But what does it mean for a

*strategy to be abstrac*t, and what does it mean to*abstract a strategy*? Dan has really thought through what it would mean for "to abstract" to mean "to remove irrelevant details," and for an abstraction to be the result of this process. But I think that there's a flipside of that, which is that when we're abstracting we're bundling. That's important enough to my argument I'm going to write that in really big letters before attempting to defend it. One second,**Abstracting as Bundling**

...and we're back! I think that a lot of what it means to abstract something is to bundle it together with a bunch of things. So, for example, you start with Bessie.

But

*then*you add Judith, Ann and Cleopatra to the mix.
One way of looking at this process is with the metaphor* of the "ladder of abstraction." I like the idea of a ladder, but it makes me sad that Ann, Cleopatra and Judith don't get to be on the ladder since they were so important to this whole deal. So I made a tree diagram that has room for them.

** More like MEAT-aphor amiright?*

“A major realization was that Polya’s descriptions of the strategies were too broad: “Try to solve an easier related problem” sounds like a sensible strategy, for example, but it turns out that depending on the original problem, there are at least a dozen different ways to create easier related problems. Each of these is a strategy in itself.”It’s great to have a top-level strategy like “solve an easier, related problem” if you have lots of experience with all the strategies that are under it. But if you're don’t, then when someone tells you “Try solving an easier problem!” it's like pointing at Bessie telling a kid “Do you see that wealth out there, chomping grass in the field?”

We’re jumping to an abstraction, and the communication fails.

When it comes to teaching the big math skills and strategies, I think that we have to move up and down this ladder of abstraction. In particular, we want to help kids organize their learning more effectively by helping students bundle their strategies into more general, more widely-applicable packages.

Consider this kid, who is getting pretty good at drawing a picture for certain kinds of multiplication problems.

In fact, she can do this for addition problems and fraction problems too. She knows how to draw pictures to help her solve problems in lots of different specific situations. This is where the teacher's work begins.

Eventually, this kid will become pro at drawing pictures to help solve arithmetic problems. Does that mean that she'll be able to draw a picture to help her solve a linear equation? Of course not! Even if she understood variables and equations, do we expect her to easily intuit that a scale is a helpful way to visually represent a linear equation?

No. She needs to learn some powerful new pictures to draw. But once she does learn this, we should encourage her to connect this to the other pictures she knows are useful to draw for algebra problems. And throughout this process we should encourage her to connect these sorts of pictures to the representations she draws for other sorts of problems. We need to encourage her to see how all these strategies are very similar, and truly belong under one bundle.

Is this enough to give our student a general-purpose problem solving strategy yet? Probably not! This process of bundling and packaging is going to continue throughout her mathematical career. At some point she'll have seen a lot of math problems where visualizations can help and, when facing something new, she'll ask herself, "Hey, I wonder if I can draw a picture here."

But the point is that there's a long development to this point, and I don't think that it's enough to just ask kids, “Have you tried drawing a picture?”

**What is a teacher to do?**

Consider this kid, who is getting pretty good at drawing a picture for certain kinds of multiplication problems.

One way that we can help this student is by encouraging her to see these three specific drawing tricks that she already knows about as a

*strategy*. We're asking her to do some bundling. Instead of Wilcox, Ann, Judith and Cleopatra, we're asking her to bundle "draw a picture to help solve a multiplication problem," "draw a picture to help solve a fraction problem" and "draw a picture to help solve an addition problem," into a single category.
Now, say this kid is stuck on a subtraction word problem and you advise her, "Try drawing a picture!" There's a decent shot, I'd argue, that this kid will be able to pick a representation for this problem. If not, that's cool, push her to think about how she'd represent this if it were an addition problem, or a division problem. Draw connections that will help her bundle her strategies and ascend higher and higher up the ladder of strategy-abstraction.

No. She needs to learn some powerful new pictures to draw. But once she does learn this, we should encourage her to connect this to the other pictures she knows are useful to draw for algebra problems. And throughout this process we should encourage her to connect these sorts of pictures to the representations she draws for other sorts of problems. We need to encourage her to see how all these strategies are very similar, and truly belong under one bundle.

But the point is that there's a long development to this point, and I don't think that it's enough to just ask kids, “Have you tried drawing a picture?”

**Questions**

- We need to teach kids lots of more specific strategies, but should that happen instead of teaching top-level strategies, or while teaching the most abstract strategies? What's most effective?
- Do we just want kids to notice "Hey! A picture I drew helped me solve all these different sorts of arithmetic problems." Or do we want them to notice similarities and connections within the
*types*of pictures that ended up helping them? - Where does this leave our posters of problem solving skills? Do these things help, in the end?
- What would it take to fully articulate the landscape of "draw a picture" or "think logically" or "work backwards" that it takes for a student to reach a serious level of expertise? Any grad students out there wanna pitch in on this?

In order for the cue, "Try drawing a picture," to be helpful to a student solving, say, a problem about adding fractions, I think it would be necessary for the teacher to have *modeled* this for the student in advance. In particular, the teacher should know that the student can represent fractions using a variety of pictures, even if the student does not know how to use those pictures to perform the addition -- now there's an interesting challenge! Also, the purpose of the lesson matters: if it's to *discover* how to add fractions, that's one thing, but if it's to *practice* adding different kinds of fractions, having been taught how to do at least some kind of fraction addition, then maybe it's another.

ReplyDeleteI wanted to comment that your idea that abstraction is bundling reminded me of when I used Bootstrap this past semester - which is a mini-curriculum (20 days) that ties algebra concepts into computer programming with the end goal of making a video game.

ReplyDeleteWhenever the students had to write a computer function to do something, such as draw a solid green triangle of a size given by the user, they always had to list examples first. So, say they wanted a triangle of size 20, so I would write

(triangle 20 "green" "solid")

And if they wanted a triangle of size 3, I would write

(triangle 3 "green" "solid")

And after writing a few examples, we would look for what was the same in all the examples and what was different and what was the same - so that we're abstracting into the function by identifying constants and variables.

(triangle size "green" "solid")

And that same process works when we wanted to, say, write an abstract linear function to represent something.

2 + 3*3

2 + 3*5

2 + 3*7

2 + 3x

Yeah! Those are great examples, I think.

DeleteSome might say: "Look, what you're doing there isn't bundling. It's zeroing in on the important information, the structure, and ignoring some of the details." And I think that's right, but that's the result of this bundling process. You can only skip the concept-building bundling process if you already know how variables work, or how functions work, or how to use high-level math strategies.

When a friend and former student was having trouble seeing the proportions involved in a shadow problem, I suggested she draw a picture.

ReplyDeleteHer picture showed more detail than mine would, and *did not use a right angle between person and shadow* because her picture was artistically more accurate than mine would be, with the shadow coming up at an angle behind her person.

That's when I realized that math pictures take certain skills. How do I show the important parts of the 3D reality on 2D paper? The right angle between person and ground, and tree and ground, makes right triangles that will be similar, and I had hoped that picture would help her then go to proportional reasoning. I'm not sure now that even the right sort of picture would have helped. There are many layers...

I love this story. What it lays clear to me is that helpful pictures are

Deletepartof mathematical understanding. To understand a concept often involves understanding how to productively represent it. The ability to draw a helpful shadow picture is part of expertise in proportional reasoning.