1. This is a really cool question.
2. Gregory Taylor says he doesn’t know what the kid was thinking. Thoughts?
1. This is a really cool question.
2. Gregory Taylor says he doesn’t know what the kid was thinking. Thoughts?
Thoughts on this division mistake? #mathmistakes http://t.co/TZ1dxsNFBH
— Chris Robinson (@absvalteaching) January 31, 2014
Or did I get this wrong?
Part of what makes learning fractions tricky is that there at least three unnatural things to learn:
I work in a third grade classroom right now and I’ve heard a bunch of kids say something like the following:
“This is a third.”
Why? There’s an enormous mushing that goes around with “fourth” and “four,” with “third” and “three.”
—
Related(?) mistake: 4/6 is equivalent to 1/3
Maybe that isn’t related, but I heard it out of a kid who thought a third was 3/4, so it’s probably connected somehow. Maybe you guys can figure out how.
I’m a big fan of Stadel’s Black Box. I think what makes it fun is that there’s something small to figure out (What does the black box do?) before figuring out the big thing (What’s the sum of those two fractions?)
I recently did this with my fourth graders, and it was a ton of fun. Here were some of their answers to 1/2 + 1/3:
3/4
3 1/2 / 4 (three and a half fourths)
7/8
2/3
5/6
10/12
Can you figure out how kids got each of these answers?
Yes, yes, kids multiply the base and the power. Here’s what’s remarkable about this:
By defining exponents in terms of multiplication while offering no other images or models for what exponentiation does, we create a default model for exponents that sticks with people forever. When mentally taxed — either with a tough multiplication, or with an unusual power — kids revert back to this default model. They’ll do this especially in high school, and they’ll get questions wrong on tests and all sorts of other things not because they’re being sloppy, but because this default model is constantly lurking in their minds.
Incidentally, I asked these kids why they think about multiplication when they see powers, and this is what they said:
I’ve written about a lot of this stuff before. See here, especially, where I shared the high school versions of this mistake.
Now that all of this has been established, the next step needs to be finding a curricular approach that doesn’t rely as heavily on the “repeated multiplication” model for exponents. We need to build a distinctive set of images and intuitions that are native to exponents so that our kids aren’t always defaulting into multiplication when they have to think hard about math.
This is work that I’ve started, in a post titled “Exponents Without Repeated Multiplication.” I’ll send you there for the details, but I stake out two major claims about exponents education:
Beyond all of this, these exponents mistakes serve as a big reminder about the nature of learning, teaching and knowledge. The big, big lesson of all of this is that knowing/not-knowing is not clean and it’s not binary. There are degrees of knowing something. Would you say that these students don’t yet understand what exponents mean? What does that even mean, given the contradictory evidence we have in front of us.
But, then, what does it mean to understand something at all?
Misconceptions surrounding fractions are so well-studied that I feel a bit ridiculous sharing anything about them. Anyway…
I was chatting with this kid who was having a bunch of trouble with written fraction notation. She had been correctly solving problems that involved language such as “shade in four out of seven pieces” or “divide this shape into eighths,” but got stuck when she reached a problem that asked her to “shade in 4/6 of the shape.”
Alice: Oh, so that’s 5.
Me: Can you explain why?
Alice: Because it’s not six sixths.
Me: So, not quite.
Alice: Oh, it’s 2. Because that’s 6-4.
Me: …
Alice: Or it’s 10?
Me: See…
Alice: I’m really confused here. What’s the answer?
There’s no puzzles or misunderstandings here. Alice thought that the fraction symbol was an operation between the numbers 4 and 6. And of course she did. Every other time that she’s seen two numbers and a symbol before she’s been asked to produce a third number. This is new ground for her.
I’ve been taking the advice of Brilliant Commenters Fawn, Jenny and Avery and using the language of “out of” to bridge the gap for this kid.