Learning new problem-solving strategies leads to changes in problem representation

Children sometimes solve problems incorrectly because they fail to represent key features of the problems. One potential source of improvements in children's problem representations is learning new problem-solving strategies. Ninety-one 3rd- and 4th-grade students solved mathematical equivalence problems (e.g., 3 + 4 + 6 = 3 + __) and completed a representation assessment in which they briefly viewed similar problems and either reconstructed each problem or identified it in a set of alternatives. Experimental groups then received a lesson about one or both of two solution strategies, the equalize strategy and the add–subtract strategy. A control group received no instruction. All children completed posttest assessments of representation and problem solving. Children taught the equalize strategy improved their problem representations more than those not taught it. This pattern did not hold for the add–subtract strategy. These results indicate that learning new strategies is one source of changes in problem representation. However, some strategies are more effective than others at promoting accurate problem representation.     

The cognitive profile of Chinese children with mathematics difficulties

This study examined how four domain-specific skills (arithmetic procedural skills, number fact retrieval, place value concept, and number sense) and two domain-general processing skills (working memory and processing speed) may account for Chinese children’s mathematics learning difficulties. Children with mathematics difficulties (MD) of two age groups (7-8 and 9-11 years) were compared with age-matched typically achieving children. For both age groups, children with MD performed significantly worse than their age-matched controls on all of the domain-specific and domain-general measures. Further analyses revealed that the MD children with literacy difficulties (MD/RD group) performed the worst on all of the measures, whereas the MD-only group was significantly outperformed by the controls on the four domain-specific measures and verbal working memory. Stepwise discriminant analyses showed that both number fact retrieval and place value concept were significant factors differentiating the MD and non-MD children. To conclude, deficits in domain-specific skills, especially those of number fact retrieval and place value understanding, characterize the profile of Chinese children with MD.     

First-grade predictors of mathematical learning disability: A latent class trajectory analysis

Kindergarten to third grade mathematics achievement scores from a prospective study of mathematical development (n = 306) were subjected to latent growth trajectory analyses. The four corresponding classes included children with mathematical learning disability (MLD, 6% of sample), and low (LA, 50%), typically (TA, 39%) and high (HA, 5%) achieving children. The groups were administered a battery of intelligence (IQ), working memory, and mathematical-cognition measures in first grade. The children with MLD had general deficits in working memory and IQ and potentially more specific deficits on measures of number sense. The LA children did not have working memory or IQ deficits but showed moderate deficits on these number sense measures and for addition fact retrieval. The distinguishing features of the HA children were a strong visuospatial working memory, a strong number sense, and frequent use of memory-based processes to solve addition problems. Implications for the early identification of children at risk for poor mathematics achievement are considered.     

Strategic development in exact calculation: group and individual differences in four achievement subtypes

This longitudinal study sought to identify developmental changes in strategy use between 5 and 7 years of age when solving exact calculation problems. Four mathematics and reading achievement subtypes were examined at four time points. Five strategies were considered: finger counting, verbal counting, delayed retrieval, automatic retrieval, and derived fact retrieval. Results provided unique insights into children's strategic development in exact calculation at this early stage. Group analysis revealed relationships between mathematical and/or reading difficulties and strategy choice, shift, and adaptiveness. Use of derived fact retrieval by 7 years of age distinguished children with mathematical difficulties from other achievement subtypes. Analysis of individual differences revealed marked heterogeneity within all subtypes, suggesting (inter alia) no marked qualitative distinction between our two mathematical difficulty subtypes.     

In pursuit of knowledge: comparing self-explanations, concepts, and procedures as pedagogical tools

Explaining new ideas to oneself can promote learning and transfer, but questions remain about how to maximize the pedagogical value of self-explanations. This study investigated how type of instruction affected self-explanation quality and subsequent learning outcomes for second- through fifth-grade children learning to solve mathematical equivalence problems (e.g., 7 + 3 + 9 = 7 + _). Experiment 1 varied whether instruction was conceptual or procedural in nature (n = 40), and Experiment 2 varied whether children were prompted to self-explain after conceptual instruction (n = 48). Conceptual instruction led to higher quality explanations, greater conceptual knowledge, and similar procedural knowledge compared with procedural instruction. No effect was found for self-explanation prompts. Conceptual instruction can be more efficient than procedural instruction and may make self-explanation prompts unnecessary.     

