Bryan Meyer claims to prefer the abstract task on the left to the concrete task on the right:

But both of those images have information that is extraneous to the question Bryan poses about them. Color information is irrelevant in both, for instance, so you get rid of it. You don’t need the names of California’s missions or the name of the ocean to its west. Both images are in need of abstraction. Therefore they are both concrete.

Or at least they’re more concrete than what we get after we ask ourselves:

- What information is important?
- How do I represent it?

This points to the highly subjective nature of the *adjectives* “abstract” and “concrete.” They’re often statements of preference (eg. “I prefer concrete to abstract” or vice versa) and they’re both relative terms. Everything is more abstract and more concrete than something else on the same ladder. But we’re looking at two different ladders here, not one.

## 14 Comments

## dan goldner

August 7, 2012 - 11:45 am -I don’t know, Dan. I was just talking with a friend about this and he was suggesting that distinction between what people like or don’t has to do with specificity vs. generality, not contextualized vs. decontextualized. I wonder if, when people identify what they don’t like as “abstract”, they may mean “general”. When I look at the last two images above, I see them as concrete and concrete: a specific iteration with a specific initial length, a specific given line and given scale. The abstract versions to me would be, How do you find the length of the nth iteration and does it converge as n increases? If you have a curve that is perturbed from a line on all scales, what could we mean by its length? I lose students when I go general, not when I decontextualize.

There’s been a lot of commentary on this thread I haven’t read – apologies if I’m retreading covered ground.

## Steve G.

August 7, 2012 - 12:59 pm -There is an issue here, Dan M, with regard to your use of the words “abstract” and “concrete” combined with your placement of items thus labeled on the Ladder Of Abstraction. The first is a dichotomy, whereas the second is a spectrum (a quantized spectrum, if you like). 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. Find a “math picture” and label it concrete or abstract (by your own sense of it, at random, or however) and some other math teacher is likely (certain?) to be able to justify labeling it the other way.

I don’t know what the solution is, but it strikes me as in important issue to grapple with as you flesh out the LOA metaphor.

## William

August 7, 2012 - 3:19 pm -@Steve G.

“concrete *with respect to some other position on the Ladder*”

The idea that abstract and concrete are relative seems pretty important. I wonder whether we’d be better off describing various representations as more or less *useful* for tackling a particular question?

In the case of the snowflake, it looks like what Dan’s calling the abstract representation has more information than the concrete representation. But in the case of the coastline, the abstract representation has less information than the concrete representation.

In both cases, the “concrete” and “abstract” representations are consistent with each other, so we can freely choose to play and work with either of them. We can also construct other consistent representations if it seems like a good idea.

It seems like the definition of “abstract” that Dan’s using is something like “containing the information relevant to answering some particular question”. It turns out that being able to construct a representation with that property is totally important. Is there a better name for that property than “abstract”?

## Bryan Meyer

August 7, 2012 - 3:27 pm -It’s been a while since I posted those two images. Thanks for helping me revisit my thoughts at that time. I think what I take away most from your post about it here is a reminder that “abstract” can be used as both an adjective and a verb and that, in both cases, language, meaning, and knowledge are both relative and subjective. The relativism actually interests me most at the moment, but that is food for another post…

My original intent with the Koch Snowflake/Coastline question was mostly to have students grapple with the notion of infinite iteration and sums of infinite sequences. At the time, I was persuaded to opt for the Koch Snowflake question because I believed (as I mentioned in my original post) that it would present a more clear opportunity for students to arrive at (what most mathematicians would consider to be) a correct answer.

As it pertains to your recent interest in the ladder of abstraction, I think I view abstraction as the process of mathematizing our experiential reality. In this way, perhaps it makes sense to start with the question about the coastline and move towards something like the Koch Snowflake as a way of having students CREATE mathematics that helps them model THEIR world. Does the model match “reality” perfectly in this case? No. But I think it offers a great opportunity for students to see that math does not exist independent of our world but is, rather, a human creation that is rooted in a desire to understand that world better. As Einstein said:

“As far as the propositions of mathematics refer to reality they are not certain, and so far as they are certain, they do not refer to reality.”

