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## [LOA] The Ladder of Abstraction, Part One Of Probably A Lot

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:

1. 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.
2. 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.

## [LOA] Hypothesis #5: Bet On The Ladder, Not On Context

#5: Kids care less about context — “real world” problems — than they do about problems that start at the bottom of the ladder. “Real world” is a risky bet.

Real World

Here is a “real world” problem:

The caterers Ms. Smith wants for her wedding will cost \$12 an adult for dinner and \$8 a child. Ms. Smith’s dad would like to keep the dinner budget under \$2,000. Ms. Smith would like to invite at least 150 guests to her wedding. How many children and adults can Ms. Smith invite to her wedding while staying within budget?

There is nothing to predict. Nothing to compare. The important information has already been abstracted. The question has been fully defined. The problem, as a whole, has been stretched tight and nailed to a board. The student’s only task is to represent the important information symbolically and then apply some operations to that representation.

And so hands go up around the room. The students attached to those hands say, “I don’t know where to start.” The task has hoisted them up to a middle rung on the ladder of abstraction and left their feet dangling in the air. Students are frustrated and disengaged in spite of the “realness” of the task.

Fake World

Meanwhile here is a “fake world” problem:

1. What are the new percents? Write down a guess.
2. Which quantities change?
3. Which quantities stay the same?
4. What names could we give to the quantities that are changing?

These questions include students in the process of abstraction. Each student guesses the new percents and is consequently a little more interested in an answer. Students aren’t just asked to accept someone else’s arbitrary abstraction [pdf] of the context. They get to make their own arbitrary abstraction of the context. (Why ABCD? Why not WXYZ?) All of these tasks prepare them to work at higher levels of abstraction later.

Solution

My preference is a combination of the two — a context that is real to students and a task that lets them participate in the abstraction of that context.

But I can’t tell you how many conversations I’ve had with teachers (veteran and new) and publishers (big and small) who tell me the fix for material that students don’t like is to drape some kind of context around the same tasks. Rather than expanding and enriching their tasks to include the entire ladder of abstraction, they insert iPads or basketballs or Justin Bieber or whatever they perceive interests students.

Real-world math is a risky bet. Bet on the bottom of the ladder. Here are some of those bets:

1. With the wedding task above, the teacher can ask students to pick any combination of children and adults they think will work. Any combination. 100 kids and 50 adults? Fine. Now tell me how much it costs. We’re all invested for a moment in a problem of our own choosing. Then we assemble student work side-by-side and notice that we’re all doing the same kind of calculations. Then we say, “All your work looks the same. What’s happening every time?” The students are participating in the symbolic abstraction.
2. Louise Wilson is using the images and videos on 101questions to give students practice just asking questions about a context. Asking questions is the assignment. Getting answers isn’t.
3. Andrew Stadel is giving his students daily practice with estimation, another task at the bottom of the ladder.

We ask our students to work most often at the top of the ladder and the result is a pervasive impression that a successful math student is a student who can memorize formulas and implement them quickly and correctly. Those are, of course, great and useful skills, but mathematicians also prize the ability to ask good questions, make good estimations, and create strong abstractions. These are skills where other students may excel. There is unrewarded excellence in our math classrooms because we have defined excellence narrowly as being good at abstract skills. You can only find (and then reward) that excellence by betting on the bottom of the ladder of abstraction.

## [LOA] Concretizing Abstraction

Let’s drop down a rung and make abstraction concrete.

You’re walking across a street. This is a photograph of what you see.

This is your context. What is its abstraction? There’s no way to know because you don’t know your purpose here, your question.

You ask yourself, “What colors do I see?” Now you have a question and you’re on the ladder of abstraction.

You start speaking very informally about the context, perhaps comparing one shade of green to another. You ask yourself, “What’s important here?” and decide it doesn’t matter whether the green thing is a car or a tree. All that matters is its greenness. This is abstraction. You’re removing aspects of the context that are inconsequential to your question.

