The present study investigated how elementary-school children solve two-digit addition problems (e.g., 34+68). To achieve this end, we examined age-related differences in children’s strategy use and strategy performance. Results showed that (a) both third and fifth graders used a set of 9 strategies, (b) fifth-grade individuals used more strategies than third-grade individuals, (c) age-related differences in the size of strategy repertoire was partially explained by age-related differences in basic arithmetic fluency, (d) how often children used each available strategy changed with problem difficulty and children’s age, as younger children tended to focus more on one or two strategies and older children used a wider range of strategies, (e) increased arithmetic performance with age varied with problem difficulty both when overall performance was analyzed and when analyses of performance was restricted to children’s favorite strategy. The present findings have important implications for our understanding of how complex arithmetic performance changes with children’s age and change mechanisms underlying improved performance with age in complex arithmetic.
The ability to solve arithmetic problems such as 8+7 or 45+38 correctly is one of the basic skills acquired during elementary school and used widely in daily life. A number of studies have documented the developmental course of how this skill is acquired during childhood (see
Previous studies found that to understand how children solve arithmetic problems, as well as age-related changes in arithmetic problem solving, it is important to determine which strategies they use. A strategy is defined as a “procedure or a set of procedures for achieving a higher level goal or task” (
The lack of knowledge about children’s strategy repertoire for solving two-digit addition problems contrasts with well-documented strategy repertoire in adults. Specifically, previous studies have found that both young and older adults have 9 available strategies to solve two-digit addition problems.
Unfortunately, no prior studies investigated in detail strategies that children might use to solve two-digit addition problems. The studies that have assessed strategies on two-digit addition problems examined small sets of problems and did not manipulate problems features like problem difficulty (
Moreover, we investigate how strategic aspects underlying children’s performance while solving two-digit addition problems change with children’s age.
The second important goal of the present study was to investigate how problem features influence strategies in complex arithmetic. Problem features have been shown in arithmetic to crucially influence both strategy use and strategy performance in children and adults (see
In the present study, we adopted the same approach as in
Children were given two tasks, an experimental task (i.e., two-digit addition problem solving task) and an independent, pencil-and-paper task (i.e., assessing arithmetic fluency on single-digit problems). In the experimental task, children were asked to solve 48 easy and 48 hard two-digit addition problems, and strategies were assessed on each problem. Effects of age and problem difficulty on performance (i.e., solution latencies and percentages of errors), strategy repertoire, and strategy selection were investigated. Given previous findings on mathematical development during childhood, children of both age groups were expected to use several strategies and to select strategies on a problem-by-problem basis. Moreover, poorer performance was expected for younger children, especially on harder problems, because these complex problems require cognitive resources, and younger children have fewer resources than older children. Also, assessing strategies on each problem enabled us to determine whether younger and older children use the same (or different) number of strategies and use available strategies with comparable frequencies on different types of problems. Finally, in addition to our experimental task of two-digit addition problem solving, we assessed basic arithmetic fluency for each child with an independent pencil-and-paper test. This enabled us to determine to what extent age-related changes in basic arithmetic skills explain age-related changes in children’s strategy use during complex arithmetic problem solving tasks.
Sixty-six children were tested. They came from French upper class urban public schools in Aix-en-Provence and Marseille (France). They were divided into two age groups: 33 third graders (18 girls) and 33 fifth graders (14 girls). In order to assess simple arithmetic fluency, children completed a paper-and-pencil arithmetic task at the end of the experiment (i.e., after the main two-digit arithmetic problem solving task). The task was to solve as quickly and accurately as possible the 81 additions of two one-digit numbers presented on a sheet of paper. Times and number of correct answers were recorded. Participants’ characteristics are summarized in
Characteristic | Third graders | Fifth graders | |
---|---|---|---|
33 (18) | 33 (14) | -- | |
Age (in months) | 106 | 128 | -- |
Range | 100-111 | 116-145 | -- |
Arithmetic fluency times (in s) | 410 (98) | 306 (106) | 17.29** |
Arithmetic fluency scores | 79.6 (1.4) | 79.7 (1.1) | 0.08 |
**
Each participant solved 96 complex addition problems. These problems were composed of two two-digit numbers with a mean sum of 96.5 (
Following previous findings in arithmetic (see see
Children were tested individually in one session, which lasted approximately 90 minutes. First, they performed the experimental task, and then the paper-and-pencil simple arithmetic fluency task.
