Single-digit Multiplication Problems Research Articles

The number of different digits determines solution and verification of multiplication problems.

The solution and verification of single-digit multiplication problems vary in speed and accuracy. The current study examines whether the number of different digits in a problem accounts for this variance. In Experiment 1, 41 participants solved all 2-9 multiplication problems. In Experiment 2, 43 participants verified these problems. In Experiment 3, 26 participants solved 10 problems that differed in shared-digit network (SDN) size and matched in problem size. In Experiment 4, 24 participants verified these matched sets. Results show faster and more accurate responses to problems that include fewer different digits relative to problems with more different digits, and faster and more accurate responses to problems whose SDN is small relative to problems whose SDN is large. We thus show that the number of different digits in a problem, including the operands and the solution, determines the speed and accuracy of its solution and verification. This parsimonious account also explains why responses to five and tie problems, which include fewer different digits relative to nonfive and nontie problems, are faster and more accurate than responses to other problems. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Development is in the details: Event-related theta oscillations reveal children and adults verify multiplication facts differently.

When verifying the correctness of single-digit multiplication problems, children and adults show a robust ERP correctness effect thought to reflect similar cognitive processes across groups. Recent studies suggest that this effect is instead a modulation of the negative-going N400 component in children, reflecting access to semantic memory, and the positive-going P300 component in adults, reflecting stimulus categorization. However, the relative difference in ERP amplitude is the same for both components, more positive for correct than incorrect solutions, presenting a challenge to ascertaining the appropriate interpretation. Time-frequency analysis (TFA) of the N400/P300 window provides an objective approach to dissociating these effects. TFA measured from solution onset during single-digit multiplication verification revealed significant modulations of event-related as theta power (3-6Hz) in both groups. Correct trials elicit less power in children (9-12 years) and more power in adults relative to incorrect trials. These findings are consistent with modulations of the N400 and P300, respectively, where opposite effects were predicted for spectral power. The ERP results further support a reinterpretation of the multiplication correctness effect. In contrast, TFA of the N400 effect elicited to a word-picture verification task revealed the same event-related theta effect in both groups, with increased power for mismatched than matched pictures. Together, these findings provide evidence for a developmental shift in cognitive processing specific to the multiplication task. Models of arithmetic should account for this overlooked difference in cognitive processing between children and adults.

When multiplying is meaningful in memory: Electrophysiological signature of the problem size effect in children

Children are less fluent at verifying the answers to larger single-digit arithmetic problems compared with smaller ones. This problem size effect may reflect the structure of memory for arithmetic facts. In the current study, typically developing third to fifth graders judged the correctness of single-digit multiplication problems, presented as a sequence of three digits, that were either small (e.g., 4 3 12 vs. 4 3 16) or large (e.g., 8 7 56 vs. 8 7 64). We measured the N400, an index of access to semantic memory, along with accuracy and response time. The N400 was modulated by problem size only for correct solutions, with larger amplitude for large problems than for small problems. This suggests that only solutions that exist in memory (i.e., correct solutions) reflect a modulation of semantic access likely based on the relative frequency of encountering small versus large problems. The absence of an N400 problem size effect for incorrect solutions suggests that the behavioral problem size effects were not due to differences in initial access to memory but instead were due to a later stage of cognitive processing that was reflected in a post-N400 main effect of problem size. A second post-N400 main effect of correctness at occipital electrodes resembles the beginning of an adult-like brain response observed in prior studies. In sum, event-related brain potentials revealed different cognitive processes for correct and incorrect solutions. These results allude to a gradual transition to an adult-like brain response, from verifying multiplication problems using semantic memory to doing so using more automatic categorization.

Fact retrieval or compacted counting in arithmetic—A neurophysiological investigation of two hypotheses.

