Supporting Algebraic Symbolization: Effects of Diagrams Are Moderated by Mathematics Ability

• In Event: 34.032 - Probing Students' Understanding With Differing External Representations

Abstract

Diagrams have been demonstrated to be both beneficial (Hembree, 1992) and harmful (Lee et al., 2009) to students’ mathematical learning. These inconsistencies may arise because different types of diagrams support problem solving in different ways (Schnotz, 2002). In this research, we consider the effects of different types of diagrams in supporting students’ abilities to symbolize the mathematical relationships in story problems. We believe different types of diagrams may differentially facilitate correct symbolization (Eitel et al., 2013). Furthermore, we expected mathematics experience to play a significant role, as suggested by Booth and Koedinger (2011), who found diagrams were helpful for low ability students.

One hundred twenty-one participants (ageM = 19) were sampled from a Midwest university in the U.S., and randomly assigned to one of three between-subject conditions. We used two types of diagrams. Both included numerical content from the story problems; the only difference was whether the diagram displayed the relational structure in the problem. One diagram displayed the relational structure (integrated diagram), while the other displayed the operations separately (discrete diagram) (See Figure 7). Math ability was measured using self-reported ACT/SAT scores.

The integrated diagram increased the probability of students’ correctly integrating both operations from the text, as compared to the discrete diagram (2(1) = -4.04, p = .04), but not compared to no diagram (see Figure 8). This effect was moderated by math experience, such that both diagrams helped low math ability students integrate operations (2(1) = 5.35, p = .02); moreover, the integrated diagram was better than the discrete diagram (2(1) = 4.71, p = .03). However, high math ability students integrated operations more with no diagram, as compared to both diagrams, 2(1) = 4.16, p = .04 (see Figure 9). In a follow-up study (with data collection ongoing [current n = 31]), the integrated diagram also increased the probability of symbolization on a more difficult transfer problem, as compared to the discrete diagram, F(1, 30) = 3.1, p = .04 (see Figure 10).

When it comes to correctly integrating operations, the integrated diagram was overall more useful than the discrete diagram, but not more useful than having no diagram. Furthermore, preliminary data suggests the same trend for symbolization on a more difficult transfer problem. However, it is important to note the role of math ability. For students of low math ability, the integrated diagram increased the probability that students successfully integrated the operations as compared to having a discrete diagram or no diagram. However, this pattern was reversed for high math ability students, for whom the probability of integration increased with no diagram.

These findings suggest that different visual representations paired with the same text differentially affect students’ mental models of a problem situation. In all, when designing materials to support performance in mathematics, it is important to consider not only how a provided visual representation may help, but also whom the visual representation is intended to help.

Authors

• Anna N. Bartel, University of Wisconsin - Madison

• Martha W. Alibali, University of Wisconsin - Madison