns is impacted upon by student’s arithmetic competence and mathematical language. Capítulo 2 - Números e Álgebra | Chapter 2 - Numbers and Algebra

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Year 4 Participants Before the commencement of the Year 4 teaching episodes none of the students had had any experience with exploring growing patterns or had participated in discussions where they were asked to recognise and talk about generalisations. The materials used in these lessons were ceramic tiles. We also had an accompanying set of tiles with magnets on the back. These materials served two purposes. They allowed us to model the pattern on the whiteboard for all to see. They also facilitated classroom discussions. Each student could quickly rearrange the tiles as they ‘talked’ through their solutions to various questions, questions such as what does the next step in the growing pattern look like? This supported public discussion and debate about whether particular solutions were correct or not. The use of models to represent both data sets in the pattern Given the difficulties we encountered with the Grade 3 class in the initial activities we ensured that students explored simple geometric growing patterns where the arithmetic relationship between the term and its position in the pattern was simple (e.g., twice the position number). It was conjectured that this assisted in reducing cognitive load and thus allowed the students to focus on the pattern generalisation as opposed to arithmetic computations. Generalisation in geometric growing patterns relies on coordinating two different sets of information, the geometric pattern itself and the position of each successive term in the pattern. The tiles were one colour and the position number was represented on cards with ordinal words, such as, 1st, 2nd, 3rd 4th. As the students constructed the pattern using the tiles they were instructed to place the appropriate position number under each term.

Thus students had two iconic representations of each geometric growing pattern, the ceramic tiles and the positional cards. The physical placement of position cards under each term was believed to assist students to begin to link the two sets of data, the term and its position in the pattern. After completing the first three terms, a card with 10th written on it was placed on the desk. Students were asked to create the pattern for the 10th position, and asked to explain why they believed that their creation for the 10th position was correct. They were then challenged to extend this thinking to larger position numbers, and finally to an unknown position number.

Explicit questions to link the position to the spatial structure of the pattern. As we continued the classroom discussions there was a continual focus on the students’ descriptions of the patterns. As they proffered explanations of what the pattern looked like at various steps, their generalisations were utilised to construct the position under consideration. For example, for the 10th position, the question was asked: What would it look like? Pam: 20 36

Patterns and relationships in the elementary classroom

Pam: Um, cause the 10th pattern is ... 20 TR: Yeah but what does it look like? Pam: 2 rows of 10 - 20 on each side.

This thinking was then extended to larger positions. We believed that this was an important step in which students needed to engage before they described the generalisation in general terms.

Generalising from the pattern in small position numbers to large position numbers. To ensure students were correctly linking the two data sets (the number of tiles and the position) discussion ensued for each new term number and eventually focussed on the uncountable term number, n. TR: What would it look like, the 1000th one? Yes. Bill: 1000 on the bottom and 1000 on the top. TR: 1000 on the top so what I want you to do is write a generalisation, it’s the nth position, what would it look like? Bill: nth on the top and nth on the bottom. TR: Do we mean nth? Think about your tiles? Bill: n tiles on the top and n tiles on the bottom.

Many struggled with the mathematical language needed to describe the pattern in general terms, particularly with the introduction of the formal use of letters of the alphabet to represent ‘any number’. Caleb: It always grows two more (inaudible) letters TR: It always grows two more? Caleb: m, p like that. Caleb: Like with the alphabet Caleb: Its like adding two more letters. TR: n,m, o ok.

Students were also unfamiliar with the usage of word column and rows, and tended to confuse ordinal language with numeric language when it came to describing the pattern for the larger position. This was particularly true when it came to the nth position. These difficulties are further illustrated from the classroom discussion with regard to the second pattern.

1st



2nd

3rd

TR: What about the 1000th? Bill: 1000 on one side and 1001 on the other. Jen: two lots of 1000 and one more on the right. TR: What about, what about, What about the nth one?

Capítulo 2 - Números e Álgebra | Chapter 2 - Numbers and Algebra

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Jan: 1 nth on one side and and nth on the other. Bill: nth and one!

As the above extract indicates, these students were capable of generalising for large uncountable numbers, a capability we refer to as reaching a ‘quasi-generalisation’. They experienced language difficulties when abstracting this thinking to more formal algebraic discussions. Conjecture 2. The coordination of ordinal language with the langue of generality needs be made explicit in classroom discourse about pattern rules. The capability of these young students to reach quasi-generalisations is best illustrated by Alex’s discussion about the following pattern.

Alex: Each side is equal to the turn. TR: What does that mean? Alex: If you have one side of 10 the other is 10. TR: What about the middle tiles (the black tiles). Alex: You add the sides together and take one off. That is the middle. TR: So for the 40th step? Alex: One side would be 40 [white tiles] and the other side would be 40 [white tiles] so you add it together and take one off which is 79. There are 79 tiles in the middle, 79 black tiles.

Using patterns with missing steps For many students recursive thinking continued to persist. We conjectured that this type of thinking was supported by continually giving the students patterns where the first three steps had been completed and then asking them to make the next two steps before constructing a large step. This sequence inadvertently reinforced thinking that looked along the pattern. Thus in the next phase students explored patterns where there were gaps in the initial steps given, for example, the initial pattern consisted of the 1st, 3rd and 6th steps. One of the patterns used was:

1st

2nd

5th

This pattern was specifically chosen because it is difficult to build from 1st term to the 2nd term and thus was believed to assist in discouraging recursive thinking. The students needed to look closely at 2nd and 5th steps, ascertain the 38

Patterns and relationships in the elementary classroom

relationship between the step number and pattern in order to successfully build the 3rd and 4th steps. Students found this to be a difficult task with many building

instead of

.

The students struggled with this pattern. We believe that this was for two reasons. First, the 1st step did not explicitly contain the structure of the pattern. There were no blue tiles and only one yellow tile. Thus it could only be used for confirming general thinking rather than assist in developing general thinking. Second, the pattern generalisation contained subtraction. The number of blue tiles in each term was 2(n-1). Patterns containing the operation of subtraction seem to be more difficult than patterns containing addition and multiplication (Warren, 1996). Conjecture 3. Explicit modelling of relationships between the two data sets and limiting the complexity of the arithmetic relationship between the sets assists in discussing pattern generalisations. In these classrooms many students experienced difficulty in ‘seeing’ the multiplicative structure in the growing patterns. They thus tended to adopt additive thinking when searching for solutions. Students were also encouraged to write their generalisations. Given that this was the first time that they had been asked to not only express generalisations in everyday language, but also write it as a sentence, their first attempts were commendable. Figure 2 shows some of the explanations when asked, If I was at any step number what would the pattern look like?

Fig. 2 Students’ explanations for the pattern

To ascertain the impact that these activities had on young students’ ability to generalise geometric growth patterns a pre and post test were administered before the commencement of the teaching phase and two weeks after the completion of the teaching. The results indicated that there was significant growth in these students’ ability to describe the pattern in general terms, that is, specifically relate the pattern to its position (co-variational thinking) (Warren & Cooper, 2008).

Year 5 Classrooms In this classroom the particular issue that we investigated was the relationship between writing a generalisation in language and writing the generalisation in symbols. The lessons consisted of five main dimensions (a) using concrete materials to represent various geometric growing patterns, (b) translating the pattern into a table to draw out further the functional relationship inherent in the pattern, (c) developing specific language to support the development of the concept of the relationship between pattern position and number of tiles, (d) sharing different ways of describing the generalizations in everyday language and symbols, and (e) encouraging students to justify their generalizations. Given the results of our work with the Year 4 students, the patterns chosen were linear and the representations were arranged so that the links between the visual, position and pattern rule were explicit. The generalization for each lesson were: Lesson 1 2n+1, Lesson 2 2n+2 and 2n+1, Lesson 3 2n+1 and 3n+1, and Lesson 4 3n+1, 2n-1 and 2n+2.

Capítulo 2 - Números e Álgebra | Chapter 2 - Numbers and Algebra

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Teaching actions in the year 5 classroom Using language and actions to relate the visual representation to the table of values. Specific teaching strategies and language were developed to not only encourage students to examine functional relationship in the pattern but also to transfer this understanding to the table of values. As students proffered their generalizations each was classified as a Growing rule or a Position rule. This classification system was also used to explicitly map the two ways of examining the visual pattern (i.e., looking along the visual pattern vs linking the pattern to its position) and the two ways of examining the table (i.e., looking down one column vs looking across the two columns). The following extract exemplifies the types of classroom conversations that ensued during the four lessons: Referring to the tables of values TR: What is a position rule? Cath: A position rule is where the step number has something to do with the pattern. TR: Yes, the step number and the pattern. So you are looking across the table that is where the pattern is. TR: I want you to think how you go from step 2 to get 6, from step 3 to get 8 (drawing arrows across the table linking the two columns and writing position rule)

And in another excerpt when referring to the visual pattern: TR: How would you describe this pattern? John: You just keep adding on 2. TR: Is that a growing rule or a position rule? Is it telling me how the pattern is growing? (drawing an arrow along the pattern and writing growing rule) Sally: Yes. It is telling us how it is growing.

Introducing specific language to assist in describing the visual pattern. Specific language was introduced to assist students to describe the pattern in general terms. It was evident that in all instances it was simpler to say “the pattern is growing by two tiles” than to say, “both rows always have the same number of tiles as the position number”. Hence language such as rows, columns, double and multiply was introduced throughout the lesson sequence. By the completion of the four lessons many students were still experiencing difficulties in expressing generalizations in language, however analysis of the videos indicated a marked improvement in the manner in which they described their generalizations. TR: Tell me what the 4th step looks like. Mary: Four greens and five reds. TR: How do you know that? Mary: The green row is always one more than the step and the red one below is one more than the green. Cath: Whatever the number is you double that number and add 2.

Explicit justification of their descriptions appeared to assist students refine their descriptions. Nick: What I did is, plus 1 to the step number then times it by two. TR: So would it work, let’s see. So for 5 what would I do? (Pointing to the step on the board) Nick: Plus 1 equals 6 times 2, equals 12 . TR: Who has a different one? Jill: Times the step by 2 add 4 and take 2. 40

Patterns and relationships in the elementary classroom

The conversation then focused on Jill justifying her description and ascertaining if it worked for a wide variety of step numbers. As students gave their verbal descriptions they were asked to express it in symbolic form. The following excerpt represents a typical conversation that occurred. TR: If we want to talk about any step how do we describe this? Carol: We call it the n step where n is any number you want it to be. TR: How many green tiles would there be? Carol: Depends on which number it is. Could be two billionth step. There would be Two billion reds and one left over and there would be two billion greens. TR: Excellent. Does anyone have another description? Tom: The n step is anyone with one left over. Henry: There are n green ones and n plus one red ones.

At the completion of the teaching phase a test comprising of three questions was administered. The questions reflected the types of activities that occurred within the lessons. Students were asked to continue each pattern and write the position rule for the pattern first in words and then in symbols. The results indicated that there was no significant difference between the level of response each student proffered for the written description and the symbolic description of the generalization for each of the three Questions. (Warren, 2006). There appears to be a tension between natural language and the impossibility of using it for the construction of symbolic representations. The results of Redden’s (1996) longitudinal study with 26 students (aged 13) indicated that for number patterns, asking students to apply their rule to an uncountable example forced them to rethink the way they expressed the rule in natural language. On the whole, it also appeared that using natural language to describe the generality of number patterns is a necessary prerequisite for the emergence of algebraic notation. Stacey and Macgregor (1995) conjectured that correct verbal descriptions are more likely to lead to correct algebraic rules and students who could find the correct functional relationship could usually articulate this relationship verbally. Kaput (1999) suggested that some students articulate ‘a hybrid of arithmetic symbolization and transliteration schema based on natural language’ (p. 135) and that these articulations do not easily translate to correct algebraic symbols. In fact over-learned natural language can inhibit appropriate algebraic symbolism, (Bloedy-Vinner, 1995). Our research suggests that for young students the relationship between the ability to express a generalisation written and in symbolic form is at the best tenuous. Conjecture 4: The relationship between writing the generalisations in words and writing the generalisation in symbols is tenuous. The second teaching experiment was conducted with two different Grade 5 classes. We were particularly interested in investigating the role of the tables of values. In both classrooms four representations were developed throughout the lesson series, namely, concrete representations of geometric growing patterns and cards indicating the position of each component in the pattern (positional cards), drawings of growing patterns, specific language to assist student description of this action and symbols used to summarise the relationship between each component of the pattern and its position in the pattern. In contrast to Julie’s class (NoTables class), Adam’s class (Tables class) also incorporated a table of values. Capítulo 2 - Números e Álgebra | Chapter 2 - Numbers and Algebra

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Some examples of the types of growing patterns explored within in the lessons are presented in Figure 5. Lesson 1

Lesson 3

Lesson 2

Lesson 4

Fig. 5 Typical growing patterns that were explored in the lessons.

Both classes were typically asked to construct the first five steps and the 10th step of the pattern using concrete materials. Under each step they placed the appropriate cards relating the components of the patterns to the relevant step number. The NoTables class was then challenged to describe the pattern from these visual representations, that is, verbalise the relationship between the components and their position in the pattern. By contrast, the Tables class was instructed to record their values in the t-chart presented on the accompanying worksheet. The table of values consisted of two columns, one headed ‘step number’ and the other headed ‘number of tiles’. Students then focused on investigating the table of values to ascertain the relevant relationship between the step number and the number of tiles. The role that tables of values played in assisting students reach generalizations appeared twofold. First, they did assist the Table students to organize their data, and reach generalizations based on number pattern recognition (Herbert & Brown, 1997). Second, they appeared to restrict the richness of the descriptions offered. Not only were the generalizations not explicitly related to the given visual pattern but also the table appeared to restrict students’ capability to give a variety of differing descriptions for each generalization. The students also exhibited a propensity to search for patterns down the table rather than across the table. This is not at odds with past research. Many of the trends found in these students’ responses mirrored previous research results from large studies with young adolescent and adult students (e.g., Redden, 1996; Warren, 1996). The analysis of the lessons indicated that explicit teaching was required to assist students make this transition. The use of the terms such as ‘growing rules’ and ‘position rules’ and the accompanying discussion and classification of student generated generalizations appeared to begin to provide these students with a language framework for classifying the rules, and in some instances aided them to switch their thinking from single variational to co-variational. The analysis of the videos and researcher’s field notes indicated that reaching generalizations in the NoTable class was a struggle. Explicit teaching focusing on breaking each pattern into its visual components appeared to assist them ‘see’ the generality in the pattern. The public sharing of successful descriptions also appeared to assist other students in the NoTable class engage with these conversations. It should also be noted that the NoTable class experienced greater success in delineating the missing steps in the pattern. Perhaps this physical engagement with the pattern 42

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assisted them to view the pattern in differing ways. Thus it is conjectured that a necessary component of reaching differing generalizations is physically constructing and deconstructing patterns and continually justifying descriptions as valid descriptions of the visuals given. As suggested by Warren (2000), this approach calls upon students to visualize spatially and complete patterns, two dimensions not commonly called on in traditional approaches to algebra, thinking that is currently not privileged within the Australian context. By contrast, once the Table class recorded their data in the table of values the conversation quickly focused on sharing rules. Some of the difficulties that the Table class experienced in reaching a second description of the general rule can be explained in terms of Duval’s theory on representations. Unlike the “NoTable” group, these students were asked to deal with two representations of the problem, the tables of values and the visual representation of the pattern. From the results it appeared that their first description was reached by examining the relationships between the values in the table. Thus once the table of values was generated students appeared to ignore the visual representation of the pattern. This could reflect the fact that working with tables of values is more in harmony with the type of mathematics with which these students commonly engage, relying on their arithmetic knowledge and ability with basic facts rather than their ability to reason spatially and complete patterns. But the second description required them to return to the visual pattern, change registers (Duval, 1999) and search for a general rule in this register, working in a register that appeared to be more cognitively challenging. They were not making links between the parallel representations. By contrast, the NoTable class was only using a single representation in order to find generalizations. These students experienced the same level of success as the Table class and were more successful in delineating a second description, beginning to reach both the goals associated with exploring visual patterns, namely, understanding variables and equivalent expressions. When looking for patterns in tables, the most common approach is to identify connections among consecutive elements in the sequences (Orton & Orton, 1999). This same trend was also exhibited by third grade students (Schliemann, Carraher, & Brizuela, 2001). Once students find the functional relationship within a table of values, they appeared reluctant to return to exploring the visual pattern for further relationships (Warren, 1996). It was conjectured that while tables of values appeared to assist students reach functional relationships between the pattern and its position, they may have restricted them from reaching equivalent functional representations of the pattern (Warren, 1996). Conjecture 5. Different representations privilege different thinking. Tables of values tend to support the process of guess and check and recursive thinking. Visual patterns tend to support the development of an array of generalisations and equivalence.

Discussion and Conclusions This research commences to not only identify teacher actions that support examining growing patterns as functional relationships between the pattern and its position, but also delineates thinking that impacts on this process. Many of the difficulties these students experienced mirror the difficulties found in past research with young adolescents. This suggests that perhaps these difficulties are not so much developmental but experiential, as these early classroom exCapítulo 2 - Números e Álgebra | Chapter 2 - Numbers and Algebra

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periences began to bridge many of these gaps. The results of the NoTable vs Table classes indicated that after the four lessons thirty seven percent of the students could successfully write a rule relating the pattern elements to their position and nineteen percent could successfully write the rule using symbols. These results are comparable with the results from studies conducted with young adolescent students (e.g., Warren, 1996). It appears that algebraic activity can occur at an earlier age than we had ever thought possible and that these experiences with appropriate teacher actions may assist more students join the conversation in their adolescent years. Presently in many instances the transition from arithmetic to algebra is too abrupt, with many young adolescents quickly moving from arithmetic to the introduction of the concept of a variable to symbolic manipulation. Phases of generalising The literature identified three phases associated with generalisation, namely, noticing a commonality, forming a general concept, and providing a rule that is applicable to all cases. Fuji & Stephens (2001) in their extension of the work by Carpenter and Levi (1999) on equations such as 78-49+49=78 suggest that there is an intermediate stage in forming the concept of a variable. They identified this as quasi-variable understanding, that is, recognising that the equation belongs to a group of equations which is true whatever number is taken away and added back. Our research suggests that this idea is extendable to generalisation, to give a notion of quasi-generalisation phase where students are able to express the generalisation in terms of specific numbers (Warren, & Cooper, 2008). We found that quasi-generalisation in the elementary context appears to be a necessary precursor to expressing the generalisation in natural language and algebraic notation. Thus we are suggesting that for young children the phases are: (a) understanding the tasks, (b) forming a generalising example from a few cases, (c) expressing this as a quasi-generalisation and (d) representing this generalisation in symbols. The structure of the quasi-generalisation leads to different symbolic expressions. For example, for the pattern used to express odd numbers, knowing that the 256th step is created by adding two rows, one containing 256 tiles and one containing 256+1 tiles leads to the symbolic notation n+(n+1), whereas the quasigeneralisation of 2x256 + 1 leads to 2xn+1. This agrees with Radford (2006). Categories of generalising Initial exploration of numeric and geometric patterns requires students to explore multi-representations, including everyday language, tiles, pointers, gestures of a pattern, tables of values and the simple use of letters. Radford (2006) refers to one of the most difficult phases with regard to generalising as the ‘the positioning problem’, entailing the identification of the pattern and its link to its position. Our research suggests that exploration of the pattern with signs that ‘point’ to both dimensions of the pattern (the index card and the geometric tiles) and allowing students to explore both by kinaesthetic activity supports this ‘positioning’ process. We would suggest that ‘positional generalisation’ is an additional category, a mapping generalisation, and exists between the contextual generalisation and symbolic generalisation as delineated by Radford. This allows students to not only map the relationship between both signs but also to begin to de-subjectify the spatial features of the pattern by relating the ‘number’ features of the pattern to the patterns position. The use of large position numbers assists this process. 44

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Strategies of generalising Two of the strategies of generalising with adolescent students entail multiplicative thinking, one with an adjustment and one without. These were referred to as ‘chunking’ and ‘whole object’. Our experience in the elementary classroom suggests that these are strategies rarely used by young children. This could reflect that for many multiplicative thinking was only just beginning to be explored in mathematics. But the two main strategies were explicit and recursive. Justification The role of tables of values in this process is questionable. We are suggesting that deleting the actions from the development of an understanding of the generalisation process may result in losing the semiotic power of the representations. The pattern may simply become hieroglyphic - like a mark. Students may no longer see the emergent algebraic expressions as a whole and its connections to the sign system from which it emerged . This results in a tension between shifting the object of knowledge and the signs that created this object and the deprivation of the spatial temporal terms that students utilised in its creation. The resultant process becomes one of simply guessing and checking to find the generalisation, referred to by Harel (2001) as a results generalisation, generalising from a few examples. The more powerful generalisation, process generalisation has the potential of being lost in the translation from the geometric pattern to the tables of values. We conjecture that the ability to deconstruct and reconstruct the spatial aspects of the pattern and relate it to different symbolic expressions is a key component of the justification process. This resonates with Steel and Johanning (2004) who suggested that drawing diagrams are powerful tools to build successful problem solving schemas. What it means to express this algebraically is an important dimension of the generalisation process (Kieran, 1989). We would suggest for young students quasi-generalisation is a substantial step towards thinking algebraically as it entails thinking about the general in a way that makes it distinctively algebraic in its form of reasoning as well as its expression. Finding the formula aspect of generality based on a trial and error heuristic, a process often utilised when finding symbolic formula in tables of values, confines algebraic notation to the status of place holders (Radford, 2006). Rules found in this way are naïve and students are making an induction rather than a generalisation. Searching for commonalities reaches beyond this. It entails recognising certain common features of the terms and generalising them to the terms that follow in the sequence, whereas the former relies on rules formed by guessing Finally, the conversations presented in this paper begin to tease out some of the key issues and learning that needs to occur in the elementary years from an early algebraic thinking perspective. From a patterning perspective these are: (a) understanding that patterns underpin functional thinking and involve the coordination of two pieces of information, (b) recognising the relationship between these two data sets, (c) expressing this relationship in a variety of equivalent ways, and finally (d) writing a pattern rule. Whether this needs to be in word form before attempting symbolic form requires further investigation. The conversations also delineated representations and teacher actions that begin to support student learning. As mentioned earlier, this research is exploratory by nature and as such commences to map the territory. It begins to sketch out a learning-teaching trajectory for early algebra. Much needs to be done to fill in the gaps. However, the findings do indicate that many of the difficulties that the literature has presented with adolescent students Capítulo 2 - Números e Álgebra | Chapter 2 - Numbers and Algebra

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are mirrored in our research in elementary classrooms, suggesting that the ability to reach generalisation may not be as closely aligned with maturity as past research has suggested.

References Ainley, J., Bills, E., & Wilson, K. (2005). Purposeful task design and the emergence of transparency. Paper presented at the Proceedings of the 20th Conference of the International Group for the Psychology of Mathematics Eduction, Melbourne, Australia Blanton, M., and Kaput, J. (2004). Elementary grades students’ capacity for functional thinking. In M. Jonsen Hoines & A. Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, Vol. 2, Oslo, Norway, pp. 135-142. Bloedy-Vinner, H. (1995) Analgebraic interpretation of algebraic expressions: functions or predicates? In L. Meira & D. Carraher (Eds.), Proceedings of PME 19 (Recife), v.2, 42-49. Carpenter, T. P., & Levi, L. (1999, April). Developing conceptions of algebraic reasoning in the primary grades. Paper presented at the annual meeting of the American Educational Research Association, Montreal, Quebec, Canada. Clements, D., (2007). Curriculum research: Towards a framework for “Research based curriculum. Journal for Research in Mathematics Education, 30, 35-70. Cobb, P. (2000). Conducting teaching experiments in collaboration with teachers. In A. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 307-333). Mahwah, NJ: Lawrenece Erlbaum Associates Inc. Confrey, J., & Lachance, A. (2000). Transformative Teaching Experiments through conjecture-driven research design. Chapter 10. In A. Kelly & R. A. Lesh (Eds.), Handbook of Research design in mathematics and science education (pp. 231-265). Mahwah, NJ: Lawrennce Erlbaum Associates, Inc. Duval, R. (1999). Representations, vision and visualization: Cognitive functions in mathematical thinking. Basic issues for learning. In F. Hitt & M. Santos (Eds.), Proceedings of the 21st conference of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol 1, pp. 3-26). Fujii, T., & Stephens, M. (2001). Fostering understanding of algebraic generalisation through numerical expressions: The role of the quasi-variables. In H. Chick, K. Stacey, J.Vincent and J.Vincent (Eds.), The Future of the Teaching and Learning of Algebra. Proceedings of the 12th ICMI study conference (Vol 1,pp. 258-64). Melbourne, Australia. Gattegno, C. (1987). The Science of Education: Part 1 Theoretical Considerations. New York: Educational Solutions. Harel, G. (2001). The development of mathematical induction as a proof scheme: A model for DNR-Based induction. Cited Feb 2007. math.ucsd.edu/harel. Herbert, K., & Brown, R. (1997). Patterns as tools for algebraic reasoning. Teaching Children Mathematics, 340-344. Kaput, J. (1999). Teaching and learning a new algebra. In E. Fennema & T. Romnberg (Eds.), Mathematics classrooms that promote understanding (pp. 133-155). Mahway, NJ: Lawrence Erlbaum. Kaput, J., and Blanton, M. (2001). Algebrafying the elementary mathematics experience. In H. Chick, K. Stacey, J. Vincent and J. Vincent (Eds.), The Future of the Teaching and Learning of Algebra. Proceedings of the 12th ICMI study conference Vol. 1, Melbourne: Australia, pp. 344-352. Kieran, C. (1989). A Perspective on Algebraic Thinking. In G Vernand, J., Rogalski, & M.Artigue (Eds). Proceedings of the 13th International Conference for the Psychology of Mathematics Education (Vol. 2, pp. 163-171). Paris, France Kruteskii, V. A. (1976). The psychology of mathematical abilities in school children. Chigago: University of Chicago Press. Lakoff, G., & Nunez, R. (2000). Where mathematics comes from: The Embodied mind brings mathematics into being. New York: Basic Books. Lannin, J., Barker, D., & Townsend, B. (2006). Algebraic Generalisation Strategies: Factors Influencing Student Strategy Selection. Mathematics Education Research Journal, 18 (3), 3-28. Mason, J. (1996). Expressing generality and roots of algebra. In N. Bednarz, C. Kieran, and L. Lee (Eds.), Approaches to algebra: Perspectives for research and teaching. Utrech: Kluwer Academic Publishers.

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Orton, A., & Orton, J. (1999). Pattern and the approach to algebra. In A. Orton (Ed.), Pattern in the teaching and learning of mathematics (pp. 104-120). London: Cassell. Queensland Study Authority (2004). MathematicsYears 1-10 syllabus. www.qsa.qld.edu.au. Radford, L. (2001). Factual, contextual and symbolic generalisations in algebra. In M. van den Heuvel-Penhuizen (Ed.), Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Eeducation. 1, 81-89. Utrech, Netherlands. Radford, L. (2006). Algebraic Thinking and the Generalization of Patterns: A Semiotic Perspective. Paper presented at the 28th annual meeting of the International Group for the Psychology of Mathematics Education (NA), Merida, Mexico. Redden, T. (1996). “Wouldn’t it be good if we had a symbol to stand for any number”: The relationship between natural language and symbolic notation in pattern description. In L. Puig & A. Gutierrez (Eds.), Proceedings of the 20th International Conference for Psychology of Mathematics Education (Vol. 4, pp. 195-202). Valencia, Spain: Program Committee for PME 20. Schliemann, A. D., Carraher, D., & Brizuela, B. M. (2001). When tables become function tables. In M. van den Heuvel-Penhuizen (Ed.), Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 145152). Utrecht: Drukkerij Wilco. Stacey, K. (1989). Finding and Using Patterns in Linear Generalising Problems. Educational Studies in Mathematics. 20(2), 147-164 Stacey, K., & MacGregor, M. (1995). The effects of different approaches to algebra in students’ perception of functional relationships. Mathematics Education Research Journal, 20, 147-164. Steele, D., & Johanning, D. (2004). A schematics-theoretic view of problem solving and development of algebraic thinking. Educational Studies in Mathematics, 57, 65-90. Van den Heuvel-Panhuizen. M. (Retrieved, 2008). Realistic Mathematics Education as work in progress http://www.fi.uu.nl/publicaties/literatuur/4966.pdf retrieved 7th May 2008. Warren, E. (1996). Interaction between instructional approaches, students’ reasoning processes, and their understanding of elementary algebra. Unpublished dissertation. Queensland University of Technology. Warren, E. (2000). Visualisation and the development of early understanding in algebra. In M. v. d. Heuvel-Penhuizen (Ed.), Proceedings of the 24th conference of the International Group for the Psychology of Mathematics Education, Vol. 4, pp. Hiroshima, Japan, pp.273-280. Warren, E., & Cooper T. J. (2008). Generalising the pattern rule for visual growth patterns: Actions that support 8 year olds’ thinking. Educational Studies in Mathematics, 67(2), 171-185. Warren, E., (2006). Teacher actions that assist young students write generalizations in words and in symbols. Novotna, J., Moraova, H., Kratka, M, & Stehlikova, N. (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Eeducation, 5, 377-384. PME: Prague.

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Improving mathematics learning in numbers and algebra (IMLNA) – a current project

Improving mathematics learning in numbers and algebra (IMLNA) – a current project1 Manuel Joaquim Saraiva [email protected], Universidade da Beira Interior and CIEFCUL RESUMO Este artigo inicia com uma breve apresentação do projecto (objectivos e organização). Depois, passa-se à apresentação de alguns resultados já obtidos em um dos três subgrupos do projecto (Tarefa 1) sobre a generalização efectuada pelos alunos. Neste processo, a maioria dos alunos constrói uma regra mental, verbaliza-a e escreve a regra em linguagem corrente, mas não escrevem a generalização simbolicamente. Os alunos entendem a letra como algo desconhecido, mas têm dificuldade em interpretá-la como um número generalizado. Finalmente, são colocadas algumas questões. Palavras-chave: pensamento algébrico; letra; sequência; equação; generalização. ABSTRACT I will begin this reading with a brief presentation of the project (aims and organization). Afterwards I will present some results already reached in one of the three subgroups of the project (Task 1) concerning students’ generalization. In this process, most of the students build a mental rule, verbalize it and write the rule in natural language, but do not get to symbolize the generalization. The students give sense to the letter as an unknown, but they have difficult in interpreting it as a generalized number. Finally, some questions are posed. Key words: algebraic thinking; letter; sequence; equation; generalization.

The Project IMLNA The aims In most countries, numbers and algebra are two fundamental topics of the school mathematics curriculum. Numbers have a decisive role in mathematics learning in early years and algebra is a key mathematical topic from the intermediate years onwards. This project aims to contribute towards a better understanding of the reasons that yield Portuguese students to have low achievement in these areas and to identify what can be done to improve their learning. This project aims to contribute towards (i) a better understanding of the difficulties of students in learning numbers and algebra and of the potential of the teaching approaches based in innovative strategies; (ii) disseminating results concerning these aspects and promoting their discussion by the national and international research community; (iii) providing suggestions, recommendations and relevant examples to those responsible for writing the official mathematics curriculum, to textbook authors and those in charge of pre-service and in-service mathematics teacher education; and (iv) promoting the professional development of project members, and those who contact closely with it or have the opportunity of participating in teacher education and dissemination sessions.

The algebraic thinking In this view, algebraic thinking includes: -- the ability to deal with algebraic computations and functions; 1

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-- the ability to deal with mathematical structures and using them in the interpretation and solving of mathematical or extra mathematical problems; -- the manipulation of symbols, using them in creative ways in describing situations and in solving problems.

So, in algebraic thinking attention is given not only to objects but also to existing relations among them, representing and reasoning about those relations in a way as general and abstract as possible. Therefore, one of the most important ways to promote this reasoning is the study of patterns and regularities.

The learning and the teaching This project presumes that students’ learning trajectories may be strongly assisted by using appropriate teaching strategies based on exploratory and investigative work, using realistic learning situations, new information and communications technologies (ICT) and adequate representation systems. The teacher has here a new educational role – stimulating students’ mathematical activity and synthesizing collective mathematical validations – along with the classical role of providing information and mathematical knowledge.

The organization The project IMLNA involves four tasks (T1, T2, T3 and T4) – three to be developed in three small groups and a fourth involving all teams. Each of the first three aims to study the comprehension and the difficulties of students from different age groups (from the 5th to the 11th grade and at university level, 2nd year) in key numerical and algebraic topics (rational numbers, proportion, functions, patterns and relationships and numerical analysis). It also seeks to analyse the potential of an approach based on strategies of problem solving, exploration, investigation, use of ICT (thus facilitating working with different representations), and using real life situations. Besides the national and international literature, in each of these tasks empirical work will be carried out. Teaching units according to these strategies will be developed including tasks about rational numbers, proportion, patterns and relationships, numerical estimation, number sense, symbol sense, functions and numerical analysis, their representations and connections. Data gathering instruments concerning classroom observations, interviewing students and reflecting with teachers will be developed. In the fourth task, involving the participation of two consultants (a Brazilian and an Italian), a global analysis will be carried out of learning trajectories in the field of numbers and algebra along the 2nd and 3rd cycles of basic education and secondary school, as well as a global evaluation of the teaching strategies applied. Working instruments will be prepared, both for the teaching experiments and for data collection and analysis and there will be discussions about each team’s reports. Additionally two seminars will be held and open to outside researchers and teachers. Another important aspect of this task is the collective discussion of papers to publish in national and international scientific journals as well as a book about the teaching units covered in the project (including the tasks developed and their role in the curriculum). Another important concern of the project is sharing ideas and results with the national and international 50

Improving mathematics learning in numbers and algebra (IMLNA) – a current project

scientific community and promoting the professional development of both researchers and teachers involved in a direct or indirect way.

Results Already Reached (T1) The letters and the generalization Pereira & Saraiva (2009) studied a class of grade 7 students. The problem of the study was to identify the learning and the difficulties that students present concerning the comprehension and the use of letters when they investigate involving generalization.

