!-ATHEMATICAL
0RIVATE %YE JI-EUN LEE
I
AND
KYOUNG-TAE KIM
F YOU ARE A CLOSE FRIEND OF SOODONG,
would you know the answer to these questions? “Do you know Soodong’s birthday?” “Do you know the age of Soodong’s mother?” “Do you know Soodong’s favorite number?” “Do you know Soodong’s telephone number?”
If you are one of Soodong’s close friends, you may be able to answer some of these questions. However, JI-EUN LEE,
[email protected], is an assistant professor
at Oakland University, Rochester, MI 48309. She teaches K–8 mathematics methods courses and is interested in developing instructional strategies for conceptual understanding. KYOUNGTAE KIM,
[email protected], is a research scientist. He is interested in the integration of mathematics and science.
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M ATH E M ATI CS T E A CH IN G IN T H E M I DD L E S C HO OL Copyright © 2007 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
if Soodong is a total stranger, is it possible to answer them correctly? This article explores an instructional idea wherein students figure out the correct answers using the power of mathematics, particularly the power of algebraic thinking—even though students do not know Soodong. The context of the problems in this article involves an individual’s personal information, which other people do not know, and ends with generalized patterns of solutions. Koirala and Goodwin (2000) introduced a simple form of a similar activity called mathmagic. In one example, students were asked to solve a simple magic problem: “Think of any number, add 7, add 3, and subtract your original number.” All students computed the same answer, 10. This context works for everyone because the final answer is independent of the original number that the student chose. In algebraic form, it can be solved as x + 7 + 3 – x = 10. In this article, we propose several advanced contexts that incorporate simple linear equations and simultaneous equations. All the problem contexts are related to the student’s personal life. This condition draws students’ attention to important characteristics of algebraic thinking. The ultimate goal of this activity is to facilitate the smooth transition from arithmetic to algebra by focusing on students’ understanding of variables, not only when the solution has a specific value but also when it has a range of values.
Weakest Link: Understanding Variables THE RECENT EMPHASIS ON ALGEBRAIC THINKING
in school mathematics supports algebra as a strand that is begun in the early grades (NCTM 2000). We hope this early exposure will help students become comfortable with algebraic symbols at the middle school and high school levels. This use of symbols is one of the obstacles that students encounter with algebra (e.g., Philipp 1992; Usiskin 1988). Algebraic symbols, in particular letters, are used in several ways: as specific unknowns (e.g., 3 + n = 10); in formulas (e.g., Area = l r w [length r width]); and to represent a range of values, as in the expression of generalized patterns (e.g., y = 3t + 6) (Kuchemann 1981; NCTM 2000; Usiskin 2000). However, Kieran (1991) pointed out that “the use of letters to represent a range of values is far more neglected in the teaching of prealgebra than their use of unknowns” (p. 49). In addition, some research studies show that many students have difficulties seeing the link between arithmetic and algebra, also known as generalized arithmetic (Lee and Wheeler 1989; Peck and Jencks 1988). NCTM (2000) suggests that students’ facility with symbol manipulation can be enhanced when it is based on extensive experience with quantities in contexts. In response to this suggestion, the
problems presented in this article provide students with contexts that clearly show the link between arithmetic and algebra, with particular attention paid to the variable as representing a range of values.
Using Personalized Contexts A GOOD PROBLEM-POSING STRATEGY HAS A
profound positive effect not only on understanding mathematical concepts but also on creating a more investigative classroom climate (Whitin 2004). According to previous research, personalizing mathematics word-problem contexts is a good problemposing strategy. This strategy draws and focuses students’ attention on the problem context and helps motivate them to solve it. At the same time, students’ attitudes are more positive when their interests are incorporated into the question asked (Mayer 1984). We designed the problems so that students could benefit from these personalized contexts to help them understand variables as representing a range of values. In the activity, an eighth grader and a sixth grader were paired as a “private eye” and a “client.” We gave prescribed directions to the private eye. The private eye asked a series of questions, which required arithmetic computations by the client. At the final stage, the private eye revealed the client’s personal information using the generalized patterns, that is, algebra. The following section presents examples that use a personalized context for creating generalized solution patterns.
From Personalized Contexts to Generalized Patterns THE EXAMPLES DESCRIBED IN THIS ARTICLE ARE
based on two instructional objectives:
• Students must formulate and reformulate generalized solution patterns out of specific cases. • In so doing, students enhance their fluency in manipulating symbolic expressions. The examples are categorized into two types, based on how the secret information is revealed. In type A problems, the client gives his or her secret number to the private eye as the answer. In type B problems, the private eye must do quick mental math or use a calculator, based on the information the client provides. The following is a pair of middle school students’ work that consists of three type A problems and four type B problems. Lyta, the eighth grader, was a private eye, and Joel, the sixth grader, was a client. At the end of each activity, they discussed why and how the private eye could reveal the secret information.
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TABLE 1 Example 1: “I Know Your Favorite Number!”
TABLE 3 Example 3: “I Know Your Mother’s Age!”
TABLE 2 Example 2: “I Know What Month You Were Born!”
TABLE 4 Example 4: “I Know Your Favorite Day of the Week!”