Algorithmic solution of arithmetic problems and operands-answer associations in long-term memory

Many developmental models of arithmetic problem solving assume that any algorithmic solution of a given problem results in an association of the two operands and the answer in memory (Logan & Klapp, 1991; Siegler, 1996). In this experiment, adults had to perform either an operation or a comparison on the same pairs of two-digit numbers and then a recognition task. It is shown that unlike comparisons, the algorithmic solution of operations impairs the recognition of operands in adults. Thus, the postulate of a necessary and automatic storage of operands-answer associations in memory when young children solve additions by algorithmic strategies needs to be qualified.     

Math in actions: Actor mode reveals the true arithmetic abilities of French-speaking 2-year-olds in a magic task

Our previous studies provide some evidence of between-language effects on arithmetic performance in 2-year-olds. French-speaking children were especially biased by the use of the word un as a cardinal value and as an article in the singular/plural opposition (1 vs. the set 2, 3, …). Here we evaluated the ability of a new action-based assessment method to avoid this bias. A total of 80 French-speaking 2- and 3-year-olds were confronted with impossible (1 + 1 = 1 or 1 + 1 = 3) and possible (1 + 1 = 2) addition problems that triggered the bias. The problems were either presented to the children by the experimenter (onlooker mode) or realized by themselves (actor mode). The 2-year-olds performed better in the actor mode than in the onlooker mode. A subtraction control with no language ambiguity (2-1 = 2 or 1) was conducted with 80 other children; both modes elicited comparable performances regardless of age. These data indicate that the actor mode is effective for assessing arithmetic ability in French-speaking 2-year-olds.     

When you don't have to be exact: Investigating computational estimation skills with a comparison task

The present study is the first systematic investigation of computational estimation skills of multi-digit multiplication problems using an estimation comparison task. In two experiments, participants judged whether an estimated answer to a multi-digit multiplication problem was larger or smaller than a given reference number. Performance was superior in terms of speed and accuracy for smaller problem sizes, for trials in which the reference numbers were smaller vs. larger than the exact answers (consistent with the size effect) and for trials in which the reference numbers were numerically far compared to close to the exact answers (consistent with the distance effect). Strategy analysis showed that two main strategies were used to solve this task—approximate calculation and sense of magnitude. Most participants reported using the two strategies. Strategy choice was influenced by the distance between the reference number and exact answer, and by the interaction of problem size and reference number size. Theoretical implications as to the nature of numerical representations in the ANS (approximate number system) and to the estimation processes are suggested.     

A test of two methods of arithmetic fluency training and implications for educational practice

Children are exposed to multiple training tasks that are intended to support acquisition of basic arithmetic skills. Surprisingly, there is a scarcity of experimental research that directly compares the efficacy of those tasks, raising the possibility that children may be spending critical instructional time on tasks that are not effective. We conducted an experiment with 1st through 6th grade children comparing two arithmetic training tasks that are widely used: answer production training and fact triangle training. Results show that answer production training produces substantial fluency gains, whereas fact triangle training does not. Further, we show that, despite theoretical considerations that suggest otherwise, fact triangle training does not produce more flexibly applicable learning. Implications for memory representation, arithmetic fluency training, and broader educational strategy are discussed.     

Cognitive representations and processes in arithmetic: inferences from the performance of brain-damaged subjects.

In this article, we present data from two brain-damaged patients with calculation impairments in support of claims about the cognitive mechanisms underlying simple arithmetic performance. We first present a model of the functional architecture of the cognitive calculation system based on previous research. We then elaborate this architecture through detailed examination of the patterns of spared and impaired performance of the two patients. From the patients' performance we make the following theoretical claims: that some arithmetic facts are stored in the form of individual fact representations (e.g., 9 x 4 = 36), whereas other facts are stored in the form of a general rule (e.g., 0 x N = 0); that arithmetic fact retrieval is mediated by abstract internal representations that are independent of the form in which problems are presented or responses are given; that arithmetic facts and calculation procedures are functionally independent; and that calculation algorithms may include special-case procedures that function to increase the speed or efficiency of problem solving. We conclude with a discussion of several more general issues relevant to the reported research.     