The state of the art Research shows that many students have great difficulties in numbers and operations. Other students obtain here a reasonable level of performance, but later come across with great difficulties in algebra. One of the reasons of these difficulties is related to the diverse subtleties and changes of meaning of symbols when one moves from one field to the other (Usiskin, 1988). Another difficulty is related to the symbolic understanding of the numerical and algebraic expressions and their connections (Schoenfeld, 2005). A student trained to answer only to algorithmic questions is hardly able to deal with questions that aim at a conceptual understanding or that involve a combination of representations. The difficulties of the students in the transition of arithmetic to algebra have been studied by numerous authors (for example, Booth, 1994; Rojano, 2002). They include (i) giving meaning to an algebraic expression; (ii) failing to see a letter as representing a number; (iii) attributing concrete meanings to letters; (iv) translating information from natural to algebraic language; (v) understanding the changes of meaning of the symbols + and = from arithmetic to algebra; and (vii) failing to distinguish arithmetic (3+5) from algebraic addition (x+3). Up to the present, school mathematics has emphasized the teaching of algorithms and computation procedures. However, teaching these algorithms when the students do not yet grasp the meaning of the operations leads to a mechanization without understanding that yields to weak performances as well as to an attitude of rejection of mathematics (Rojano, 2002). Important curriculum initiatives acknowledge these problems. For example, the influential NCTM (2000) document regards both numbers and algebra as a basic part of the school mathematics curriculum. Algebra involves strong symbolization. In fact, symbolization begins in arithmetic. In recent years, symbolism has been downplayed. However, symbolism is an essential part of mathematics that cannot be excluded. In fact, on the one hand, symbols have great value since they agglutinate ideas in compact aggregates, transforming them in information easy to understand and manipulate (Sfard & Linchevski, 1994). On the other hand, symbolism leads easily to formalism when we lose of sight the meanings that the symbols represent and only give attention how to manipulate them (Davis & Hersh, 1995), thus hampering the learning process. It is necessary, therefore, to find a road in teaching and learning numbers and algebra that provides an accessible and productive entrance both to mathematical language and to mathematical understanding. Capítulo 2 - Números e Álgebra | Chapter 2 - Numbers and Algebra

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Usiskin (1988) says that the school algebra has to do with the understanding of “letters” (today we usually call them variables) and their operations, and

considers the five following equations to illustrate the different meanings to the

letters: 1. A = LW 2. 40 = 5x 3. sin x = cos x • tan x 4. 1 = n • (l/n) 5. y = kx

To him, each of these equations has a different feel. We usually call (1) a formula, (2) an equation (or open sentence) to solve, (3) an identity, (4) a property, and (5) an equation of a function of direct variation (not to be solved). To Usiskin, these different names reflect different uses to which the idea of variable is put. In (1), A, L, and W stand for the quantities area, length, and width and have the feel of knowns. In (2), we tend to think of x as unknown. In (3), x is an argument of a function. Equation (4), unlike the others, generalizes an arithmetic pattern, and n identifies an instance of the pattern. In (5), x is again an argument of a function, y the value, and k a constant (or parameter, depending on how it is used). Only with (5) is there the feel of “variability,” from which the term variable arose. To Mason, Graham & Wilder (2005), the algebraic thinking, particularly the recognition and articulation of generality, is within reach of all learners, and vital if they are to participate fully in society (p. ix). Yet, every learner who starts school has already displayed the power to generalize and abstract from particular cases, and this is the root of algebra. To those authors, the expressing generality is entirely natural, pleasurable, and part of human sense-making. Algebra provides a manipulative symbol system and language for expressing and manipulating that generality (p. 2). However, many students have great difficulties in algebra, particularly in problem solving involving symbolic generalizations. The meaning of the symbols students make is frequently without sense – it is only a memory process. Many authors, such as Rojano (2002), say that the generalization process has a first moment of perception of generality, which consists, for example, in the recognizing of a pattern in a numeric sequence – it is a mental process, and it happens, for instance, when the students are able to get any term of a sequence without the necessity to extend the terms of the sequence to that order; a second moment of the expression of generality, elucidating a general rule, verbal or numeric, to generate a sequence – it is a mental rule presented in natural language, or numerically; a third moment which is the symbolic expression of generality, yielding a formula corresponding to the general rule; and a forth one of the manipulation of the generality, solving problems related to the sequence. This is a cycle that must be seen in a flexible way, but it contains the essential moments in the generalization process. As Ponte (2006) indicates, the analysis of the mathematics curriculum of Portugal and other countries, in numbers and algebra, raises questions regarding its intuitions and basic models, main structural concepts, basic representations, study of algorithms and role of technology. Kaput & Blanton (2005) suggest that is necessary to experiment curricula that combines (i) promoting representation and thinking processes that seek generalization whenever possible; (ii) treating numbers and operations algebraically, giving attention to existing relations (and not just to the numerical values) as 52

Improving mathematics learning in numbers and algebra (IMLNA) – a current project

formal objects for algebraic thinking; and (iii) promoting the study of patterns and regularities, from as early as possible. On the other hand, the algebraic structure and symbolism can be building from the mathematical experience with numbers, emphasizing the intuitive and the strategic aspects (NCTM, 2000; Guzmán, 1996). Also, an investigative approach, including the visualization and the manipulation of figures, is considered a good support to the generalization, because it can allow the students to the building of an algebraic formula (Kieran, 2006).

Pedagogical proposal The pedagogical proposal was elaborated with tasks that promote the generalization, the resolution of equations, and problem solving including equations. Investigative, exploratory, problems and exercises (these one from the text-book) tasks were proposed to the students. Some tasks were formulated in pure mathematical terms and others were semireal (in the sense of Skovsmose, 2000) – real at first sight but, in practice, conditioned to a didactic contract established with the students about the acceptance of the conditions of the statement of the tasks that are relevant to the resolutions; so, there are many real characteristics of the objects referred in the statement that are not considered as in a pure real task. By this way, the first task, an investigative one, John’s birthday, is put in the Sequences theme. The second one, Discovering the value of the letters, an exploratory task, is put in the Equations theme. The third task, The test of evaluation (TE), with the investigative task The tower of the odd numbers, aims to think over the students’ learning, and it is put in the Sequences theme. The fourth task, a problem, The money-pots, is put in the Problem solving including equations theme.

Methodology In this study a qualitative and interpretative approach was followed (Bogdan e Biklen, 1991). The teacher, Magda Pereira, simultaneously assumed the role of teacher and investigator. The empirical work was realized during the second period of the school year of 2007/2008, with a class, grade 7, fifteen students, during 11 classes with 90’ each one. The students worked in group (three of them with four and one group with three students), in the mathematical Numbers and Calculus theme.

The data collected: i) the students’ resolutions of the sequenced tasks proposed by the teacher; ii) the dialogues in the class, between teacher and students, recorded by the teacher; iii) the students’ resolutions of the test of evaluation (TE), and iv) the interviews made in group (two groups – GA and GB – with three students each one) in the end of the teaching of the Numbers and Calculus theme. To GA was proposed the exploratory task The three twins, and The problem of the ages; to GB was proposed the exploratory task The messages of mobile, and The problem of the three brothers. The data analysis: They were considered two categories of analysis – the ability to generalize; and the meaning of the letters.

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Results The ability to generalize In the beginning of the study, the students represent their reasoning by schemes, establishing relations between the data. In the first task (Jonh’s birthday): John organizes at his home a party in the day of his birthday. We don’t know how many friends go to the party. However, we know that John will be at the door of his home to receive his friends - while they will be bringing near, they will greet John with a handshake, as well as each one of the friends who have already arrived. Only when they all are joined they enter at home. How many handshakes will be there before John and his friends enter at home? the students started to handshake one each other, and recording what happened when the number of friends was increased (figure 1)

Fig. 1 The scheme Jonh’ s birthday (Class, Group A)

By the scheme, the students identify and record the number of the handshakes to a specific number of friends, but they are not able to generalize it, even in the natural language. Also the students are not able to build any symbolic expression modeling the situation. The teacher, in the whole class discussion, suggested the following table (figure 2):

Fig. 2 Table with values written by the teacher on the blackboard (2008/01/16) 54

Improving mathematics learning in numbers and algebra (IMLNA) – a current project

The whole class discussion: Teacher: What happens inside the circles? We can obtain 1, on the right, because we have, on the left… Marco: I know! But, I don’t know! I thought in repeated values, but that is only for the first situation. The values inside the squares are not as I was thinking. Manuel: I have already figured it out. I figured out one thing, I think! Teacher: Please say, Manuel. Manuel: 3×4 are 12 and 12 ∕ 2 are 6. It happens the same to the values that are inside the triangles. And it is the same to the others. Teacher: Very well. And with a very big number of friends? How can we think? Manuel: By the same way. For example, if we have 1000 friends, to know the number of handshakes … is, and then we will divide by 2. Teacher: And if we will have an any number of friends? António: So, it is that number times the number before it, and after we will divide by 2. (Classroom, 16/01/2008)

With the teacher’s help, the students separate from the small numbers of their initial schemes to the number 1000. They are able, even, to say with their own words how to calculate the number of handshakes. To the teacher’s question about the eventual big number of friends, as big as we wish, Antonio says that the process is the same. However, the students are not able to get a symbolic expression of generality. In this study, the students make evident their difficulties to get a symbolic expression of generality, yielding a formula corresponding to the general rule of the situation. In the lesson number 6 of this theme, students’ evidence difficulties to write a symbolic expression of generality to the task The tower of the odd numbers (Consider the table of numbers 1 1 3 1 3 5 1 3 5 7 1 3 5 7 9 1 3 5 7 9 11 … Say the value of the sum of the numbers of a line of this triangle according to the number of the line (descending order)). For instance, we present Rosas’ answer (figure 3): We go always multiplying the number of the line. For example, if I want to know the result about the line number 18, I must do 18×18=324.  

 

Fig. 3 Rosa’s answer

Rosa just gives a rule in her natural language, giving an example to the case 18. She doesn’t present a symbolic expression of generality. Even in the end of the teaching of the Numbers and Calculus theme, the students evidence some resistance to get a formula corresponding to the general rule of the situation. Most of them are satisfied with a schema, as we can see in the answer given by Group B, in the interview, to the question The messages of mobile (Carla, Mário Capítulo 2 - Números e Álgebra | Chapter 2 - Numbers and Algebra

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and Bia met, today, in a bakery. They drink tea and, in the end of their meeting, they decide, when they arrive to their homes, to send messages, by mobile, to invite some more friends to go with them to the cinema. They invited some friends, but we don’t know how many of them will go to the cinema. We know that Carla, Mário and Bia send, each one, one message to the same group of friends. We also know that these friends exchanges one message between them. How many messages are sended?).

Fig. 4 The scheme presented by Group B (interview)

The students, by themselves, don’t sense the generality. They consider essentially specific cases. An intervention of the teacher is necessary to the students give the step forwards the symbolic expression of generality.

The meaning of the letters In the resolution of first task (John’s birthday), and after the students arrived to the number 1000, the use of the symbols only appears with the teacher’s help, as we can see in the next report (Whole class discussion): Teacher: And if we have an any number of friends? António: So, it is any number times the number before it, and after we will divide by 2. Teacher: So, and in a simple way, if we say that that number is n, how can we will find the number of handshakes? Manuel: The number of handshakes is any other one, like y, for example. Teacher: Well, that way, we don’t know anything. I say that n friends went to the party, and you say that there were y handshakes. By this way, do we get any information? Nelson: And with n we also don’t know anything, because we need to know how many friends are n. Teacher: How did we think of 1000 friends? Can we think the same way for n friends? António: Even without knowing how much is n? Rosa: I think I know! It is n times the number before it, that is… António: So, can we multiply numbers without knowing their values? And, afterwards, how do we know the result? Rosa: The result depends of how many friends are n. (Classroom, 16/01/2008)

The students’ difficulty is concerns the value of n – they don’t know it. How can they reason with n without knowing its value? In the task The tower of the odd numbers, the most part of the students explain by own words what happened in a certain line using the natural language. However some of them solved the task appointing the number of any line of the tower through a letter, seeming that they had built a symbolic relation with mathematical meaning (figure 5):

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For example, in line 20 is: 19 x 20 + 20 = 400

Fig. 5 Manuel’s answer(TE)

Manuel needed to present a specific case to confirm his symbolic expression. Besides, the process he uses, as well as the symbolic expression, is similar to the John’s birthday one, revealing a transposition of the processes used previously – which emphasizes the usefulness of the own mathematical experience. Because in a generalized expression the letter represents a number – it can assume various values, and for that very reason the same to the result of the expression – is an obstacle to the students. They have an inclination to consider the letter as a static and unique unknown value, exactly the same as the letter in an equation. The next small report of the interview (GA) – discussing again the task The messages of mobile – is an example of this: Teacher: So, using the letter n, what is the result of that reasoning? António: I would like to find a way to build an equation, but I don’t know what the unknown is. I think that the unknown is the number we go always increasing to the number of messages, as the number of friends increases, but I don’t know. Teacher: Please think a little more. You have a table. What does happen as the number of friends increases? Marco: If we have n friends, …, no, …, n messages. I don’t know what we are looking for. (...) Mara: I think I have one way. It is similar to a task we solved in class, some weeks ago. For example: 5× 4 = 20 . Afterwards 20 ÷ 2 are 10. But we want 7, so we must subtract 3. Teacher: So, and to an any number of friends?

€ of friends increases, and António: Ok, that is what I want, but I don’t know which the unknown is, because the number € also the number of messages, but they don’t increase in the same way. What is the unknown? (...) Group A (interview)

The students make evident a strong resistance to the building of a generalization. The difficulty seems to be in the interpretation that the students make of a letter in the various situations. Along the study the students promote more familiarity with the letter as an unknown – as an equation, the letter is the unknown value that we must calculate – than as a generalized number.

Conclusions The students make sense of the letter when it assumes the role of unknown. When the letter is inserted in a functional context they manifest lots of difficulties, especially to write a symbolic expression to generalize the general term of a sequence. However, the students are able to calculate the first terms of a sequence, and they are able to verbalize, and to write, in their natural language the general rule of the generalization. In the process of generalization, most of the students build a mental rule, verbalize it and write the rule in natural language – but they do not get to symbolize the generalization. The students stay in the second stage of the generalization referred by Rojano (2002). Perhaps on account the ambiguity of the symbols. In fact, n, individually, not being a natural Capítulo 2 - Números e Álgebra | Chapter 2 - Numbers and Algebra

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number represents all of them (Caraça, 1998). The schemes and the tables inserted in the resolution of tasks with an exploratory and investigative nature, and with a discussion students/students and students/teacher, promote the calculation of the first terms of a sequence, as well as the description of the general rule of the generalization using the preceding term. However, they did not promote the writing of the symbolic generalized expression of a sequence.

Final Notes In this point, we pose some questions that are a challenge for our future research: -- What is the role played by the schemes and by the visualization on the reasoning for generalization? What kind of table is more useful? -- Why do students believe more in the meaning of the letter as an unknown than as a generalized number? -- In what sense are the different meanings of the sign “=” related with the difficulties that the students evidence to interpret the letter as a generalized number in a symbolic expression context?

References Bogdan, R. & Biklen, S. (1991). Investigação Qualitativa em Educação – Uma introdução à Teoria e aos Métodos. Colecção Ciências da Educação. Porto: Porto Editora. Booth, L. R. (1984). Algebra: Children’s strategies and errors. Windsor: Nfer-Nelson. Caraça, B. J. (1998). Conceitos fundamentais da Matemática. Lisboa: Gradiva. Davis, P., & Hersh, R. (1995). A experiência matemática. Lisboa: Gradiva. Guzmán, M. (1996). El Rincón de la Pizarra: Ensayos de Visualización en Análisis Matemático. Madrid: Ediciones Pirámide. Kaput, J., & Blanton, M. (2005). Algebrafying elementary mathematics in a teacher-centered, systemic way. (Retried 30-06-2005 from http://www.simcalc.umassd.edu/downloads/AlgebrafyingMath.pdf) Mason. J., Graham, A. & Wilder, S. (2005). Developing thinking in algebra. The Open University. NCTM (2000). Principles and standards for school mathematics. Reston, VA: NCTM. Pereira, M. & Saraiva, M. (2009). The symbols and the generalization. (submitted). Ponte, J. P. (2006). Números e álgebra no currículo escolar. In I. Vale et al.(Eds.), Números e álgebra (pp. 5-27). Lisboa: SEM-SPCE. Rojano, T. (2002). Mathematics Learning in the Junior Secondary School: Students’ Access to Significant Mathematical Ideas. In L. English, M. B. Bussi, G. A. Jones, R. A. Lesh &D. Tirosh (Eds.), Handbook of international research in mathematics education (vol. 1, pp. 143-161). Mahwah, NJ: Lawrence Erlbaum. Sfard, A., & Linchevski, L. (1994). The gains and piftalls of reification: The case of algebra. Educational Studies in Mathematics, 26, 191-228. Schoenfeld, A. (2005). Curriculum development, teaching and assessment. In L. Santos et al. (Eds.), Educação matemática: Caminhos e encruzilhadas (pp. 13-41). Lisboa: APM. Skovsmose, O. (2000). Cenários para Investigação. Bolema, Ano 13, Nº14 (pp. 66-91). Usiskin, S. (1988). Conceptions of school algebra and uses of variables. In A. F. Coxford & A. P. Schulte (Eds.), The ideas of algebra. Reston, VA: NCTM 58

Exploração de Padrões e Pensamento Algébrico

Exploração de Padrões e Pensamento Algébrico António Borralho [email protected], Centro de Investigação em Educação e Psicologia da Universidade de Évora

Elsa Barbosa [email protected], Centro de Investigação em Educação e Psicologia da Universidade de Évora RESUMO A passagem da Aritmética para a Álgebra é uma das grandes dificuldades dos alunos e os professores devem diversificar estratégias permitindo, aos seus alunos, desenvolver o pensamento algébrico e o sentido de símbolo (Arcavi, 2006). Este estudo pretendeu compreender o significado da utilização, na sala de aula, de padrões num contexto de tarefas de investigação de forma a melhorar o desenvolvimento do pensamento algébrico. O ponto de partida assentou em quatro aspectos: (1) imagem da Matemática; (2) conexões matemáticas; (3) raciocínio algébrico; e (4) comunicação matemática. Palavras-Chave: Ensino de matemática, tarefas de investigação, padrões, raciocínio matemático, Álgebra, pensamento algébrico ABSTRACT The passage from arithmetic to algebra is one of the major difficulties that students face and teachers should diversify strategies in order to allow their students to develop algebraic thinking and the sense of symbol (Arcavi, 2006). The definable goal of this research lead to the understanding of the use of patterns in class, in a context of investigation tasks, in order to develop algebraic thought. One of the attempts of dealing with this set of problems has been done within four aspects: 1) the image of Mathematics; 2) mathematical connections; 3) the understanding of Algebra; 4) mathematical communication. Key words: mathematics education, investigations tasks, patterns, mathematic reasoning, Algebra, algebraic thought

Introdução A maioria dos professores sente uma grande dificuldade em fazer a passagem da aritmética para a Álgebra no que diz respeito à aprendizagem dos alunos. A introdução das variáveis é sempre confusa e na maioria dos casos descontextualizada (Vale, Palhares, Cabrita e Borralho, 2006). Sendo o ensino-aprendizagem da Álgebra essencial à comunicação matemática, a Álgebra deve ser introduzida como uma parte útil, apetecível e atractiva que facilite os procedimentos empíricos indutivos frente ao formalismo dedutivo (Socas, Camacho, Palarea e Hernandez, 1989). As actividades algébricas podem ser, como afirmam Brocardo, Delgado, Mendes, Rocha e Serrazina (2006) e Soares, Blanton e Kaput (2005), geradas a partir de actividades numéricas. A forma como o problema é apresentado, pode transformar um simples problema aritmético em algébrico. Facilmente se transformam problemas com respostas numéricas simples em novas situações onde os alunos têm a possibilidade de conjecturar, construir padrões, generalizar e justificar factos e relações matemáticas. Assim, a utilização dos padrões no ensino da Matemática pode ajudar os alunos a aprender uma matemática significativa e/ou a envolver-se na sua aprendizagem facultando-lhes um ambiente que tenha algo a ver com a sua realidade e experiências.

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Problema e enquadramento do estudo Pressupõe-se que a procura de padrões e regularidades permite formular generalizações em situações diversas, particularmente em contextos numéricos e geométricos, o que contribuirá para o desenvolvimento do raciocínio algébrico do aluno. Será que a utilização, na sala de aula, de padrões num contexto de tarefas de investigação permite um melhor desenvolvimento do pensamento algébrico por parte dos alunos? O trabalho desenvolvido procurou algumas respostas a este tema, através das seguintes questões mais específicas: 1. Os padrões, num contexto de tarefas de investigação, permitem o estabelecimento de conexões matemáticas? 2. Como é que a análise de padrões e regularidades, envolvendo números e operações elementares, contribui para o entendimento da Álgebra? 3. De que modo é que os padrões, num contexto de tarefas de investigação permitem promover a aptidão para discutir com outros e comunicar descobertas e ideias matemáticas através do uso de uma linguagem, escrita e oral, não ambígua e adequada à situação?

A passagem dos números para um maior grau de abstracção parece ser uma das etapas mais complicadas do ensino-aprendizagem da Matemática. Assim, é essencial a escolha de estratégias adequadas que permitam, aos alunos, desenvolver a compreensão da linguagem algébrica. Os problemas de raciocínio algébrico podem ter múltiplas soluções, o que permite aos alunos explorar diferentes caminhos de resolução. É aqui que os professores têm um papel fundamental, é a eles que lhes cabe incentivar a explorar diferentes resoluções, ou seja ajudando-os a desenvolver o pensamento algébrico. Orton e Orton (1999) afirmam que os padrões são um dos caminhos possíveis quando pensamos em introduzir a Álgebra e, consequentemente, desenvolver o pensamento algébrico. Segundo Bishop (1997), quando um aluno relaciona quantidades com padrões está a adquirir conceitos matemáticos muito importantes, como por exemplo, o conceito de função. Está a aprender a investigar e a comunicar algebricamente. A resolução de tarefas de investigação que envolvam padrões, por um lado salientam a exploração, investigação, conjectura e prova, por outro, não menos importante, são interessantes e desafiadoras para os alunos (Vale e Pimentel, 2005) e, finalmente, promovem a comunicação de ideias matemáticas (Barbosa, 2007). Poder-se-á afirmar que a abordagem dos padrões permite promover as competências matemáticas dos estudantes na medida em que se interliga com actividades de exploração e de investigação.

A exploração de padrões num contexto de tarefas de investigação e o pensamento algébrico A competência em Álgebra é bastante útil para o estudante na sua vida de todos os dias e para prosseguimento de estudos. Quem não tiver uma capacidade razoável de trabalhar com números e suas operações e de entender e usar 60

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a linguagem abstracta da Álgebra fica seriamente limitado nas suas opções escolares profissionais e no seu exercício de cidadania democrática (Ponte, 2006, p. 5). O NCTM (2000) indica, claramente, que todos os alunos devem aprender Álgebra desde os primeiros anos de escolaridade. No entanto, o seu estudo está fortemente ligado à manipulação simbólica e à resolução de equações. Mas a Álgebra é mais do que isso: os alunos precisam de entender os conceitos algébricos, as estruturas e princípios que regem as manipulações simbólicas e como estes símbolos podem ser utilizados para traduzir ideias matemáticas. Como Ponte (2006) afirma, a melhor forma de explicitar os objectivos do estudo da Álgebra, ao nível escolar, é dizer que se pretende desenvolver o pensamento algébrico dos alunos. Segundo Arcavi (2006), o pensamento algébrico inclui a conceptualização e aplicação de generalidade, variabilidade e estrutura (p.374). O autor defende ainda que o principal instrumento da Álgebra é o símbolo. Apesar do pensamento algébrico e dos símbolos terem muito em comum, não significam exactamente a mesma coisa. Pensar algébrico consiste em usar os instrumentos simbólicos para representar o problema de forma geral, aplicar procedimentos formais para obter um resultado, e poder interpretar esse resultado (...) ter “symbol sense” implica (...) questionar os símbolos em busca de significados, e abandoná-los a favor de outra representação quando eles não proporcionam esses mesmos significados (p. 374). Por forma a melhorarmos o desenvolvimento do pensamento algébrico, tem que se desenvolver o sentido do símbolo, uma condição necessária para que tal aconteça é a utilização de práticas de ensino apropriadas onde todo o trabalho seja desenvolvido através de tarefas de natureza investigativa e exploratória, onde os alunos tenham a oportunidade de explorar padrões e relações numéricas e a possibilidade de explicitar as suas ideias e onde possam discutir e reflectir sobre as mesmas. Muitas das dificuldades sentidas ao nível da Álgebra resulta da não compreensão do sentido de variável. Pode-se afirmar que a utilização de actividades que envolvam o estudo de padrões e regularidades são um dos caminhos privilegiados para desenvolver o pensamento algébrico. Os padrões ajudam, os alunos, a perceber a “verdadeira” noção de variável, que para a maioria é apenas vista como um número desconhecido (Star, HerbelEisenmann e Smith, 2000). A exploração de padrões num contexto de tarefas de investigação permite desenvolver a capacidade de os alunos, partindo de situações concretas, generalizarem regras, ou seja, ajuda os alunos a pensar algebricamente. Em suma poder-se-á afirmar que a integração de tarefas de investigação com padrões, no currículo da Matemática escolar, assume um papel de destaque na abordagem à Álgebra, e nos primeiros anos de escolaridade, de base ao pensamento “pré-algébrico”. (Vale, Palhares, Cabrita e Borralho, 2006).

Proposta pedagógica O grande desafio do ensino da Álgebra é o desenvolvimento do “sentido do símbolo”. Ganhar este desafio “obriga” a que os símbolos sejam introduzidos de forma contextualizada, num quadro de actividades que mostrem de forma natural aos alunos o poder matemático da simbolização e da formalização (Ponte, 2005, p. 40). Assim, a unidade escolhida para desenvolver o trabalho foi a unidade temática “Números e Cálculo” (DGEBS, 1991). Quanto ao Currículo Capítulo 2 - Números e Álgebra | Chapter 2 - Numbers and Algebra

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Nacional do Ensino Básico (ME-DEB, 2001), os domínios trabalhados foram Números e Cálculo e Álgebra e Funções e o tema escolhido Equações. As competências matemáticas que se pretenderam desenvolver foram: -- a predisposição para procurar e explorar padrões numéricos em situações matemáticas e não matemáticas e o gosto por investigar relações numéricas, nomeadamente em problemas envolvendo divisores e múltiplos de números ou implicando processos organizados de contagem (p. 60); -- a predisposição para procurar padrões e regularidades e para formular generalizações em situações diversas, nomeadamente em contextos numéricos e geométricos; -- a aptidão para analisar as relações numéricas de uma situação, explicitá-las em linguagem corrente e representá-las através de diferentes processos, incluindo o uso de símbolos; -- a aptidão para construir e interpretar tabelas de valores, gráficos, regras verbais e outros processos que traduzam relações entre variáveis, assim como para passar de umas formas de representação para outras, recorrendo ou não a instrumentos tecnológicos (p. 66); -- o significado de fórmulas no contexto de situações concretas e a aptidão para usá-las na resolução de problemas (p. 67). Deste modo, foram preparadas nove tarefas de investigação, quatro delas preliminares, e duas aulas de exploração de conteúdos. Todas as tarefas foram resolvidas em pequenos grupos, no final de cada tarefa era obrigatório a realização de um relatório. Não houve limite de tempo para a realização das tarefas, cada grupo trabalhava ao seu ritmo. Todas as tarefas foram realizadas em sala de aula, a investigadora recolhia os enunciados, mesmo que os grupos ainda não tivessem realizado o trabalho na íntegra, no final de cada aula. Os relatórios realizados pelos alunos eram todos analisados e comentados. As aulas de exploração de conteúdos tiveram como objectivo proporcionar e promover “a ponte” entre as tarefas de investigação realizadas e os conteúdos abordados. Acima de tudo, tentou-se sempre manter um bom ambiente de trabalho, descontraído e construtivo, onde os alunos se sentissem à-vontade para colocar qualquer tipo de dúvida ou questão. É ainda de referir que esta proposta de trabalho foi desenvolvida tendo em conta que sempre que se justificasse, deveria ser adaptada, reformulada e reorientada.

Metodologia Atendendo às questões do estudo, foi adoptada uma abordagem de investigação qualitativa e interpretativa, onde uma turma foi a unidade de análise no que diz respeito às tarefas de investigação envolvendo padrões. Tuckman (2000) de acordo com Bogdan e Biklen (1994) afirma que, a investigação qualitativa apresenta as cinco características principais que se seguem: 62

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1. A situação natural constitui a fonte dos dados, sendo o investigador o instrumento-chave da recolha de dados. 2. A sua primeira preocupação é descrever e só secundariamente analisar os dados. 3. A questão fundamental é todo o processo, ou seja, o que aconteceu, bem como o produto e o resultado final. 4. Os dados são analisados indutivamente, como se reunissem, em conjunto, todas as partes de um puzzle. 5. Diz respeito essencialmente ao significado das coisas, ou seja, ao “porquê” e ao “o quê”.

As características atrás referidas mostram-se de acordo com a essência das questões do estudo que foi realizado. Atendendo a que se pretendeu responder a questões de natureza explicativa, que não se desejou exercer qualquer tipo de controlo sobre a situação, que se pretendeu estudar uma turma de 8º ano e dois alunos da mesma, correspondendo a critérios definidos e que se visou obter um produto final de natureza descritiva e analítica, a opção metodológica desta investigação recaiu na realização de um estudo de caso qualitativo e analítico. O ano de escolaridade escolhido foi o 8º ano pelo facto de ser neste ano que a Álgebra começava a ser verdadeiramente introduzida (formalmente) no currículo dos alunos. A turma enquanto objecto de investigação trabalhou em pequenos grupos. De entre os alunos da turma foram escolhidos dois alunos, um aluno com pior desempenho e outro com melhor. Os dois alunos escolhidos pertenceram a grupos distintos. O objectivo desta escolha prendeu-se com o facto de se ter pretendido confrontar as posições de alunos com desempenhos distintos. A recolha dos dados foi feita através de um questionário, duas entrevistas semi-estruturadas, observação directa de aulas e análise de documentos produzidos pelos alunos.

Sofia A Sofia define-se, sem grande convicção, como uma aluna média. A única disciplina que, em todo o seu percurso escolar, obteve nível de classificação inferior a três foi a Matemática. Não gosta das aulas de Matemática quando são muito expositivas. O tipo de tarefas que mais gosta é as de investigação. Ela considera que este tipo de actividades são um desafio constante que obrigam os alunos a raciocinar e que 1

promove uma maior aprendizagem. A Sofia afirma que faz bem puxar pelo raciocínio (E2S) .

A contribuição dos padrões e regularidades no desenvolvimento do pensamento algébrico Durante a realização das primeiras tarefas, a Sofia, foi uma aluna desinteressada e pouco confiante nos seus raciocínios e, inclusive demonstrou dificuldade em perceber a lei de formação dos diferentes padrões. No entanto, ao longo da realização das tarefas a sua evolução foi evidente. Nas últimas tarefas já era ela que avançava com as fórmulas solicitadas nas questões propostas e, facilmente, explicava os seus raciocínios. A realização da tarefa A Mol1



(E2S) significa que os dados foram retirados da 2ª entrevista da Sofia. Capítulo 2 - Números e Álgebra | Chapter 2 - Numbers and Algebra

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dura foi um desses exemplos, onde a Sofia esteve sempre a “puxar” pelo grupo, experimentou diversas estratégias, apresentou algumas conjecturas, procurou a generalização e explicitou os seus raciocínios aos colegas de grupo e à professora. O relatório escrito no final da realização da tarefa mostra a forma como todo o trabalho se desenvolveu:

Fig. 1 Extracto do relatório da tarefa A Moldura

Assim, é possível afirmar que, o grupo em geral e a Sofia em particular, conseguiram: (i) identificar e generalizar relações; (ii) representá-las simbolicamente; (iii) “tomar consciência” que as relações simbólicas representam informações dadas ou desejadas; e (iv) “tomar consciência” da importância da verificação dos resultados. Nas aulas de Álgebra revelou facilidade na aquisição de conteúdos como monómios e polinómios (coeficiente, parte literal e grau) e operações com polinómios (adição algébrica e multiplicação). Contudo, ainda mostrou alguma dificuldade na simplificação de expressões algébricas. A sua evolução foi evidente e no fim do estudo a Sofia conseguia: (i) simplificar fórmulas; (ii) distinguir monómio de polinómio; (iii) identificar expressões equivalentes; e (iv) ter algum sentido do símbolo. Este último aspecto é evidenciado no excerto de entrevista que se apresenta:

Invest: Como é que sabemos que esta igualdade é verdadeira 2×C+ 2×L – 4= =(C+C+L+L) – 4? Posso dizer que esta igualdade é verdadeira ou não? Sofia: Sim. Invest: Porquê? Sofia: Porque juntamos na mesma duas vezes o comprimento e duas vezes a largura. São só duas maneiras diferentes de representar a mesma fórmula. Invest: Na fórmula a que chegaram, ou seja, na vossa generalização, consegues dar exemplos de monómios e de polinómios? [...] Sofia: O 2C. [...] Invest: [...] Então e um exemplo de um polinómio? Sofia: [silêncio] Invest: Retirado dessa expressão. Sofia: 2C + 2L. Invest: Muito bem. Podemos adicionar variáveis com significados diferentes? Sofia: Não. Invest: Porquê? Podíamos adicionar o 2C com o 2L? 2 64

Em anexo

Exploração de Padrões e Pensamento Algébrico

[...] Sofia: Porque significam coisas diferentes. Invest: Simplifica esta expressão: (2×n) + (n – 2). Enquanto simplificas fala comigo. [...] Sofia: 3n – 2. (E2S)3

José O José confia no seu trabalho e define-se como um aluno médio pois, segundo ele, só os alunos que obtêm nível cinco de classificação é que são bons. Na aula é muito participativo pois, como ele próprio afirma, gosta de expressar a sua opinião. Gosta de tudo nas aulas de Matemática: gosto de [aprender] coisas novas sobre a matemática, [quanto ao] que gosto menos [...], acho que [...] 4

[posso afirmar que não há] nada [que não goste] (E1J) . O tipo de tarefas que mais gosta são as de investigação pois, 4

para ele, este tipo de tarefas são as que permitem aprender mais de uma determinada matéria. (E1J)

A contribuição dos padrões e regularidades no desenvolvimento do pensamento algébrico Desde o início que o grupo do José mostrou facilidade em perceber a lei de formação de cada um dos padrões apresentados mas só após muito esforço é que conseguiam descobrir a generalização. No entanto, é de salientar as capacidades de argumentar sobre ideias algébricas, como o raciocínio dedutivo e indutivo, a representação, a igualdade 5

e a variável. Um bom exemplo disso é o relatório da tarefa As escadas , por eles apresentado:

Fig. 2 Extracto do relatório da tarefa As escadas

A análise deste relatório permite afirmar que os alunos conseguiram: (i) identificar e generalizar relações; (ii) representálas simbolicamente; (iii) transformar expressões noutras equivalentes; e (iv) “tomar consciência” de que as relações simbólicas representam informações dadas ou desejadas. 3 4 5

(E2S) significa que os dados foram retirados da 2ª entrevista da Sofia. (E1J) significa que os dados foram retirados da 1ª entrevista do José. Em anexo Capítulo 2 - Números e Álgebra | Chapter 2 - Numbers and Algebra

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O José conseguiu perceber com facilidade a lei de formação dos padrões mas, segundo ele, teve alguma dificuldade em descobrir o enésimo termo. Apesar disso, mostrou facilidade em entender e manipular as fórmulas encontradas:

Invest: Tens dificuldades quando trabalhas com fórmulas matemáticas? Entendes o que elas significam quando estás a resolver uma tarefa? José: Sim, tive algumas dificuldades em encontrar [a fórmula] mas depois consigo percebe-la. […] Invest: Para que serve o enésimo termo? Qual é o seu significado? José: […] Aquela forma que estão a pedir serve para qualquer um, por exemplo num espelho pode servir para qualquer espelho. Invest: “Num espelho pode servir para qualquer espelho” O que queres dizer? Nessa tarefa o que te era pedido? José: Por exemplo, o número de azulejos em redor. Invest: Portanto, permite-te prever? É isso? Corrige-me se estou enganada. Permite-te prever o número de azulejos para um espelho de qualquer tamanho? José: Sim. (E2J)6

Nas aulas de Álgebra, o José foi um dos alunos mais empenhado e participativo. Durante as aulas evidenciou: (i) ter adquirido o sentido do símbolo; (ii) dominar o conceito de monómio e polinómio (coeficiente, parte literal e grau); (iii) saber operar com polinómios; (iv) ter aprendido a simplificar expressões algébricas; (v) ter adquirido o conceito de expressões equivalentes. Sobre este último aspecto o excerto de entrevista que se apresenta é bastante esclarecedor:

Invest: Olha para esta igualdade: 2×C + 2×L – 4 = (C+C+L+L) – 4. O que significa? Como é que sabemos que esta igualdade é verdadeira? José: Então, os menos quatro equivalem aos quatro azulejos comuns em todos os espelhos, e tá duas vezes C e noutro lado está C+C porque pode se pode substituir o C+C por 2×C ou dois C e a mesma coisa para a largura o L, que L+L substitui-se por 2×L, duas vez a largura. […] Invest: Na generalização que vocês encontraram na última tarefa, consegues dar-me exemplos de monómios? E de polinómios? José: Sim. Invest: Então diz-me lá um monómio. José: A. Invest: E um polinómio? José: A+ A+ C+C. [...] Invest: […] Podemos adicionar variáveis com significados diferentes? José: Não. Invest: Porquê? […] José: Porque como têm significados diferentes não se podem adicionar, nem subtrair também. (E2J)6

Conclusão A exploração de padrões num contexto de tarefas de investigação permitiu o desenvolvimento do pensamento algébrico ou, mais especificamente, o sentido do símbolo, ao proporcionar que os alunos utilizem diferentes representações, identifiquem e generalizem relações, analisem os seus significados e tomem consciência da importância da verificação 6 66

(E2J) significa que os dados foram retirados da 2ª entrevista do José.

