http://www.nytimes.com/2012/07/29/opinion/sunday/is-algebra-necessary.html?smid=pl-share

I see some points in that argument that your post answers… unfortunately, many of his critiques stand because ‘in practice’ the teaching of algebra doesn’t connect the representational and abstract.

The first example (which I like more and more) is baseball statistics. The question is which baseball players are worth paying to play for your team. The “concrete” context for the question is the historical performance of all the players. [1] On top of that concrete data you can construct abstract representations. One batter might have a batting average of 0.325, another 0.221. One pitcher might have an ERA of 2.2 and another 4.5. We compare baseball players at that level of abstraction. If we construct other abstractions (say, OPS or VORP, or WHIP, or whatever new contraption we use to talk about fielding skill), we’ll make different decisions about which baseball players to hire.

@Gary, your talk about physics reminded me of another important example: mathematical astronomy. The question is “how do the things in the sky move around?” The context is observations of the things in the sky. The ancienct mathematical astronomers collected surprisingly good concrete data about the positions of the various celestial objects. They then constructed an abstract representation of the world to explain the movements of those things. The abstraction they chose was one with the Earth at the center of the universe and a stack of celestial spheres rotating around it. It turns out that that’s a surprisingly good abstraction to pick. You can do fruitful mathematical work in it and it allows you to predict where the celestial objects will be really well. You pay a price in scary geometrical theorems that are very hard to read today, but the work stands. The Copernican revolution was, in large part, a change in abstracted representation. The Copernican astronomers were trying to answer the same question, and used much of the same data as the Ptolemaic astronomers, but they constructed a different set of abstractions on top of the concrete data. Tycho Brahe proposed an abstraction where Mercury and Venus orbited the Sun while the sun itself orbited the Earth. Copernicus proposed a set of heliocentric abstractions. [2]

To take a Nolanesque turn to the meta, I’m suggesting that we have a concrete context of students in math classes and a question, “how do we effectively teach them to enjoy mathematics?” One abstracted representation of how we do mathematics is traversing a ladder of abstraction. I think that produces a particular pedagogy that’s way, way better than most. So, as far as it goes, I think the ladder of abstraction is a good systemic metaphor. I’d suggest, though, that if you abstract the process of mathematics differently, say as constructing and traversing a cyclic, undirected graph of representations or as traversing a lattice of abstractions, you’ll get a subtly (!) different pedagogy that will better serve students. [3] You’ll get a pedagogy that not only teaches students to move up and down the ladder (hugely important and generally missing from our curricula), but also teaches them to deftly choose which abstractions will get them from where they are to where they want to go.

[1]It turns out that this too is an abstracted representation. When you keep score for a baseball game, you record particular, concrete events using a symbolic language that regularizes them. You convert the batter ripping a hard ground ball down the left field line past the diving third baseman to “2B”. You convert the pitcher buckling the knees of the batter with a 103 mile per hour fastball to “K”. That translation itself requires significant synthetic reasoning.

[2] The same thing happened with Newton and Einstein. Relativity is a new abstracted representation of the concrete context of mechanics and the Michelson Moreley experiment. Atomic theory, too, has had its fair share of different abstractions.

[3] You definitely don’t want to tell students that you’re teaching them to traverse an undirected graph in so many words, not least because mathematicians brutally equivocate “graph”.

I have discovered that in our school we have never laid this fundamental foundation of essentially how math and science are related. I think you have found a very good way of describing how science asks questions, and then we have to find a way of isolating the dimensions that interact that truly define the condition that we are curious about.

It is this process of abstraction where we make the transition from the question that is perhaps qualitative in nature to framing the question in a quantitative manner using mathematics.

Thanks for a great post. I’m already thinking about asking my physics students to bring a photo of an event that interests them and then asking them through a series of abstractions to reduce their point of interest to a relationship between two defining dimensions (one of the seven fundamental SI base units).

Yes, and I am thinking that perhaps some formed of worked examples may be useful.

“What set of abstractions you use to answer your question will hugely influence the answer you get. ”

Using some examples do demonstrate your point above (with wildly different answers depending on the abstractions chosen) could be a good case. Now I just need to think of some to use!!!

“45 + 32 is another way of representing 87. But I could also represent that information as 40 + 30 + 7 or 0b1010111 or as a picture of forty-five apples stacked next to thirty-two pears, or as a point on the number line, or as a pile of actual skittles.”

A number is all the ways you can name it, is a good thing to teach. And this could be part of the lessons on abstraction. For instance 3 is fine saying your age or counting how many toy bears you have, 1/2 of 6 is good when you are are sharing 6 cookies between two people or cutting a 6 foot board in half. 10-7 is good when we want to know how much we will have left after spending 7 of our 10 dollars.