Shy children’s coping with a social conflict: The role of personality self-theories

Shy children’s risk for psychosocial difficulties may result partly from ineffective coping with social stressors. Little is known about which shy children are most susceptible to maladaptive coping styles. Personality self-theories may be one source of individual differences in shy children’s coping with social stressors. The purpose of this study was to examine whether the links between shyness and coping were moderated by personality self-theories. Participants were 175 children (M_age = 10.11 years), who completed self-report assessments of shyness, personality self-theories, and coping strategies. Self-theories moderated links between shyness and coping, sometimes differentially for boys and girls. For example, shyness was most strongly related to internalizing coping among entity-oriented children. However, shyness was most strongly negatively related to approach coping (social support, problem-solving) among incrementally-oriented boys and entity-oriented girls.     

MENTAL ARITHMETIC IN AFRICA AND AMERICA: STRATEGIES,PRINCIPLES, AND EXPLANATIONS*

Mental arithmetic abilities were studied among unschooled African adults and American college students. A set of problems tested the use of the four basic arithmetic operations. Performance was analyzed for: strategies, implicit arithmetic principles, and explicit explanations of the principles. Both groups showed accurate mental arithmetic strategies related to the base ten structure of their native counting systems. The American students' mental strategies additionally made use of algorithms based on written place values. Several principles were implicit in the calculation strategies of both groups, and both used these principles to short-cut calculations. Though the African subjects did not, the American subjects did describe these abstract principles in explicit common language or algebraic expressions. Relationships between schooling and cognition are discussed.     

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.

     

The development of category learning strategies: What makes the difference?

Category learning can be achieved by identifying common features among category members, distinctive features among non-members, or both. These processes are psychologically and computationally distinct, and may have implications for the acquisition of categories at different hierarchical levels. The present study examines an account of children’s difficulty in acquiring categories at the subordinate level grounded on these distinct comparison processes. Adults and children performed category learning tasks in which they were exposed either to pairs of objects from the same novel category or pairs of objects from different categories. The objects were designed so that for each category learning task, two features determined category membership whereas two other features were task irrelevant. In the learning stage participants compared pairs of objects noted to be either from the same category or from different categories. Object pairs were chosen so that the objective amount of information provided to the participants was identical in the two learning conditions. We found that when presented only with object pairs noted to be from the same category, young children (6 ? YO ? 9.5) learned the novel categories just as well as older children (10 ? YO ? 14) and adults. However, when presented only with object pairs known to be from different categories, unlike older children and adults, young children failed to learn the novel categories. We discuss cognitive and computational factors that may give rise to this comparison bias, as well as its expected outcomes.     

Effects of strategy sequences and response–stimulus intervals on children’s strategy selection and strategy execution: A study in computational estimation

The present study investigates how children’s better strategy selection and strategy execution on a given problem are influenced by which strategy was used on the immediately preceding problem and by the duration between their answer to the previous problem and current problem display. These goals are pursued in the context of an arithmetic problem solving task. Third and fifth graders were asked to select the better strategy to find estimates to two-digit addition problems like 36 + 78. On each problem, children could choose rounding-down (i.e., rounding both operands down to the closest smaller decades, like doing 40 + 60 to solve 42 + 67) or rounding-up strategies (i.e., rounding both operands up to the closest larger decades, like doing 50 + 70 to solve 42 + 67). Children were tested under a short RSI condition (i.e., the next problem was displayed 900 ms after participants’ answer) or under a long RSI condition (i.e., the next problem was displayed 1,900 ms after participants’ answer). Results showed that both strategy selection (e.g., children selected the better strategy more often under long RSI condition and after selecting the poorer strategy on the immediately preceding problem) and strategy execution (e.g., children executed strategy more efficiently under long RSI condition and were slower when switching strategy over two consecutive problems) were influenced by RSI and which strategy was used on the immediately preceding problem. Moreover, data showed age-related changes in effects of RSI and strategy sequence on mean percent better strategy selection and on strategy performance. The present findings have important theoretical and empirical implications for our understanding of general and specific processes involved in strategy selection, strategy execution, and strategic development.     

Fostering Taiwanese preschoolers’ understanding of the addition–subtraction inverse principle

The present research involved gauging preschoolers’ learning potential for a key arithmetic concept, the addition–subtraction inverse principle (e.g., 2 + 1 − 1 = 2). Sixty 4- and 5-year-old Taiwanese children from two public preschools serving low- and middle-income families participated in the training experiment. Half were randomly assigned to an experimental group; half, to a control condition. Participants were tested for an understanding of inversion before and after intervention. One-third of the 5 year olds from both groups performed at the marginally or reliably successful levels before the intervention, and three quarters of them did so in the posttest. Only one of the 4 year olds was marginally successful before the intervention and 4 year olds in the experimental group somewhat benefited from the intervention. Significant social class effect were evident.     

