Children's Multiplication Research Articles

Gray matter volume in left intraparietal sulcus predicts longitudinal gains in subtraction skill in elementary school

Although behavioral studies show large improvements in arithmetic skills in elementary school, we do not know how brain structure supports math gains in typically developing children. While some correlational studies have investigated the concurrent association between math performance and brain structure, such as gray matter volume (GMV), longitudinal studies are needed to infer if there is a causal relation. Although discrepancies in the literature on the relation between GMV and math performance have been attributed to the different demands on quantity vs. retrieval mechanisms, no study has experimentally tested this assumption. We defined regions of interests (ROIs) associated with quantity representations in the bilateral intraparietal sulcus (IPS) and associated with the storage of arithmetic facts in long-term memory in the left middle and superior temporal gyri (MTG/STG), and studied associations between GMV in these ROIs and children's performance on operations having greater demands on quantity vs. retrieval mechanisms, namely subtraction vs. multiplication. The aims of this study were threefold: First, to study concurrent associations between GMV and math performance, second, to investigate the role of GMV at the first time-point (T1) in predicting longitudinal gains in math skill to the second time-point (T2), and third, to study whether changes in GMV over time were associated with gains in math skill. Results showed no concurrent association between GMV in IPS and math performance, but a concurrent association between GMV in left MTG/STG and multiplication skill at T1. This association showed that the higher the GMV in this ROI, the higher the children's multiplication skill. Results also revealed that GMV in left IPS and left MTG/STG predicted longitudinal gains in subtraction skill only for younger children (approximately 10 years old). Whereas higher levels of GMV in left IPS at T1 predicted larger subtraction gains, higher levels of GMV in left MTG/STG predicted smaller gains. GMV in left MTG/STG did not predict longitudinal gains in multiplication skill. No significant association was found between changes in GMV over time and longitudinal gains in math. Our findings support the early importance of brain structure in the IPS for mathematical skills that rely on quantity mechanisms.

Capturing Children's Multiplication and Division Stories

Learn the value of having students create their own stories and pictures to represent number sentences as classroom assessments.

Children's multiplication and division shortcuts: Increasing shortcut use depends on how the shortcuts are evaluated

The goals of this study were to investigate whether children's use of two arithmetic shortcuts based on the understanding of the operations of multiplication and division could be increased and to examine how children's evaluations of the shortcuts interacted with the brief task aimed at promoting shortcut use. Grade 6, 7, and 8 students solved two sets of inversion (e.g., d×e÷e) and associativity (e.g., d×e÷f) problems. Children with a good understanding of the relation between multiplication and division can use conceptually-based shortcuts to solve both types of problems. Students were also given a brief task demonstrating the inversion shortcut (the answer is the first number and no calculations are required) and the associativity shortcut (dividing first and then multiplying) and were asked to compare each shortcut to a left-to-right procedure (i.e., multiplying and then dividing). Half the participants were given the demonstration task between the problem sets and the other half after the sets. Inversion and associativity shortcut use increased by Grade 8 and improved across problem sets for both the Demonstration Middle and Demonstration Last groups. The demonstration task successfully promoted subsequent shortcut use but the participants who positively evaluated the shortcuts compared to the left-to-right procedure in the demonstration task had greater subsequent shortcut use. Conceptually-based shortcut use on multiplication and division problems was increased using a brief demonstration task but children's subsequent use of shortcuts depended on their evaluations of the shortcuts.

The Contribution of Numerical Magnitude Comparison and Phonological Processing to Individual Differences in Fourth Graders’ Multiplication Fact Ability

Although numerical magnitude processing has been related to individual differences in arithmetic, its role in children’s multiplication performance remains largely unknown. On the other hand, studies have indicated that phonological awareness is an important correlate of individual differences in children’s multiplication performance, but the involvement of phonological memory, another important phonological processing skill, has not been studied in much detail. Furthermore, knowledge about the relative contribution of above mentioned processes to the specific arithmetic operation of multiplication in children is lacking. The present study therefore investigated for the first time the unique contributions of numerical magnitude comparison and phonological processing in explaining individual differences in 63 fourth graders’ multiplication fact ability (mean age = 9.6 years, SD = .67). The results showed that children’s multiplication fact competency correlated significantly with symbolic and nonsymbolic magnitude comparison as well as with phonological short-term memory. A hierarchical regression analysis revealed that, after controlling for intellectual ability and general reaction time, both symbolic and nonsymbolic magnitude comparison and phonological short-term memory accounted for unique variance in multiplication fact performance. The ability to compare symbolic magnitudes was found to contribute the most, indicating that the access to numerical magnitudes by means of Arabic digits is a key factor in explaining individual differences in children’s multiplication fact ability.

