Operation-specific encoding in single-digit arithmetic

Solving simple arithmetic problems involves three stages: encoding the problem, retrieving or calculating the answer, and reporting the answer. This study compared the event-related potentials elicited by single-digit addition and multiplication problems to examine the relationship between encoding and retrieval/calculation stages. Results showed that the operation effect appeared as early as the encoding of the first operand and continued to the retrieval/calculation stage: compared to addition, multiplication elicited larger negative potentials in the left anterior electrodes and larger positive potentials in the right posterior electrodes. The consistency of this operation effect across the first two stages of arithmetic processing suggests that encoding of arithmetic problems can be modulated by the nature of representation of the to-be-retrieved arithmetic facts, and thus these two stages are additive rather than interactive.     

Interactive effects of numerical surface form and operand parity in cognitive arithmetic

In Experiment 1, adults (n = 48) performed simple addition, multiplication, and parity (i.e., odd-even) comparisons on pairs of Arabic digits or English number words. For addition and comparison, but not multiplication, response time increased with the number of odd operands. For addition, but not comparison, this parity effect was greater for words than for digits. In Experiment 2, adults (n = 50) solved simple addition problems in digit and word format and reported their strategies (i.e., retrieval or procedures). Procedural strategies were used more for odd than even addends and much more for word than digit problems. The results indicate that problem encoding and answer retrieval processes for cognitive arithmetic are interactive rather than strictly additive stages.     

What affects strategy selection in arithmetic? The example of parity and five effects on product verification

The parity effect in arithmetic problem verification tasks refers to faster and more accurate judgments for false equations when the odd/even status of the proposed answer mismatches that of the correct answer. In two experiments, we examined whether the proportion of incorrect answers that violated parity or the number of even operands in the problem affected the magnitude of these effects. Experiment 1 showed larger parity effects for problems with two even operands and larger parity effects during the second half of the experiment. Experiment 2 replicated the results of Experiment 1 and varied the proportion of problems violating parity. Larger parity effects were obtained when more of the false problems violated parity. Moreover, all three effects combined to show the greatest parity effects in conditions with a high proportion of parity violations in problems containing two even operands that were solved during the second half of the experiment. Experiment 3 generalized the findings to the case of five rule (i.e., checking whether a false product ends in 5 or 0), another procedure for solving and verifying multiplication problems quickly. These results (1) delineate further constraints for inclusion in models of arithmetic processing when thinking about how people select among verification strategies, (2) show combined effects of variables that traditionally have been shown to have separate effects on people's strategy selection, and (3) are consistent with a view of strategy selection that suggests a bias either in the allocation of cognitive resources in the execution of strategies or in the order of execution of these strategies; they argue against a simple, unbiased competition among strategies.     

Developing symbolic capacity one step at a time

The present research examines the ability of children as young as 4 years to use models in tasks that require scaling of distance along a single dimension. In Experiment 1, we found that tasks involving models are similar in difficulty to those involving maps that we studied earlier (Huttenlocher, J., Newcombe, N., & Vasilyeva, M. (1999). Spatial scaling in young children. Psychological Science, 10, 393-398). In Experiment 2, we found that retrieval tasks, where children indicate the location of a hidden object in an actual space are substantially more difficult than placement tasks, where children put a visible object in a particular location in an actual space. We discuss possible implications of the differential difficulty of retrieval and placement tasks for the understanding of symbolic development.     

Spatial congruency between stimulus presentation and response key arrangements in arithmetic fact retrieval

It is known that number and space representations are connected to one another in numerical and arithmetic abilities. Numbers are represented using the metaphor of a mental number line, oriented along horizontal and vertical space. This number line also seems to be linked to mental arithmetic, which is based partly on arithmetic fact retrieval. It seems that number representation and mental arithmetic are linked together. The present study tested the effect of spatial contextual congruency between stimulus presentation and response key arrangements in arithmetic fact retrieval, using number-matching and addition verification tasks. For both tasks in Experiment 1, a contextual congruency effect was present horizontally (i.e., horizontal presentation of stimuli and horizontal response key alignments) but not vertically (i.e., vertical presentation of stimuli but horizontal response key alignments). In Experiment 2, both tasks showed a contextual congruency effect for both spatial conditions. Experiment 1 showed that the interference and distance effects were found in the horizontal condition, probably because of the spatial congruency between stimulus presentation and response key arrangements. This spatial congruency could be related to the activation of the horizontal number line. Experiment 2 showed similar interference and distance effects for both spatial conditions, suggesting that the congruency between stimulus presentation and response alignment could facilitate the retrieval of arithmetic facts. This facilitation could be related to the activation of both horizontal and vertical number lines. The results are discussed in light of the possible role of a mental number line in arithmetic fact retrieval.     

