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 for x, you might see:

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?

In addition to the problem you describe, we then build onto it huge confusion about the differences among equations, functions and expressions. Those students don’t stand a chance!

Force students to use ; for separate steps

So 4×3=12-2=10 becomes

4×3=12;

12-2=10;

Mind you, modern calculators like the CASIO Graphical calculators have EXE buttons. I think it has more to do with enforcing and explaining the mathematical and technological “vocabulaire”, and especially the difference between equivalence (pun intended) and the operator(s).

@John: Yes. Many students think the terms “equation”, “function”, “expression”, and even “number” and “graph” are all synonymous. Language is a huge issue- just like with spoken languages, it seems that if you don’t take pains to get the usage right very early on — as in, elementary school — then it only leads to intractable issues later.

@Christian: I had forgotten about the Casios. Thanks. And, being a MATLAB person, I like the use of the semicolon to mark a new step (or end of the current step). Although in MATLAB the semicolon suppresses the output.

Ah, this was one of the first things that our high school teacher emphasised. Since the repeated use of = as shown makes “false statements”, he encouraged the use of => as:

3x-2=10

=> 3x = 12

=> x = 4

I actually commented on George Woodbury’s blog (Thoughts on Equal Signs http://georgewoodbury.wordpress.com/2010/08/11/thoughts-on-equal-signs/) before reading yours, but you could reasonably argue the = is an operator in that a non-trivial equation will have two different sides — there is some kind of transformation, albeit not of value…

The key point for me is there should be two and only two sides to an equation (at least to begin with). I have to say I still have a little ‘cringe’ when I see:

2(x-7)+4 = 2x – 14 + 4 = 2x – 10

even though it is technically correct (unlike the example you provided!).

The calculator-key issue seems to me to come from assigning “=” to “is equal to the answer” rather than “has a value of”.

Colin

Finally got around to reading this post. My conclusion: Reverse Polish Notation, FTW!

More seriously, I think I buy the argument here. I certainly find it frustrating when students misuse the equals sign in the way you describe in your post.

I’m reminded of another use of the equals sign, the “assign equal to” command used in programming. Of course, that use is often denoted with a “:=” instead of a “=” for a bit more clarity.

I agree with your comments here. I’d just posted on this earlier today:

http://gasstationwithoutpumps.wordpress.com/2010/09/03/equal-sign/

As a long-time user of an HP-15C, I want to make a minor correction to your first example (3+5) – the keystrokes are: 3, ENTER, 5, +

The ENTER is used to separate the input values, so the operator “+” can clearly be seen as actually doing the work of adding the two values and giving the result.

There really is NO equivalent to the = key on an RPN-only calculator.

As far as explaining correct usage of = in equations, the analogy usually given in my long-ago classes was a balance — both sides were in some sense “the same”, and as long as you do the same thing to both sides, the equality remains.

(Just to complicate things, in many common computer languages, “=” actually IS an operator – assignment – with strict rules on what can be in the “left hand side” and “right hand side” of an assignment statement. This doesn’t help explaining equals in a mathematical sense at all.)

Thanks for the clarification Andy. Was/is this the syntax on later HP models? I used to use an HP-28 and later an HP-48 — neither of them work now, but what I wrote was my recollection of the keystroke sequences for each. But I could be recollecting wrongly.

Andy is quite correct. I still use an HP calculator and have since the days of the HP-45 (my current model is a 32SII). The ENTER is used to put things on the stack, the +-*/ operators to pop the top 2 elements off the stack, combine them, and put the result back, so 3 ENTER 5 + would result in 8 on the stack. There are no parentheses or “=” on a proper RPN calculator. (HP makes some non-RPN calculators and some bastardized ones that confusingly try to mix RPN and conventional notation.)

I have this issue, as well. On the other hand, I’ve seen the opposite problem. Students put various expressions all over the paper in a “random” way and then somewhere in the middle write “x = 4.” I guess it’s more technically correct with the rest of the stuff just being thought process written in the margins, but it makes grading those papers hard to follow.

The biggest place I see this misused is when you have limits and students don’t write the “lim” part each time.

We cannot blame students for getting confused about what ‘=’ means.

We tell them it means one thing in calculus, then we tell them it means something completely different in MATLAB.