What is this, you ask? A question from a fifth-grade PARCC exam that was given to Massachusetts students last spring. I read through the whole test recently — fractions, volume, perimeters — and when I reached this one, I went through a multistage process.
Stage one: Panic. “Area model what-huh?” Stage two: Frustration. Can’t a poor kid just do long division and be done? Stage three: Extrapolation. Is this the Common Core? The future of education? To the barricades!
In the pitched debates over whether to replace Massachusetts’ MCAS exams with the Common Core-aligned PARCC tests — and whether to keep the Common Core or blow up education, again — new math litters the landscape like spent ammunition. “Partial sums addition.” “Area models.” The “lattice method,” which turns a basic multiplication problem into what looks like a Sudoku puzzle from hell. Parental panic translates to political will.
One thing to remember about new math is that it predates the Common Core curriculum standards — and it probably won’t disappear if the Common Core falls. It’s part of a broader philosophy that the Common Core embraces: that kids should know not just the standard algorithms, but the concepts behind them. It’s not just whether you can do long division; it’s do you get long division? Can you explain it in words?
This can feel like a ridiculous task, like finding words to describe the color “blue.” But some people actually like this way of mathematical thinking. I showed the “area model” question to a colleague who sat down calmly, figured it out, and was impressed. To him, the old long-division algorithm had always felt like magic; this was breaking down the math into its component parts, the way I use blocks to help my first-grader with addition.
Teaching math this way has consequences. Richard Phelps, a critic of the Common Core and the author of several books about testing, notes that once you get to higher-order math, you can’t be reinventing the wheel with every problem. He compares the standard algorithm to learning to type with a QWERTY keyboard: Eventually, it becomes part of your muscle memory.
In fact, the Common Core says students should learn standard algorithms, too. So PARCC raises a different philosophical question: What should a test be testing? If you understand what division does, how much more do you need to know? When I use a calculator, after all, what matters is the inputs, not the code that the machine uses to get the right answer.
That’s one of Phelps’s critiques of PARCC: The questions tend to be more complicated, involving some wordy new formats — like presenting a math question that a fictional kid answers wrong, then asking students to explain why he’s incorrect.
“I think the fairest test questions,” Phelps said, “are the ones where the format is the most straightforward.”
Which brings us back to that “area model” question. I asked Jeff Nellhaus, PARCC’s chief of assessment, why it was on the test. Nellhaus, it should be noted, helped to develop MCAS back in the 1990s. He said this question is meant to measure one of the Common Core standards: how much students understand the connection between multiplication and area. A student who can do long division will get partial credit, he said. And he imagines that a question like this could provoke a big classroom discussion.
“Think about math and why it’s so boring: Students get worksheets where they have to solve 50 division problems,” he told me. “This kind of problem actually makes them think and gets them engaged.”
Well, yes, but so does a classic word problem, the kind that doesn’t require a whole new vocabulary. I’m all for encouraging students to think rather than simply regurgitate formulas. But the developers of PARCC need to be careful, and not just because a new-math zinger can scuttle political support for a test that’s largely quite good.
There’s a difference between teaching methods and testing skills — a difference that might mean everything to a stressed-out kid on a timed test. For that kid’s sake, PARCC ought to be tough. But it also should bend over backwards to be clear and fair.
The answers to the above question are:
M = 6400
N = 70
P = 3
Q = 24
R = 2