Multiplication Fact Retrieval Research Articles

No influence of masked priming on the multiplication fact retrieval in a result verification task

In three experiments, we used a masked prime in a verification task to investigate the processing stages occurring during multiplication fact retrieval. We aimed to investigate the retrieval process by overlapping its execution with the processing of a masked prime consisting of a number. Participants evaluated the correctness of multiplication equations, where the result was preceded by a masked prime (presented for 30 ms, with different stimulus onset asynchrony between the operands and the prime). Decade consistency and relatedness of the prime were manipulated. For example, given the equation 4 x 7 = 28, the prime could be: a neighbor either decade consistent (24) or inconsistent (32), or an unrelated number either decade consistent (23) or inconsistent (31). We expected that the feature of the prime (relatedness or decade consistency) that generates interference depends on the processing stage reached when the prime is processed. Although Experiment 1 showed promising results, Experiments 2 and 3 suggest that the pattern found in Experiment 1 was a false positive. Overall, the paradigm used in this study (i.e., masked prime with a verification ask) does not seem to produce a stable interference during the retrieval process.

Language does arithmetic: linguistic differences in children’s place-value processing

The representation and retrieval of multiplication facts is dependent on linguistic specificities such as number word inversion (i.e., 23 is spoken dreiundzwanzig in German which translates to three and twenty). Previous research has evaluated these language influences in adults. Now this study aims to follow-up on earlier findings and takes a closer look at inversion-related effects on place-value processing during multiplication in children. In a task of choice 46 children, either German- or Italian-speaking, had to pick the right answer out of two options for a given multiplication problem. Already established effects in adult participants such as decade-consistency and table-relatedness were also present in elementary school children, but different between the language groups. For decade-consistent items the effect of table-relatedness was larger for Italian-speaking students than for German-speaking. This indicates that the inversion property in the German language leads to those children putting less emphasis on the tens digit when solving multiplication problems, than Italian-speaking children.

Oscillatory electroencephalographic patterns of arithmetic problem solving in fourth graders

Numerous studies have identified neurophysiological correlates of performing arithmetic in adults. For example, oscillatory electroencephalographic (EEG) patterns associated with retrieval and procedural strategies are well established. Whereas fact retrieval has been linked to enhanced left-hemispheric theta ERS (event-related synchronization), procedural strategies are accompanied by increased bilateral alpha ERD (event-related desynchronization). It is currently not clear if these findings generalize to children. Our study is the first to investigate oscillatory EEG activity related to strategy use and arithmetic operations in children. We assessed ERD/ERS correlates of 31 children in fourth grade (aged between nine and ten years) during arithmetic problem solving. We presented multiplication and subtraction problems, which children solved with fact retrieval or a procedure. We analyzed these four problem categories (retrieved multiplications, retrieved subtractions, procedural multiplications, and procedural subtractions) in our study. In summary, we found similar strategy-related patterns to those reported in previous studies with adults. That is, retrieval problems elicited stronger left-hemispheric theta ERS and weaker alpha ERD as compared to procedural problems. Interestingly, we observed neurophysiological differences between multiplications and subtractions within retrieval problems. Although there were no response time or accuracy differences, retrieved multiplications were accompanied by larger theta ERS than retrieved subtractions. This finding could indicate that retrieval of multiplication and subtraction facts are distinct processes, and/or that multiplications are more frequently retrieved than subtractions in this age group.

Can the interference effect in multiplication fact retrieval be modulated by an arithmetic training? An fMRI study

Single-digit multiplications are thought to be associated with different levels of interference because they show different degrees of feature overlap (i.e., digits) with previously learnt problems. Recent behavioral and neuroimaging studies provided evidence for this interference effect and showed that individual differences in arithmetic fact retrieval are related to differences in sensitivity to interference (STI). The present study investigated whether and to what extent competence-related differences in STI and its neurophysiological correlates can be modulated by a multiplication facts training.Participants were 23 adults with high and 23 adults with low arithmetic competencies who underwent a five-day multiplication facts training in which they intensively practiced sets of low- and high-interfering multiplication problems. In a functional magnetic resonance imaging (fMRI) test session after the training, participants worked on a multiplication verification task that comprised trained and untrained problems.Analyses of the behavioral data revealed an interference effect only in the low competence group, which could be reduced but not resolved by training. On the neural level, competence-related differences in the interference effect were observed in the left supramarginal gyrus (SMG), showing activation differences between low- and high-interfering problems only in the low competent group. These findings support the idea that individuals’ low arithmetic skills are related to the development of insufficient memory representations because of STI. Further, our results indicate that a short training by drill (i.e., learning associations between operands and solutions) was not fully effective to resolve existing interference effects in arithmetic fact knowledge.

