趋势解读：Understanding systems，聚焦形式化数学证明能力

Some time ago I read an article on what makes a good tutor. 1 1 I cannot find the article again, but in the process of writing this I came across the book Improving Academic Achievement , which has a chapter that may have inspired that article: The Wisdom of Practice: Lessons Learned from the Study of HIghly Effective Tutors ; Lepper and Woolverton; Academic Press; 2002. It explicated many of the things I do when tutoring, so obviously I thought it was a great article. When I had a side gig as a private tutor, I covered mostly maths and physics, so that’s how I’ll frame things in this article. The same thing applies to other fields too, but it might be harder the further away from maths they are . The main thrust of the lost article (as I remember it) was that effective tutors are highly empathetic to the level of motivation of their student, and they quickly adjust the lesson to that. That’s it. That’s the main thing good tutors do differently. If motivation decreases, they switch to lighter content, or even transition into non-lesson conversation. If motivation increases, they ramp up the difficulty of the lesson. Tutoring is, say, 80 % motivation management. Okay, but that undersells it a little. Lesson difficulty is not fixed for any topic; it depends on the student. Annoyingly, it even depends on the student’s level of motivation! The tutor must somehow know what is going to be difficult and what is going to be easy for their student, in every specific situation. Here’s how I figured it out when I was tutoring. A lesson with me consisted basically of me repeatedly (a) selecting an exercise for the student from their book, and then (b) watching the student work through it. 2 2 You can guess why motivation management is a big part of this! It sounds very monotonous and boring unless done right. Selecting an exercise for the student is a really fun activity. There needs to be some thematic variety to break the monotony of the lesson. But then the exercise should also be just at the limit of the student’s abilities at that moment, and ideally it should also end up revealing a flawed mental model of theirs. This meant I could only tutor students in subjects I was good at, because I had to quickly skim the exercises and visualise the steps to solve them, to find one in which a flawed mental model would be exposed. I knew which flawed mental models the student had because that’s what happened in the watching-them-work step. As the student performs the motions, they continuously emit clues as to the mental models running in their head. Sometimes there is something subtly weird about what they do – even if the result works out in the end – and that’s a potential flawed mental model. If I observed something weird, I would choose the next exercise to bring that specific mental model to the forefront. Most of the time, that exercise reveals there was nothing wrong at all, but sometimes the student does something very wrong in that exercise, confirming the suspicion. 3 3 This is also why, with new students, I would start out by going through a bunch of different exercises. Open a random page in the book, do one exercise there, then switch to another random page. The goal of this is for me to calibrate catalogue of mental models the student uses, and find out which are good and which need improvement. The student needs to find out about their flawed mental model too, of course. Many of my students had a learned response to check the solutions in the back of the book as soon as they had attempted to solve an exercise. I really, really wanted to tear out the solutions pages from their books and throw them in the rubbish bin. Looking at the solutions is not a good way to learn. When the student had attempted to solve an exercise, I would ask them if they believed their answer was correct. Whatever they answered, I would then ask them to verify their own solution. 4 4 Then why did I ask? The strength of their belief indicates how mu