Many math education researchers come from one of two camps: (a) math teachers who want to know more about the psychology of the student, or (b) psychologists who want to know more about how students learn math. When these groups of researchers work together, good things can happen.
The University of Wisconsin has long been known as one of the best universities anywhere for math education research. Historically this tradition has included names like Henry Van Engen, Tom Romberg, Tom Carpenter, and many others. More recently, a new generation of researchers have been making their mark. In this five-author article, you see two math educators (Eric Knuth and Ana Stephens) and three math-specializing psychologists (Martha Alibali, Shanta Hattikudur, and Nicole McNeil) teaming up to do a longitudinal study of middle school students' algebraic reasoning.
Previous research had indicated two key ideas: (a) a proper understanding of equals and equivalence is key to success in algebra, and (b) the equal sign and equivalence is misunderstood by students at all ages. While some of this previous research is very good in its own right, the longitudinal aspect of this study helps it stand out. Commonly, and incorrectly, students hold an operational view of the equals sign. To those students, they see the "=" sign as meaning "do something." When students encounter a problem like \(3+5=\) they think the equal sign is a prompt to "write the answer," which in this case is 8. Unfortunately, some of those same students will see a problem like \(3+5=x+2\) and still think \(x=8\). Not knowing what to do with the 2, they might also think \(x=10\) (because they're just adding all the numbers they see) or they think it's okay to write \(3+5=8+2=10\). Statements like this with multiple equal signs should look familiar to any math teacher who has watched students show their work for a multi-step problem, such as showing work for order of operations. This only makes sense to students who have an operational view of the equal sign, as to them it just means, "the answer to this step is." But that's incorrect. Instead, we want students to understand the equal sign as a relational symbol, one that is neither prompting action nor implying a direction to that action. Without this, solving equations in algebra has very little meaning.
For this part of their study, Alibali et al. studied 81 middle school students (62% white, 25% African American, 7% Asian, 5% Hispanic) from 6th grade through 8th grade. The middle school used the Connected Mathematics curriculum and introduced solving linear equations in grade 7. The students were asked to explain what they thought the "=" sign meant, and to understand their use of that sign they were given an interesting set of tasks. For example:
Is the value of n the same in the following two equations? Explain.
\( 2 \times n + 15 = 31 \) and \(2 \times n + 15 -9 = 31 - 9\)
Here the researchers apply what they call an "atypical transformation," and they look carefully at how students find n. Many students would solve by "doing the same thing to both sides" for both equations, a procedure they can follow whether they had a solid understanding of equals and equivalence or not. But by subtracting 9 from each side in the second equation -- something mathematically "legal" despite not being all that helpful in finding n -- you can more easily identify which students break with "standard" procedure and show an understanding of equivalence. Those students won't treat the second equation like a new problem and instead quickly see that whatever they found for n in the first equation must also be n in the second.
Not surprisingly, Alibali et al. found that students' understanding of the equal sign got better over time. Also not surprisingly, students who have the correct, relational view of equals are more likely to see equivalence relations and solve equations correctly, and the earlier they understand it, the better. At the beginning of 6th grade, about 70% of students had an operational view and only 20% had a relational view. (10% of students held some other view that didn't fit in these two conceptions of equals.) By the end of 8th grade, that balance had almost flipped: only about 30% still held an operational view while 60% had a relational view. That's a lot of improvement, but that improvement took a long time (3 years) and still 40% of students didn't have a correct and meaningful understanding of the equals sign by the end of 8th grade. Also, students who showed a relational view of equals sometimes slipped back into an operational view. Almost a quarter of the students in the study used a less sophisticated strategy sometime after using a better one. Lastly, even when students consistently defined the equal sign as a relational symbol, they didn't always recognize equivalence in problems such as the one above. It's these types of caveats that make teaching equals and equivalence a tricky business.
So if you're a teacher with students having trouble with the equal sign, what can you do? More research needs to be done in this area, but one thing you can do is be more aware of your students' "compulsion to calculate" (a clever term used by Stacey & MacGregor, 1990, p. 151, as cited by Alibali et al., 2007, p. 245). Try giving students a task like the one above, ask them to evaluate the task for a minute or two without touching their pencils, and then find n. Afterwards, have students describe their strategies and solutions. Also, if you want to avoid the complications of using a variable, you can give students a number of statements and see if they can spot the ones that are equivalent. (Alibali et al. suggest statements like 9 + 5 = 14, 9 + 5 - 3 = 14 - 3, and 9 + 5 - 3 = 14 + 3). Also, try putting the unknown to the left of the equals sign. If you ask a student to solve \( \underline{\hspace{0.25in}} = 3 + 4 \) and they tell you the problem is "backwards," then you know they struggling with an operational view of equals. Giving those students more problems where the "answer" doesn't come "last" (to the right or at the bottom) will help the student expand their understanding of what equals really means.
References
Alibali, M. W., Knuth, E. J., Hattikudur, S., McNeil, N. M., & Stephens, A. C. (2007). A longitudinal examination of middle school students’ understanding of the equal sign and equivalent equations. Mathematical Thinking and Learning, 9(3), 221-247.