Difficulties in retrieval multiplication facts: The case of interference to reconsolidation

Many students have difficulties in retrieving multiplication facts from memory. The aim of the present study was to test the difficulty in retrieval of multiplication facts from the perspective of the reconsolidation of long-term memory phase, which has been found to be sensitive to interferences. Students learned multiplication facts and then received a reminder, which led to reactivation and reconsolidation. After the reminder, additional multiplication facts (interference) were learned and memory was tested. Students who received both a reminder and interference during reconsolidation showed no significant improvement in retrieving multiplication facts from memory, whereas Students who received either a reminder or additional multiplication facts (interference) exhibited a better performance in retrieval. These results indicate, for the first time, that the reconsolidation phase is sensitive to interferences in mathematical declarative memory content. The findings indicate additional possible causes for difficulties in retrieval of multiplication facts in class.

Interference during the retrieval of arithmetic and lexico-semantic knowledge modulates similar brain regions: Evidence from functional magnetic resonance imaging (fMRI)

Single-digit multiplications are mainly solved by memory retrieval. However, these problems are also prone to errors due to systematic interference (i.e., co-activation of interconnected but incorrect solutions). Semantic control processes are crucial to overcome this type of interference and to retrieve the correct information. Previous research suggests the importance of several brain regions such as the left inferior frontal cortex and the intraparietal sulcus (IPS) for semantic control. But, this evidence is mainly based on tasks measuring interference during the processing of lexico-semantic information (e.g., pictures or words). Here, we investigated whether semantic control during arithmetic problem solving (i.e., multiplication fact retrieval) draws upon similar or different brain mechanisms as in other semantic domains (i.e., lexico-semantic).The brain activity of 46 students was measured with fMRI while participants performed an operand-related-lure (OR) and a picture-word (PW) task. In the OR task participants had to verify the correctness of a given solution to a single-digit multiplication. Similarly, in the PW task, participants had to judge whether a presented word matches the concept displayed in a picture or not.Analyses showed that resolving interference in these two tasks modulates the activation of a widespread fronto-parietal network (e.g., left/right IFG, left insula lobe, left IPS). Importantly, conjunction analysis revealed a neural overlap in the left inferior frontal gyrus (IFG) pars triangularis and left IPS. Additional Bayesian analyses showed that regions that are thought to store lexico-semantic information (e.g., left middle temporal gyrus) did not show evidence for an arithmetic interference effect. Overall, our findings not only indicate that semantic control plays an important role in arithmetic problem solving but also that it is supported by common brain regions across semantic domains. Additionally, by conducting Bayesian analysis we confirmed the hypothesis that the semantic control network contributes differently to semantic tasks of various domains.

Attitudes Toward Math Are Differentially Related to the Neural Basis of Multiplication Depending on Math Skill.

Attitudes towards math (ATM) predict math achievement. Negative ATM are associated with avoidance of math content, while positive ATM are associated with exerting more effort on math tasks. Recent literature highlights the importance of considering interactions between ATM and math skill in examining relations to achievement. This study investigated, for the first time, the effects of the interaction between math skill and ATM on the neurocognitive basis of arithmetic processing. We examined the effect of this interaction using a single-digit multiplication task in 9- to 12-year-old children. Results showed that higher math skill was correlated with less activation in the left inferior frontal gyrus (IFG), and positive ATM were correlated with less activation in the left IFG. The relation between ATM and the neural basis of multiplication varied depending on math skill. Only among children with lower math skill, positive ATM were associated with greater activation of the left IFG. The results suggest that positive ATM in low skill children might encourage them to more fully engage the neurocognitive systems underlying controlled effort and retrieval of multiplication facts. Our results highlight the importance of examining the role of both attitudinal and cognitive factors on the neural basis of arithmetic development.

The neural substrates of the problem size and interference effect in children’s multiplication: An fMRI study

Within children’s multiplication fact retrieval, performance can be influenced by various effects, such as the well-known problem size effect (i.e., smaller problems are solved faster and more accurately) and the more recent interference effect (i.e., the quality of memory representations of problems depends on previously learned problems; the more similar a problem is to a previously learned one, the more proactive interference impacts on storing in long-term-memory). This interference effect has been observed in behavioral studies, and determines a substantial part of performance beyond problem size. Unlike the problem size effect, the neural basis of the interference effect in children has not been studied. To better understand the underpinning mechanisms behind children’s arithmetic fact retrieval, we aimed to investigate the neural basis of both effects in typically developing children. Twenty-four healthy 9- to 10-year-olds took part in a behavioral and fMRI scanning session, during which multiplication items had to be solved. Data were analyzed by manipulating problem size and interference level in a 2 × 2 factorial design. Concurring with previous studies, our results reveal clear behavioral effects of problem size and interference, with larger and high interfering items being solved significantly slower. On the neural level, a clear problem size effect was observed in a fronto-parietal and temporal network. The interference effect, however, was not detected; no clear neural distinctions were observed between low and high interfering items.