Exploração de Padrões e Pensamento Algébrico

de dados. Além disso, é importante salientar o desempenho que os alunos tiveram durante as aulas de Álgebra. Foi notória a facilidade com que aprenderam a noção de monómio e polinómio (coeficiente, parte literal e grau), operações com polinómios (adição algébrica e multiplicação), simplificação de expressões algébricas e expressões equivalentes, atingindo a generalidade dos objectivos propostos. Em suma, à semelhança dos resultados obtidos no estudo de Orton e Orton (1999), é possível concluir que o estudo da Álgebra pode ser iniciado através da exploração e generalização de padrões. Mas, em simultâneo, é necessário mudar práticas de ensino, deixar para trás um ensino “tradicionalista” que promove a rotina e, consequentemente, a aprendizagem “isolada” de conteúdos, para passarmos a ter práticas de ensino que desenvolvam aprendizagens significativas por parte dos alunos.

Referências Arcavi, A. (2006). El desarrolo y el uso del sentido de los símbolos. Em I. Vale, T. Pimental, A. Barbosa, L. Fonseca, L. Santos e P. Canavarro (Org), Números e Álgebra na aprendizagem da Matemática e na formação de professores (pp. 29-48). Lisboa: Secção de Educação Matemática da Sociedade Portuguesa de Ciências da Educação. Bogdan, R. e Biklen, S. (1994). Investigação qualitativa em Educação. Porto: Porto Editora. Barbosa, E. (2007). A Exploração de Padrões num Contexto de Tarefas de Investigação com Alunos do 8ºano de Escolaridade. Lisboa: APM. Bishop, J. W. (1997). Middle School students’ Understanding of Mathematical Patterns and Their Symbolic Representations. Chicago: Annual Meeting of the American Educational Research Association. Brocardo, J., Delgado, C., Mendes, F., Rocha, I. e Serrazina, L. (2006). Números e Álgebra: desenvolvimento curricular. Em I. Vale, T. Pimental, A. Barbosa, L. Fonseca, L. Santos e P. Canavarro (Org), Números e Álgebra na aprendizagem da matemática e na formação de professores (pp. 65-92). Lisboa: Secção de Educação Matemática da Sociedade Portuguesa de Ciências da Educação. Matos, J. F. e Carreira, S. P. (1994). Estudos de caso em Educação Matemática – Problemas actuais. Quadrante, 3(1), 19-53. ME-DEB (2001). Currículo Nacional do Ensino Básico: Competências Essenciais. Lisboa: Ministério da Educação, Departamento de Educação Básica NCTM (2000). Principles and Standards for School Mathematics. Reston: NCTM. Orton, A. e Orton, J. (1999). Pattern and Approach to Algebra. Em A. Orton (Ed.), Pattern in the Teaching and Learning of Mathematics (pp. 104-124). Londres. Cassel. Ponte, J. P. (2005). Álgebra no currículo escolar. Educação e Matemática, 85, 36-42. Ponte, J. P. (2006). Números e Álgebra no currículo escolar. Em I. Vale, T. Pimental, A. Barbosa, L. Fonseca, L. Santos e P. Canavarro (Org), Números e Álgebra na aprendizagem da matemática e na formação de professores (pp. 5-27). Lisboa: Secção de Educação Matemática da Sociedade Portuguesa de Ciências da Educação. Soares, J., Blanton, M. e Kaput, J. (2005). Thinking Algebraically across the Elementary School Curriculum. Teaching Children Mathematics, Dezembro, 228-235. Socas, M., Camacho, M., Palarea, M. e Hernandez, J. (1989). Iniciación al Algebra. Madrid: Editorial Sintesis, S. A. Star, J. R., Herbel-Eisenmann, B. A. e Smith J. P. (2000). Algebraic Concepts: What`s Really New in New Curricula? Mathematics Teaching in the Middle School, 5(7), 446-451. Tuckman, B. W. (2000). Manual de investigação em educação. Lisboa: Fundação Calouste Gulbenkian. Vale, I., Palhares, P., Cabrita, I., e Borralho, A. (2006). Os padrões no Ensino-Aprendizagem da Álgebra. Em I. Vale, Capítulo 2 - Números e Álgebra | Chapter 2 - Numbers and Algebra

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T. Pimental, A. Barbosa, L. Fonseca, L. Santos e P. Canavarro (Org), Números e Álgebra na aprendizagem da matemática e na formação de professores (pp. 193-212). Lisboa: Secção de Educação Matemática da Sociedade Portuguesa de Ciências da Educação. Vale, I., Pimentel, T. (2005). Padrões: um tema transversal do currículo. Educação e Matemática, 85, 14-22.

Anexos As escadas7

1. Desenha a escada seguinte e descobre o número de setas usadas em cada escada. 2. Explica como é que se pode obter o número de setas necessárias para fazer uma escada com 30 degraus. 3. Será que consegues prever o número de setas necessárias para fazer uma escada com n degraus?

A Moldura8 A Moldarte faz molduras em espelhos rectangulares formadas por azulejos quadrados, como mostra a figura.

1. Quantos azulejos são necessários para fazer o espelho representado na figura anterior? 2. Desenha espelhos de várias dimensões. Explica por palavras tuas, recorrendo a números, a tabelas, etc., o número de azulejos que são necessários para colocar à volta de um espelho com quaisquer dimensões. 3. Tenta encontrar uma fórmula que permita saber o número de azulejos necessários à construção de qualquer espelho.

7 8 68

Adaptado de Vale e Pimentel, 2005 Adaptado de Vale e Pimentel, 2005

Padrões num Contexto de Formação Inicial de Educadores

Padrões num Contexto de Formação Inicial de Educadores Isabel Cabrita [email protected], CIDTFF – Universidade de Aveiro RESUMO Neste artigo, descreve-se e reflecte-se sobre parte de uma experiência desenvolvida no âmbito da disciplina de Matemática na Educação de Infância de um curso de Licenciatura em Educação de Infância de uma instituição do Ensino Superior. Mais concretamente, evidencia-se e discute-se o conhecimento que os alunos revelaram sobre padrões – conceito, termos associados, tipos e resolução de problemas - e as perspectivas de abordagem dessa temática. O fraco desempenho que os discentes manifestaram reforça a opção de se ter planeado e implementado a disciplina envolvendo os alunos na concepção e implementação de projectos sobre padrões, envolvendo crianças do pré-escolar, bem como na sua reformulação à luz da reacção e desempenho das crianças. Palavras-Chave: Formação inicial, Educação de infância, Padrões, Conhecimento, Experiências de aprendizagem ABSTRACT In this paper we describe and reflect upon a part of an experience carried out within the scope of the curriculum unit of Matemática na Educação de Infância (Pre-school Maths Education) in a Degree in Pre-school Teacher Education from an institution of Higher Education. More specifically, we shall highlight and discuss the knowledge revealed by students concerning patterns – concept, associated terms, types and problem solving – as well as the different approaches used to address this issue. The low performance evidenced by students strengthens our option of having planned and implemented the above-mentioned curriculum unit, in which students were lead to design and implement (directly with preschool children) projects on patterns as well as to reformulate them according to children’s task performance. Key words: Teacher training; Kindergarden; Patters; Knowledge; Teaching experiments

Introdução A importância da abordagem da matemática, designadamente centrada nos padrões, desde o pré-escolar está referida em literatura vária incluindo as orientações curriculares para esse nível em Portugal (ver, por exemplo; Barody, 2002; Cabrita e Moderno, 2003; Castro e Rodrigues, 2008; Clements, Sarama e DiBiaise, 2004; Clements, Sarama, 2007; Curcio & Schwartz, 1997; Greenes, Ginsburg e Balfanz, 2004; ME-DEB, 1997; Moreira e Oliveira, 2003; NCTM, 2007; Palhares e Mamede, 2002; Vale et al, 2009). Para que os Educadores a possam fazer e da melhor forma, é fundamental que comecem a vivenciar tais experiências desde a Formação Inicial. Neste contexto, decidiu-se dar um lugar de destaque ao estudo dos padrões, enquanto contexto de abordagem de qualquer outro tópico matemático e do desenvolvimento de capacidades como a resolução de problemas, o raciocínio, a comunicação (Arcavi, 2006; Devlin, 2002; NCTM, 2000; Steen, 1988; Vale et al, 2009), na disciplina de Matemática na Educação de Infância (MEI) do curso de Licenciatura em Educação de Infância. Era uma disciplina obrigatória, semestral e com uma carga horária de 2h Teóricas e 2 Teórico-Práticas semanais e constituia-se a única oportunidade, formal, dos futuros Educadores lidarem com questões da matemática e da educação em matemática ao longo do curso. Essa disciplina foi palco de uma experiência que se descreve a seguir. Capítulo 2 - Números e Álgebra | Chapter 2 - Numbers and Algebra

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Questões de Investigação e objectivos A questão de investigação que norteou a experiência a desenvolver é Qual o impacte da disciplina MEI numa abordagem da matemática centrada nos padrões? Como principais objectivos, definiram-se: -- reestruturar o programa da disciplina em função -- das mais recentes orientações para a Matemática -- do conhecimento que os alunos revelam sobre padrões e da forma como perspectivam a abordagem do tema -- implementá-lo proporcionando que os futuros educadores se envolvam activamente na construção de conhecimento e na concepção, implementação, avaliação e reformulação de experiências de aprendizagens matemáticas centradas nos padrões -- avaliar -- as aprendizagens construídas -- a importância que atribuem a essa disciplina imediatamente após o seu términus e após a conclusão da PP -- se proporcionaram (ou não), a crianças do pré-escolar no âmbito da PP, experiências de aprendizagens baseadas nos padrões, porquê e, em caso afirmativo -- com que impacto (ao nível dos orientadores, das crianças, dos próprios e dos colegas de estágio).

Este artigo incide, principalmente, sobre o conhecimento que os alunos revelaram e sobre perspectivas de abordagem da temática dos padrões, com base nos quais se reformulou e se implementou o plano inicialmente pensado.

Desafios à formação inicial e padrões na Educação Pré-Escolar O trabalho com e sobre padrões, por apelar mas, numa perspectiva dinâmica, também desenvolver o pensamento algébrico e geométrico, e estar intimamente relacionado com a resolução de problemas e a comunicação, incluindo formas diversificadas de representação (Blanto & Kaput, 2005; Jacobs et al., 2007; Lubinski & Otto, 2002; Orton, 1999; Vale et al. 2009; Warren & Cooper, 2005), assume um lugar de destaque nos currículos dos mais diversos países. Até porque, segundo diversos autores, atraem as crianças, apelam ao seu sentido estético e permitem desenvolver a criatividade (Hardy, 2002; McGettrick, 2008). Também em Portugal, as instituições ministeriais estão consicentes da importância da abordagem desta temática desde o pré-escolar. Consequentemente, os Educadores devem dominar o tema padrões bem como formas de o abordar com as crianças. A Formação Inicial, entendida como uma de várias etapas do desenvolvimento profissional (Marcelo, 1999; Nóvoa, 70

Padrões num Contexto de Formação Inicial de Educadores

1992; Pacheco, 1995), desempenha um papel crucial na consecução de tais objectivos. Mas tal formação não se pode desenvolver há luz de uma matriz tradicional/artesanal, condutista ou mesmo personalista ou, se quisermos, seguindo uma orientação académica, tecnológica ou prática. Antes, deve inscrever-se num paradigma orientado para a indagação e num quadro reconstrucionista social tal como preconizado por Zeichner (1983) e Feiman-Nemser (1990). Também se deve desenrolar no pressuposto de que, a par de muitas certezas, as ciências desvendam, todos os dias, muitas incertezas, como defende Morin (2000). De facto, nunca como agora se viveram momentos de contemporaneidade tão marcadamente caracterizados pela complexidade e pela imprevisibilidade. É neste cenário que se devem entender as ‘actuais’ orientações para a Educação Pré-escolar. Tal como já sintetizamos noutro momento (Vale et al., 2008), em Portugal, as Orientações Curriculares para a Educação Pré-Escolar, consignadas no Despacho nº 5220/97 (2ª série), publicado a 4 de Agosto de 1997, no Diário da República nº 178, II série, não constituem um currículo – “por incluírem a possibilidade de fundamentar diversas opções educativas, portanto, vários currículos” (ME, 1997: 13) - nem um programa – “pois adoptam uma perspectiva mais centrada em indicações para o educador do que na previsão da aprendizagem a realizar pelas crianças” (id: ib). Antes, assumem-se como um importante referente para todos os profissionais do nível pré-escolar. Contemplam três Áreas de Conteúdo - Área de formação pessoal e social; Área do Conhecimento do Mundo e Área de Expressão e Comunicação – em torno das quais se devem desenvolver experiências de aprendizagem ricas e significantes. Os domínios das expressões motora, dramática, plástica e musical; da linguagem oral e abordagem à escrita e da matemática estruturam a última Área referida. Embora de uma forma mais intensa e explícita no domínio da matemática, tais Orientações legitimam um trabalho sério e diversificado com padrões perpassando os restantes domínios, vertentes e Áreas conteúdais. Por exemplo, no domínio das Expressões fala-se, explicitamente, num trabalho com padrões rítmicos. Relativamente à matemática, surge o termo padrão, no contexto da medida, mas com acepções diferentes daquelas que interessam explorar no âmbito deste artigo - “(…) comparação entre as alturas das crianças, organização do espaço da sala, medições dos espaços com um padrão não convencional (pau, fita, corda, etc) ou com referência ao metro como medida padrão (fitas métricas, réguas graduadas)” (77). Mas é também considerado como uma sequência que tem regras lógicas que importa descobrir, actividade facilitadora da criação de novos padrões e do desenvolvimento do raciocínio lógico. Nesta perspectiva, procurar padrões aparece como um elo natural entre a matemática e: -- O domínio da Linguagem e abordagem à escrita - “A linguagem é também um sistema simbólico organizado que tem a sua lógica. A descoberta de padrões que lhe estão subjacentes é um meio de reflectir sobre a linguagem e também de desenvolver o raciocínio lógico” (id: 78). No mesmo sentido, vão afirmações apresentadas anteriormente, que sublinham a descoberta de relações e introduzem o termo norma – “Esta aprendizagem (da linguagem) baseia-se na exploração do carácter lúdico da linguagem, prazer em lidar com as palavras, inventar sons, e descobrir relações” (id: 67) e “Começando a perceber as Capítulo 2 - Números e Álgebra | Chapter 2 - Numbers and Algebra

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normas da codificação escrita, a criança vai desejar reproduzir algumas palavras. Por exemplo, aprender a escrever o seu nome, que tem sentido afectivo para a criança e lhe permite fazer comparações entre letras que se repetem noutras palavras, o nome dos companheiros, o que o educador escreve” (id: 69); -- A área do Conhecimento do Mundo – “Como forma de pensar sobre o mundo e de organizar a experiência que implica procurar padrões, raciocinar sobre dados, resolver problemas e comunicar resultados, a Matemática está directamente relacionada com a área do Conhecimento do Mundo.” (id: ib). No domínio da matemática, privilegiam a exploração de padrões repetitivos e de padrões não repetitivos e, dentro destes, os crescentes. Estabelece-se, assim, conexões com a seriação ou a ordenação de objectos de acordo com as diferentes gradações que determinada qualidade – como a altura, tamanho, espessura, luminosidade, velocidade, altura ou intensidade do som, duração - pode admitir. Falam ainda na ordenação de materiais a usar no âmbito da Expressão Plástica (id: 62) e na ordenação de dados na parte consignada à Área do Conhecimento do Mundo e, mais concretamente, a propósito do método científico que se deve ir introduzindo desde a mais tenra idade – “A organização (de) dados levará provavelmente à necessidade de usar formas de registo que permitam classificá-los e ordená-los – desenhos, gráficos, descrição escrita do processo” (id: 83). Refere-se, ainda, nas Orientações Curriculares, a vivência e/ou explicitação de diversas sucessões temporais, privilegiando a narração de histórias como um contexto para a sua abordagem – “A narração de histórias é um meio de se apropriar da noção do tempo, pois corresponde a uma sucessão temporal marcada por ligações de continuidade traduzidas habitualmente pela expressão ‘e depois’. Recontar a história oralmente ou através de uma série de desenhos, seriar imagens, tendo como suporte uma pequena história, relaciona-se com a construção da noção do tempo e também com a linguagem” (id: 77).

Aspectos metodológicos A experiência assumiu contornos de um estudo qualitativo e desenvolveu-se num contexto de investigação-acção. Envolveram-se 26 alunos que frequentavam a disciplina em causa, que estiveram presentes na primeira sessão da disciplina e que responderam ao questionário e a um mini-teste aplicado. Atendendo às questões às quais se pretendia dar resposta, inquiriram-se os alunos sobre: o que entendiam por padrão; que termos lhe estavam associados e que tipos de padrões se podem considerar. Numa outra parte, pedia-se que descrevessem uma situação de aprendizagem a proporcionar a crianças do pré-escolar e que envolvesse padrões. Finalmente, propôs-se a resolução de problemas. A informação recolhida foi alvo de uma análise de conteúdo, orientada por categorias definidas à priori, reformuladas à luz das produções dos alunos. Os dados são apresentados preferencialmente de forma descritiva, evidenciando-se as afirmações com digitalizações ou transcrições de respostas dos alunos. Quando possível, ainda se sintetizam e apresentam alguns dado sobre a forma de tabela.

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Resultados e discussão Este ponto estrutura-se em 3 pontos principais, de acordo com as opiniões e produções dos alunos – conceitos associados a padrão; experiências de aprendizagem e resolução de problemas.

Conceitos associados a padrão Para tentarem uma aproximação à definição de padrão, seis alunos recorreram só a texto e 20 alunos serviram-se de uma representação mista - texto e imagem. Dois alunos deram respostas muito imprecisas, vagas ou confusas, como se evidencia na figura seguinte.

Fig. 1 Tentativa de definição de padrão apresentada por um aluno

Só 1 aluno associou padrão a módulo, considerando que “Padrão é a base de uma série de imagens, por exemplo, que podem ser criadas a partir dele. Este repete-se sempre podendo variar e que é depois criado a partir dele”. No que respeita ao contexto que enquadrou a tentativa de delimitação do conceito de padrão, distinguem-se respostas que remetem para um contexto cultural, focalizadas no dia-a-dia e do âmbito da matemática. Transcreve-se uma das duas respostas que se enquadra no primeiro parâmetro – “O conceito de padrão associa-se a várias áreas, tanto sociais como económicas e até religiosas. A padronização de vários sistemas sociais, ou até de comportamento humano e/ou social pode ser definido como o elo de ligação entre os vários cenários analisados”. A contextualização no dia-a-dia remete para – “camisola às riscas”; “tapetes de Arraiolos”; “azulejos”; “calçada portuguesa”; “cestos africanos”; “padrões musicais”; “lengalengas”. Relativamente ao contexto matemático: -- um aluno refere-se ao numérico; -- dois discentes consideraram qualquer área – “para mim padrão significa imagem, objecto ou algarismo que se repete formando um todo” ; -- a maioria (18 alunos) reporta-se ao contexto geométrico.

Mas a ideia geral de padrão que prevalece está associada a repetição. Um aluno refere – “É um acontecimento que se repete regularmente. Por exemplo 5/10/15/20 … (ou seja soma-se sempre 5 neste caso)”. Na figura seguinte apresenta-se a resposta de outro aluno, mais centrada no tema da geometria.

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Fig. 2 Definição de padrão associada a regularidade apresentada por um aluno

Em resposta à solicitação de que explicitassem termos associados a padrão, cinco alunos não responderam. Dos restantes alunos, nove reportaram-se a regularidade e a sequência; quatro a desvio-padrão e dois a simetria (ver figura seguinte). Para além de termos mais vastos como cultura, música e tonalidade, pode formar-se outra família de constelações com os termos lógica, base, módulo, ordem e irregularidade. Ainda se considerou outro grupo do âmbito das transformações geométricas – rotação e translacção. De uma forma mais lata, ainda indicaram imagem e figura. 1 cultura

1 música

9 regularidade sequência

1 ordem

1 módulo

1 irregularidade

1 rotação

2 simetria

1 figura

1 lógica

1 base

4 desvio-padrão 1 imagem

1 tonalidade

1 translacção

Fig. 3 Termos associados a padrão referidos pelos inquiridos

No que concerne aos tipos de padrões, nove alunos não explicitaram qualquer um. Dez inquiridos referiram-se a padrões geométricos, como se representa na figura seguinte. Curiosamente, três alunos referiram-se a círculo, quadrado e rectângulo como tipos de padrão. Seis alunos indicaram padrões numéricos, tendo um aluno escrito ‘conjunto’. Três alunos nomearam padrões estatísticos. Com uma única ocorrência ainda surgiram como tipos de padrões: matemático; desvio-padrão; sinais e elementos. 1 Círculo

10 geométricos 1 Conjunto

1 Desviopadrão 1 sinais

1 elementos

1 quadrado

6 numéricos

3 estatísticos

1 matemático

Fig. 4 Tipos de padrões referidos pelos inquiridos

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1 rectângulo

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Experiências de aprendizagem Relativamente à descrição de uma situação de aprendizagem envolvendo padrões, passível de ser tratada a nível do pré-escolar, sete inquiridos não apresentaram qualquer sugestão e cinco alunos apresentaram ideias consideradas sem sentido como a ilustrada a seguir e: -- “desenhar por exemplo uma casa, ou até mesmo uma flor” (um aluno); -- “aprender numerais” (dois alunos); -- “com um carimbo pode fazer uma imagem e assim já usa a noção de padrão” (um aluno).

Fig. 5 Situação de aprendizagem envolvendo padrões apresentada por um aluno

Três alunos apresentaram ideias muito vagas. Nesta categoria englobou-se: -- “visita à fábrica azulejos” ; -- “desenhar formas geométricas e pintar”; -- “cortar quadrados e formar conjuntos diferentes”. Quatro inquiridos focaram-se mais directamente na temática, mas revelaram ideias muito gerais: “mostrar diferentes padrões”; “identificar, discutir e criar diferentes padrões”; “explicar como construir a partir do módulo”; “construir padrões com carimbos … ou módulos”. Dois alunos referiram-se à “visualização do caleidoscópio” e outros três explicitaram padrões do tipo ababab…. Um dos alunos referiu-se à carimbagem, actividade tão do agrado das crianças – “Utilizando carimbos com a forma de animais (por exemplo: um gato e um cão), pedir à criança que carimbe numa folha branca um gato e um cão, alternadamente. Desta forma, formaram um padrão através do divertimento”. Dois alunos explicitaram a construção de uma grelha. Na figura seguinte, exemplifica-se uma dessas respostas. Notese o facto do aluno ter generalizado a construção da grelha (com uma configuração muito habitual nestes níveis de escolaridade - 4x4) recorrendo a f1 e f2 para representar dois elementos diferentes a usar como módulo (situação também muito comum) pelas crianças. Também é interessante verificar o pormenor da sugestão para que tal construção se aproveite para fazer uma capa, provavelmente para as crianças guardarem as suas produções. Capítulo 2 - Números e Álgebra | Chapter 2 - Numbers and Algebra

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Fig. 6 Situação de aprendizagem envolvendo padrões do tipo ababab….

Resolução de problemas Foi ainda proposta a resolução de problemas. A seguir, sintetiza-se o desempenho dos alunos a duas dessas tarefas. O enunciado do primeiro problema apresenta-se a seguir:

Continua a sequência 2

5

10

17

26

…

Indica a lei de generalização.

Como se verifica pela análise do quadro seguinte, a esmagadora maioria dos alunos não continua a sucessão.

Nº de respostas continua lei gen.

certas

erradas

Não responde

7

1

18

18

8

Tabela 1 Número de respostas certas ou erradas ao primeiro problema apresentado

Um aluno apresentou uma resposta considerada errada (ver figura seguinte) e sete respondentes apresentaram a resposta correcta.

Fig. 7 Resposta de um aluno à questão da continuação do termo seguinte de uma sucessão

No que respeita à generalização, não se considerou qualquer resposta correcta. Dois alunos apresentaram como solução “quadrados perfeitos” e um aluno registou “x + 3”. 76

Padrões num Contexto de Formação Inicial de Educadores

Quinze alunos tentaram uma generalização por recursividade, mas ficaram muito aquém de uma lei de generalização aceitável, como se ilustra a seguir. Apesar de terem percebido que os números ímpares desempenhavam um papel importante na evolução dos termos da sucessão, note-se a dificuldade que os alunos manifestaram em comunicar as suas ideias, quer matematicamente quer mesmo ao nível da língua materna.

Fig. 8 Resposta de alunos à questão da lei da generalização da sucessão

Outra das tarefas propostas foi o ‘problema das tendas’:

Quantos caminhos são necessários para ligar, entre si, 3 tendas (não alinhadas)? E 4? E n tendas?

Pela análise da tabela seguinte, pode verificar-se que a maior parte dos alunos respondeu correctamente à primeira questão da tarefa, sobre o número de caminhos necessários para ligar entre si 3 tendas não alinhadas.

Nº de respostas nº de tendas

certas

erradas

Não responde

3

21

4

8

4

6

16

4

18

8

n

Tabela 2 Número de respostas certas ou erradas ao problema das tendas

Os três alunos que erraram essa questão deram como resposta dois ou quatro caminhos. Curiosamente, oito alunos não deram qualquer resposta. Relativamente à questão seguinte, a maior parte dos alunos (16) responde que são necessários três ou quatro caminhos para ligar entre si 4 tendas. Só seis alunos é que deram a resposta correcta. Relativamente à generalização para n caminhos, oito alunos não deu qualquer resposta e a maior parte dos restantes referiu “n” ou “n-1” caminhos. A seguir, ilustra-se a resposta de um aluno que evidencia o que foi dito.

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Fig. 9 Resposta de um aluno às duas primeiras questões do problema das tendas

Só um aluno é que tentou uma generalização um pouco mais elaborada, como se ilustra a seguir. Mais uma vez, vejase a dificuldade do aluno em expressar, correctamente, o raciocínio.

Fig. 10 Resposta de um aluno ao problema das tendas

Conclusões Os alunos envolvidos na experiência apresentaram muitas lacunas no conhecimento sobre padrões e em formas de abordar o tema. Estes resultados só reforçam a necessidade e pertinência de se ter perspectivado e implementado a disciplina de forma a se envolver, activa e colaborativamente, os futuros Educadores de Infância na: construção de conhecimento e na concepção, implementação, avaliação e reformulação de experiências de aprendizagens matemáticas centradas nos padrões. Tal reformulação foi justificada com base nos desempenhos efectivos das crianças a quem as futuras Educadoras implementaram os projectos concebidos. Esperemos que o forte envolvimento e de qualidade que os alunos manifestaram na disciplina de Matemática na Educação de Infância venha a ter repercussões positivas ao nível da Prática Pedagógica, que se espera acompanhar no ano lectivo 2009/10.

Referências Arcavi, A. (2006). El desarrolo y el uso del sentido de los números. Em Vale, I. et al. (org.), Números e álgebra. Lisboa: SPCE (29-48). Baroody, A. (2002). Incentivar a aprendizagem matemática das crianças. In B. Spodek (Org). Manual de Investigação em Educação de Infância. Lisboa: Fundação Calouste Gulbenkian. (333-390).

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Blanto, M. and Kaput, J. (2005). Characterizing a classroom practice that promotes algebraic reasoning. Journal For Research In Mathematics Education, vol 36, nº 5, 412-446. Cabrita, I. & Moderno, A. (Coord) (2003). Imagens da interculturalidade na educação de infância - Nós e os outros. Lisboa: ME/DEB Castro, J. P., & Rodrigues, M. (2008). Sentido do número e organização de dados. Textos de apoio para educadores de infância. Lisboa: Ministério da Educação, DGIDC. Clements D.; Sarama, J. & DiBiaise, A.(Eds.) (2004). Engaging Young Children in Mathematics - standards for early childhood mathematics education. Mahwah, NJ: Lawrence Erlbaum. Clements, D. H. & Sarama, J. (2007). Early childhood mathematics learning. In Frank K. Lester (Ed.) Second Handbook of Research on Mathematics Teaching and Learning. Charlotte, NC: Information Age Publishing (461-556).

Curcio, Frances R.; Schwartz, Syndey L. (1997). What does algebraic thinking look like and sound like with preprimary children? Teaching Children Mathematics, February 1, Devlin, K. (2002). Matemática: a ciência dos padrões. Porto: Porto Editora. Feiman-Nemser, S. (1990). Teacher preparation: structural and conceptual alternatives. In W. Houston (ed.), Handboobk os research on teacher education. New York: MacMillan (212-233). Greenes, C., Ginsburg, H., & Balfanz, R. ( 2004). Big Math for little kids. Early Childhood Research Quarterly, 19, (159-166). Jacobs, V.; Franke, M.; Levi, T and Battey, D. (2007). Professional development focused on children’s algebraic reasoning in elementary school. Journal for Research In Mathematics Educations, vol 38, nº 3, 258-288. Lubinski, C and Otto, A. (2002). Meaningful mathematical representations and early algebraic reasoning, Teaching Children Mathematics, October, 76-80. Marcelo, C. (1999). Formação de professors. Para uma mudança qualitativa. Porto: Porto Editora. McGettrick, Bart (2008). Creativity and Enjoyment in Education. Comunicação apresentada no ETEN 2008. ME-DEB (1997). Orientações Curriculares para a Educação Pré-Escolar. Lisboa: Ministério da Educação, Direcção-Geral do Ensino Básico. Moreira, D., & Oliveira, I. (2003). Iniciação à Matemática no Jardim de Infância. Lisboa: Universidade Aberta Morin, E. (2000). Os sete saberes necessários à educação do futuro. São Paulo: Cortez Editora. NCTM (2000). Principles and Standards for School Mathematics. Reston:NCTM. Nóvoa, A. (1992). Os professores e a sua formação. Lisboa: D. Quixote. Orton, A. (1999) (ed). Pattern in the Teaching and Learning of Mathematics. London: Cassell. Pacheco, J. (1995). Formação de professores. Teoria e praxis. Braga: Universidade do Minho Palhares, P. e Mamede. E. (2002). Os padrões na matemática do pré-escolar. Educare/Educere, 11, 115-131 Vale, I.; Barbosa, A.; Borralho, A.; Barbosa, E.; Cabrita, I.; Fonseca, L. e Pimentel, T. (2009). Padrões no Ensino e Aprendizagem da Matemática - Propostas Curriculares para o Ensino Básico. Viana do Castelo: Escola Superior de Educação do Instituto Politécnico de Viana do Castelo. ISBN: 978-989-95980-2-7 Vale, I.; Fonseca, L.; Barbosa, A.; Pimentel, T.; Borralho, A. E Cabrita, I. (2008). Padrões no Currículo de Matemática: Presente e Futuro. Actas dos XII SEIEM – XIX SIEM – XVIII EIEM, Badajoz, 3 al 6 de septiembre 2008 Warren, E. and Cooper, T. (2005). Introducing functional Thinking in Year 2: a case study of early algebra teaching, Contemporary Issues in Early Childhood, Volume 6, Number 2, 2005, 150-162 Zeichner, K. (1983). Alternative paradigms of teacher education. Journal of Teacher Education, 34(3), 3-9.

