Category: loa

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[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:

Here are questions you can ask at the bottom of that task’s ladder:

  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.


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] Hypothesis #4: Right Question / Right Rung

#4. Choose the right question for the right rung.

This is a high-level abstraction of cities on a map:

I don’t think it’s easy to start so high up on the ladder and answer questions like:

  1. “Can you guess where they should put the new cell tower?” or
  2. “What information will be important to know here?” or
  3. “How should we represent that information?”

Guessing, it seems to me, is a task that is easier to perform at lower level of abstractions. (Like this one.) Meanwhile, it’s impossible for the student to consider the lower-level question, “What information will be important to know here?” when the important information has already been selected. (The relative locations of the cities.) It’s impossible to consider the question, “How should we represent it?” when the representation has already been selected. (A coordinate plane.)

Likewise, it’s impossible to ask a student to “Calculate the location of the new cell tower” when they’re looking at a low-level abstraction of the context. Calculation is a task that’s made possible by higher levels of abstraction.

Again, we find a limitation of print-based curricula. The authors choose to show a single level of abstraction of a context and then ask all their questions about it, whether or not they’re the right questions for that rung.

[LOA] Hypothesis #3: Test Your Abstractions

#3: Let your students test their abstractions.

The Google team has one hell of an abstraction on their hands. They’ve distilled the complicated process of driving a car and its infinite judgment calls, muscle twitches, and cursing into a finite set of variables. That set of variables is so finite, in fact, they say that a computer can compute it in real-time and drive a car by itself.

That just isn’t plausible. At various points, I’ll wager the Google team didn’t think their abstraction was plausible either.

I’ll put any sum of money on this: the team wanted to know if their abstraction was any good. They’d thrown away so much data for the sake of a manageable abstraction. Did they throw away too much? Could they have thrown away more? Is the abstraction just right? They could have turned to existing theory and models in artificial intelligence and said, “Well, the literature says the model should work.” But no one would have walked away from that conversation satisfied.

People like to test their abstractions. They want to see the driverless car drive.


The thing is, you and I are in on the joke. A lot of our abstractions are flimsy. If the basketball falls off the plane defined by the player and the hoop, the model falls apart. We abstract runners into particles moving at constant speed — no acceleration or deceleration. Try that abstraction with Playing Catch-Up. It falls apart. But it falls apart interestingly, and we win twice over. First, our students work on the abstractions we need them to work on. But we get a discussion about the limitations of those abstractions as a bonus. How much error should we tolerate? Are there ways we could improve the abstraction?

So for all these reasons and because there’s very little downside, give students the opportunity to test out and refine their abstractions.

[LOA] Hypothesis #2: Paper Is A Problem

#2. Print-based tasks often obscure the process of abstraction.

I advanced a hypothesis in my last post that we don’t clue students into the everyday abstractions that come so easily and subconsciously to their teachers. We find it easy to represent contexts (applied or pure) with symbols, tables, line drawings, and coordinates, so we often glide over those processes, obscuring them in the process.

So it’s helpful to give students a concrete context and explicitly show them how to climb the ladder of abstraction at every rung.

For example, when I worked with teachers on Popcorn Picker last week, a task that starts without any mathematical abstraction whatsoever, just a video, I marveled at different times at our work on the board and on their papers. “There’s no popcorn here,” I’d say. “Where’s the popcorn? You took that video and said, ‘The color of the wall doesn’t matter. The actual items filling the cylinders doesn’t matter. The guy filling the cylinders doesn’t matter. This is all that matters.”

It should go without saying that if the contexts in your textbook are predigested with those symbols, tables, line drawings, and coordinates, we’re already in trouble. The context has already been abstracted and we can only hope that every student already understood how to apply that abstraction.

My hypotheses here is that this predigestion is a fundamental condition of print-based curricula and very hard to counteract. For example, here again is Pearson’s cell phone tower problem with a presentation that conceals the ladder of abstraction.

Let me offer a presentation that would reveal the ladder. We would start with the satellite view of the cities, a low level of abstraction.

Then we’d move up to the ladder to a road map, clear-cutting forests, damming streams, getting rid of information that isn’t relevant to our question.

Then we’d abstract away most of that information, leaving behind three points on a plane with their labels.

To talk about the location of those points, we’d put a coordinate plane beneath them.

We’d consider each of those four frames separately. We’d move on to the next frame only after we had discussed the abstraction required to get us there.

With digital media, those four frames cost nothing but a few extra bits on a hard drive. But if I print each of those frames out on its own page and then bind those pages into a book and then mass produce that book, those four frames become very expensive and very heavy very quickly.

So instead, print-based curricula compress all those frames into one. They default to a very high level of abstraction and hope that everyone is already comfortable working at that level. It’s an expensive problem to fix in print.

This isn’t to say that print-based curricula isn’t great for a lot of things. This is just to say that making the ladder of abstraction clear to students isn’t one of those things.

2012 Sep 20. Brian Stockus starts a series on the issue.

Featured Comment

Brian Stockus, who works in the industry:

The other issue with print-based materials is that they can’t control the release of information very well. Generally a problem and any accompanying pictures or sample work are printed right next to each other. What if the person who wrote that problem wants the students to look at the sample work *after* they solve the problem for themselves? Well, it’s hard to make students do that when a sample solution is mere centimeters away from the problem.

Or in the case of the four images you suggest to use to help students abstract the cell tower problem. If they’re printed on the same page, the students can see all four steps at once which is inherently different than revealing one at a time and discussing each in turn.

This makes a case for the added value of digital tools in education. Not only is a video more dynamic than text, but you also have the ability to pause and rewind. Even using software like Powerpoint is more powerful than print in this regard because you can control when information is revealed. The same holds true for asking students questions on the computer using some sort of educational software. The instructional designer can control student movement so that an abstraction can be revealed in steps or after students have had the opportunity to think it through on their own first.

[LOA] Hypothesis #1: Be Explicit

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.

Featured Comment


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.