Children’s strategies in complex arithmetic

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Abstract

Strategies used to solve two-digit addition problems (e.g., 27 + 48, Experiment 1) and two-digit subtraction problems (e.g., 73 – 59, Experiment 2) were investigated in adults and in children from Grades 3, 5, and 7. Participants were tested in choice and no-choice conditions. Results showed that (a) participants used the full decomposition strategy more often than the partial decomposition strategy to solve addition problems but used both strategies equally often to solve subtraction problems; (b) strategy use and execution were influenced by participants’ age, problem features, relative strategy performance, and whether the problems were displayed horizontally or vertically; and (c) age-related changes in complex arithmetic concern relative strategy use and execution as well as the relative influences of problem characteristics, strategy characteristics, and problem presentation on strategy choices and strategy performance. Implications of these findings for understanding age-related changes in strategic aspects of complex arithmetic performance are discussed.

Introduction

Research in arithmetic aims at understanding how people accomplish arithmetic problem solving tasks and how arithmetic skills change with participants’ age. Examining determiners of participants’ performance has helped to build models of arithmetic processing. The current study aimed at further understanding how children and adults solve complex, two-digit arithmetic problems. To achieve this end, we investigated strategic aspects of their performance. Before outlining the logic of the current work, we briefly review previous findings on complex arithmetic problem solving.

Most researchers agree that when participants solve complex arithmetic problems, they need to encode digits, find arithmetic facts in long-term memory (LTM) or count, temporarily hold partial results, and add multiple digits. Above and beyond these elementary processes, previous research on complex addition and subtraction problems showed several important findings that guided the current work. First, both adults and children use several strategies. A strategy can be defined as “a procedure or a set of procedures for achieving a higher level goal or task” (Lemaire & Reder, 1999, p. 365). To solve multidigit addition and subtraction problems, both children and adults use several strategies such as direct retrieval of the correct solution in LTM, several different transformation strategies (e.g., to solve 45 + 39, they can do 45 + 40 – 1, 40 + 40 + 5 – 1, or 50 + 34), or several decomposition strategies (e.g., to solve 45 + 39, they can do 40 + 30 + 5 + 9, 45 + 30 + 9, 40 + 39 + 5, or 5 + 9 + 30 + 40). Such strategy variability has been observed for both addition and subtraction problems in children and adults of all ages (e.g., Arnaud et al., 2008, Beishuizen, 1993, Beishuizen et al., 1997, Lemaire and Lecacheur, 2001, Lucangeli et al., 2003). Two of the complex arithmetic strategies have been the focus of investigation in several past studies. These are the full and partial decomposition strategies that both children (as young as 7 years) and adults use. In the full decomposition strategy, both addends are split into tens and units, which are added separately and then are combined again (e.g., 53 + 46: 50 + 40 = 90, 3 + 6 = 9, 90 + 9 = 99). In the partial decomposition strategy, only the second addend is split up and the tens and units are added onto the unsplit first addend (e.g., 34 + 63: 34 + 60 = 94, 94 + 3 = 97). The current study also focused on these two strategies.

Complex arithmetic strategies have been found to be used with different proportions in different age groups. For example, Lucangeli and colleagues (2003) found that third, fourth, and fifth graders used different strategies to solve multidigit addition, subtraction, multiplication, and division problems. Regarding the two strategies that are the focus of the current work, third graders used the full decomposition strategy less often than fourth and fifth graders (14, 31, and 28%, respectively, for addition problems and 2, 8, and 13%, respectively, for subtraction problems) and the three age groups used the partial decomposition strategy equally often (3% for addition problems and 9% for subtraction problems). The full decomposition strategy was favored by second graders, whereas third graders used both the full and partial decomposition strategies equally often (Beishuizen et al., 1997, Blöte et al., 2001, Fuson, 1990).

The second robust empirical finding relevant to the current project concerns the role of problem characteristics in participants’ performance and strategy use such as size of the operands and whether problems involve carryover or not. For example, Green, Lemaire, and Dufau (2007) found that adults used the unit strategy (i.e., adding units first, then decades, and then hundreds) more often with carry problems than with no-carry problems while solving three-digit addition problems. Imbo, Vandierendonck, and De Rammelaere (2007) found that participants’ performance decreased as the number and value of carries increased in multidigit addition problem solving.

The third empirical finding that is relevant to the current project is that arithmetic processes and performance are influenced by a number of different situational constraints. For example, participants change their strategies and have different levels of performance when they are tested under varying time–accuracy pressure conditions (e.g., Campbell & Austin, 2002), when they are or not asked verbal protocols regarding how they solved problems (Kirk and Ashcraft, 2001, LeFevre et al., 2006), and when problems are presented under different formats (e.g., Campbell & Fugelsang, 2001; LeFevre et al., 2001, Noël et al., 1997). For example, LeFevre and colleagues (2001) observed a smaller problem size effect with auditory stimuli than with Arabic stimuli while participants were solving simple multiplication problems. The role of these situational constraints has been much less investigated in children, although existing data suggest that such constraints also affect their arithmetic strategies and performance. For example, Lucangeli and colleagues (2003) found that 8- to 10-year-olds did not use the same strategies to solve mental versus written addition and subtraction problems.

Previous research on strategic aspects of children’s complex arithmetic has been limited in several respects. First, no studies have examined age-related changes in how often children use complex arithmetic strategies when strategies are assessed on a trial-by-trial basis. Therefore, we do not know whether the frequency with which children use different strategies changes with development for complex arithmetic. To determine how often each strategy is used by individuals as well as age-related differences in distributions of strategies, it is necessary to assess strategy use on each trial.

