# Reflecting on the ‘equals’ sign

Last December I tweeted this from the MAV Annual Conference (the Mathematical Association of Victoria):

What did you get for the gaps and for the final result?

If you said ‘4+7 = 8+3 = 1+10 = 11’, you are, of course, right. But plenty of people don’t get 11. Stop for a moment and wonder what they do instead. (I’ll tell you soon.) Unfortunately I didn’t record the source of the problem, or the data supporting the claim that plenty of teachers get it wrong, so I committed on Twitter to try it with my primary and middle pre-service teachers this year.

Four weeks ago I got a chance. I used the question as an opener for a lecture on equations. From 20 pre-service teachers, I got only two responses:

• Two-thirds got the right answer of 11.
• One-third got an incorrect answer of 24.

(Actually, the proportion of students who got it right was higher than I expected!)

So what goes wrong to get an incorrect answer of 24? It stems from the interpretation of the equals sign. Many students — at all levels — interpret ‘=’ as ‘find answer to previous expression and keep working’. This gives: 4+7 = 11+3 = 14+10 = 24.

But this post is not about why this misconception exists. Rather it is about how we might help students make sense of this misunderstanding for themselves.

Recently, at NCTM 2016 in San Francisco I recounted this story separately to Andrew Stadel, Jana Sanchez and Tracy Zager (clang!). I idly wondered what the response might have been if I had instead asked: ‘4+7 = 3+_ = 10 + _ = ?’. My conjecture is that many more students would get this revised problem right. I suspect that finding a number already in the spot where you plan to write ‘the result so far’ is just enough to cause you to pause and reflect. In that moment of reflection, the ‘=’ sign becomes more meaningful.

Andrew asked what happened when I asked the students to try this revised problem. Good question! I hadn’t tried it because I only thought of it while reflecting on my experience out loud (that’s another blog post right there).

We quickly sketched out a plan:

1. Ask students the original problem: ‘4+7=__+3=__+10=?’.
2. Without any discussion, ask students the revised problem: ‘4+7=3+_=10+_ =?’.
3. Ask if anyone wants to revise their answer to the original problem.
4. Share experiences and discuss what happened with their thinking.

This simple revision to the lesson plan, from ‘try, correct, move on’ to ‘try, revise, reflect’, means students are more reflective about their own learning. I’m excited to try it out next time I introduce equations, which will probably not be for at least another year.

This misconception is well-documented. But, in wondering, discussing and reflecting on it with others, it became a whole lot more meaningful to me.

So, have you had experience with this misconception? Where do you think this misconception comes from? Do you have ideas how to help students correct this misunderstanding? I’d love to hear if you try the sequence outlined above. Please share in the comments.

## 11 thoughts on “Reflecting on the ‘equals’ sign”

1. goodnewmusic says:

Interesting issue. Will it when I get a chance and let u know.

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• Amie says:

Thanks for posting the first comment on my blog. If you try it out, let me know how it goes!

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2. Deb Downes says:

Do you think that the misconception about “=” also relates to a misconception of brackets? Or what I might call the “parcelling up” of pieces of information. When I look at that problem my process is 4+7=(something+3)=(something+10)=? So, therefore 4+7=11, replace the appropriate “somethings” to provide the solution. Possible to introduce BODMASS as well as clarifying “=”? I’m sorry, I’m not sure about what age students you’re targeting here:)

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Hey Deb! Thanks for adding your thoughts. My students are primary and middle pre-service mathematics teachers. I’m primarily aiming to address their misconceptions, but also to expose potential misunderstandings of their students. (I’m a little terrified of their tiny students though; the concepts might look deceptively simple but require careful, deliberative planning to teach.)

I suspect that your suggested use of brackets is enough of a visual disruption to cause students to pause and reflect.

You’ve added another layer that I’d like to try out!

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3. Liz says:

My only comment is that I wouldn’t change the numbers, I’d have ‘4+7=__+3=__+10=?’ and then ‘4+7= 3 + __ = 10 + __=?’ with the only difference being the order. Less ambiguous.

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Thanks for picking up my mistake; that’s what I meant to write! I’ve updated the post.
* 4+7=__+3=__+10=?
* 4+7=3+__=10+__=?
and
* 4+7=__+3=__+10=?
* 4+7=8+__=1+__=?
or other variations.

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4. millikinjax says:

I’ve always rewritten the equivalent equation below. However, more and more students are coming to me writing them to the right on the paper as if they are typing or writing a letter. I wonder if rewriting equivalent equations below was something I was taught to do or if it is what I picked up (the same can be said for my students). Why would we want students to reason with lots of equals signs in the same sentence?

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• Amie says:

‘Why would we want students to reason with lots of equals signs in the same sentence?’

That is a very good point. I can’t think of a reason — except, perhaps, to expose this misconception.

I was taught ‘one equals sign per line’. But I see lots of equals signs wandering across the page, where ‘=’ is used to mean ‘now I did this’. I’m not sure where they pick it up (I teach University students). I should dig into this more.

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5. Christine Newell (@MrsNewell22) says:

Great post, Amie!

‘Why would we want students to reason with lots of equals signs in the same sentence?’

Such a good question. It makes me think about some of the work we’ve been doing with number talks in elementary classrooms. We talk to teachers about recording student thinking with one equal sign per line to attend to precision. If a student says they “multiplied 3 by 4 which is 12 and added 5 to get 17” when looking at a dot image, many teachers are apt to write 3 x 4 = 12 + 5 = 17… a “run-on sentence” in math. This is neither true, nor precise.

I think there is value in exposing students to multiple equal signs in the same sentence to really drive home the idea that the equal sign means that all expressions are equivalent, not just “here’s the answer to what came before it”. We start this work in 1st grade in the Common Core State Standards where 1st graders are expected to “understand the meaning of the equal sign”. Deep stuff.

Great posts this week! We met briefly at the Math Forum booth at NCTM I believe, when you were working with Andrew Stadel on clothesline equations. Good stuff!

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• Amie says:

Very thoughtful comment, Christine; thank you. This insight is very helpful.

How good was NCTM, and the Maths Forum booth. Glad to have met you there — I can see I need to set aside some time to have a good read through your blog!

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