Two Strategies are Better than One: Multiple Strategy Use in Word Problem Solving

Kenneth R. Koedinger and Hermina J. M. Tabachneck

Abstract

Much research on skill and skill acquisition in math and science has focused on the formal representations and procedures that are the stuff of traditional textbooks. However, a number of more recent concerns have shifted their focus to informal representations and strategies involved in perceptual intuition, situated cognition or other types out-of-school learning. There is little question among educational researchers and practitioners that formal procedures and representations are difficult for students to learn. Furthermore, they are often not used outside of school, and those people who do effectively solve mathematical problems in everyday situations often employ common sense strategies that are different than those taught in school. Although some might intend it to, the trend toward legitimatization of informal strategies should not be read as a claim that they are the mainstay of skill and that the formal representations and procedures of traditional schooling are unnecessary. Instead, we agree with the position that it is through the integration of common sense and school-taught knowledge that the greatest understanding of mathematics is achieved. This paper provides verbal protocol evidence in the domain of mathematical word problem solving that the integrated use of formal and informal strategies is more effective than the strict use of a single strategy. We provide a theoretical interpretation of why this is so and are working on a computer simulation of this theory.

In looking in detail at the strategies of college students solving simple algebra word problems, we observed a high frequency of three unschooled, non-algebra strategies. In contrast to both opposing positions on the importance of unschooled strategies, we found that no single strategy, schooled or unschooled, is significantly more effective than any other. Instead, the key finding was that the use of multiple strategies within a single problem is significantly correlated with success. We analyzed the strengths and weaknesses of each strategy and claim that the unschooled strategies¹ strengths lay mainly in the processes of comprehension and translation, and the schooled strategies¹ strength lay mainly in the calculation process. Thus, schooled and unschooled strategies can complement each other in the process of solving a single problem.

We are developing a cognitive model of multiple strategy use in the ACT-R production system to better understand how and why multiple strategy use provides an advantage. Computer modeling forces a "discipline of detail" that requires a characterization of multiple strategy use that is precise and unambiguous. In addition, computer models provide a sufficiency proof that each strategy can produce correct solutions. By inspecting the resulting models, we can more directly see the unique features and strengths of the various strategies.