Hi folks. Blogging was light in July. But I'm back with a 6-part series on math teaching.
1. Elizabeth Green wrote a Sunday piece about math teaching, called Why Do Americans Stink At Math?
It is excerpted from her new book. You can pre-order it here. She's an excellent writer, so good chance it's a best-seller.
The book features several friends you've seen on this blog before, including Deborah Ball at U Michigan and Doug Lemov at Uncommon, as well as Magadalene Lampert of Boston Teacher Residency.
Today I'll briefly snip from her math teacher article. Then I append Dan Willingham's response here from Real Clear Education. I encourage you to read both in full.
2. I've spoken so far to 4 of my favorite math teachers. Earth to three more favorites: Veronica, Erica, and Jen S -- would you love your thoughts on the article too!
So I'll collect these "math teacher responses" and share with you in coming days.
Got it? Good.
(Um, beware: We have a SPAM filter issue on the blog that someone is fixing).
3. Possible Bonus: I will also try via my colleague Sean to get John Mighton's thoughts on the article.
Mighton is to math teaching/curriculum as Bill Belichick is to football coaching/strategy. Except: just like Belichick was mostly hidden from public view before 2000, and then has proven himself to be the best football mind of all time, Mighton remains mostly hidden from public view. Mighton is known to some Canadians, and he's got a math RCT in one of the large USA urban districts, just as Belichick was admired when he was an assistant to Bill Parcells. But few appreciate what pure freaking math awesomeness Mighton has cooked up at JUMP Math.
Jump has also created Common Core friendly stuff. So if you're with a charter CMO and you're not at least looking at his curriculum,
you are your kids are missing out.
Hey Jay Mathews - if you're looking to tell a story about Jaime Escalante #2, fly up to Toronto.
3. Okay, that was an aside. Here is Elizabeth:
The trouble always starts when teachers are told to put innovative ideas into practice without much guidance on how to do it. In the hands of unprepared teachers, the reforms turn to nonsense, perplexing students more than helping them. One 1965 Peanuts cartoon depicts the young blond-haired Sally struggling to understand her new-math assignment: “Sets . . . one to one matching . . . equivalent sets . . . sets of one . . . sets of two . . . renaming two. . . .” After persisting for three valiant frames, she throws back her head and bursts into tears: “All I want to know is, how much is two and two?”
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.
Left to their own devices, teachers are once again trying to incorporate new ideas into old scripts, often botching them in the process. One especially nonsensical result stems from the Common Core’s suggestion that students not just find answers but also “illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.” The idea of utilizing arrays of dots makes sense in the hands of a skilled teacher, who can use them to help a student understand how multiplication actually works.
For example, a teacher trying to explain multiplication might ask a student to first draw three rows of dots with two dots in each row and then imagine what the picture would look like with three or four or five dots in each row. Guiding the student through the exercise, the teacher could help her see that each march up the times table (3x2, 3x3, 3x4) just means adding another dot per row. But if a teacher doesn’t use the dots to illustrate bigger ideas, they become just another meaningless exercise.
And here is a snip from Dan W:
Green’s take is that if you hand down a mandate from on high “teach this way” with little training, and hand it to people with a shaky grasp of the foundations of math, the result is predictable; you get the fuzzy crap in classrooms that’s probably worse than the mindless memorization that characterizes the worst of the “I, We, You” method.
But I think there are other factors that make improving math even tougher than Green says.
First, the “You, Y’all, We” method is much harder, and not just because you need to understand math more deeply. It’s more difficult because you must make more decisions during class, in the moment. When a group comes up with a solution that is on the wrong track, what do you do? Do you try to get the class to see where it went wrong right away, or do you let them continue, and play out the consequences of the their solution? Once you’ve decided that, what exactly will you say to try to nudge them in that direction?
As a college instructor I’ve always thought that it’s a hell of a lot easier to lecture than to lead a discussion. I can only imagine that leading a classroom of younger students is that much harder.
There are also significant cultural obstacles to American adoption of this method. Green notes that Japanese teachers engage in “lesson study” together, in which one teacher presents a lesson, and the others discuss it in detail. This is a key solution to the problem I mentioned; teachers discuss how students commonly react during a particular lesson, and discuss the best way to respond. That way, they are not thinking in the moment in the classroom, but know what to do.
The assumption is that teachers are finding, if not the one best way to get an idea across, then a damn good one. As Green notes, that often gets down to details such as which numbers to use for a particular example. An expectation goes with this method; that everyone will change their classroom practice according to the outcome of lesson study. This is a significant hit to teacher autonomy, and not one that American teachers are used to. It’s also noteworthy that there is no concept here of honoring or even considering differences among students. It’s assumed they will all do the same work at the same time.
And if you want a little more, here's an interview with Elizabeth about her article.
I'll follow up with some teacher reaction.