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Pupils’ perception of shape, pattern and transformations

Pupils’ perception of shape, pattern and transformations Jean Orton [email protected], University of Leeds (retired) RESUMO Neste artigo é feita uma reflexão sobre os resultados da investigação acerca da capacidade, de alunos de 9-16 anos, para reconhecerem formas congruentes em diferentes orientações e diferentes contextos. São focados os resultados que dão informações sobre a percepção dos alunos de transformações comuns e sobre a utilização de quadros de referência. Consideram-se as palavras usadas pelos alunos para descrever os padrões que vêem, sendo ainda incluídas observações recentes referentes à utilização de um padrão disposto numa grelha, com alunos com bom desempenho. A questão de saber se uma abordagem visual é sempre útil, ou se pode ser uma distracção, será levantada em relação aos padrões numéricos e à generalização. Palavras-chave: padrão, forma, congruência, reflexão, rotação. ABSTRACT This paper reflects on research results about the ability of pupils aged 9 – 16 to recognise congruent shapes in different orientations and in different contexts. It focuses on results giving information about children’s perception of common transformations and their use of frames of reference. It considers the words used by children to describe patterns which they see, and includes recent observations from using an unusual grid pattern with able students. The question of whether a visual approach is always helpful, or whether it can be a distraction, will be raised in relation to number patterns and generalisation. Key words: pattern, shape, congruence, reflection, rotation.

Introduction Claims have been made in the past about children’s ability to appreciate visual patterns. For example, Wells (1982) states that ‘Visual relationships, especially visual patterns, are a superb source of ideas of structure because almost all pupils find it so easy to ‘see’ the pattern’ and Merttens (1986) suggests that children who are not very fast or efficient at computation or whose memories are poor can yet become very good at detecting patterns. The purpose of my research work at Leeds was to investigate how easily pupils do ‘see’ geometrical patterns. It was also interesting to explore what students perceive as pattern.

What is a geometrical pattern? Patterns used on fabrics or wallpaper suggest the idea of repetition is important. A pattern seems to need regular repetition to distinguish it from other designs. Indeed, ‘Designing a pattern means repeating something’ were the opening words of a British mathematics television programme about patterns. Another perspective of pattern is: -- ‘any kind of regularity that can be recognised by the mind’ (Sawyer, 1963) -- ‘a configuration consisting of several elements that somehow belong together’ (Zusne, 1970, in Reed, 1973). This use of ‘pattern’ is more like the idea of pattern as a model, or template to be recognised or copied, as with a knitting pattern. Capítulo 3 - Geometria | Chapter 3 - Geometry

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If pattern involves repetition or copying of shapes it will only be recognised as pattern if the constituent shapes can be recognised as ‘the same’. Because of this my study (0rton, 1999) had to include a study of the recognition of congruent shapes. The following question was given to nearly 300 students aged 9 – 16 to see how well they could identify an embedded shape:

Shade in the triangle which could be placed on top of the triangle Z, if you cut it out.

Fig. 1 Question about identifying an embedded triangle

The results were: Age group

9-10

11-12

13-14

15-16

% correct

61

73

95

92

Table 1 Results for the identification of an embedded triangle

The results suggest a definite progression with age (at least from age 9 to 14) in identification of the correct shape. Why has the triangle not always been correctly identified?

Because the triangle is embedded? Ghent’s work (1956) using drawings of overlapping figures concluded that ‘young children have difficulty in perceiving a given boundary as simultaneously belonging to more than one form’. It was suggested that this could be due to a ‘narrow perception span’.

Because children focus on the outside boundary? The theory of Gestalt psychologists (see, for example, Resnick and Ford, 1984) suggests that whole configurations are perceived first and then separate elements. ‘Gestalt’ is roughly translated as ‘shape’, ‘form’ or ‘whole’ and it is emphasized that perception is always more than separate elements of sensory data.

Because the students could not cope with the necessary mental transformation? Theories of pattern perception suggest that a mental transformation involves an image but is the image pictorial or propositional? Has a picture of the triangle Z been rotated mentally or does the mental activity use a set of propositions (expressing the properties of triangle Z)? The use of visual images is suggested by template-matching theories and Selfridge and Neisser (1960 in Seamon, 1980) suggests ‘normalising operations’ take place so that unidentified stimuli are always rotated to adjust the longest 82

Pupils’ perception of shape, pattern and transformations

axis to the vertical position to aid recognition. Clements (1982) gives a summary of theories supporting and opposing visual or pictorial images and Cooper (1990) provides more evidence in support of mental representations of 3-D objects. Krutetskii’s distinction between different types of mathematical mind (Krutetskii, 1976) suggests that the thinking of some students (‘analytic’ ones) may be without pictures or entirely non-visual. The use of propositions is suggested by theories of feature analysis (see Rosch in Roth and Frisby, 1986). Features such as size, length, or orientation are considered to be extracted from the shape and compared to lists of defining features in the memory, or in this case associated with the visual picture of the triangle Z. Features used in this question might include: -- the size of the triangle; -- the sides of equal length; -- the sharpness of the angles. This approach suggests several reasons why students may fail to recognise the congruent triangle: -- poor feature extraction from the triangle that is being considered; -- inability to match the extracted features with the features of the triangle Z; -- inaccurate representation of the triangle Z; -- a wrong decision about categorising the triangle observed.

Do students see parts within a whole? Feature analysis emphasizes the features or parts of a shape but ignores the relation between the parts. Reed suggests that people store patterns as parts together with rules for linking them up (see Roth and Frisby, 1986). His experimental work with the diagram in Figure 2 shows how some parts, like the rhombus, were easier to identify than others, like the parallelograms.

Fig. 2 Shape identification

A good activity with students might be to give them tasks of finding and drawing shapes on isometric paper, to provide experience of shape identification (see Figure 3).

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Fig. 3 Identifying shapes on isometric paper

Van Hiele (1986) proposes levels of spatial development and at the first level children are not aware of the relationship between parts of a shape. Figures are only distinguished by their individual shape as a whole. Only at Level 2 would a student recognise that an isosceles triangle, like triangle Z, has two equal sides and two equal angles. In my research children’s ability to recognise congruent shapes was explored in other questions where the comparative difficulty of mental transformations would feature. The first task [question 6(a)] involved only rotation. See Figure 4.

Which of the shapes could be cut out and placed on top of shape Q? Fig. 4 Shape recognition and mental manipulation involving rotation

The results show the percentage of boys, girls, and all students giving the correct response at each age level. The actual numbers of students is represented by n. Age

9/10

11/12

13/14

15/16

% correct (boys)

61.4 (n=43)

68.1 (n=45)

84.3 (n=50)

66.7 (n=25)

% correct (girls)

44.4 (n=34)

56.4 (n=38

62.2 (n=35)

91.3 (n=22)

% correct (all)

53.8 (n=77)

62.8 (n=83)

75.0 (n=85)

78.0 (n=47)

Table 2 Results for question 6(a)

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Pupils’ perception of shape, pattern and transformations

There was a slight suggestion of improvement with age for all students but boys gave more correct responses than girls in the younger age groups. The most common wrong response was shape F.

The second task, [6(b)], involved reflection as well as rotation. A few students during individual interviews, however, asked whether turning over was allowed, suggesting that reflection might have been ignored by some students.

6(b) Which shapes could be cut out and placed on top of shape X? Fig. 5 Mental manipulation using reflection and rotation.

The results are shown in Table 3. Age

9/10

11/12

13/14

15/16

% correct (boys)

63.2 (n=43)

57.5 (n=45)

77.7 (n=50)

54.1 (n=25)

% correct (girls)

55.6 (n=34)

70.3 (n=38)

54.6 (n=35)

53.0 (n=22)

% correct (all)

59.8 (n=77)

63.3 (n=83)

67.9 (n+85)

53.6 (n=47)

Table 3 Results for question 6(b)

Shapes B, D and K involve reflection and featured amongst the shapes missed by quite a few students. It was not clear whether this was because reflection was discounted, or because the mental manipulation involved was too demanding. Another problem was that, despite the wording of the question, some students seemed to think that only one shape was required. Researchers are always battling with the sad fact that students’ understanding of what is intended in a question frequently fails to match the intentions of the question designer. Shape G, involving only a slight rotation was rarely missed.

The next part [6(c)] used more complex shapes, with the necessary mental transformations being reflection and rotation.

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6(c) Which shapes could be cut out and placed on top of shape Y? Fig. 6 Reflection and rotation with more complex shapes

The results are shown in Table 4. Age

9/10

11/12

13/14

15/16

% correct (boys)

52.3 (n=43)

53.8 (n=45)

73.4 (n=50)

53.4 (n=25)

% correct (girls)

50.4 (n=34)

62.3 (n=38)

51.7 (n=35)

47.2 (n=22)

% correct (all)

51.4 (n=77)

57.6 (n=85)

64.3 (n=85)

50.6 (n=47)

Table 4: Results for Question 6(c)

Students avoiding, or having problems with, reflection would be expected to omit shapes E, F, H and K. However, exclusive omission of all these shapes rarely happened, suggesting the influence of other factors. Surprisingly, shape B, which only required translating without any reflection or rotation, was sometimes missed. The final part of the question [6(d)] used triangles.

6(d) Which triangles could be cut out and placed on top of the shape W? Fig. 7 Reflection and rotation question using triangles

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Pupils’ perception of shape, pattern and transformations

Age

9/10

11/12

13/14

15/16

% correct (boys)

45.8 (n=43)

52.5 (n=45)

64.7 (n=50)

51.9 (n=25)

% correct (girls)

51.9 (n=34)

54.7 (n=38)

53.6 (n=35)

52.2 (n=22)

% correct (all)

48.5 (n=77)

53.5 (n=83)

60.0 (n=85)

52.0 (n=47)

Table 5 Results for Question 6(d)

The scores for this part were generally lower. Shapes G and M which required reflection were sometimes omitted but rarely exclusively. It had not been expected that shape E would be missed, but sometimes it was. The scores were probably less meaningful than had been hoped, because of some students only giving one answer. During individual interviews the students were timed as they wrote down their answers to these questions and there was some evidence that the length of time to answer the question decreased with age.

Pattern recognition and general ability Tables 1-5 reveal some surprisingly low scores for the 15/16 year-olds. The reason for this was that the students available during a busy examination term tended to be those of lower ability. It had been anticipated that there would be high correlation between students’ general ability and their ability to perceive pattern and this was explored using scores of general ability derived from the AH4 test (Heim, 1970) and total scores (or sub-total scores) from my pattern recognition test. The relationship was not as simple as expected. There seemed to be more correlation in the lower half of the ability range than in the upper half. Although correlation was evident over the whole ability range the significant value of r would not have been obtained without the lower ability students. The results suggest that this correlation might not be detected in higher ability setted classes. They do, however, explain the comparatively low results for 15/16 year-olds.

Frames of Reference Question 4 was invented to explore the possible influence of a frame of reference when transforming a shape.

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Fig. 8 Question to explore use of frames of reference

This was not an easy question to analyse. The drawings on the backs of the flags made it difficult to interpret what transformation the students were using. For example, one transformed flag in 4(a) could be a reflection of the front of the flag in a horizontal axis or a rotation of the back of the flag. The other transformed flag in 4(a) could be a rotation of the front of the flag or a reflection of the back of the flag in a diagonal axis. It would also be possible for students using the outline of the flags as a frame of reference to complete 4(a) and 4(b) without any mental transformation. Indeed many of the students in the individual interviews moved their answer paper round to align the flagpoles with the ‘vertical’ of their desktop.

The table shows the percentage of the total marks available scored by boys (M), girls (F) and all (A) students in each part of the question at the different age levels. The sample sizes were the same as in question 6. Age

9/10

11/12

13/14

15/16

4(a)

M 85.2

M 94.7

M 91.2

M 87.0

4(a)

F

54.2

F

79.5

F

94.6

F

91.3

4(a)

A

71.3

A

87.8

A

92.8

A

89.0

4(b)

M 55.7

M 78.7

M 85.3

M 94.4

4(b)

F 33.3

F 66.7

F 87.8

F 84.8

4(b)

A 45.6

A 73.3

A 86.4

A 90.0

4(c)

M 54.6

M 72.3

M 76.5

M 75.3

4(c)

F 36.1

F 53.0

F 72.1

F 76.8

4(c)

A 46.3

A 63.6

A 74.6

A 76.0

Table 6 Results of Question 4

Question 4(b) was generally found to be more difficult than 4(a). Younger students, in particular, found the asymmetrical figure on the flag more problematic. It required direction to be considered. Older students seemed to make more errors with 4(c) than with the other parts.

There was no frame of reference to use. Also the orientation of the handle was

often ignored, and all transformations taken as rotations. It had been anticipated that reflection in the horizontal would be found easier than rotation but there was no evidence for this in either 4(a) or 4(c). 88

Pupils’ perception of shape, pattern and transformations

It was not surprising that a frame of reference simplified the question. Bryant (1974) claimed that the difficulty young children have in distinguishing between / and | is removed when they can use a perceptual framework like the side of a page, and Küchemann (1980) reported on the helpful effect of a grid in questions involving reflection and rotation.

Comparative difficulty of transformations It was hoped that my research would throw more light on the comparative difficulty of mental transformations. According to Piaget et al (1960) children find reflections easier than rotations. Perham (in Dickson et al, 1984) has distinguished between horizontal, vertical and oblique transformations and found that the last caused considerable difficulty. Schultz (in Dickson et al, 1984) found similar results and noted that if young children did attempt an oblique translation the image was often given in the direction of the translation [see (A) in Figure 9]. With reflections and rotations the tendency was to turn the image so that it faced the direction of reflection or turn [see (B) in Figure 9]. Küchemann (1980) found that many 14 year-olds ignored the slope of an oblique mirror line and reflected vertically or horizontally [see (C) in Figure 9] and the first APU survey (APU,1980) showed that only 14 per cent of eleven year-olds could reflect a shape in an oblique line.

(A)

(B)

(C)

Fig. 9 Common misconceptions

Küchemann suggested that children were working with reflection at different stages: global – the object was considered as a whole and reflected as a single object; analytic – reducing the object to points and reflecting these, before drawing in lines; semi-analytic – reflecting part of the object and then drawing in the rest. With rotation 14 year-olds have been shown (Hart, 1981) to have problems if the centre of rotation is not on the object and only 17 per cent could successfully give the image after rotation when the object was obliquely orientated. Research by Bell (1989) highlights two common misconceptions: -- that horizontal and vertical objects always have horizontal or vertical images [cf. (C) in Figure 9]; -- that a line dividing a shape into two parts of equal area must be a line of symmetry [ e.g.

]

Thomas (in Dickson et al., 1984) worked with children aged nine, twelve, fifteen and seventeen years. They were asked to imagine the result of reflecting or rotating a cardboard square with a letter of the alphabet on it and to choose from Capítulo 3 - Geometria | Chapter 3 - Geometry

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four possible images. The main findings were: -- All children found it difficult to rotate a letter like S and N which already has rotational symmetry; -- Horizontal reflection with the non-symmetric J was difficult; -- The children aged nine scored much lower on all transformations; -- No striking difference was found between horizontal and vertical reflection, nor between rotations in different directions.

The ability to select an image from several alternatives has been found to precede the ability to construct the image (Perham in Dickson et al, 1984). In my work I decided to use a question where children were required to select from five possible images to explore the relative difficulty of mental transformations (see Figure 10). Each part of the question starts with a model shape which is transformed. Table 7 shows the mean scores for each part of the question at the different age levels. (a) Ref |

(b) Rot Õ 90˚

(c) Rot Ö 45˚

(d) Ref ___

(e) Ref /

(f) Rot 180˚

(g) Ref

9/10

0.70

0.31

0.43

0.40

0.04

0.33

0.48

11/12

0.76

0.52

0.65

0.53

0.07

0.47

0.70

13/14

0.97

0.63

0.86

0.69

0.08

0.69

0.75

15/16

0.96

0.64

0.90

0.58

0.00

0.62

0.72

Table 7 Results for Question 10.

Part (a) [a vertical reflection] was clearly the easiest part and part (e) [an oblique reflection] was the most difficult. Part (g) was also an oblique reflection but much easier than (e) as the shape to be transformed in (g) had a vertical line corresponding to the vertical line of the model shape, so it was possible to guess the orientation of the image from the orientation of the transformed model shape. Students had, in effect, been provided with a frame of reference.

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Fig. 10 Question to explore mental transformations

The strategy of using the orientation of the transformed model shape also seemed to have been used, erroneously, in (e). The table below, showing the frequency of wrong responses, gives the most popular wrong responses for (e) as image 3 and image 4 (the images whose orientation matched the orientation of the transformed model shape).

10

(a)

10

(b)

Image

1

2

3

4

1

2

3

5

No. of responses

15

3

3

4

21

76

18

4

10

(c)

10

(d)

Image

1

2

4

5

1

3

4

5

No. of responses

11

28

20

3

25

58

20

3

10

(e)

10

(f)

Image

1

2

3

4

2

3

4

5

No. of responses

22

26

93

114

67

18

27

3

10

(g)

Image

1

3

4

5

No. of responses

13

32

10

15

Table 8 Frequencies of wrong responses for Question 10 (Figure 10)

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Indeed this strategy could well have been used in parts (c) and (f) where the model shape and the shape to be transformed also had corresponding vertical lines. It would work well with (c) giving image 3 as the only shape whose orientation matched the orientation of the transformed model shape, but in (f) it would leave four possible images. This might explain why (c) a rotation question was not found as difficult as expected but (f) was found more difficult. Part (d), a horizontal reflection was not found easy, as expected. The most popular wrong response was image 3 suggesting that horizontal reflection had been recognised but the mental transformation had been inaccurate. The relative complexity of the shape might have been a factor, as suggested above by Thomas. Part (b), a rotation was not expected to be easy. Some students might have imagined a vertical line through the S to compare with the model (‘standing upright, moving on to its side’ as one student explained it) or rotation could have been recognised but inaccurately performed. Image 2 would have been the expected response in both cases, and this proved to be the most common error. Individual interviews revealed that oblique reflection was not generally recognised. The transformation was seen as a combination of rotation and reflection: ‘Gone round and then over’; ‘Move round and then flip over’; ‘Turn and then tilt’; ‘Reflection in what would be the x-axis and then gone anti-clockwise a bit’. From that perspective, (e) is an unreasonable question. The size of rotation becomes important and none of the possible images show the correct angle of rotation. It was not clear from the students’ explanations, however, whether they were using mental transformations, or whether they used a less-visual analysis.

Student descriptions In another part of my study other questions were used during individual interviews to further explore how students describe what they ‘see’. For example, students were shown the shape in Figure 11 and asked whether they saw any pattern in it.

Fig. 11 One of the additional questions used during individual interviews

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All students claimed to see pattern. Their explanations included: -- mention of repeated shapes “There’s a shape at the bottom and three circles above it … in each corner” -- mention of pictures “Like a torch and balls … and a track”; “A swastika” -- recognition of turning “There’s an L going round each time”; “It turns like a clock”; “Sort of arranged in four parts turned round each time” -- mention of symmetry “It’s a balanced shape” -- occasionally “It’s a rotational pattern”

Responses showed that: -- Some children saw it as a pattern without apparently recognising any turning or rotational symmetry; -- Only the older students had the necessary language to mention rotational symmetry.

Geometric Patterns and Number Patterns The relationship between number patterns and geometric patterns can be fascinating. I should like to mention one pattern which I have used with clever students aged 12 who have attended our Mathematics Masterclasses at the University. I first met this pattern of numbers in the journal ‘Mathematics in School’. It was given without explanation as shown in Figure 12.

Fig. 12 An interesting number pattern

Where have the numbers come from? How have they been generated? The students are given this pattern in a session entitled Fibonacci Fun so it does not take long for them to recognise the numbers 1, 1, 2, 3, 5, 8, as the start of the Fibonacci sequence. The next numbers 3, 1, 4, 5 … are usually described as the units digits of the rest of the numbers. Mathematicians might mention numbers modulo 10. The arrangement Capítulo 3 - Geometria | Chapter 3 - Geometry

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of the numbers enables many patterns to be spotted. Students usually first comment on individual numbers: the position of the zeros, the square made by the four 5s in the centre or the rectangle made by the other four 5s. Then they mention pairs of equal numbers or pairs of numbers whose sum is 10. When students are exploring the pattern on their own, lines are drawn spontaneously to show the relationships that have been discovered (see Figure 13).

Fig. 13 Some of the patterns noticed by students

I am grateful to my former colleague William Gibbs who investigated the numbers further by taking pairs of consecutive numbers and using each pair as co-ordinates which can be recorded on a grid (Gibbs, 1999). I give this activity to the students in the Masterclass, asking them to use a square of their grid to represent each co-ordinate pair, to shade the squares and to search for pattern. Figure 14 shows the resulting shaded squares.

Fig. 14 Shaded squares showing co-ordinate pairs

Initially students seem to comment on the shapes they see, the lines of squares that are shaded, or the square ‘holes’ surrounded by shading. Some recognise the repetition of shapes, and those who ‘see’ the ‘square with tails surrounding a blank square’ being repeated, sometimes eventually ‘see’ the rotation and are able to identify the centre. It is easier to appreciate the rotational symmetry without the border of squares alongside the axes (see Figure 15). 94

Pupils’ perception of shape, pattern and transformations

Fig. 15 Separating the rotational and line symmetry

It is interesting that this border of squares possesses line symmetry. The axis of symmetry is the line y=x. Students seem to appreciate this when shown, but I have not yet met a student who has recognised the line symmetry from their completed shading.

Using shape patterns to ‘explain’ number patterns Patterns of shapes have been linked with number patterns in many school text books. Indeed we hope they will help students’ appreciation of the number patterns. My research work and the results of others, who have highlighted the gradual development and misconceptions of students as they work with shapes, patterns and transformations, raises the question whether this is always a good approach. Let us consider some examples.

It is natural to illustrate the square numbers using squares of dots (see Figure 16)



1x1

2x2

3x3

4x4

5x5



1

4

9

16

25

Fig. 16 Square numbers

It is a good illustration for linking the numerical and shape meanings of ‘square’. Connecting the borders helps young children to recognise the squares but may mask some of the basic numerical properties! The illustration can be adapted to help students to ‘see’ that the square numbers can be formed by the addition of odd numbers (see Figure 17).

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x 1

v

v

v

v

+

+

+

+

+

+

v

o

o

o

o

+

o

o

+

v

x

o

x

o

+

x

o

+

v

1+3=2x2

1+3+5=3x3

1+3+5+7=4x4

Fig. 17 Square numbers as the sum of odd numbers

The generalisation ‘The sum of n odd numbers = nxn’ may follow, though the algebraic notation



2

will

€

The triangle numbers can also be illustrated using dots (see Figure 18)



∑ (2r − 1) = n r =1

be beyond most students.



r =n

1

1+2

1+2+3

1+2+3+4

1+2+3+4+5

1

3

6

10

15

Fig. 18 Triangle numbers as the sum of natural numbers

To continue further, using right angled triangles, and gain insight into summing the natural numbers may seem to be a productive extension, but it will not be helpful to students who cannot recognise the congruence of the two right angles triangles (see Figure 19).

o

o

1=½(1x2)

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

o

3=½(2x3)

6=½(3x4)

10=½(4x5) r =n

Fig. 19 Visual approach towards the generalisation

∑ r = 1/ 2n(n + 1) r =1

The use of geometric shapes, which give rise to number patterns and thus provide the opportunity to express general€ ity have become popular in text books. What is not clear is whether students find the shapes themselves helpful in coming to ‘see’ the relationship between the variables involved. My own research (Orton, 1997 and Orton et al, 1999) has involved individual interviews with 30 students aged nine to thirteen using matchstick patterns to investigate linear patterns. The first three shapes were built for the students (see Figure 20) and then they were encouraged to build the next shape in the pattern themselves. 96

Pupils’ perception of shape, pattern and transformations

Fig. 20 Matchstick patterns

Students were asked questions about the numbers of matches used, and whether they could predict the numbers of matches that would be needed for the twentieth, the hundredth and the nth shapes. The main focus of the research was on the use of the concrete support but it did confirm the results of others (e.g. Stacey, 1989) including: -- the automatic use of differencing used by students in their first attempt to ‘see’ the pattern of numbers; -- the common error of using a product of the difference to predict another term; -- the common error of making a ‘short cut’ (e.g. finding the twentieth term by multiplying the fourth term by five). Several children showed success in generalising with numbers and some managed to give algebraic answers but the matchsticks were not always used. Once the common difference had been found several students chose not to use the matches. The conclusion had to be that the matches provided concrete support, not to all, but to some, students some of the time. To broaden the experience of learners and provide help to some in spotting the multiplicative relationship, further examples of matchstick patterns were suggested. It was hoped that the structure of the matchsticks would suggest the structure of the formula.

I have recently been using a question (Edexcel, 2001) based on a sequence of hexagons shapes (see Figure 21)

Fig. 21 Hexagon pattern

The question supplies a table showing the number of dots in each of the first 4 shapes in the sequence: Shape number (n)

1

2

3

4

Number of dots (d)

7

13

19

25

Table 9 Text book table

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Students are asked: -- to write down the number of dots for shape number 8 -- to explain how they worked out the answer -- to write down a formula which can be used to calculate the number of dots, d, in terms of the shape number, n. I have frequently observed students -- being tempted to use the ‘short-cut’ method of doubling the result for shape number 4 to obtain the number of dots for shape number 8; -- using the difference 6 to correctly work out the number of dots for shape number 8 but then failing to make further progress; -- ignoring the hexagons and studying only the table of numbers. (Indeed the structure of the question almost suggests that this is the correct approach.) Because I hope the visual representation will help students to understand the algebraic expression I usually try to refocus attention on the hexagons. Unfortunately students do not always see that six new dots are added when each new shape is created from the previous one. Some students see the seven dots of the first shape as so significant that each subsequent shape is related to this. The second shape is seen as two lots of 7, minus one dot, because of the overlap; and the third as three lots of 7, minus 2 overlapping dots, etc. Of course, this can be generalised as 7n – (n-1) but the journey to this expression is a lot more complex for struggling students.

In this question and others dealing with linear relationships a much more successful strategy for students seems to be to relate the numbers to the multiplication tables. In this case, because the difference is six, the table can be extended to show the 6-times table: Shape number (n)

1

2

3

4

Number of dots (d)

7

13

19

25

6-times table

6

12

18

24

….

8

n

Table 10 Extended table

The focus on the multiplication table draws students away from the use of differences and helps them to use the multiplicative relationship which can then often be successfully generalised. This strategy of ‘looking for multiplication tables’ has been discussed by Hargreaves et al (1999).

98

Pupils’ perception of shape, pattern and transformations

Another pattern problem involving hexagons has been discussed by Warren (1992).

Fig. 22 Flower beds, surrounded by hexagonal paving slabs

Students are required to find a formula to give the number of slabs needed for any number of flower beds. Since no table is given, students will be forced to study the hexagons. They need to be able to separate the hexagons into parts which relate to the question. Students who can mentally separate the hexagonal tiles as shown in Figure 23 then need to see that the number of tiles needed for n flower beds is represented by the expression 2 + 4n.

Fig. 23 Separating the hexagonal tiles

Students who focus on pairs of vertical hexagons may see n+1 pairs of adjacent tiles (o) and n pairs separated by the flower beds (x) which would be represented by the expression 2(n+1) + 2n.

Fig. 24 Seeing pairs of hexagons

Students who focus on the 6 tiles round each bed and try to subtract the overlapping hexagons would need to be able to cope with the expression 6n – 2(n-1). There may be valuable opportunities for clever students to appreciate the algebraic equivalence of these expressions. The algebraic demands on weaker students may mean that they fail to generalise at all.

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Would the strategy of looking for multiplication tables help?

No.of flower beds

1

2

3

4

n

No.of slabs

6

10

14

18

22

? - times table Table 11 Searching for a multiplicative relationship

Counting is not usually a problem so we hope the second row of the table will be completed. Noticing the common difference of 4 may then lead to comparison with the 4 times table and the multiplicative relationship being ‘seen’.

If this approach is adopted, however, are we sacrificing what can be an enriching experience for some students, and an opportunity to develop flexible thinking for others? Students who can link their generalisation and algebraic representation to the given visual data will have a deeper understanding and appreciation of the power of algebra. Teaching can often be a compromise! We aim to help all students to enjoy some measure of success, and therefore introduce successful strategies. At the same time we must not ignore the skills of the few natural visualisers in our classes or deny others the opportunities of perhaps developing visual thinking.

References APU (1980) Mathematical Development: Primary Survey Report Number 1. London: HMSO. Bell, A. (1989) Teaching for the test. Times Educational Supplement, 27 October. Bryant, P. (1974) Perception and Understanding in Young Children. An experimental approach. London: Methuen. Edexcel (2001) GCSE Mathematics Intermediate Course. Oxford: Heinemann Educational Publishers. Clements, K. (1982) Visual imagery and school mathematics. For the Learning of Mathematics. 2(2), 2-9. Cooper, L.A. (1990) Mental representation of 3-dimensional objects in visual problem-solving and recognition. Journal of Experimental Psychology: Learning Memory and Cognition 16, 1097-1106. Dickson, L., Brown, M. and Gibson, O. (eds) (1984) Children Learning Mathematics. London: Cassell. Ghent, L. (1956) Perception of overlapping and embedded figures by children of different ages. American Journal of Psychology, 69, 575-86. Gibbs, W. (1999) Pattern in the classroom. In A. Orton (ed.) Pattern in the Teaching and Learning of Mathematics. London: Cassell. 207-220. Hargreaves, M., Threlfall, J., Frobisher, L. and Shorrocks-Taylor, D. (1999) Children’s strategies with linear and quadratic sequences. In A. Orton (ed.) Pattern in the Teaching and Learning of Mathematics. London: Cassell. 67-83. Hart, K.M. (ed.) (1981) Children’s Understanding of Mathematics 11-16. London: John Murray. Heim, A.W. (1970) AH4 Group Test of General Intelligence. Windsor: NFER. Krutetskii, V.A. (1976) The Psychology of Mathematical Abilities in School Children. Chicago: University of Chicago Press. Küchemann, D. (1980) Children’s difficulties with single reflections and rotations, Mathematics in School, 9(2), 12-13. 100

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Merttens, R. (1986) Introducing Patterns. Junior Education. August 1986. Topic Pack. Orton A. (ed.) (1999) Pattern in the Teaching and Learning of Mathematics. London: Cassell. Orton, J. (1999) Children’s perception of pattern in relation to shape. In A. Orton (ed.) Pattern in the Teaching and Learning of Mathematics. London: Cassell. 149-167. Orton, J. (1997) Matchsticks, pattern and generalization. Education 3-13, 25(1), 61-65. Orton, J., Orton A. and Roper, T. (1999) Pictorial and practical contexts and the perception of pattern. In A. Orton (ed.) Pattern in the Teaching and Learning of Mathematics. London: Cassell. 121-136. Piaget, J., Inhelder, B. and Szeminska, A. (1960) The Child’s Conception of Geometry. London: Routledge and Kegan Paul. Reed, S.K. (1973) Psychological Processes in Pattern Recognition. New York: Academic Press. Resnick, L.B. and Ford, W.W. (1984) The Psychology of Mathematics for Instruction. Hillsdale, NJ: Lawrence Erlbaum. Roth, I. and Frisby, J.P. (eds) (1986) Perception and Representation: a Cognitive Approach. Milton Keynes: Open University Press. Sawyer, W.W. (1963) Prelude to Mathematics. Harmondsworth: Penguin Books. Seamon, J.G. (1980) Memory and Cognition. Oxford: Oxford University Press. Stacey, K. (1989) Finding and using patterns in linear generalising problems. Educational Studies in Mathematics, 20, 147-64. Van Hiele, P.M. (1986) Structure and Insight. London: Academic Press. Warren E. (1992) Beyond manipulating symbols. In A. Baruto and T. Cooper (eds), New Directions in Algebra Education. Red Hill, Australia: Queensland University of Technology. Wells, D. (1982) Three Essays on the Teaching of Mathematics. Bristol: Rain Publications.

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Wallpaper Patterns in Origami

Wallpaper Patterns in Origami Luís António Teixeira de Oliveira [email protected], Faculdade de CIências da Universidade do Porto e Centro de Matemática da Universidade do Porto1 RESUMO Este artigo refere-se a um projecto desenvolvido por um grupo de três futuros professores e cujo objectivo era construir um modelo em origami para cada um dos grupos de simetria dos dezassete padrões de papel de parede. O projecto visava desenvolver o conceito de simetria, a capacidade de reconhecer simetrias, e a capacidade de estabelecer conexões entre diferentes simetrias num objecto. Este artigo descreve os desafios matemáticos enfrentados ao longo deste projecto e como foram superados. Palavras-chave: padrões de papel de parede, pavimentação, grupo de simetria, restrição cristalográfica, origami. ABSTRACT This paper relates to a project developed by a group of three student teachers and whose goal was to construct a model in origami for each one of the seventeen symmetry groups of wallpaper patterns. The project aimed at developing the notion of symmetry, the ability of recognizing symmetries, and the capacity of establishing connections amongst different symmetries in an object. This paper reports the Mathematical challenges faced during this project and how they were overcome. Key words: wallpaper pattern, tessellation, symmetry group, crystallographic restriction, origami.

Wallpaper patterns Wallpaper patterns (WP) are geometric plane patterns with two translation symmetries associated with linearly independent vectors u and v. Further, all other translation symmetries must be associated with vectors that are linear combinations of u and v with integer coefficients. In any WP, it is possible to choose the vectors u and v such that: a) no translation symmetry is associated with a vector of length less than the length of u, and b) no translation symmetry associated with a vector linearly independent from u has its vector length less than the length of v. From now on, we consider u and v as a pair of vectors in the conditions just described. Given a point in a WP, consider all its images under the WP translation symmetries. The set of lines in the direction of u or v that contain these images forms a grid called a wallpaper pattern net. Any such net divides the WP into small equal tiles with the shape of a parallelogram whose sides have the direction and length of the vectors u and v. These tiles are called unit cells and they tile the WP. Thus, WPs are special tessellations.

 

Fig. 1 A wallpaper pattern with a dark green wallpaper pattern net.

