I'm going to lay out five hypotheses over the next five days that will be the current tally of my writing, reading, thinking about the ladder of abstraction this summer. These should all be tested, contested, and generally kicked around.
#1: Teachers need to be explicit about the ladder of abstraction.
We represent towns with coordinates when our question concerns their location.
We represent data with tables because it keeps the data organized and sometimes reveals patterns.
We give points one capital letter and line segments two because it make them easier to talk about.
We turn real-world phenomena like trees and their shadows into right triangles when the tree-ness of the tree and the shadow-ness of the shadow don't matter, when their height and length and and included angle are all we care about.
We climb the ladder of abstraction all the time. We teachers are good at that climb. We aren't often explicit about the motivations and methods for making that climb.
We turn trees into line segments and cities into coordinates without so much as a word about that weird, violent stripping away of context. All of those implicit, elided abstractions in someone's teenage years contribute to her adult sense that math is hopelessly abstract. We need to make these motivations and methods explicit.
"Let's talk about these cities here. All we really care about is their location. Coordinates are a useful way of representing locations. Let's lay down a grid so we can put numbers to those coordinates."
Does it matter where you set the origin? Ask them. Then talk about it. I realize these kids are in ninth grade and should be totally adept at that kind of abstraction but let's not assume that about them. Particularly when it just cost you an extra minute to have that conversation and make the abstraction explicit.
2012 Sep 18. Great line here from Frorer, et al, (1997):
And yet while abstraction in mathematics has some additional qualities or meaning, we rarely find it explicitly discussed let alone defined. You can pick up a book entitled Abstract Algebra and not find a real discussion of abstraction as a process, or of abstractions as objects.
I do not think use of the ladder metaphor is an admission that there is only a single way to get to certain understanding. I picture the ladder to mean that there is a path that I cognitively take to move along the spectrum of abstraction. This would allow room to climb a particular ladder to higher levels of abstraction and climb down another. The fact that there are multiple ladders to reach the same point does not invalidate the use of the ladder.
On another note, I think being explicit is enormously important especially if your goal as a teacher is to eventually make yourself useless to the students. Without revealing the undercurrents of your decision-making and assumptions, I think that you do not fully prepare them for life without you.