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.

8 responses to “A few thoughts on integer operations”

1. “Zero is not empty” — per http://www.youtube.com/watch?v=Wre9P3n7TNY eh?
Many of my adult students *do* understand debt/credit analogies really well, but generally I don’t assume they will and prefer opposites like up and down, in and out, and cold and hot. Around here, we do go below zero enough for that to have *some* meaning, though I’d want the actual thermometer to look at. What were some of the “clear and logical opposites” that were suggested?

• Christopher

Re: Zero pair of eyes…Oh my!

2. Did Pettis & Glancy suggest models, then? This makes me think of charges (like electron/proton) – pretty abstract, and changes in quantity (+/-) – also pretty abstract. But I like the framework as a means to explain what is tough about the models we use.

What always amazes me is how many models for teaching integer operations just repeat the same arbitrary instructions in the new context.

3. suevanhattum

Great discussions. I wonder if I could get pre-calc students to question why a negative times a negative should be positive.

4. Aran Glancy

Thanks, Christopher, for a) the shout out and b) summarizing our presentation so clearly and succinctly!

@xiousgeonz and @John Golden: really good analogies are hard to come by but we did identify a few that we thought were pretty good. We’d love it if people could come up with more! Protons & electrons definitely work, if kids know the science content, but the one we’ve spent the most time with is “floats” (positive analogs) and “anchors” (negative analogs). When you put something heavy in a boat (i.e. an anchor), the boat sinks a little to compensate. Conversely, if you remove something heavy (i.e. subtract a negative), the net effect on the boat is that it rises up a little relative to the water.

A number sentence like 4 – (-3) = 7 actually uses the negative sign in two different ways: the first is a binary operation indicating subtraction, while the second is a unary operation indicating a position or direction. Many analogies we use to teach integer operations don’t allow for this distinction to be clear. In the ‘floats & anchors’ analogy, the binary operation corresponds to removing things from the boat, while the unary operation corresponds with the anchors themselves. In the temperature model, for example, the distinction is not so clear. -3 degrees could refer to a temperature 3 below zero or to the action (operation) of dropping 3 degrees. In this case, I would argue that although -3 degrees and 3 degrees are in some sense “opposite” they don’t form “zero-pairs,” which means that this model does not support (very well) the idea that zero is not empty.

We worked the floats and anchors metaphor (and lefts and rights on the number line–another good one) into some instructional games. These games and the slides from our presentation can be found at https://sites.google.com/a/umn.edu/integer-games/

All this being said, the original question was about (-1)(-1), or multiplying negatives, which is very different from subtracting negatives…

Thanks Christopher and Dan for the posts. I really like the distinctions you both drew in the approaches toward these concepts.

5. Amy Tanner

I’ve been teaching and creating curriculum for mathematics courses for elementary and secondary education for years now, and I find integers the most challenging topic to teach. There are many unsatisfactory or problematic models out there, and I find it difficult to move the instruction from telling to asking.

Because I am a course coordinator for our department, I have a lot of say on when particular topics are taught in our mathematics sequence, and I like to position integers near the end of the course (or sequence of courses where el ed is concerned) so that students are better prepared to discuss not just the mathematics, but to evaluate and critique different integer models. We do have some very interesting conversations, but I wouldn’t say I’m satisfied yet with my integer units and what my students glean from them.

I’ve looked repeatedly at the research that is out there, but integers have received so little attention compared to, say, fractions or whole number operations. I’m very interested in the article you’re writing, and what you might have to say about integers.

6. Thank you, thank you, thank you. I’m in the trenches and this is an excellent conversation. I also want to thank Aran for sharing his presentation and website. They’ll be put to good use.