Tom, in his fourth year of his PHD, very kindly came to the Big Smoke from leafy Cambridge, for a morning’s problem-solving with my Further Maths Class. STEP questions are just beautiful.
We started with an exploration of being stuck. Tom told us the story of his struggles:
- He was stuck for 8 months. 8-month-Tom is his new name.
- His teacher (supervisor) did not understand what he was doing, but was still able to re-motivate him at their weekly meetings.
- He took a 2 month break from the problem, returning with fresh vigour and a new perspective
- When he finally broke through the problem in a café, he went for a walk in a park, dancing and singing for joy. What a perfect image.
- Only 3 people in the world understand what he did.
Top tips for students:
- Cultivate the art of being bored. Take breaks from doing Maths, and from doing anything. Checking your phone or playing basketball are not acceptable, you need to let your mind wander. We went outside and each student wandered around the freezing playground in silence. Only Tom had a breakthrough in that time – fluke or because he has trained his mind so well?
- When you are stuck, just keep on working. It is tough, but guess, plug in numbers, try easy cases. Failing is still learning. Cultivate the art of being stuck.
- Don’t leave your homework till last evening. Give yourself time to get stuck and for the answer to come to you when your mind is empty.
- Tom noticed that students would continue down a dead-end for longer than he would. Constantly change tack, attack from new directions, to maximise your chances of success.
Observations from the lesson:
- Tom and I were both completely stuck at times, on the problems. The kids loved this.
- “Sir can I have a two month holiday so I can solve this STEP problem?”
- I need to help the students with their formal understanding of logic.
- Some groups stayed just on the first part of the question, and dived into the rabbit-hole. Generating questions is such an important art.
- Highlights from video: Tom pacing up and down waving his hands “Yeah it is true, I think it is true … … Yup, I think it is true” (that pause lasted 30 seconds). “O so now we are proving, not disproving! Saucy” exclaimed Wintana.
Implications for university:
- Problem-solving does take place in student kitchens or in communal study areas – students struggling through example sheets, going over lecture notes together.
- What is lacking is a sense of autonomy. Students in general are not learning to generate questions, to learn which questions are useful/beautiful/interesting. They tackle questions given to them, for which they know there is an achievable answer. There is no real ownership of problems. If you continue to a PHD, then this skill is essential. If you leave the world of academic Maths, then this skill is essential. Therefore, this skills is essential for all.
- Maybe I should go and do a PHD, to reflect on whether my problem-solving skills have improved through helping others develop theirs?