Do multiplication and division strategies rely on executive and phonological working memory resources?

Rational number representation by the approximate number system

The approximate number system (ANS) enables organisms to represent the approximate number of items in an observed collection, quickly and independently of natural language. Recently, it has been proposed that the ANS goes beyond representing natural numbers by extracting and representing rational numbers (Clarke & Beck, 2021a). Prior work demonstrates that adults and children discriminate ratios in an approximate and ratio-dependent manner, consistent with the hallmarks of the ANS. Here, we use a well-known "connectedness illusion" to provide evidence that these ratio-dependent ratio discriminations are (a) based on the perceived number of items in seen displays (and not just non-numerical confounds), (b) are not dependent on verbal working memory, or explicit counting routines, and (c) involve representations with a part-whole (or subset-superset) format, like a fraction, rather than a part-part format, like a ratio. These results vindicate key predictions of the hypothesis that the ANS represents rational numbers.

Strategy variability in computational estimation and its association with mathematical achievement

Computational estimation requires a breadth of strategies and selection of the relevant strategy given a problem's features. We used the new Test of Estimation Strategies (TES), composed of 20 arithmetic problems (e.g., 144 x 0.38), to investigate variability in strategy use in young adults. The TES targets the five estimation strategies that adults use most frequently, which fall into two Classes. The three Class One strategies are general-purpose and taught in schools. Proceed Algorithmically entails applying an algorithm (e.g., shifting a decimal place). Round One and Round Two are defined as rounding one or both operands, respectively. The two Class Two strategies are more advanced, requiring application of conceptual knowledge of mathematics. Known-and-Nice is used when a participant relies on a well-known mathematical fact (e.g., 25 × 4 = 100) to form an estimate. Fractions uses a fraction or percentage in the estimation process (e.g., 943 x 0.48 is about 50% or half of 900). We divided our sample of adult participants into two groups (i.e., high, average) based on their estimation performance on the TES. The high-performance group used a broader range of strategies and more frequently applied the most relevant strategy given a problem's features. Overall estimation accuracy was correlated with mathematical achievement, as were strategy breadth and strategy relevance. However, none of these associations survived first controlling for verbal achievement. Participants' strategy reports suggested that the TES problems were generally successful in eliciting the five target strategies and provided evidence for a new strategy, Partitioning. These findings provide a basis for future instructional studies to improve students' computational estimation.

Estimation Strategy Selection Is Modulated by Snapshot Emotional Priming, but Not Math Anxiety

The present study explored the role of snapshot emotional priming and math anxiety in estimation strategy selection. Participants were asked to complete a two-digit multiplication estimation task (e.g., 34 × 67) under explicit (Experiment 1) and implicit (Experiment 2) snapshot emotional priming conditions by freely choosing to use DU (down-up, e.g., doing 30 × 70 = 2100 for 34 × 67) or UD (up-down, e.g., doing 40 × 60 = 2400 for 34 × 67) strategies to arrive as close as possible to the correct answer. In Experiment 1, individuals' estimation performance was positively influenced by explicit happy priming (shorter RT (reaction time)), while not affected by explicit fear priming. In Experiment 2, individuals' estimation ACC (accuracy) when using the UD strategy was negatively affected by both implicit happy and fear priming, but their RT when using DU and UD strategies was positively impacted by implicit happy priming. In both experiments, the correlations between math anxiety and estimation performance (ACC, RT, and strategy selection adaptivity) was not significant. The present study suggests that fear priming was not always detrimental to individuals' estimation performance, and happy priming did not always universally improve individuals' estimation performance. Additionally, estimation strategy selection was not influenced by math anxiety.

