Eric writes in with this really cool mistake.
Do you see the problem?
Of course you don’t, because you don’t know the original question.
Eric writes, “The problem was originally 12 2/3 + 8.”
One of the joys of teaching fractions for the first time is seeing mistakes from my students that, prior to now, I’ve only read about.
(Does that make me a dork?)
(Yes, Michael, that makes you a dork.)
Anyway, the first mistake is an absolute classic. With numbers that show any sort of structural complexity, students regularly treat the components individually. See, decimals, complex numbers, polynomials, and other things I’m sure. (If you have links or suggestions of other sorts of common mistakes that fit this paradigm, drop a comment in the comment hole.)
The second mistake pushes on their part-whole understanding in a direct way. I’m not sure how this mistake plays itself out beyond the geometric context. Thoughts, people?
The third mistake is an interesting one, and one that I suspect is about being used to seeing number lines only of unit length before. Other than that, I don’t have much interesting to say about that. But maybe you do?
Lots of good stuff going on here. But I don’t think I entirely understand where 1/8 came from, though I get how that gets turned into 5.8.
[I never know whether to include all the mistakes from a class set or just a few. I feel as if it's helpful to include more mistakes, but sometimes overwhelming. My solution today is to post one especially cool mistake largely, and the others smallerly. Let me know whether that works.]
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!)
Offered to you without comment. Say something interesting in the comments.
Summer’s over. Let’s get back to work here.
A sprinkling of thoughts:
- Nicora, on a recent post, came out against using “x out of y” to introduce fractions. She writes: ”One of the most common ways to introduce fractions to young students is to talk about ¾ as “3 out of 4. And it’s part of the reason why students have so much trouble with fractions later on.” I wonder whether this is the sort of mistake that you’d see more of if you use this sort of language in class.
- I bet that this student would have trouble if asked to place these fractions on the number line.
- I don’t really get the task. Are these supposed to add up to 1? What’s the unit here?
- I don’t think that the student believes that 3/5 is more than 2/3. I’d bet that this student just isn’t paying care to the relative sizes of boxes. This aspect of the representation just doesn’t strike her as significant.
- I also notice, though, that within each activity there’s an attempt to make the boxes the same size.
I think that this last observation might be a way into a line of questioning that could help. I’d point to a shaded in box (maybe the kickball one) and ask, “What does this mean?” And then I’d point to another box and ask the same question. This would force us to bring out the unit, and the comparisons between the shaded boxes would force us to have a conversation about the relative amount of time spent at each activity. This would naturally bring us into ranking, which I think would be a good follow-up activity.
There are a bunch of interesting things here — please comment on them — but one moral I’ll take out of this is that learning math often involves becoming sensitive to nuances that would otherwise seem irrelevant.
I’ve been reading some constructivist stuff lately, so…
What resources does this student have for refining their understanding of fractions?
(Thanks for the submission, Dionn!)
What’s the fastest way to help this 9th grade student?
How did the student make the decision to plot 1/4? Tell a story.
Source: Chris Robinson