Activation of Real-World Knowledge in the Solution of Word Problems

This article describes two studies that examine factors influencing children's access to real-world knowledge during the solution of word problems. In the first study, based on work in Brazil by Carraher, Carraher, and Schliemann (1987), children were asked to solve arithmetic problems presented in three contexts: (a) as word problems, (b) in simulated store situations, and (c) as symbolic computations. Brazilian children were both more successful and more likely to use mental, informal strategies when solving word problems than when solving symbolic computations. We did not find the same results with our U.S. sample; no effects of context were found in either strategy use or success. Comparison of U.S. and Brazilian children's responses suggested that children may tend to access real-world content when the numbers in a word problem match the problem content, and a second study was conducted to test this interpretation. Children were presented with word problems in which the problem content either matched or did not match the numbers in the problem. It was found that when the numbers matched the problem content, children were more successful in solving the problems and more likely to access their domain knowledge during problem solution, as evidenced by the strategies they used to solve problems in the matched condition. These findings suggest ways in which activation of real-world knowledge might be facilitated during the solution of word problems in school.     

元认知监测与算术知识制约小学儿童心算策略运用能力的发展:一年纵向考察

为探究元认知监测与算术知识对儿童心算策略运用能力的影响如何随个体发展而变化，采用计算机任务与纸笔测量的方法，对85名小学三、五年级儿童进行了历时一年的纵向追踪研究。研究发现：（1）两组儿童的元认知监测和算术知识均呈增长趋势，算术知识的增长速度五年级显著快于三年级，且元认知监测增长速度与算术知识增长速度显著相关；（2）两组儿童中，元认知监测与算术知识增长速度更快的个体策略执行反应时与错误率的减少速度也更快；（3）五年级儿童的算术知识在元认知监测影响策略选择发展中起着完全中介作用。     

Conceptual Knowledge,Procedural Knowledge,and Metacognition in Routine and Nonroutine Problem Solving

When, how, and why students use conceptual knowledge during math problem solving is not well understood. We propose that when solving routine problems, students are more likely to recruit conceptual knowledge if their procedural knowledge is weak than if it is strong, and that in this context, metacognitive processes, specifically feelings of doubt, mediate interactions between procedural and conceptual knowledge. To test these hypotheses, in two studies (Ns = 64 and 138), university students solved fraction and decimal arithmetic problems while thinking aloud; verbal protocols and written work were coded for overt uses of conceptual knowledge and displays of doubt. Consistent with the hypotheses, use of conceptual knowledge during calculation was not significantly positively associated with accuracy, but was positively associated with displays of doubt, which were negatively associated with accuracy. In Study 1, participants also explained solutions to rational arithmetic problems; using conceptual knowledge in this context was positively correlated with calculation accuracy, but only among participants who did not use conceptual knowledge during calculation, suggesting that the correlation did not reflect “online” effects of using conceptual knowledge. In Study 2, participants also completed a nonroutine problem-solving task; displays of doubt on this task were positively associated with accuracy, suggesting that metacognitive processes play different roles when solving routine and nonroutine problems. We discuss implications of the results regarding interactions between procedural knowledge, conceptual knowledge, and metacognitive processes in math problem solving.     

Contextualization in Perspective

In this article, we examine the hypothesis that problem comprehension and computational processes interact during the solving of an arithmetic word problem. Two experiments tested subjects on a series of addition and subtraction word problems, the content of which varied on the basis of problem type and on the magnitude of the numbers involved (problem size). Performance data are presented and analyzed in terms of solution reaction times and error patterns. Results confirmed the main effects of problem size and problem type as factors determining reaction times but failed to show any significant interaction between the two factors. These results suggest that the cognitive processes involved in understanding an arithmetic word problem and in performing the required computations are best explained by a serial processing model. The absence of an interaction between problem comprehension and computational processes questions the notion that automatized retrieval facilitates problem solving and assertions suggesting that increasing computational requirements can interfere with problem-solving performance.     

