At the close of last year, I was working with my fifth grade students on equivalence. I gave them the following question after seeing something similar in an NCTM monthly article:

My hope with this question was to understand how my students interpreted the equal sign. As I had guessed, they had two understandings equality and “the answer.” In 5th grade, I expected these two responses and we worked together to clarify the meaning.

As this school year began, I was curious how my 12th grade Calculus students would answer the same question. I gave my seniors the same prompt and I got one of two responses below.

It was extremely interesting to me that both 5th graders and 12th graders had the same misunderstanding. It made me think about how many other mathematical misunderstandings students acquire through grade school without being corrected and as teachers what are we doing to help prevent these subtle, but critical misunderstandings.

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For my chemistry students, there’s always a few I have to fight regarding equals signs versus arrows when writing chemical equations. The two sides of chemical equations are NOT equal, it’s a process!

Is there just some sort of comfort in using equals signs, since they’ve been in math for their entire school career?

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I’m always fascinated by the way technology sometimes reinforces misunderstanding. For example, the “4 + 23 = 27 + 6 = 33” misunderstanding makes perfect sense if the student uses a calculator that behaves that way.

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Yet another reason to prefer RPN calculators that do not have an = sign.

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Also, any student who has done any programming has probably seen something like:

counter = counter + 1

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mantap gan perkembangan teknologi sekarang, , klw kita gak ikuti bisa ketinggalan kereta , Aerith

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While all the suggestions above are interesting, I really think this is explicitly about what exposure kids have. Much of what we teach is geared toward teaching kids the process of adding, or the process of mutliplying, etc, etc. If we have more opportunities to talk about equivalent expressions, and not just teaching kids how to compute, then they’d have real exposure to equations with multiple terms on the left and one on the right (3+4=7), or multiple terms on both sides (3+4=2+5), or even multiple equal signs (7=1+6=2+5).

Where are there opportunities to discuss equivalence in your curriculum? We should tag them with a post-it to remind us to strike while the iron’s hot…

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A fascinating debate which can be resolved by giving the kids some MATHELONA problems to solve.

Visit Amazon.com/dp/B017DTRDTQ for some brilliant challenges.

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