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

September 21st, 2012 by Dan Meyer

**#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:

- What are the new percents? Write down a guess.
- Which quantities change?
- Which quantities stay the same?
- 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:

- 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.
- 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.
- 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.

on 21 Sep 2012 at 6:57 am1 Michael PWhat is starting at the bottom of the ladder good for? It’s good for three things, if I’ve got you right:

1. Students will be less “frustrated and disengaged” with a task.

2. It will prepare students “to work at higher levels of abstraction later,” with that same task.

3. Students will have more opportunities to excel at crucial mathematical skills, besides for memorizing/using formulas.

And, even though this isn’t in the post, I’ll add a fourth thing that I (think that I) have seen tossed around this blog:

4. Students will be more adept at moving up and down the LOA on their own.

And I think it’s important to note how independent of each other these 4 things are. One might be sold on the increased engagement but skeptical that it improves a kid’s ability to perform the higher abstractions of this task. One might be sold on how starting at low rungs helps kids work on higher abstractions with a given task without buying the idea that they’ll be better at moving up and down the ladder on their own.

Each of these four things above is its own separate hypothesis.

on 21 Sep 2012 at 7:51 am2 a different DaveI realize it’s late in the game to ask this question, but…do students ever get tired of discussions that begin with asking them to “guess”?

As a reader of this blog, I’ve never tired of it. :) I expect the answer to be “no”, because it seems to hit on the kind of exploratory learning/thinking that our brains are built to love. Since I don’t spend as much time in the classroom as I imagine most commenters here do, I wanted to put the question out as a sanity check.

on 21 Sep 2012 at 11:34 am3 ErikThe thin veneer that is often pasted on and called context is troubling in its messages about the role of a learner. By giving questions (especially those neatly packaged ones Dan aruges against), students see learning as a passive activity where the initial stages of investigation and problem solving are completed for them. Framing a problem, challenge, or mess can be one of the more difficult yet impotant skills, but curricula typically removes that from the students. However, in my mind, it further entrenches this mindset when context is artificially applied because the interest in the learning is fortified not by the content but the novel context. (And, in traditional curricula, this context is broadly applied using assumptions as to what the students might be interested in.)

on 21 Sep 2012 at 10:14 pm4 Bowen KerinsI think it’s important when giving students a task that involves “guessing” to focus on overseeing the process, with a goal in mind of solving the problem through abstraction. As Dan describes it here, that’s the assembly of student work side by side.

Without this focus, students often go into “guess, guess better, guess all day” mode without doing much thinking about the guesses, or only thinking about the guesses (100, that’s too much, 70 that’s too little, 85 that’s too much…). This especially happens if “guess and check” has been taught as a way to solve problems. Guessing is good, estimation is good, abstracting is better. In the classroom I got around this by frequently giving problems that didn’t work out nicely, and the abstractors solved them more quickly than the guessers.

Care must be taken when putting student work side by side to abstract: good student work on the same problem can look a lot different. In the problem Dan gives, one student may operate on the number of kids while another operates on the number of adults while another operates on the budget, etc., etc. It’s not a tough thing to work around, and actually benefits students even more as they can then compare and contrast methods and recognize that there is more than one good way to solve a problem.

on 23 Sep 2012 at 7:31 am5 Bob HansenMichael P wrote…

1. Students will be less “frustrated and disengaged” with a task.

Yes, that is why teachers do this. I get that.

2. It will prepare students “to work at higher levels of abstraction later,” with that same task.

The only way to work at higher levels is to WORK AT HIGHER LEVELS. I do not see that here. In fact, that is my main complaint. This is in fact avoiding higher levels, and it is done because of point 1.

3. Students will have more opportunities to excel at crucial mathematical skills, besides for memorizing/using formulas.

Whether you avoid higher levels by staying at the bottom of the ladder or by memorizing formulas, the result (or lack of) is the same. If your point is that some students should be given more time with arithmetic reasoning because it is better to really get arithmetic reasoning than to continue forward into algebra and get nothing (including arithmetic reasoning), then I whole heartedly agree.

4. Students will be more adept at moving up and down the LOA on their own.

Once you are successful with mathematics (you get it) there is no moving up and down the ladder. All levels are in play at the same time.

It seems to me that the main gist here is to avoid teaching algebra. Instead of all of this algebra avoidance (and that is indeed what this is) why not speak out more against pushing students into algebra before they are ready? In three years of watching this thread and all of its claims of teaching algebra better, I have not seen any actual algebra. At what point do you get past the guessing and foreplay and get on with algebra?

on 23 Sep 2012 at 9:25 am6 Dan MeyerBob:No one is advocating staying at the bottom of the ladder. But algebra teachers who can manipulate algebraic abstractions are a dime a dozen. Teachers who can motivate and explain those abstractions are less common. My goal is to help develop those teachers.

Historically, you’ve offered only one suggestion towards that goal: students who don’t like how you prefer to teach algebraic abstraction (ie. as much of it as possible as soon as possible) have no business taking algebra. It never occurs to you to reconsider your preference. That kind of determinism has no place in the education of children and I find it pretty boring around here. Please find a new line or a new place to peddle the old one.

on 23 Sep 2012 at 10:55 am7 JonathanI’ve had a similar question for a while. That is, building inquiry, learning how to ask questions, slowly peeling back the onion are all fantastic, yet where is the math?

