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 R^{d} 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.

The closed/open problem is an instance where mathematicians named something badly. I’m not sure what I’d have called them, but if you use “closed” and “open” in the same context, every English speaker is going to be inclined to think that they are disjoint properties. I’m not sure what a better name would be, but that linguistic problem requires careful pedagogy to overcome.

I have noticed that many students make mistakes like forget a square root or accidentally writing a plus instead of minus and vice versa. Of course we all are human and we make mistakes.

The problem unfortunately is that many students that I have encountered made the mistake of not trusting their abilities to do math. They would calculate answers and solve mathematical problems on the right way, but they would scratch over their calculations because they believe that they made a mistake and that they are wrong.

Another big problem is notation. Many students have the right idea but when they write it down its is incorrect because of notation. Some students struggle with geometry due to the fact that they know how to calculate the answers but they find it difficult to show how they got their answers.

It is important for educators to look at the mistakes students make because it gives them an idea of how the students think. They can then try to adjust their classes to ensure that what they are explaining are carried over correctly to the students.

This is an interesting post, thank you for sharing this. Obviously we realize when students make mistakes its because they don’t understand the bigger concept. The mistakes that drive me crazy as a tutor, is when students mess up which formula to use. I’ve noticed with low performing math students, that they try to use formulas so much but never really understand what they are doing which results in the slope intercept form and the standard form being used interchangeably because they don’t understand what the variables represent.