Working memory and children's use of retrieval to solve addition problems

This study tested the hypothesis that children with high working memory capacities solve single-digit additions by direct retrieval of the answers from long-term memory more often than do children with low working memory capacities. Counting and reading letter span tasks were administered to groups of third-grade (mean age=107 months) and fourth-grade (mean age=118 months) children who were also asked to solve 40 single-digit additions. High working memory capacity was associated with more frequent use of retrieval and faster responses in solving additions. The effect of span on the use of retrieval increased with the size of the minimum addend. The relation between working memory measures and use and speed of retrieval did not depend on the numerical or verbal nature of the working memory task. Implications for developmental theories of cognitive arithmetic and theories of working memory are discussed.     

An evaluation of sex and cultural differences in arithmetic retrieval-induced forgetting

Retrieval practice of arithmetic facts (e.g. 2?×?3) can interfere with retrieval of other, closely related arithmetic facts (e.g. 2?+?3), increasing response time (RT) and errors for these problems. Here we examined potential sex and culture-related differences in arithmetic retrieval-induced forgetting (RIF). This was motivated by re-analyses of several published arithmetic RIF data sets that appeared to show that the effect occurred for women but not men. Experiment 1 (n?=?72) tested for possible sex differences in a diverse but predominantly Canadian university sample. Experiment 2 (n?=?48) examined potential sex differences in native Chinese participants, which previous research indicated may not be susceptible to the RIF effect for a particular subset of small addition problems (sum?≤?10). In Experiment 1, we found no evidence that the addition RIF effect differed between male and female adults. In Experiment 2, the Chinese adults showed RIF for tie problems (e.g. 2?+?2, 3?+?3, etc.) regardless of sex, but neither sex presented RIF for small non-tie addition problems. The results indicated that the RIF effect is not gender specific, and there might not be strong memory retrieval competition between addition and multiplication facts for non-tie problems in Chinese adults.     

Influences of problem format and SES on preschoolers’ understanding of approximate addition

Recent studies suggest that 5-year-olds can add and compare large numerical quantities through approximate representations of number. However, the nature of this understanding and its susceptibility to environmental influences remain unclear. We examined whether children's early competence depends on the canonical problem format (i.e., arithmetic operations presented on the left side). Sixty children from middle-to-high-SES backgrounds (Experiment 1) and 47 children from low-SES backgrounds (Experiment 2) viewed events that required them to add and compare large numbers. Events were shown in a canonical or noncanonical format. Children from both SES backgrounds performed above chance on the approximate addition tasks, but children from middle-to-high-SES backgrounds performed significantly better. Moreover, children from middle-to-high SES backgrounds performed better when problems were presented in the canonical format, whereas children from low-SES backgrounds did not. These results suggest that children's understanding of approximate number is affected by some of the same environmental factors that affect performance on exact arithmetic tasks.     

Verifying simple arithmetic sums and products: Are the phonological loop and the central executive involved?

In two experiments, we investigated the role of the phonological loop and the central executive in the verification of the complete set of one-digit addition (Experiment 1) and multiplication (Experiment 2) problems. The focus of the present study was on the contradictory results concerning the contribution of the phonological loop in the verification of true problems (e.g., 8 + 4 = 12 or 4 x 6 = 24) reported until now. The results revealed that this slave system is not involved in verifying simple arithmetic problems, in contrast to the central executive. Furthermore, our results indicated that the split effect is due to the use of two different arithmetic strategies.     

Sequential difficulty effects during strategy execution

In two experiments, we tested the hypothesis that strategy performance on a given trial is influenced by the difficulty of the strategy executed on the immediately preceding trial, an effect that we call strategy sequential difficulty effect. Participants' task was to provide approximate sums to two-digit addition problems by using cued rounding strategies. Results showed that performance was poorer after a difficult strategy than after an easy strategy. Our results have important theoretical and empirical implications for computational models of strategy choices and for furthering our understanding of strategic variations in arithmetic as well as in human cognition in general.     

