HEA course for new maths lecturers

Last week I had the pleasure of attending a course for new maths lecturers run by the Maths, Stats and Operations Research discipline of the Higher Education Academy (HEA/MSOR). I was pleased that it dispelled several myths for me, in particular the myth that these courses never cater for mathematicians' needs. All the talks were given by experienced mathematics lecturers or people who have spent a considerable amount of time undertaking educational research specific to university-level maths. With such good quality input, and with the high level of engagement discussion amongst the participants, I learned a lot. Here are a couple of ideas I took away (not necessarily maths-specific!).

• It's important to let people figure things out by themselves. Maybe this advice sounds obvious, but I realised during the course that my natural inclination when I see someone struggling with a problem for more than about a minute is to jump in and suggest or hint at the next step. (e.g. "If you don't believe the statement, maybe you should just try and write down a counterexample?") If someone figures out the answer with my help, that's not as valuable to them as figuring out for themselves how to even approach the question in the first place. So in future I will try and sit back and listen and let my students shine.
• Get students to construct examples. I liked this extremely practical piece of advice. Instead of giving them a list of examples and saying "Check which ones have certain properties," you can instead ask them to get their hands dirty and construct examples. You can even do this during a lecture to give them time to absorb the definitions you've just told them (or to realise that they didn't understand something). Much like a radio or an atomic bomb, when you construct a mathematical object for yourself you understand it much better than if it's handed to you with a set of how-to-use instructions.
• There's a distinction between problems and exercises, and students need both to thrive. Problems being problematic and involving thought, exercises being repetitive (even if hard). The point was also made that exercises can conceivably be automated and that there is software (STACK, due to Chris Sangwin) which interacts with Moodle (a popular "virtual learning environment") which allows students to respond in mathematics which is then parsed by a computer algebra system and the answer is checked. For example, the student is asked "Give an example of an odd function", they input \(sin(x)\) and the computer checks that they got it right. This has the advantage that they can do many exercise and get immediate feedback on how they're doing. By contrast, imagine having to mark a student's answers to an indeterminate number of randomly generated integrals... isn't there something better you could be doing with your time? You might argue that university level maths involves more thought and proof than mindless computation, and maybe this automation is more useful for pre-university students. That's probably true, but my Methods course this term will involve plenty of computation, mindless or otherwise.
• New lecturers are overambitious with content. Having lectured two graduate-level courses and struggled to say what I wanted to say in the time I had to say it, I can testify to this. But can I put this observation to use when lecturing this year? My second years had better hope so...

A useful piece of software I encountered was GeoGebra (for producing geometric diagrams, etc.). While it's true that the same results (and many more) could be obtained with Maple or Sage, it's also true that GeoGebra is free and intuitive, and it's extremely quick to produce diagrams of the sort it's good at (not as code-heavy as Maple). I'm told there's a 3d version and this would have been useful for producing these diagrams.

I can highly recommend this HEA course for new maths/stats lecturers.