Characterizing the intuitive representation in problem solving: Evidence from evaluating mathematical strategies

Two experiments were conducted to investigate the nature of the intuitive problem representation used in evaluating mathematical strategies. The first experiment tested between two representations: a representation composed of principles and an integrated representation. Subjects judged the correctness of unseen math strategies based only on the answers they produced for a set of temperature mixture problems. The distance of the given answers from the correct answers and whether the answers violated one of the principles of temperature mixture were manipulated. The results supported the principle representation hypothesis. In the second experiment we manipulated subjects’ understanding of an acid mixture task with a brief paragraph of instruction on one of the principles. Subjects then completed an estimation task intended to measure their understanding of the problem domain. The evaluation task from the first experiment was then presented, but with acid mixture instead of temperature mixture. The results showed that intuitive understanding of the domain mediates the effect of instruction on evaluating problems. Additionally, the results supported the hypothesis that subjects perform a mapping process between their intuitive understanding and math strategies.

[1]  Stephen K. Reed,et al.  A structure-mapping model for word problems. , 1987 .

[2]  Stephen K. Reed,et al.  Learning functional relations: A theoretical and instructional analysis , 1987 .

[3]  S. Siegel,et al.  Nonparametric Statistics for the Behavioral Sciences , 2022, The SAGE Encyclopedia of Research Design.

[4]  C. Moore,et al.  Development of intuitive and numerical proportional reasoning , 1992 .

[5]  Lotfi A. Zadeh,et al.  A COMPUTATIONAL APPROACH TO FUZZY QUANTIFIERS IN NATURAL LANGUAGES , 1983 .

[6]  J. Mestre,et al.  The relation between problem categorization and problem solving among experts and novices , 1989, Memory & cognition.

[7]  Madan M. Gupta,et al.  Fuzzy automata and decision processes , 1977 .

[8]  Mark H. Ashcraft,et al.  Evidence for retrieval and decision processes in mental addition , 1978 .

[9]  W Kintsch,et al.  Understanding and solving word arithmetic problems. , 1985, Psychological review.

[10]  Johan de Kleer.  Multiple Representations of Knowledge in a Mechanics Problem-Solver , 1977, IJCAI.

[11]  L E Krueger,et al.  Why 2 + 2 = 5 looks so wrong: On the odd-even rule in sum verification , 1984, Memory & cognition.

[12]  Stephen E. Newstead,et al.  Context and the interpretation of quantifiers of frequency , 1987 .

[13]  John F. Lemmer,et al.  Causal Modeling , 1993, UAI.

[14]  M. Perlmutter,et al.  Cognitive arithmetic: comparison of operations. , 1984, Journal of experimental psychology. Learning, memory, and cognition.

[15]  L. Zadeh.  A COMPUTATIONAL APPROACH TO FUZZY QUANTIFIERS IN NATURAL LANGUAGES , 1983 .

[16]  Kate Ehrlich,et al.  Knowledge and processes in the comprehension of computer programs. , 1988 .

[17]  Stephen E. Newstead,et al.  Fuzzy quantifiers as an explanation of set inclusion performance , 1984 .

[18]  Eric A. Jenkins,et al.  How Children Discover New Strategies , 1989 .

[19]  Alf Zimmer.  A Common Framework for Colloquil Quanti ers and Probability Terms , 1988 .

[21]  Dominic W. Massaro,et al.  Integration of featural information in speech perception. , 1978 .

[22]  Norman H. Anderson,et al.  Cognitive Algebra: Integration Theory Applied to Social Attribution1 , 1974 .

[23]  R. Siegler.  Three aspects of cognitive development , 1976, Cognitive Psychology.

[24]  Beth A. Haines,et al.  Components of Understanding in Proportional Reasoning: A Fuzzy Set Representation of Developmental Progressions , 1991 .

[25]  Amnon Rapoport,et al.  Measuring the Vague Meanings of Probability Terms , 1986 .

[26]  R. Maki.  Categorization and distance effects with spatial linear orders. , 1981 .

[27]  Brian R. Gaines,et al.  Foundations of fuzzy reasoning , 1976 .

[28]  Dominic W. Massaro,et al.  A pattern recognition account of decision making , 1994, Memory & cognition.

[29]  Norman H. Anderson,et al.  Function knowledge: Comment on Reed and Evans. , 1987 .

[30]  D. Massaro,et al.  Integration of featural information in speech perception. , 1978, Psychological review.

[31]  B. Mellers,et al.  Preferences, prices, and ratings in risky decision making. , 1992 .

[32]  Dorothea P. Simon,et al.  Expert and Novice Performance in Solving Physics Problems , 1980, Science.

[33]  Robert S. Siegler,et al.  A featural analysis of preschoolers' counting knowledge. , 1984 .

[34]  F. Restle.  Speed of Adding and Comparing Numbers. , 1970 .

[35]  K. Crowley,et al.  Constraints On Learning in Nonprivileged Domains , 1994, Cognitive Psychology.

[36]  T. Pearson,et al.  Direct comparison of the efficacy of intuitive and analytical cognition in expert judgment , 1987, IEEE Transactions on Systems, Man, and Cybernetics.

[37]  R. Stavy,et al.  U-shaped behavioral growth , 1982 .

[38]  M. Fayol,et al.  When plausibility judgments supersede fact retrieval: The example of the odd-even effect on product verification , 1995, Memory & cognition.

[39]  M. Riley,et al.  Conceptual competence and children's counting , 1984, Cognitive Psychology.

[40]  Diane J. Briars,et al.  An integrated model of skill in solving elementary word problems cognition and instruction , 1984 .

[41]  L E Krueger,et al.  Why 2×2=5 looks so wrong: On the odd-even rule in product verification , 1986, Memory & cognition.

[42]  C. Moore,et al.  The Developmental Role of Intuitive Principles in Choosing Mathematical Strategies. , 1996 .

[43]  J. Kleer.  Qualitative and Quantitative Knowledge in Classical Mechanics , 1975 .

[44]  G. Oden.  Dependence, independence, and emergence of word features. , 1984, Journal of experimental psychology. Human perception and performance.

[45]  M. Ashcraft,et al.  Menatal addition: A test of three verification models , 1981, Memory & cognition.

[46]  Michelene T. H. Chi,et al.  Expertise in Problem Solving. , 1981 .