It took me a while, but I’ve finally figured out that Dan does not advocate skipping the problem sets and the higher level problems. What he’s doing is encouraging people to build stronger beginnings that will help the students find context in the problem sets they’re doing later.

Example: Graphing Linear Equations

Approach 1: Linear Equations can take the form y = mx + b, or ax + by = c, or y = m(x – x1) + y1. Let’s calculate a slope using m = (y2 – y1)/(x2 – x1). Here are your grids, let’s plot the y-intercept and determine the next point using the slope, etc.

Approach 2: Present a picture of a mountain. Pose the questions: what’s the steepest part of the mountain? Which side looks the safest to climb? Assist by segmenting the mountain profile and having them guess. How can we prove who is right? What would we need? Can I create a point of reference? What now?

Overlay a grid. Give the parts of the mountain coordinates. Ask how to find slope. Determine the winner.

Present them with the same problem set you were going to do in Approach 1.

In the end, the same Algebra gets done. But which version is more engaging? That’s the focus. We all know how to hand out problem sets.

on 23 Sep 2012 at 11:51 am8 Michael Paul GoldenbergBob Hansen wrote: “It seems to me that the main gist here is to avoid teaching algebra. Instead of all of this algebra avoidance (and that is indeed what this is) why not speak out more against pushing students into algebra before they are ready? In three years of watching this thread and all of its claims of teaching algebra better, I have not seen any actual algebra. At what point do you get past the guessing and foreplay and get on with algebra?”

Ah, yes, “algebra avoidance,” Wayne Bishop’s favorite epithet for any approach to teaching algebra that represents something more than what is dreamt of in his philosophy.”

If you’ve got to channel a mathematician, try George Polya. The guy who wrote “Let Us Teach Guessing.” He knew a good deal more about both mathematics and its teaching than Wayne or you ever will.

Having recently had the pleasure of viewing in its entirety the film of Polya’s legendary mid-’60s class at Stanford in which for some strange reason this world-class mathematician starts by asking to students to guess how many regions of three-space are formed by 5 planes, I’m confident that he’d have been very comfortable with the direction Dan and others are taking things these days. And that he would have looked at the Bishop-Hansen model of instruction as “not very interesting.” Not in 1965 and not now.

on 23 Sep 2012 at 12:50 pm9 Erik V.Bob Hansen wrote: Once you are successful with mathematics (you get it) there is no moving up and down the ladder. All levels are in play at the same time.

Then, the questions becomes “how do you get kids ‘to get’ math?” Understanding abstractions is key to this. And, more importantly, we do not teach algebra; we teach students, and making abstractions and the process of concretizing/abstracting explicit is paramount to teaching students to think mathematically.

on 23 Sep 2012 at 1:10 pm10 Michael Paul GoldenbergErik: some people seem to believe that they’re teaching little clones of themselves and that is what they should be paid for. These folks aren’t interested in pedagogy: pedagogy is irrelevant because they view education as a process of passing information they got from teachers much like themselves to students much like themselves. Anyone who doesn’t like it can get the heck off the bus.

In this particular world-view, there are “mathy” people and everyone else. The mathy ones learn math (we can discuss what exactly that means to them, but I’d say it is precisely the mathematics they’ve mastered. Anything they haven’t learned, don’t know about, struggled with in school, etc., doesn’t matter), and the rest are. . . unworthy. So why bother to think about different ways of teaching? After all, there’s a nifty little syllogism that involves some very self-serving premises: Whatever I know comprises what’s worth learning. I learned it in the way I learned it. Those who cannot learn what I’ve learned the way I learned it are. . .

well, you finish it, Bob. After all, it’s what keep you convinced you have something valuable to contribute to the world of education.

Unfortunately, that world-view leaves a rather sizable number of folks thrown off the bus of mathematics very early on in the game, and with little or no opportunity to get back on. People like me, who slept through most of high school math thanks to teachers who thought much as Bob does, were clearly far too “other” to be invited to rethink their inevitable dislike of what they’d been misled into thinking was the entire mathematical story.

Luckily, I knew some pretty bright people, read some interesting articles and books, and became motivated to find a more inviting bus to get on. It still contained real mathematics. It simply wasn’t being run by elitists.

It’s remarkable to see how threatened some people are by that, and by people who are working to make the next generation of buses much more likely to keep a lot more riders than the line Bob operates.

Erik, you make a crucial point: we teach students. Not every student is at the same place as every other one at a given point in his/her development. Those who are quite content to see those who aren’t at some arbitrary point at some arbitrary age left behind for good (see the so-called Common Core State Standards for the latest idiotic blueprint) have no business teaching or telling others how to teach.

on 23 Sep 2012 at 3:42 pm11 blaw0013I believe this conversation is confused somewhat by our inadequate language; we are failing to distinguish between the narrow (and outdated / dead) symbolic manipulation component of Algebra, and the broader Algebraic Thinking–something I conjecture is actually shut down by most ALL pedagogical practices in the modern high school classroom.

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