Spatial and contextual factors in human performance on the travelling salesperson problem

The travelling salesperson problem (TSP) provides a realistic and practical example of a visuo-spatial problem-solving task. In previous research, we have found that the quality of solutions produced by human participants for small TSPs compares well with solutions from a range of computer algorithms. We have proposed that the ability of participants to find solutions reflects the natural properties of human perception, solutions being found through global perceptual processing of the problem array to extract a best figure from the TSP points. In this paper, we extend the study of human performance on the task in order to understand further how human abilities are utilised in solving real-world TSPs. The results of experiment 1 show that high levels of solution quality are maintained in solving larger TSPs than had been investigated previously with human participants, and that the presence of an implied real-world context in the problems has no effect upon performance. Experiment 2 demonstrated that the presence of regularity in the point layout of a TSP can facilitate performance. This was confirmed in experiment 3, where effects of the internality of point clusters were also found. All three experiments were consistent with a global, perceptually based approach to the problem by participants. We suggest that the role of perceptual processing in spatial problem-solving is an important area for further research in both theoretical and applied domains.     

Children’s strategies in complex arithmetic

Strategies used to solve two-digit addition problems (e.g., 27 + 48, Experiment 1) and two-digit subtraction problems (e.g., 73 – 59, Experiment 2) were investigated in adults and in children from Grades 3, 5, and 7. Participants were tested in choice and no-choice conditions. Results showed that (a) participants used the full decomposition strategy more often than the partial decomposition strategy to solve addition problems but used both strategies equally often to solve subtraction problems; (b) strategy use and execution were influenced by participants’ age, problem features, relative strategy performance, and whether the problems were displayed horizontally or vertically; and (c) age-related changes in complex arithmetic concern relative strategy use and execution as well as the relative influences of problem characteristics, strategy characteristics, and problem presentation on strategy choices and strategy performance. Implications of these findings for understanding age-related changes in strategic aspects of complex arithmetic performance are discussed.     

Individual differences in children's understanding of inversion and arithmetical skill

Background and aims. In order to develop arithmetic expertise, children must understand arithmetic principles, such as the inverse relationship between addition and subtraction, in addition to learning calculation skills. We report two experiments that investigate children's understanding of the principle of inversion and the relationship between their conceptual understanding and arithmetical skills. Sample. A group of 127 children from primary schools took part in the study. The children were from 2 age groups (6–7 and 8–9 years). Methods. Children's accuracy on inverse and control problems in a variety of presentation formats and in canonical and non‐canonical forms was measured. Tests of general arithmetic ability were also administered. Results. Children consistently performed better on inverse than control problems, which indicates that they could make use of the inverse principle. Presentation format affected performance: picture presentation allowed children to apply their conceptual understanding flexibly regardless of the problem type, while word problems restricted their ability to use their conceptual knowledge. Cluster analyses revealed three subgroups with different profiles of conceptual understanding and arithmetical skill. Children in the ‘high ability’ and ‘low ability’ groups showed conceptual understanding that was in‐line with their arithmetical skill, whilst a 3rd group of children had more advanced conceptual understanding than arithmetical skill. Conclusions. The three subgroups may represent different points along a single developmental path or distinct developmental paths. The discovery of the existence of the three groups has important consequences for education. It demonstrates the importance of considering the pattern of individual children's conceptual understanding and problem‐solving skills.     

Adults' strategies for simple addition and multiplication: verbal self-reports and the operand recognition paradigm

Accurate measurement of cognitive strategies is important in diverse areas of psychological research. Strategy self-reports are a common measure, but C. Thevenot, M. Fanget, and M. Fayol (2007) proposed a more objective method to distinguish different strategies in the context of mental arithmetic. In their operand recognition paradigm, speed of recognition memory for problem operands after solving a problem indexes strategy (e.g., direct memory retrieval vs. a procedural strategy). Here, in 2 experiments, operand recognition time was the same following simple addition or multiplication, but, consistent with a wide variety of previous research, strategy reports indicated much greater use of procedures (e.g., counting) for addition than multiplication. Operation, problem size (e.g., 2 + 3 vs. 8 + 9), and operand format (digits vs. words) had interactive effects on reported procedure use that were not reflected in recognition performance. Regression analyses suggested that recognition time was influenced at least as much by the relative difficulty of the preceding problem as by the strategy used. The findings indicate that the operand recognition paradigm is not a reliable substitute for strategy reports and highlight the potential impact of difficulty-related carryover effects in sequential cognitive tasks.     

