Interesting report here (via Reidar Mosvold) about American students’ misunderstanding of the “equals” sign and how that understanding might feed into a host of mathematical issues from elementary school all the way to calculus. According to researchers Robert M. Capraro and Mary Capraro at Texas A&M,
About 70 percent of middle grades students in the United States exhibit misconceptions, but nearly none of the international students in Korea and China have a misunderstanding about the equal sign, and Turkish students exhibited far less incidence of the misconception than the U.S. students.
Robert Capraro, in the video at the link above, makes an interesting point about the “=” sign being used as an operator. He makes a passing reference to calculators, and I wonder if calculators are partly to blame here. After all, if you want to calculate 3+5 on a typical modern calculator, what do you do? You hit “3″, then “+”, then “5″… and then hit the “=” button. The “=” key is performing an action — it’s an operator! In fact, I suspect that if you gave students that sequence of calculator keystrokes and asked them which one performs the mathematical operation, most would say “=” rather than the true operator, “+”. The technology they use, handheld calculators, seems to be training them to think in exactly the wrong way about “=”. What we have labelled as the “=” key on a calculator is really better labelled as “Enter” or “Execute”.
In fact, the old-school HP calculators, like this HP 33c, didn’t have “=” buttons at all:
That’s because these calculators used Reverse Polish Notation, in which the 3 + 5 calculation would have been entered “3″, then “5″, “+”, then “Enter” — and then you’d get an answer. What HP calculators label as “Enter”, on a typical modern calculator would be labelled “=”, and in that syntax lies a lot of the problem, it seems.
The biggest problem I seem to encounter with “=” sign use is that students use it to mark a transition between steps in a problem. For example, when solving the equation \(3x – 2 = 10\) for x, you might see:
\(3x – 2 = 10 = 12 = x = 4\)
The thought process can be teased out of this atrocious syntax, but clearly this is not acceptable math — even though the last bit of that line (x=4) is a correct statement. If the student would just put spaces, tabs, or even a semicolon between the steps, it would be a big improvement. But many students are so trained to believe that the right answer — the ending “4″ — is all that matters, they have little experience with crafting a good solution, or even realizing that a mathematical solution is supposed to be a form of communication at all.
What are some of the student misconceptions you’ve seen (or perpetrated!) with the “=” sign? If you’re a teacher, how have you approached mending those misconceptions?