In a fascinating article "Why Do Americans Stink at Math," Elizabeth Green explores, among other things, the recent history of math education in the US, and talks a little about a fairly well-known traditional teaching technique known as "I, We, You."
Most American math classes follow the same pattern, a ritualistic series of steps so ingrained that one researcher termed it a cultural script. Some teachers call the pattern “I, We, You.” After checking homework, teachers announce the day’s topic, demonstrating a new procedure: “Today, I’m going to show you how to divide a three-digit number by a two-digit number” (I). Then they lead the class in trying out a sample problem: “Let’s try out the steps for 242 ÷ 16” (We). Finally they let students work through similar problems on their own, usually by silently making their way through a work sheet: “Keep your eyes on your own paper!” (You).
By focusing only on procedures — “Draw a division house, put ‘242’ on the inside and ‘16’ on the outside, etc.” — and not on what the procedures mean, “I, We, You” turns school math into a sort of arbitrary process wholly divorced from the real world of numbers. Students learn not math but, in the words of one math educator, answer-getting. Instead of trying to convey, say, the essence of what it means to subtract fractions, teachers tell students to draw butterflies and multiply along the diagonal wings, add the antennas and finally reduce and simplify as needed. The answer-getting strategies may serve them well for a class period of practice problems, but after a week, they forget. And students often can’t figure out how to apply the strategy for a particular problem to new problems.
How could you teach math in school that mirrors the way children learn it in the world? That was the challenge Magdalene Lampert set for herself in the 1980s, when she began teaching elementary-school math in Cambridge, Mass. She grew up in Trenton, accompanying her father on his milk deliveries around town, solving the milk-related math problems he encountered. “Like, you know: If Mrs. Jones wants three quarts of this and Mrs. Smith, who lives next door, wants eight quarts, how many cases do you have to put on the truck?” Lampert, who is 67 years old, explained to me.
She knew there must be a way to tap into what students already understood and then build on it. In her classroom, she replaced “I, We, You” with a structure you might call “You, Y’all, We.” Rather than starting each lesson by introducing the main idea to be learned that day, she assigned a single “problem of the day,” designed to let students struggle toward it — first on their own (You), then in peer groups (Y’all) and finally as a whole class (We). The result was a process that replaced answer-getting with what Lampert called sense-making. By pushing students to talk about math, she invited them to share the misunderstandings most American students keep quiet until the test. In the process, she gave them an opportunity to realize, on their own, why their answers were wrong.
The problem with "I, We, You" is that it provides students with a pre-packaged solution (e.g., long division) right away, and then provides a bunch of (often contrived) problems which that solution can be used to solve. There's not much actual thinking required. After all, the teacher already provided the solution; all the students have to do is start using this new solution on a bunch of similar problems with slightly different numbers, a process known to many as plug-and-chug.
The "You, Y'all, We" approach starts with a problem (hopefully an intriguing problem) and asks, "Can you figure this out?" This can then lead to all kinds of interesting, and mathematically valuable, outcomes: What kind of approaches will the students come up with? Will they all get the same answer? Is there a single right way to solve the problem? If someone uses a different approach and gets a different answer, was there something wrong with their approach? What exactly was wrong with their approach? If I'm not able to figure it out on my own, can I figure it out with a partner? After I figure out the solution to this problem, can I then use my approach to solve similar problems? If not, why not?
It's not too hard to see that the "You, Y'all, We" approach tends to encourage and support actual problem-solving and critical thinking, or, if you prefer, sense-making, while "I, We, You" tends to encourage answer-getting. So if "You, Y'all, We" is so much better, why don't more math teachers use it? Green explores a lot of fascinating reasons in her article, but I'll note two reasons based on my own classroom experience:
1. "You, Y'all, We" is harder for the teacher
The fact is, we math teachers already know how to do long division, and solve word problems, and interpret graphs of functions. And the simplest way to explain to someone else how to do those things is, well, to just tell them. "Here's step 1, here's step 2, and here's step 3." Even if a teacher recognizes that this is not always the most effective method, coming up with ways to teach math concepts using the "You, Y'all, We" method is tougher than you might think. Plus, it almost always takes more class time, which means we might not get to cover everything we're supposed to cover for this class.
2. "You, Y'all, We" is harder for the students
"You, Y'all, We" may in fact be a much better way to learn and understand math, and it may actually be more interesting and engaging for the students, but it does have one major drawback: it takes some mental effort on the part of the student. And if you happen to be the kind of student who is looking to do the least amount of mental work that you can possibly get away with, that's a big drawback. Or maybe you're the kind of student who's not necessarily opposed to putting forth a little effort in your math class, but you've spent the past eight or nine years learning math using the "I, We, You" approach, and that's pretty much the only way you know. Even if you're willing to try this better method, it may take a while for you to get used to it. Which is more time that doesn't get spent covering other material, and you can see where this is going.
I love the "You, Y'all, We" approach, though I'd never heard it called that until I read Green's article; it's similar to many other approaches you may have heard of (discovery learning, Dan Meyer's "Three Acts of a Mathematical Story," etc.), and I try to use it as much as I can in my classes. It really is a good approach to teach real understanding of math concepts and not just answer-getting. But it's not a simple thing to do, either for the teacher or for the students.
(Hat tip to Dan Meyer for his blog post that pointed me to Green's article.)