A meta-analysis of the relation between math anxiety and math achievement

Meta-analyses from the 1990s previously have established a significant, small-to-moderate, and negative correlation between math achievement and math anxiety. Since these publications, research has continued to investigate this relation with more diverse samples and measures. Thus, the goal of the present meta-analysis was to provide an update of the math anxiety-math achievement relation and its moderators. Analyzing 747 effect sizes accumulated from research conducted between 1992 and 2018, we found a small-to-moderate, negative, and statistically significant correlation (r = -.28) between math anxiety and math achievement. The relation was significant for all moderator subgroups, with the exception of the relation between math anxiety and assessments measuring the approximate number system. Grade level, math ability level, adolescent/adult math anxiety scales, math topic of anxiety scale, and math assessments were significant moderators of this relation. There is also a tendency for published studies to report significantly stronger correlations than unpublished studies, but overall, large, negative effect sizes are underreported. Our results are consistent with previous findings of a significant relation between math anxiety and math achievement. This association starts in childhood, remains significant through adulthood, is smaller for students in Grades 3 through 5 and postsecondary school, is larger for math anxiety than for statistics anxiety and for certain math anxiety scales, and is smaller for math exam grades and samples selected for low math ability. This work supports future research efforts to determine effective math achievement and math anxiety interventions, which may be most helpful to implement during childhood. (PsycInfo Database Record (c) 2021 APA, all rights reserved).

Individual Differences in Math Ability Determine Neurocognitive Processing of Arithmetic Complexity: A Combined fNIRS-EEG Study

Some individuals experience more difficulties with math than others, in particular when arithmetic problems get more complex. Math ability, on one hand, and arithmetic complexity, on the other hand, seem to partly share neural underpinnings. This study addresses the question of whether this leads to an interaction of math ability and arithmetic complexity for multiplication and division on behavioral and neural levels. Previously screened individuals with high and low math ability solved multiplication and division problems in a written production paradigm while brain activation was assessed by combined functional near-infrared spectroscopy (fNIRS) and electroencephalography (EEG). Arithmetic complexity was manipulated by using single-digit operands for simple multiplication problems and operands between 2 and 19 for complex multiplication problems and the corresponding division problems. On the behavioral level, individuals with low math ability needed more time for calculation, especially for complex arithmetic. On the neural level, fNIRS results revealed that these individuals showed less activation in the left supramarginal gyrus (SMG), superior temporal gyrus (STG) and inferior frontal gyrus (IFG) than individuals with high math ability when solving complex compared to simple arithmetic. This reflects the greater use of arithmetic fact retrieval and also the more efficient processing of arithmetic complexity by individuals with high math ability. Oscillatory EEG analysis generally revealed theta and alpha desynchronization with increasing arithmetic complexity but showed no interaction with math ability. Because of the discovered interaction for behavior and brain activation, we conclude that the consideration of individual differences is essential when investigating the neurocognitive processing of arithmetic.

Understanding Strategy Change: Contextual, Individual, and Metacognitive Factors

Learning, development, and response to instruction often involve changes in the strategies that learners use to solve problems. In this chapter, our focus is on mathematical problem solving in both children and adults. We offer a selective review of research on three classes of factors that may influence processes of strategy change in mathematical problem solving: contextual factors, individual factors, and metacognitive factors. Contextual factors involve information that learners encounter in the learning context, such as feedback about prior strategies and examples of alternative strategies. Individual factors involve the abilities, dispositions, and knowledge that learners bring to the learning context. Metacognitive factors involve knowledge about strategies and factors that affect the application of strategies-including perceptions of problem difficulty, confidence in the strategies one already knows, and judgments about the qualities of alternative strategies. These factors operate both independently and in combination to influence learners' behavior. Therefore, we argue that scientific progress in understanding strategy change will require comprehensive conceptual models that specify how different factors come together to explain behavior. We discuss several such models, including vulnerability-trigger models, cumulative risk models, and dynamic systems models. Research guided by such models will contribute to greater progress in understanding processes of strategy use and strategy change.

Children's Strategy Choices on Complex Subtraction Problems: Individual Differences and Developmental Changes

We examined how children's strategy choices in solving complex subtraction problems are related to grade and to variations in problem complexity. In two studies, third- and fifth-grade children (N≈160 each study) solved multi-digit subtraction problems (e.g., 34–18) and described their solution strategies. In the first experiment, strategy selection was investigated by means of a free-choice paradigm, whereas in the second study a discrete-choice approach was implemented. In both experiments, analyses of strategy repertoire indicated that third-grade children were more likely to report less-efficient strategies (i.e., counting) and relied more on the right-to-left solution algorithm compared to fifth-grade children who more often used efficient memory-based retrieval and conceptually-based left-to-right (i.e., decomposition) strategies. Nevertheless, all strategies were reported or selected by both older and younger children and strategy use varied with problem complexity and presentation format for both age groups. These results supported the overlapping waves model of strategy development and provide detailed information about patterns of strategy choice on complex subtraction problems.