The set of all symmetries of a WP with the usual composition of transformations constitute an algebraic structure called 1 This work was partially supported by Fundação para a Ciência e a Tecnologia (FCT) through the Centro de Matemática da Universidade do Porto (http://www.fc.up.pt/CMUP/) Capítulo 3 - Geometria | Chapter 3 - Geometry

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a wallpaper symmetry group (WSG). There are exactly seventeen non-isomorphic WSGs. A proof for this result can be found in Martin’s book (Martin, 1982). We will use a standard code notation for these groups. The explanation for these codes can be found in http://en.wikipedia.org/wiki/Wallpaper_group.

The project Origami is the Japanese art of paper folding. There are two basic styles of Origami: single paper origami and modular origami. Single paper origami is an origami made from a single sheet of paper, while modular origami is an origami made from several modules, each module made from a single sheet of paper or from other modules, that are put together without following apart. Usually, modular origamis are sturdy “3D objects” due to friction amongst the modules. It is often the case that modular origami only stays together after the last module is put in its place. Tomoko Fuse has developed some work on “planar modular origami”. The technique used in this book Origami quilts (Fuse, 2001) for joining the modules is different from the usual techniques in modular origami as it allows us to lock two modules in a way that we can build a plane origami that does not fall apart even when we move it. This paper reports on a project whose goal was to study the possibility of building a model in origami for each one of the seventeen WSGs based on the constructions presented in Fuse’s book (Fuse, 2001). The project was developed by a group of three student teachers as part of their work for the student teaching internship. With this project we aimed at developing the student teachers’ visualization skills and their capacity for recognizing geometric patterns, helping them in better identifying symmetries and in understanding how different symmetries interact with each other.

The crystallographic restriction The rotation center A of a rotation symmetry of a WP is called an n-rotation center if n is the largest integer such that the WP has a rotation symmetry with rotation center A and rotation angle 360º/n. One of the major results in WP symmetry theory is the so-called crystallographic restriction. This result tells us that any rotation center of a WP is a 2, 3, 4 or 6-rotation center. For example, a 120º rotation symmetry has a 3-rotation center. A proof for the crystallographic restriction can be found in Martin’s book (Martin, 1982). The crystallographic restriction can be, in fact, strengthened by a deeper analysis on rotation symmetry composition. A WP with an n-rotation center and an m-rotation center must have a k-rotation center for the least common multiple k of n and m. In particular, no WP has simultaneously a 4-rotation center and a 3 or 6-rotation center. We can now divide the WSGs into five classes according to classification of their rotation centers: a) with no rotation centers; b) with a 6-rotation center; c) with a 4-rotation center; d) with 3-rotation centers only; and e) with 2-rotation centers only. The project was developed using this five classes division. The WPs with symmetry groups in the same class shared many characteristics. As we shall see, most models presented for WSGs in the same class have the same underlying origami structure. They usually differ only on the choice of color or orientation of the modules used. Before addressing the WSGs in each of the five classes, we need to analyze the information stored in the unit cells. 104

Wallpaper Patterns in Origami

Unit cells All symmetries of a WP have their imprint on any unit cell and, therefore, unit cells give important information about the symmetry structure of a WP. For instance, all information about the translation symmetries of a WP is stored in the sides of a unit cell since they define the vectors u and v. Basic facts about composition of rotation and translation symmetries allow us to conclude that a WP with a rotation symmetry f and a translation symmetry associated with a vector w also has a translation symmetry associated with the vector f(w). If we take now into account the properties of the vectors u and v, we immediately conclude that u and v must have the same length if the WP has a rotation center that is not a 2-rotation center. We can say, in fact, a bit more: a) if the WP has a 4-rotation center, then u and v are also perpendicular and the unit cells are squares; and b) if the WP has 3 or 6-rotation centers, then the angle between u and v is either 60º or 120º, and the unit cells are rhombuses with 60º and 120º internal angles. Note, however, that, in case b), if the angle between u and v is 120º, then we can consider v’ = u + v instead of v and the angle between u and v’ is 60º. Therefore, in case b), we can always choose v such that the angle between u and v is 60º. A WP has many different unit cells since we can build wallpaper pattern nets from any point on the WP. For WPs with rotation symmetry we shall only consider unit cells given by wallpaper pattern nets constructed from a maximum nrotation center (the greatest possible n such that there is an n-rotation center). We make this choice since all these unit cells have the same placement for their rotation centers. The next figure identifies all rotation centers in the unit cells according to the WP maximum n-rotation center. The red hexagons represent 6-rotation centers, the blue squares represent 4-rotation centers, the yellow triangles represent 3-rotation centers, and the yellow diamonds represent 2-rotation centers. Observe that all WSGs with rotation symmetries belonging to the same class have the same rotation centers’ placement scheme. For the WPs with 2-rotation centers only, we can have many parallelogram shapes for unit cells. However, as we shall see, their shape will influence the possible reflection symmetries. 6-rotation centers

 

4-rotation centers

 

3-rotation centers only

 

2-rotation centers only

 

Fig. 2 Rotation centers’ placement scheme for WPs

A simple analysis of reflection symmetry allows us to conclude that a WP with two crossing reflection or glide reflection axes also has rotation centers. Thus, all reflection or glide reflection axes on a WP with no rotation centers must be parallel lines. Capítulo 3 - Geometria | Chapter 3 - Geometry

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For wallpaper patterns with 3, 4 or 6-rotation centers, we can indicate all possible reflection or glide reflection axes on a unit cell. In the following figure, we represent in green a unit cell for a WP and in gray circles maximum n-rotation centers. The red lines represent the possible reflection or glide reflection axes, while the dashed blue lines represent the possible glide reflection axes only. With 4-rotation centers

With 3 or 6-rotation centers

 

 

Fig. 3 Possible reflection or glide reflection axes in wallpaper patterns.

For WPs with 2-rotation centers only, the analysis of reflection symmetries depends on the parallelogram shape of the unit cells. If this shape is neither a rectangle nor a rhombus, then the WP has no reflection symmetry. The following figure addresses the unit cells’ shape of a rectangle and of a rhombus (or both, that is, a square). Again, the red lines are possible reflection or glide reflection axes and the blue dashed lines are possible glide reflection axes only. Rectangle

 

Square

Rhombus

 

 

Fig. 4 Possible reflection or glide reflection axes in wallpaper patterns with 2-rotation centers only.

Wallpaper patterns with no rotation centers The class of WPs with no rotation centers has four symmetry groups and we already saw that the reflection and glide reflection axes are all parallel lines. In Fuse’s book (Fuse, 2001) there is a construction for a 4-star origami module by joining four “kite” modules together (Figure 5). We can choose the color for each kite module. There is also another origami module (in black in Figure 5) that allows us to join together 4-star modules. Thus, we can construct several origami WPs by repeating a 4-star origami module.

Fig. 5 Kite modules to construct a 4-star module and a module to join 4-star modules.

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The existence of rotation or reflection symmetry in these WPs depends entirely on the existence of rotation or reflection symmetry, respectively, on the 4-star origami module. Such 4-star origami modules have no rotation symmetry precisely when at least two opposite kites have different colors. A quick analysis tells us there are five different types of 4-star origami modules with no rotation symmetry. However, the corresponding five origami WPs models only model three of the four WSGs with no rotation centers (there are two pairs that model the same symmetry group). From the five types of 4-star modules, two of them have no reflection symmetry. They originate models for the so-called p1 symmetry group that has only translations as symmetries. Two others have a pair of opposite kites with the same color and a diagonal reflection axis. They originate models for the cm symmetry group which has translations, reflections and glide reflections with glide reflection axes that are not reflection axes. Finally, the last 4-star module has two consecutive pairs of kites with the same color and has therefore a non-diagonal reflection axis. This last 4-star module originates a model for the pm symmetry group. This group has translation and reflection symmetries (glide reflection axes are reflection axes too). cm

  pg

p1

 

  pm

 

Fig. 6 Origami models for the WSGs with no rotation centers.

The fourth WSG with no rotation centers is the pg symmetry group that has only translation and glide reflection symmetries. We cannot construct a model for this group using only one 4-star origami module. We overcome this problem by considering a block formed by two 4-star modules side-by-side such that one is the horizontal reflection of the other. If the 4-star modules have neither rotation nor reflection symmetry, then the block has neither rotation nor reflection symmetry. Using this block as a unit cell we construct a model for the pg symmetry group.

Wallpaper patterns with 4-rotation centers or with 2-rotation centers only We could think about 4-star modules with some rotation symmetry and about the origami WP models obtained from them. We would obtain models for some WSGs with 4-rotation centers and for some WSGs with 2-rotation centers only. In fact, if we allow using blocks of 4-star modules as for the pg symmetry group, we can construct an origami model for each WSG with 4-rotation centers and for each WSG with 2-rotation centers only. However, some of the Capítulo 3 - Geometria | Chapter 3 - Geometry

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blocks would need many 4-star modules (not just two as for the pg symmetry group). Although still based on Fuse’s book (Fuse, 2001), we decided to use different modules to construct origami models for the WPs with 4-rotation centers or with 2-rotation centers only. We decided to address these two classes together since there was no relevant strategic difference in the construction of the origami modules for both classes. However, there was an important concept that made a difference when constructing origami models for WSGs within each class, the concept of orientation of an object. One of the modules constructed in Fuse’s book (Fuse, 2001) is the “windmill module” (Figure 7). This has clearly rotation symmetry, but no reflection symmetry since any reflection changes its orientation. These oriented objects were useful in constructing origami models that kept most rotation symmetries but eliminated some or all reflection symmetries.

Fig. 7 Windmill origami module.

There are five WSGs with 2-rotation centers only and three WSGs with 4-rotation centers. In the following figure we present an origami model for each one of these symmetry groups. However, we do not present a description for the symmetries in each one of them since this description can be found in (Martin, 1982).

With 2-rotation centers only p2

With 4-rotation centers

pmm

 

 

pmg

p4

 

pgg

 

 

p4m

 

p4g

cmm

 

 

Fig. 8 Origami models for WSGs with 2-rotation centers only and with 4- rotation centers. 108

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Wallpaper patterns with 3-rotation centers only The origamis built for the three previous classes of WSGs use modules with 90º or 180º rotation symmetry. For the WSGs with 3-rotation centers only we need to consider modules with 120º rotation symmetry instead. Thus, we need to look for modules with a triangular shape. Fortunately, we can find many such modules in Fuse’s book (Fuse, 2001). There are three WSGs with 3-rotation centers only: p3, p3m1 and p31m. The p3 symmetry group has translation and rotation symmetries only. The other two symmetry groups also have reflection symmetries and glide reflection symmetries with glide reflection axes that are not reflection axes. The symmetry groups p3m1 and p31m are distinguished by the placement of their rotation centers: all rotation centers of a p3m1 model are on reflection axes, while some rotation centers of a p31m model are not on reflection axes. We can easily draw a standard model scheme for each of these three WSGs using colored triangles. In the next figure we present such schemes and we include the corresponding origami models we constructed. Observe that we used two distinct origami constructions for these models. However, by the standard model scheme, we see that we can build models for these three symmetry groups using any one of these two origami constructions. p3m1

p31m

 

 

p3  

Fig. 9 Model schemes for WSGs with 3-rotation centers only and the corresponding origami models.

Wallpaper patterns with 6-rotation centers A natural way to construct models for WSGs with 6-rotation centers is by using hexagon like objects. Unfortunately, Fuse’s book (Fuse, 2001) has no useful such origami module. We knew from the beginning that this class was the most problematic one. We overcome this problem when we realized that if we put four unit cells together (Figure 10), then a 6-rotation center (in red) is the center of a regular hexagon whose vertices are 3-rotation centers (in yellow). Thus, we started thinking in terms of these hexagon vertices instead.  

Fig. 10 Four unit cells together in WPs with 6-rotation centers. Capítulo 3 - Geometria | Chapter 3 - Geometry

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There are two WSGs with 6-rotation centers: p6 and p6m. The difference between them is that p6 has no reflection symmetry. Thus, we had to come up with a model for p6 with some kind of orientation to avoid reflection symmetry. We should say that the two origami models we constructed are not indicated in Fuse’s book (Fuse, 2001). We had to look at the different module combinations presented in the book and come up with our own module combination. In the next figure, we present the origami models constructed for those two symmetry groups. p6

p6m

 

 

Fig. 11 Origami models for the WSGs with 6-rotation centers.

Conclusion When the project began the student teachers evidenced a clear difficulty in identifying the unit cells. They showed the usual tendency to look at horizontal or vertical directions for the vectors u and v, not considering any other cases. The student teachers’ awareness for other translation symmetry directions improved and, by the end of the project, they could easily identify the unit cells for new WPs. The most problematic topic was the recognition of glide reflection symmetries. The student teachers had difficulties in visually identifying glide reflection axes and, even when doing so, they were unsure about their conclusions. At some point, they started using information about rotation centers and reflection axes to search for or confirm the existence of some glide reflection axes. This suggested some improvement in the student teachers’ awareness about the connections amongst different symmetries in WPs. In my opinion, this project developed the student teachers’ overall visualization skills. They became more proficient in recognizing patterns and symmetries in plane geometric objects. I believe they now have a better understanding of many relationships amongst different plane symmetries.

References Fuse, T. (2001). Origami quilts. Tokyo, Japan: Japan Publications Trading Company. Martin, G. E. (1982). Transformation geometry: An introduction to symmetry. NY: Springer-Verlag. http://en.wikipedia.org/wiki/Wallpaper_group, accessed on September 19, 2009.

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Looking for patterns in geometric transformations with pre-service teachers Lina Fonseca [email protected], School of Education of the Polytechnic Institute of Viana do Castelo RESUMO A matemática é considerada por vários autores como a ciência dos padrões (Biggs & Shaw, 1985; Devlin, 2002; Goldenberg, 1998; Mottershead, 1985; Orton, 1999). Procurar aspectos que se repetem, que se mantêm invariantes, pode ser um incentivo para o estudo da geometria, em particular para o estudo das transformações geométricas. Orton (1999) refere que os padrões geométricos estão presentes no dia-a-dia. Quando os alunos procuram este tipo de padrões descobrem transformações geométricas, como as translações, reflexões, rotações, reflexões deslizantes e suas composições. É um desafio descobrir o(s) tipo(s) de transformações geométricas subjacentes aos padrões analisados, bem como o motivo mínimo necessário para gerar o padrão. Este artigo refere-se a uma experiência desenvolvida no âmbito da formação inicial de professores do 1º e 2º ciclos da escolaridade básica. Palavras-chave: Padrões geométricos; Transformações geométricas; Invariante; prova. ABSTRACT Mathematics is considered the science of patterns (Biggs & Shaw, 1985; Devlin, 2002; Goldenberg, 1998; Mottershead, 1985; Orton, 1999). Looking for features that repeat, that maintain invariable, may be an incentive for the study of geometry, in particular of geometric transformations. Orton (1999) refers that geometric patterns are present in everyday life. When students are looking for this kind of patterns they discover geometric transformations like translation, reflexion, rotation, glide reflection and their compositions. It’s a challenge for them to discover what kind(s) of geometric transformations underlie the pattern analysed and what is the minimal motif needed to generate the entire pattern. This paper refers to an experience developed in a teacher training course with pre-service teachers (grade 1 to 6). Key words: Geometric patterns; geometric transformation; invariant; teacher training; proof.

Geometric Patterns Geometry is a topic that integrates Portuguese and foreign educational plans and it is essential in schools. According to Duval (1998), its interest is granted by cognitive complexity because it helps develop the capacity of visual representation and reasoning, by the formulation of a conclusion and its justification and by being a mean which helps understand the world that surrounds us. Geometry’s deductive character was emphasized for many years, being approached by theory and was the mean in which many students contacted with a mathematical demonstration. At the end of the last century we started to attribute more attention to the practical aspects of geometry and a decrease occurred in the deductive tonic aspects. While studying geometry it is important to establish connections between practical geometry and deductive geometry and for that it is necessary to develop the student’s geometric eye (Fujita & Jones, 2002, relating to Godfrey, 1910) which is the capacity to see geometric properties that are highlighted in a figure of speech. When working with a given context, students look for objects that repeat themselves, they understand the means that produce repetition, they observe relations that maintain constant when everything else around modifies itself, they try to understand and explain the reasons underlying those relations, students develop a more profound comprehension of hard-working matters, they create a more dynamical conception of mathematics and they verify that the patterns are underlying to mathematics, particularly to geometry. Orton (1999) refers to geometric patterns that surround us as Capítulo 3 - Geometria | Chapter 3 - Geometry

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day to day objects, which we can detect in craftwork, embroideries, forged iron, tiles, pavements, moulding, wallpaper, gardens, etc., but there is a necessity to learn how to see those patterns. The repetition idea is very much associated with geometric patterns. We can see geometric patterns when an object repeats itself regularly and according to determined rules, that geometric transformations are: translations, rotations, reflections, slippery reflections or its compositions. Mean while, even when we analyse these geometric patterns in greater depth, and we focus and study the composition of geometric transformations it is possible to detect the existence of aspects which maintain invariants, this is the presence of fixed relations when all around is changing, those are the relations that give origin to mathematical properties. According to Mason (2005) the notion of invariant is important in mathematics, so looks for invariants and the justification construction should be a central issue in mathematical learning (Goldenberg, 1998). Many theorems and many techniques are thought of as a base that remains constant in conditions of change. The invariance doesn’t make sense without the change and for the change to be observed it is necessary for some invariance. Observe the invariants, try, test, formulate and reformulate conjectures, generalize, justify the conjectures, these are the essential processes to develop in mathematics and in particular with geometric patterns. Goldenberg, Cuoco and Mark (1998) state that geometry is an adequate matter that helps students develop their thinking in ways in which mathematical facts are obtained. Geometry is an intellectual territory ideal for experiments, developing reasoning based on visualization, learning to find invariants and using those and other types of reasoning, creating constructive arguments (pp 4-5). In this text the word “pattern” is used to refer the design got under the repetition of a minimum motif by the influence of geometric transformations, for example, in the wallpaper patterns, as to refer to the existence of an invariant when investigating geometric properties.

Teacher training The teachers’ ideas about teaching and learning mathematics sometimes becomes an impediment to the demanding changes as a result of educative retirements orientated by principles like the ones defended by the National Council of Teachers of Mathematics for the mathematical education and shaped, for example, in the new mathematical programme for basic education (M.E., 2007). So that future teachers interiorize beginnings of the type already mentioned in the new programme “they must experience them in their own education – it is not enough to acknowledge them only mentally” (Hefendehl-Hebeker, 1998, p.122). Therefore, it is important that teacher training applies the principles that it defend. The training, whether mathematical or didactic, needs to substantiate its analysis and explanations in practical inquiries in education. Active apprenticeship is necessary and possible in all levels (Hefendehl-Hebeker, 1998). Two pillars where we should place the mathematical apprenticeship are the mathematical experiences and the reflection about that experience. In the classroom we should value the problems, the tasks of exploration and the investigation. 112

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The world in which we live today is in constant change and teacher training needs to accompany these changes, so teachers should be in permanent training throughout life. In spite of this, the initial teacher training can and should, provide experiences that are not definite, but they should be enriching and challenging and a willingness of continued study. Being that the teachers are the mediators between the knowledge and the students, if we wish that they work with a determined theme, for example patterns, then they should study and work that theme in a way that they are able to explore it with their students and should work with it in the same way as they would to see it done in the classroom. Fernandes (1989) opinion maintains itself that actual: teachers should be prepared to utilize methods, techniques and materials that they would like to be used with students. Other authors (e.g. Carillp & Contreras, 2000, Chapman, 2000, Cooney, 1994, Hiebert & Carpenter, 1992, Kloosterman & Mau, 1997, Thompson, 1985, Villani, 1998) also relate that teachers are more apt to teach how they learned, instead of how they were told to teach, being, therefore, the most important are teaching methods instead of the contents that being dealt with. Defending the students, they should actively participate in the construction of their knowledge whether the constructive theory of learning is constant in the classroom. Following Cobb (1996) in relation to mathematical learning “it (constructive learning theory) should be considered both as an active individual constructive process and a process of acculturation with the mathematical experience of society in general”. (free translation, pp. 60-61). Learning is not only the effort of ones individual mind trying to adapt itself in an environment, it cannot even be reduced to a process of acculturation of a pre-establish culture. In a mathematical class, the individual construction of significances has its place in the interactions with the culture in the classroom, meanwhile at the same time it contributes to the construction of that culture (Sierpinska & Lerman, 1996).

The study An exploratory study was developed that claimed to describe and analyse the work done by future teachers when they looked for patterns in the context of geometric transformations. In particular we pretend: (a) to promote the development of the mathematical knowledge about geometric transformations, (b) to develop the ability to observe, to notice patterns, to formulate, test and refine conjectures and to justify them. The study took place in the 2008/2009 school year at a School of Education with pre-service teachers of Elementary School (grade 1 to grade 6), of the variants of Mathematics and Science. There was a natural classroom context developed in the discipline of Geometric Transformations in the 4th year course. For pre-service teachers to be able to work with their students with geometric transformations they need to develop competence in that area. We notice that this matter passed integrating explicitly the new programme of Mathematics of Basic Education (M.E., 2007). In the referred discipline the work environment was organized in a way that pre-service teachers would increase their mathematical knowledge about the matter, having had contact with definitions and exploring properties of geometric transformations, resolved tasks of investigation, participated in discussions, formulated, tested, evaluated and proved conjectures. They also used an application of dynamic geometry (Geometer’s Sketchpad) Capítulo 3 - Geometria | Chapter 3 - Geometry

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(GSP) to explore tasks. They also had the opportunity to become aware that patterns and geometric transformations are around you in everyday life, in the city: in the tiles, the pavements, monuments, embroideries, ceramic, gardens, etc. Conscious of the importance in the methods of teaching, according to those defended by various authors (e.g. Carrillo & Contreras, 2000; Chapman, 2000; Cooney, 1994; Hiebert & Carpenter, 1992; Kloosterman & Mau, 1997; Thompson, 1985; Villani, 1998) was important to decide how to attack, for example, the study of the composition of geometric transformations. In a previous study the work strategy with pre-service teachers started as an investigation about the composition of geometric transformations – Investigate the composition of geometric transformations – having used GSP, formulated conjectures and constructed proofs for it. Then the students studied the patterns’ characteristics of wallpaper and friezes and looked for examples in their cities. Meanwhile, it was detected that in this second moment students did not have present recurrent results anymore of the first exploration that they had had. For that reason, in the present study exploration we decided to use another strategy. We presented an initial task that consisted of the construction of patterns (of wallpaper or friezes) and the obtained compositions of geometric transformations were studied. When students look for these types of patterns they can discover geometric transformations subjacent to the pattern and what the minimum necessary motif to generate it. The proof of the formulated conjectures rises, more naturally, as a good mean of understanding and explaining the mathematical relations involved. The data was retrieved by resource the assignments involving geometric transformations, observations and interviews, the data analysis was holistic, descriptive and interpretative.

Examples of tasks It was suggested to students the following initial task. “Imagine you are a designer of a wallpaper factory. A client would like a new wallpaper pattern and brought the following motif. Present a proposal of a wallpaper pattern for the client to chose”.

The students worked individually on the task, but would talk with their peers so that they would not repeat the patterns that they were constructing. Many proposals emerged like those presented in the following photographs. After the patterns were built it was important to study them and the following assignment came about: “In every pattern created: …….1. Identify the geometric transformations involved. …….2. Identify the composition of the geometric transformations. …….3. Formulate conjectures about the composition of the geometric transformations. …….4. Justify the conjectures that you formulated.

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This second task had a first moment of individual work where pre-service teachers observed and analysed the patterns that were constructed and identified the simple geometric transformations or the compositions. Then at a second moment there was a discussion between pairs to discuss doubts, clarify some of the conjectures gathered and share the conjectures that, still in inceptive manner, were being presented. The colleagues support was essential to reinforce confidence in the proposals that were gaining form, as well as, help the ones that still had doubts. While the initial task was considered “very interesting and amusing” the second part was considered “interesting, but complicating and difficult”. “I never thought that I would be able to find (in a pattern that was being analysed) so much mathematics here” (L).

Not all patterns constructed by future teachers were wallpaper patterns, but yes repetitions of friezes that, by translation, made a pattern, which is illustrated in the following image. In all the proposals were being identified translations, reflections of parallel and/ or perpendicular axis, various amplitude rotations and glide reflections. The composition of geometric transformations was arising with the analysis of the patterns constructed. Look at the following two situations that were explored.

A. Composition of reflections of parallel axis. In the following pattern students analysed the compositions of two reflections of parallel axis. Paying attention to the superior triangle that is coloured, verify that it has a reflection according to the first horizontal axis and then after a new reflection. Comparing the triangle in the initial position with the triangle (coloured) in the final position, future teachers concluded that Capítulo 3 - Geometria | Chapter 3 - Geometry

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Conclusion The composition of two parallel axis reflections is a translation associated with a vector going towards the first axis to the second, the direction is perpendicular to the reflexive axis because the axis is the mediatrix of the line segment [AA’] and the second axis is the mediatrix of the line segment [A’A”] (by the reflection definition) so the two axis are perpendicular to the vector [AA”] and the intensity is double to the distance between the two parallel axis. The possibility that the composition of two transformations of the same type could bring different transformations was amazing. They experienced on GSP and accepted easily the translation because the experiences created always showed a slide of the considered figure, a triangle, but needed to understand why it was like this. They looked for a justification. Some pre-service teachers tried to justify with numeric relations created by GSP. Those relations are convenient, but do not explain the reason of its existence. A general explanation was necessary, as well as, resistant to all the experiences.

Looking to Figure 1, 1) As we create two reflections, with parallel axis, r and s, we could se a translation, between triangles [A B C] and [A” B” C”] where the vector is doubled to the axis distance r and s, and the direction is perpendicular to the axis. According to the reflection definition, the distance of A to the axis r is the same to the distance of A’ to the axis r, According to the refection definition, the distance of A’ to the axis s is the same to the distance of A” to the axis s,

The distance of A and A” is doubled in length of the axis r and s.

Fig. 1

CC'' = 7,15cm

€

€ AR = 1,41cm

CC'' = 2,00 d(r,s)

RA' = 1,41cm

AR + RA' + A' S + SA'' = 7,15cm € €

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€

A' S = 2,16cm

SA'' = 2,16cm

RS = 3,57cm

Looking for patterns in geometric transformations with pre-service teachers

AR + RA' + A' S + SA'' = 2,00 d(r,s)

€

In spite of the inaccurate, the justification introduced shows reasons of conclusion with regards to the length of the translation vector. The reasons that justify its direction and way were not explicit in this case. It must be referred that pre-service teachers did not explore, of is proper one, the composition of more than two reflections with parallel axis in too much detail. Even while the discussion regarding this situation was put forth and even though the experiences were created, they gathered conclusive results.

B. Composition of two half turns of distinct centres. In the following pattern students identified the presence of rotations of 90º, 180º and 270º and also the existence of half turns that could be applied sequentially. Therefore they analysed the composition of two half turns with different centres. Taking a look at the pattern, the two coloured squares, one being the initial figure and the other the transformed image in the composition of two half turns, it looked like they appeared again in a translation. They formulated the following conjecture. Conjecture: The composition of two half turns with different centres is a translation where the vector is doubled from the distance between the two centres, with the same direction and way of the centre of the first half turn to the centre of the second half turn.

B'

C

A

They experienced with GSP and easily accepted it as a translation because the experiences created always showed a slide, but they

P

needed to understand why it was so. A'

F

C'

B

Q

They did not have previous knowledge to justify this situation. They had to look for a justification. They started by building a figure that

C'' A''

represented a composition of two half turns, having initially used one triangle. B''

Fig. 2 Capítulo 3 - Geometria | Chapter 3 - Geometry

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Using the Figure 2 and reasoning about it they constructed a justification to the conjecture. Triangles [A A’ A”] and [PA’Q] have an angle in common with vertex A’. Side [AA’] contains point P as medium point, this is the centre of the half turn that transformed A in A’. The same happened with side [A’A”] that contains point Q that is the medium point, it is the centre of the half turn that transformed point A to point A”. So AP = PA' and A''Q = QA' .

€ [A A’ A”] and € [A’ PQ] have and angle in common and the sides that they are made up by are proporBecause triangles tional – the triangles are similar. We can even say that [A”A] is parallel to [QP], the straight line segment that joins the two medium points of the two sides of one triangle are parallel to the third side of that triangle. So [AA”] has the same direction as [QP]. The triangle’s sides that oppose each other to the common angle are proportional (in similar triangles, the same angles oppose proportional sides). So the vector length of the translation is double the length of the defined segment by the centres of the half turns, the direction is visual (of the first half turn to the second). Once again they explored the possibility of composing three half turns with different centres, but did not have conclusive results. This same wallpaper pattern was used to explore the composition of two glide reflections of parallel axis.

Students’ Results The participants in this study: (a) showed interest in researching geometric transformations in the context of the patterns that they built because “… it is a creative task … a challenging assignment that can also be used with our students” (E). The interest which the participants in this study manifested was much more significant than by those other students who participated in previous studies, where they first created the study of geometric transformations and then the two compositions and only after analysed wallpaper patterns provided by the teacher. The participants in previous studies, still in the presence of theoretical results, revealed that they had knowledge of it but they did not identify them in the context of the patterns. For this reason, the results gathered in this study indicate more adequacy in the methodology followed, as one participant defends. We also need to devote ourselves more while completing this and … we have some difficulties and I also think that by experiencing difficulties … I have an incentive for the after learning, because we do not like to feel the difficulties and it helps us to be more interested in what we are doing (N). (b) they learned how to recognize, in a more diverse way, geometric transformations and their compositions 118

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The patterns can look different while being the same (having the same subjacent geometric transformations) and that is interesting to analyse and discuss (L). (c) they revealed some difficulties with the visualization (c1) difficulties visualizing the subjacent geometric transformations to the patterns, because they need to firstly recognize figures/polygons that repeated themselves, after identifying congruent figures/polygons and finally identifying the subjacent geometric transformation(s) “I had a lot of difficulties in the beginning because I have always had difficulty with geometry. I have difficulty mentally visualizing and to help myself out I had to cut out the square to see the rotation, or the reflection, any transformation, to see if it was what I imagined that it would be or to see if it would fit there and many times didn’t. I had a lot of difficulties with that part” (N), (c2) difficulties in visualizing the figures or parts of the figures that were needed to construct proofs, because most of the time these were embedded in more complex figures. This difficulty that they manifested is referred to Orton (in press) being an obstacle that students have to overcome in order to be able to analyse figures and decide which parts they should give attention to when constructing mathematical reasonings.

(d) they sometimes revealed some difficulties in formulating conjectures about the composition of geometric transformations, in constructing proofs about its validity. This difficulty had its purpose with the fact that in some cases the composition of two geometric transformations is a geometric transformation of the same type; when this would not happen pre-service teachers feeling confused. The use of GSP helped overcome this situation because it allowed them to test out many particular cases. Although some participants revealed difficulties initially in the construction of the proofs, they considered this phase essential in their learning. I think it is very important (to prove) because it obligates us to think a little more, it is not, and sometimes we tell ourselves, we respond to the question but if we do not justify it, we do not go a little further and when we justify what we have done sometimes we even say “man this is difficult, it is not quite …. how am I going to explain this?”. It obligates us to think more. (…) As I justify, I find the difficulty or even another solution, look this is like this, but also could be like this, finding another way…I try to do that (N).

I agree (that creating proofs is important) to understand, but I have difficulty in justifying because sometimes I am not able to relate the situations (L). (e) they considered that they could use a learning sequence that they followed with their future students in basic education, besides permitting to work with the theme of geometric transformations it permits to develop the transversal capacities (M.E., 2007) of communication and reasoning. Capítulo 3 - Geometria | Chapter 3 - Geometry

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Students are going to like making these patterns. It is a funny way to work with geometry, for example, reflections. Working this way allows students to be more motivated and learn better. I think like this … when they walk down the streets they see mathematics. It is a practical situation … although I had difficulties in the beginning … it was stimulating my learning why we do not like to have difficulties and to overcome them you had to work hard. (…) Everything needs to be well spoken, discuss the reasoning, especially with children because they constitute… they think that mathematics is memorizing the formulas, memorizing all these little things and really it is not, it is spoken reasoning, communication, I think … I really link mathematics to communication (N).

The way we worked (learning sequences) was good because we start with a simple task about patterns and geometric transformations and then we did more complex tasks with the compositions… and proofs are a mean to verify if the students understand (L).

Final Conclusion In synthesis, we can say that from the work carried out with pre-service teachers in this subject they obtained mathematical knowledge about geometric transformations, knowledge that they applied in the task resolution that was proposed, whether the submitted or others. In the exploration of the constructed patterns they identified the geometric transformations that generated the pattern, they formulated conjectures about the composition of two geometric transformations of the same type and, the majority, constructed proofs and almost proofs (Fonseca, 2004) that they pretended to guarantee the validity of the formulated conjectures, having missed small aspects to be considered complete. They revealed visualization difficulties, concordant with the afore said by Orton (2009). They considered that they can explore with their future students the theme of geometric transformations and that the following learning sequence can be “surpassed” for the basic teaching, moreover permitting working the matter of geometric transformations, in addition it permits to develop the transverse abilities, pointed by the new mathematics programme (M.E., 2007), of communication and reasoning. As we related this exploratory study had a preliminary edition, in the 2007/2008 school year, where another work sequence was carried out. From the obtained results the sequence carried out in this study seems to be adequate to tackle the geometric patterns theme in the initial teacher training.

