Proofs are an important part of mathematics. In many courses, proofs will be important in two ways:
- reading proofs – e.g. to understand new ideas in the course through the proofs of important results, and to see applications of earlier concepts or theorems,
- writing proofs – e.g. to show understanding of ideas from the course by being able to apply them to solving “unseen” problems, including proving results that go a bit beyond what was covered in the course.
This post is focused on the first of these. Developing students’ abilities to read proofs is something that is not often done explicitly – there may be an assumption that students will pick it up by osmosis. There is some research into how to help students to develop these abilities (e.g., Hodds et al.. 2014), and a key part of this is having a good way to measure students’ level of comprehension of a given proof.
Proof comprehension framework
Mejia-Ramos et al. (2012) give a framework for assessing proof comprehension, with 7 different types of questions that can be asked:
You can see some more detail about these different categories in a recent talk by Pablo.
The framework is helpful when trying to write questions to assess students’ understanding of a given proof, as it gives ideas for different types of questions you can ask.
A few years ago, I used this framework to put together some multiple-choice proof comprehension questions for our Year 3 course, Honours Analysis.
My experience of these is that students found them quite hard – the mean score was around 75%, so they are not trivial for students to answer.
Hodds, M., Alcock, L., & Inglis, M. (2014). Self-Explanation Training Improves Proof Comprehension. Journal for Research in Mathematics Education, 45(1), 62. https://doi.org/10.5951/jresematheduc.45.1.0062
Mejia-Ramos, J. P., Fuller, E., Weber, K., Rhoads, K., & Samkoff, A. (2012). An assessment model for proof comprehension in undergraduate mathematics. Educational Studies in Mathematics, 79(1), 3–18. https://doi.org/10.1007/s10649-011-9349-7