Unfortunately, most students haven't been taught mathematics well at any point in their lives. The week before the fall semester starts I teach an algebra review class to incoming freshmen, and I'm amazed at how poor some of their math skills are (for example, confused why 1/x + 1/y is not 1/(x + y), or reducing (5 + x) / (5 + y) to x / y). It's a travesty that students can get this far in life without really understanding what's going on, but I don't think it's all their fault either. Fortunately, they have you! Better late than never, as they say. So here's my advice on teaching mathematical rigor to people: 1. Be as excited as you can be. Rigorous argument is not necessarily the most enthralling of things, but people pick up on whether you care about something and that can make a big difference in their opinion of the topic. 2. Be patient. You've, perhaps subconsciously, spent years developing the understanding you have of the subject; they have not. That can change, but it won't happen right away. 3. Ask them questions. Building connections between ideas might come naturally to you, but it does not to everyone. Try to nudge them to see relationships between things, even if you/they don't fully explore that relationship and why it exists right away. Walk them through your process of seeing why a solution is right or wrong. 4. Think about what guides your intuition for problems and explain what you can of that process.
This is so important. My precalc instructor was boring AF and I remember none of it. My DiffEQ instructor got so carried away that the prof next door would come over and tell him to stop banging on the chalkboard. As a result, I think DiffEQ is the fundamental magic that makes the world go 'round.1. Be as excited as you can be. Rigorous argument is not necessarily the most enthralling of things, but people pick up on whether you care about something and that can make a big difference in their opinion of the topic.
It's funny because mine was math so pure that the final was "Imagine a function that does xxxxxxxxxxx. What would happen if A) yyyy? B) zzzz? C) ppp ddd qqq?" No numbers. Fuckin' essay-style with roll-your-own functions and derivation. It was eye-poppingly hardcore. The rate of attrition in that class was such that 15% of the people who started it finished it. I got a 25% on the one assignment he gave all quarter and immediately asked him what I could do to resurrect my grade. he said "relax! You were in the top 20%!" When all of us were heads down and desperate on that final he said "if you take it home and study I will give you an A." Fuckin' A I'll bet I could teach Diff EQ even now and it's been 20 years. That class was goddamn amazing.
Maybe it was my professor - he was basically like "here's a Laplace and this is how you calculate a Wronskian so go do that now okthxbai". He never explained why things were done that way properly, so I had to resort to fucking Khan Academy more than once to understand what was happening.
Thanks. I'm trying. Believe me, I'm doing my best here. I've been tutoring people for quite a while and I agree with all that you've said. I'm excited, doing what I can to be patient and try to prompt them where and what questions one should ask. There's some improvement already. It's just tiring. But there is a massive problem where I don't know if I can give them a boost: my track was thrown into the deep water from the start. From the day one, I was solving problems that were too hard for me by design. I once described to you what I was doing as a freshman: Tackling problems on your own, without having to be prompted, takes years of practice and some mental independence. I can give them the gist, and I will, but it's an entirely different way of thinking about tasks. Sorry for being on a Star Trek reference binge, but this whole assignment feels to me like Kobayashi Maru test. The teacher must experience futility and yet remain optimistic. Do note that I have never said I'm giving up on them. It just seems like no matter what I do it feels half-arsed and exhausting.
Yep, it is exahusting especially if, like me, you're an introvert and any kind of interaction with people takes energy. On top of that, it's unlikely that you'll really see the fruits of your work -- the foundation you build with them now won't really begin to show until they build on it in higher-level classes (and maybe even projects/work after they graduate). Since I stuck around at the same university for undergrad and grad school, I've had the pleasure of running into senior students that I tutored or taught when they were freshmen and having them tell me just how much they benefited from my efforts. That's one of the best feelings in the world. So, keep at it! Things may seem futile now, but in the (very) long run they will be better people because of you.
Do they not teach the trick to expand 5x to xxxxx? I still use that whenever I'm in doubt. A few years ago I tutored high school kids in math and physics and I wholly agree with your points. I had a few strategies that worked well depending on the kind of student. One was to break a problem into its smallest constituents, to ask basic questions about those problems (which they usually got right) and to then assemble it into a bigger picture. Another is to attack a problem like you're Sherlock: what do we have here, and what do you know about problems that look like this? In my opinion, getting a student from doing math to understanding math is by asking the why question again and again and patiently teaching them the underlying fundamental principles.