I just finished grading my first midterms of the semester, and I’m learning a lot about how my students think through the mistakes they made. (With apologies to Tolstoy, I’m definitely experiencing a bit of “correct solutions are all alike; every incorrect one is incorrect in its own way.”)
Last semester, the most frustrating (at least to me) mistake my students made on their first midterm was saying that if a set was open, then it wasn’t closed, and vice versa. They sometimes even came to the conclusion that Rd was neither open nor closed because it was both open and closed! That mistake taught me a lot about how language was influencing my students’ understanding of mathematical definitions, and I wrote about it last September on my other blog. This semester, my students largely avoided that mistake (maybe they read my post about it??), but they have been making other mistakes that I did not expect.
Michael Pershan is a math educator who collects and shares interesting Math Mistakes so he and other teachers can try to figure out what their students are thinking. One recent post came about because a lot of his students were writing things like three and a half fourths, instead of simplifying fractions the normal way. I also enjoy his continuing crusade to get people excited about exponent mistakes.
Pershan has the mistakes tagged based on where they fit into the common core standards, which is probably a helpful way for math teachers to see some real examples of mistakes their students might make, along with some possible reasons why. If you have an interesting mistake to share (elementary through high school level math), send it his way.
Mistakes can help us understand human thinking, but they can also show us how human thinking differs from the way computers calculate. Over at the aperiodical, Christian Perfect and David Cushing noticed a mistake in a Wolfram Alpha regression that can help us understand what computers are doing when they compute. And Patrick Honner had an interesting discussion with his class when the free online graphing calculator Desmos didn’t handle a removable discontinuity very well.
Ideally, I would not be learning so much about my students’ thinking on tests via their mistakes. It would be nice to be able to diagnose misunderstandings earlier. My weekly student problem sessions, group work in class, and one-on-one talks with students in my office give me a glimpse into their thinking, but I still don’t catch everything I’d like to before test time.
My teaching is of course a work in progress, and I am trying to figure out better ways to structure my classes and conduct assessments of student learning. The idea of using standards-based grading intrigues me, but I’m not quite ready to take the plunge. Joshua Bowman, who blogs at thalestriangles, has some reflections on what worked for him in standards-based grading. I’ve also been reading the standards-based grading posts at Bret Benesh’s and Kate Owen’s blogs. All of these blogs have given me a lot to think about as I reflect on how I want to organize my classes.