Seeing language learning inside the math: Cognitive analysis yields transfer

Seeing Language Learning inside the Math: Cognitive Analysis Yields Transfer Kenneth R. Koedinger (koedinger@cmu.edu) Human-Computer Interaction Institute, Carnegie Mellon University 5000 Forbes Avenue, Pittsburgh, PA 15217 USA Elizabeth A. McLaughlin (mimim@cs.cmu.edu) Human-Computer Interaction Institute, Carnegie Mellon University 5000 Forbes Avenue, Pittsburgh, PA 15217 USA (Heffernan & Koedinger, 1997; 1998). Table 1 shows examples of symbolization problems, which ask students to translate a story problem into an algebraic expression. The obvious potential connection between language learning processes and this task is in learning to read and comprehend story problems. While such learning is indeed a significant challenge for elementary students (Cummins, Kintsch, Reusser, & Weimer, 1988), our past data provided evidence that comprehending story problems is no longer a major sticking point for most beginning algebra students. This claim can be illustrated by an analogy to foreign language translation: Translating a story problem to algebra is like translating English to Greek. For an English speaker, the difficulty in translating to Greek is not comprehending the English, but generating the Greek. Similarly, the challenge for older students in a beginning algebra course is much less in understanding the English in which the story problems are written and more in being able to express that understanding algebraically, that is, in the language of algebra. One indication that comprehension of algebra story problems is not a major sticking point for beginning algebra students comes from Heffernan and Koedinger’s (1998) data showing that students can solve story problems (produce a value for the dependent or “y” variable when a value for the independent or “x” variable is given) much more accurately (63% correct) than they can symbolize (write an equation relating x and y) a story problem (18% correct). Since solving requires comprehension of the story, the performance difference is suggestive that symbolizing is problematic for students in ways beyond the demands of sentence comprehension. A second indication presents a contrast with a difficulty experienced by Artificial Intelligence systems programmed to solve story problems, namely that of understanding the arithmetic relationships between quantities described in the story (Bobrow, 1968). We created problems where natural implicit descriptions of such relationships (e.g., “Ms. Lindquist teaches 62 girls. Ms. Lindquist teaches b boys.”) are supplemented (Heffernan & Koedinger, 1997) or replaced (Koedinger, Alibali, & Nathan, 2008) with explicit descriptions (e.g., “The number of students Ms. Lindquist teaches is equal to the number of boys plus the number of girls.”), which are much easier for a program to process. We found, however, that providing such explicit descriptions does not Abstract Achieving and understanding effective transfer of learning requires a careful analysis of the hidden knowledge and skills to be transferred. We present an experiment that tests a subtle prediction of such an analysis. It concluded that a critical difficulty in students’ learning to translate algebra story problems into symbolic expressions is in learning the grammar of such expressions. We hypothesized that exercises requiring students to substitute one algebraic expression into another would enhance students’ algebraic grammar knowledge. This hypothesis led to a counter-intuitive prediction that learning to symbolize story problems could be better enhanced through practice on dissimilar looking substitution exercises than through practice on more similar looking story problems. We report on an experimental comparison involving 303 middle school students that supports this prediction. We discuss how having learners externalize a uniform abstract form and get interactive feedback on it may be important factors in enhancing transfer. Keywords: cognitive task analysis; transfer; grammar learning; mathematics education. Introduction Humans learn language before they have a language to use to learn. Might the learning processes that make this amazing feat possible, like the capability to learn grammatical structures through experience without explicit instruction, be useful for other kinds of learning tasks? Once children have acquired language, are the cognitive functions employed in language learning no longer useful? For instance, as students take courses in complex academic topics, like algebra, does all that brain matter for language learning have nothing to do? Or is it possible that some of the same implicit learning mechanisms employed in language learning are useful for learning math and science? This paper does not aim to provide conclusive answers to these questions, however, it does provide a compelling demonstration that grammar learning processes may be important in learning mathematics. Students may engage in such learning without explicit awareness and such implicit learning may be more prevalent in academic learning than is generally recognized (e.g., Alibali & Goldin-Meadow, 1993; Landay & Goldstone, 2007). In earlier work, we performed a cognitive task analysis of the important task domain of “symbolization”, that is, the ability to model problem situations or “story problems” in algebraic symbols