There is broad consensus on the assumption that adults solve single-digit multiplication problems almost exclusively by fact retrieval from memory. In contrast, there has been a long-standing debate on the cognitive processes involved in solving single-digit addition problems. This debate has evolved around two theoretical accounts. Proponents of a fact-retrieval account postulate that these are also solved through fact retrieval, whereas proponents of a compacted-counting account propose that solving very small additions (with operands between 1 and 4) involves highly automatized and unconscious compacted counting. In the present electroencephalography (EEG) study, we put these two accounts to the test by comparing neurophysiological correlates of solving very small additions and multiplications. Forty adults worked on an arithmetic production task involving all (nontie) single-digit additions and multiplications. Afterward, participants completed trial-by-trial strategy self-reports. In our EEG analyses, we focused on induced activity (event-related synchronization/desynchronization, ERS/ERD) in three frequency bands (theta, lower alpha, upper alpha). Across all frequency bands, we found higher evidential strength for similar rather than different neurophysiological processes accompanying the solution of very small addition and multiplication problems. In the alpha bands, evidence for similarity was even stronger when operand-1-problems were excluded. In two additional analyses, we showed that ERS/ERD can differentiate between self-reported problem-solving strategies (retrieval vs. procedure) and between very small n × 1 and n + 1 problems, demonstrating its high sensitivity to cognitive processes in arithmetic. The present findings support a fact-retrieval account, suggesting that both very small additions and multiplications are solved through fact retrieval. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

Playing Games to Promote Fluency: Making Products to Make Gains

ABSTRACT Mathematical fluency supports maintenance of mathematical skills (Shurr et al., 2019). Multiple interventions support fluency, including the use of games. However, limited research exists exploring games as a means of increasing student fluency in mathematics. In this single case design study, researchers examined the relation between four elementary students’ fluency with single-digit multiplication problems and playing the Product Game, which is a game focused on whole number multiplication facts. Researchers found a functional relation between the intervention of playing the game for 10 min and student accuracy with digits and answers on a 1 min multiplication fluency probe. Overall, students also maintained higher digit and answer rates.

Individual differences in the development of children’s arithmetic fluency from grades 2 to 3.

In the present research, we provide empirical evidence for the process of symbolic integration of number associations, focusing on the development of simple addition (e.g., 5 + 3 = 8), subtraction (e.g., 5 - 3 = 2), and multiplication (e.g., 5 × 3 = 15). Canadian children were assessed twice, in Grade 2 and Grade 3 (N = 244; 55% girls). All families were English-speaking, and parent education levels ranged from high school to postgraduate, with a median of community college. In Grade 2, children completed general cognitive tasks (i.e., receptive vocabulary, working memory, nonverbal reasoning, and inhibitory control). In both grades, children completed single-digit addition and complementary subtraction problems. In Grade 3, they completed single-digit multiplication problems and measures of applied mathematics, specifically, word-problem solving, algebra, and measurement. We found that addition and subtraction were reciprocally related (controlling for cognitive skills). Subtraction fluency predicted multiplication in Grade 3, whereas addition fluency did not. In Grade 3, both subtraction and multiplication fluency were predictors of applied mathematics, with multiplication partially mediating the relation between subtraction and applied mathematics performance. These findings support the view that learning arithmetic associations is a hierarchical process. As students practice each new skill, individual differences reflect the integration of the novel component into the developing associative network. (PsycInfo Database Record (c) 2021 APA, all rights reserved).

Retrieval priming in product verification: Evidence from retrieval-induced forgetting

The conditions under which multiplication verification (3 × 6 = 12, true or false?) involves product retrieval and comparison or familiarity-based recognition judgements has not been clearly established. In two experiments examining verification of single-digit multiplication problems, we used Retrieval-Induced Forgetting (RIF), a signature of retrieval use, as an index of product retrieval in multiplication verification. In Experiment 1, 72 adults practiced multiplication either in a production format or in a verification format and then were tested on corresponding addition and control problems. The results showed RIF (i.e., slower answer production for addition problems whose multiplication counterparts had been practiced) in both the production-practice and the verification-practice groups, but RIF was stronger following true than false verification. Experiment 2 tested verification with related-false and unrelated-false products. Related-false equations produced longer RTs than unrelated false equations. Practice of true, related-false and unrelated-false multiplication equations all produced RIF of the addition counterparts but, overall, related-false multiplication equations produced relatively weak RIF. The results indicated that product retrieval mediates multiplication verification even when false answers are weak associative lures and suggest that a retrieve-and-compare process is the default strategy when false answers are at least plausible. We conclude that the presented answer in verification equations act as retrieval-priming stimuli with true equations priming correct answer retrieval and related-false answers interfering with correct answer retrieval.

