It’s a familiar scene for a math teacher. You’re chatting with a stranger at a party or the guy giving your hair a quick trim or anyone else. Conversation comes around to occupations. You mention you’re a math teacher. No one has a neutral reaction to “math teacher.” You take the tension head-on and ask, “What did you think about math as a kid?” The majority opinion on childhood mathematics is often negative and you notice the same adjective crops up over and over again: “abstract.”
“I liked Geometry. Algebra was too abstract.”
“Math was too abstract. I liked working with my hands more.”
“I liked Algebra. Geometry was too abstract.”
I’m going to try to pound in some fenceposts around the terms “abstract,” “abstraction,” and specifically, “the ladder of abstraction.”
That last term has its deepest roots in the fields of language and rhetoric (Hayakawa, 1940) though Bret Victor recently knocked it out of the park with an interactive essay describing its applications in mathematics and computer science. This fencepost-pounding process may require only a few months and a few blog posts (if you’re lucky) or a few years and a dissertation (if I’m lucky). However long it takes, you should help me interrogate the term. Does it mean anything? Does its meaning have any implications for the workings of a math classroom? If we understand it, can we counteract the perception that math is too abstract, or at least understand that perception well enough to manage it?
I’ll finish this brief introduction by describing the personal appeal of the ladder of abstraction:
- Self-study. In the best classroom experiences I’ve witnessed or orchestrated, I could describe the students as “ascending the entire ladder of abstraction.” I want to know more about that.
- It ties a lot of good pure and applied math instruction together. I’ve done an excellent job pigeonholing myself as some kind of zealot for applied mathematics but some of my favorite experiences in the classroom haven’t involved any applied context at all. Common to all of them (and common to my applied math methods) is their origin at the bottom of the ladder of abstraction. I didn’t hoist students to a higher rung until they’d worked on the rungs below.
- There might be a dissertation & career in there somewhere. Implementing the ladder of abstraction in the classroom requires multiple media. Don’t misunderstand me. I’m not saying, “Good math instruction requires digital media — photos and videos, etc.” I’m saying that it’s difficult to fully exploit that ladder if you design a task using only one medium. (Paper, in particular, limits your tasks to exactly one rung on the ladder, depending on how strictly we define our terms.) The task I linked above — graph: 3x + 2y = 12 — required two media, none digital: 1) voice, 2) a collaborative writing surface. Tasks that work up and down the entire ladder of abstraction don’t require digital media, but holy cow does a digital platform make those tasks easier to implement. As I look ahead to (fingers crossed) finishing this PhD and getting a job doing I have no idea what, I think I could contribute some value to our field by helping people create tasks that ascend the entire ladder. Provided I understand it.
Thanks in advance for your help.