Am I allowed?
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?
- To solve problems confidently and correctly, and to feel that, having taken my course, they are now in command of an array of mathematical techniques which can help them. If someone has to ask "Am I allowed?" then they can't feel in control of the techniques. It's more like they're borrowing my tools and are afraid of breaking them. Even if they break something, they learn more about their tools by doing so than by hesitating to try.
- To make connections across the syllabus to other things they have learned. If they're worried about losing marks for using tools from one course to solve problems in another then they'll actively avoid doing this.
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.