Could we then say that abstraction consists of part physical and part mathematical generalization? Then abstraction is no longer one ladder but rather a grid representing two dimensions. As an example: “Anna invests $50 at 4% interest compounded yearly for 12 years” is low on both dimensions, while “x increases or decreases by y percent z times” is rather high on both dimensions, and “50 increases by 4% 12 times” is high on physical generalization but low on mathematical.

>Removing the information also can make the problem easier because students no longer need to filter the important from the unimportant.

This strikes me as Dan (M)’s main focus from the beginning of this thread (and, really, from near the beginning of his blog): how do students develop into people who can filter the important from the unimportant on their own?

That seems as important, and maybe a prerequisite to, generalizing or not. I am confused about the distinction as well (obviously, from my first comment) …. Could they actually be entirely orthogonal to one another — completely separate activities? And if so then why do they get jumbled so easily?

I think generalization is the key to understanding the coastline example progression as well. By removing the names of the places the problem is no longer about California’s coastline but rather about any coastline. But why make it a coastline and not just a curve? Why use miles instead of just units? why specify the scale? All those elements could be removed to generalize/abstract the problem further.

By contrast, “removing unimportant information” such as colors and the Koch picture, doesn’t strike me as abstracting at all, unless that removal of information somehow makes the problem less specific. Removing the information also can make the problem easier because students no longer need to filter the important from the unimportant.

I was going to write something about the relationship between generalization and decontextualization, but after a few attempts it hit me that I have no idea how to make that distinction or even if I can properly define the term context. Ugh. Someone, help.

One of the things I really appreciated about the video of the Japanese classroom was how the teacher asked the students why the ordered table was a better representation than the unordered table. They responded that it let them see the relationships between the number of pens and price better. So the teacher had them evaluating representations based on how elegant and useful they were. Pretty cool.

Being able to determine which representations are useful for understanding a particular question is a crucial, crucial skill to teach. Usually I do it by trial and error, letting the kids represent things in different ways and talking about what the advantages and disadvantages are. Gradually, kids (and grown-ups?) pick up an intuition about what representations will be useful for particular sorts of questions. There’s a lot to think about in understanding how that intuition works, I suspect.

You yourself have indicated that to call something, for example, concrete, is to call it concrete with respect to some other position on the Ladder. The fact that you and Bryan Meyer label the same picture as being on opposite sides of the dichotomy, while you and Dan G. do the same thing on another pair of images suggests that there may be an inherent problem with labeling anything as one or the other.

The image that labels them both “concrete” may be misleading. I’m saying that calling either of those states “concrete” or “abstract” obscures more than it reveals. They are on separate ladders and can’t be compared all that effectively along that dimension. I tried to make that clear in my last paragraph.

William:

I wonder whether we’d be better off describing various representations as more or less useful for tackling a particular question?

Yeah, very interesting. Here’s another developing thesis we can hack away at: the persistent emphasis on multiple representations of the same context (ie. verbal / graphical / tabular / symbolic) may be a mistake. It’s important that students become fluent in all of them, of course, but they aren’t all equally good for every task. Emphasizing all representations for every context deemphasizes the fact that some of them may be less useful for some tasks. The emphasis on multiple representations doesn’t let students practice selecting between them.