Elsevier

Computers & Education

Volume 62, March 2013, Pages 308-319
Computers & Education

Analysis of peer learning behaviors using multiple representations in virtual reality and their impacts on geometry problem solving

https://doi.org/10.1016/j.compedu.2012.10.005Get rights and content

Abstract

Learning geometry emphasizes the importance of exploring different representations such as virtual manipulatives, written math formulas, and verbal explanations, which help students build math concepts and develop critical thinking. Besides helping individuals construct math knowledge, peer interaction also plays a crucial role in promoting an understanding of mathematics and geometric problem solving. In this research, an Interactive Future Mathematics Classroom (IFMC) based on the Collaborative Virtual Reality Learning Environment (CVRLE) was proposed to facilitate three-dimensional (3-D) geometric problem solving. Two representational tools, the virtual manipulative and a white board, were integrated into the IFMC to help students with the following activities: to synchronously review peers' solving processes; to individually or collaboratively manipulate 3-D objects using the virtual manipulative; and to give comments on peers' white boards for future queries and discussions. One eight-week experiment was conducted and the results showed that the experimental group using the IFMC performed significantly better than the control group on geometric learning achievement. Further analysis showed that the peer learning behaviors of the experimental group in the two kinds of geometric problems, volume and surface area calculation, were different due to the problems' varying difficulty levels. Moreover, various peer learning behaviors with multiple representations lead to different types of strategies for geometric problem solving in the IFMC. Therefore, peer learning behaviors in the IFMC were found useful to facilitate geometric problem solving by sharing ideas and exploring multiple representations.

Highlights

► Interaction of peers learning behaviors in IFMC facilitates students' geometry problem solving. ► We refine students' strategies for geometry problem solving in the virtual reality. ► Students using IFMC perform better in the post-test of geometry problem solving. ► Students express exploring peers learning with multiple representations is enjoyable and useful for geometry problem solving.

Introduction

Geometric education has students use geometry within the process of problem solving to understand and explain the physical world around them (Baki, Kosa, & Guven, 2009). To fully appreciate the real world and other fields such as computer graphics, engineering, and architecture, exploration of 3-D geometry (including moving, positioning, orienting, constructing, building, and communicating 3-D objects) is an important topic that should be exercised by students in a geometric learning curriculum (Yeh, 2007).

Some researchers conceived the nature of mathematics as the outcome of social process and math knowledge, which is thus understood to be fallible and eternally open to revision both in terms of its proofs and its concepts (Ernest, 1999). In addition to individual exploration of multiple representations for constructing geometric knowledge, which is urged by the development of constructivism, math reform gradually advocates that the interaction of sharing, assessing, and collaborating for discovering the solution with peers and their representations must be implemented in a student's problem solving process (NCTM, 2000; Wilkins, 2008; Wood, Williams, & McNeal, 2006). That is, students interact with the instructor, peers, and learning materials to share their thoughts and to verify solutions from multiple viewpoints. Obviously, communicating math concepts through mutual observation and discussion with peers often helps students identify unforeseen perceptions (Sheffield, 2000).

Due to the lack of examination of the impacts on students' three-dimensional (3-D) geometry concepts learning and problem solving caused by synchronous interaction with multiple representations among peers in the Collaborative Virtual Reality Learning Environment (CVRLE), this research proposed the Interactive Future Mathematics Classroom (IFMC), based on the CVRLE to facilitate students solving geometry problems and afterwards to study the effect of peer learning behaviors to learning achievement.

Section snippets

Geometric problem solving with multiple representations

Goldin (1998) indicated that there were external representations in the real world and internal representations in the mind. Psychologists view the representation as the process of modeling concrete objects in the real world into abstract concepts or internal representations, which consist of mental components such as metaphorical, visual-spatial, and structural knowledge (Jonassen & Carr, 2000). For the mathematical context, Nakahara (2008) articulated five kinds of representations used in

Design and development of the IFMC

The IFMC is a 3-D CVRLE, which includes several functionalities implemented as “modules” based on the architecture of the Open Wonderland platform. Open Wonderland is a pure Java-based open source toolkit developed by Sun Micro System for creating a collaborative 3-D virtual world. Hence, with the interactive capabilities and simultaneous communication tools provided by the IFMC, students are free to create, express, and share their ideas individually or collaboratively via different types of

Results and discussion

In addition to discussing the statistical details, the interview content was also analyzed for further interpreting the subject's learning behaviors. Therefore, five students attended an interview after the experiment. Among five interviewees, three of them were chosen due to their highest or lowest ratings of the questionnaire, and the other two were selected because of their largest or least numbers of annotations left on the white board. We would like to study deeply the reasons why they had

Conclusions

First, promoting peer learning or interaction in the IFMC should be valued during mathematical problem-solving activities. The immersion-based CVRLE with multiple representations such as the IFMC in this research motivates students to have fun interacting with peers and share great learning experiences about geometric problem solving. Through meaningful peer learning, students with different abilities can be inspired from observing peers' manipulations; therefore, they can deepen their

References (37)

  • C. Christou et al.

    Developing the 3DMath dynamic geometry software: theoretical perspectives on design

    International Journal for Technology in Mathematics Education

    (2006)
  • D.H. Clements et al.

    Constructivist learning and teaching Retrieved from NCTM

    (2009)
  • V.A. Debellis et al.

    Affect and meta-affect in mathematical problem solving: a representational perspective

    Educational Studies in Mathematics

    (2006)
  • P. Ernest

    Is mathematics discovered or invented

    Philosophy of Mathematics Education Journal

    (1999)
  • A. Gagatsis et al.

    Ability to translate from one representation of the concept of function to another and mathematical problem solving

    Educational Psychology

    (2004)
  • T.L. Good et al.

    Grouping for instruction in mathematics: a call for programmatic research on small-group processes

  • Hancock, G., & Lane, C. (2010). Case study: the effectiveness of virtual manipulatives in the teaching of primary...
  • J.W. Heddens

    Improving mathematics teaching by using manipulatives

    (2005)
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