Non-adaptive strategy selection in adults with high mathematical anxiety

Participants with mathematical anxiety (MA) tend to show particular difficulty in mathematical operations with high working memory (WM) demands compared to operations with lower WM demands. Accordingly, we examined strategy selection to test the cognitive mechanism underlying the observed weakness of high MA participants in mathematical operations with high WM demands. We compared two groups of college students with high or low MA, in the solution of simple non-carry addition problems (e.g., 54 + 63) and complex carryover addition problems (e.g., 59 + 63). The results indicated that high MA participants showed particular difficulty in the harder carry condition. Testing the strategy selection mechanism among high MA participants, we found in the carry condition 1) they used the common strategy less often compared to low MA participants and 2) employed unusual strategies more often compared to low MA participants. Therefore, high MA participants were less efficient in their strategy selection, which may be due to weaker spatial representations, numerical difficulties, or less experience solving complex problems. These primitive representations are not adaptive, and can negatively impact performance in math tasks with high WM demands.

When is working memory important for arithmetic? The impact of strategy and age

Our ability to perform arithmetic relies heavily on working memory, the manipulation and maintenance of information in mind. Previous research has found that in adults, procedural strategies, particularly counting, rely on working memory to a greater extent than retrieval strategies. During childhood there are changes in the types of strategies employed, as well as an increase in the accuracy and efficiency of strategy execution. As such it seems likely that the role of working memory in arithmetic may also change, however children and adults have never been directly compared. This study used traditional dual-task methodology, with the addition of a control load condition, to investigate the extent to which working memory requirements for different arithmetic strategies change with age between 9–11 years, 12–14 years and young adulthood. We showed that both children and adults employ working memory when solving arithmetic problems, no matter what strategy they choose. This study highlights the importance of considering working memory in understanding the difficulties that some children and adults have with mathematics, as well as the need to include working memory in theoretical models of mathematical cognition.

What Is Specific and What Is Shared Between Numbers and Words?

Reading and spelling performance have a significant correlation with number transcoding, which is the ability to establish a relationship between the verbal and Arabic representations of numbers, when a conversion of numerical symbols from one notation to the other is necessary. The aim of the present study is to reveal shared and non-shared mechanisms involved in reading and writing of words and Arabic numerals in Brazilian school-aged children. One hundred and seventy-two children from second to fourth grades were evaluated. All of them had normal intelligence. We conducted a series of hierarchical regression models using scores on word spelling and reading single words and Arabic numerals, as dependent variables. As predictor variables we investigated intelligence, the phonological and visuospatial components of working memory (WM) and phonemic awareness. All of the writing and reading tasks (single word spelling and reading as well as number reading and number writing) were significantly correlated to each other. In the regression models, phonological WM was specifically associated to word reading. Phonemic awareness was the only cognitive variable that systematically predicted all of the school skills investigated, both numerical and word tasks. This suggests that phonemic awareness is a modular cognitive ability shared by several school tasks and might be an important factor associated to the comorbidity between dyslexia and dyscalculia.

Working memory, worry, and algebraic ability

Math anxiety (MA)-working memory (WM) relationships have typically been examined in the context of arithmetic problem solving, and little research has examined the relationship in other math domains (e.g., algebra). Moreover, researchers have tended to examine MA/worry separate from math problem solving activities and have used general WM tasks rather than domain-relevant WM measures. Furthermore, it seems to have been assumed that MA affects all areas of math. It is possible, however, that MA is restricted to particular math domains. To examine these issues, the current research assessed claims about the impact on algebraic problem solving of differences in WM and algebraic worry. A sample of 80 14-year-old female students completed algebraic worry, algebraic WM, algebraic problem solving, nonverbal IQ, and general math ability tasks. Latent profile analysis of worry and WM measures identified four performance profiles (subgroups) that differed in worry level and WM capacity. Consistent with expectations, subgroup membership was associated with algebraic problem solving performance: high WM/low worry>moderate WM/low worry=moderate WM/high worry>low WM/high worry. Findings are discussed in terms of the conceptual relationship between emotion and cognition in mathematics and implications for the MA-WM-performance relationship.