Disentangling Neural Sources of Problem Size and Interference Effects in Multiplication.

Multiplication is thought to be primarily solved via direct retrieval from memory. Two of the main factors known to influence the retrieval of multiplication facts are problem size and interference. Because these factors are often intertwined, we sought to investigate the unique influences of problem size and interference on both performance and neural responses during multiplication fact retrieval in healthy adults. Behavioral results showed that both problem size and interference explained separate unique portions of RT variance, but with significantly stronger contribution from problem size, which contrasts with previous work in children. Whole-brain fMRI results relying on a paradigm that isolated multiplication fact retrieval from response selection showed highly overlapping brain areas parametrically modulated by both problem size and interference in a large network of frontal, parietal, and subcortical brain areas. Subsequent analysis within these regions revealed problem size to be the stronger and more consistent "unique" modulating factor in overlapping regions as well as those that appeared to respond only to problem size or interference at the whole-brain level, thus underscoring the need to look beyond anatomical overlap using arbitrary thresholds. Additional unique contributions of interference (beyond problem size) were identified in right angular gyrus and subcortical regions associated with procedural processing. Together, our results suggest that problem size, relative to interference, tends to be the more dominant factor in driving behavioral and neural responses during multiplication fact retrieval in adults. Nevertheless, unique contributions of both factors demonstrate the importance of considering the overlapping and unique contributions of each in explaining the cognitive and neural bases of mental multiplication.

More than simple facts: cross-linguistic differences in place-value processing in arithmetic fact retrieval.

Linguistic specificities such as the inversion property of number words (e.g., in German 43 is spoken dreiundvierzig, literally threeand forty) moderate Arabic number processing. So far, cross-linguistic studies have mostly focused on inversion-related effects on simple (e.g.,number comparison) and calculation-based (e.g.,multi-digit addition) magnitude processing of numerical information. Despite the assumption that multiplication facts are represented in verbal format, not much attention has been paid to inversion-related influences on multiplication fact retrieval. Accordingly, the current study evaluated inversion-related effects on the processing of place-value information in multiplication. In a verification paradigm, the decade consistency effect (i.e., more errors when the decade of a solution probe shares the decade digit with the correct solution) was larger for English- than German-speaking participants for table-related probes. Processing of decade digits might be prioritised in English-speaking participants because the decadedigit is named first in English number words, whereas in German number words the unitdigit is named first. Our results indicate that (1) the influence of specificities of a verbal number word formation on place-value processing generalise to arithmetic fact retrieval and (2) inversion of number words might even be advantageous in specific cases.

Involvement of working memory in mental multiplication in Chinese elementary students

ABSTRACTThe authors' aim was to examine the relation between two-digit mental multiplication and working memory. In Study 1, involving 30 fifth-grade students, we used digit span backward as an abbreviated measure of working memory. In Study 2, involving 41 fourth-grade students, working memory comprised measures of phonological loop, visuospatial sketchpad, and central executive (i.e., updating) based on A. Miyake et al. (2000). In Study 1, working memory measured through the use of digit span backward explained unique variance in response time. In Study 2, participants' accuracy and response time were most susceptible to phonological loop influence. The findings support the argument that multiplication facts are stored in a verbal form and the retrieval of multiplication facts relies more on verbal modality. Unique features of Chinese mathematics instruction of mental multiplication were discussed. The findings suggested that the involvement of different subcomponents of working memory in mental arithmetic may be subject to instructional, contextual, and linguistic factors.

