For many years, cognitive psychologists have studied the calculation strategies that are used to solve arithmetic problems (Ashcraft & Guillaume, Reference Ashcraft, Guillaume and Ross2009), focusing their attention on the factors that determine strategy selection and efficiency. Strategy selection refers to the selection of one of the available strategies to solve a given problem, while strategy efficiency refers to how fast and accurately this strategy leads to the solution (see Imbo & Vandierendonck, Reference Imbo and Vandierendonck2008a, Reference Imbo and Vandierendonckb). The efficiency of the different strategies available seems to be the basis of strategy selection (Campbell & Alberts, Reference Campbell and Alberts2009), which may vary according to the level of arithmetic skill. This study aimed to investigate whether arithmetic skill favors the selection of more efficient calculation strategies or whether it enhances the efficiency of these strategies.

It is generally agreed that individuals rely on two different strategies when performing calculations. The first —referred to as retrieval— consists of retrieving the result directly from long-term memory. For example, in the addition 2 + 3, the result 5 just pops into your head. The second —referred to as procedures— consists of using procedural solutions such as counting or transformation. For example, the problem 7 + 8 can be solved by counting eight units on from seven, or by transforming it into 8 + 2 + 5.

In order to study the kind of strategies used, a combination of verbal reports and behavioral measures (Siegler, Adolph, & Lemaire, Reference Siegler, Adolph, Lemaire and Reder1996) is usually applied. Specifically, problems are grouped on the basis of verbal reports and then analyzed using standard behavioral measures. These studies have found that problems in which participants report having used a retrieval strategy are solved faster and with fewer errors than those in which the use of procedures is reported. This has been attributed to the fact that procedures use requires more steps to reach the solution than retrieval strategy. Although the validity of verbal reports as an indicator of the underlying cognitive processing has been questioned (Cooney & Ladd, Reference Cooney and Ladd1992; Kirk & Ashcraft, Reference Kirk and Ashcraft2001; Russo, Johnson, & Stephens, Reference Russo, Johnson and Stephens1989), recent studies both with behavioral (LeFevre, DeStefano, Penner-Wilger, & Daley, Reference LeFevre, DeStefano, Penner-Wilger and Daley2006; Seyler, Kirk, & Ashcraft, Reference Seyler, Kirk and Ashcraft2003; Smith-Chant & LeFevre, Reference Smith-Chant and LeFevre2003) and psychophysiological measures (Grabner & De Smedt, Reference Grabner and De Smedt2011; Grabner et al., Reference Grabner, Ansari, Koschutnig, Reishofer, Ebner and Neuper2009) have shown that, when properly used, they can be a valid source of information. Accordingly, behavioral differences when comparing trials in which the same strategy has been reported may be a signature of strategy efficiency.

More recently, strategy selection has been investigated by means of the operand-recognition paradigm. This paradigm is based on the assumption that the algorithmic computation performed during the use of procedures impairs the traces in memory of the operands involved in the problem (Thevenot, Barrouillet, & Fayol, Reference Thevenot, Barrouillet and Fayol2001; Thevenot, Fanget, & Fayol, Reference Thevenot, Fanget and Fayol2007). The operands’ recognition time after a calculation task and after a number comparison task is compared, since no computation is required in the latter. If procedures have been used, it will be harder to recognize a previously presented number in a calculation task than in a comparison task. But if the calculation task is solved by retrieving the solution from the long-term memory, then operand recognition will be equally easy after comparison and after calculation. Nevertheless, the validity of these inferences has also been challenged, as we discuss below.

Based on these measures, a variety of studies have suggested that strategy selection and strategy efficiency depend on the problem size, the arithmetic operation, and the level of arithmetic ability. The problem-size effect has been widely reported and refers to the fact that response time and errors increase as the size of the operands increases (Groen & Parkman, Reference Groen and Parkman1972; Zbrodoff & Logan, Reference Zbrodoff, Logan and Campbell2005). Campbell and Xue (Reference Campbell and Xue2001) proposed three sources of the problem-size effect related to strategy selection and strategy efficiency. First, small problems (e.g., 2 + 4) may be solved by retrieval and large problems (e.g., 8 + 6) may be solved by procedures: hence the increase in response time and error rate found in the latter. Verbal reports have confirmed this explanation, with more frequent retrieval use reported for small than for large problems (Campbell & Xue, Reference Campbell and Xue2001). Second, it has been suggested that retrieval may be more efficient — faster and less error-prone — for small than for large problems, so the problem-size effect may be the consequence of differences in the accessibility of the results from long-term memory (Ashcraft & Christy, Reference Ashcraft and Christy1995). This means that retrieval may be used in all problems, but since large problems are less frequent, they are less activated and thus are more difficult to retrieve. And third, procedural strategies could be less efficient — slower and more error-prone — in large than in small problems (LeFevre, Sadesky, & Bisanz, Reference LeFevre, Sadesky and Bisanz1996). This means that all problems may be solved by procedures, although these strategies become less efficient when the size of the problem increases. This may be due to the fact that procedures tend to be more complex (or require more steps to reach the solution) for large problems. In short, three alternative explanations have been proposed for the problem size effect, and latency or error rates do not distinguish between them. Hence, other complementary measures must be used. In the experiment reported, we combined behavioral measures with verbal reports.