References Biggs, E. & Shaw, K. (1985). Maths Alive! London: Cassell. Carrillo, J. & Contreras, L. (2000). El amplio campo de la resolución de problemas. In J. C. Yánez & L. C. Contreras (Eds.), Resolución de problemas en les albores del siglo XXI: Una visión internacional desde múltiples perspectivas y niveles educativos (pp.1338). Huelva: Hergué, Editora Andaluza Chapman, 2000. Cobb, P. (1996). Onde está o espírito? Uma coordenação de perspectivas construtivistas socioculturais e cognitivas. In C. Fosnot (Ed.), Construtivismo e educação. Teoria, perspectivas e prática (pp.59-84). Lisboa: Instituto Piaget. Conney, T. (1994). On the application of science to teaching and teacher education. In R. Biehler, R. Schlz, R. Stäßer & B. Winkel120

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mann (Eds.), Didactics of mathematics as a scientific discipline (pp.103-116). London: Kluwer Academic Publishers. Devlin, K. (2002). Matemática. A ciência dos padrões. Porto: Porto Editora. Duval, R. (1998). Geometry from a cognitive point of view. In C. Mammana & V. Villani (Eds.), Perspectives on the teaching of geometry for the 21st Century (pp.37-51). Dordrecht: Kluwer Academic Publishers. Fernandes, D. (1989). Aspectos metacognitivos na resolução de problemas. Educação e Matemática, 8, 3-6. Fonseca, L. (2004). A formação inicial de professores: A demonstração em geometria. (Colecção teses). Lisboa: APM. Goldenberg, P. (1998). Hábitos de pensamento: um princípio organizador para o currículo (II). Educação e Matemática, 48, 37-44. Goldenber, E. P., Cuoco, A. e Mark, J. (1998). A role for geometry in general education. In. R. Leher e D. Chazan (eds.), Designing learning environments for developing understanding of geometry and space (pp. 3-42). London:LawrenceErlbaun Associates Publishers. Hefendehl-Hebeker, L. (1998). The practice of teaching mathematics: Experimental conditions of chance. In F. Seeger, J. Voigt & U. Waschescio (Eds.), The culture of the mathematics classroom (pp.104-126). Cambridge: Cambridge University Press. Hiebert, J. & Carpenter , T. (1992). Learning and teaching with understanding? In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp.65-100). New York: MacMillan Publishing Company. Kaput, J. (1999). Teaching and learning a new algebra. In T. Romberg e E. Fennema (Eds.), Mathematics classroom that promotes understanding (pp.133-155). Hillsdale: Lawrence Erlbaum. Kloosterman, P. & Mau, S. (1997). Is this really mathematics? Challenging the beliefs of preservice primary teachers. In D. Fernandes, F. Lester, Jr., A. Borralho & I. Vale (Coords.), Resolução de problemas na formação inicial de professores de matemática. Múltiplos contextos e perspectivas (pp.217-248). Aveiro: GIRP. Mason, J. (2005). Usando ecrãs mentais e electrónicos. Educação e Matemática, 81, 33-36. Ministério da Educação (M.E.) (2007). Programa de Matemática do Ensino Básico. Lisboa: DGIDC. Mottershead, L. (1985). Investigations in Mathematics. Oxford: Basil Blackwell. Orton, J. (1999). Children´s perception of Patterns in Relation to Shape. In A. Orton (ed.) Pattern in the Teaching and Learning of Mathematics. London: Cassel. Orton, J. (no prelo). Pupils’ perception of shape, pattern and transformations. Actas do the International Meeting on Patterns. Viana do Castelo 1, 2 May 2009. Polya, G. (1957). How to solve it (2nd ed.). Princeton: Princeton University Press. Sierpinska, A. & Lerman, S. (1996). Epistemologies of mathematics and of mathematics education. In A. Bishop, K. Clements, C. Keitel, J. Kilpatrick & C. Laborde (Eds.), International handbook of mathematics education (pp.827-876). Dordrecht: Kluwer Academic Publishers. Thompson, A. (1985). Teacher´s conceptions of mathematics and the teaching of problem solving. In E. A. Silver (Ed.), Teaching and learning mathematical problem solving: Multiples research perspectives (pp.281-294). N.Y.: Lawrence Erlbaum Associates. Villani, V. (1998). The way ahead. In C. Mammana & V. Villani (Eds.). Perspectives on the teaching of geometry for the 21st century (pp.319-328). Dordrecht: Kluwer Acdemic Publishers.

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Visuoalphanumeric Mechanisms that support Pattern Generalization

Visuoalphanumeric Mechanisms that support Pattern Generalization1 Ferdinand D. Rivera [email protected], San Jose State University, USA RESUMO Neste artigo, inspiro-me em duas fontes de conhecimento pertinentes à actividade com padrões. A primeira é uma síntese interpretativa (reflexiva) da investigação sobre os padrões com base em estudos realizados com alunos do ensino básico e secundário, em diferentes países e diferentes contextos, durante, pelo menos, os últimos vinte anos. A segunda fonte é um estudo empírico longitudinal, com a duração de três anos, que conclui recentemente e que envolveu dois grupos de alunos do ensino básico (idades compreendidas entre os 10 e os 13 anos) de um contexto urbano. Foram analisados os seguintes aspectos: a natureza dos padrões; significado do contexto de generalização de padrões e modelos visuais. Destaca-se a importância conceptual da unidade estrutural, da abdução e do raciocínio aditivo e multiplicativo no desenvolvimento da generalização de padrões. ABSTRACT In this paper, I draw on two sources of knowledge relevant to patterning activity. The first source is an interpretive (reflective) synthesis of research knowledge on patterns on the basis of studies done with elementary and high school students in different countries and in different contexts at least in the last twenty years. The second source is the empirical three-year longitudinal study that I recently completed that involved two cohorts of middle school students (ages 10 to 13 years) in an urban context. We pursue the following aspects: nature of patterns; meaning context of pattern generalization, and visual templates. We highlight the conceptual significance of structural unit, abduction, and additive and multiplicative thinking in the development of pattern generalization.

I recently concluded a longitudinal study on patterning and generalization in an urban middle school school in Northern California. Since patterning activity is prevalent in almost all elementary mathematics classrooms, K-5 teachers may benefit from learnings I have obtained in my three-year study. In fact, the progression in cognitive proficiency involving pattern generalization among my middle school students parallels the recommended California state standards in mathematics on patterning at the elementary level. First graders learn about simple repeating patterns involving rhythm, numbers, color, and shape. In second grade, they begin the process of extending a linear pattern from a given sample of a few initial stages (see, for e.g., the circle fan pattern in fig. 1 with five stages). In third grade, they describe their generalizations in words. In fourth grade, rules in words are replaced by variable-based direct expressions. In my own study, I used linear patterning and generalization in helping my middle school students to deepen their understanding of algebra.

This reflective essay draws on findings I obtained from two cohorts of middle school students who participated in my three-year study. In Fall 2005, the first cohort explored linear patterning and generalization for almost eight weeks following a teaching experiment on integers and integer operations, which was taught using a set-theoretic model (i.e., with binary chips). The first cohort consisted of twenty-nine 6th grade students who were predominantly AsianAmericans. The students’ mathematical proficiency at the beginning of the study fell under basic or below proficient by 1 The research that is reported in this article has been funded by the National Science Foundation under DRL 0448649. The opinions expressed are those of the author and do not convey the views of the foundation. Capítulo 4 - Visualização | Chapter 4 - Visualization

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state and departmental assessments. Clinical interviews also showed little to no understanding of variables and pattern recognition and generalization. When they were asked to obtain generalizations for linear and nonlinear patterns, most of their rules were additive (for e.g., “keep adding x”) and were expressed in words. In Fall 2006, three students in the first cohort moved to a different school and were replaced with six new seventh-graders. Again, linear patterns and generalization were pursued after a second round of teaching experiment on integer and integer operations, which was taught using a number-line model. In Fall 2007, only fifteen students from the first cohort were allowed to participate in the study. They were then mixed with a new cohort of nineteen 7th- and 8th-graders. Both cohorts pursued the state-mandated Algebra 1 curriculum with me for one full school year, and I used linear patterning and generalization in exploring relevant algebraic concepts such as variables, multiple representations, equivalence of algebraic expressions, functions, domain and range, function rules, and linear functions. In the first two years of the study, I collaborated with the assigned classroom teachers. In the Year 3 study, I taught the class with the assigned teacher taking the role of observer. Another recent event further encouraged me to write this essay. A few months ago, I was asked to analyze a videotape session of an exemplary elementary teacher who was working with her 5th-grade class on a linear figural pattern, that is, a geometric pattern that consists of stages such as the one shown in fig. 1. While there were similarities between her instructional approach and my own, I realized that we teachers oftentimes approach patterning activity with the generous assumption that our students would be in possession of a sophisticated cognitive mechanism that would enable them to easily and naturally transition from rules in words to rules in variable form. The assumption might hold with some children, but it is not the case with all students. Hence, in the 5th-grade teacher’s class, as well as my own, I saw a few students who felt overwhelmed and eventually refused to participate. I focus on four key findings that I refer to as teacher-lessons. Some of the findings are subject to argument, but I encourage elementary teachers to keep an open mind and to reflect on their own instructional practices in terms of how they teach patterns and generalizations in their own contexts. Due to shared curriculum requirements in elementary mathematics across states, all the examples I use in this article involve linear patterns.

Lesson 1. There is more to generalizing figural patterns than simply obtaining a rule. When we ask students to deal with figural patterns such as the one shown in fig. 1, we initially ask them to extend the patterns by either describing or drawing the next few stages. Extending a pattern should be aimed at having them articulate their assumptions about the overall structure of the pattern on the basis of what they see and how they imagine the succeeding stages in the pattern. In Year 3 of my study, I have found open-ended patterning tasks such as the one shown in fig. 2 to be powerful in helping students to see the necessity of articulating their assumptions about their patterns because the assumptions provide the initial and necessary step in justifying their formulas. Fig. 3 shows a range of responses that my students in Year 3 of the study articulated in relation to the fig. 2 task. The instructional and psychological intent of open tasks such as fig. 2 is to confront a common student misconception that a pattern behaves 124

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(i.e., increases or decreases) in only one way on the basis of the given stages. Thus, constructing a direct formula for a given pattern should take into account assumptions that students explicitly articulate about the pattern. Further, their assumptions depend on what they perceive to be meaningful and relevant to them.

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Lesson 2. Asking students to determine what stays the same and what changes in a pattern is not an easy task for them. Patterning activity at the elementary level introduces students to structures and structural analysis. Mathematics, after all, is a science of structures. One recommended method in helping students to develop a structure for a figural pattern is to have them investigate parts or features in the pattern that stay the same and those that change as well. These tasks assist students in developing structure sense and I refer to them as structuring questions. However, students’ approaches and responses on such tasks depend on a number of factors that are significantly influenced by developmental changes or social learning and experience. So, it is normal for them to perceive different structures. For example, Chloe, 8th Grader in Year 3, saw “the gray circle in the middle” as staying the same from one instance to the next and “the white circles around” the gray circle as changing (refer to fig. 4). Delilah perceived the same structure as Chloe but saw “three arms” that grew from one stage to the next (see fig. 5). On the other hand, Earl, 7th Grader in Year 3, saw stage 1 of the pattern as embedded in the succeeding stages (see fig. 6).

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What is oftentimes difficult for many students, which I have observed in my own class, involves the process of translating verbal descriptions relevant to the structuring questions in algebraic form. Referring to figures 4, 5, and 6, the verbal descriptions of Chloe and Delilah were certainly easier to translate algebraically than Earl’s. This translation issue is discussed in greater detail in Lesson 3. What I would like to point out, however, is the implication of the structuring questions on students’ developing understanding of the concept of a unit. Knowing what comprises a unit, in fact, is fundamental in understanding fractions, polynomials, rational algebraic expressions and, more generally, multiplicative reasoning. A unit can either be a single object or a group of objects (“composite unit”). For example, in Dienes blocks arithmetic, one long as a composite unit is equivalent to 10 single units. In algebra, a unit may take the form of a variable 1 2 1 in the case of + (a rational exx x x pression addition problem). In the case of patterns, it makes much more sense to talk about the idea of a structural unit. expression such as x in the case of 3x - x (a polynomial subtraction problem) or

A structural unit emerges when students are able to identify a consistent invariant property or feature across the instances in a figural pattern and then uses it to establish a valid general rule in algebraic form (i.e., the direct formula). For example, both Chloe and Delilah used their structural unit (the middle gray circle and a growing set of three arms) in stating a possible direct formula for the pattern in Fig. 1, which took the form C = 3n + 1. Fig. 7 shows the figural extension of Karen, 7th Grader in Year 3, in relation to the task in Fig. 2 and the manner in which she justified her formula, S = 2n – 1. Karen’s structural unit came to her when she saw the same invariant structure of horizontal and vertical sides that define the growing L-shaped pattern that she constructed from one stage to the next. She then used her structural unit in articulating her formula, as follows: I visualized it in groups. So like for the 2n – 1, you take the stage number which are these two [i.e., the two circled groups in every stage] and then you subtract 1 because there’s an overlap.

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Earl in fig. 6 clearly articulated a structural unit but failed to use it in establishing the correct general rule, which should have taken the form C = 4 + 3(n – 1), where C refers to the total number of circles and n the stage number. But rules of this particular kind are especially difficult among older children and obviously even more so among elementary children. When Earl found it difficult to apply his structural unit in developing a general formula, he then used a numerical method (with the aid of a table) in generating a direct formula that then enabled him to establish the same structural unit that Chloe and Delilah saw. Lesson 4 deals with tensions between visual and numerical modes of generalizing. The structural unit activity in fig. 8 is meant to introduce beginning students to patterning activity. The subtasks target different aspects of patterning activity, as follows: structural unit analysis (1A, 1B); generating “near” (1C) and “far” (1D, 1E) extensions; possibility of equivalence (2A, 2B), and; free construction.

Lesson 3. Generalizing figural patterns is greatly facilitated when students understand the concept of multiplication of whole numbers. In the thirty-minute videotape episode of the 5th grade teacher that I briefly mentioned in the introduction, she clearly wanted her students to shift their thinking from their initial additive observation (“this pattern keeps adding two triangles”) to one that involves the multiplicative form 2x. In her written reflective essay, she pointed out the difficulty that too many of her students oftentimes experience when they have to transition from the verbal description (additive) to the algebraic form (multiplicative). What I found surprising was the fact that she did not include the concept of multiplication on her list of required prior mathematical knowledge that plays an important role in constructing a direct formula. The core cognitive characteristics of arithmetical thinking among young children are naturally additive and recursive. 128

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Hence, when they deal with, say, increasing linear figural patterns for the first time, the common expected response is to “keep adding x.” Even among my students who already have acquired years of training in various operations involving whole numbers, the entry point of their generalization was mostly additive in both form and content. But the underlying structure of most textbook-based linear patterns such as the one shown in Fig. 1 involves the concept of multiplication and, thus, needs to be explicitly taught. Two common approaches to teaching multiplication of whole numbers involve the grouping model and the rectangular array model (fig. 9), and both models would suffice in dealing with most types of simple linear figural patterns.

For example, the pattern in fig. 1 can be generalized by using the grouping model, while many of the responses in fig. 3 can be generalized by interpreting the stages as arrays of rows and columns. From the grouping model of multiplication, it is important for students to know that the symbolic expression a x b in concrete terms means a groups of b objects. Prior to any patterning activity, students can benefit from activities such as the ones shown in figures 10 and 11. In fig. 10, they learn the correct way of writing an expression involving multiplication. They need to see that even if the commutative property of multiplication preserves part-whole relationships (i.e., the whole is the same regardless of the order in which the parts are being multiplied), grouping by a x b is not the same as grouping by b x a.

In fig. 11, I share samples of student work by 7th and 8th graders in my class on a task that asked them to find and justify equivalent mathematical expressions that produced the same total number of circles. At the elementary level, students would benefit from activities that allow them to link their multiplicative expressions with the expected gradelevel counting strategy. For example, in California, 2nd grade students are expected to be proficient in counting by 2s, 4s, 5s, and 10s. Hence, set-counting activities can be framed within grade-level expectations. Capítulo 4 - Visualização | Chapter 4 - Visualization

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When students understand the meaning of multiplication of two whole numbers, then the translation issue (from verbal description to algebraic formula) is dealt with in a more systematic manner. The sample guided activity in fig. 12 is meant to initially assist students developing a generalization process, which involves three steps.

First, they need to verbally articulate a structural unit. My students, like Chloe, Earl, and Karen in figures 4, 6, and 7, respectively, have found it useful to circle groups consistently from stage to stage. Second, they set up a structuring 130

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table of values that takes into account all the information acquired from the structural unit. This step is critical because the process of constructing a direct formula involves inductively producing a general mathematical expression that oftentimes involves multiplication. Third, once a direct formula is established, they should then be able to justify it. The follow up activity in fig. 13 is a more difficult task since students need to take into account overlapping parts. In my class, I used the activity in fig. 14, which revisits Earl’s initial structural unit in fig. 6, in helping the students to see that the formula he established in item 2 in fig. 6 actually conveyed a different structural unit.

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The activity in fig. 15 asks students to visually justify two equivalent direct formulas for the same pattern. The closure activity in fig. 16 asks students to generate and justify their own direct formulas.

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Lesson 4. Obtaining a direct formula for a linear figural pattern numerically using the method of common difference does not promote deep mathematical thinking and justification. The manner in which my students in Year 1 of the study dealt with the pattern in fig. 17 is worth discussing because it provided me with some insights into why many students might have difficulty with pattern generalization in the long term.

The students initially constructed the differencing table below and then used arrows to establish the fact that the dependent terms were constantly increasing by 2. I was pleasantly surprised when Anna’s group suggested a numerical strategy

Stage Number n

Total Number of Circles C

1

5

2

7

3

9

4

11

5

13

+2 +2 +2 +2

that aligned with the institutional mathematical practice we oftentimes refer to as the method of finite differences. This particular classroom event provides powerful evidence that learners can generate valid mathematical practices when they are provided with an opportunity to explore either on their own or in the context of collaborative work. In the classroom episode below, Anna shared with the class how her group thought about their direct expression. Anna: We made up a formula. Like we got the figures until figure 5, and we tried it with other ones. We got n x 2 + 3, where n is the figure number and timesed it by 2. So 5 x 2 equals 10, plus 3, that’s 13. So for figure 25, it’s 53. FDR: I like that formula. So tell me more. So your formula is? Anna: n x 2 + 3. FDR: So how did you figure this out? Anna: First we were like making the numbers to 25. We kept adding 2 and for figure 25, it was 53. FDR: Wait. So you kept adding all the way to 25? Anna: Yeah…. Then we used our chart. Then finally we figured out that if we timesed by 2 the figures and plus 3, that would give us the answer. FDR: Does that make sense? Students: Yes. FDR: So what Anna was suggesting was that if you look at the chart here, Anna was suggesting that you multiply the figure number by 2, say, what’s 1 x 2? Tatiana: 2. FDR: 2. And then how did you [referring to Anna’s group] figure out the 3 here? Anna: Because we also timesed it with figure number 13. FDR: What did you have for figure 13? Capítulo 4 - Visualização | Chapter 4 - Visualization

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Anna: That was 29. And then 13 x 2 equals 26 plus 3. FDR: Alright, does that work? So what they were actually doing is this. They noticed that if you look at the table, it’s always adding by 2. You see this? [Students nodded.] They were suggesting that if you multiply this number here [referring to the common difference 2] by figure number, say figure number 1, what’s 1 x 2? Students: 2. FDR: Now what do you need to get to 5? What more do you need to get to 5?

[Some students said “3” while others said “4.”]

FDR: Is it 4 or 3? Students: 3. FDR: It’s 3 more. So what is 1 x 2? Students: 2. FDR: Plus 3? Students: 5.

One positive result of the above episode was the high level of success that all the students in our class achieved insofar as direct formula construction mattered. When they were presented with a linear pattern that either had figural stages or numbers in a table, they used the above numerical strategy. Even in Year 2 of the study when they dealt with decreasing linear patterns and in Year 3 when they dealt with tables of values with fractional common differences, the numerical strategy worked much to their delight. Unfortunately, the above numerical generalization strategy prevented them from further developing their mathematical ability to justify. Many of them, in fact, thought that the numerical method was a form of justification. There were times, too, when they tried to simply fit the expression n x a + b on the stages in a pattern without linking it with their initial structural unit analysis. For example, Frank’s pattern in relation to the task in fig. 2 is shown in fig. 3A. He obtained the direct formula, S = 2n – 1, numerically but the manner in which he justified it ignored the structural unit he initially saw in his pattern (see fig. 18). Fig. 19 shows the written work of Marla and Jenny, 7th Graders in Year 3 of the study, in relation to the pattern in fig. 1. They initially found the above numerical method of common difference easier to accomplish than the tabling method in Lesson 3. The justification of their direct formula, C = n x 3 + 1, however, reflected a rather unstable process that I refer to as a mere appearance match.

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Thus, there is a significant conceptual difference between the structuring tables in Lesson 3 and the above differencing table. Inherent among structuring tables such as the ones shown in Lesson 3 is a continuous relationship between direct formula construction and justification on the basis of a perceived structural unit. In the case of differencing tables that merely record outcomes and are assessed for common difference, direct formula construction and justification are viewed as two separate activities.

Conclusion The Principles and Standards for School Mathematics (NCTM, 2000) wrote the following two following recommendations in helping Pre-K-2 students understand patterns: -- Encourage them to explore and model relationships using language and notation that is meaningful for them; -- Help them see different relationships and make conjectures and generalizations from their experiences with numbers. (NCTM, 2000, p. 92)

In the same document, pattern activity from Grades 3 through 5 should encourage students to: -- Analyze the structure of the pattern and how it grows or changes, organize this information systematically, and use their analysis to develop generalizations about the mathematical relationships in the pattern; -- Think about how to articulate and express a generalization; -- Make generalizations by reasoning about the structure of the pattern. (NCTM, 2000, pp. 158-159)

While the two sets of recommendations above appear in separate places in the document, they should not be interpreted separately or independent from each other. On the basis of learnings that I have drawn from my three-year study, patterning activity can foster the development of algebraic thinking with its primary focus on structures and ways of representing them. In this article, each lesson draws on a particular aspect of patterning and generalization. LesCapítulo 4 - Visualização | Chapter 4 - Visualization

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son 1 emphasizes the subjective and interpretive nature of perceiving patterns, which explains the necessity of having students to articulate their assumptions about them. Lesson 2 focuses on the significance of a structural unit of a pattern, which lays the foundation for multiplicative thinking in upper-level mathematics. Lesson 3 underscores the role of the concept of multiplication in the construction of a direct formula, which focuses on grouping objects in a particular way. Further, tables constructed out of this deep understanding of the relationship between multiplication and grouping provide a much more meaningful route to constructing and justifying direct formulas. Lesson 4 is meant to convey the view that some taken-as-shared mathematical practices on generalization may still not be appropriate to teach at the elementary level.

References Beckmann, S. Activities Manual: Mathematics for Elementary Teachers. Boston, MA: Pearson and Addison Wesley, 2008. National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Mathematics. Reston, VA: NCTM, 2000. Roebuck, Jay. “Coloring Formulas for Growing Patterns.” Mathematics Teacher 98 (March 2005): 472-75.

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Exploring generalization with visual patterns: tasks developed with pre-algebra students

Exploring generalization with visual patterns: tasks developed with pre-algebra students Ana Barbosa [email protected], School of Education of the Polytechnic Institute of Viana do Castelo

Isabel Vale [email protected], School of Education of the Polytechnic Institute of Viana do Castelo

Pedro Palhares [email protected] , Institute of Education, University of Minho RESUMO Este artigo refere-se a um estudo desenvolvido com 54 alunos do 6.º ano de escolaridade. O principal objectivo passava por analisar o seu desempenho na resolução de tarefas que envolviam a generalização de padrões visuais. De forma a compreender este problema centramo-nos nas seguintes temáticas: tipo de estratégias de generalização utilizadas; dificuldades que emergiram do seu trabalho; e o papel da visualização no seu raciocínio. Neste artigo são apresentados alguns resultados relativos à implementação de duas tarefas. Palavras-chave: padrões, generalização, visualização. ABSTRACT This paper refers to a study developed with fifty-four 6th grade students. The main goal was to analyse their performance when solving tasks involving the generalization of visual/figural patterns. In order to better understand this problem we focussed on the following features: type of generalization strategies used; difficulties that emerged from students’ work; and the role played by visualization on their reasoning. On this paper we present some results related to the implementation of two tasks. Key words: patterns, generalization, visualization.

Introduction The 80’s represent a landmark for school mathematics. At that time, profound curricular changes were made and problem solving became an integral part of all mathematics learning (NCTM, 2000). This idea is still present in the recent curricular guidelines of several countries, where the ability to solve problems is mentioned as one of the main goals of learning mathematics. In spite of the relevance given to this theme, some international studies (SIAEP, TIMSS, PISA) have shown that Portuguese students perform badly when solving problems (Ramalho, 1994; Amaro, Cardoso & Reis, 1994; OCDE, 2004). Pattern exploration tasks may contribute to the development of abilities related to problem solving, through emphasising the analysis of particular cases, organizing data in a systematic way, conjecturing and generalizing. For instance, the Principles and Standards for School Mathematics (NCTM, 2000) acknowledges the importance of working with numeric, geometric and pictorial patterns. This document states that instructional mathematics programs should enable students, from pre-kindergarten to grade 12, to engage in activities involving understanding patterns, relations and functions. Work with patterns may also be helpful in building a more positive and meaningful image of mathematics and contribute to the development of several skills, in particular related to problem solving and algebraic Capítulo 4 - Visualização | Chapter 4 - Visualization

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thinking (Vale et al, 2006). On the other hand, Geometry is considered a source of interesting problems that can help students develop abilities such as visualization, reasoning and argumentation. Visualization, in particular, is an important mathematical ability but, according to some studies, its role hasn’t always been emphasized in students’ mathematical experiences (Healy & Hoyles, 1996; Presmeg, 2006). Although the usefulness of visualization is being recognized by many mathematics educators, in Portuguese classrooms teachers privilege numeric aspects over geometric ones (Vale & Pimentel, 2005). Considering it all, we think that more research is still necessary concerning the role images play in the understanding of mathematical concepts and particularly in problem solving. This study intends to analyse the performance of 6th grade students (11-12 years old) when solving problems involving visual/figural patterns. The tasks used in the study require pattern generalization and students of this age have not yet had formal algebra instruction, thus the importance of analysing the nature of their approaches. This study attempts to address the following research questions: 1. Which difficulties do 6th grade students present when solving pattern exploration tasks? 2. How can we characterize students’ generalization strategies? 3. What’s the role played by visualization on students’ reasoning?

Theoretical Framework Patterns in the teaching and learning of mathematics Many mathematicians share an enthusiastic view about the role of patterns in mathematics, some even consider mathematics as being the science of patterns (Steen, 1990). This perspective highlights the presence of patterns in all areas of mathematics, considering it a transversal and unifying theme. In particular, the search for patterns is seen by some investigators as a way of approaching Algebra since it is a fundamental step for establishing generalization, which is the essence of mathematics (Mason, Johnston-Wilder & Graham, 2005; Orton & Orton, 1999; Zazkis & Liljedahl, 2002). The mathematics curricula of many countries contemplate significant components like: searching for patterns in different contexts; using and understanding symbols and variables that represent patterns; and generalizing. Portuguese curriculum mentions the importance of developing abilities like searching and exploring numeric and geometric patterns, as well as solving problems, looking for regularities, conjecturing and generalizing (DEB, 2001; ME, 2007). These abilities are directly related to algebraic thinking and support the development of mathematical reasoning as well as the connection between mathematical ideas (NCTM, 2000).

Patterns and generalization There has been considerable research concerning students’ generalization strategies, from pre-kindergarten to secondary school. We proceeded to an adaptation of some frameworks proposed by different investigators (Lannin, 2003; Lannin, Barker & Townsend 2006; Orton & Orton, 1999; Rivera & Becker, 2005; Stacey, 1989; Swafford & Langrall, 2000) and came up with the following categorization: 138

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Strategy

Description

Counting (C)

Drawing a figure and counting the desired elements. No adjustment (W1)

Considering a term of the sequence as unit and using multiples of that unit.

Numeric adjustment (W2)

Considering a term of the sequence as unit and using multiples of that unit. A final adjustment is made based on numeric properties.

Visual adjustment (W3)

Considering a term of the sequence as unit and using multiples of that unit. A final adjustment is made based on the context of the problem.

Recursive (D1)

Extending the sequence using the common difference, building on previous terms.

Rate - no adjustment (D2)

Using the common difference as a multiplying factor without proceeding to a final adjustment.

Rate - adjustment (D3)

Using the common difference as a multiplying factor and proceeding to an adjustment of the result.

Whole-object

Difference

Discovering a rule, based on the context of the problem, that allows the immediate calculation of any output value given the correspondent input value.

Explicit (E)

Guessing a rule by trying multiple input values to check its’ validity.

Guess and check (GC)

Table 1 Generalization Strategies Framework

Analyzing previous research and drawing on personal experiences, we recognized that these strategies often emerge through different types of reasoning and we think that it’s fundamental that students understand the potential and limitations of each approach. Depending on the type of task, some of these strategies may lead students to difficulties or even incorrect answers. This fact is reported on a variety of studies: -- an incorrect application of the direct proportion method, mainly when attempting to generalize linear patterns (Lannin, Barker & Townsend 2006; Rivera & Becker, 2005; Sasman, Olivier & Linchevski, 1999; Stacey, 1989). Students often operate exclusively on numeric contexts, manipulating variables without recognizing their meaning; -- the fixation on a recursive approach that, although being useful in solving near generalization tasks, doesn’t contribute to the understanding of the structure of a pattern (Orton & Orton, 1999). Frequently, when dealing with far generalization, students apply a recursive rule based on multiples of the common difference, neglecting to correct the result through the adjustment to the context of the problem (Lannin, Barker & Townsend 2006; Sasman, Olivier & Linchevski, 1999); -- guess and check is a well know strategy for solving problems, often encouraged in our classrooms. This approach is considered a good problem solving numerical tool, but it can lead students to incorrect conclusions if all conditions aren’t considered (Rivera & Becker, 2005);

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-- functional reasoning proves to be very complex for many students, especially those from elementary levels. Some of these difficulties are due to: using improper language to describe relations; the extensive use of recursive reasoning and the guess and check strategy; and the inability to visualize or explore patterns spatially (Warren, 2008).

Thinking preferences and generalization Patterning activities can be developed in a variety of contexts (numeric, geometric, concrete and figural) and through the application of different approaches. Gardner (1993) claims that some individuals recognize regularities spatially or visually, while others notice them logically or analytically. In fact, it is common, in mathematical activities, that different individuals process information in different ways. Many students favour analytic methods while others have a tendency to reason visually. The relation between the use of visual abilities and students’ mathematical performance constitutes an interesting area for research. Many investigators stress the importance of the role visualization plays in problem solving (Presmeg, 2006; Shama & Dreyfus, 1994), while others claim that visualization should only be used as a complement to analytic reasoning (Goldenberg, 1996; Tall, 1991). In spite of some controversy, these visions reflect the importance of using and developing visual abilities in mathematics but teachers tend to present visual reasoning only as a possible strategy for problem solving in an initial stage or, when necessary, as a complement to analytic methods (Presmeg, 1986). Several studies point to the potential of visual approaches for supporting problem solving and mathematical learning. The reality of our classrooms, however, tells us that students display frequently reluctance to exploit visual support systems (Dreyfus, 1991) and tend not to make links between visual and analytical thought (Presmeg, 1986). Some studies report the impact of the nature of the approaches used by students to generalize, referring, in particular, to the relevance of visual abilities. García-Cruz & Martinón (1997) developed a study aiming to analyse the processes of generalization developed by secondary school students. They classified generalization strategies according to their nature: visual, numeric and mixed. If the drawing played an essential role in finding the pattern it was considered a visual strategy, on the other hand, if the basis for finding the pattern was the numeric sequence then the strategy was considered numeric. Students who used mixed strategies acted mainly on the numeric sequence and used the drawing as a means to verify the validity of the solution. Results of this research have shown that the drawing played a double role in the process of abstracting and generalizing, in both cases fundamental. It represented the setting for students who used visual strategies in order to achieve generalization and acted as a means to check the validity of the reasoning for students who favoured numeric strategies. In a more recent study, Becker and Rivera (2005) described 9th grade students work after they were asked to perform generalizations on a task involving linear patterns. They tried to analyse successful strategies students used to develop an explicit generalization and to understand their use of visual and numerical cues. The researchers found that students’ strategies appeared to be predominantly numeric and identified three types of generalization: numerical, figural and pragmatic. Students using numerical generalization employed trial 140

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and error with little sense of what the coefficients in the linear pattern represented. Those who used figural generalization focused on relations between numbers in the sequence and were capable of seeing variables within the context of a functional relationship. Students who used pragmatic generalization employed both numerical and figural strategies, seeing sequences of numbers as consisting of both properties and relationships.

Method Fifty four sixth-grade students (11-12 years old), from three different schools in the North of Portugal, participated in this study over the course of a school year. The study was divided in three stages: the first corresponded to the administration of a test focusing on pattern exploration and generalization problems; second stage, which went on for six months, involved all students in each classroom solving patterning tasks, in pairs; and, on the third, students repeated the test in order for us to examine changes in the results. These students were described by their teachers as being of average ability and had no prior experience with this kind of tasks. Over the school year all students involved in the study solved seven tasks and two pairs from each school were selected for clinical interviews. Students’ activity when solving the tasks was videotaped and transcribed for further analysis. The tasks applied along the study required near and far generalization and featured increasing and decreasing linear patterns as well as non linear ones. In the selection process we tried to privilege tasks whose structure could lead to the use of multiple strategies, allowing students to find patterns in either numeric or visual contexts. In this paper we report some results from the application of two of the tasks.

Preliminary Results The Pins and Cards task One of the selected tasks was called Pins and Cards (figure 1). This was the first task solved by the students in this study and represents an increasing linear pattern, presented in a figural form. The first question called for near generalization, by finding the 6th term of the sequence, while the last two questions required far generalization.

Fig. 1 Pins and Cards task Capítulo 4 - Visualização | Chapter 4 - Visualization

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The analysis of students’ work allowed us to identify a diversity of generalization strategies, as well as some difficulties approaching particular questions. We start to present in table 2 the number of pairs of students that used a given strategy, based on the categories described on the Generalization Strategies Framework (table 1). In some cases we couldn’t categorize students’ answers so those cases appear in the last column of the table, not categorized (NC). This table allowed us to analyze not only the approach used to solve each of the questions of the task, but also compare it with the level of generalization involved (near or far). C

W1

W2

W3

W

D1

D2

D3

D

E

GC

NC

1.

16

8

-

1

9

1

1

-

2

-

-

-

2.

-

3

2

1

6

3

1

-

4

12

-

5

3.

-

2

1

-

3

-

4

3

7

9

-

8

Table 2 Summary of the strategies used by the students

Near generalization strategies The first question of this task required near generalization. This type of questions can easily be solved by making a drawing of the requested term of the sequence and counting its elements, using the counting strategy. Figure 2 represents an example of this approach, to find the number of pins, presented by a pair of students.

Fig. 2 Example of the counting strategy.

