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• W267077672 abstract "Since Polya, Wertheimer, and Hadamard's descripticns of qualitative reasoning strategies used by scientists and mathematicians, very little data has been collected on whether these strategies are act, ly used by experts This study use.; video-taped thinking-aloud interviews U. examine the problem solving strategies of professors and advanced graduate students in technical fields. Evidence from these interviews documents the use of analogies, visual transformations, extreme cases, creative partitioning, and other plausible reasoning strategies used by experts. Preparation of this paper vas supported in part by a grant from the Natioral Science Foundation program for Research in Science Education, SED80-16662. NON-FORMAL REASONING IN EXPERT'S SOLUTIONS TO MATHEMATICS PROBLEMS TABLE OF CONTENTS Expert on a Physics Problem 1 Analogical in a Mathematics Problem 3 Results: Analogies 5 Analogies 5 Analogy Generation Methods 6 Evaluating the Cylinder Conjecture 7 Other Insightful Reasoning Processes 6 Extreme Cases 8 Partitioning and Symmetry Arguments Reassembl, of a Partition 9 Embedding 10 Spatial 10 Discussion 11 Conclusion 12 References 14 BEST COPY AVAILABLE U S DEPARTMENT OF EDUCATION Office of Educational Research andjmprovement EDUCATIONAL RESOURCES INFORMATION i 'Me document (ERIC) nee been reproduced as eceived from the person or organization originating it C' Minor changes have be, ..r., .o improve reproduction quality Points of view or opinions stated in t..is document do not necessarily represent official OE RI position or policy ISSION TO REPRODUCE THIS AL HAS BEEN GRANTED BY TO THE EDUCATIONAL RESOURCES INFORMATION CENTER (ERIC). NON-FORMAL REASONING IN EXPERTS' SOLUTIONS TO MATHEMATICS PROBLEMS Considering helpful analogous and extreme cases, breaking problems into analyzable parts, and performing simplifying spatial transformations are key reasoning processes in solving non-trivial problems. These processes allow talented scientists to attack problems outside the domain of familiar problems fot which they hae established algorithmic procedures They allow them to attack problems they have never seen before, giving them a degree of problem solving power and scope that is truly impressive. In previous reports, (1,2,3,4), I have documented the fact that these qualitative reasoning processes are used by expert scientists in solving physics problems. This paper shows that empirical evidence for these processes can also be collected in the case of experts' solutions to mathematics problems. I will first briefly describe results from the earlier physics problem study, ane then describe those from the current mathematics problem study. EXPERT REASONING ON A PHYSICS PROBLEM In the previous study, ten expert subjects were asked to solve the spring problem shown in Fig. 1. All subjects were advanced doctoral candidates or professors in technical fields. The study concentrated most on documenting and analyzing the use of analogies. Some examples of analogies generated for this problem are as follows: 2 loaded with a weight at the other end. He felt that a long blade would bend more easily than a short one, and this indicated to him that the wider spring might stretch more. Other examples of proposed analogies were that a longer horizontal hairpin shaped vire would extend more than a shorter one (see Fig 2), and that a larger dingle square would stretch more than a smaller one. Another subject examined the relationship between coil diameter, coiling angle, and wire length by thinking about mountain roads winding up narrow and wide mountains. The correct answer to the problem is that the wide spring will stretch farther (the stretch in fact increases with the cube of the diameter). This seems to correspond to most peoples' initial intuition about the problem. However, explaining why the wide spring stretches more (and explaining exactly where the stretch of the spring comes from), is a much more difficult task when taken seriously. Some of the findings from this study were as follows: Spontaneously generated analogies were observed to play a significant role in problem solutions of scientifically trained subjects. Seven of the ten subjects generated at least one salient analogy. The subjects generated a large variety of analogous cases. Not all of the analogies were to situations familiar to the subject. Some were novel cases in the form of Gedanken experiments that appeared to be invented by the subject. In addition to the initial process of pen rating an analogy, there is second process that is just as important in expert problem solving, that of critically evaluating the validity of the analogy. Analysis of the transcripts indicated that there was more than one type of analogy generation method used Two of these methods are the associative leap, and the generative transformation. The svb)ect using an associative leap jumps to an analogous One sabject thought about horizontal saw blade held fixed at one end and situation that differs in many ways from the original problem." @default.
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• W267077672 date "1984-03-25" @default.
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• W267077672 title "Non-Formal Reasoning in Experts' Solutions to Mathematics Problems." @default.
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