Tag Archives: integers

A few thoughts on integer operations

Responding to a request here.

The short version is that Dan got it right (go ahead and give it a read if you haven’t already).

I do have a bit to add to the conversation, though. Dan pointed to the difference between being a teacher who views engagement through the lens of being useful in the world outside the math classroom and one who views it through the lens of curiosity. That’s a really nice distinction.

I want to point to an additional subtlety.

The formal mathematical view says that the distributive and associative properties of multiplication and addition ought to carry over to integers, and deduces the relevant result. The exact train of thought, and where it begins depends on the grade level. Consider Hung-Hsi Wu’s thoughts on the matter:

The key step in the correct explanation lies in the proof of (-1)(-1) = 1 (as asserted in the grade 7 standard). Pictorially, what this equality says is that multiplying (-1) by (-1) flips (-1) to its mirror image 1 on the right side of 0. A more expansive treatment of this topic in accordance with the CCSMS [sic] would show that, more generally, multiplying any number by (-1) flips it to its mirror image on the other side of 0.

The approach suggested by Brian is formal as well, but it’s different from the one above. It doesn’t tell it asks.

I cannot overemphasize the importance of this difference.

Of course it is appropriate to tell kids stuff sometimes. Of course it is. But there is far too much of that going on in classrooms already. Wu is concerned with steering this telling in a mathematically correct direction. That’s fine.

But I don’t want Griffin and Tabitha‘s mathematical educations to depend on better telling. I want them to explore and to wonder. I want them to commit to their ideas and see what the consequences of those ideas are, and to revise their thinking when their present ideas are not good enough to explain what’s going on in the world.

And what I want for my own children is no different from what I want for my students, and no different from what I want for all children.

There is a place for good, mathematically correct explanations. I want kids to experience those when they’re the right move.

More importantly, I want them to learn to think for themselves.

Appendix 1

I am working this summer on an article about integer operations that I’ll submit for publication. If you have an interest in such things, keep an eye on Twitter; I’ll be looking for a couple of critical readers in a few weeks.

appendix 2

I attended a lovely session on integer operations at the Minnesota Council of Teachers of Mathematics spring conference back in April. Two University of Minnesota grad students, Christy Pettis and Aran Glancy, presented a useful framework for characteristics of good integer contexts. I now pass it along to you.

Clear and logical opposites. Integer contexts should have clear and logical opposites. An important point here is that money and debt are not clear and logical opposites for many kids. If I have 3 dollars and owe 2 dollars, it is not obvious to kids that this is the same as having 1 dollar. Indeed in many respects it is not the same. Credit and debt are logical opposites, but more abstract. This may be inherent in working with integer operations.

Net value. The context needs to be able to support the idea of numbers as being net values. Kids should be able to reason in the context about 2 as 3—1, or 1+1, or 3+(–1), etc. Not all contexts support this (cf: chip boards).

Zero is not empty. This follows from the net value idea, but emphasizes the special role of zero in the integer system. In particular, the context needs to support seeing zero as the state of the existence of an exact set of opposites.

Guess the temperature

Griffin and I play a little game called Guess the Temperature. It goes about how you would expect. We step outside on the way to his bus. I ask him to guess the temperature. If I don’t already know, I get to guess after he does. If I do already know, I don’t cheat; we just remark on how close his guess was.

In Minnesota, this means we get to study integers.

Me: Griff, guess the temperature.

Griffin (eight years old): Two below zero.

Me: It’s three degrees above.

G: So I was off.

Me: Not by much, though. How much were you off by?

G: [muttering to himself, then loudly] Five degrees!

Me: How did you know that?

G: It’s two degrees up to zero, then three more.

Let’s pause for a moment here. You know how I just won’t shut up about CGI (Cognitively Guided Instruction)? It’s because they’re right. Children know mathematics before it is formally taught.

Consider the grade 6 (for 11-year olds) Common Core Standard 6.NS.C.5

Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

Griff pretty much has this nailed down and is making progress on grade 7. But no one has formally taught him how to subtract integers. He reasons his way through a problem by making sense of the relationships in the context. He can find 3-(-2) without knowing keep-change-change.

But it’s not just Griffin. CGI demonstrated that children—all children—develop mathematical models of their worlds that precede instruction, and that instruction sensitive to these mathematical models is better than instruction that ignores them.

Back to our conversation.

Me: So what if it 10 degrees out, and you guessed 3?

G: [quickly] I’d be seven off.

Me: Right. How do you know that?

G: Ten minus three is seven.

Me: Nice. Subtraction. Do you know that you can always express the difference between your guess and the actual temperature with subtraction?

