So here is a task that occupies maybe two rungs on the ladder of abstraction:
You bought some 40-cent pencils and some 70-cent pens. There were 10 of them in total for 460 cents. How many pencils and ballpoint pens did you buy?
At this altitude, it's difficult to apply your intuition to a solution. The important information has already been identified. You'll decide how to represent that information (symbolically, perhaps, with variables and two simultaneous linear equations) and what to do with that representation (manipulate those equations, solving for both unknown variables). But that's the extent of the task ladder.
So watch and marvel, as I have, at video of a skilled Japanese teacher reconstructing that task ladder to include more rungs above and below what few currently exist. I have watched it five-or-so times since Steve Leinwand introduced it in his April talk at NCTM. I encourage you to do the same. For the sake of our conversation here, though, I have created a smaller excerpt of just the teacher's questions:
Here are those questions and some commentary. The questions are listed in chronological order, like a script, but I'm sure you understand we're starting at the bottom of the ladder:
- This is what we're going to study today. What is this?
- What's this?
- How much are they? Guess.
- So how much is a ballpoint pen?
I prefer tasks that make more intentional use of guesses, prediction, and intuition. But I don't know what else he could do with this task except ask his students if they know what a ballpoint pen is and check to see if they have enough sense of that context to guess at its cost.
- You bought some 40 yen pencils and some 70 yen ballpoint pens. There were 10 of them in total for 460 yen. How many pencils and ballpoint pens did you buy?
The task is introduced.
- A table?
- Do you think the table would help you?
- Do you understand what the table is?
- Rio, do you think the table would be useful?
- Well then, if we're going to make a table, we need some labels, don't we?
- What kind of table?
- Do we need a label like this?
- Do you understand this?
- What else do we need as a label?
- Do we still need something else?
- How about this? "The total price."
- What do you mean 0-10?
- You mean no pencils?
- And how many ballpoint pens?
This is one of my favorite moves in the lesson. A similar task in the United States might explicitly direct students to create a table. It might provide a table with several rows pre-filled. In both cases, the problem would be abstracting itself. It would be answering the question, "How do I represent the information that's important?" for the student. It would be locking the process of abstraction inside a black box, far from the student's view. And then twenty years later we'll marvel when that student, now grown, complains that "algebra always seemed too abstract."
But the Japanese teacher isn't sending his students off to discover the abstraction. Yes, the students are making most of the crucial choices here — from the selection of a table to the labels of the table to the domain of the table — but the teacher steps in after each of those choices and makes them totally explicit.
The choice to include "0 pencils" in the domain of the table, for one example, is far from obvious. So when a student suggests it, the teacher underscores that suggestion over and over again, doing everything but sending a skywriter up in the air to write "YOU MEAN NO PENCILS?!" above the school in smoke.
- Well there are many combinations of the numbers right?
- Seika, can you tell me any examples of the combination?
- 3 pencils and 7 pens. Did you make the same combination?
- Is it correct?
- Are there any mistakes on the board, please tell me?
- But before we see that, there already is an answer, isn't there?
- What is the answer?
- This pair?
- Is it similar with your notebook?
The teacher asks his students to calculate the table by hand. In another bright move, he passes out the table's 11 columns to different students who calculate and return them. Now those columns are pinned to the board, arranged in no particular order.
He notes that they have solved the problem. One of the columns contains 460 yen. That's the combination of pens and pencils they're after. The task seems complete.
But the ladder continues upward.
- Can someone arrange the cards and make the table easier to see?
- Why did you arrange the cards like this?
- But why?
- It's easier to see?
- The order of numbers?
- For example, if I change the cards like this, is it wrong?
- This is the answer, is that all?
He's still picking at the abstraction!
Two girls come up and arrange all those haphazard columns into ascending order in a table. And he asks them, "Why did you arrange the cards like this?" He doesn't let up. He takes one column and sets it out of order. "Is it wrong?"
No, it's just less useful. A student says, "It's easier to see."
What's easier to see?
The student explains that when we arrange the table in order, the change in total price is hard to miss. When they're out of order that change is hard to see.
- How does our method match the Easy / Fast / Accurate rule?
- Is it fast?
- It is easy?
- Is it accurate?
- How can we solve the problem more quickly?
Now we're abstracting over new variables: speed, ease, and accuracy. The students decide unanimously that their method is accurate. They're mixed on ease and no one thinks it's fast. So now we're ready for a faster, easier method for a new problem.
- You bought some 60 yen colored pencils and some 80 yen markers. There were 12 of them all together. The price they cost was 820 yen. How many colored pencils and markers did you buy?
- What is the title of today's lesson?
What amazing work from such meager beginnings.
A few final notes before we dive into your commentary:
- Titles. What's with titling the lesson at the end of the day? Anyone know what's going on there?
- Technology. Blackboard. Chalk. Teacher voice. Student voice. Paper. Pencil. It's multimedia in the most literal sense of the word. (Tentative hypothesis: it's very difficult to work on the ladder of abstraction if your tasks are limited to one medium.)
- Tables. A table is a representation of a context's most important information. It's an element of an abstraction with which math teachers are extremely comfortable. I think we find it easy to assume, particularly at the secondary level, that what's obvious to us about these representations is also obvious to our students. I love how every single aspect of the table in the video is up for negotiation and debate — its labels, the inclusion of zero values, the fact that you probably oughtta put something in ascending order. Nothing is taken for granted.
- Time. At the end of class, the teacher notes that, wow, that took a lot of time. It turns out a deep understanding of abstraction doesn't come for free.
BTW: Michael Pershan offers an excellent analysis of Khan Academy as it relates to high-performing Japanese classrooms like this one. He closes with an alternate vision for Khan Academy that's provocative. You'll especially enjoy it if you find this #mtt2k business too mean-spirited. Commentary about that video belongs on that video's YouTube page, though. All comments about it here will be shot on sight.
Totally at the effectiveness of leaving the title to the end. The students not only see it because the content of the lesson is played out if front of them, but there’s a pretty clear agreement that the title of the lesson fits. One of the students even says, “Cool.” How often do you get a student saying cool when you *start* a lesson by writing the title on the board.