The role of physical digit representation and numerical magnitude representation in children’s multiplication fact retrieval

Arithmetic facts, in particular multiplication tables, are thought to be stored in long-term memory and to be interference prone. At least two representations underpinning these arithmetic facts have been suggested: a physical representation of the digits and a numerical magnitude representation. We hypothesized that both representations are possible sources of interference that could explain individual differences in multiplication fact performance and/or in strategy use. We investigated the specificity of these interferences on arithmetic fact retrieval and explored the relation between interference and performance on the different arithmetic operations and on general mathematics achievement. Participants were 79 fourth-grade children (Mage=9.6 years) who completed a products comparison and a multiplication production task with verbal strategy reports. Performances on a speeded calculation test including the four operations and on a general mathematics achievement test were also collected. Only the interference coming from physical representations was a significant predictor of the performance across multiplications. However, both the magnitude and physical representations were unique predictors of individual differences in multiplication. The frequency of the retrieval strategy across multiplication problems and across individuals was determined only by the physical representation, which therefore is suggested as being responsible for memory storage issues. Interestingly, this impact of physical representation was not observed when predicting performance on subtraction or on general mathematical achievement. In contrast, the impact of the numerical magnitude representation was more general in that it was observed across all arithmetic operations and in general mathematics achievement.

Contribution of working memory in multiplication fact network in children may shift from verbal to visuo-spatial: a longitudinal investigation.

Number facts are commonly assumed to be verbally stored in an associative multiplication fact retrieval network. Prominent evidence for this assumption comes from so-called operand-related errors (e.g., 4 × 6 = 28). However, little is known about the development of this network in children and its relation to verbal and non-verbal memories. In a longitudinal design, we explored elementary school children from grades 3 and 4 in a multiplication verification task with the operand-related and -unrelated distractors. We examined the contribution of multiplicative fact retrieval by verbal and visuo-spatial short-term and working memory (WM). Children in grade 4 showed smaller reaction times in all conditions. However, there was no significant difference in errors between grades. Contribution of verbal and visuo-spatial WM also changed with grade. Multiplication correlated with verbal WM and performance in grade 3 but with visuo-spatial WM and performance in grade 4. We suggest that the relation to verbal WM in grade 3 indicates primary linguistic learning of and access to multiplication in grade 3 which is probably based on verbal repetition of the multiplication table heavily practiced in grades 2 and 3. However, the relation to visuo-spatial semantic WM in grade 4 suggests that there is a shift from verbal to visual and semantic learning in grade 4. This shifting may be induced because later in elementary school, multiplication problems are rather carried out via more written, i.e., visual tasks, which also involve executive functions. More generally, the current data indicates that mathematical development is not generally characterized by a steady progress in performance; rather verbal and non-verbal memory contributions of performance shift over time, probably due to different learning contents.

Strategy Instruction and Maintenance of Basic Multiplication Facts through Digital Game Play

Formative instruction on multiplication primarily focuses on rote memorization. This leads to factual fluency, but also develops a narrow view of multiplication and hinders the development of conceptual understanding. Theory and research recommend the concurrent development of conceptual understanding and factual fluency during the initial stages of learning about multiplication. Woodward (2006) conducted a field study to investigate this instructional approach and found a significant difference between those who received instruction on multiplicative properties and timed-drills of multiplication facts on a conceptual measure than those who only spent time on timed drills. This study investigated the efficacy of integrating the same approach within a digital game. There was a significant decrease between pre- and post- measures of participants timed retrieval of multiplication facts, but no differences were found between conditions on pre- and post-measures of conceptual understanding. These findings indicate that special attention must be paid to intrinsic integration of instructional content in order to address conceptual understanding through digital game play.

Retrieval-induced forgetting of arithmetic facts.

Retrieval-induced forgetting (RIF) is a widely studied phenomenon of human memory, but RIF of arithmetic facts remains relatively unexplored. In 2 experiments, we investigated RIF of simple addition facts (2 + 3 = 5) from practice of their multiplication counterparts (2 × 3 = 6). In both experiments, robust RIF expressed in response times occurred only for high-strength small-number addition facts with sums ≤ 10, indicating that RIF from multiplication practice was interference dependent. RIF of addition-fact memory was produced by multiplication retrieval (2 × 3 = ?) but not multiplication study (2 × 3 = 6), supporting an inhibitory mechanism of RIF in arithmetic memory. Finally, RIF occurred with multiplication practiced in word format (three × four) and addition tested later in digit format (3 + 4), which provides evidence that digit and written-word formats for arithmetic accessed a common semantic retrieval network. The results support the view that addition and multiplication facts are stored in an interrelated semantic network and that RIF of competing addition facts is an intrinsic process of multiplication fact retrieval.

Neurophysiological markers of retrieval-induced forgetting in multiplication fact retrieval

Event-related potential (ERP) counterparts of practice effects in multiplication fact retrieval were examined. Participants performed a multiplication verification task after having practiced a specific problem set. Practice was either active (retrieval of solutions to multiplication problems) or passive (reexposure to the same operands plus the correct result). Behavioral data showed retrieval-induced facilitation for practiced items and retrieval-induced forgetting for related, unpracticed items, irrespective of practice type. ERPs revealed that, for the active practice group, forgetting was reflected in a reduced N100 component time-locked to result onset. Irrespective of practice type, forgetting was also reflected in a reduced result-locked P350 component, whereas facilitation was associated with an increased amplitude of the same component. These results suggest that beneficial and detrimental effects of practice may be mediated by partially distinct processes.

