Academic paper

Linear Representations of Two-digit Numbers Promote First Graders' Numerical Estimation

Yu Zhang & Yukari Okamoto University of California Santa Barbara

Linear Representations of Two-digit Numbers Promote First Graders' Estimation Objectives Previous studies indicated that it is not until about second grade that numerical estimations of two-digit number magnitudes show a linear correspondence between estimated and actual magnitudes (Siegler & Booth, 2004). This is in line with the developmental progression depicted by Case and Okamoto (1996). Both the empirical evidence and theoretical prediction suggest that children in first grade are beginning to encode magnitudes of two-digit numbers. Thus it becomes important to provide effective instruction to assist first graders in this learning process. The objective of the present study is to examine the effectiveness of three instructional strategies to promote first graders’ acquisition of numerical magnitudes of two-digit numbers. Teaching children magnitudes of decades (e.g., 10, 20, and 30) in a linear fashion was predicted to be more effective than the alternative representations. Theoretical Perspectives Children’s understandings of numerical magnitude between 0 −100 improve with age. The first graders are typically poor at estimating numbers between 0 and 100 (Siegler & Booth, 2004), where smaller numbers are often overestimated and larger numbers are underestimated. The developmental trajectory is assumed to progress from logarithmic to linear representations. Kindergartners, for example, might estimate 9 to be close to 30. They might also estimate 69 to be close to 50. Older children, on the other hand, tend to more accurately estimate numbers. By second grade, most children’s estimates correspond to actual numerical magnitudes, showing a linear function for two- digit numbers (Case & Okamoto, 1996; Booth & Siegler, 2006).

Recent studies with preschool-age children from low-income families suggest that they can learn the magnitudes of single-digit numbers on a number line (Ramani & Siegler, 2008). Siegler and Ramani argued that this is the type of skill children from most middle- income families naturally develop in preschool years through playing board games such as Chutes and Ladders and Candyland. Playing games like these, however, may not necessarily be part of family interactions for children from low-income families. As Ramani and Siegler showed, playing a simple board game with single-digit numbers for 15 minutes a day for 4 days improved kindergartners’ accurate estimates of numerical magnitudes (Ramani & Siegler, 2008). More recently, Laski and Siegler (2014) extended this line of work and showed that playing a 10 × 10 matrix board game promoted kindergartners’ estimation of numbers on a 0−100 number line. They further noted that learning occurred only when the activity was introduced in a context of a board game. That is, children in the context of a board game can encode numbers by connecting them “to visual, auditory, kinesthetic, and temporal magnitude cues...” (p. 861). We acknowledge that board games provide a meaningful context for children to encode numbers. However, we believe there are other ways to achieve the same. In this study, we tested an approach that, if successful, could be easily implemented as part of mathematics instruction in the classroom. Our approach to promoting encoding of twodigit numbers draws ideas from the work of Miura and Okamoto (1988) as well as Siegler and his colleagues (e.g., Ramani & Siegler, 2008). Miura and Okamoto (1988) reported that children who represent two-digit numbers as a precise combination of 10 and unit blocks (e.g., two 10 blocks and four unit blocks for 24) show robust understanding of place value. In turn, those children who

represent numbers with unit blocks only (e.g., 24 unit blocks for 24) lack place value understanding. Thus their findings suggest that teaching children to represent numbers in terms of 10 and unit blocks provides a meaningful context to encode two-digit numbers. Ramani and Siegler's (2008) study showed that moving the token in a linear fashion from left to right effectively models increasing numerical magnitudes on a number line. Combing the two lines of work, we created three instructional conditions for first graders to encode two-digit numbers. Children were instructed to construct two-digit numbers in a linear fashion using: multiple 10 and unit blocks, unit blocks and only one 10 block, and unit blocks only (see Appendix A for sample representations). We predicted that children learning to construct numbers linearly using a precise combination of 10 and unit blocks would learn to encode two-digit numbers more accurately. Methods and Data Sources Participants Participants were 31 first graders (12 boys and 19 girls, mean age = 7 years 1 month, SD = .79, ranging from age 6 years to 8 years 6 months), recruited from an afterschool program of a public elementary school and a parochial primary language school in Southern California. These schools served children from predominantly low to lowermiddle class families. All students from the public school received free/reduced meals under the Federal School Lunch program. The ethnic composition of the participants was 55% Latino/Hispanic, 32% Asian, and 13% Caucasian. 87% of the participants were bilingual Spanish/English or Chinese/English speakers. Materials and Procedures

All participants met individually with the experimenter for four 15-minute sessions (pretest, two training sessions, and posttest, in this order) over a 4-week period. Teachers brought participating students to the pretest session one at a time. The participants were assigned to one of the three conditions in the order they were brought to the experimenter. Pre- and post-test measures. We used a closed number line task (Siegler & Booth, 2004). As in previous studies, a 25-cm line was drawn on each sheet with “0” just below the left end and “100” just below the right end of the number line. The experimenter pointed to the 0 and 100 positions and said, “if this is where 0 goes and this is where 100 goes, where will N go?” No feedback was provided. Children worked on 26 estimation problems presented in a random order. The numbers used were: 2, 3, 6, 7, 11, 14, 15, 19, 21, 23, 24, 28, 32, 36, 44, 47, 51, 58, 63, 69, 72, 76, 84, 87, 91, and 98. The same numbers were used in the pretest and posttest. As in previous studies, we computed each child’s percent absolute error (PAE) as an absolute value of (estimate – estimated quantity) / scale of estimates, multiplied by 100. For instance, if a child was asked to estimate 25 on a 0-100 number line and placed (!"!!")

the mark at the location of 15, then PAE is |

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| x 100, which is 10% (Siegler &

Booth, 2004). A regression equation (i.e., y = ax + b) was applied to examine the estimated numbers fitted by a linear function. Coefficient R2 and slope "a" were used to measure linearity. For instance, if Rlinear2 > Rlog2, the number line is linearly dominated, and vise versa. The ideal linear function is y = ax + b with Rlinear2 = 1.00 and a = 1.00. Training sessions. Children in the unit block only condition used unit blocks only (Ross, 1986). The experimenter first demonstrated how to construct a number using unit

blocks only. She did so by placing the correct number of unit blocks in a linear fashion. Children were then asked to construct five numbers (i.e., 16, 22, 33, 41, and 56) in a linear fashion using unit blocks only. The order of presentations of the numbers was randomized. The training session for the unit blocks and one 10 block condition was identical to the unit block only condition except that children were allowed to used only one 10 block. The experimenter gave an additional demonstration of how to use only one 10 block and unit blocks to construct a number linearly (See Appendix A). Children were then asked to construct the same five numbers linearly as the way she demonstrated. The training session for the multiple 10 and unit blocks condition was identical to the unit blocks and one 10 block condition, except that the experimenter used a correct combination of 10 and unit blocks for her demonstration. For instance, to demonstrate how to construct 37, the experimenter counted out three 10 blocks and then seven unit blocks. She placed them in a line to analogy to the 0−100 number line. Children were then asked to use as many as 10 blocks and the leftover unit blocks to construct the same five numbers linearly. Results Preliminary analyses found no gender differences in any of the indices. Thus data from female and male students were combined for subsequent analyses. First, we carried out a repeated-measures multivariate analysis of variance. The results indicated a significant difference among the three groups, Wilks’ λ = .10, F(6, 52) = 19.34, p < .001, ŋp2 = .69. The effect of time was also significantly different, Wilks’ λ = .13, F(3, 26) = 59.72, p < .001, ŋp2 = .87, as well as significant condition by time interaction, F(4, 54) = 13.16, p