As we can see from table 2 counting over a drawing was the predominant strategy in near generalization, always leading to a correct answer. The whole-object strategy also emerged from the work of some of the pairs. This approach is associated to direct proportion situations and this particular problem does not fit this model. Nevertheless, eight pairs of students used proportional reasoning, duplicating the number of pins associated to the three cards. For this strategy to be adequate, students had to make a final adjustment based on the context. Only one of the pairs felt the need to adjust the result obtained in the duplication of the number of pins of the three cards (figure 3).

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Exploring generalization with visual patterns: tasks developed with pre-algebra students

3+3=6 cards 10+10=20 pins

These two groups share 1 pin. 20-1=19 pins

Fig. 3 Example of the W3 strategy.

According to the existing literature (e. g. Orton & Orton, 1999; Stacey 1989) this type of tasks can promote the use of recursive thinking, especially when near generalization is involved. Curiously only one pair of students extended the sequence using the common difference to solve this question. We registered one other case, in which the difference strategy was employed but in an incorrect way. To obtain the number of pins necessary to hang 6 cards, these students used a multiple of the common difference without adjusting the result, as happened in other cases with the whole-object strategy. The explicit and guess and check strategies were not applied to solve this question.

Far generalization strategies Although both questions 2 and 3 require far generalization, the third question of the task had a different structure, involving reverse thinking. When approaching far generalization students revealed more difficulties and that can be seen by the increasing number of not categorized answers, that represent imperceptible reasoning or no answer at all (table 2). We noticed in table 2 that students dropped the counting strategy when solving these two questions. Some pairs did start by using it but gave up along the way, claiming that “there were too many cards”. Instead, the application of explicit strategies prevailed. Those who relied on this approach, using the context to identify an immediate relationship between the two variables, presented a high level of efficacy. Some students “saw” that each card needed three pins and the last one would need four, deducing that the rule was 3(n-1)+4, n being the number of cards. Other pairs “saw” the pattern differently considering that each card had three pins adding one more pin at the end. Here the rule was 3n+1. In fact, research on pattern and generalization shows that individuals might see the same pattern differently (Rivera & Becker, 2007), originating equivalent expressions. The whole-object strategy continued to appear as in the previous question (1), but this time a new approach emerged. Some students considered multiples of known terms of the sequence and adjusted the result based only on numeric properties (figure 4). 6 cards 12 cards 18 cards 24 cards 30 cards 36 cards -

20 pins 40 pins 60 pins 80 pins 100 pins 120 pins

120-4=116 pins 36-1=35 cards Fig. 4 Example of the W2 strategy.

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The example given on figure 4 shows that students rely on proportional reasoning to determine the number of pins and, when adjusting the result they don’t consider the context of the problem, only numeric properties, obtaining an incorrect answer. Comparing the first question with the last two, we can see that the use of the difference strategy increases. Some students gave up on counting, as the order of the term became far, and started basing their reasoning on the common difference between terms. In the third question of the task, we noticed that three pairs of students applied a strategy that hasn’t been used before. 3×200=600 pins But we need one more pin for the last card, so she only can hang 199 cards. Fig. 5 Example of the D3 strategy

The difference between consecutive terms is of three pins, so, in this case, students used that fact to approach the number of pins available. Knowing the structure of the pattern, they were able to criticize the result, adjusting it.

The Sole Mio Pizzeria task This task was solved four months later. The problem is similar to the one presented on the previous task, exhibiting an increasing linear pattern and contemplating near and far generalization questions.

Fig. 6 Sole Mio Pizzeria task

In order to compare the strategies, selected by students in this task, with the strategies used in the Pins and Cards task, we organized the categories in the following table:

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Exploring generalization with visual patterns: tasks developed with pre-algebra students

C

W1

W2

W3

W

D1

D2

D3

D

E

GC

NC

1.

21

-

-

-

-

4

-

-

4

2

-

-

2.

1

-

-

-

-

3

-

1

4

22

-

-

3.

-

-

-

-

-

2

3

-

5

14

5

3

Table 3 Summary of the strategies used by the students

One of the most obvious facts is the lack of preference for the whole-object strategy. Being a linear pattern, the use of proportional reasoning is not adequate, unless an adjustment based on the context is made. We believe that, in this case, the adjustment was more complex than in the previous problem which could justify the absence of this approach.

Near generalization strategies Counting is once again the privileged strategy in near generalization. It is applied by the majority of the students and this preference has increased compared to the previous task. Other strategies emerged but only a minority of students used them. Four pairs used recursive reasoning to extend the sequence to the 10th term and two pairs applied an explicit reasoning. In the first task, explicit strategies only appeared when students were dealing with far generalization so it’s surprising that they used it at this stage, showing that they immediately discovered the structure of the pattern.

Far generalization strategies As in the Pins and Cards task, when dealing with far generalization, students do not recognize the usefulness of counting and that’s why it has no expression on table 3, as we progress to far generalization. On the other hand, explicit reasoning prevails being implemented by even more students in a successful way. Curiously all of them described the pattern as 2n+2, n being the number of pizzas. They frequently referred that “in front of each pizza are two people and one more at each end of the table”. Some students chose a safe path going with a recursive approach, through the extension of the sequence using the common difference. Similarly to what happened in the previous task, there were three pairs that considered multiples of the common difference but neglected to adjust the result, showing that their work was merely based on number relations. We also noticed the use of a new strategy, guess and check, that was only applied in far generalization when reverse thinking was involved. 24+24+2=50 25+25+2=52 26+26+2=54 27+27+2=56 28+28+2=58

He would have to order 28 pizzas.

Fig. 7 Example of the guess and check strategy

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This example shows that the students identified the relation between the two variables in the case that the independent variable was the number of pizzas and the independent variable was the number of people. Based on that rule they tried some numbers until they achieved the result wanted.

Difficulties emerging from students work When solving the first task some students struggled with cognitive difficulties that led to incorrect answers. Some pairs made false assumptions about the use of direct proportion. In these cases attention tended to be focused only on numeric attributes with no appreciation of the structure of the sequence. This happened with strategies W1 and W2, where the only concern was to satisfy numeric relations. The use of strategies based on recursive reasoning wasn’t always made correctly, especially when far generalization questions were involved. The recursive approach through the use of D2 lacked a final adjustment based on the context of the problem, because students only considered a multiple of the common difference, forgetting to add the last four pins or the last pin, depending on the interpretation. Also, when they used explicit strategies, the model wasn’t always correctly applied. In some cases, students added pins and cards in the end. We are convinced that these errors are linked to the extensive experience of students in manipulating numbers without meaning, making no sense of what the coefficients in the linear pattern represent. Analyzing tables 4 and 5 we verify that the level of efficacy presented by students increases on the second task. There is more awareness on the selection of the proper strategies to use in each case, for example, the inadequate use of direct proportion is no longer observed. In spite of these differences, we still notice that students experience difficulties when reverse thinking is involved, being also clear that this type of questions provokes a shift on the type of approaches used by them.

Pins and Cards Near generalization % of efficacy Far generalization % of efficacy Far generalization (reverse thinking) % of efficacy

C 16 100% -

W 9 11% 6 0%

D 2 50% 4 75%

E 12 100%

GC -

-

3

7

9

-

-

0%

43%

89%

-

Table 4 Level of efficacy per strategy

The Sole Mio Pizzeria Near generalization % of efficacy Far generalization % of efficacy Far generalization (reverse thinking) % of efficacy

C 21 100% 1 100%

W -

D 4 100% 4 100%

E 2 100% 22 100%

GC -

-

-

5

14

5

-

-

40%

100%

100%

Table 5 Level of efficacy per strategy

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The role of visualization in students’ reasoning According to Presmeg (1986) a strategy is considered visual if the image/drawing plays a central role in obtaining the answer, either directly or as a starting point for finding the rule. In this sense we believe that the following strategies are included in this group: counting, whole-object with visual adjustment, difference with rate-adjustment and explicit. Counting was always a successful strategy but only useful in solving near generalization questions. Drawing a picture of the object required and counting all the elements is an action used in near generalization questions and does not lead to a generalized strategy. Strategy W3 was only used by one pair of students, when solving the first task. They’ve only applied correctly in near generalization. We think that this kind of reasoning involves a higher level of abstraction in visualization, difficult to attain. In spite of not being one of the most frequent strategies, students who used D3 always reached the correct answer. This fact reflects once more the relevance of understanding the context surrounding the problem, making the relation between variables clearer. Finally, the application of an explicit strategy lead to a high level of efficacy. Students based their work on the structure of the sequence, making reference to the relation between the variables reported on the problem. We registered only a few cases that, along the way, disconnected from the context and mixed different variables.

Discussion In this research, the main purpose of using pattern exploration tasks was setting an environment to analyse difficulties presented by students, strategies emerging from their work and the impact of using visual strategies in generalization. As for the research questions outlined earlier in this paper, we can now make some observations: (a) students achieved better results in near generalization questions than on far generalization questions and, even with some experience with patterning activities, reverse thinking was still complex for many of them; (b); a variety of strategies were identified in the work developed by students, although some were more frequent than others, like counting (mostly on near generalization) and explicit (more frequent on far generalization); (c) some of the pairs worked exclusively on number contexts using inadequate strategies like the application of direct proportion, using multiples of the difference between two consecutive terms without a final adjustment and mixing variables. Along the study, this tendency was gradually inverted as most students understood the limitations of some of those strategies; (d) visualization proved to be a useful ability in different situations like making a drawing and counting its elements, to solve near generalization tasks, and “seeing” the structure of the pattern, finding an explicit strategy to solve far generalization tasks. So we think that it’s important to provide tasks which encourage students to use and understand the potential of visual strategies and to relate number context with visual context to better understand the meaning of numbers and variables. It is our strong belief that evidence about the ways children work with patterns may contribute to significant teaching decisions, about the ways to increase mathematical knowledge in our students and particularly of algebraic thinking.

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References Amaro. G., Cardoso, F. & Reis, P. (1994). Terceiro Estudo Internacional de Matemática e Ciências, Relatório Internacional, Desempenho de alunos em Matemática e Ciências: 7.º e 8.º anos. Lisboa: IIE. Becker, J. & Rivera, F. (2005). Generalization strategies of beginning high school algebra students. In H. Chick & J.L. Vincent (Eds.), Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 121128). Melbourne, Australia: University of Melbourne. Departamento do Ensino Básico (DEB). (2001). Currículo Nacional do Ensino Básico. Competências Essenciais. Lisboa: Editorial do Ministério da Educação. Dreyfus, T. (1991). On the status of visual reasoning in mathematics and mathematics education. In F. Furinghetti (Ed.) Proceedings of the 15th International Conference of the International Grouo for the Psychology of Mathematics Education. (Vol. 1, pp. 33-48). Genova, Italy. García Cruz, J. A. & Martinón, A. (1997). Actions and Invariant Schemata in Linear Generalizing Problems. In E. Pehkonen (Ed.) Proceedings of the 21th Conference of the International Group for the Psychology of Mathematics Education. (Vol. 2, pp. 289-296). University of Helsinki. Gardner, H. (1993). Multiple Intelligences: The Theory in Practice. New York: Basic Books. Goldenberg, E. P. (1996). “Habits of Mind” as an organizer for the curriculum. Journal of Education, 178(1), 13-34. Healy, L., Hoyles, C. (1996). Seeing, doing and expressing: An evaluation of task sequences for supporting algebraic thinking. In L. Puig & A. Gutierrez (Eds.). Proceedings of the 20th International Conference of the International Group for the Psychology of Mathematics Education. (Vol. 3, pp. 67-74). Valencia, Spain. Lannin, J. (2003). Developing algebraic reasoning through generalization. Mathematics Teaching in the Middle School, 8(7), 342-348. Lannin, J., Barker, D. & Townsend, B. (2006). Algebraic generalization strategies: factors influencing student strategy selection. Mathematics Education Research Journal. Vol. 18, (3), pp. 3-28. Mason, J., Johnston-Wilder, S. & Graham, A. (2005). Developing Thinking in Algebra. London: Sage (Paul Chapman). ME-DGIDC (2007). Programa de Matemática do Ensino Básico. Lisboa: Ministério da Educação, Departamento de Educação Básica. NCTM (2000). Principles and Standards for School Mathematics. Reston, VA: NCTM. OCDE (2004). PISA 2003: Relatório Internacional, O rendimento dos alunos em Matemática. Lisboa: Santillana-Constância. Orton, A. & Orton, J. (1999). Pattern and the approach to algebra. In A. Orton (Ed.). Pattern in the teaching and learning of mathematics, pp. 104-120. London: Cassel. Presmeg, N. (1986). Visualization and mathematical giftedness. Educational Studies in Mathematics, 17, pp. 297-311. Presmeg, N. (2006). Research on visualization in learning and teaching mathematics: Emergence from psychology. In: A. Gutiérrez, & P. Boero (Eds.), Handbook of research on the psychology of mathematics education (pp. 205–235). Dordrecht: Sense Publishers. Ramalho, G. (1994). As nossas crianças e a Matemática. Caracterização da participação dos alunos portugueses no “Second International Assessment of Educational Progress”. Lisboa: DEPGEF. Rivera, F. & Becker, J. (2005). Figural and numerical modes of generalizing in Algebra. In Mathematics Teaching in the Middle School, 11(4), pp. 198-203. Rivera, F. & Becker, J. (2007). Abduction in pattern generalization. In Woo, J. H., Lew, H. C., Park, K. S. & Seo, D. Y. (Eds.). Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education, Vol. 4, pp. 97-104. Seoul: PME. Sasman, M., Olivier, A., Linchevski, L. (1999). Factors influencing students’ generalization thinking processes. In O. Zaslavski (Ed.). Proceedings of the 23th International Conference for Psychology of Mathematics Education. (Vol. 4, pp. 161-168). Haifa, Israel. Shama, G. & Dreyfus, T. (1994). Visual, algebraic and mixed strategies in visually presented linear programming poblems. Educational Studies in Mathematics, 26, pp. 45-70.

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Stacey, K. (1989). Finding and Using Patterns in Linear Generalising Problems. Educational Studies in Mathematics 20(2), pp. 147164. Steen, L. (1990). On the shoulders of giants: New approaches to numeracy. Washington, DC: National Academy Press. Swafford, J. & Langrall, C. (2000). Grade 6 students’ preinstructional use of equations to describe and represent problem situations. Journal for Research in Mathematics Education, 31(1), 89-112. Tall, D. (1991). (Ed). Advanced Mathematical Thinking, Kluwer Academic Publishers: The Netherlands. Vale, I & Pimentel, T. (2005). Padrões: um tema transversal do currículo. Educação e Matemática, n.º 85, pp.14-20. Vale, I., Palhares, P., Cabrita, I. & Borralho, A. (2006). Os padrões no ensino aprendizagem da Álgebra. In I. Vale, T. Pimentel, A. Barbosa, L. Fonseca, L. Santos, P. Canavarro (Orgs.), Números e Álgebra na aprendizagem da matemática e na formação de professores (pp. 193-213). Lisboa: SPCE. Warren, E. (2008). Generalising the pattern rule for visual growth patterns: Actions that support 8 year olds’ thinking. Educational Studies in Mathematics, 67, pp. 171, 185. Zazkis, R. & Liljedahl, P. (2002). Generalization of patterns: The tension between algebraic thinking and algebraic notation. Educational studies in mathematics, 49, 379-402.

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Visual Pattern Tasks with Elementary Teachers and Students: a Didactical Experience

Visual Pattern Tasks with Elementary Teachers and Students: a Didactical Experience Isabel Vale [email protected], School of Education of the Polytechnic Institute Viana do Castelo, Portugal

Teresa Pimentel [email protected], School of Education of the Polytechnic Institute Viana do Castelo, Portugal RESUMO Estamos envolvidas num projecto1 cuja finalidade é analisar o impacto de tarefas com padrões no ensino e aprendizagem da matemática. Este artigo incide numa investigação em curso com professores do ensino básico envolvidos num programa de formação contínua e com os respectivos alunos. Se “ver” é uma componente importante da generalização uma das nossas principais preocupações deve ser o estudo sobre tarefas com padrões figurativos/visuais que possam conduzir à generalização. Apresentaremos alguns dos resultados obtidos por aplicação das tarefas na experiência didáctica e algumas conclusões. Palavras-chave: padrões, generalização, visualização, pensamento algébrico

ABSTRACT We are involved in a project1 in which our aim is to analyze the impact of pattern tasks in the teaching and learning of mathematics. This paper reports on a part of an ongoing research with in-service teachers and their elementary students. If “seeing” is an important component of generalization our main concern is focusing on figurative pattern tasks that can be related to generalization. We will present the results obtained with the implementation used in a didactical experience and some preliminary conclusions. Key words: patterns, generalization, visual, algebraic thinking

Introduction If algebra is considered a tool for expressing generalities, exploring growing patterns in elementary levels lays in the foundation for the algebraic reasoning (Blanton & Kaput, 2001; Usiskin, 1999) School teachers traditionally have a tendency to explore more numerical than visual patterns which in many situations can be a problem to reach generalization, and consequently to get an algebraic expression/formula. Pattern tasks provide a context through the development of meaning of formal representations and can reduce the dificulties associated with instruction that focuses on manipulating symbols without connection to the meaning behind these symbols (Lannin, 2005). On the other hand, previous work with visual contexts can help students to get generalization easier, particularly counting tasks can be a way to develop some basic skills that provide students to translate visual patterns into numerical expressions and later to reach far generalization - an important feature in mathematics construction. We expect students generalize the pattern using words and gradually begin to write it using algebraic expressions. There is yet insufficient research at our country with patterns as well as with algebra in the elementary years so it was important for us as mathematics educators to find ways to contribute for the learning and teaching of patterns. 1 The project Mathematics and patterns in elementary school: perspectives and classroom experiences of students and teachers is supported by the Science and Technology Foundation FCT with the reference PTDC /CED/69287/2006 Capítulo 4 - Visualização | Chapter 4 - Visualization

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Theoretical framework In this work we highlight two main areas: teaching and learning mathematics and teacher training in elementary school. Several researchers (e.g. Raymond, 1997; Remillard & Bryans, 2004) propose that teachers’ knowledge, beliefs and attitudes influence theirs actions in the classroom and their interactions with pupils. We strongly believe that one of the main goals of teacher education is to develop teachers’ knowledge and practice competence, in order to promote dynamic, interactive and reflective teachers (Biehler, 1994). Towards a mathematics preparation, Ma (1999) claimed that teachers must have a deep understanding of elementary mathematics to provide teaching and learning processes. Teachers with more explicit and organized knowledge tend to use with their students more conceptual connections, appropriate representations and active student discourse (Warren, 2006). Professional development is nowadays considered a priority (Sowder, 2007). One of the main reasons is that a successful change in students’ performance depends upon teachers’ ability of incorporating new approaches into their instructional practices that increase the mathematical power of their students. For this to happen teachers must be involved in the same kind of activities that they are supposed to propose and develop with their own students. In this way, patterns are important in a mathematics classroom because, through problem solving and pattern searching, students can experience the power and utility of mathematics and also acquire new mathematical concepts. Pattern tasks give students the opportunity to observe and verbalize generalizations and/or translate them in a formal way (English & Warren, 1998; Blanton & Kaput, 2005). If teachers are not in the habit of getting students to work at expressing their own generalizations, then mathematical thinking is not taking place (Mason & Johnston-Wilder., 2004). The development of algebraic thinking in the earlier grades requires the development of particular ways of thinking that result from analyzing relationships between quantities, noticing structure, studying change, generalizing, problem solving, modeling, justifying, proving, and predicting (Kieran, 2004). Number patterns, the relationship between variables and generalization are considered important components of algebra curricula reform in many countries. Those curricula often use generalized number patterns as an introduction to algebra. We are particularly interested in visual/figural contexts because seeing a pattern is a necessary first step in pattern exploration (Lee & Freiman, 2006) so mathematics learning must include problems that compel students to think visually (Tripathi, 2008). The ability to develop and use visual representational forms is valuable enough to become an integral part of mathematical learning (Stylianides & Silver, 2004). Besides, the ability to find and describe patterns found in numbers, operations, charts, geometric figures and so on, is important for the development of a deep understanding of mathematics in general and algebra in particular. Actually, teachers still believe that algebraic thinking specially relates to variables and equations and the study of patterns only relates to geometric transformations. So, professional development programs must develop these important features with teachers in different ways in order to promote their awareness that this work is possible with their own students.

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Methodology We started this study with some assumptions: (1) school teachers tend to explore more the numerical than the figural patterns, which, in many situations, can be too difficult to reach far generalization and, eventually, to get an algebraic expression; (2) it is possible to work a generalized arithmetic and algebra since very early; and (3) teachers need to propose challenging tasks that focus generalization. But before working with patterns it will be useful teaching students to see, as an important component of mathematics activity.

Our goal is starting with problem tasks with a simple numerical answer (arithmetic) and enriches them in order to generate opportunities for pattern building, conjecturing, generalizing and justifying mathematical facts and relations algebrafying activity (Blanton & Kaput, 2003). This way, a diversified set of educational materials was produced based on tasks that were tried in practice and later refined. This exploratory study seeks to understand in what way a didactical experience grounded on figural patterns is a suitable context to get expression of generalization and can contribute to foster mathematical learning, particularly to approach algebraic thinking. We are interested in answering to the question: “What prior experiences and knowledge would students need to engage with growing pattern tasks successfully?” that can be oriented by the following sub questions. Q1. How can we characterize students’ responses about pattern tasks involving the expression of generalization?; Q2. How can a didactical experience with patterns, with a focus on visualization, help to get generalization (verbal or numeric and algebraic expressions)?; and Q3. What is the impact of this didactical experience with patterns to reach generalization in students’ understanding of algebraic ideas?

We adopted a qualitative approach. In the first part of the study we followed a class of elementary pre-service teachers of a School of Education where the didactical experience was implemented, and then six in-service teachers were observed during the teaching experiment with their own 1-4 grade students. The data was collected in a holistic, descriptive and interpretative way through observations, questionnaires and documents (e.g. worksheets, tests, individual works). This paper focuses only the 1-4 grade students and the main data sources are the students’ solutions to some pattern tasks.

The didactical experience with examples The goal of this didactical experience was to promote a meaningful pattern approach to algebraic thinking through figural pattern tasks. We are interested in exploring the ways in which generalization is related to visual contexts. In particular, we intend to translate between growing figural patterns and number patterns using different representations: objects, drawings, words, numbers and formulas. We use the term growing figural/figurative/visual patterns when in a sequence the numbers, drawings or shapes exhibit relationships among one another that allow us to continue it in a predictive way. Fig. 1 shows an example of a growing pattern. Capítulo 4 - Visualização | Chapter 4 - Visualization

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Fig. 1 The growing T pattern

More than to develop students’ skills to get a formula it is important that they understand the meaning of that formula or rule and reason in such a way to convince themselves and the others of the validity of the rule or formula (generalization) through numerical or visual methods. This experience has a strong figurative component, based in a problem solving context where we mainly promote communication through different representation modes: objects, drawing, verbal, numerical or algebraic, to get near and far generalization (e.g. Stacey, 1989). Mason (1996) speaks in local and global generalization. We mean near generalization when, in a sequence, we can extend the pattern to terms near of those that are given, by the analysis of the previous terms, using a drawing or a recursive method (going step by step). We mean far generalization when the terms are in such a position that it is difficult or impossible to get them recursively and we need to look for other strategies as seeking a relationship between the term (e.g. the number of elements that compose the figure) and its order, or position, in the sequence. This type of generalization requires a more sophisticated reasoning: to recognize the global structure of the pattern. It is a functional generalization that can be verbalized but normally requires symbols. Within the didactics of mathematics we are interested in what Simon calls hypothetical learning trajectory. It reflects its roots in a particular constructive perspective that includes the learning goal, the learning activities and the thinking and learning in which the students might engage (Simon, 1995). So we propose a set of tasks that constitutes a lesson series or a teaching sequence and our framework is divided in four main category step tasks: previous visual counting; visual counting in different contexts; sequences; and problem tasks. This set of tasks is crossed by an unifying construct - generalization - that involves the mathematics concepts and properties that can represent a rich background for algebraic thinking in the elementary grades. (1) Previous visual counting - Patterns recognition to develop subitizing, by the recognition of basic pattern configurations (e.g. the five pattern dispositions in cards, dominoes, conventional dice), as a fundamental skill to number sense and consequently to other contexts; (2) Visual counting in different contexts - Numerical tasks of counting have their foundations on the recognition of patterns. Counting tasks can be a way to develop some skills that provide students to translate visual patterns into numerical expressions. This also contributes to the development of number sense: counting, mental computation, properties of and relations between operations, writing and recognition of numerical expressions. All these provide students with flexible, intuitive thinking about numbers; (3) Sequences - To look for patterns in sequences (concrete, figurative, numeric), and to get generalization through rules that students can formulate, allows algebra learning in a gradual manner. We expect that, when students are extending sequences, they can reach near and far generalization, by representing their mathematical discoveries in different ways. Near generalization will give students an opportunity to get recursive reasoning while far generalization is an opportunity to get functional rea154

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soning; and (4) Problems - To look for a pattern is a powerful strategy of problem solving. In these tasks, students have to construct their own sequences to discover the pattern to reach generalization and consequently to get the solution. Far generalization can be reached either through formulas or verbally depending on the level of the student. Figure 2 resumes our learning and teaching framework.

Fig. 2 Learning and teaching framework

The following shows the different steps of the didactical experience with some classroom examples.

Previous visual counting tasks. To establish the necessary relationships between order and term in growth patterns conducting to far generalization, it is crucial the capacity of seeing, namely decomposing the figure in several ways and recognizing patterns. Visual counting tasks can be a prior experience to develop some skills that provide students to translate visual patterns into numerical expressions. We argue that we can teach students to see providing them with this type of tasks. This work gave the basis for working counting in context more complex, but for getting success this previous work is essential. Figure 3 shows several modes of seeing the number six by first graders. In the sample task first grade students relate dots and empty spaces using the ten frame – they work the complementary to ten. Getting familiar with several visual number patterns, they simultaneously work with numbers and operations and their relationships.

Fig. 3 Several modes of seeing six and working the complementary

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Visual counting tasks in other contexts. We also propose other, more elaborated, visual tasks for elder students, in order to promote a grasp of visual counting skills to help the exploration of growing patterns in sequences. In dealing with counting, mental computation, properties of and relations between operations, writing and recognition of numerical expressions, students acquire number sense in developing flexible, intuitive thinking about numbers. Among the visual counting tasks proposed to the students we show the following in Figure 4: The pizza How many slices of tomato and pepper has this pizza? Find a quick process to count them

Fig. 4 The Pizza

Students began working in groups of four. Figure 5 shows some of the solutions of elementary students of 3rd/4th grade. Students had to promptly count in various ways the pieces of tomato and pepper in the pizza and then write the corresponding numerical expression.

Fig. 5 Some of the solutions of pizza counting problem

Figure 6 shows the work of students on the pizza counting problem, and each group presentation of their solution at the end, as can be seen, since we specially value the development of communication skills. In fact, in sharing with their colleagues their ways of thinking, students deepen their own comprehension, and, on the other hand, one’s explanation may be helpful to others to get the first way or more ways of seeing the situation.

Fig. 6 Students work and presentation of the solutions 156

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Sequences – discovering, extending, generalizing. To look for patterns in sequences (concrete, numeric, figurative), and to get generalization through rules that students can formulate, allows algebra learning in a gradual manner. We expect that when students extending sequences can reach near and far generalization, representing their mathematical discoveries in different ways. Different solutions can be got through different ways of seeing that match to different algebraic expressions and all are equivalent. We chose the example in Figure 7 to illustrate one of the tasks involving sequences. Many other tasks were explored with repeating patterns and growing patterns.

The “fives” dots sequence 1. Analyze the growing pattern. 2. If you need you can continue the sequence using plastic balls. 3. How many dots have each five? As you did visual counting, try to seek numerical expressions to the counting of the dots. 4. Together with your group, find a rule that relates the number of dots of each “five” with its position in the sequence. Fig. 7 The “fives” dots sequence

In the beginning of the task, the teacher stressed the importance of visual counting of the dots, as they did previously with the pizza counting problem. To this problem five different strategies were got in the groups of students of the class of 3rd/4th graders. Two of them were numerical and the other three were visual. We will present all of them through the following Figures 8 to 12.

1. Numerical strategy Students did recursive thinking, using later trial and error of numerical expressions that could fit the numbers 6, 11, 16, etc. Nevertheless, they couldn’t find a “good” expression and had to get help to achieve far generalization.

Fig. 8 1st numerical strategy

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2. Numerical strategy The students decomposed 11 in 10+1, 16 in 15+1, ... and then they saw the “five table” in the first addend.

Fig. 9 2nd numerical strategy

3. Visual strategy Students saw five equal groups and one more dot. Then they could make a far generalization, noticing that each of the five groups had the same number of dots than the number of the figure. Students achieved formal generalization using the variable n since the seeing process they used allowed getting the algebraic expression easily.

Fig. 10 1st visual strategy

4. Visual strategy Students saw three rows and two columns, discovering that each row had one more dot and each linking column one less dot than the number of the figure. They achieved far generalization to the required figure 100, and then, to show they perfectly understood the process, they did it to figure 99. They also could verbalize this understanding. In this case students didn’t achieve an algebraic expression because the way of seeing would conduct to a too complex expression at this level. Fig. 11 2nd visual strategy

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5. Visual strategy They saw two columns in the borders and some columns of three in the middle. This was the most difficult to generalize, but one of the students perfectly explained to the class that each column had two more dots then the number of the figure and each of the linking rows had one less dot then the number of the figure. This allowed them to generalize to the figure 100.

Fig. 12 3rd visual strategy

Problems. To look for a pattern is a powerful strategy of problem solving. In these tasks, students have either to construct their own sequences to discover the pattern to reach generalization and consequently to get the solution, or identify patterns that generate numerical properties. To solve problems in a critical and flexible way is one of the most important goals of school mathematics. Figure 13 shows one of the problems used and the solution of a 4th grader.

The rumor problem At 9 o’clock, John gave a good new to two colleagues: tomorrow there is a great movie at school. In the following five minutes each of the students that heard the new told it, the next five minutes, only to two more friends and then didn’t tell it to anyone else. At nine and a half how many students knew the new?

Fig. 13 Solution of the rumour problem

This student began with a tree diagram and the construction of a numerical sequence. When he perceived the pattern, quickly abandoned the diagram and, using recursive thinking, got the solution in a more efficient way.

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Teacher’s reflections We present some comments in the teacher portfolio, which reveal an accurate perception of the importance of this type of work with children. -- This figure [Fig 14] – as well as other similar ones made by students in their sheets – shows that the previous work of visual counting was actually applied in this exploration.

Fig. 14 Presented by the teacher in her portfolio

-- When I prepared the lesson, I worried since I realized that my interpretation didn’t exhaust every possibilities of analysis. This could make difficult my comprehension of students reasoning or condition them to my strategies. -- On the other hand, I was afraid that children didn’t achieve generalization alone (...). Anyway, I decided to give them the opportunity to surprise us. -- After all, they exceeded my expectations in -- motivation and engagement; -- approaches sharing; -- reflection; -- using of several strategies; -- reasoning in the exploration and generalization of the pattern; -- translating their thinking into mathematical expressions.

Concluding remarks We intended to illustrate that a patterning work is a way to: look for different ways of seeing when working with numerical sequences and problems; promote different strategies for counting; give sense to numerical expressions in relating them with visual representation; allow near and far generalization; motivate students; develop mathematical knowledge and communication skills; and construct algebraic thinking. This didactical experience allows us to conclude that pattern tasks, in addition to motivate students to math, since they 160

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challenge curiosity and creativity, also allow in some cases that young students can reach far generalization, translated by expressions with variables, in a natural way. They justify their algebraic generalizations, most of the times, reasoning on the figures, through verbal and writing communication. In particular, in the growing sequences tasks, the previous work about counting in figurative contexts gave opportunities to students for choosing the ­numerical, visual or mixed approach that was more convenient to solve the proposed task. This was also a learning process for the teachers involved, since their related knowledge was weak and this is a new topic that is now taking an important place in the new curriculum. These results show that mathematics learning must include problems that compel students to think visually and talk about those different seeing using different representations for translate them. So teachers must be aware of the power of these tasks in the mathematics classes since early years, that must be integrated in their teaching, and also notice that this only happens if they ask students not only to solve the tasks but mainly to communicate and to justify their reasoning.

References Biehler, R. (1994). Teacher education and research on teaching. Em R. Biehler, R. Scholz, R. Sträber e B. Winkelmann (Eds.), Didactics of mathematics as a scientific discipline (pp. 55-60). Dordrecht: Kluwer Academic Pubishers. Blanton, M. & Kaput, J. (2001). Algebrafying the Elementary Mathematics Experience. Part II: Transforming Practice on a District-Wide Scale. Proceedings of the ICMI-Algebra Conference. Melbourne, Australia, Dec.2001. Blanton, M. & Kaput, J. (2003).Developing Elementary Teachers’ “Algebra Eyes and Ears”. Teaching Children Mathematics, 10(2), 70-77. Blanton, M. & Kaput, J. (2005). Characterizing a classroom practice that promotes algebraic reasoning. Journal for Research in Mathematics Education, 36(5), 412-446. English, L. & Warren, E. (1998). Introducing the variable through pattern exploration, Mathematics Teacher, 91(2), 166-170. Kieran, C. (2004). Algebraic Thinking in the Early Grades: What is it?. Mathematics Educator, Vol.8, No.1, 139-151. Lannin, J. (2005). Generalization and Justification: the challenge of introducing algebraic reasoning through patterning activities. Mathematical Thinking and Learning, 7(3), 231-258. Lee, L. (1996). An initiation into algebraic culture through generalization activities. In N. Bednarz, C. Kieran and L. Lee (Eds.) Approaches to Algebra: Perspectives for Research and Teaching (pp. 87-106). Dordrecht: Kluwer Academic Publishers. Lee, L. & Freiman, V. (2006). Developing algebraic thinking through pattern exploration, Mathematics Teaching in the Middle School, 11(9),428-433. Ma, L. (1999). Knowing and teaching mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum. Mason, J. & Johnston-Wilder, S. (2004). Fundamental Constructs in Mathematical Education. London: Routledge-Falmer and The Open University. Mason, J. (1996), Expressing Generality and Roots of Algebra. In N. Bednarz, C. Kieran and L. Lee (Eds.), Approaches to Algebra, Perspectives for Research and Teaching (pp. 65-86). Dordrecht: Kluwer Academic Publishers. Raymond, A. (1997). Inconsistency between a beginning elementary school teacher’s mathematics beliefs and teaching practices. Journal for Research in Mathematics Education, 28, 550-576. Remillard, J. & Bryans, M. (2004). Orientation Toward Mathematics Curriculum Materials. Journal for Research in Mathematics Education, 35, 352-388. Rivera, F. & Becker, J. (2008). Middle school children’s cognitive perceptions of constructive and deconstructive generalizations involvCapítulo 4 - Visualização | Chapter 4 - Visualization

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ing linear figural patterns. ZDM Mathematics Education, 40(1), 65-82. Simon, M. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114-145. Sowder, J. (2007). The mathematical education and development of teachers. In Frank Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning, Vol.1 (pp.157-224). Reston: NCTM. Stacey, K. (1989). Finding and Using Patterns in Linear Generalizing Problems. Educational Studies in Mathematics, 20(2), pp. 147164. Stylianides, G. J. & Silver, A. (2004). Reasoning and proving in school mathematics curricula: An analytic framework for investigating the opportunities offered to students. In D. McDougall & J. Ross (Eds.), Proceedings of the 26th Annual Meeting of the North American Chapter of the International Group for the PME (Vol. 2, pp. 611-619). Toronto, Canada: OISE/UT. Tripathi, P. (2008). Developing Mathematical understanding through multiple representation. Mathematics Teaching in the Middle School, 13(8), 438-445. Usiskin, Z. (1999). Doing Algebra in Grades K-4. In Barbara Moses (Ed.), Algebraic thinking, grades K-12 – Readings from NCTM’s School-Based Journals and Other Publications. Reston: NCTM. Warren, E. (2006). Supporting learning in early algebra: a model of professional learning. Proceedings of MERGA 29, Mathematics Education Research Group of Australasia (pp.535-542), Camberra, Australia.

