Equations Are Effects: Using Causal Contrasts to Support Algebra Learning

Equations are Effects: Using Causal Contrasts to Support Algebra Learning Jessica M. Walker (jessicamariewalk@gmail.com) Patricia W. Cheng (cheng@lifesci.ucla.edu) James W. Stigler (stigler@psych.ucla.edu) Department of Psychology, University of California, Los Angeles, CA 90095-1563 USA Abstract U.S. students consistently score poorly on international mathematics assessments. One reason is their tendency to approach mathematics learning by memorizing steps in a solution procedure, without understanding the purpose of each step. As a result, students are often unable to flexibly transfer their knowledge to novel problems. Whereas mathematics is traditionally taught using explicit instruction to convey analytic knowledge, here we propose the causal contrast approach, an instructional method that recruits an implicit empirical-learning process to help students discover the reasons underlying mathematical procedures. For a topic in high-school algebra, we tested the causal contrast approach against an enhanced traditional approach, controlling for conceptual information conveyed, feedback, and practice. The causal contrast approach yielded remarkably greater success, especially on novel problems, across students with varying levels of mathematical competence. Keywords: comparison; mathematics education; causal induction; concepts, knowledge transfer Introduction Mathematics experts use general goal-directed reasoning to solve problems. Students, in contrast, are commonly observed to use procedures without understanding, often just following a memorized sequence of steps to solve a problem (Stigler, Givvin, & Thompson, 2010). Because they have not connected procedures to goals or concepts, students lack the flexibility of experts: they often are unable to decompose a procedure and apply the constituent operations successfully to a novel problem for which the learned sequence is inadequate (Cooper & Sweller, 1987). This lack of goal-directed reasoning in mathematics may be a major cause of the poor performance of U.S. students (PISA, 2009, 2013; Stigler & Hiebert, 1999; TIMSS, 2007). A critical prerequisite for goal-directed reasoning is learning cause-and-effect relations. People, including children and even infants, have a natural capacity for learning causal relations (Cheng, 1997; Gopnik et al, 2004; Leslie & Keeble, 1987). Much like language learning, causal induction is a universal learning process that is shared by virtually all humans. Because of its universality, causal induction can be recruited in nearly all learners to support goal-directed reasoning. Once an effect of interest is identified, most learners will naturally seek to discover causes of that effect. Moreover, they will succeed if the requisite information for the discovery is readily available. For example, imagine you wake up one morning, disappointed to discover that your digital video recorder (DVR) failed to record a special television show last night (your goal). You would probably think back to occasions on which your DVR successfully recorded, and compare them to the failed attempt. If a presetting to record a regular show is the only feature that differs between your failed and successful attempts, you would readily determine the cause of the failure -- the presetting interfered with your new setting. This process enables you to discover a new causal relation and better understand how your DVR works. Causal induction is the process whereby we come to know how the empirical world works; it is what allows us to predict, diagnose, and intervene on the world to achieve an effect. Based on the premise that action-outcome relations in mathematics are no less causal than those in other domains, we ask: Can we 1) activate students’ natural capacity for causal induction during mathematical problem- solving, inducing them to formulate the goals of a mathematical procedure, and 2) promptly have the requisite information available, allowing students to discover the actions that would achieve those goals, and thereby to understand the causal structure of the solution to a problem? If so, students may become better able to generalize their mathematical knowledge. Although previous researchers have noted the importance of applying causal knowledge to solve mathematical problems (e.g., Anderson, 1990), to our knowledge the simple but powerful process of causal induction has never been systematically recruited in the acquisition of the requisite causal knowledge in mathematics education. Causal induction may at first appear irrelevant to mathematical learning, as mathematics is analytic whereas causation is empirical. Moreover, the causal induction process is implicit; neither its requisite input nor its operation is open to introspection. Thus, the potential use of this process in mathematics learning does not readily present itself. Traditional instructional approaches teach students analytically and explicitly the rules and steps for solving specific types of problems. For example, a student presented with this equation (Eq. 1) x 2 – 13x – 30 = 0 is taught to first factor the expression on the left, then determine the possible values of x using the zero-product property (ZPP) (if a • b = 0, then a = 0 or b = 0). For a substantial fraction of students, as evidenced by their failure to flexibly generalize their learning to novel problems, this