October 23, 2006 | Volume 4 | Number 10 Middle School Math Textbooks: A Simplistic Approach to the Equal Sign?

Middle School Math Textbooks: A Simplistic Approach to the Equal Sign?

Dan Laitsch

The Question

How do the ways in which mathematics textbooks present the equal sign interact with students' operational understanding of “equals”?

The Context

The equal sign is an important mathematical symbol that represents a variety of interpretations depending upon the way it is taught to students. At its most simplistic, the sign can be interpreted as a symbol requiring completion of an operation (e.g., 7 + 3 =__ means add 7 and 3). Mathematical instruction in elementary school typically approaches the equal sign in this manner, firmly entrenching in students an operational definition of the sign. As students engage in more complex mathematics, such as algebra, it becomes much more important that they understand the actual meaning of the equal sign—as representing relational equivalence. In other words, students need to understand that the equal sign is a relational symbol that implies an equivalence relationship between the quantities on both sides of the symbol (e.g., 5 + 6 = 3 + 8).

Historically, the assumption has been that young children cannot developmentally grasp the sign as a relational symbol. Recently, however, researchers have begun to question whether this is indeed the case, arguing instead that understanding is also affected by context, and that given the appropriate context (e.g., 4 = 4), even young children can generate a relational understanding of the symbol. Textbooks—and the ways in which they present the equal sign—are an important aspect of the broader discussion of context, because it is through texts that students most frequently interact with mathematical operations.

The Details

Nicole McNeil, Laura Grandau, Eric Knuth, Martha Alibali, Ana Stephens, Shanta Hattikudur, and Daniel Krill conducted the study highlighted in this issue of
ResearchBrief (see
below for full citation). The researchers wanted to examine how the equal sign was presented in middle school mathematics texts and how context related to student understanding of equivalence. The researchers looked at four texts, all from major publishers, for grades 6–8. Two were based on National Council of Teachers of Mathematics and National Science Foundation standards, while the other two employed a more skills-based approach that emphasized computation. The researchers created a random sample consisting of 50 percent of the pages from each book and then categorized the ways in which the equal sign was presented on each page on the basis of two top-level categories—an “operations equals answer” context (e.g., 5 + 4 = 9) and a “nonstandard” context (e.g., 6 = 6; 5 = 3 + 2; 1 ft. = 12 in.)—as well as two subcategories for the “nonstandard” context: one that included operations on both sides of the symbol (e.g., 5 + 4 = 3 + 6 or 3x + 6 = 2x) and one simply called “other” nonstandard usage. The “other” nonstandard usage was then further broken into three categories: operations on the right side of the equation (e.g., 4 = 3 + 1 or y = 2x); equations without operations (e.g., 12 in.=1 ft., 7 = 7); and nonequations (e.g., “Complete each statement with =, >, or <.”). Interrater reliability^{*}
was established on a randomly selected 20 percent sample of equations and ranged from 96–99 percent, depending on the context.

The skills-based texts were more likely to represent the equal sign in an “operations equals answer” context than were the standards-based texts; however, such presentation decreased as the grade level increased (for all four texts). The likelihood of seeing an equal sign in the operations-on-both-sides context did not differ across the four texts, although it did increase with grade level. Even so, the equal sign appeared in the operations-on-both-sides context only 5 percent of the time (on average across grades), with a maximum of 9 percent (in one of the grade 8 texts). The majority of occurrences across all texts, however, occurred in the nonstandard usage category.

Researchers were interested in examining the effect of such nonstandard use on student understanding of the equal sign as a relational symbol of equivalence. As a result, they designed two experiments to determine how much of a difference, if any, context made in student understanding. Both experiments drew students from the same public middle school in the Midwest. In the first experiment, 110 students in grade 6, 117 students in grade 7, and 93 students in grade 8 participated. At each grade level, the students were randomly assigned to view the equal sign in one of three contexts: operations equals answer; operations on the right side; and reflexive (e.g., 7 = 7). They were then asked to identify and interpret (via written response) the meaning of the equal sign. Student responses were coded as showing a relational understanding (e.g., “the amounts are the same” or “they are equivalent”); an operational understanding (e.g., “adds up to,” “add up the numbers,” “the total”); an unspecified interpretation (e.g, “equal”); or other (e.g., “I don't know”). Students were more likely to attach a relational interpretation to the equal sign when viewing the nonstandard equations, and students viewing the nonstandard presentations (right-side or reflexive) were equally likely to interpret the sign as a symbol of relational equivalence. As they aged, students were also more likely to see the equal sign as representing equivalence.

The second experiment paralleled the first and included 97 students in grade 6, 107 in grade 7, and 106 in grade 8. In this experiment, the students looked at the equal sign in only one of two contexts: a “right-side operation” context or an “operations on both sides” context. Students viewing the operations-on-both-sides context were much more likely to exhibit a relational equivalence understanding than were students viewing the nonstandard, right-side operation context.

The Bottom Line

Current middle school textbooks are much more likely to show the equal sign used in a nonstandard context than in a context that has operations on both sides of the sign. Although such nonstandard use is better at eliciting an understanding of the equal sign as representing relational equivalence than a standard presentation, the operations-on-both-sides context is a more effective method for helping students understand the concept of relational equivalence.

Who's Affected?

Middle school students in grades 6–8 were the focus of this study.

Caveats

The findings were consistent across the four textbooks analyzed, but there was significant variation between textbooks. Although the textbooks were from major publishers, it is impossible to generalize results to other textbooks, so teachers interested in this research should examine how the equal sign is presented within their own resources and practice.

^{*}
Interrater reliability refers to the extent to which two or more people observing the same situation are likely to exhibit the same interpretation of that situation.

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All comments regarding ReseachBrief should be sent to RBfeedback@ascd.org. To speak directly with an ASCD staff member, please contact us.

Dan Laitsch serves as ASCD's consultant editor for ResearchBrief. Laitsch is an assistant professor in the faculty of education at Simon Fraser University in British Columbia, Canada, and is coeditor of the International Journal for Education Policy and Leadership.

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