For the last month or so since I read this article, I’ve been unable to shake the idea of “mathematical worlds”:

Rather than describing the challenges of integer learning in terms of a transition from positive to negative numbers, we have arrived at a different perspective: We view students as inhabiting distinct mathematical worlds consisting of particular types of numbers…Proficient students and adults may also inhabit multiple mathematical worlds from moment to moment.

This metaphor completely fits my experience with children. There is often doubt about what “world” we’re operating in. Are fractions allowed? Are we using *i *right now, or does x^2 = -9 have no solutions?

(The other relevant metaphor is of a *game*, as in “I didn’t realize fractions were allowed.” Is there a connection between the metaphors or *world *and of *game*?)

One thing this metaphor helped me realize is that my students might bring up negatives in my class more often if I simply gave them permission to. How do you give permission? I decided to bring them up right away with my 4th Graders. (I culled the questions from here.)

I handed out a piece of paper to each student. I asked them to work completely on their own. The first question was 9 – 10 = __. Here were the responses:

- 1 (7 kids)
- -1 (4 kid)
- ? (1 kid)

Next I asked, 3 – 5 = ___, and received similar responses. So far, kids were pretty confident.

The next question led to protests, though: 6 + __ = 4. Some protests, some talk of -2. (*Wait, are we allowed to use negatives?*) I collected their papers and, since kids seemed to have what to say about this, I asked volunteers to share their thinking.

First went E, who explained that since 6 – 2 = 4, this had to be -2. I wasn’t sure exactly what she meant by this, but then H objected to this whole thing in an incredibly helpful way:

H: “Wait, I don’t think that makes sense. How can we say that we’re doing 6…plus minus 2?”

Me: I have no idea what you mean so I’m going to offer a belabored explanation of how we’re not adding “minus 2,” rather we’re adding “negative 2.”

H: “I mean like I don’t see how you can add minus like that.”

Me: Wait, I think I understand what you’re trying to say. You’re saying it’s like if we said **3 times divide 2**, like, what would *that* mean?

H: Yeah, that doesn’t make any sense.

But then a bunch of other kids jumped in. They were *totally *able to make sense of “6 plus minus 2.” They said you were **adding a minusing of 2**. This is subtly different than anything I’d ever before understood kids to say. They weren’t treating “-2” as a point on the number line or even as a single number at all. They were making sense of -2, in this context, as an *action*. The *action *of subtracting 2.

All the while this was making me nervous. Middle and high school students have trouble distinguishing between the three slightly different ways in which we use the minus sign in mathematics. (See: “High School Students’ Conceptions of the Minus Sign”)

Should I have said something to keep kids from thinking of “6 + -2” as “6 plus a minusing of 2”?

The more I thought about it, though, the more confident I became that this was an incredibly productive way to start thinking about adding a negative. Kids who think about 6 + -2 as “adding a minus 2” are already along the road towards thinking of the *process *of subtracting as an *object* (*a* subtracting). There is more work for such a thinker to do — they have to come to think of this *subtracting of 2* as the sort of thing that you can operate on. (What does it mean to add 3 to a minusing of 2, as in -2 + 3?) But there is a clear conceptual path for these students to take.

(I’m thinking here of Anna Sfard’s idea of interiorization, condensation, and reification.)

This all is a brand new idea to me, a way of making sense of negative arithmetic that I’d never understood before. I continue to be blown away by the variety of ways that kids make sense of arithmetic with negatives — each problem situation seems to allow for slightly different conceptions. Really making sense of negative arithmetic might involve negotiating many different ways of making sense of many different types of problems. It’s only once kids have a few different mental models that they can start uniting their thoughts into one **big **idea of negative numbers.

We’re going to keep this up in 4th Grade. Actually, this all happened yesterday, and today kids ended up digging into the number line, its mirror-like symmetry and the “realm of the negatives” (which J connected to Minecraft and Portal, god save the children.)

Kids seem to be having fun exploring this realm, and they’re doing good math. Why not give kids permission to enter this world?

The worst thing in arithmetic/algebra is the “binary operation”, as in “2 + 3 is the sum of 2 and 3”. The more logical and more common sense view is “add 3 to 2”, and “multiply 2 by 3”. Then the process is “start with 2 and add 3 to it”. This was dealt with in the 60’s using a bar or a mini-minus to indicate the negative number, and the minus sign to indicate subtraction. There are difficulties with this, so my view is to separate the signs in 2 + -3 and read it as 2 + (- 3), which can formally be written as 2 – 3.

See Whitehead from 1911:

https://howardat58.files.wordpress.com/2015/07/whitehead-numbers-2b.png https://howardat58.files.wordpress.com/2015/07/whitehead-numbers-3a.png

Have some fun with this!!!!

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Oh my! So much to process!

I am coming to realise that my own understanding of this seems to be different to other people. Right now I don’t see the single “-” next to a number as an operation at all! I see it as a way to name something. “-2” is the name of a number that has a special relationship to 2. “- -2” is the name of a number that has a special relationship to “-2”, which itself has a special relationship to 2. It’s never been an issue to me to call this number “minus 2”, since I have always been aware that words have different meanings in different contexts.

The most amazing thing to me to realise is that I may have originally seen 6 + -2 as “a minusing of 2”. Whenever I’m called upon to explain why this works, my first instinct is to explain -2 as a number which has the effect of subtracting 2 when you add it. In that sense, I seem to be seeing -2 as an objectification of a minusing of 2.

Thank you to your kids for this amazing insight, and to you for listening to them.

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I wonder if there’s a distinct learner profile that tends to get caught in worlds more often. I’m guessing that students with a performance orientation get caught by worlds more than those with a learning orientation. That is, if a student thinks a correct answer is one that the teacher will accept, they’ll naturally want to ask the teacher what game is currently being played. But if the student thinks the correct answer is the one that satisfies the constraints of the problem itself, they’re less likely to be trapped in one limited world. Everyone gets trapped in worlds sometimes. Just wondering if the extent of trappiness is correlated with learner orientation.

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I enjoyed reading about this process of students prodding and poking at the idea of adding a negative. Forgive me for being greedy for more: I will be hoping and waiting to hear about their conversations around subtracting a negative number.

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