So in that last example, you subtracted your guess from the actual temperature. You could do that with your real guess today.

So three minus negative 2 is five.

G: [silent]

By this time we were nearing the bus stop. I had offered this tidbit as an intellectual nugget to chew on, rather than a lesson I expected him to absorb. But that is what it means to have instruction be sensitive to children’s mathematical models.

Back in the saddle

It’s summertime here at OMT. After an initial flurry of posting back at beginning of June, things have slowed down. But only on the blog. Behind the scenes, we’ve been busy as beavers. The interns have been trained in, and we’re rarin’ to go.

In particular, I’ve had a number of really interesting (to me) conversations by email. You’ve heard of email right? Kind of like Twitter, only no character limit, so most people use full sentences and spell words correctly.

Except one colleague of mine who sends the following sort of crap to all faculty (he is referring to Rate My Professor, and I wish I were making this up):

I dfer 2 yor sensibilities in this matter as I’v not gone there in yrs b/c I feel that things lyk worrying about how ‘hot’ students find us contribute 2 grade inflation—wich is a horse I’ve ben riding 4 sum tym, now

I digress.

Email conversations. Right.

I’d like to tell you about a few of these over the next week or so. Like the one I had with Justin Yantho as we hashed out whether the following video represents good teaching (with thanks to Frank Noschese for alerting us to it).

In a sign of summer torpor (and of the OMT interns’ inexperience), the following is copied and pasted mostly verbatim from one of my replies in the conversation.

That video is “training”, not “teaching” in my view. Effective training. But training nonetheless. I think of this analogously to training dolphins to jump through hoops. Stimulus (hoop), response (jump), reward (fish). Self-contained system, disconnected from other behaviors.

I’ve only watched four minutes or so of video. But it’s offered as “exemplary”-both in the sense of being an example of what’s being promoted, and in the sense of being very good. And as an example of what teaching should look like? I’m opposed.

To be sure, I’m opposed to a lot of what I’ve seen in other sorts of classrooms (including all too frequently in my own!)

What bothers me the most here is related to my reaction to another video Frank Noschese sent around recently in which “Integers are important because they’re a state standard.”

In the former, teacher says, “Tell your neighbor about the four operations”. In the latter, the teacher says, “In a pair/share, talk about why is it important for you to understand integers?”

In both instances, we’re setting the standard that “talking about math” equates with “repeating previously stated information” rather than with “exploring ideas, wondering or processing”.

Now, I get that I’m drawing gross generalizations based on small sample size (short video snippets). But each of the videos is purporting to demonstrate an aspect of good practice. These people want us to learn from the teaching we are seeing; they want teachers to emulate the model. That makes it fair to pick the examples apart, I think. And I’m totally ready to eat my words if you find videos in either one of these sites that pushes kids to really think about mathematics.

But I’ve been in a lot of math classrooms over the years. Lessons rarely move from this sort of rote opening into a mode involving rich thinking and dialogue. Not never, but it’s rare.

This puts me in the mind of The Teaching Gap (a book I cannot recommend highly enough). The authors of that book draw on evidence from a well-designed international video study to outline important differences in classroom practice in three countries: the US, Japan and Germany.

The connection here is that the teaching we see in these videos is an extreme example of how US teachers spend their class time-recitation and practice, in stark contrast to how Japanese teachers spend their class time-problem solving and discussing ideas.

And I haven’t even addressed the error(s), right? Why does the set of “order of operations” have six elements if there are only four operations? Is exponentiation not an operation? Why does the opening example involve an operation about which she does not speak?

Thanks to Justin for giving me permission to reference our conversation. As you can see, I didn’t really let him get a word in edgewise. He did a fabulous job of arguing back, though. If you’re not following him on Twitter, do so now.


The New York Times highlighted an elementary curricular innovation today. It was my first encounter with Jump. The authors of the materials are reporting phenomenal success with students. They attribute the success to “breaking down math to its component parts”. At first blush, this sounds like the Saxon drill and skill rhetoric.

But then the example suggests otherwise:

Take the example of positive and negative integers, which confuse many kids. Given a seemingly straightforward question like, “What is -7 + 5?”, many will end up guessing. One way to break it down, explains [Jump math founder] Mighton, would be to say: “Imagine you’re playing a game for money and you lost seven dollars and gained five. Don’t give me a number. Just tell me: Is that a good day or a bad day?”

There is a subtle but important difference between this example and a standard classroom technique. It’s the directive, “Don’t give me a number.” If this example is representative of the spirit of the materials, then sense-making is an important part of the curriculum. More to the point, using students’ intuitions about their lived experience in the real world to draw mathematical conclusions seems to be important.

I’m eager to learn more.