Stability and change in children's division strategies

Age-related changes in children's performance on simple division problems (e.g., 6/2, 72/9) were investigated by asking children in Grades 4 through 7 to solve 32 simple division problems. Differences in performance were found across grade, with younger children performing more slowly and less accurately than older children. Problem size effects were also found in that children were faster and more accurate on small problems than on large problems. Two strategies changed across age, with children in Grade 4 relying heavily on the strategy of "addition" (adding the divisor until the dividend was reached) to solve the problems and children in Grades 5 through 7 relying primarily on the strategy of "multiplication" (recasting the division problem as a multiplication problem) to solve the problems. Surprisingly, the frequency of direct retrieval (retrieving the answer directly from memory) did not increase across grade and never became the dominant strategy of choice. Reasons for why retrieval use remains infrequent and age invariant are discussed. Overall, the results suggest that division is a unique operation and that the continued study of division may have implications for further understanding of how procedural and conceptual knowledge of arithmetic develops.     

The relationship between phonological processing and arithmetic in children with learning disabilities

Phonological processing skills have not only been shown to be important for reading skills, but also for arithmetic skills. Specifically, previous research in typically developing children has suggested that phonological processing skills may be more closely related to arithmetic problems that are solved through fact retrieval (e.g., remembering the solution from memory) than procedural computation (e.g., counting). However, the relationship between phonological processing and arithmetic in children with learning disabilities (LDs) has not been investigated. Yet, understanding these relationships in children with LDs is especially important because it can help elucidate the cognitive underpinnings of math difficulties, explain why reading and math disabilities frequently co-occur, and provide information on which cognitive skills to target for interventions. In 63 children with LDs, we examined the relationship between different phonological processing skills (phonemic awareness, phonological memory, and rapid serial naming) and arithmetic. We distinguished between arithmetic problems that tend to be solved with fact retrieval versus procedural computation to determine whether phonological processing skills are differentially related to these two arithmetic processes. We found that phonemic awareness, but not phonological memory or rapid serial naming, was related to arithmetic fact retrieval. We also found no association between any phonological processing skills and procedural computation. These results converge with prior research in typically developing children and suggest that phonemic awareness is also related to arithmetic fact retrieval in children with LD. These results raise the possibility that phonemic awareness training might improve both reading and arithmetic fact retrieval skills.

Research Highlights

• Relationships between phonological processing and various arithmetic skills were investigated in children with learning disabilities (LDs) for the first time.
• We found phonemic awareness was related to arithmetic involving fact retrieval, but not to arithmetic involving procedural computation in LDs.
• The results suggest that phonemic awareness is not only important to skilled reading, but also to some aspects of arithmetic.
• These results raise the question of whether intervention in phonemic awareness might improve arithmetic fact retrieval skills.

     

How is phonological processing related to individual differences in children’s arithmetic skills?

While there is evidence for an association between the development of reading and arithmetic, the precise locus of this relationship remains to be determined. Findings from cognitive neuroscience research that point to shared neural correlates for phonological processing and arithmetic as well as recent behavioral evidence led to the present hypothesis that there exists a highly specific association between phonological awareness and single‐digit arithmetic with relatively small problem sizes. The present study examined this association in 37 typically developing fourth and fifth grade children. Regression analyses revealed that phonological awareness was specifically and uniquely related to arithmetic problems with a small but not large problem size. Further analysis indicated that problems with a high probability of being solved by retrieval, but not those typically associated with procedural problem‐solving strategies, are correlated with phonological awareness. The specific association between phonological awareness and arithmetic problems with a small problem size and those for which a retrieval strategy is most common was maintained even after controlling for general reading ability and phonological short‐term memory. The present findings indicate that the quality of children’s long‐term phonological representations mediates individual differences in single‐digit arithmetic, suggesting that more distinct long‐term phonological representations are related to more efficient arithmetic fact retrieval.     

Children’s strategies in complex arithmetic

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.     

Practice effects on strategy selection and strategy efficiency in simple mental arithmetic

Two experiments were conducted to investigate the effects of practice on strategy selection and strategy efficiency in mental arithmetic. Participants had to solve simple addition or multiplication problems, after having received 0, 3, or 6 practice sessions (Experiment 1), and before and after having received 3 practice sessions (Experiment 2). Strategy selection was measured by means of trial-by-trial strategy reports, whereas strategy efficiency was measured by means of response latencies. Results showed significant practice effects on retrieval frequency, procedural frequency, retrieval efficiency, and procedural efficiency. However, practice effects on strategy efficiency appeared to be both strategy-specific (i.e., only for procedural strategies) and operation-specific (i.e., only for multiplication problems). Implications of the present results for mathematic cognition and its modeling are discussed.     