Second, because strategies are used unequally often by participants of different groups and on different types of problems, strategy use and execution are confounded. This makes it impossible to compare strategy use and execution across different groups of children and to know whether relative strategy performance stems from true differences in strategy speed and accuracy or from strategy selection (over items or participants) artifacts. To examine relative strategy efficiency and how it changes with age, it is important that the frequencies of strategy use and the problems on which each strategy is used are the same across age groups. We did this in the current study.

Third, no studies have investigated the determiners of strategy choices, so that we do not know whether, like in simple arithmetic, complex arithmetic strategies are influenced by problem characteristics (e.g., size of operands) and/or strategy characteristics (e.g., strategy speed). We also do not know how effects of problem and strategy characteristics on strategy use change with children’s age. Here we tested whether the use of complex arithmetic strategies is influenced by problem and/or strategy characteristics and how the effects of these factors change with children’s age.

Fourth, although previous studies have shown that problem features affect children’s strategy use and their arithmetic performance, one problem feature has received much less attention—carryover. No studies have investigated systematically the role of carries on children’s strategy use, how children execute carry processes, and how carry processing changes with age. Because carry processing puts a large burden on working memory resources (see DeStefano & LeFevre, 2004, and LeFevre, DeStefano, Coleman, & Shanahan, 2005, for detailed discussion and review of the role of working memory in arithmetic performance), and because children’s working memory resources increase with age (see Hitch, 2006, for a review), we tested the hypothesis that carry processing becomes more efficient as children age. More specifically, we predicted that the differences in performance to solve carry and no-carry problems decrease as children’s age increases.

Finally, very few studies have investigated how task constraints affect children’s arithmetic performance and processes. Therefore, we do not know how such situational constraints influence children’s arithmetic and how these effects of task constraints change with children’s age. For example, no studies have compared children’s performance in conditions where problems are presented vertically versus horizontally while each participant used both strategies equally often and on the same problems. We compared strategy use and strategy execution for horizontally and vertically presented problems. Such a comparison was important not only to understand the role of task constraints in children’s arithmetic but also because most studies of children’s arithmetic have tested horizontally presented problems. Thus, the current study offered the opportunity to generalize conclusions from these studies to vertically presented problems.

The major goal of this project was to contribute to our further understanding of age-related changes in children’s complex arithmetic. We adopted a perspective in which we studied performance by controlling the strategies used by children and adults to solve multidigit arithmetic problems.

One group of adults and two groups of children solved two-digit addition problems (Experiment 1) or subtraction problems (Experiment 2). Based on previous studies (e.g., Beishuizen, 1993, Beishuizen et al., 1997, Lucangeli et al., 2003), third and fifth graders solved addition problems (Experiment 1) and fifth and seventh graders solved subtraction problems (Experiment 2). All participants were tested in both the choice and no-choice conditions (Siegler & Lemaire, 1997). In the choice condition, participants were allowed to choose between the full and partial decomposition strategies on each problem. In the no-choice condition, participants solved all problems first with one of the two strategies and then with the other strategy.

The data collected in this study enabled us to address the following issues regarding age-related changes in strategic dimensions of complex arithmetic performance: Do participants have preferences among full and partial decomposition strategies? Are strategy use and execution influenced by participants’ age, problem features (carryover and size of operands), strategy characteristics (relative strategy performance), and/or task constraints (presentation format)? How do effects of problem type, strategies, and task constraints change with children’s age? How does processing carries change with children’s age?

We hypothesized that participants’ performance in the no-choice conditions would reflect their preferred strategy in the choice condition. In particular, we expected that the strategy most often chosen by an individual would also be the faster and more accurate one in the no-choice condition. Furthermore, we expected that participants would perform better while solving no-carry problems than while solving carry problems and would use full decomposition more on carry problems than on carry problems because it is harder to execute on the latter. It was hard to make specific predictions regarding effects of presentation format because no previous studies had compared problems presented both horizontally and vertically. Finally, age-related differences in strategy use and execution predict different strategy preferences in each age group, increased strategy performance in older participants, and age-related differential influences of problem features (i.e., decreased differences in performance between carry and no-carry problems) and task constraints (i.e., different effects of presentation format as a function of children’s age).

Section snippets

Participants

A total of 90 individuals—30 adults (20 women and 10 men), 30 fifth graders (16 girls and 14 boys), and 30 third graders (16 girls and 14 boys)—participated in this experiment. The adults were undergraduate college students who received course credit for participating in the experiment. Their mean age was 21 years 4 months (range = 18 years 1 month to 22 years 5 months). The fifth graders had a mean age of 10 years 6 months (range = 9 years 9 months to 11 years 1 month), and the third graders had a

Participants

A total of 56 individuals—20 adults (15 women and 5 men), 18 seventh graders (10 girls and 8 boys), and 18 fifth graders (10 girls and 8 boys)—participated in the study. The adults were undergraduate college students who received course credit for participating in the experiment. Their mean age was 23 years 5 months (range = 18 years 0 months to 30 years 1 month). The seventh graders had a mean age of 12 years 3 months (range = 12 years 0 months to 13 years 1 month), and the fifth graders had a

General discussion

The current experiments documented children’s age-related differences in complex arithmetic strategy use and execution. They replicated previous findings regarding children’s complex arithmetic skills and documented age-related changes in strategic aspects unknown before. Two sets of interesting findings, one each concerning strategy use and execution, have implications for further understanding age-related changes in strategic behaviors, in general, and in complex arithmetic, in particular.

Acknowledgment

This research was supported in part by the CNRS (French NSF) and a grant from the Agence Nationale de la Recherche (BLAN07-1_196867).

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