Now you have to decide how to represent the consequential aspects. You could represent them with words:

A lot of grays on the street and sidewalk. Light blue in the sky. Red on the curb. Different shades of green in the trees and on a car.

Different representations are more useful for different purposes. This representation might work if you were writing some prose about the colors. If you wanted a more precise representation, though, you might turn to a histogram of the red, green, and blue values.

Now if the question changes, the entire ladder changes. If your question is, instead, “How do I get home from here?” different predictions are useful, different information becomes consequential, and the representations of that information will look nothing like the histogram we used to examine color.

A useful abstraction of this scene would be an overhead view of the terrain.

Of course, we only care all that much about the roads, not the trees or houses in between them, so we abstract all that away.

If our purpose here is to create some kind of enormous geolocation system, we don’t really care whether or not a road curves. We just care whether or not the road connects one intersection to another, or, abstracting those terms a little, we care whether or not an edge connects one vertex to another in a graph:

An array would be a representation of the graph that’s friendly to manipulation by a computer, though as a human, I miss a lot of the visual information we’ve abstracted away.

Great. But not perfect. This representation will only tell us whether or not it’s possible to get from one point to another — whether a route exists. If we want to find the shortest route, we add another useful variable, “abstracting over distance,” at it’s said.

If we want to find the fastest route, we’ll also need to abstract over the speed limits of each of those edges.

That’s a concrete example of the process and ladder of abstraction. The adjectives “concrete” and “abstract” just aren’t all that useful here. Everything is concrete if you think about the rungs above it and everything is abstract when you think about the rungs below it. The photograph that kicks off the post is more concrete than everything that comes after it but it’s also more abstract than the full-bleed, full-audio, moving panorama you experienced as you walked across the street. What matters isn’t the rung itself but how deftly you can move between all the rungs above and below you on the ladder of abstraction.

Checking For Understanding: Give an example of abstraction as it exists in your own life, in the problems you or other people try to solve. Two examples to kick off our list:

• Airplanes landing at night don’t care about the color of the tarmac or the grass on either side. All they care about are the margins of the landing strip, which are therefore lit up by lights.
• Google’s self-driving cars abstract away a metric ton of data that your senses usually take in while driving — the color of the sky, the music in the car, the humidity outside, etc. It also retains a metric ton of data, of course, and the quality of Google’s abstraction of the roadway will determine whether these things will kill us or let us (once again!) text while driving.

Featured Abstractions

Should I call or text? If the message is short and quick, I could just shoot a text, but what is my data plan like, and are there financial considerations? Or I could call, and risk a phone conversation which I perhaps don’t have the time or want for at this point in my life? How important is the message? Or will I possibly see this person soon anyway and it’s all a moot point? Or maybe I could just tweet it? But how sensitive is the message?

One way I can see abstractions is to think of domains where I’m not an expert. Is it an abstraction when two kids who are really into video gaming communicate their solutions to challenges in terms of button pushes rather than the story on the screen? As in, I’m likely to say, “Gee, I wish I could make Mario jump up and do a flip in the air to get that gold coin without being hit by the hammer.” Whereas a Mario expert is likely to tell a fellow expert, “That level’s easy. It’s just right-right-A-left” (or whatever).

When looking at my email inbox, I disregard most of the information presented there. During the busy workday I “see” only the messages that are highlighted as unread and which were sent to me individually rather than schoolwide. This level of abstraction sometimes presents problems, so if I’ve read an important message but haven’t yet resolved it, I have to star it AND mark it as unread, tricking it back into the category that registers.

Featured Comment

One thing a great context / question also gives you is the experience of figuring out what information is important and what sort of abstraction is most useful for extracting and using the right information thoughtfully. And that’s a skill a lot more adults will use than factoring.

Richard Bitgood, via e-mail:

Transformations is one step higher than functions because it is the abstraction of our abstractions.