## gasstationwithoutpumps

August 8, 2012 - 9:43 am -I have trouble with the final pair. The Koch snowflake is quite concrete (a very specific instance), but not visual. The coastline is quite visual, but not concrete (I have no idea how the curve is defined, nor what resolution it is to be examined at, which makes a difference to what the right answer is).

## Dan Meyer

August 8, 2012 - 3:21 pm -Steve G.: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: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.

## William

August 8, 2012 - 7:40 pm -It’s really important to acknowledge that different representations are useful for different things. Some representations are just unuseful. Roman Numerals are sort-of useful for writing literature and making monuments. They’re terrible for everything else; even the Romans used base-10 for their abacus. Throughout the history of math, we’ve used (invented? discovered?) new ways of representing things, and added them to our quiver.

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.

## Julia Tsygan

August 9, 2012 - 3:31 pm -Dan Goldner’s point about the major difficulty for students being to more from specific to general really hits home with me. This is of course also a ladder of sorts – 52nd iteration -> nth iteration -> general summation and convergence -> general iterated processes using arbitrarily defined operations over some algebraic structures such as fields is one example of such a ladder. Sometimes finding the general is the key to finding the specific, but the difficulty students have is often to even imagine what a general case might mean. I have students that are happy calculating how much their $42 investment will be worth with compound interest over 500 years, but balk if I ask “what about A dollars over n years?” In this case this is a problem using symbols for unknown/variable quantities – for all I know the major problem students have moving into algebra. Why some students resist this move I can’t say. How often do kids think about the general case of anything outside the mathematics classroom?

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.

## dan goldner

August 9, 2012 - 3:57 pm -@Julia:

>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

canfilter 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?

## Dan Meyer

August 9, 2012 - 5:08 pm -Yeah, I’m stumbling a bit trying to link up the concrete/abstract and specific/general axes, but it seems important. You can’t go from the specific to the general without abstracting. That’s all I have at the moment.

## louise

August 9, 2012 - 5:41 pm -Yes, yes, that’s the ability we have to “learn”. We always talk about “recognize patterns”, but I don’t know how to teach that, I only know how to present examples, and give opportunities. This is the difficulty my daughter had with math (algebra at age 10) although by age 17 it was no problem. Maybe some kids would be better off doing something else for a while, and then when their brains grow into it, do the abstraction.

My guess is that most math teachers can do this abstraction easily – I cannot remember a time ever not being able to do it – so we don’t know what the “not-seeing” is like, and therefore don’t know how to assist.

## Julia Tsygan

August 9, 2012 - 5:49 pm -@Dan G

I agree with what you wrote in the first paragraph. Regarding the rest, I’m thinking that generalizing must include (or be equal to) decontextualizing – or at least being able to transfer between different contexts. (By the way, would you say that students develop transfer before or after they develop decontextualization?) So I’d rather say that generalization and decontextualization mean the same thing, and that the major distinction we are discussing is about whether this generalization applies to physical (“real life”) or mathematical contexts.

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.

## Fawn Nguyen

August 10, 2012 - 6:10 pm -I also see all images in this post as concrete. A student can come up with numerical answers to the Koch snowflake and that of coastline. It seems to me that the idea of spatial dimensions (length and perimeter being one dimensional) would help me climb that abstraction ladder related to the task. Learning to combine like terms may make more sense to kids if they learn that they can’t add km to km^2 because one is a measure of coastline and one is of area of land.

## Elizabeth Legg

August 19, 2012 - 11:43 am -Dan, I didn’t get to read all the content on your blog, but I especially liked the idea of, when students get to challenging idea, that they wrestle with it and play with it, instead of just giving them the instructions first (from your Angry Birds analogy). Thanks for the input!