In terms of the arithmetic operation, it has traditionally been claimed that adults predominantly use retrieval to solve addition and multiplication and procedures to solve subtraction and division (Barrouillet, Mignon, & Thevenot, Reference Barrouillet, Mignon and Thevenot2008; Campbell & Xue, Reference Campbell and Xue2001; LeFevre et al., Reference LeFevre, DeStefano, Penner-Wilger and Daley2006; Seyler et al., Reference Seyler, Kirk and Ashcraft2003). Moreover, adults use more retrieval in multiplication than in addition; this is probably because multiplication is learnt by rote at school, which strengthens the association between a problem and its answer in the long-term memory. Whereas addition and multiplication have been investigated in depth, subtraction and division have received less attention among cognitive psychologists. Focusing research on multiplication and addition prevents a comprehensive understanding of the use of strategies, since these well-practiced operations are more frequently solved by retrieval. Therefore, subtractions or divisions are better suited to investigate differences in the efficiency of procedures. To help to solve this gap our study investigated the strategies used and their efficiency specifically during subtraction solving.

With regard to arithmetic ability, it has been suggested that highly skilled individuals (HS) rely more frequently on retrieval than less highly skilled individuals (LS; Imbo, Vandierendonck, & Rosseel, Reference Imbo, Vandierendonck and Rosseel2007; LeFevre et al., Reference LeFevre, Sadesky and Bisanz1996; LeFevre & Kulak, Reference LeFevre and Kulak1994). Indirect support for this proposal can be found in the reduction of the problem-size effect in HS individuals when solving additions and multiplications (Imbo et al., Reference Imbo, Vandierendonck and Rosseel2007; Núñez-Peña, Gracia-Bafalluy, & Tubau, Reference Núñez-Peña, Gracia-Bafalluy and Tubau2011; Smith-Chant & LeFevre, Reference Smith-Chant and LeFevre2003). Specifically, LS individuals solved small-size problems faster than medium-size ones. The authors interpreted this result as evidence that LS individuals used retrieval only in small-size problems. HS individuals, on the other hand, did not differ between small and medium-sized problems; this finding led the authors to suggest that they used retrieval in both cases (Núñez-Peña et al., Reference Núñez-Peña, Gracia-Bafalluy and Tubau2011). However, the correspondence between behavioral measures and strategy selection is not as clear as initially considered, and some recent studies have concluded that higher skill in arithmetic problem-solving might also lead to the use of more efficient procedures. In a study with expert arithmetical solvers, Fayol and Thevenot (Reference Fayol and Thevenot2012) found that presenting the arithmetical sign before the operands (e.g., + followed by 8 + 6 = ?) facilitated the solving process compared to a situation in which the whole of the operation was presented at once. This effect was found for additions and subtractions, but not for multiplications; the finding was interpreted as evidence that a procedure was pre-activated by the “+” and “-” signs but not by the “x” sign. The authors concluded that solving additions and subtractions, even small ones, through procedures was much more common in this highly skilled population than previously believed (see also Barrouillet & Thevenot, Reference Barrouillet and Thevenot2013 and Campbell, Chen, & Maslany, Reference Campbell, Chen and Maslany2013). High and low skilled participants were not directly compared, although the authors reported that they had not found this effect when the whole population of Psychology students was considered.

In the context of subtractions, to our knowledge only Thevenot, Castel, Fanget, and Fayol (Reference Thevenot, Castel, Fanget and Fayol2010) have explored the relationship between arithmetic skill, problem size and strategy selection. In their first experiment, HS and LS arithmetic problem solvers were asked to perform an operand-recognition task after a calculation task and after a comparison task, and were presented with small (e.g., 9–4), medium (e.g., 11–7) and large subtractions (e.g., 41–27). Comparing the performance on the operand-recognition task after the calculation and comparison tasks, the results for small subtractions showed equivalent recognition after both tasks in the two groups, suggesting that both used retrieval. Moreover, performance on the recognition task was worse after calculation than after comparison in large subtractions in both groups, suggesting the use of procedures. Finally, in medium-sized subtractions, recognition was better after comparison than after subtraction in the less-skilled group, but no differences were found in the highly skilled group; the authors interpreted this as evidence of differences in strategy selection by the groups in these subtractions: retrieval for the HS group, and procedures for the LS group. However, Thevenot et al. were unable to find evidence of this differential selection of strategies in the participants’ verbal reports (second and third experiments). Individuals, whatever their ability, reported mainly using retrieval in small subtractions, procedures in large subtractions, and a similar percentage of retrieval and procedures in medium-sized subtractions.