Children solved eight training problems similar to (but different from) experimental problems to familiarize themselves with apparatus, procedure, and task. Problems were presented in 64-point Courier New bold font in the center of a 15-inch computer screen. Each trial was preceded by a fixation point (“*”) in the center of the screen for 1000 ms. The problem was then displayed horizontally in the center of the screen in the form of
At the end of the experiment, individuals’ arithmetic fluency was assessed with an independent paper-and-pencil arithmetic task (i.e., participants were instructed to solve as quickly and as accurately as possible the 81 addition problems of two one-digit numbers).
Results are reported in three main parts, each examining effects of age and problem difficulty on strategy repertoire, on strategy selection, and on performance (i.e., response times and percentages of errors). In all results, unless otherwise noted, differences are significant to at least
Strategies used by children were analyzed from verbal protocols. The experimenter who tested the children coded which strategy was used on each problem from written protocols after the experiment was ran. Another coder coded independently classified which strategy was used on 100 randomly selected problems from different children. The two raters agreed on 98% of problems. Analyses of verbal protocols revealed that, at the group level, both third and fifth graders used the same nine strategies that Lemaire and Arnaud found in adults (see
Strategy | Example (12 + 46) |
---|---|
Rounding the first operand down | (10 + 46) + 2 |
Rounding the second operand down | (12 + 40) + 6 |
Rounding both operands down | (10 + 40) + (2 + 6) |
Columnar retrieval | (6 + 2) + (40 + 10) |
Rounding the first operand up | (20 + 46) - 8 |
Rounding the second operand up | (12 + 50) - 4 |
Rounding both operands up | (20 + 50) - 12 |
Borrowing units | 18 + 40 |
Direct retrieval | 58 |
Children’s most often used strategy was columnar retrieval, which was used on 64.5% of all problems, followed by rounding both operands down (19.4%), direct retrieval (8.8%), and borrowing units (3.5%). The other strategies were used on less than 2% of trials. Although this order of strategy preferences was the same in third and fifth graders, note that third graders tended to focus on one strategy (i.e., columnar retrieval) that they used on 83% of problems on average; they used rounding both operands down on 8% of problems, direct retrieval on 5% of problems and the other strategies on 4% of problems. Fifth graders showed a more even strategy distribution, as they used columnar retrieval on 46% of problems, rounding both operands down on 31% of problems, direct retrieval on 13% of problems, borrowing units on 5% of problems, and the other strategies on 6% of problems.
To determine if the mean number of strategies used by individual children varied with participants’ age and problem difficulty, analyses of variance (ANOVAs) were performed on the mean number of strategies used by individuals with a mixed design, 2 (Age: third vs. fifth graders) x 2 (Difficulty: easy vs. hard problems), with repeated measures on the last factor. Fifth-grade individuals used significantly more strategies than third grade individuals (3.1 and 1.9 strategies respectively),
We next assessed how age-related changes in arithmetic fluency mediated effects of age on mean number of strategies. To achieve this end, we compared the proportion of variance (reflected in increments of
Group | Number of strategies |
|||||||
---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
Third graders | 11 | 11 | 8 | 1 | 1 | 0 | 0 | 1 |
Fifth graders | 2 | 11 | 5 | 2 | 9 | 2 | 2 | 0 |
Third graders | 13 | 11 | 7 | 0 | 1 | 0 | 0 | 1 |
Fifth graders | 3 | 11 | 8 | 3 | 5 | 1 | 2 | 0 |
Third graders | 17 | 9 | 6 | 0 | 0 | 1 | 0 | 0 |
