R. Lesh and H. M. Doerr (Eds.), Beyond constructivism: A models & modelling perspective on mathematics problem solving, learning & teaching. Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.

@Max, no complaints at all. I agree with your second-to-last graf, in particular, about the implications of the ladder metaphor and its uses in teaching, curriculum development, etc. The framing of abstraction as a geodesic dome or a web just isn’t useful to me. It leaves me twiddling my thumbs thinking, “So now what?”

Here’s the first:

Any individual, in solving a math problem or considering a mathematical situation, makes choices to move back and forth between making the situation more concrete and making it more abstract. She considers, for example, the real terrain, then a map of the terrain, then a lattitude/longitude grid, then makes a hypothesis about the grid which she tests on the map, then she studies the real terrain a bit more to add another layer to her grid, etc.

In that case, the ladder’s an apt metaphor (although abstractions that don’t turn out fruitful may occasionally lead to an imaginary “branched” ladder). It’s allowed people to have a conversation about the ways that abstraction and concretization (if that’s a word) are always available at any stage in problem-solving and that there are really important skills that aren’t about reaching the optimum rung but are instead about being facile climbers, I think the ladder draws attention nicely to the existence and availability of multiple stages and the requirement to get good at going up and down — for this thinker, on this problem.

The second way the ladder metaphor seems to be being used (and is not, I think, the way Dan or others who’ve used the ladder intended it) is as a way of organizing the whole domain of abstractions and concretizations available to math learners or that could be involved in thinking about a particular math situation.

It’s certainly not the case that multiplication is the top (or middle) rung of a single path of increasing abstractions from counting groups of beans. Or that Dan is advocating the answer to a math problem comes on some rung of a single ladder.

I think a geodesic dome or web or network may be a more apt latter for the space of all representations of mathematical situations, and they can vary in all sorts of dimensions.

But an individual’s path through that space while working in a single mathematical situation tends to move “up and down” along the abstraction axis, and can be (ha!) abstracted by squishing the other axes we aren’t paying attention to into a vertical path. A ladder, if you will.

And that abstraction is useful to me when thinking about, for example, how to represent the situation to the learner just thinking about abstraction for this task for my student — where on the ladder do I want them to start? What tools for concretization do I want them to have available? What tools for abstraction?

Maybe that will help people define more whether they have a problem with the ladder metaphor for an individual problem-solving path, abstracted to one dimension, or whether they were thinking of the whole collection of representations for big mathematical concepts.

Isn’t this just Piaget’s formal operations we’re talking about?

@Dan: No.

Mathematical abstraction starts in pre-K with counting. There are all sorts of understandings that need to be reached before a child can be said to “count”, like —

* Knowing that the sequence is always the same.
* Knowing that the last number named is equivalent to the size of the set.
* Knowing that each element of the set is counted once and only once.
* Knowing that the number of items in a set is invariant if the set does not change.

Each one of these (except for arguably the first) reflects the child building an abstraction. Usually it is formed by the child encountering lots of instances until the idea is realized — for example, counting lots of “4” sets until the idea that the “four” at the end of the sequence somehow corresponds to the number of the things in the set.

Forming the first step up the ladder, here, is then not done off a single problem but a network of interconnected problems. According to Piaget the early grade level students will not be able to think about the abstraction without some concrete instance, but even if that is so (Piaget’s sort of fuzzy on number sense to begin with) an abstraction must be formed nonetheless for a student to know that “five” is five.

As a later example, the same sort of principle applies for models of multiplication. Here is where the idea of a ladder starts to be dangerous, because the grade 2+ teachers dealing with it often get the impression there is only one way up the ladder for each problem, and that multiplication is only conceivable abstractly in one way (for example, repeated addition). This causes issues in later grades with problems that ought to be parallel, but from the student’s perspective aren’t (for example, the gas problem quoted in the link above).

So, again — for us we can handle the idea of the ladders having multiple starting points and multiple ending points, but K-5 teachers will (from what I have seen in practice) take such analogies more literally. Because of this I personally would feel uncomfortable using the ladder analogy as a teacher training tool.

Mind you, I’d *like* this thing to be hashed out. I’m not trying to rain on the party here, given I have attempted (and had trouble with) training in the exact thing being discussed here. I just think (if nothing else) you might want to test your analogy on some live subjects who are less math-inclined before you roll with it.

I’m not sure how often you visit K-5 classrooms, but I have known many teachers, myself included, who spend time with realistic problems and abstractions. The difference at the elementary age is the amount of scaffolding and support students need from the teacher and each other. There may be an issue of transfer, of maintaining a skill, or of consistency between grade levels with regards to students’ flexibility with moving between situations and abstractions, but it is not fair to generally characterize lower grade teachers as deficient in working on these skills.

Regardless, a metaphor about abstraction is going to be most applicable in grades where they do the most abstraction. Less so in primary.

If primary school kids dealt more with realistic problems, not just mechanics, they wouldn’t have so much trouble with abstraction when they get older. To accomplish that, K-5 teachers would need to be more comfortable with the ladder/web/geodesic dome of abstraction.