Error Adaptation in Mental Arithmetic

Until now, error and conflict adaptation have been studied extensively using simple laboratory tasks. A common finding is that responses slow down after errors. According to the conflict monitoring theory, performance should also improve after an error. However, this is usually not observed. In this study, we investigated whether the characteristics of the experimental paradigms normally used could explain this absence. More precisely, these paradigms have in common that behavioural adaptation has little room to be expressed. We therefore studied error and conflict adaptation effects in a task that encounters the richness of everyday life's behavioural adaptation—namely, mental arithmetic, where multiple solution strategies are available. In accordance with our hypothesis, we observed that posterror accuracy increases after errors in mental arithmetic. No support for conflict adaptation in mental arithmetic was found. Implications for current theories of conflict and error monitoring are discussed.

Passive hand movements disrupt adults' counting strategies

In the present study, we experimentally tested the role of hand motor circuits in simple-arithmetic strategies. Educated adults solved simple additions (e.g., 8 + 3) or simple subtractions (e.g., 11 − 3) while they were required to retrieve the answer from long-term memory (e.g., knowing that 8 + 3 = 11), to transform the problem by making an intermediate step (e.g., 8 + 3 = 8 + 2 + 1 = 10 + 1 = 11) or to count one-by-one (e.g., 8 + 3 = 8…9…10…11). During the process of solving the arithmetic problems, the experimenter did or did not move the participants’ hand on a four-point matrix. The results show that passive hand movements disrupted the counting strategy while leaving the other strategies unaffected. This pattern of results is in agreement with a procedural account, showing that the involvement of hand motor circuits in adults’ mathematical abilities is reminiscent of finger counting during childhood.

The representation of multiplication and division facts in memory

Recently, using a training paradigm, Campbell and Agnew (2009) observed cross-operation response time savings with nonidentical elements (e.g., practice 3 + 2, test 5 - 2) for addition and subtraction, showing that a single memory representation underlies addition and subtraction performance. Evidence for cross-operation savings between multiplication and division have been described frequently (e.g., Campbell, Fuchs-Lacelle, & Phenix, 2006) but they have always been attributed to a mediation strategy (reformulating a division problem as a multiplication problem, e.g., Campbell et al., 2006). Campbell and Agnew (2009) therefore concluded that there exists a fundamental difference between addition and subtraction on the one hand and multiplication and division on the other hand. However, our results suggest that retrieval savings between inverse multiplication and division problems can be observed. Even for small problems (solved by direct retrieval) practicing a division problem facilitated the corresponding multiplication problem and vice versa. These findings indicate that shared memory representations underlie multiplication and division retrieval. Hence, memory and learning processes do not seem to differ fundamentally between addition-subtraction and multiplication-division.

Strengths and Weaknesses of the Choice/No-Choice Method in Research on Strategy Use

About 10 years ago, Siegler and Lemaire (1997 ) introduced the choice/no-choice method as a means of obtaining unbiased estimates of performance characteristics of cognitive strategies. They also illustrated the possibilities of this method and discussed its potential extensions for cognitive (developmental) research. The present article provides a critical discussion of the strengths and weaknesses of the choice/no-choice method in research on strategy choice and strategy change, starting from an overview of the studies that have been done with this new method. We end with a general evaluation and some issues for further research.

Effects of problem size, operation, and working-memory span on simple-arithmetic strategies: differences between children and adults?

Adult's simple-arithmetic strategy use depends on problem-related characteristics, such as problem size and operation, and on individual-difference variables, such as working-memory span. The current study investigates (a) whether the effects of problem size, operation, and working-memory span on children's simple-arithmetic strategy use are equal to those observed in adults, and (b) how these effects emerge and change across age. To this end, simple-arithmetic performance measures and a working-memory span measure were obtained from 8-year-old, 10-year-old, and 12-year-old children. Results showed that the problem-size effect in children results from the same strategic performance differences as in adults (i.e., size-related differences in strategy selection, retrieval efficiency, and procedural efficiency). Operation-related effects in children were equal to those observed in adults as well, with more frequent retrieval use on multiplication, more efficient strategy execution in addition, and more pronounced changes in multiplication. Finally, the advantage of having a large working-memory span was also present in children. The differences and similarities across children's and adult's strategic performance and the relevance of arithmetic models are discussed.