Fifth graders | 6 | 10 | 5 | 7 | 4 | 0 | 1 | 0 |
Strategy | Third Graders |
Fifth Graders |
||
---|---|---|---|---|
Easy Problems | Hard Problems | Easy Problems | Hard Problems | |
Rounding-down the first operand | 0.3 | 0.1 | 1.2 | 1.3 |
Rounding-down the second operand | 0.1 | 0.1 | 2.6 | 1.0 |
Rounding-down both operands | 7.6 | 9.1 | 28.4 | 32.6 |
Columnar retrieval | 81.7 | 84.4 | 42.4 | 49.6 |
Rounding-up the first operand | 0.9 | 0.6 | 2.0 | 1.3 |
Rounding-up the second operand | 0.3 | 0.4 | 1.3 | 0.9 |
Borrowing units | 2.1 | 1.8 | 4.9 | 5.2 |
Direct retrieval | 6.9 | 3.4 | 17.3 | 7.8 |
Rounding-up both operands | 0.1 | 0.1 | 0.1 | 0.2 |
The main effect of strategy was significant,
Grades | Easy Problems | Hard Problems | ||
---|---|---|---|---|
Third Graders | 13625 | 16743 | 15184 | 3118 |
Fifth Graders | 9799 | 11552 | 10675 | 1753 |
11712 | 14147 | 12929 | 2435 | |
3826 | 5191 | 4509 | ||
Third Graders | 14.0 | 21.5 | 17.7 | 7.5 |
Fifth Graders | 11.6 | 18.6 | 15.1 | 6.9 |
12.8 | 20.0 | 16.4 | 7.2 | |
2.4 | 2.9 | 2.7 |
Fifth graders were faster than third graders (10675 ms vs. 15184 ms;
We next compared children’s performance for easy and hard problems for columnar retrieval that both age groups used most often. Thus, mean solution latencies and percent errors (see
Mean Solution Times (and Error Rates) on Easy and Hard Problems When Third and Fifth Graders Used Columnar Retrieval.
The present study aimed at studying how elementary-school children solve two-digit addition problems, and how strategic aspects of two-digit addition problem solving performance change with children’s age/grade. Previous works in arithmetic showed that children as young as 7 year-old solve two-digit arithmetic problems with several strategies and that their performance depend on which strategies are used (e.g.,
The present experiment showed effects of age and problem difficulty on children’s performance, strategy repertoire, and strategy selection while solving two-digit addition problems. With respect to solution times and error rates, fifth graders had an advantage over third graders for both easy and hard problems, but especially for the hard problems. Such age-related improvement concurs with comparable improvements found in previous studies in both simple and complex arithmetic (see
It is difficult to know whether the larger number of strategies used by fifth graders to solve two-digit addition problems here stems from a strategy repertoire becoming larger as children grow older, from increased processing resources with age, or from schooling effects. The increasing number of strategies with children’s age may not come from increasing size of strategy repertoire between third and fifth grades because the same repertoire of nine strategies was observed in both age groups. Rather, it is possible that, given increased processing resources (i.e., working memory, processing speed, and executive control resources) in older children, fifth graders were more able than third graders to keep active a larger number of strategies while selecting a strategy on each problem. Such active maintenance enabled them to use a larger number of strategies across the 96 problems. Note that it is not necessary for children to keep active all the 9 available strategies on each problem before selecting a strategy. They can activate a sub-set of these strategies, and different sub-sets can be activated for different problems. It is enough that older children activate larger sub-sets for them to have more chances to use a larger strategy repertoire across the 96 problems. This is possible for older children with more processing resources available. As we did not collect any measures of processing resources, we could not test the hypothesis that increased processing resources led older children to use more strategies than younger children, a hypothesis that future research may test more directly.
