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Thank you for this - I'll be teaching physics this fall, and I will be using some variation of this mini-quiz to pinpoint these reasoning errors of the students.

Thank you for this very interesting article. It drives me insane when my students write something like 3+5=8-7=1. I think we take for granted that they will understand the meaning of "=", when according to the figures quoted above they certainly do not! I might use those five questions at the start of the article with my students this year.

This is clearly a 'hot topic' at moment! I agree with you about precision in the use of language (actually the impact of teaching language on the learning of Mathematics is my main research interest!). I have already made comments about this on two other blogs: Robert Talbert on "calculator syndrome" (as I call it) and George Woodbury's thoughts on simplification of algebraic expressions. Your readers may find them interesting too.

Oh, and since you mentioned speed... does it annoy you when people talk about "driving at a high rate of speed"? Surely they mean acceleration, don't they? Or, do they just mean driving very fast...


Colin: Yes! I am an advocate for talking simple: "I was driving fast" is accurate and works just fine.

I like: "I was turning the corner and I wasn't even accelerating..." and anything Rusty Wallace says when he's doing commentary during a race.

I wouldn't characterize this as a difficulty with understanding equals so much as a difficulty with operator precedence. If you placed parentheses, making, for example, (_+3)=(5+7), I'll bet the failure rates change dramatically.

As a long-time programmer and computer scientist, whenever I write code that is meant to evaluate such an equation, I put in the parentheses, it's just simpler. I will furthermore add that the sort of thing that cox_dan points out above makes perfect sense if you are working on a calculator. It's exactly the steps you might take in a calculation: you might press '3' then '+' then '5' then '=', showing '8' on screen, then '-' then '7' then '=' again showing '1'.

Actually the translation for et al. into Texan is et y'all.

not all of us are observant enough to notice this, if you're noticing what I think you're noticing. the difference between boolean equals and the other kind. boolean equals is noted as "==" or an equals with three lines instead of two. this should be a simple and obvious topic because it has been studied for how many decades or hundreds of years? but it certainly isnt common curriculum. the most basic definitions of our most basic math operator.

I agree with Doctor Jay, the problem seems to be understanding of syntax, not semantics.

I was coaching a boy who had a bit of trouble with equations. So I got him to stand up, arms outstretched, and pretend to be a balance.

Then I asked him to add, say +5, to one side, and then asked what's happening? Then to ask him to add +5 to the other side, and asked again. This seemed to help.

An online friend sent me this example of abuse of the equals sign from a textbook. This problem is propagating right from the top.

I guess that I had good math teachers in high school because they always taught that the "=" symbol read "true" not equals.

Yes! This couldn't be more important. Thank you. Having tutored HS math students, many read = as the enter key on a calculator--no understanding of the symbol having meaning.

Interesting, but the conclusion about the 160= part really baffles me. Mind you: i'm just somebody who finished high school and who does nothing with maths. When I did the 'test' I also finished it with something like 160=120+40. Not because I expect some kind of operation after the blank, but because I thought 'well of course 160=160! That is so bloody obvious. But this test obviously requires me to do some addition tricks, so, if the testers want me to do a trick, I'll do it'

I think it's extremely logical to fill it out like that. When you do a test, you're always second-guessing the people who put it together. It's not because I thought the equal sign requires an operation, it's because I thought the testers required an operation. It has nothing to do with understanding the equal sign.

My answers:
A. False
B. False
C. False
D. True
E. False

Did I get them right?

[Note: Given that the author of this paean to precision amusingly failed to actually ask a question, I substituted in my own:
"Assume __ = 0. Are the following equations true or false?"]

Maybe I'm just weird, but when I saw __+3=5+7=__, I initially assumed that the definition of + was what was really at stake, that some kind of cyclic group or modular arithmetic or something was involved. (Upon further reflection, that doesn't work either, because it would be mod 3, but there are numbers larger than 3 in the equation.) It did not occur to me that the same value (represented in the problem's symbology by an underscore) could stand in for different numbers on the same line of equation.

Of course, it's possible the problem's instructions (which I didn't see) hinted at that, with wording like "fill in each blank with whatever number will make the equation true", or somesuch.

2+3=5+7=12 makes perfect sense to someone who is using a calculator. Since American kids use calculators for even the most trivial arithmetic, it's no wonder they would see this as a reasonable math "sentence". Kids in many places around the world still do arithmetic manually, and seldom see this kind of strung-together expression.

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