After attending the middle school data analysis meeting, I decided to do some analysis of my own. I started with the inverse function problem on the PreCalculus midterm since it was the only topic students did well on during the semester but struggled with on the exam.
What types of mistakes do you think students would make?
Have a hypothesis? Okay, now you can view the exact mistakes my students made at the Math Mistakes blog.
Back? Great! So here’s what I noticed:
Many students made small errors in basic algebra that may have been due to fatigue (this was question 23 of 25 and most students had to stay past the scheduled 90 minutes to finish the exam).
It was a “check if these are inverses” question but many students attempted to check by finding the inverse themselves, rather than composing the functions. When their solution didn’t look exactly like the given inverse they said “not inverses” and moved on. In fact, the inverse many students found was equivalent to the one given (they found ).
Next time I teach/assess this:
In class/homework I need to give more examples of equivalent functions that don’t look equivalent so students are compelled to check more carefully.
I wonder if giving the functions in reverse order would change the results for the better (so students who found the inverse themselves would start with and work from there). I’m inclined to think I would see more algebra mistakes but fewer students who found the inverse correctly and didn’t see that their solution was equivalent to the given one.
My co-worker and I found it difficult to come up with a problem that requires students to solve the inverse since many of our students could immediately “see” the inverse without having to write anything down. I will need to spend more time finding good problems with interesting inverses that aren’t too complex.
Requests for feedback:
Did the mistakes my students made match the ones you expected to see? What would you do to remediate this year or prevent such mistakes next year?