The modulation of the problem-size effect may also be due to differences in the efficiency of the strategies. In our view, the discordance between data from verbal reports and those from the operand-recognition paradigm reported by Thevenot et al. (Reference Thevenot, Castel, Fanget and Fayol2010) may be attributed to the fact that HS individuals use procedures more efficiently than their less-skilled peers. As a matter of fact, Thevenot et al. compared the subtraction solution times of both skill groups according to the strategies used and found that LS individuals were slower than their HS peers in both retrieval and procedural trials, although no measures of significance were reported. Furthermore, despite the fact that procedures were used in medium-sized problems by a similar percentage in both groups, a detailed look at the kind of procedures used (see their Table 1) shows that the LS group made more use of costly procedures such as counting or decomposition (Baroody, Reference Baroody1983). This choice of procedures may have obliged LS subjects to recall more numbers during the operand-recognition task, and may account for their poorer results. In contrast, HS individuals used more efficient procedural strategies such as transformation or addition reference (Imbo & Vandierendonck, Reference Imbo and Vandierendonck2008). Notice that these data also corroborate the proposal raised by Fayol and Thevenot (Reference Fayol and Thevenot2012) that becoming expert in arithmetic solving implies an ability to use more sophisticated procedures. However, these are all indirect indicators and more investigation is required.

Table 1.

Means and standard errors (in brackets) for response time (in ms) and hit rate (in %) for small and large subtractions in the LS and HS groups

The aim of the present study was to examine whether the differences in the problem-size effect in highly-skilled and less highly-skilled arithmetic problem solvers are due to differences in strategy selection or strategy efficiency. Most previous studies using additions or multiplications attributed the differences between high and low skilled groups to an increase in retrieval use with ability (although see Campbell et al., Reference Campbell, Chen and Maslany2013 for a possible increase in the use of fast procedures in high skilled participants). We wondered whether this would also be the case in subtraction, an operation considered to be more frequently solved through procedures than multiplication or addition. To this end, highly skilled and less highly skilled arithmetic problem solvers were presented with small and large problems in a subtraction verification task and were asked on a trial-by-trial basis which strategy they had used to solve each problem. Based on previous studies on additions and multiplications (Núñez-Peña et al., Reference Núñez-Peña, Gracia-Bafalluy and Tubau2011; Smith-Chant & LeFevre, Reference Smith-Chant and LeFevre2003), we expected that, as the problem size increased, the response time and errors would increase more for LS individuals than for HS.

By also analyzing performance according to participants’ self-reports in each problem size and for each skill group, we expected to shed more light on the sources of the performance differences between groups. As suggested above, these differences may be due to three sources: more use of retrieval by HS individuals in large-sized problems, or more efficient use of either retrieval or procedures by this participants’ group.

Results

Solution times and accuracy

Median ^{Footnote 2} response time (RT) for correctly solved trials and the percentage of correct responses were analyzed with analyses of variance (ANOVAs), taking Problem size (small, large) and Proposed solution (correct, incorrect) as within-subject factors and Group (HS, LS) as the between-subjects factor. The F value, the degrees of freedom, the probability level, and the η _p ² effect size index (Kirk, Reference Kirk1996) are given. Whenever an interaction reached significance, pairwise comparisons were conducted, and the Bonferroni adjustment was used to control for the increase in Type I error. Only significant effects (p ≤ .05) are reported.

In terms of response time, the main effects of Problem size (F(1, 39) = 21.26, p < .001, η_p ² = .35) and Proposed solution (F(1, 39) = 53.06, p < .001, η_p ² = .58) were statistically significant. Small subtractions were verified faster (660 ms) than large subtractions (765 ms) and incorrect problems were solved slower (762 ms) than correct ones (663 ms). The interaction Problem size x Group was also significant (F(1, 39) = 5.91, p = .02, η_p ² = .13). When problem sizes were compared in each group, results showed that both groups were slower in large subtractions than in small subtractions (see Tables 1 and 2). However, analysis of the difference in response time between large and small subtractions in both groups showed that the increase in response time was larger for the LS group (160 ms) than for the HS group (50 ms) (F(1, 39) = 5.91, p = .02, η_p ² = .13).

