The Importance of Multiple Representations of Mathematical Problems: Evidence from Chinese Preservice Elementary Teachers’ Analysis of a Learning Goal

Abstract

This article describes a study of how Chinese preservice teachers unpacked a learning goal pertaining to adding fractions and understanding the concepts underlying the operation. Based on work in the USA by Morris, Hiebert, and Spizter (Journal for Research in Mathematics Education, 40(5), 491–529, 2009), 50 Chinese preservice teachers completed a task, anticipating an ideal student response, in the context of four representations: (1) fraction pieces, (2) graph paper, (3) common denominator algorithm, and (4) pennies. Like the US-based study, this study shows that Chinese preservice teachers’ ability to unpack a learning goal was highly influenced by how the problem was represented. The pennies and graph paper problems provided more supportive contexts for unpacking the learning goal; the algorithm problem provided the least supportive context. The main difference between the preservice teachers from these two countries was the US preservice teachers chose the pennies and graph paper problems as having the most potential for revealing students’ understanding of the learning goal; while the Chinese preservice teachers chose the algorithm problem despite the problem’s unsupportive context. Chinese preservice teachers’ preference suggests that they privileged algorithmic/symbolic representations over pictorial/concrete representations. Based on our results, we argue that it is time for cross-cultural comparative research to refocus on the common barriers, challenges, and benefits as related to mathematics teacher preparation and professional development. A cooperative instead of competitive orientation will lead to richer and deeper dialogues among mathematics educators.