Differences in the number of strategies between younger and older children may also be the result of schooling effects. Such schooling effects on age-related changes in arithmetic in general and in two-digit arithmetic (subtraction) problems has been found in numerous previous studies (e.g.,
It was interesting to find here that basic arithmetic fluency mediated age-related changes in the number of strategies. It is thus possible that older children mastered basic arithmetic facts and this higher level of basic arithmetic fluency freed processing resources which were used to activate more available strategies to choose among on each problem. Note that age-related changes in arithmetic fluency did not explain all the age-related variance in mean number of strategies, as age had a remaining significant unique effect after statistically controlling for arithmetic fluency. This suggests that other factors contributed to increased number of strategies with age. It is possible that in addition to basic arithmetic fluency, increased processing resources helped older children to use more strategies (e.g.,
Interestingly, and probably as a result of younger children’s using fewer strategies, strategy distributions were not the same in each grade. Younger children favored columnar retrieval that they used on 83% of the problems. The other two strategies that they used on 8.3% and 5.1% of the problems were rounding both operands down and direct retrieval. Interestingly, fifth graders also favored columnar retrieval, but used rounding-down on 31% of problems and direct retrieval on 13% of problems. With more experience, older children were able to retrieve more correct sums from long-term memory. They were also more able to use transformation strategies like rounding both operands down which was probably easier for them to execute on many problems than columnar retrieval.
One limitation of the present study that future study may address concerns strategy execution. We assessed children’s performance (latency and accuracy) and found age-related differences therein, even when we focused on the most favorite strategy (i.e., columnar retrieval) of the two groups of children. However, we could not assess strategy performance in unbiased way. Given strategy selection effects, it was impossible to compare across grades relative strategy efficacy without this efficacy being uncontaminated by strategy use. For example, even if we restricted comparison third and fifth graders’ performance to their favorite strategy, columnar retrieval, it was used on an unequal number of trials in each age group. Younger children used it much more often than older children. Also, younger children used it equally often on easy and hard problems, whereas older children used it more often on hard than on easy problems. This made it impossible to determine, for example, whether age-related differences are larger for some strategies than for others or if, when they use some strategies, younger and older children obtain comparable arithmetic performance. This would have also enabled us to address other important issues, like how relative strategy performance predicts strategy choices. It is possible that younger children used columnar retrieval on most problems because, on these problems, this is the fastest/most accurate strategy for them. It is possible that older children used rounding-both operands down on problems for which it would take much more time to execute columnar retrieval. Such issues can be addressed only when strategy performance is uncontaminated by strategy selection. Such strategy selection biases result in each strategy being used on a different number of problems, on different types of problems, and in different proportions by each age group. Unbiased measures of strategy execution can be obtained via the choice/no-choice method proposed by
At a more general level, the present findings speak to two general issues, each concerning the development of complex arithmetic and concerning strategic aspects of cognitive development. No previous data described the exact and specific strategies that children use to solve complex addition problems. Solving two-digit arithmetic problems entails retrieving basic sums, coordinating these partial results, managing carries and rounding processes, and choosing which operands to round to which closest decades. As revealed by this study, it also involves some very specific strategies (e.g., rounding one operands up is not the same as rounding both operands down) that are differently affected by children’s age and problem characteristics. If they are to be computationally implemented, theoretical models of complex arithmetic need this type of detailed description of strategies, and how their use changes with children’s age.
Age-related similarities and differences found here also speak to broader issues in cognitive development. Siegler’s work (e.g.,
All in all, the present research is based on the lack of detailed and specific knowledge on how third and fifth graders solve two-digit arithmetic problems, in contrast to previous findings on other complex arithmetic problem solving (like subtraction). We found that children used a set of 9 strategies, that older children use more strategies than younger children, and that this age difference is partly the result of increase arithmetic fluency, that both young and older children’s strategy use and strategy performance were influenced by problem difficulty. These are important findings for both theoretical reasons (e.g., models of complex arithmetic need to be based on which strategies children actually use when they solve two-digit addition problems) and for practical reasons (e.g., efficient educational practices for teaching complex arithmetic during elementary school need to know how children actually solve complex arithmetic problems).
The underlying data for this article can be found at
This research was supported in part by the CNRS (French NSF), a grant from the Agence Nationale de la Recherche (Grant # BLAN-1912-01), and a grant from La Fondation de France.
We would like to thank two anonymous reviewers for helpful comments on a previous version of this manuscript.
The author has declared that no competing interests exist.