Children with mathematical learning disability fail in recruiting verbal and numerical brain regions when solving simple multiplication problems

Greater skill in solving single-digit multiplication problems requires a progressive shift from a reliance on numerical to verbal mechanisms over development. Children with mathematical learning disability (MD), however, are thought to suffer from a specific impairment in numerical mechanisms. Here we tested the hypothesis that this impairment might prevent MD children from transitioning toward verbal mechanisms when solving single-digit multiplication problems. Brain activations during multiplication problems were compared in MD and typically developing (TD) children (3rd to 7th graders) in numerical and verbal regions which were individuated by independent localizer tasks. We used small (e.g., 2 × 3) and large (e.g., 7 × 9) problems as these problems likely differ in their reliance on verbal versus numerical mechanisms. Results indicate that MD children have reduced activations in both the verbal (i.e., left inferior frontal gyrus and left middle temporal to superior temporal gyri) and the numerical (i.e., right superior parietal lobule including intra-parietal sulcus) regions suggesting that both mechanisms are impaired. Moreover, the only reliable activation observed for MD children was in the numerical region when solving small problems. This suggests that MD children could effectively engage numerical mechanisms only for the easier problems. Conversely, TD children showed a modulation of activation with problem size in the verbal regions. This suggests that TD children were effectively engaging verbal mechanisms for the easier problems. Moreover, TD children with better language skills were more effective at engaging verbal mechanisms. In conclusion, results suggest that the numerical- and language-related processes involved in solving multiplication problems are impaired in MD children.

Facts, rules, and strategies in single-digit multiplication: evidence from event-related brain potentials

It has been hypothesized that zero vs. nonzero operands in single-digit multiplication problems invoke distinct solution strategies. We studied such problems in an implicit production task with event-related brain potentials (ERPs) recorded from 61 scalp positions in 18 participants. The topography of a slow negative wave, which accompanied the implicit production of the multiplication result, varied with problem type. In comparison to small problems, larger problems evoked a stronger negativity over fronto-central and right temporal sites, and zero problems evoked a left anterior negativity. These topographic differences indicate not only that zero and small nonzero problems are solved by means of distinct strategies—most likely rule application vs. fact retrieval—but also that larger, less practiced problems invoke other processes than pure fact retrieval. Moreover, ERPs showed a positive deflection around 450 ms with a centro-parietal topography (P300), whose amplitude reflected differences in anticipated problem difficulty.

Comparing arithmetic and semantic fact retrieval: effects of problem size and sentence constraint on event-related brain potentials.

Event-related potentials were recorded with 61 electrodes from 16 students who verified either the correctness of single-digit multiplication problems or the semantic congruency of sentences. Multiplication problems varied in size and sentence fragments in constraint. Both semantic and arithmetic incongruencies evoked a typical N400 with a clear parieto-central maximum. In addition, numerically larger problems (8x7), in comparison to smaller problems (3x2), evoked a negativity starting at about 360 ms whose maximum was located over the right temporal-parietal scalp. These results indicate that the arithmetic incongruency and the problem-size effect are functionally distinct. It is suggested that the arithmetic and the semantic incongruency effects are both functionally related to a context-dependent spread of activation in specialized associative networks, whereas the arithmetic problem-size effect is due to rechecking routines that go beyond basic fact retrieval.

Doing as they are told and telling it like it is: self-reports in mental arithmetic.

Adults (n=64) solved single-digit multiplication problems under both speed and accuracy instructions. Half also provided self-reports of their solutions to the problems. The participants with relatively low levels of arithmetic fluency were most influenced by instructional requirements. They responded more slowly and accurately when asked to provide descriptions of their solution procedures, whereas the performance of the participants with high and average levels of arithmetic fluency did not change. Furthermore, the performance of the low-fluency participants was more affected by speed and accuracy demands than was that of the other individuals, but only when the low-fluency participants were also required to provide self-reports. Accordingly, models of mental arithmetic will need to include roles for individual differences and situational factors.

Multiplication by eye and by ear for Chinese-speaking and English-speaking adults.

English-speaking (n = 32) and Chinese-speaking adults (n = 32) solved single-digit multiplication problems. In one condition, problems were presented as visual digits (e.g., 8 x 9). In the other condition, problems were presented as auditory number words in the participant's first language (e.g., /eit/ /taimz/ /nain/). Chinese-speaking adults made proportionately more operand-intrusion errors (e.g., 4 x 8 = 24) than English-speaking adults. Both groups made more operand-intrusion errors with auditory than with visual presentation. These findings are similar to those found when participants solve problems presented as visual number words (e.g., eight x nine), suggesting that in both cases the activation of phonological codes interferes with processing.