Applied Problem Solving in Children with ADHD: The Mediating Roles of Working Memory and Mathematical Calculation

The difficulties children with ADHD experience solving applied math problems are well documented; however, the independent and/or interactive contributions of cognitive processes underlying these difficulties are not fully understood and warrant scrutiny. The current study examines two primary cognitive processes integral to children’s ability to solve applied math problems: working memory (WM) and math calculation skills (i.e., the ability to utilize specific facts, skills, or processes related to basic math operations stored in long-term memory). Thirty-six boys with ADHD-combined presentation and 33 typically developing (TD) boys aged 8–12 years old were administered multiple counterbalanced tasks to assess upper (central executive [CE]) and lower level (phonological [PH STM] and visuospatial [VS STM] short-term memory) WM processes, and standardized measures of mathematical abilities. Bias-corrected, bootstrapped mediation analyses revealed that CE ability fully mediated between-group differences in applied problem solving whereas math calculation ability partially mediated the relation. Neither PH STM nor VS STM was a significant mediator. When modeled together via serial mediation analysis, CE in tandem with math calculation ability fully mediated the relation, explained 79% of the variance, and provided a more parsimonious explication of applied mathematical problem solving differences among children with ADHD. Results suggest that interventions designed to address applied math difficulties in children with ADHD will likely benefit from targeting basic knowledge of math facts and skills while simultaneously promoting the active interplay of these skills with CE processes.     

You’ll see what you mean: Students encode equations based on their knowledge of arithmetic

This study investigated the roles of problem structure and strategy use in problem encoding. Fourth-grade students solved and explained a set of typical addition problems (e.g., ) and mathematical equivalence problems (e.g., or ). Next, they completed an encoding task in which they reconstructed addition and equivalence problems after viewing each for 5 s. Equivalence problems of the form overlap with a perceptual pattern found in traditional arithmetic problems (i.e., answer blank in final position), and students’ encoding was poorest on problems of this type. Individual differences in encoding the equivalence problems were related to variations in strategy use. Some students solved blank-final equivalence problems using the standard arithmetic strategy of performing all given operations on all given numbers. These students made more errors in encoding problem structure, but fewer errors in encoding the numbers, than did students who solved the problems using correct or other incorrect strategies. Moreover, students who expressed many strategies for solving the blank-final equivalence problems made fewer errors in encoding problem structure, but more errors in encoding the numbers, than did students who expressed only a single strategy. Results highlight that encoding is intended to guide action and that prior experience can simultaneously facilitate and interfere with accurate encoding.     

Preschoolers doing arithmetic: the concepts are willing but the working memory is weak.

The study of early mathematical development provides important insights into young children's emerging academic competencies and, potentially, a basis for adapting instructional methods. We presented nonverbal forms of two- and three-term arithmetic problems to 4-year-olds to determine (a) the extent to which certain information-processing demands make some problems more difficult than others and (b) whether preschoolers use arithmetic concepts spontaneously when solving novel problems. Children's accuracy on simple arithmetic problems (a + b and a - b) was strongly related (r2 = .88) to representational set size, the maximum number of units that need to be held in working memory to solve a given problem. Some children also showed spontaneous use of procedures based on the arithmetic principle of inversion when solving problems of the form a + b - b. These results highlight the importance of identifying information-processing and conceptual characteristics in the early development of mathematical cognition.     

Computer-assisted intervention for children with low numeracy skills

We present results of a computer-assisted intervention (CAI) study on number skills in kindergarten children. Children with low numeracy skill (n = 30) were randomly allocated to two treatment groups. The first group played a computer game (The Number Race) which emphasized numerical comparison and was designed to train number sense, while the other group played a game (Graphogame-Math) which emphasized small sets of exact numerosities by training matching of verbal labels to visual patterns and number symbols. Both groups participated in a daily intervention session for three weeks. Children's performance in verbal counting, number comparison, object counting, arithmetic, and a control task (rapid serial naming) were measured before and after the intervention. Both interventions improved children's skills in number comparison, compared to a group of typically performing children (n = 30), but not in other areas of number skills. These findings, together with a review of earlier computer-assisted intervention studies, provide guidance for future work on CAI aiming to boost numeracy development of low performing children.