# Should you show me the money? Concrete objects both hurt and help performance on mathematics problems

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5 N.M. McNeil et al. / Learning and Instruction 19 (2009) 171e Procedure Students were randomly assigned to one of two conditions. In the experimental condition, students were given a stack of bills (major denominations up to \$50) and a small plastic bag containing pennies, nickels, dimes, and quarters. The bills and coins were designed to activate students real-world knowledge of U.S. currency, and they looked similar to real money (see Fig. 1, left panel). Students were told that they should use the bills and coins to help them solve the problems. However, we did not prescribe ways for students to use the materials because one of the major criticisms of manipulatives is that students are often required to handle them according to rules designated by the teacher, and this practice deters students from constructing their own understandings (Gravemeijer, 2002). We wanted students to use the money in whatever way made the most sense to them. Students in the experimental condition also had the paper-and-pencil test in front of them at all times, so they were able to use both bills and coins and traditional written strategies. We hoped that students would use the bills and coins as tools for problem solving, but our primary goal was to examine how their presence would affect students performance. In the control condition, students solved the same word problems involving money, but without the presence of bills and coins. Experimenters observed students solving the problems and noted whether those in the experimental condition used the money when solving the problems. Because children were tested in a large group setting, it unfortunately was not possible for experimenters to monitor each child s use of the bills and coins on a problem-by-problem basis. However, experimenters were assigned to monitor specific children throughout the testing session (approximately 5e8 children per experimenter). They checked on these children several times throughout the session and noted whether or not the children were interacting with the bills and coins (e.g., looking, touching, counting, etc.). Most children in the experimental condition (71%) were seen interacting with the bills and coins on a consistent basis. To illustrate how students could use the bills and coins, consider Problem #6 in which Charles buys a gumball and a candy bar for \$0.20 and \$1.15 and gets back change from \$5.00. To calculate the correct change using the bills and coins, students could have placed three one-dollar bills on the table, along with two dollars worth of change (e.g., four quarters, eight dimes, and four nickels). Then, they could take away two dimes (to represent the \$0.20) and one dollar, one dime, and one nickel away (to represent the \$1.15). Finally, they could count the money left on the table (\$3.65). In Fig. 1. Examples of the perceptually rich bills and coins used in Experiments 1 and 2 (left panel) and the bland bills and coins used in Experiment 2 (right panel).

8 178 N.M. McNeil et al. / Learning and Instruction 19 (2009) 171e184 h 2 ¼ To test our hypothesis that perceptually rich bills and coins hinder performance, we performed a set of orthogonal contrasts using Helmert coefficients. Consistent with the results of Experiment 1, perceptually rich bills and coins hindered performance on the problems. Students in the perceptually rich condition solved fewer problems correctly (M ¼ 4.08, SD ¼ 2.29) than students in the other two conditions (M ¼ 5.50, SD ¼ 2.34), F(1, 76) ¼ 6.34, p ¼ 0.01, partial h 2 ¼ The number of problems solved correctly by students in the bland condition (M ¼ 5.56, SD ¼ 2.20) did not differ significantly from the number of problems solved correctly by students in the control condition (M ¼ 5.45, SD ¼ 2.49), F(1, 76) ¼ 0.03, p ¼ Thus, it was not the presence of objects per se that hindered performance on the problems. We also examined performance on a problem-by-problem basis to make sure that the observed differences between conditions were not driven by any one problem, in particular. As shown in Table 1, the patterns observed on a problemby-problem basis were consistent with the results of the overall analysis. The percentage of students who solved any given problem correctly was the lowest (or approximately tied for lowest) in the perceptually rich condition across all problems. To gain a deeper understanding of whether the conditions affected the way students approached the problems, we performed a detailed analysis of students written work on Problem #1 in which Maggie is saving her \$15.00 weekly allowance to buy a new CD player that costs \$95.99 and needs to calculate how many weeks it will take to buy it. We chose to examine students work on this problem because (a) it showed one of the largest discrepancies in terms of correctness between students in the perceptually rich condition and students in the other two conditions (see Table 1), (b) students showed a good deal of work on the problem, and (c) there were systematic individual differences in the way students approached the problem. One of the systematic differences from student-to-student on this problem was whether, or not, the student used a repeated addition strategy vs. a multiplication or division strategy. That is, some students added \$15.00 in column format repeatedly until they reached the desired amount, whereas other students tried to incorporate multiplication or division into their approach. Students who used the repeated addition strategy (N ¼ 40) were more likely to solve the problem correctly than were students who tried to incorporate multiplication or division into their approach (N ¼ 35), 65% vs. 40%, c 2 (1, N ¼ 75) ¼ 4.69, p ¼ Although the (more successful) repeated addition strategy was used by about 53% of children overall, it was used by only 18% of students in the perceptually rich condition. Students in the perceptually rich condition were more likely to try to incorporate multiplication or division into their approach than were students in the other conditions, c 2 (2, N ¼ 75) ¼ 15.77, p < This finding was not predicted a priori, so it should be interpreted with caution. However, it is consistent with the prediction that the presence of the perceptually rich bills and coins influences how students approach the problems Conceptual errors Results thus far are consistent with Experiment 1. The presence of perceptually rich bills and coins hindered performance on word problems involving money. However, the perceptually rich bills and coins may have facilitated other aspects of performance. As evident in the analysis of student work on Problem #1 in which Maggie is saving for a CD player, the presence or absence of bills and coins may affect how students approach problems. Because they are designed to cue real-world knowledge, the perceptually rich bills and coins may have helped students conceptualize the problems. To address this hypothesis, we need to look beyond correctness. Another way to look at the data is to examine the types of errors made by students in each condition. Of particular interest is the proportion of errors that were conceptual errors because these errors reflect conceptual misunderstandings about the problems. We conducted an ANOVA with condition (perceptually rich, bland, control) as independent variable and proportion of errors that were conceptual errors as the dependent measure. Three students were excluded from this analysis Table 1 Percentage of students who solved each problem (1e10) correctly in each condition in Experiment 2 Condition Problem Perceptually rich Bland Control

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