2012 Aug 26. Of course, money is an abstraction of the value we provide society.

## [LOA] Abstracting Abstraction

or: Titles Guaranteed to Murder Your PageRank

As I mentioned previously, I find the verb “abstract” way more interesting than the adjective “abstract.” The adjective is often used critically and defensively (“Ugh. Algebra. Too abstract.”) whereas the verb represents a milestone in human cognitive development and a skill that grows more and more precious in the modern workforce. How many skills have a shelf life of thousands of years? (A: Not typesetting.)

So let’s define the verb:

Abstraction is a process or result of generalization, removal of properties, or distancing of ideas from objects.

To take away; remove.

Let’s stipulate, then, that abstraction requires a context and a question.

If you’re going to remove stuff, there has to be stuff to remove. (A context.) If you’re going to remove stuff, you have to have some purpose that tells you to remove this stuff but not that stuff. (A question.)

Then you’re on the ladder. As you go up the ladder you turn the context into something that excludes the noisy richness of the context but which is much more useful for answering your question. As I looked at all the times my students have abstracted in math class, I saw that the tasks and questions we confront and their order look a lot like this:

We debate the context on the level of experience and intuition. We make predictions. We compare different examples of the context until we understand which of its aspects are common and consequential to our question and which aren’t. We give those aspects names. We decide how to represent them. We decide what to do with those representations. And then we abstract other things in the same context.

If my mind were “light and deft and beautiful” as a monkey in a tree, I’d stop abstracting abstraction here, step down a few rungs on the ladder, and concretize this abstraction of the process of abstraction with an example. That’s next.

2012 Sep 26. William Carey passes along this succinct definition from Barry Mazur (2007):

This issue has been with us, of course, forever: the general question of ab-straction, as separating what we want from what we are presented with. It is neatly packaged in the Greek verb aphairein, as interpreted by Aristotle in the later books of the Metaphysics to mean simply separation: if it is whiteness we want to think about, we must somehow separate it from white horse, white house, white hose, and all the other white things that it invariably must come along with, in order for us to experience it at all.

## Watch an Expert Math Teacher Put Three Kinds of Knowledge to Work in the Same Class

Lisa Bejarano’s post Two Kinds of Simplicity offers a useful idea about teaching complex fractions, but much more interesting to me are the three kinds of knowledge she puts to work in her class.

Lisa has read widely from sources online and offline and has a great memory. So when she asks herself, “How am I going to teach [x]?” she can quickly summon up all kinds of helpful posts, essays, books – even the mental recording of previous classes she’s taught on [x].

I stopped to think about how this would work with my class.

Lisa has taught long enough and knows her students well enough that she can test each of those resources out in her head, all during the lunch break before class. You can see her swiping right and left on each of them – “Yeah, maybe this idea. Definitely not that one.” – as she sees her students in her imagination. I’m sure Lisa is open to the possibility that her flesh-and-blood students will differ in surprising and awesome ways from her mental model of those students. I wouldn’t bet against her intuition, though.

She ultimates decides to start her precalculus students with the elementary school analog of their lesson, turning an abstract fraction division problem into a more concrete one.

Then, as her students acquaint themselves again (or in some cases for the first time) with helpful models for that division, she builds back up to the abstract version of her task.

Lisa is only able to move up and down the ladder of abstraction like this because she knows a lot of math – specifically where it builds from and towards. If she doesn’t know that math, her options for helping her students basically shrink down to “let’s solve a few together.”

Finally

I don’t know if it’s possible to practice what Lisa is doing here. It’s knowledge, the tightly connected kind you get when you spend thousands of hours in math classes, reflect on those observations, write about them, talk with other people about them, and then use them to inform what you do in another math class.

It’s possible, even easy, to spend the same number of hours without acquiring that tightly connected knowledge.

It’s something special to see it all put to use.

BTW. My guess is a lot of those knowledge connections were tightened because Lisa is a dynamite blogger. On that theme, let me recommend The Positive Effects of Blogging on Teachers, an article which does a great job describing ten reasons why teachers should think about blogging.