Relational Thinking in Elementary Mathematics

• In Event: 52.031 - Relational Reasoning in STEM Domains: What Empirical Research Can Contribute to the National Dialogue

Abstract

Purpose/Theoretical Framework

Relational thinking is core to mathematical thinking and generalizable learning, in the form of representing problems or concepts as systems of relationships, yet relational thinking is seriously underutilized in U.S. classroom teaching (Hiebert et al, 2003; Richland, Zur & Holyoak, 2007), and many students graduate from k-12 instruction with explicit beliefs that mathematics is a discipline of memorization as opposed to reasoning (Givven, Stigler & Thompson, 2011).
We describe key pedagogical strategies for using relational thinking in classroom practice for drawing connections: comparing students' solutions to one problem (Ball et al, 2008; Stein, et al, 2008). A common strategy for leading classroom discussions in which solutions to a single problem are compared draws in part on the Japanese instructional model and the conceptual change model (see Stigler & Hiebert, 1999, Vosniadou & Brewer, 1987). In this routine, the teacher selects multiple students to present their solutions, typically beginning with a student demonstrating a common misconception, followed by students who use variations of valid solutions. However, we previously found that when a student misconception was presented first and inadequately compared with more appropriate strategies, students were likely to continue or begin using the misconception strategy (Begolli & Richland, 2013).
A set of experiments tested whether the comparison between solution strategies was best organized by beginning or ending with the misconception, and whether the challenges of ignoring a presented misconception at the beginning of a lesson is more difficult for students with low inhibitory control and executive function skills.

Method

In all experiments, fifth and sixth-grade students individually interacted with videotaped classroom instruction on rate/ratio. A whole class lesson was videotaped and broken into segments that were presented to students in a new school using an interactive computer program. Participants were administered a pretest, immediate posttest, and delayed posttest with validated procedural and conceptual subscales. In experiment 1, the misconception and valid solutions were presented in different orders, randomized within classroom, with either misconception shown first followed by valid (n = 47), or the opposite (n = 48). Participants also completed a battery of working memory and executive function measures. Experiment 2 replicated Experiment 1 with additional cognitive measures and a larger sample on the misconception first condition. Experiment 3 replicated the effect of presentation order.

Results

A median split design was used to examine the relations among instructional order, cognitive skills and posttest scores. Analyses controlled for pretest so to minimize relations between prior knowledge and cognitive skills. See table 3.1 for information on significance tests.
In sum, comparing solution strategies can be a powerful way to support students in developing procedural and conceptual knowledge, but only when they have adequate processing resources or pedagogical support to ensure that they make the comparisons. If not, presenting a misconception before valid instruction may lead to greater use of the misconception at posttest.

Significance

With careful support, relational thinking is a powerful tool for promoting flexible procedural and conceptual knowledge in mathematics education.

Authors

• Lindsey E. Richland, University of Chicago

• Kreshnik Nasi Begolli, Temple University

• Rebecca Frausel, University of Chicago