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Discussion Panel: Patterns in Teacher Training Rosa Antónia Tomás Ferreira [email protected], Faculdade de CIências da Universidade do Porto e Centro de Matemática da Universidade do Porto1

Every contribution to this International Meeting on Patterns focused the importance of patterns and patterning in the learning of mathematics, mostly at the elementary level of education but also at other levels. This panel was intended to shift the Meeting participants’ attention to another important factor for the success of the teaching and learning process – teacher education – given that this topic had not been directly addressed by the Meeting’s speakers. I moderated the panel whose members were Anthony Orton and Jean Orton, from the University of Leeds, United Kingdom, Ferdinand Rivera, from San José State University, United States of America, and Elizabeth Warren, from the Australian Catholic University, in Australia. In this text, I make a short overview of the main aspects that caught my own attention not only based on the panel members’ interventions but also on the discussions that emerged for the duration of the whole panel. Despite the fact that the panel members had contributed to this Meeting with many and profound insights regarding the learning of mathematics, all of them had vast and diversified experience in working in teacher education contexts, Thus, I launched them the challenge of sharing with the audience some of their experiences in which one could see the importance and role of patterns in teacher education programs or in professional development initiatives. I proposed the following questions to all four members of the panel, leaving them the choice of addressing whatever issue they would prefer or find more relevant for the discussion: 1) How can patterns help pre-service teachers or inservice teachers develop their mathematical and didactical knowledge? 2) Are there any mathematical content topics more favorable to address patterns in mathematics teacher education or professional development programs? If so, what are they and why is that so? and 3) How can we think about the role of patterns in teacher education or professional development programs for teachers who teach at the secondary level (or even higher levels)? Jean Orton was the first panel member to address the discussion questions. Her intervention was based on one book whose main goal is to help pre-service and practicing elementary teachers understand the role of patterns and patterning in the learning of mathematics: Learning to teach shape and space (Frobisher, Frobisher, Orton, & Orton, 2007). According to Jean Orton, the ideas of pattern and patterning permeate the entire book, with an intentional focus on classroom use of these two concepts. One major concern of the book’s authors was to make it appealing to teachers, without losing its main purpose of engaging them in thinking about and recognizing the role of patterns in mathematics learning. Thus, besides the many pictures, diagrams, and photographs included in the book, the text was arranged in small sections, all of them clearly linked to the British National Curriculum. This feature is an important one given that, in England, complying with the National Curriculum has constituted one of teachers’ major concerns at the classroom 1 This work was partially supported by Fundação para a Ciência e a Tecnologia (FCT) through the Centro de Matemática da Universidade do Porto (http://www.fc.up.pt/CMUP/) Capítulo 5 - Painel e Síntese | Chapter 5 - Panel and Synthesis

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level. The authors found the use of concept maps (see figure 1) appropriate to help teachers better understanding how patterns are related to many mathematical topics, at all levels of education, not just at the elementary level.

Fig. 1 Concept map (in Frobisher et al., 2007)

One other important issue stressed by Jean Orton was the need to inform teachers, again, at all levels of education, about research results. It is well known teachers’ typical resilience to read research reports not only because they are usually quite long but perhaps mostly because of the technical language in which they are written, more often than not inaccessible to teachers. Yet, it is of utmost importance to have teachers reading and understanding research results so that they can better comprehend their students’ thinking and learning difficulties, and improve their own teaching. Therefore, Frobisher and friends’ book included several references to research results, from a pragmatic point of view. The analysis of students’ actual work on tasks dealing with patterns and patterning can also constitute a means of focusing teachers’ attention on important issues regarding the learning of mathematics. Thus, the book Learning to teach about shape and space (Frobisher et al., 2007) also includes extensive examples of students’ work, as well as pictures of students involved in mathematical activity based on patterns and patterning, which can help teachers in better comprehending students’ thinking and better conducting classroom discussions. The teachers are also encouraged to engage in mathematical explorations or investigations around patterns, as figure 2 illustrates.

Fig. 2 Examples of explorations and investigations for teachers (in Frobisher et al., 2007) 164

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Anthony Orton followed his wife’s intervention and based his speech on how to help preparing mathematics teachers to use patterns and patterning in their teaching, using the English context as a referent. Anthony Orton made a short overview of how secondary mathematics teachers are recruited in England, calling special attention to the consequences of the fact that most secondary mathematics teachers do not hold an undergraduate degree in mathematics. In addition, they take only a short teacher training course which cannot offer them the necessary preparation to start facing the many challenges of the profession. These teachers are expected to develop all the knowledge on teaching mathematics necessary to do well in the classroom, including how to use patterns to enhance students’ mathematical understanding. Yet, patterns are not definitely a priority in those teacher training courses, despite being mentioned in the English National Curriculum. Despite some differences, there are also some similarities in the Portuguese reality. The mathematical experiences of most British secondary mathematics teachers, throughout their undergraduate programs, make them users of mathematics as opposed as constructors of mathematical ideas. As such, most secondary mathematics teachers do not develop a state of mind that allows them to see mathematics as it really is, especially as a science of patterns. Therefore, it is very difficult for them, if at all possible, to give patterns the importance they should have in the experiences teachers must provide to their students in the classroom. The issue of the need of textbooks for teacher education courses came back to the table, given that Jean Orton had already touched the topic when describing the book Learning to teach shape and space (Frobisher et al., 2007). The need (or no need) of textbooks for teacher education courses emerged in connection with teachers’ typical use of school textbooks as sources of ideas for classroom use, or even as reflections of the curriculum itself. This is so regardless of the level of education, whether elementary of secondary school teaching. Like in many other countries, in England there are many school textbooks, some with great quality and others quite poor. At the elementary level, most textbooks do address patterns, but this is not the case with secondary textbooks. Given the typical textbook-dependence of teachers, a good textbook might actually work as an effective way to have teachers addressing patterns in their classrooms. However, it still remains unknown how elementary or secondary teachers view the importance of patterns for the learning of mathematics. Other important differences between elementary and secondary educational levels were referred in Anthony Orton’s intervention – while both teachers and teacher educators at the elementary level seem to share a significant enthusiasm for patterns and patterning, the reality changes when secondary education is concerned. Furthermore, teachers’ beliefs about the importance of patterns for the learning of mathematics seem reflected in their lack of enthusiasm for enrolling in professional development programs focused on patterns or patterning. The rest of Anthony Orton’s intervention was based on the illustration of how another book for teachers (Frobisher, Monaghan, Orton, Orton, Roper, & Threlfall, 1999), different from that mentioned by Jean Orton in her intervention, addresses patterns as a transversal issue in the teaching and learning of elementary mathematics. All mathematical topics are contemplated with references to patterns and to their importance in the learning of various topics, from number systems and operations, to algebra, to geometry. In addition, several learning processes related to patterns are stressed in Capítulo 5 - Painel e Síntese | Chapter 5 - Panel and Synthesis

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this book, such as pattern spotting, repeating patterns, cyclic patterns, and generalizing. Figure 3 illustrates this idea, already present in the book which Jean Orton referred in her intervention earlier.

Fig. 3 Concept map (in Frobisher et al., 1999)

During some discussion around the idea of textbooks for teacher education courses or other professional development programs, a Portuguese example came to the shore: The mathematical experience at basic education (Boavida, Paiva, Cebola, Vale & Pimentel, 2008). This book, available for download at the Ministry of Education website, was essentially written based on the work developed in a nationwide project aimed at improving the mathematical knowledge of teachers teaching 1st through 6th grade – the Professional Development Program in Mathematics for Teachers. However, this project is classroom-based, giving it a different character from more typical and more academic professional development endeavors in Portugal. The book The mathematical experience at basic education fulfills a series of characteristics mentioned by Jean and Anthony Orton as important in order to have teachers understanding the role of patterns (and many other concepts) in the learning of mathematics. Furthermore, the book also contributed to bridging the gap between theory and practice and to illustrating how research results do have a pedagogical outreach. According to Anthony Orton and, once again, following his wife’s opinion, the book Learning to teach number (Frobisher et al., 1999) is expected to help elementary mathematics teachers in better understanding the role of patterns in the school mathematics curriculum. Yet, there is no research information so far about how the book has been used in teacher education programs or on how teachers’ beliefs about patterns in the teaching and learning of mathematics have evolved. Ferdinand Rivera’s contribution to the panel was based on his paper, presented earlier at the Meeting. He stressed the importance of analyzing students’ work and valuing students’ alternative strategies to solve problems involving pat166

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terns or patterning. This demands from the teachers a wide and solid knowledge of mathematics. One other aspect emphasized by Ferdinand Rivera was the various ways of reaching a generalization and the importance of visualization in that process. Furthermore, justifications and argumentations must also be valued at the classroom level as they allow teachers to better access students’ thinking and help students in constructing their knowledge based on what they already know and understand. According to Rivera, and reinforcing many ideas presented before by Anthony and Jean Orton, all these aspects must have a deeper attention in current teacher education programs, as well as professional development endeavors, because they are not widely addressed nor developed. The last panel member to speak was Elizabeth Warren. She reinforced many suggestions already offered by the other panel members, and mentioned, during her intervention, a very recent experience she had had with a group of Portuguese teachers, working at junior high or secondary school levels. The seminar she had given to this group of teachers, and the work she had done with them, was not contextualized in any teacher education or professional development program, but rather occurred during one of the regular monthly meetings that this group of teachers has with me. The teachers with whom Elizabeth Warren worked were, at the time, working on a nationwide project designed to improve the teaching and learning of mathematics and, consequently, to promote students’ success in mathematics. The project was called the Plan of Mathematics, started in the school year of 2006/07, and lasted for three years. In a nutshell, all public schools teaching students from grades 5 to 9 (basic compulsory education in Portugal goes through 9th grade) designed their own project, based on the diagnosis of the learning difficulties of their own students and on the schools’ human resources and characteristics of the surrounding communities. There were more than 1000 public schools involved in this wide project, each one with its own project, objectives and strategies, but all of them converging to the goal of students’ success in mathematics and benefitting from some kind of support from the Portuguese Ministry of Education. A group of teachers was chosen to provide scientific and pedagogical support to the development of those projects in schools, and my job was to monitor and coordinate the subgroup of teachers who worked in the northern region of Portugal. Elizabeth Warren’s seminar to the aforementioned group of teachers was on patterns in the teaching and learning of mathematics, a seminar that was also open to other attendants – pre-service and practicing teachers, and university researchers. None of the attendants had any experience in teaching at the elementary level (1st through 4th grade). Though many of the teachers had taught several mathematical topics making use of patterns, the insights gained from looking at patterns from a basic level and seeing the high-level mathematics that is involved and with which very young children can work were overwhelming. Elizabeth Warren’s hands-on approach to address important issues related to patterns engaged the teachers in serious mathematical thinking and helped them in better understanding students’ thinking and some of their learning difficulties. The mention Elizabeth Warren made to her recent experience related to the Plan of Mathematics made a perfect link to the set of questions I had prepared for the second part of the panel, aimed at the audience. I knew that many people currently involved in the implementation of several nationwide projects in Portugal would be attending the Meeting. Capítulo 5 - Painel e Síntese | Chapter 5 - Panel and Synthesis

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Thus, I prepared some questions ahead to pose them, hoping that the panel members, based on their experience and expertise, would give some feedback about all those projects. I wanted to know whether patterns had been addressed within the context of the Plan of Mathematics, and, if so, how schools had integrated patterns in the work they were developing with the students. Some contributions to the Meeting were contextualized within the Professional Development Program in Mathematics for Teachers which I referred earlier when I stressed the importance of the book The mathematical experience at basic education (Boavida et al., 2008). I wanted to know how important patterns were within the Professional Development Program in Mathematics for Teachers at a national level, not just at the level of the particular experiences that had been shared earlier in the Meeting. Last but not least, I wanted to know what the piloting teachers of the New Mathematics Program for Basic Education (DGIDC, 2007), for grades 1 through 9, were thinking about the role of patterns for the learning of mathematics, and what challenges they were facing when emphasizing patterns in the classroom. Furthermore, I wished to know what professional development programs were being designed to support teachers in better understanding the role and importance of patterns in the light of the New Mathematics Program. With this new set of questions, the discussion moved away from the panel members to the audience, and many people (those directly involved in the projects as well as those who had presented communications at the Meeting) contributed to the discussion by sharing their experiences, exchanging ideas, and discussing perspectives. Giving feedback to the ideas that were being put to discussion was not an easy task for the panel members, as they were lacking some contextualization of the many projects that were being addressed. At the same time, with such an enthusiastic participation of the audience in the discussion, my role as a moderator was also quite difficult. However, in the end, everybody learned something with each other, stressing the need of complementarity amongst all the projects that were currently going on in Portugal.

References Boavida, A. M., Paiva, A. L., Cebola, G., Vale, I., & Pimentel, T. (2008). A experiência matemática no ensino básico [The mathematical experience at basic education]. Lisboa: DGIDC. DGIDC (2007). Programa de Matemática do Ensino Básico [Mathematics program for basic education]. Available at http://www.dgidc. min-edu.pt/matematica/Documents/ProgramaMatematica.pdf Frobisher, L., Frobisher, A., Orton, A., & Orton, J. (2007). Learning to teach shape and space. Cheltenham: Nelson Thornes Frobisher, L. Monaghan, J., Orton, A., Orton, J., Roper, T., & Threlfall, J., (1999). Learning to teach number. Cheltelnham: Stanley Thornes.

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Uma agenda para investigação sobre padrões e regularidades no ensino-aprendizagem da Matemática e na formação de professores João Pedro da Ponte [email protected], Instituto de Educação da Universidade de Lisboa RESUMO O trabalho com padrões e regularidades tem conhecido uma atenção crescente, tanto em Portugal como em muitos outros países. É manifesto que se trata de uma área com importantes potencialidades para a aprendizagem dos alunos, merecedora de grande atenção nos programas escolares e de um lugar de relevo na formação inicial e contínua de professores. A investigação em educação matemática deve dar-lhe, por isso, especial atenção, estudando as suas diversas vertentes. Nesta nota final faço algumas reflexões em relação a estes pontos, tendo por base as apresentações feitas e as questões discutidas durante o encontro. ABSTRACT Work with patterns and regularities has known increased attention, both in Portugal and many other countries. It is clear that this is an area with significant potential for student learning, worthy of great attention in the school curriculum and an important place in pre-service and in-service teacher training. Therefore, research in mathematics education should give patterns special attention, studying its various aspects. In this final note I make some reflections about these aspects, based on the presentations and questions discussed during the meeting.

O que são padrões e regularidades? As noções de padrão e regularidade estimulam a nossa curiosidade e imaginação. O termo “padrão” levanta, no entanto, diversas dificuldades. Em primeiro lugar, temos desde logo uma dificuldade de tradução. A literatura em inglês usa abundantemente o termo “pattern” e a primeira coisa que nos ocorre é traduzir este termo por “padrão”. No entanto, como nos mostra o dicionário, para além de alguns significados próximos nas duas línguas, como “modelo” ou “motivo”, os dois termos têm também significados muito divergentes – é impensável, por exemplo, traduzir “Padrão dos Descobrimentos” como “Pattern of the Discoveries”... Em segundo lugar, e certamente mais complicado, ainda não se conseguiu uma definição consensual de “padrão”. Isso acontece porque a noção de padrão não é uma noção matemática propriamente dita, inserida num campo da Matemática bem definido, mas sim uma noção “meta-matemática”, transversal aos mais diversos campos. Podemos falar de padrões em todas as áreas da Matemática e envolvendo os mais diversos objectos – na Geometria, na Teoria dos Números, na Álgebra, etc. Em cada caso, esses padrões adquirem configurações e propriedades próprias, sendo difícil identificar o que é comum a todos eles. Podemos ir um pouco mais longe e perguntar se precisamos realmente de uma “definição” de padrão. O que faríamos com tal definição? Em Matemática, usamos as definições dos conceitos para deduzirmos as suas propriedades. A educação matemática, no entanto, não é uma teoria dedutiva como a Matemática, e o conhecimento em educação não se constrói nem se valida por dedução. Por isso, tal definição apenas poderia servir para usar como critério para decidir se um dado objecto é ou não um padrão. No entanto, a verdade é que qualquer objecto pode ser visto como um padrão – tudo depende do sistema em que está integrado e é para as características desse sistema que temos de voltar a nossa atenção. Capítulo 5 - Painel e Síntese | Chapter 5 - Panel and Synthesis

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Um termo que na língua portuguesa aparece frequentemente associado a “padrão” é “regularidade”. Ao passo que “padrão” aponta sobretudo para a unidade de base que eventualmente se replica, de forma exactamente igual ou de acordo com alguma lei de transformação, “regularidade” remete sobretudo para a relação que existe entre os diversos objectos, aquilo que é comum a todos eles ou que de algum modo os liga. Padrões e regularidades são, por isso, dois pontos de vista complementares. No trabalho dos alunos podemos dar mais ênfase a um ou a outro, conforme os objectivos que estiverem em causa. Uma outra questão importante no trabalho com padrões e regularidades diz respeito às representações. Podemos falar de representações pictóricas, geométricas, numéricas, tabulares, algébricas, verbais, através de objectos materiais… Precisamos de saber qual o papel e a importância a atribuir a estas representações. Cada uma delas remete, naturalmente, para um certo campo da Matemática, mas muitas vezes, em vez de se ficar encerrado num só domínio, é mais frutuoso estabelecer ligações entre vários campos. Notemos ainda que, curiosamente, uma mesma situação surge umas vezes designada por “padrão pictórico” (ou “figurativo”) e outras vezes por “padrão geométrico”. No entanto, sabemos bem que “figura” e “figura geométrica” não são a mesma coisa. Em Matemática, ao trabalharmos com um conjunto de objectos, precisamos de saber quais as propriedades que estamos a considerar, e as figuras geométricas têm propriedades diferentes das figuras em geral.

Padrões e regularidades no currículo de Matemática Padrões e regularidades aparecem no trabalho com numerosos tópicos dos programas de Matemática. Como ficou documentado neste encontro, eles surgem, por exemplo, na Geometria (no estudo da simetria, das transformações, das pavimentações, dos papéis de parede ou wallpapers, etc.), na Análise Infinitesimal (nos gradientes, na integração, etc.) e na Álgebra (envolvendo as noções de pensamento algébrico, adição e multiplicação, etc.). Em Portugal, os padrões e regularidades surgem de forma explícita ou implícita em todos os ciclos do ensino básico e também no ensino secundário. É importante continuar a discutir o papel e os objectivos que assumem em cada ciclo. Os padrões e regularidades têm muitas potencialidades para o ensino e a aprendizagem mas também têm um reverso. A sua valorização nos programas de Matemática dá origem a problemas específicos. Na verdade, há algumas incompreensões que facilmente surgem, como, por exemplo, os alunos (e os professores) pensarem que há apenas uma forma correcta de continuar um dado padrão sequencial dado por um número finito de termos, quando, na verdade, existem múltiplas formas de o fazer. Outro possível problema é o trabalho dos alunos com padrões acabar por assumir uma forma mecânica. Se as questões com padrões forem relativamente pobres e tendencialmente sempre muito semelhantes, os alunos rapidamente se apercebem que se trata de um “certo tipo de exercício”, em que o objectivo é determinar “o termo seguinte” ou o “termo geral” e mecanizam estratégias para responderem sem ter muito que pensar. Outro problema, ainda, tem a ver com a tendência para valorizar a descoberta de padrões e regularidades e não prestar atenção às demonstrações que se podem fazer acerca deles. Na verdade, é interessante descobrir generalizações mas, em Matemática, o trabalho só fica completo quando conseguimos demonstrar aquilo que descobrimos. 170

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Por isso, sempre que possível, devemos considerar a possibilidade de levar o trabalho dos alunos da generalização à demonstração, usando argumentos apropriados baseados nas propriedades das figuras geométricas, dos números e das operações e das relações e estruturas algébricas.

A aprendizagem e o raciocínio dos alunos A área de investigação mais evoluída no que respeita a padrões e regularidades diz respeito ao estudo dos problemas da aprendizagem e do raciocínio dos alunos. Especial atenção tem sido dada à formulação de estratégias por parte dos alunos e à análise das suas dificuldades em diversos tipos de tarefa, em particular no que respeita ao estabelecimento de generalizações. É claro que existem padrões e regularidades com diferente nível de dificuldade, desde os padrões repetitivos com uma unidade simples ou composta, aos padrões de crescimento lineares e não lineares, passando pelos padrões mistos, com parte repetitiva e parte não repetitiva. Em todos eles as dificuldades variam enormemente e também as estratégias que os alunos usam para os resolver. Os padrões numéricos lineares estão entre os mais acessíveis. Mas os padrões mais interessantes são aqueles onde é possível conjugar elementos geométricos (simetrias, repetições) e aspectos numéricos, de modo a descobrir uma lei geral de formação, ou seja, tirando partido de raciocínios onde a representação visual desempenha um papel importante. É preciso perceber melhor quais são as estratégias de raciocínio dos alunos no trabalho com diversos tipos de padrões e regularidades e também como é que este trabalho pode ajudar a promover todas as outras aprendizagens em Matemática. Outra questão importante tem a ver com a linguagem em que são formuladas estas generalizações. Numa primeira fase, não há muita alternativa senão usar a linguagem natural. A pouco e pouco, no entanto, será possível começar a introduzir elementos simbólicos, contribuindo para desenvolver nos alunos o domínio da linguagem algébrica. A investigação já realizada mostra que os alunos têm dificuldades no seu trabalho inicial com padrões e regularidades, mas também mostra que os alunos aprendem com relativa rapidez e que, uma vez já à vontade, conseguem ir muito mais longe na análise de situações do que aquilo que se poderia pensar. Deste modo, os limites da capacidade dos alunos no trabalho com este tipo de situações estão longe de se encontrarem perfeitamente identificados.

O trabalho na sala de aula O trabalho na sala de aula depende de forma decisiva das tarefas que o professor propõe aos alunos. Muitas das tarefas propostas com padrões e regularidades constituem tarefas de exploração ou investigação, as primeiras mais acessíveis à generalidade dos alunos e as segundas com maior complexidade. Nestas questões, os alunos deparamse com situações matematicamente ricas, acerca das quais se podem colocar diversas perguntas, cabendo-lhes formular de forma mais precisa os aspectos a estudar. A discussão no encontro mostrou a importância dos contextos em que as questões são formuladas e das características que assumem. Assim, é importante que as situações apresentadas possam dar origem a modelos ou conceitos Capítulo 5 - Painel e Síntese | Chapter 5 - Panel and Synthesis

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matemáticos importantes, permitindo aos alunos colocar questões e identificar regularidades, compreendendo o que estão a fazer. Para isso, é necessário usar contextos que sejam familiares ou, pelo menos, compreensíveis para os alunos, ajudando-os desse modo no seu trabalho de reinventar a Matemática. Questões que também merecem a nossa atenção dizem respeito à natureza das tarefas e ao modo como são organizadas entre si. Por exemplo, as tarefas podem ser apresentadas de forma isolada ou é melhor em construir cadeias de tarefas? Que vantagens há em terem maior estrutura ou em serem mais abertas? É preferível colocar apenas uma ou duas questões a propósito de cada situação, ou é mais vantajoso explorá-la de modo mais aprofundado? Qual o papel para tarefas “não-standard”? Até que ponto se deve diversificar as tarefas a propor? Em que momentos e circunstâncias faz sentido propor tarefas de “consolidação”? Questões igualmente fundamentais dizem respeito ao modo como as tarefas são exploradas na sala de aula. Por exemplo, o que se pode esperar do trabalho dos alunos em grupo e o que se pode esperar de uma discussão geral com todos os alunos da turma? Como aproveitar a discussão geral de uma tarefa para a institucionalização de novo conhecimento? E durante a aula, como gerir o tempo e os diferentes ritmos de trabalho dos alunos? Um ponto acerca do qual se faz bastante confusão diz respeito ao propósito das tarefas. Uma tarefa para avaliação dos conhecimentos dos alunos deve ter certamente características muito diferentes uma tarefa cujo objectivo é servir de base à aprendizagem. Essas diferenças devem ser visíveis na formulação das questões, na extensão da tarefa e na sua organização interna. No entanto, usa-se frequentemente uma mesma tarefa tanto com um propósito como com outro, e, por vezes, nem se percebe bem qual o propósito associado à realização de uma certa tarefa. Outra questão diz respeito ao cuidado a ter para manter num patamar elevado o nível cognitivo de uma tarefa. Associado a esta questão é o apoio a prestar aos alunos com dificuldades. Não apoiar estes alunos, quando estes o solicitam ou quando é visível que não conseguem progredir e tendem a desligar-se da tarefa, pode implicar a sua desmotivação para o trabalho matemático presente e futuro. Apoiá-los, ajudando a ultrapassar o obstáculo, pode contribuir para reforço de hábitos de trabalho pouco produtivos e da sua dependência em relação ao professor. Tratase de uma questão que não é fácil, nem tem necessariamente sempre a mesma resposta em todas as circunstâncias.

Formação de professores Um outro campo importante quando se pensa no trabalho com padrões e regularidades no ensino da Matemática diz respeito à formação de professores. Vale a pena determo-nos um pouco no termo “formação de professores”. Na língua portuguesa, ele não se afigura problemático. Quanto o usamos, assumimos que todos lhe atribuímos o mesmo significado. No entanto, os nossos colegas de língua inglesa usam dois termos, com significados bastante diferentes, “teacher training” e “teacher education”. O “teacher training” remete para uma aquisição de competências e capacidades muito específicas. A ideia que lhe está subjacente é que os formadores sabem razoavelmente bem o que o professor deve fazer e treinam-no para desempenhar esse papel. O “teacher education” sugere que o objectivo é proporcionar um crescimento do professor, nos seus conhecimentos, compreensões, capacidades e modos de actuar. 172

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Reconhece-se implicitamente que o professor deve aprender certas coisas, mas é ele que na sua situação profissional terá que tomar as decisões críticas – A que objectivos educacionais dar prioridade? Que tarefas escolher? Como as apresentar aos alunos? Como apoiar o trabalho destes? Como promover a partilha do trabalho feito? Reconhece-se que, embora existam orientações gerais acerca do modo como isto se pode fazer, é em última análise ao professor que cabe decidir como actuar em cada aula, tendo em conta as características dos seus alunos. Ou seja, está em causa saber se o trabalho do professor é de natureza essencialmente técnica, de aplicação de orientações e conhecimentos produzidos externamente, ou se envolve de forma decisiva uma vertente profissional. Também é preciso saber como se concebe a aprendizagem por parte do professor. Considera-se que o professor aprende, em termos cognitivos, reconhecendo conceitos e instâncias da sua concretização nas práticas de ensinoaprendizagem, ou reconhece-se que, para além disso, existe igualmente um papel das influências de ordem social (desde a origem social dos alunos, às pressões da comunicação social sobre os professores), profissional (incluindo grupos associativos, movimentos pedagógicos e outros grupos colaborativos de natureza informal) e institucional (relativas à organização do sistema educativo, do sistema de avaliação, da escola e da sua cultura)? Se se assumir que a investigação em educação, “básica” ou “aplicada”, é capaz de identificar conhecimentos e orientações suficientemente fortes para perspectivar todo o trabalho do professor, então, conceber um programa de formação de professores é basicamente seleccionar, “didactizar” e distribuir esses conhecimentos (alguns dos quais relativos a padrões e regularidades) por um conjunto de disciplinas do plano de estudos. Outra perspectiva, é assumir que o trabalho do professor é vincadamente profissional e que, por isso, a formação de professores tem de ter uma lógica própria, mais complexa, não se reduzindo a servir de simples veículo para os conhecimentos desenvolvidos na investigação “básica” ou “aplicada”. Também neste caso o trabalho com padrões e regularidades terá o seu lugar. Não pode é ser visto como o ponto de partida para se pensar toda a formação do professor. Outra questão central na formação de professores é saber se os padrões e regularidades devem ser abordados como um tópico isolado ou em conjunto com outros tópicos matemáticos. Uma questão relacionada com a anterior, é saber se o trabalho de cunho “mais matemático” com padrões e regularidades deve ser feito em separado do trabalho mais centrado no seu ensino e aprendizagem, ou se há vantagem em integrar os dois aspectos e, caso afirmativo, de que modo.

Questões para trabalho futuro A investigação futura tem de continuar a dar um lugar central ao estudo dos fenómenos da aprendizagem dos alunos quando trabalham com padrões e regularidades. A compreensão do modo como os alunos trabalham, os resultados que conseguem, as suas estratégias e dificuldades, têm de continuar a merecer um lugar central. É a partir daí que tudo mais ganha o seu sentido. Um outro campo de trabalho prioritário diz respeito ao desenvolvimento curricular. É importante considerar os documentos oficiais e outros materiais de apoio que já começam a existir, sendo necessário perceber como é que os Capítulo 5 - Painel e Síntese | Chapter 5 - Panel and Synthesis

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professores trabalham com eles e o que pode ser feito para os melhorar. É importante, também, analisar os manuais escolares no modo como trabalham com padrões e regularidades, que tarefas propõem, que exploração fazem, que ligação estabelecem com outros temas matemáticos, como são entendidos pelos alunos e pelos professores e a que actividade dão origem na sala de aula. Interessante seria também pensar noutros tipos de materiais para os alunos, envolvendo estes conceitos, nomeadamente tirando partido das tecnologias. No que se refere à formação de professores, torna-se importante conceber formas de trabalhar os padrões e regularidades no âmbito dos planos de estudo actuais e avaliar os respectivos resultados. O que é que os professores aprendem quando este estudo se insere estudo em diversas unidades curriculares da formação inicial? E quando se insere numa disciplina que aborda especificamente este tema? De que modo os formandos concretizam essas aprendizagens na sala de aula? Em particular, seria interessante saber como é que os professores que participaram no “Programa de Formação Contínua”, do 1.º e 2.º ciclo, trabalham com padrões e regularidades na sua prática, e também como o fazem antigos alunos da formação inicial de professores e outros professores envolvidos em projectos centrados neste tema. É importante perceber que compreensão sobre este assunto desenvolveram, que prática assumem e que factores a influenciam. Especial atenção deve merecer a prática real nas escolas, em especial nos casos em que os professores já tiveram uma formação substancial no que se refere ao trabalho com padrões e regularidades. É preciso perceber de que modo a aula de Matemática pode evoluir quando se trabalha com este tópico numa perspectiva exploratória e investigativa. Como é que o trabalho começa em cada aula? Como se promove o envolvimento dos alunos em trabalho matemático rico e estimulante? Como é que os professores monitorizam o trabalho dos alunos? Que problemas e impasses mais frequentes têm de ultrapassar? Como se negoceiam significados? Como se estimula a argumentação e a justificação? Como se institucionaliza o novo conhecimento?

Conclusão Ouve-se, com frequência a afirmação, “a Matemática é a ciência dos padrões”, de onde se conclui, algo apressadamente, “logo, o currículo de Matemática deve ser organizado através do estudo de diferentes tipos de padrões”. É verdade que a Matemática é a ciência dos padrões, mas a Matemática é muitas outras coisas – é a ciência das estruturas e dos modelos, é a ciência que estuda a quantidade e a forma, e é também uma actividade humana que envolve abstrair, generalizar, provar… O estudo de padrões e regularidades, devidamente perspectivado, pode ser um enriquecimento muito importante do currículo de Matemática. No entanto, encarado como um tema em si mesmo, isoladamente dos restantes, e com vocação hegemónica, pode dar origem a diversos mal entendidos e incompreensões. Também a valorização do trabalho com padrões e regularidades na formação inicial e contínua de professores é um contributo importante na formação destes profissionais. No entanto, perspectivado em competição com outros elementos tão ou mais importantes para o trabalho do futuro docente, pode assumir um papel contraproducente. É 174

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preciso encontrar o seu lugar próprio e integrá-la no contexto mais amplo das orientações curriculares e para a formação de professores. Finalmente, é preciso perceber melhor a compreensão que um professor precisa de ter, e as condições institucionais que tem de dispor, para que o trabalho com padrões e regularidades possa ser um elemento chave da sua prática profissional. Tanto na sala de aula como na formação de professores é preciso investigar e experimentar, com entusiasmo, investindo naquilo que nos parece importante, mas sem perder o sentido crítico, mostrando capacidade de distinguir o que de facto funciona do que parecia uma boa ideia mas não resulta. Este encontro mostrou que muito já se avançou no estudo do papel dos padrões e regularidades no ensino da Matemática em Portugal e noutras partes do mundo. Resta fazer votos para que este campo – que obteve amplo reconhecimento nos programas de Matemática do ensino básico de 1997 – possa continuar a desenvolver-se, assumindo plenamente as suas possibilidades no trabalho de alunos, professores e formadores de professores de Matemática.

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