Apologies, Kent, for not posting my response to your piece yet. It’s in the works. Part of the reason why I haven’t responded is (sick baby/jewish holidays/new school year/baby gets sick again) your normal life stuff. But I also feel like my understanding of integers is changing every few days right now. I need to capture a bit of learning I’ve had over the past week or so, to remember what this sort of learning about teaching math can be like.

This is a story about how I learned something new about how kids can think about subtracting negatives.

It started two weeks ago. I asked my Algebra 1 kids to answer some questions about integer addition/subtraction (taken from the Shell Center).

I was talking about this in the math office, and two other Algebra I teachers decided to give the assessment to their classes and (knowing the nerd that I am) share the results with me. There was one thing that I was truly surprised to find. I liked it so much, I tweeted about it:

(I think it’s really important to this story that I was genuinely surprised by this.)

I explained this to myself as a fairly sophisticated understanding of negative numbers. When I imagined what it would look like in class, I imagined relating the idea that (-8) – (-3) = (-5) to the idea that you can multiply both sides of an equation by -1. I thought to myself, hmm, that would be tricky to bring up in class, but it’s a very cool, sophisticated way to think about it.

I moved on, though the idea was still in the back of my head.

(I think it’s really important to this story that I am actively engaged in an attempt to better understand teaching/learning negative arithmetic with Kent. If I wasn’t engaged in this project, I might never have gone any further.)

The next step was murky. I was teaching my kids negative numbers (a little bit here and there, never our main curricular focus) and I was also reading articles and swapping emails with Kent.

Last night I threw out a question on twitter:

Kevin Moore, a researcher, replied with some names of researchers who work on understanding how kids think about negative arithmetic. This pointed me to Project Z, a big CGI-style research project about how kids think about negative arithmetic before and after formal instruction.

After a few hours of poking around (I got really into it!) I landed on this video of a kid (“Jacob”) thinking about -7 – x = -5.

Here’s the key part of his thinking, the thing that excited and surprised me:

- He takes 5 cubes, counts them. “Pretend this is negative 5.”
- Puts 2 more cubes on top. “Plus 2.
- Then he says a bunch of interesting, incomplete sentence stuff.
- “7 – 2 = 5.”
- After some thinking, he writes, cautiously, “(-7) – (-2) = (-5).”

That made sense with something I had seen the Project Z researchers describe in a handout. They’d call this “analogical reasoning,” where the analogy is between negative numbers and whole numbers.

Now, I’m not saying this is some genius clicking on my part. But I was genuinely and honestly excited and surprised by all this. And it made me realize that the thing I had seen the kid do on the Shell Center assessment probably wasn’t the sophisticated equation manipulating that I saw. Probably, instead, they were describing some analogical thinking.

All this is making me realize that I’d never encouraged or helped my students think about negative numbers in this way. Would it work for every case? No, of course not, but thinking about (-8) – (-3) is hard enough for kids that having a way to think about it would be immensely helpful.

Then I connected this with a feeling I’ve had for a while. When looking at contexts for negative numbers with Kent I’ve felt that there really can’t be just one approach, that at best we’d need some sort of case-by-case instruction. Different models, analogies, strategies are helpful for different types of integer problems.

I had thought this through for contexts, but I hadn’t connected it with contextless thinking about negative numbers. With this, I was able to make a connection to what I know about teaching whole number arithmetic in early grades. I tweeted about it:

This has implications for my teaching of integers. I have a new strategy I can throw into the mix, new strategy I can keep my eyes out for in student work. I have a new focus for integer subtraction.

There’s always the meta-question: what allowed this learning to happen? Here’s what I’m seeing as significant in this story:

- Having a project helped.
- That the project was about student thinking about a particular topic was important for a lot of reasons. First, it’s the sort of thing different teachers on twitter can reasonably talk about. Second, student thinking about particular topics has obvious implications for teaching. Third, it’s something that researchers also work on, so I was able to be connected to Project Z by Kevin.
- It was important, I think, to be surprised. To be surprised you have to have expectations, though. I think my expectations were formed by looking at the large collection of student responses to the Shell Center task. That made this kid’s work stand out.
- I didn’t really understand the research until I watched the video of Jacob working on -7 – x = -5. I was surprised by that because I had tried to explain the surprising thing before.
- Being on twitter helped for two reasons in this process. First, twitter helped me to mark a that initial observation of (-8) – (-3) = (-11) as surprising. I used twitter as a sort of bookmark for an interesting idea. Second, without twitter Kevin wouldn’t have shared Project Z with me.

This story is about changing the way I see student learning, but it’s supporting the way I had been thinking about my own personal learning. This story increases my confidence in the potential of focusing on understanding student thinking for improving my teaching.

I was writing lessons for systems of equations, and had thoughts similar to your idea of multiplying the entire equation by -1. That’s exactly what you do to solve by elimination – but I’m putting that lesson off until later. I realized trying to explain to my kids that you can treat the whole equation like a number is going to be a whole thing on it’s own.

I mean think about it – I can not only multiply a number by 3, but I can multiply a WHOLE PROBLEM by 3. And the subtract it from another problem.

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Yeah, that’s a pretty sophisticated idea!

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I think something your first tweet and Jacob’s strategy have in common is that “negative” becomes sort of a unit. 7 − 2 = 5, so 7 negatives − 2 negatives = 5 negatives. By similar analogy, 7 halves − 2 halves = 5 halves, or 7

x&minus 2x= 5x.This type of subtracting a negative is easy to model physically, as in Borenson’s Hands-On Equations or with algebra tiles. But it becomes much less intuitive (more contrived) when you try to model positive minus negative, or even low-absolute-value negative minus higher-absolute-value negative.

I’m glad you shared the Project Z materials. It will take me quite awhile to get through them all, but the glimpses into student thinking are fascinating!

For instance, Violet’s understanding of negatives as positions on the number line where you can start or stop counting (but not as numbers that you can do operations with) matches up with most of the answers on your “Directed Numbers” worksheet above—enough so that I wonder whether the title term “directed” influenced the child’s mistakes. That is, you start on the number line position of the first number. The plus or minus sign gives your

direction, the way you will count. And then you count the (absolute value) second number of spaces in that direction.But answer (e) doesn’t fit that pattern. I would love to hear how the child explained that answer.

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Your “unit” point is such a great way of putting it.

FWIW I don’t think “directed” made that big of a difference. I don’t think my kids knew what “directed” meant.

Sadly, we have to guess how she would explain (e) for now. I’ll try to follow up.

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I think the particular representation of the negative depends on the context of the problem, too. I teach Algebra 1, 2 and geometry, so as I’ve been developing definitions of basic operations with each class, I see that it depends not only on the algebra/geometry distinction, but also on the particular representation. On the number line, subtraction means distance. 9 – 5 means I am now at 9, and started at 5. So I walked 4 feet. But if I end at 5 but started at 9, I also walked 4 feet (hey absolute value!). The negative tells me only the direction. But then what about subtracting a negative? 8 – (-3)…I am at 8, and I walked from -3. So I made it 11 feet! Same thing with (-3) – 8.

But that analogy is harder when we starting talking about using negatives as direction in money problems where + means income and – means debt/payments. Unless every number comes signed (which they do implicitly, but we may want to do more explicitly).

I’ve used 8 – (-3) as P + 8 – (-3), where P is my pre-existing money. So I made 8, and lost a debt of 3. Now I’m $11 richer!

Again, it all comes down (in my first 3 weeks of school) to defining not only values, but even the operations themselves, in different contexts.

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