A Longitudinal Examination
of Middle School Students’
Understanding of the Equal Sign
and Equivalent Equations
Martha W. Alibali, Eric J. Knuth, and Shanta Hattikudur
University of Wisconsin-Madison
Nicole M. McNeil
University of Notre Dame
Ana C. Stephens
University of Wisconsin-Madison
This longitudinal study investigated (a) middle school students’ understanding of the
equal sign, (b) students’ performance solving equivalent equations problems, and (c)
changes in students’ understanding and performance over time. Written assessment
data were collected from 81 students at four time points over a 3-year period. At the
group level, understanding and performance improved over the middle school years.
However, such improvements were gradual, with many students still showing weak
understanding and poor performance at the end of grade 8. More sophisticated under-
standing of the equal sign was associated with better performance on equivalent
equations problems. At the individual level, students displayed a variety of trajecto-
ries over the middle school years in their understanding of the equal sign and in their
performance on equivalent equations problems. Further, students’ performance on
the equivalent equations problems varied as a function of when they acquired a so-
phisticated understanding of the equal sign. Those who acquired a relational under-
standing earlier were more successful at solving the equivalent equations problems at
the end of grade 8.
MATHEMATICAL THINKING AND LEARNING, 9(3), 221–247
Copyright © 2007, Lawrence Erlbaum Associates, Inc.
Correspondence should be sent to Martha W. Alibali, Department of Psychology, University of
Wisconsin-Madison, 1202 W. Johnson St., Madison, WI 53706. E-mail: mwalibali@wisc.edu

Algebra continues to receive substantial attention in the mathematics education
community, particularly as questions concerning its nature and role in school
mathematics are (re)considered. Traditionally conceived as the domain of an
eighth- or ninth-grade mathematics course, recent reform efforts (e.g.,
Lacampagne, Blair, & Kaput, 1995; National Council of Teachers of Mathematics
[NCTM], 2000; National Research Council, 1998) as well as research efforts (e.g.,
Bednarz, Kieran, & Lee, 1996; Carpenter, Franke, & Levi, 2003; Kaput, Carraher,
& Blanton, 2007; RAND Mathematics Study Panel, 2003) have led to a
reconceptualization of algebra as a continuous K–12 strand. Instantiating this con-
ceptualization of algebra as a continuous K–12 strand requires bringing out the al-
gebraic character of early mathematics activities (Blanton & Kaput, 2005;
Carraher, Schliemann, Brizuela, & Earnest, 2006), and coherently connecting fun-
damental concepts of algebra across the elementary and secondary school years
(Kaput, 1998).
One concept that is both fundamental to algebra understanding and ubiquitous
in school mathematics at all levels is that of equality and, in particular, the equal
sign. Developing an understanding of the equal sign has typically been considered
mathematically straightforward—after its initial introduction during students’
early elementary school education, little, if any, instructional time is explicitly
spent on the equal sign in later grades. Yet, research suggests that many students at
all grade levels have not developed an adequate understanding of the meaning of
the equal sign. The results of such research highlight that “the notion of ‘equal’ is
complex and difficult for students to comprehend” (RAND, 2003, p. 53) and,
moreover, that it should be developed throughout the curriculum (NCTM, 2000).
In this article, we report results from a longitudinal study that examined middle
school students’ understanding of the equal sign, the relationship between their un-
derstanding and their performance on problems using the equal sign, and the na-
ture of changes in their understanding and performance over the course of their
middle school years. The study is part of a larger, 5-year effort that seeks to under-
stand and cultivate the development of middle school students’ algebraic reason-
ing (Nathan & Koellner, this issue; see also http://labweb.education.wisc.edu/
knuth/taar/).
RESEARCH ON STUDENT UNDERSTANDING OF THE
EQUAL SIGN AND PERFORMANCE SOLVING
EQUIVALENT EQUATIONS PROBLEMS
Student understanding of the equal sign has received considerable research atten-
tion in recent years (e.g., Alibali, 1999; Baroody & Ginsburg, 1983; Behr,
Erlwanger, & Nichols, 1980; Falkner, Levi, & Carpenter, 1999; Kieran, 1981;
McNeil & Alibali, 2005). A primary result emerging from these studies is that
222 ALIBALI ET AL.