It's really a bit malformed, if you think about it. Mathematically what's really happening when you hit that button is a request to "simplify" the numerical expression that you've typed in so far. So perhaps it would be better if the button were labelled "simplify" or "evaluate" -- or maybe a "total" button like on cash registers, or some abbreviation along those lines.

Possibly the malformed understanding of the equals symbol (thinking that a simplified number always goes on the right side) is due to the hundreds and thousands of times that students have used a calculator "=" button by the time the issue matters in algebra?

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Interesting, I had similar thoughts a little while back: http://blog.mathed.net/2011/09/you-can-press-enter-but-think-twice.html and then later summarized that research (which is more prominent, I believe, than the Texas A&M research you mentioned in a previous post) at http://blog.mathed.net/2011/11/rysk-alibali-et-als-longitudinal.html. I think understanding of equals is something many teachers take for granted, but when misunderstanding rears its ugly head, things can spiral out of control pretty quickly.

ReplyDeleteRaymond, that's great! I would love to see the experiment you describe in your blog carried out. :-)

ReplyDeleteA secondary thing I would point to is the frequency with which some teachers write problems assuming an operational use of equals, like "3x + 5x = ?" (with no verb or direction). This pops up usually once even at my college's department-written remedial algebra finals. That drives me nuts.

And then, as a result, I get a weaker student in my stats class asking "will you always tell us what to find in a test problem?" (happened last week), as they have the impression that any dump of symbols has an implied action you have to take on them.

Delta, we completely agree Re. implied operations to be carried out. At least in my grade school education we were coerced, lest points be deduced, to supply full-sentence answers to most math problems. You had to write, formulaically, something like: Ans.: The sole value of x that solves the equation is 5.

ReplyDeleteIn line with that expectation on the students, the problems were usually written out in full, and there was little there that was implied. Even simple, "obvious" problems, had a prelude imploring "carry out the arithmetic operations as indicated". This was all 30 years ago, so I might be, perhaps, providing a rose-colored retrospective. You can never be sure, even though I hated school at the time, so at least my internal bias would be, I think, more negative than positive.

^ Similar anecdote from this past week: On the first day of my classes, I give out a one-page handout of all the definitions introduced in the class. I have one student approach me at the end, thanking me profusely. "My last teacher, I begged him to tell us the definitions of things, and he said don't worry about it, just do the problems", which just about broke my heart.

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