1 divided by two thirds

I’m not sure that I agree with the teacher’s diagnosis that this sort of mistake is procedural. Presumably, she means that the mistake was the result of remembering some sort of algorithm, a wrong algorithm. But what sort of division algorithm would lead you to multiply 2 and 3? in this situation?

On twitter, someone suggested (was it you?) that the issue here was that 1 divided by 1/3 is 3. And then twice that would be 6.  This would then be an instance of a more common pattern of error, the “it’s always linear” error. (Was it Dave who said this? Maybe this was Dave.)

This actually fits pretty well with the student’s explanation. It’s not a bad take.

Do you agree? The first three times that I wrote this post I said something like “The ‘linearity’ hypothesis is a pretty good one, but it doesn’t quite fit with what the student said. I’d suggest that the student had an association between the numbers 1, 2, 3 and 6, and it’s that instant judgement that her explanation is aiming to justify.” But now I’m not sure if I can really find any fault with the linearity hypothesis.

People who have more experience with fractions than I have: is the linearity explanation one that resonates with your experience of kids learning to divide fractions?

(P.S. I’ve got 4th and a 5th grade assignments, along with my high school classes this year. I’m excited to bulk up my understanding of little person stuff. Any of your submissions would help me test the theory that analyzing student work is a solid way to help build pedagogical content knowledge. SUBMIT!)

  • One thing that could be happening is that the student is having a hard time keeping track of the meaning of the two parts of the fractions and what division means — it’s high cognitive demand to reason through “Here is one portion. How many 2/3 of a portion are in your one portion?”

    It’s brilliant to first think of unit fractions — relate it to a simpler problem! There are 3 1/3’s in 1 portion, that’s obvious!

    Um, now what do I do with the 2, the fact that there are 2 thirds. Well, 2/3 is twice 1/3 so maybe there are twice as many 2/3s as 1/3s in 1.

    I think it’s analogous to this mistake: 90 – 11 = 81: take off the 1 from 11, do the subtraction, then add the 1 back in.

    Some possible reasons that 6 is more attractive and doesn’t send up red flags:
    – it’s nice to have a whole number answer, rather than that there are 1 1/2 two-thirds in 1.
    – 1, 2, 3, and 6 are a nice fact family
    – I’m thinking of numbers rather than some visual estimation of quantity.

    Also, I would agree that the student’s *explanation* is mostly procedural: it’s I did this, I did that, except for the one almost “why” which is “one-third goes into one three times.” We don’t hear, “and 2 * 3 is 6 and I did 2 * 3 because 2/3 is 2 thirds,” or “And then I wasn’t sure what to do so I just multiplied.” It’s the “what” rather than the “why.” But I agree that her work is not algorithmic necessarily, it seems to be sensible enough, and falling down at the “how do I know my simpler problem is valid/works like I expect it to?” that so many simpler problem methods fall down at.

    Of course, the linearity piece is also compelling and I’ve no idea how to test which explanation is right? Thoughts?

  • Speaking of high cognitive load, getting through that post and reply…whew. So some of this has been said (the breaking down into sub-problems part), but there’s at least a possibility that the mistake is more algorithmically deliberate and less cognitively loaded in the first place. Decomposing 2/3 into 1/3 * 2 can be a pretty useful thing to do sometimes. From that point, it might just be a failure of operator precedence. Essentially, 1 ÷ 2/3 = 1 ÷ (1/3 * 2), which is no problem, but if you fail to remember that 2/3 is actually a composed unit (i.e. that the multiplication actually takes precedence over the division), then you might think of it as 1 ÷ 1/3 * 2, in which case you come up with 6 pretty easily. Am I restating the linearity hypothesis here? At any rate, I think Max is projecting his intellect with the “there are 2 times as many 2/3 as 1/3 in 1.” I don’t see any evidence of that kind of reasoning. I think it’s essentially mentally dropped parentheses.

    The fact family thing is easy to test: just offer a problem with less beautifully related members. Alternatively, it might be interesting to see what Leslie thinks 2 ÷ 1/3 is. If she ends up with 6 for that one as well, that’s a point in Max’s column, because there really are twice as many 1/3 in 2 as there are in one. So if she’s thinking that way, here’s her chance to show me.

  • Chris’s idea makes me think of asking about things like 3/4 ÷ 1/2 and 3/4 ÷ 1/4. If she does something like cross multiplying for 3/4 ÷ 1/4 then she’s being more algorithmic and sloppy/confused. If she reasons about how many 3/4 are in 1/4 that’s a better sign (but could also be a recalled fact). 3/4 ÷ 1/2 would put that to more of a test — and at least seeing what algorithm she uses would help. Does she get 3/8? 3/2? Something else?

  • I thnk that one of the learners in Jo Boaler’s HOw to Learn Math class did something like this live and in person. She multiplied all the digits together. The video had the kiddo explaining her thinking, too… but I”m not finding the right one to link it.
    It made “perfect” sense visually when she wrote out what she did on the board. I think she even flipped the fraction. Lots of my students have constructed strictly visual-kinesthetic procedural frameworks for doing problems that have nothing to do with what things mean…

  • I like the idea of analyzing student work to build pedagogical content knowledge! I’m impressed with this blog. Les Steffe and his team at University of Georgia have done of of the best work I know of in analyzing student work in fractions-though I have to admit, when I was teaching I may not have had time to dig into some of their advanced thinking due to time constraints. The rational number project http://www.cehd.umn.edu/ci/rationalnumberproject/ has some nice free fraction lessons and a lot of free papers analyzing student work in fractions and division as well. But again, I find that sometimes researchers write things that are really longer and more complicated to read than necessary-especially if you are under the demands of lesson planning, teaching and grading. I do admit that given the time to sit down and read their work i have benefited immensely.