Characterizing the growth of one student’s mathematical understanding in a multi-representational learning environment

Highlights

• Treatments and conversions supports the growth of mathematical understanding.

• Manipulatives support the movements involving different registers of representations.

• Multiple representations in learning environments are the source of interventions.

• Multiple representations are powerful tools for acting and expressing activities.

Abstract

The purpose of this study was to characterize the growth of one student’s mathematical understanding and use of different representations about a geometric transformation, dilation. We accomplished this purpose by using the Pirie-Kieren model jointly with the Semiotic Representation Theory as a lens. Elif, a 10^th- grade student, was purposefully chosen as the case for this study because of the growth of mathematical understanding about dilation she exhibited over time. Elif participated in task-based interviews before, during and after participating in a variety of transformation lessons where she used multiple representations, including physical and virtual manipulatives. The results revealed that Elif was able to progress in her mathematical understanding from informal levels to the formal levels in the Pirie-Kieren model as she performed treatments and conversions, movements involving different registers of representations. The results also showed numerous examples of Elif’s mathematical understanding based on folding back activities, complementary aspects of acting and expressing, and interventions. Using the two theories together provides a powerful and holistic approach to a deeper understanding of mathematical learning by characterizing and articulating the growth of mathematical understanding and the way of mathematical thinking.

Introduction

An important idea in mathematics education is creating teaching and learning environments that allow students to learn mathematics with understanding (Carpenter & Lehrer, 1999). The mathematics education community suggests that one way to provide learning environments that support mathematical understanding is to promote the effective use of multiple representations of mathematical ideas. Including multiple representations in instructional environments to help students develop a deep understanding of mathematics has been highly recommended in the literature (see Ainsworth, Bibby, & Wood, 1998; Gagatsis & Shiakalli, 2004; Kaput, 1998; Lesh, Post, & Behr, 1987; Ng & Lee, 2009). The fundamental idea behind the encouragement of dealing with multiple representations is that each representation emphasizes some parts of the mathematical object by ignoring the others and students may develop a stronger understanding of these objects by taking advantage of the knowledge each representation offers (Ainsworth, 1999). In this way, students construct connections among representations of a mathematical concept in order to gain the objectified/embodied and generalized mathematical knowledge of that concept (Goldin, 1987; Hiebert, 1988).

However connecting multiple representations offered in mathematical learning environments is not an easy task for students (Ainsworth et al., 1998). Research emphasizes that students only take advantage of the benefits of a multi-representational approach if they can construct the links among different representations (Dreher & Kuntze, 2015; Renkl, Berthold, Große, & Schwonke, 2013). Teachers’ use of multiple representations during instruction are not magic by itself; it is the students’ sense making and reasoning activities that is important while they are studying mathematical tasks that involve different representations (Flores, Koontz, Inan, & Alagic, 2015). The aforementioned recent studies state that there is a need for greater clarification on how a learning environment, enriched with multiple representations of related concepts, can lead to students’ mathematical understanding. We focus on this issue by characterizing one 10^th-grade student’s growth of mathematical understanding about one of the important geometric transformations, dilation, in an environment where multiple representations were used to support growth.