A Cognitive Task Analysis of Using Pictures To Support Pre-Algebraic Reasoning

Abstract

A Cognitive Task Analysis of Using Pictures To Support Pre-Algebraic Reasoning Kenneth R. Koedinger (koedinger@cmu.edu) Human-computer Interaction Institute, Carnegie Mellon University 5000 Forbes Ave. Pittsburgh, PA 15213-3890 Atsushi Terao (atsushi@cs.cmu.edu) Human-Computer Interaction Institute, Carnegie Mellon University 5000 Forbes Ave. Pittsburgh, PA 15213-3890 USA Abstract We present an analysis of hypothesized advantages of pictorial representations for improving learning and understanding of pre-algebraic quantitative reasoning. We discuss a Picture Algebra strategy that has been used successfully by 6 th grade students as part of a new middle school mathematics curriculum. This strategy supports students in sense making both as they construct pictorial representations and as they use them to cue appropriate computations. Although we demonstrate that 6 th grade students can use this strategy to successfully solve algebra-level problems, our detailed production rule analysis revealed limitations in our instructional approach and targeted areas for improvement. Introduction As part of a larger effort to develop a 6 th grade mathematics course including both a textbook and Cognitive Tutor software (cf., Koedinger, Anderson, Hadley, & Mark, 1997), we have been exploring the use of pictorial representations to support student reasoning and learning (Rittle-Johnson & Koedinger, 2001). Here we investigate the claim that pictorial representations can help students gain early entry into algebraic reasoning and build a foundation that will facilitate more effective learning of formal algebra. Why might using pictures or diagrams be advantageous? Cognitive scientists have presented arguments and experiments for the advantages of diagrams (e.g., Cheng, 1999; Larkin & Simon, 1987). According to Larkin and Simon (p. 98), a diagram can be superior to a verbal description for solving problems for three reasons. First, diagrams reduce problem-solving search by providing localized groupings of relevant information. Second, diagrams reduce the need for matching symbolic labels. Third, diagrams support perceptual inferences that are often easier than corresponding symbolic inferences. Others have presented arguments for the use of diagrams for mathematics instruction in particular. The mathematics standards of the National Council of Teachers of Mathematics (NCTM, 2000) recommends use of pictures to support students in developing a conceptual understanding of mathematics. Pictorial representations are used extensively in Asian curricula (cf., Singapore Ministry, 1999). This usage may be a factor in the success of Asian countries on international mathematics assessments (TIMSS, 1996). Despite these arguments for the advantages of pictures, there is also reason for caution. One argument for the use of alternative representations, like pictures, is that traditional instruction focuses too much on error- prone rote learning. However, students may also acquire rote procedures when learning to use alternative representations. Further, learning an alternative representation takes time that might be better spent learning the standard representation. In this paper we introduce the Picture Algebra strategy, present student data on the use of it, and discuss a production rule model of the strategy and implications for transfer and instructional design. Picture Algebra Try to solve the Cans problem shown in Table 1 and reflect on the strategy that you use to do so. We have informally observed that many adults do not directly infer what arithmetic operations are needed to solve this problem. Instead, most begin by translating the problem statement to one or more algebraic equations, for instance, x + (x + 9) + (x + 17) = 227. They then perform transformations on the equation to arrive at a solution. Although a few use other means (cf., Hall et al., 1989), most find this problem difficult without the use of algebraic equations. In other words, this is arguably an algebra problem that we might expect to be out of reach of students without algebra instruction, for instance, 6 th graders. Figure 1A shows a 6 th grade student’s solution to this problem using a Picture Algebra strategy that was taught to students as part of our middle school mathematics curriculum. Like other problem-solving strategies, Picture Algebra can be described in two phases: a representation phase and a solution phase. In the representation phase, the student first translates the