There's a fundamental tension in teaching one-on-one that applies whether someone's trying to solve a math problem or write a paper. They will get stuck or be unsure or wrong or ineffective, and they'll need your help. How do you respond? Do you show them how it's done and hope some of the spaghetti sticks, to brandish one extreme of the map, or do you ask them lots of questions and try like Socrates to guide them to an inner wisdom they'd always concealed from themselves?
Given enough time, my preference is the latter. About 2,500 years ago, a few years before Socrates developed his famous method of investigating and teaching, and thousands of miles away, Confucius wrote, "Every truth has four corners: as a teacher I give you one corner, and it is for you to find the other three." He also wrote that he would refuse to work with a student who wouldn't work hard to bring in the other corners, which by today's standards is going a bit far.
With math especially, I've noticed that many tutors and teachers lean in the other direction, possibly using the justification that "worked examples" are well known to be very effective, at least for what are called "well-defined problems" like math questions with correct answers. This becomes untrue as you go further into math and the questions get open-ended, but it holds in most math classes.
Most students prefer to have someone just tell them the damn answer.
That's not my style, though. Oh, I'll tell students all kinds of answers. I love to beat their calculators at arithmetic. Sometimes I write out the algebra. Often I give very-mini-lectures. There is no one correct approach to teaching. But I believe in their intelligence, and I push it to grow. Sometimes this frustrates them. Often they think I'm stupid or incompetent. Often they're right about that. But there's a method to my stupidity and incompetence.
If you want to solve a difficult problem, you will need tolerance for the feeling that you're working on a difficult problem and not really prepared to and not really good at this.
That's the number one thing you need, actually.
That's what I try to show students how to handle. Questions come up that I don't know how to answer. Mostly I do know. Experience has taught me. But I resist taking my "authority," such as it is, either from innate intelligence or from having memorized a solution. My "authority" comes from a greater comfort with uncertainty and embarrassment and confusion and inadequacy than my students have. That lets me help them get things right when they need to be right, or original when they need to be original.
Sometimes I won't be able to figure out how to answer something, and I'll tell a student, "Ask your teacher." People respond very differently to this. Many are very accepting and are even encouraged that I'm human. They might give an indication that they feel embarrassed for me (I still do get embarrassed) but let it go. It's often the very smart ones who find this unacceptable. They'll go and complain that I had to look things up. They'll say they feel uneasy and uncertain about everything else when I don't know my stuff. That's understandable. It's a feeling that's familiar to all of us, and part of my job is to avoid that. At the same time, the key insight to learn from Wikipedia's success is not that it's unreliable because everyone can edit it. Instead, we should notice that Wikipedia is shockingly reliable given how it works, and we should take away the idea that every source, including the scientific papers, our textbooks, our professors and parents and friends, experts on television and at conferences and in labs, all of these have a level of unreliability we should not actually ignore. Most students who complain because they want me to know everything work with me again. I don't know everything; I'm not always the best source, but I'm often a good one.
Not just young adults, but many people haven't crossed a certain river yet.
Your intelligence isn't what you know.
The best teachers don't always tell you the answer.
The best teachers may not know the answer.
The most valuable work anywhere is in this tension between questions and answers, between answers and actions, and sometimes most importantly, the tension between uncertainty and the first brave question.
Does that sound a little dry? All the world's wrapped up in it. Give it a spritz and put it in the microwave.
