Students’ opposing conceptions of equations with two equal signs

ABSTRACT

Despite many studies on students’ understanding of the equal sign, little attention has been paid to whole-class discourse on their conceptions of equations with two equal signs. This study analyzed the classroom discourse of 30 fourth graders in Korea who participated in a 5-week classroom teaching experiment. A series of discussions on specific equations with two equal signs showed that the students had subtle but significantly different conceptions of the equal sign, even though they often shared the common approach of saying “is the same as” when referring to the equal sign – these findings differ from previous studies. Classroom discourse exposed the simultaneous coexistence of various conceptions in a single student’s mind, including both operational and relational conceptions of the equal sign. The study’s findings argue for and demonstrate the value of learning opportunities to explore equations with two equal signs in a systematic and explicit way.

KEYWORDS:

• Equal sign
• equation with two equal signs
• operational conception
• relational conception
• classroom discourse
• classroom teaching experiment

Acknowledgments

The authors would like to thank Carolyn Kieran at Université du Québec à Montréal and Woong Lim at Yonsei University for their warm encouragement to develop this manuscript. The authors are indebted to anonymous reviewers and the editor who provided us with insightful comments.

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Notes

1. Here the equal sign is viewed as meaning “can be substituted for.” For example, to transform 31 + 40 into a single numeral, we can change 31 + 40 to 30 + 1 + 40 using 31 = 30 + 1. We make a substitution of 30 + 1 + 40 using 30 + 1 = 1 + 30 and then change 1 + 30 + 40 to 1 + 70 using 30 + 40 = 70. These substitutions can be repeated until a single numeral, 71, is achieved. (Jones & Pratt, 2012)

2. Currently, there is no national standardized test to evaluate elementary school students’ mathematical achievements in Korea. Elementary school teachers often test their students’ achievements after teaching each unit of the mathematics textbook by using the problems that are widely shared among them. In such tests, the students participating in this study achieved an average of 70 points out of 100, which can be regarded as average in the Korean context.

3. In fact, what the student wanted to say might be as follows: 182 × 20 = (200 – 18) × 20 = (200 × 20) – (18 × 20) = 4000 – 360 = 3640.

4. The students vocalized the equal sign in two different ways. In this study, the words “is” and “is the same as” were used to distinguish them. In most cases, “is” meant to calculate, and “is the same as” meant that the quantities on both sides of the equal sign were the same. However, when the students said “is the same as,” they often did not specify what would be the same. Note that classroom discourse based on multiple equations with two equal signs revealed subtle but significantly different conceptions of the equal sign, even though the students commonly said “is the same as.”

5. Before this teaching experiment was conducted, the students participating in this study had learned the concept of equilibrium in science lessons using a platform balance with plates on two sides.

Additional information

Notes on contributors

Jiyoung Lee

Jiyoung Lee has been working as an elementary school teacher since 2004. She has also been teaching pre-service and in-service teachers at various universities in Korea since 2012. She earned an Ed.D. for Mathematics Education at Korea National University of Education. Her research focuses on students’ understanding of fractions and fraction operations, early algebraic thinking, and problem solving.

JeongSuk Pang

JeongSuk Pang is Full Professor of Mathematics Education at Korea National University of Education, Korea. She has been actively involved in the development of the national mathematics curriculum and its concomitant textbook series for elementary school students, professional development for mathematics teachers, and the analysis of the mathematics classroom culture. While her earlier research interests were in diverse aspects of elementary mathematics education, her recent publication deals with early algebraic thinking, values and valuing perspective in mathematics education, and the Korean mathematics education research trend.