Phonological processing and arithmetic fact retrieval: Evidence from developmental dyslexia

The triple-code model, cognitive neuroimaging and developmental behavioral data suggest a specific association between phonological processing and arithmetic fact retrieval. Accordingly, individuals with deficits in phonological processing, such as individuals with developmental dyslexia, are expected to show difficulties in arithmetic fact retrieval. The present study tested this proposal in 25 adults with developmental dyslexia and 25 matched controls by examining strategy use during single-digit multiplication and subtraction and its associations with phonological processing. Findings revealed that individuals with dyslexia retrieved fewer arithmetic facts from memory and were less efficient in doing so. At the same time, they showed deficits in phonological processing. Phonological processing, particularly phonological awareness, was related to arithmetic fact retrieval. This association was especially prominent in multiplication, indicating that fact retrieval in multiplication rather than subtraction is mediated by phonological processes. These data provide ground for future neuroimaging studies, who should examine the neural overlap between phonological processing and multiplication fact retrieval in the same sample of participants.

Oscillatory EEG correlates of an implicit activation of multiplication facts in the number bisection task

Neuroimaging evidence points towards the left inferior parietal cortex to be crucial for the representation and retrieval of multiplication facts. However, to date studies allowing a functional interpretation of neuroimaging data are still scarce. In the current study we aimed at evaluating the functional involvement of left inferior parietal cortex areas in the implicit retrieval of multiplication fact knowledge in a number bisection task by examining event-related desynchronization (ERD) in the upper alpha band. Upper alpha ERD is generally agreed to be modulated by processes of memory retrieval. It was observed that upper alpha ERD decreased for multiplicative triplets (e.g. 3_6_9) but not for non-multiplicative (e.g. 2_5_8) triplets at left parietal electrodes but increased at left prefrontal electrodes. These results are interpreted to suggest that after multiplicativity has been recognized further magnitude evaluations in the left hemisphere may be abated by prefrontal processes of executive control.

Event-related brain potentials uncover activation dynamics in the lexicon of multiplication facts

The present study was designed to ascertain whether the cortical dynamics underlying multiplication fact retrieval can be modulated as a function of arithmetic relatedness. To this end, event-related brain potentials (ERPs) were recorded from 22 participants engaged in a number-matching task. The task was to decide whether a single probe number had been shown as part of a previously presented pair of cue numbers. Probes in the non-matching condition (50% of total trials) were either the product of the cue numbers (strong cue-probe association), the multiple of either cue numbers (weak cue-probe association), or an arithmetically neutral number with respect to the cue numbers (no cue-probe association). Behavioral data showed a clear interference effect (LeFevre interference), in that performance in the non-matching condition was significantly better when the probe number was arithmetically neutral compared to when it was arithmetically related to the cue digits either strongly (i.e., through the product) or weakly (i.e., through its neighbor in the multiplication table). LeFevre interference was not statistically different in magnitude for the two arithmetically related conditions. In contrast, ERPs allowed us to dissociate cortical processing dynamics for these latter conditions: whereas initially (250–350 ms) both product and neighbor probes evoked a relatively more pronounced positivity compared to neutral probes, in a later interval (350–450 ms, N400-like component), neighbors critically diverged from products, showing relatively more negative values, similar to those of neutral probes. The observed dissociation in ERP measures is interpreted as evidence of activation spreading in the network of multiplication facts, with the short-lasting response elicited by neighbors likely reflecting activation dissipating over time because of a weaker association to the cue digits.

A microgenetic study of simple division.

How simple division strategies develop over a short period of time was examined with a microgenetic study. Grade 5 students (mean age = 10 years, 3 months) solved simple division problems in 8 weekly sessions. Performance improved with faster and more accurate responses across the study. Consistent with R. S. Siegler's (1996) overlapping waves model, strategies varied in their use. Direct retrieval increased, retrieval of multiplication facts remained steady, and addition facts, derived facts, and special tricks marginally decreased. Consistent with previous research, multiplication fact retrieval was the most common strategy, although it was slower and more error prone than direct retrieval. Strategy variability within and across individuals was striking across all of the sessions and underscores Siegler's (1996) assertion that development is in a constant transitional state.