How does a child solve 7 + 8? Decoding brain activity patterns associated with counting and retrieval strategies

Cognitive development and learning are characterized by diminished reliance on effortful procedures and increased use of memory-based problem solving. Here we identify the neural correlates of this strategy shift in 7-9-year-old children at an important developmental period for arithmetic skill acquisition. Univariate and multivariate approaches were used to contrast brain responses between two groups of children who relied primarily on either retrieval or procedural counting strategies. Children who used retrieval strategies showed greater responses in the left ventrolateral prefrontal cortex; notably, this was the only brain region which showed univariate differences in signal intensity between the two groups. In contrast, multivariate analysis revealed distinct multivoxel activity patterns in bilateral hippocampus, posterior parietal cortex and left ventrolateral prefrontal cortex regions between the two groups. These results demonstrate that retrieval and counting strategies during early learning are characterized by distinct patterns of activity in a distributed network of brain regions involved in arithmetic problem solving and controlled retrieval of arithmetic facts. Our findings suggest that the reorganization and refinement of neural activity patterns in multiple brain regions plays a dominant role in the transition to memory-based arithmetic problem solving. Our findings further demonstrate how multivariate approaches can provide novel insights into fine-scale developmental changes in the brain. More generally, our study illustrates how brain imaging and developmental research can be integrated to investigate fundamental aspects of neurocognitive development.     

“2 x 3” primes naming “6”: Evidence from masked priming

It is a common assumption for multiplication-solving models that single-digit multiplications are automatically retrieved. However, the experimental evidence for this is based on paradigms under suspicion. In this research, we employed a new procedure with the aim of assessing the automatic retrieval of multiplication more directly. In two experiments, multiplication automatism was studied using briefly presented primes (stimulus onset asynchrony = 48 msec) in a number-naming task. In Experiment 1, in the congruent conditions, the target and the prime were the same numbers (e.g., prime, 6; target, 6) or the target was the solution to the multiplication prime (e.g., prime, 2×3=; target, 6). In the incongruent conditions, no relationship existed between the primes and the targets (e.g., prime, 32; target, 6; or prime, 4×8=; target, 6). Experiment 2 explored the relevance of the equal sign for the multiplication-priming effect. Data showed that naming was faster when the solution of the multiplication prime matched the target, as compared with the incongruent condition (multiplication-priming effect), and that these effects were found irrespective of the presence of the equal sign. The fact that this priming effect was found even though the participants were unaware of the presentation of the primes supports the automatic character of single-digit multiplication. We conclude by arguing that this procedure is highly valuable for exploring the mechanisms involved in simple arithmetic solving.     

The influence of problem features and individual differences on strategic performance in simple arithmetic

The present study examined the influence of features differing across problems (problem size and operation) and across individuals (gender, amount of daily arithmetic practice, calculator use, and arithmetic skill) on simple arithmetic performance. Regression analyses were used to investigate the role of these variables in both strategy selection and strategy efficiency. Results show that more skilled and highly practiced students used memory retrieval more often and executed their strategies more efficiently than did less skilled and practiced students. Furthermore, calculator use correlated with both retrieval and procedural strategy efficiency but not with strategy selection. Only very small associations with gender were observed, with boys retrieving slightly faster than girls. Implications of the present findings for models of mental arithmetic are discussed.     

The role of cue familiarity in adults’ strategy choices for simple addition

Are adults’ decisions to use direct memory retrieval for simple addition influenced by the familiarity of problem operands? We manipulated the familiarity of a subset of operands by having adults repeatedly practise specific additions (two+five=?; Experiment 1) or magnitude comparisons (two five, choose the larger; Experiment 2). Both experiments provided evidence that pre-exposure to single-digit operands increased reported use of direct retrieval for new combinations of the familiarised operands. RT and error patterns across experiments also supported the conclusion that increased use of retrieval facilitated performance. These results show that operand familiarity potentially plays a significant role in adults’ strategy choices for simple addition.     

Production,verification, and priming of multiplication facts

In the arithmetic-verification procedure, subjects are presented with a simple equation (e.g., 4 × 8 = 24) and must decide quickly whether it is true or false. The prevailing model of arithmetic verification holds that the presented answer (e.g., 24) has no direct effect on the speed and accuracy of retrieving an answer to the problem. It follows that models of the retrieval stage based on verification are also valid models of retrieval in the production task, in which subjects simply retrieve and state the answer to a given problem. Results of two experiments using singledigit multiplication problems challenge these assumptions. It is argued that the presented answer in verification functions as a priming stimulus and that on “true” verification trials the effects of priming are sufficient to distort estimates of problem difficulty and to mask important evidence about the nature of the retrieval process. It is also argued that the priming of false answers that have associative links to a presented problem induces interference that disrupts both speed and accuracy of retrieval. The results raise questions about the interpretation of verification data and offer support for a network-interference theory of the mental processes underlying simple multiplication.