小学生表征数学应用题策略的实验研究

通过一个2(成功与否)×2(提示与否)×2(题型)的混合实验设计,对小学五年级学生解决和差应用题的表征策略进行了研究.结果表明:(1)与比较应用题的表征相类似,小学生对和差应用题的表征也存在着直译策略和问题模型策略;(2)不成功组解题者在表征和差应用题时倾向于运用直译策略,而成功组的解题者更倾向于运用问题模型策略,这导致了成功者与不成功者在列式上的差异,特别是在不一致题型上表现得更明显;(3)在读题前给以“请注意理解这道题的意思”这样简单的提示,对不成功的解题者对和差问题的正确表征并不能起到作用;(4)成功的和差应用题解题者和不成功的解题者在列式正确性的自我评价上存在显著差异.     

Initial and concurrent planning in solutions to well-structured problems

Two experiments are reported, which consider planning behaviour in the context of a well-structured problem. One question in the problem-solving literature is to what extent planning a solution to a problem takes place before attempting that problem and whether this takes precedence over planning while solving a problem, hereafter referred to as “concurrent planning”. An additional question is whether the adoption of one mode of planning confers a performance advantage and under what circumstances one strategy is adopted in preference to others. The studies reported here set out to investigate the effects on performance of adopting different modes of planning and whether there is any relationship between the adoption of different strategic approaches and problem-solving performance. The results of these studies suggest that initial planning can enhance problem-solving performance, but only when problems remain relatively simple. As problem complexity increases the effects of initial planning appear to have little or no effect upon performance. In conclusion it is suggested that strategy use depends upon the interactions between individual preference for a given strategy, problem complexity, and the stage that one has reached in the development of a solution to a problem.     

Do people combine the parity- and five-rule checking strategies in product verification?

The basic question of the present experiment was whether people use a combination of arithmetic problem solving strategies to reject false products to multiplication problems or whether they simply use the single most efficient strategy. People had to verify true and false, five and non-five arithmetic problems. Compared with no-rule violation problems, people were faster with (a) five problems that violated the five rule (i.e., N×5=number with 5 or 0 as the final digit; e.g., 15 × 4=62), (b) problems that violated the parity rule (i.e., to be true, a product must be even if either or both of its multipliers is even; otherwise, it must be odd; 4 × 38=149), and (c) problems that violated both the parity and five rules (e.g., 29 × 5=142). Finally, people were equally fast and accurate when they solved two-rule violation problems than when they solved five-rule violation problems, and faster for those two types of problems than for parity-rule violation problems. Clearly, people use the single most efficient strategy when they reject false product to multiplication problems. This result has implications for our understanding of strategy selection in both arithmetic in particular and human cognition in general. Received: 18 October 1999 / Accepted: 27 January 2000     

论数学问题解决中情境模型与问题模型的关系

情境模型与问题模型是数学问题解决研究中的两个重要概念,前者是对问题所述情境的日常化的定性表征,后者是基于图式知识对问题关键变量的数量关系表征。本文介绍了两种模型的发展历史以及目前存在的争议,并提出了未来研究需要解决的问题。     

Patterns of problem‐solving in children's literacy and arithmetic

Patterns of problem‐solving among 5‐to‐7 year‐olds' were examined on a range of literacy (reading and spelling) and arithmetic‐based (addition and subtraction) problem‐solving tasks using verbal self‐reports to monitor strategy choice. The results showed higher levels of variability in the children's strategy choice across Years 1 and 2 on the arithmetic (addition and subtraction) than literacy‐based tasks (reading and spelling). However, across all four tasks, the children showed a tendency to move from less sophisticated procedural‐based strategies, which included phonological strategies for reading and spelling and counting‐all and finger modelling for addition and subtraction, to more efficient retrieval methods from Years 1 to 2. Distinct patterns in children's problem‐solving skill were identified on the literacy and arithmetic tasks using two separate cluster analyses. There was a strong association between these two profiles showing that those children with more advanced problem‐solving skills on the arithmetic tasks also showed more advanced profiles on the literacy tasks. The results highlight how different‐aged children show flexibility in their use of problem‐solving strategies across literacy and arithmetical contexts and reinforce the importance of studying variations in children's problem‐solving skill across different educational contexts.     