Extensive Practice in Mental Arithmetic and Practice Transfer Over a Ten-month Retention Interval

Participants were trained on nine single-digit multiplication problems over three sessions. Transfer of practice was tested with nine different multiplication problems in the fourth session, for half the participants. Retention of problems was examined 10 months later. In addition, memory ratings were taken after practice and before retention. Response initiation time (RTinitiate), a measure of mental calculation time, decreased during the first two practice sessions, becoming asymptotic in the third. The problem-size effect decreased with practice but was not eliminated even at asymptotic response times. Response execution time (RTexecute), a measure of motor response execution, was not affected by problem size, indicating a dissociation between both measures. No overall transfer advantage from practised to new problems was observed for RTinitiate. However, learning curves within practice and transfer sessions suggested positive transfer at the beginning of the transfer session and negative transfer at t...

Interactions among Encoding, Calculation, and Production Processes in the Multiplication Performance of Chinese-speaking Adults

Thirty-two Chinese-speaking adults solved single-digit multiplication problems. As compared to samples of English-, French-, and Dutch-speaking adults from other studies, the Chinese adults in the present study made more errors that indicated interactions between phonological codes activated at encoding and production of answers. For example, Chinese-speaking individuals made more naming errors (e.g. 4 × 8 = 48) and more operand-intrusion errors (e.g. 4 × 6 = 42) than speakers of Indo-European languages. The results are not consistent with strictly modular views of numerical processing in which encoding, calculation, and production processes are independent.

The Role of Experience in Numerical Skill: Multiplication Performance in Adults from Canada and China

Adults educated either in the People s Republic of China or in Canada solved single-digit multiplication problems. Chinese adults were faster and made fewer errors than Canadian adults and showed smaller increases in latency and errors as problem size increased. Chinese adults transformed problems with the larger operand on the left e.g. 9 6 by reversing the digits and solving the complementary problem e.g. 6 9. Only the Canadian adults showed a substantial advantage in latencies and errors for problems with operands of 5. Although both groups showed a latency advantage on ties e.g. 3 3 as compared to other problems, the advantage was much larger for the Canadian than for the Chinese adults. These findings were only partially attributable to overall differences in skill; patterns of differences persisted when groups were equated on multi-digit arithmetic performance. Chinese adults made more errors that reflect verbal-production processes that may occur after retrieval, whereas Canadian adults made more errors that reflect retrieval processes. The results are consistent with Siegler s 1988 experiential model of multiplication.

Multiple routes to solution of single-digit multiplication problems.

Researchers have assumed that adults solve simple arithmetic problems by retrieving answers from a network of stored facts. In 2 studies, undergraduates described their solutions of single-digit multiplication problems. They reported direct retrieval on approximately 80% of trials, but also reported rules (e.g., anything times 0 is 0), repeated addition (e.g., 2 × 4 = 4 + 4), number series (e.g., 3 × 5 = 5, 10, 15), and derived facts (e.g., 6 × 7 = [6 × 6] + 6). Participants were slower to retrieve problems that were most likely to be solved by nonretrieval procedures and faster to retrieve problems that were usually solved by retrieval. These results indicate that direct-retrieval models are incomplete accounts of adults' performance and support a continuing influence of learning and experience on the mental representation of simple multiplication problems.

Multiple routes to solution of single-digit multiplication problems.

Multiple routes to solution of single-digit multiplication problems.

Acquisition of Mental Multiplication Skill: Evidence for the Transition Between Counting and Retrieval Strategies

Strategies used by third and fourth graders to perform simple multiplication problems were examined. Children participated in an interview and then completed a timed production task in which they solved single-digit multiplication problems. Analyses of children's verbal reports and of solution latency data were found to support the view that the acquisition of mental multiplication begins with the use of counting strategies. By the fourth grade, however, there was a marked transition toward the use of a retrieval strategy. Children in both grade levels were found to apply rules to solve problems that involved multiplication by 1 or 0; however, not all students reported using these rules, and those students who did report the use of these rules were not consistent in the application of the rules. Comparisons between groups' and individual subjects' performance revealed that some important individual differences were obscured when the group served as the unit of analysis. Although several discrepancies were noted between the analyses of verbal reports and the analyses of chronometric data, correlations between the two methods were moderately high.

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.