Table 2.

Means and standard errors (in brackets) for response time (in ms) and hit rate (in %) for small and large subtractions in the LS and HS groups for correct and incorrect solutions

In terms of hit rate, differences were found between groups (F(1, 39) = 22.7, p < .001, η_p ² = .37) showing that LS individuals made less hits (89%) than their more skilled counterparts (97%). The main effect of Problem size (F(1, 39) = 43.1, p <.001, η_p ² = .53) was also significant, showing that small subtractions were verified more accurately (97% hits) than large subtractions (89% hits). The interaction Problem size x Group was also significant (F(1, 39) = 20.1, p < .001, η_p ² = .34). In order to study this interaction in more detail, problem sizes were compared in each group. Both groups made more errors in large than in small subtractions (F(1, 19) = 32.28, p < .001, η_p ² = .63 for the HS group and F(1, 20) = 11.29, p = .003, η_p ² = .36 for the LS group; see Tables 1 and 2). However, analysis of the difference in hit rate between small and large subtractions in both groups showed that the drop in hit rate was larger for the LS group (13 %) than for the HS group (2%) (F(1, 39) = 20.1, p < .001, η_p ² = .34).

Self-report analysis

The percentage of retrieval use was calculated for each problem size and for each participant, and was analyzed using ANOVA, taking Problem size (small, large) and Proposed solution (correct, incorrect) as within-subject factors and Group (HS, LS) as the between-subjects factor. Data from four HS individuals were not taken into account in this analysis because of technical problems during the recording of their self-report answers. The analysis revealed a significant Problem size effect (F(1, 35) = 36.5, p < .001, η_p ² = .51), showing that the percentage of retrieval use was higher in small subtractions than in larger ones (70% vs 49% respectively). Neither the main effect of Proposed solution, nor the main effect of the Group, nor the interactions between them were significant. Crucially, groups did not differ in the percentage of retrieval use for each problem size (71% and 69% in small subtractions for the LS and HS groups respectively; 49% in large subtractions for both LS and HS groups). Table 3 shows the means and standard errors for percentage of strategy use as a function of problem size and arithmetic ability.

Table 3.

Means and standard errors (in brackets) of percentage of use, response time (in ms) and hit rate (in %) as a function of self-reported strategy, problem size and arithmetic ability

Solution times and accuracy as a function of self-reported strategy

We analyzed median response times and hit rates as a function of self-reported strategy, taking Strategy (retrieval, procedures), Problem size (small, large) and Proposed solution (correct, incorrect) as within-subject factors and Group (HS, LS) as the between-subjects factor. Hit rates were calculated by dividing the number of trials correctly answered for a given combination of size and strategy between the number of trials in which the participant reported having used that strategy ^{Footnote 3} . For example, if a participant reported using retrieval in 25 small subtraction problems and had 23 hits in small subtraction problems in which he reported retrieval use, then the hit rate was 0.92. In the following we discuss only the main effect of strategy and its interaction with other factors, when they reached statistical significance, because the other effects have been reported in the previous analysis.

In terms of response time, the main effect of strategy was significant (F(1, 35) = 20.10, p < .001, η_p ² = .37); individuals were slower when they reported having used procedures (829 ms) than when they reported having used retrieval (636 ms), and the interaction Strategy x Group was also significant (F(1, 35) = 5.26, p = .028, η_p ² = .13). Groups did not differ when they reported having used retrieval, but LS individuals were slower (939 ms) than HS individuals (719 ms) when they reported having used procedures (F(1, 35) = 7.79, p = .008, η_p ² = .18). The interaction Strategy x Problem size was also significant (F(1, 35) = 9.05; p = .005, η_p ² = .21). The Strategy effect was significant in small (F(1, 35) = 9.08; p = .005, η_p ² = .21) and large subtractions (F(1, 35) = 17.16; p < .001, η_p ² = .33), showing that retrieval was faster than procedures in both sizes (658.5 ms vs 722 ms in small subtractions; 731 ms vs 935.5 ms in large subtractions). Mean response times and standard errors as a function of self-reported strategy, problem size and arithmetic ability are given in Table 3.

In terms of hit rate, the effect of strategy (F(1, 35) = 14.97, p < .001, η_p ² = .30) was significant, showing that procedures were more error-prone (88% of hits) than retrieval (94% of hits). This effect was modulated by the interaction Strategy x Group (F(1, 35) = 6.87, p = .013, η_p ² = .16), which showed that the strategy effect was only significant for LS individuals (F(1, 19) = 13.8, p = .001, η_p ² = .42; 91% and 81% for retrieval and procedures respectively) but not for their HS peers (96% and 93% for retrieval and procedures respectively).