This one is on how it’s good when parents don’t understand their kids homework, and the usual folderol about how students used to learn via “tricks” and now it’s all about “understanding”.

They pulled out the old chestnut about the fractional division rule of invert and multiply. It starts out the same as it always does:

**“Back when you were in school and when I was in school, the way we learned mathematics — and I’ll talk about the division of fractions — we all learned the trick. You flip (the fraction) over, then you multiply and that’s how you come up with the answer,” said Principal Fernando Hernandez. “It worked, but that didn’t mean that you understand the concept.**

I have my own ideas about that but what caught my interest (and stopped further reading) was what came after:

**“So something we would ask the students to do now is we might actually give them the answer. ‘One divided by two-thirds is one-half. Please justify that, prove to me that that is true.'”**

In case you’re wondering, 1 divided by 2/3 = 3/2 not 1/2, but aside from that, what they’re trying to say is that they want students to be able to show that just as in dividing whole numbers, you can reverse the process and multiply the quotient by the divisor to get the dividend. Which doesn’t really explain why the invert and multiply rule works. But as I’ve said many times, if you give me two students of whom one knows why the rule works and the other doesn’t, but both can solve a word problem that requires fractional division, I can’t tell which one knows why the rule works.

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Nor does it matter for the most part. It would be nice if all understood, but few students care terribly. I feel as a mathematician that I’m on a holy mission to help people understand why things work and ultimately that’s what the academic discipline of math is about. But FUNCTIONAL understand generally suffices.

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I should have added — RC (since I’m using the WISE Math avatar here; I don’t know if Anna would agree 100% on this particular point)

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I don’t think Anna would mind if I quoted her here. She wrote something to me that addresses this issue, and –bottom line–she agrees with what you have said:

“With the consultants I’ve met, who always push this stuff and insist that kids aren’t fluent unless they can explain everything to you, it seems that they themselves just figured out that there are reasons behind procedures in math as adults. Then they’re angry that their teachers (supposedly) didn’t explain all these things to them. They’re certain that they would have liked math more and done better if only their teachers would have focused on understanding. So, their mission is to make sure that all kids are forced to explain their thinking at every step. Pure torture, really. Funny thing is, that the understanding piece is a lot more difficult for students. They generally don’t like it and it’s something that really comes with much experience and mathematical maturity. It won’t make students like math more if we spend more time on understanding…it will just confuse and frustrate them more. In my experience, I’ve found that students like step-by-step procedures and algorithms more than anything else.”

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They use “invert and multiply” as the poster child for justifying a change to how classes are run – not for understanding. They never accept the fact that we were actually taught that understanding in “traditional” classes. Take the denominator, invert it, and divide it by itself (a/a = 1). Then multiply that by the original fraction (a*1=a). The denominator becomes one and you have invert and multiply. But how do you justify or prove multiplying fractions? Is it OK that kids learn to do that by rote? Whatever. They want to justify the change to a student-led, mixed-ability classroom format. The understanding argument is just cover. They waste a lot of class time on a few ideas, but fail to create the required breadth of understanding by enforcing individual mastery of homework sets. Skills require understanding, and they can evolve to higher layers of understanding. Conceptual understanding with few skills is nowhere and can’t evolve into anything.

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