Am I allowed?

[2019-05-21 Tue]

I notice more and more these days that students ask me "Am I allowed to..." For example, "am I allowed to use row-swapping to solve this question?" or "am I allowed to use cross products to find this vector?". Maybe cross products were covered in a different course, or maybe I introduced row-swapping after the other row operations to avoid talking about signs in determinants early on.

The question really means: "Will I lose marks for doing this?" which is a perfectly legitimate question. However, it is a clear example of educational "backwash", where the means of teaching or assessing has a negative impact on the way students learn. The students are worried about being penalised for using techniques which are correct but which are somehow "out of order" in a particular context. Of course, mathematics is not ordered, but when we teach it, it is not only ordered but compartmentalised into different subjects.

What do I want my students to be able to do?

It seems that I should make sure these two learning outcomes are built into the syllabus for any course I'm teaching, so that students know they're empowered to use whatever they know wherever it works.

Sometimes there is a good pedagogical reason for getting someone to do something without using a particular tool. For example, let's suppose you're trying to get people to understand the basic ideas underlying a particular theorem: it can help to work out the details of a special case of the theorem where the proof simplifies or which illustrates the idea of the proof. In that case, a student might want to say "this is an easy special case of Theorem X" and you'd want to say "yes, but show me that you understand what's going on by explaining how it works in this special case". This is fine, but perhaps it needs to be made explicit (a) exactly what you're looking for and (b) exactly why you're looking for it. For example, you could say explicitly "Theorem X implies Y as a special case. In fact, the proof becomes simpler in this case. In order to show that you understood how Theorem X works, explain how the proof of Theorem X works in case Y, making it as simple as you can."

However, I feel that this should be the exception, not the rule. The rule should be: you use whatever mathematical techniques you know to solve this problem. Whether you learned them from me or from someone else, it doesn't matter: they're your techniques now. If you spot a way of solving something using a technique from a different course, that shows initiative on your part, and it shows you're making connections between the different subjects you have learned. That's more important than making you jump through hoops.

I will try and make this explicit whenever I teach anything in future, and, crucially, build it into the learning outcomes so that the students can be confident that using the knowledge they've built up won't harm them.

Comments, corrections and contributions are very welcome; please drop me an email at j.d.evans at if you have something to share.

CC-BY-SA 4.0 Jonny Evans.