Responding to a request here.
— Chris Robinson (@absvalteaching) June 20, 2013
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.
— Bryan Meyer (@doingmath) June 17, 2013
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.
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.
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.