Type A: Client’s final answer = client’s secret information (examples 1–3) Example 1: “I know your favorite number!” Everyone has something favorite. How can we know each individual’s favorite number? (See the directions in table 1.) When Lyta, the private eye, asked Joel, the client, for the final answer, he gave his favorite number. Example 2: “I know what month you were born!” The directions in table 2 are designed to end with the original starting number that indicates the person’s secret birth month (1 = January, 2 = February, and so on). Example 3: “I know your mother’s age!” Since a mother’s age is two digits, the directions in table 3 require two variables: the tens place digit and the ones place digit. Joel’s final answer directly indicated his mother’s age, since the final form, 10x + y, is the expanded form of two-digit numbers. 408
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Type B: Client’s final answer = clue for the secret information (examples 4–7) In the following examples, the shaded section of each table represents the private eye’s extra work. Example 4: “I know your favorite day of the week!” In this example, the client’s final answer does not provide the client’s favorite day of the week. (See table 4.) To find this information, Lyta, the private eye, did more work by subtracting 10 from the client’s final answer and dividing it by 10. Example 5: “I know what numbers you are thinking of! Three at a time!” Although this example asks for three numbers, the private eye needs only one variable since the numbers are consecutive (see
TABLE 5 Example 5: “I Know What Numbers You Are Thinking Of! Three at a Time!”
TABLE 7 Example 7: “I Know the Last Three Digits of Your Telephone Number!”
TABLE 6 Example 6: “I Know What Your Birthday Is!” asked to provide the sum of two single-digit numbers three times. However, Lyta had to do extra work to reveal the last three digits of Joel’s phone number.
Reflections THE STUDENTS’ DISCUSSION AFTER WORKING
table 5). When Lyta asked for the final answer, Joel provided a clue for the middle number, x. Using this clue, Lyta divided Joel’s final answer by 3. Then she easily figured out that it was the middle number. The first and the last numbers were found by subtracting and adding 1 to it. Example 6: “I know what your birthday is!” This example requires solving simultaneous equations (see table 6). Lyta asked for two values (60x + y and 60x – y) to solve the simultaneous equations. By knowing two values, she found out Joel’s birthday. Example 7: “I know the last three digits of your telephone number!” This is another example of finding three numbers at a time; however, the numbers are not consecutive. The directions in table 7 were not complicated for Joel, because he was simply
through the examples described above showed their interest in doing each activity, reasoning how it works, and creating their own versions of games. Joel was more interested in type B problems. In example 7, he made some conjectures on the shaded directions, listing the possible pairs of numbers in each case. Lyta and Joel had an extensive discussion on this approach. Table 8 shows the summary of their discussion. It was not the procedure we originally intended, but it worked. Joel also mentioned, “I think I can make another problem like this.” Lyta was familiar with algebraic equations, which helped her successfully perform the role of private eye. She commented, “I have a better understanding of simultaneous equations after this game.” This comment implies that Lyta rediscovered the meaning of her mathematical actions through this activity.
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TABLE 8 Joel’s Conjectures on the Shaded Directions in Example 7 FIRST NUMBER
SECOND NUMBER
THIRD NUMBER
0
6
5
1
5
6
2
4
7
3
3
8
4
2
9
5
1
10
6
0
11
Explanations: 1. First, Joel produced six ordered pairs based on the information: the sum of the first two numbers is 6. 2. Joel figured the possible third numbers based on the information: the sum of last two numbers is 11. 3. Joel eliminated “10” and “11” from the possible third number list since these are not single-digit numbers. 4. Joel found that one combination met the last information: the sum of the first and the last number is 13. 5. The last three digits of his phone number are 4, 2, and 9.
Closing Thoughts OUR GOAL IN THIS ARTICLE IS TO HELP STUDENTS
understand the link between arithmetic and algebra. The students’ work supports the point that students’ real-life information raises the level of interest and engagement in the transition from arithmetic to algebra. We supplied the prescribed directions and asked students to reason about the generalized patterns behind our rules. However, as readers may notice, many variations are possible for each example. This activity could be extended by asking students to develop their own directions. As educators, we should allow and encourage students like Joel to make various conjectures and explore other possibilities. We hope that students will see and experience the power of generalizations as an important tool for mathematical problem solvers while they are working through this activity.
References Kieran, Carolyn. “Helping to Make the Transition to Algebra.” Arithmetic Teacher 38 (March 1991): 49–51. Koirala, Hari, and Phillip M. Goodwin. “Teaching Algebra in the Middle Grades Using Mathmagic.” Mathematics Teaching in the Middle School 5 (May 2000): 562–66. Kuchemann, Dietmar. “Algebra.” In Children’s Understanding of Mathematics: 11–16, edited by Kathleen M. Hart, pp. 102–19. London: John Murray, 1981.
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Lee, Lesley, and David Wheeler. “The Arithmetic Connection.” Educational Studies in Mathematics 20 (1) (1989): 41–54. Mayer, Richard E. “Aids to Text Comprehension.” Educational Psychologist 19 (1) (1984): 30–42. National Council of Teacher of Mathematics (NCTM). Principles and Standards for School Mathematics. Reston, VA: NCTM, 2000. Peck, Donald M., and Stanley M. Jencks. “Reality, Arithmetic, Algebra.” Journal of Mathematical Behavior 7 (1988): 85–91. Philipp, Randolph A. “The Many Uses of Algebraic Variables.” Mathematics Teacher 85 (7) (October 1992): 557–61. Usiskin, Zalman. “Conceptions of School Algebra and Uses of Variables.” In The Ideas of Algebra, K–12, edited by A. F. Coxford and A. P. Shulte, pp. 8–19. Reston, VA: National Council of Teachers of Mathematics, 1988. |||. “Doing Algebra in Grades K–4.” In Algebraic Thinking, Grades K–12: Readings from NCTM’s SchoolBased Journals and Other Publications, edited by Barbara Moses, pp. 5–6. Reston, VA: National Council of Teachers of Mathematics, 2000. Whitin, David. “Building a Mathematical Community through Problem Posing.” In Perspectives on the Teaching of Mathematics, edited by Rheta N. Rubenstein and George W. Bright, pp. 129–40. Reston, VA: National Council of Teachers of Mathematics, 2004. L
