The Role of Analogy in Teaching Middle-School Mathematics

Abstract

The Role of Analogy in Teaching Middle-School Mathematics Lindsey K. Engle (Lengle@psych.ucla.edu) Department of Psychology, University of California, Los Angeles Los Angeles, CA 90095-1563 Keith J. Holyoak (Holyoak@lifesci.ucla.edu) Department of Psychology, University of California, Los Angeles Los Angeles, CA 90095-1563 James W. Stigler (Stigler@psych.ucla.edu) Department of Psychology, University of California, Los Angeles Los Angeles, CA 90095-1563 Abstract Analogies produced in twenty-five US eighth-grade mathematics classroom lessons were analyzed according to their frequency and structure. Frequency findings suggest that analogies are a common part of mathematics classroom learning, and a component analysis revealed regular structural patterns in the way these analogies are produced. Teachers tended to organize the analogies by producing the target, source, and mapping steps before students become active participants. Students were most likely to then make inferences, adapt them to the target context, and solve target problems. Student participation was either independent or co- constructed with a teacher or other students. Findings address an important correlate with experimental research on analogical reasoning. The United States’ educational system is presently struggling to find teaching programs that facilitate mathematical understanding that goes beyond algorithmic knowledge. Standardized test results recently released in California indicate that despite improvements, the educational system remains far below state goals in mathematics (STAR, 2001). One major component of this difficulty is a lack of knowledge about how to teach abstract concepts so that students are able to transfer this learning across contexts. Systematic use of analogy may be integral to a teaching program that meets that goal. Analogy is a comparative structure that highlights abstract structural relations (Gentner, 1983), and facilitates schema acquisition and transfer across problems (Gick & Holyoak, 1983). Learning mathematics requires development of generalized concept representations that can be applied across contexts (Bransford, Brown & Cocking, 1999). Cognitive scientists have argued for decades that analogy plays a central role in human cognition, learning and problem solving (e.g., Holyoak, Gentner & Kokinov, 2001; Kolodner, 1997; Holyoak & Thagard, 1995; Gentner & Toupin, 1986; Piaget, 1950). However, there have been paradoxical findings in analogy research. While analogy has been demonstrated to be used in several everyday contexts (e.g. Dunbar 1995, 2001), most laboratory studies show low rates of spontaneous noticing and use of analogies for problem solving (e.g. Gick & Holyoak, 1980, 1983). It is necessary to understand this discrepancy between observed patterns of analogical reasoning in the laboratory and in everyday contexts, termed the analogical paradox (Dunbar, 2001), in order to design meaningful interventions to promote educational usage of analogy. We suggest that in order to clarify the paradoxical findings concerning analogy use, detailed analysis of everyday analogy usage is essential, because important aspects of analogy use can only be understood through online analysis of the pragmatics governing analogy production. The current study uses discourse analytic techniques to explore analogy production in the context of teaching mathematics in eighth-grade mathematics classrooms. Methods Sample and Coding Twenty-five videotaped eighth-grade mathematics lessons were analyzed to examine analogy activities. The lessons were randomly selected from a larger random probability sample collected as part of the Third International Mathematics and Science Study (TIMSS) directed by Jim Stigler (see Stigler, Gonzales, Kawanaka, Knoll, & Serrano, 1998). All selected classrooms were videotaped on one occasion. The classrooms were selected from US public, private, and parochial? schools in both urban and rural areas, and videotaping was conducted throughout the school year. The lesson content was not constrained, but most lessons drew from number theory, geometry, or algebra domains. Teaching styles similarly were not constrained and thus reflected a range of techniques and perspectives. Lessons were analyzed using V-Prism, a computer software package designed to allow simultaneous viewing of a digitized video and its typed transcript on a computer screen. In the program, the video’s transcript is time-linked so that the lines of text move temporally with the video.