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?
Here’s a short mistake that I came across today that I found interesting.
I was chatting with a 5th Grader. The question was, “What do you think is your top speed?”
Her: I don’t know how fast I run.
Me: Well, you know here is how fast I walk. [Walks.] I think that’s about 3 miles per hour.
Her: OK, well maybe I can run 6 miles an hour.
Other Kid: You can run way faster than that. You can run 15 miles an hour.
Her: Well, yeah, for a little bit. But I couldn’t run 15 miles in one hour. I’d get tired.
I don’t give enough thought to miles per hour. It’s really an abstraction of realistic rates, rates that you could actually use. Like, if it takes me 3.9 seconds, on average, to add a paperclip to a chain, then I can use that to realistically figure out how many paperclips I could chain together in 5 minutes. But miles per hour — at least in the context of running — isn’t realistic in that way. It’s a concept that imagines a world that pays attention to my current speed but strips away all the reality of exhaustion and physical limits.
In the future I’m going to try to be more sensitive and explicit about this when talking about miles per hour with little kids.
Thoughts about rates and the units we use would be very, very welcome. Share interesting anecdotes in the comments, please.
A nice mistake with the place value here, where kids are adding “.5″ to the “.25.”
This is an awfully common mistake. What are some of the curricular approaches that help kids avoid these sorts of things?
(Thanks to Chris for the submission!)
How did the student get from 0.8 times 1.6 to 8.0?
What tendency is this an example of? (Or is the mistake unique to the context?)
How would you test your theories?
Thanks to Chris Robinson for the work.
What’s the next step for this student? How would you help?
Thanks to Tina C for the submission.
The question below asked kids to state whether the given sentence is always, sometimes or never true. Then, they were asked to justify their response.
Suppose someone said this to you. How would you respond?
Many thanks to Tina for the submission!