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.     

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.     

Working Memory Capacity and Self‐Explanation Strategy Use Provide Additive Problem‐Solving Benefits

The present study examined the impact of working memory capacity (WMC) on college students' ability to solve probability problems while using a self‐explanation strategy. Participants learned to solve probability problems in one of three conditions: a backward‐faded self‐explanation condition, an example problem pairs self‐explanation condition, or a control (no self‐explanation) condition. Even when accounting for the impact of WMC, learning to problem‐solve using self‐explanation led to superior problem‐solving performance. Conditions that prompted self‐explanation during problem‐solving resulted in significantly better problem‐solving performance than the control condition. These findings provide insight into the influence of individual differences on problem‐solving when strategies are provided, as well as information about the effectiveness of the self‐explanation strategy during mathematical problem‐solving. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

How is phonological processing related to individual differences in children’s arithmetic skills?

While there is evidence for an association between the development of reading and arithmetic, the precise locus of this relationship remains to be determined. Findings from cognitive neuroscience research that point to shared neural correlates for phonological processing and arithmetic as well as recent behavioral evidence led to the present hypothesis that there exists a highly specific association between phonological awareness and single‐digit arithmetic with relatively small problem sizes. The present study examined this association in 37 typically developing fourth and fifth grade children. Regression analyses revealed that phonological awareness was specifically and uniquely related to arithmetic problems with a small but not large problem size. Further analysis indicated that problems with a high probability of being solved by retrieval, but not those typically associated with procedural problem‐solving strategies, are correlated with phonological awareness. The specific association between phonological awareness and arithmetic problems with a small problem size and those for which a retrieval strategy is most common was maintained even after controlling for general reading ability and phonological short‐term memory. The present findings indicate that the quality of children’s long‐term phonological representations mediates individual differences in single‐digit arithmetic, suggesting that more distinct long‐term phonological representations are related to more efficient arithmetic fact retrieval.     

Metacognition for strategy selection during arithmetic problem-solving in young and older adults

We examined participants’ strategy choices and metacognitive judgments during arithmetic problem-solving. Metacognitive judgments were collected either prospectively or retrospectively. We tested whether metacognitive judgments are related to strategy choices on the current problems and on the immediately following problems, and age-related differences in relations between metacognition and strategy choices. Data showed that both young and older adults were able to make accurate retrospective, but not prospective, judgments. Moreover, the accuracy of retrospective judgments was comparable in young and older adults when participants had to select and execute the better strategy. Metacognitive accuracy was even higher in older adults when participants had to only select the better strategy. Finally, low-confidence judgments on current items were more frequently followed by better strategy selection on immediately succeeding items than high-confidence judgments in both young and older adults. Implications of these findings to further our understanding of age-related differences and similarities in adults’ metacognitive monitoring and metacognitive regulation for strategy selection in the context of arithmetic problem solving are discussed.     

The effects of monitoring environment on problem-solving performance

While effective and efficient solving of everyday problems is important in business domains, little is known about the effects of workplace monitoring on problem-solving performance. In a laboratory experiment, we explored the monitoring environment’s effects on an individual’s propensity to (1) establish pattern solutions to problems, (2) recognize when pattern solutions are no longer efficient, and (3) solve complex problems. Under three work monitoring regimes—no monitoring, human monitoring, and electronic monitoring—114 participants solved puzzles for monetary rewards. Based on research related to worker autonomy and theory of social facilitation, we hypothesized that monitored (versus non-monitored) participants would (1) have more difficulty finding a pattern solution, (2) more often fail to recognize when the pattern solution is no longer efficient, and (3) solve fewer complex problems. Our results support the first two hypotheses, but in complex problem solving, an interaction was found between self-assessed ability and the monitoring environment.