@Dan
“What information is important? How do you represent it? You can’t do the latter without deciding on the former. I can’t think of a counterexample. So the linearity of the ladders seems true to me. Resolving the question, “How do you represent the important information?” requires someone to consider the different possible representations, their advantages and disadvantages.”

Great questions. As for a counter example to the ladder metaphor (which I still like as a starting point for kids, no metaphor is perfect) I am still thinking, but using your questions, depending on what is important,you could get different representations which at the same time hide/reveal (or perhaps allow/disallow control over) different characteristics that matter for your purpose. That said, I’m stumped right now and can’t think of a counter example.

So another way I have been thinking about abstraction, (and how have taught it) is more programming and lego related. Ie: how do you design the proper set of pieces that can be re-used to accomplish multiple tasks. One way I teach this is using parts of Barry Newell’s Turtle Confusion puzzles. Where you have kids program the turtle (or Scratch cat or Etoys object) to draw a square, triangle, and pentagon. You then ask them to look at what they just wrote and ask “Whats the same and whats different?” Every time I have done this the kids get it and see the common patterns and how they can create one set of blocks that can draw any regular polygon. Part of what can help make this work is hiding some of the complexity and providing them a limited set of programming blocks which visible for use. So the question “What’s important” is very useful in deciding what to hide and what to show when designing an activity.

If you base your answer on mathematical abstractions (as Oakland did), you can wildly outperform people who don’t base their answer on those mathematical abstractions.

Then things get interesting. What set of abstractions you use to answer your question will hugely influence the answer you get. If you think that batting average is the most appropriate abstraction, you’ll hire certain players. If you think that OPS is the most appropriate abstraction, you’ll hire other players. If you’re a VORP fan, that’ll drive decisions too. There’s not broad agreement among teams as to what abstractions generate the best answers to the question.

One frustrating reality for the modern student is how often she’s left out of this conversation. The abstraction must be accepted without any of the abstracting.

I think this is correct. There are a lot of reasons why curricula and instruction in secondary levels exclude students from the process of abstraction – many valid, many invalid, most related to the limitations of the print medium. I’m downcast on a lot of implementations of technology in the math classroom but digital technologies are uniquely suited to pull students back into that process. Those are the posts I’m most excited to get to.

Michael P:

You’re on the (a?) ladder of abstraction before then. “Colors” is an abstraction. You can’t ask the question without already being able to see the world through the lens of color.

You’re only on the ladder of abstraction once you ask the question. As you’ve pointed out, you’re not on the lowest rung of that ladder, though.

Michael P:

I don’t get how an aerial shot is an abstraction of the street scene. If by “abstract” you mean “the removal of information,” then an abstraction should never contain more information. But the aerial shot contains way more information that the on-the-ground scene.

This is fair. If the rung beneath it were every possible visual view of the area surrounding you, then the ladder makes more sense. Rather than taking every possible view of that scene, we’re getting rid of all other views but the one immediately above the street.

Michael P:

Same issue with the way you present the move graph+distance as a higher abstraction. That’s adding information.

This is another weak part of the essay. I don’t really know how to conceptualize the process of generalization or the addition of more variables. Intuitively I want to place those tasks at the top of the ladder (the sequence of events makes sense that way, for one reason) but the metaphor gets thinner the higher it goes.

William:

45 + 32 is another way of representing 87. But I could also represent that information as 40 + 30 + 7 or 0b1010111 or as a picture of forty-five apples stacked next to thirty-two pears, or as a point on the number line, or as a pile of actual skittles.

What information is important? How do you represent it? You can’t do the latter without deciding on the former. I can’t think of a counterexample. So the linearity of the ladders seems true to me. Resolving the question, “How do you represent the important information?” requires someone to consider the different possible representations, their advantages and disadvantages. You exemplify that well with 87. I don’t think that example contradicts (or even complicates) the metaphor, though.

Santosh:

As you mine these ideas further, I wonder if it is time to also start laying out the similarities and differences between a ‘representation’ and an ‘abstraction’ (the product of abstracting).

We’ve thus far tackled (lightly) the verb “abstract” and the adjective “abstract.” The noun “abstraction” has been out of scope, but that’s an oversight. Totally off the cuff, I’ll say I think the noun “representation” has a static quality that “abstraction” lacks. “Abstraction” calls attention to that which was abstracted in a way that “representation” doesn’t. I probably oughtta check a dictionary, though. Anybody else?

as you mine these ideas further, I wonder if it is time to also start laying out the similarities and differences between a ‘representation’ and an ‘abstraction’ (the product of abstracting).