Developmental change and individual differences in children's multiplication.

Age-related change and patterns of individual differences in children's knowledge and skill in multiplication were investigated for students in Grades 4 and 6 (approximately ages 9 and 11, respectively) by examining multiple measures of computational skill, conceptual knowledge, and working memory. Regression analyses revealed that indexes reflecting probability of retrieval and special problem characteristics overshadow other, more general indexes (problem size and frequency of presentation) in predicting solution latencies. Some improvement in the use of conceptual knowledge was evident between Grades 4 and 6, but this change was neither strong nor uniform across tasks. Finally, patterns of individual differences across tasks differed as a function of grade level. The findings have implications for understanding developmental change and individual differences in mathematical cognition.

There is more than one way to solve a problem: Evaluating a learning environment that supports the development of children's multiplication skills

Interpretation of the nature of mathematical understanding has changed recently. These changes have prompted calls for different instructional methods in the primary classroom. COPPERS is a mathematical learning environment which explores how such goals should be implemented computationally. Two experiments have examined how system components have advanced children's understanding that multiplication problems can have many different correct solutions. Different numbers of decompositions, learner's choice of strategy and feedback in either tabular or place value representations were all found to significantly affect learning. Theoretical interpretations of these results are considered in terms of Vygotskian approaches to scaffolding learning and current research on external representations.

Research into Practice: Children's Multiplication

The purpose of this article is to discuss how children give meaning to situations which, from an adult's perspective, could be solved by multiplying or dividing (Steffe and von Glasersfeld 1985). We have found in our research that a child's understanding of these situations may be significantly different not only from those of the teacher but also from those of other children in the same classroom. Thus our teaching decisions must be based on knowledge of the children's understanding of mathematics.

Strategy choice procedures and the development of multiplication skill.

Many intelligent strategy choices may be accomplished through relatively low-level cognitive processes. This article describes a detailed model of how such "mindless" processes might lead to intelligent choices of strategies in one common situation: that in which people need to choose between stating a retrieved answer and using a backup strategy. Several experiments testing the model's applicability to children's single-digit multiplication are reported. These include tests of predictions about when different strategies are used and how early experience shapes later performance. Then, the sufficiency of the model to generate both performance at any one time and changes in performance over time is tested through the medium of a running computer simulation of children's multiplication. The simulation acquires a considerable amount of multiplication knowledge, and its learning and performance parallel those of children in a number of ways. Finally, several implications of the model for understanding cognitive self-regulation and cognitive development are discussed.

Strategy choice procedures and the development of multiplication skill.

Many intelligent strategy choices may be accomplished through relatively low-level cognitive processes. This article describes a detailed model of how such "mindless" processes might lead to intelligent choices of strategies in one common situation: that in which people need to choose between stating a retrieved answer and using a backup strategy. Several experiments testing the model's applicability to children's single-digit multiplication are reported. These include tests of predictions about when different strategies are used and how early experience shapes later performance. Then, the sufficiency of the model to generate both performance at any one time and changes in performance over time is tested through the medium of a running computer simulation of children's multiplication. The simulation acquires a considerable amount of multiplication knowledge, and its learning and performance parallel those of children in a number of ways. Finally, several implications of the model for understanding cognitive self-regulation and cognitive development are discussed.

Teaching LD Children Multiplication Tables

A special technique for remediating difficulty in simple multiplication

An expert system for diagnosing children's multiplication errors

We describe a microcomputer-based system for diagnosing children's multiplication errors. The system incorporates the knowledge base of all known systematic errors which children make. In order to cope with the complexity of designing this computer-assisted learning package, we use a modular approach. Chaining the modules allows us to fit the overall system on microcomputers with limited memory capacity available at schools.