Socratic. I know too much, so if I start in a random location there's a 90% chance that it will not apply to any given student -- instead I need to ask them "hey, where are you?" and "why do you think that ?" until we drill down to something concrete where I can help them understand what's going on.

Usually I like to dive right in, but there should always be some time setting expectations and figuring out what their goals are.

Most of my clients have needed some form or another of mathematics help. I think that within a high school context it is nigh impossible to create independent learning: the way we teach mathematics in high school is frankly soul sucking. I think the only hope is to give a "there's a light at the end of the rainbow" vibe, to tell them that professional mathematicians can't calculate a tip to save their life, and possibly prove to them that there are different sizes of infinity.

I think this comes down to paying attention. Every student who is struggling with some subject is running into a wall where their attention quickly disperses as their confusion increases. In addition it's common to see self-deprecation, "I will never be able to do this." For the first case a good tutor need to have moments of "I think we need to take a break from this!" and for the second a good tutor needs to have genuine empathy, "I also struggled hard with this."

Two things. First, as computing pioneer Alan Kay once put it, "perspective is worth 80 points of IQ." What he meant is that if you have the right perspective on how to understand a problem, you can solve with an average 100-IQ brain problems that without that perspective, require a brilliant 180-IQ brain. So my first thing is to get a student the right perspective. Secondly, there is a lesson from juggling: the single most important generic-goal for learning any skill is getting to the point where you can self-correct. If you can make mistakes and then see "oh, that was a mistake, let me do it again without making that mistake," then it's like exponentiation, you're raising your learning to the power of self-correction.

In a technical field like math or science the biggest struggles with reading comprehension are: (a) skimming too fast through the problem statement, (b) unfamiliarity with the jargon and (c) unfamiliarity with the basic expectations of the field. If a student is skimming, then we need to rehearse test questions with an emphasis on not sprinting-out-of-the-gate. If a student doesn't know the jargon, then we can fix that in an afternoon or so if we start filling out index cards with key phrases and the student's self-description of what they mean, with the discipline of "I'm going to read over your shoulder, each time one of these phrases comes up, I want you to stop and pick up the index card and make sure you remember what it means." Reading past a word that you don't understand can totally kill comprehension, this invites people to stop right when they hit a word they don't understand and think about what it could mean. For the last problem, that is much harder; there are no short fixes and you just have to pursue a broad education in the field of study.

Let them lead at first. Ask them what their problems are, quiz them on what they know, get used to how they see the world, and try to add certain distinctions that help them see the world more precisely.

It depends on the subject, and on the reason they're struggling. For example, if you're struggling with differential equations then one plausible way to go is to show the many commonalities with linear algebra -- that will work if their course emphasizes practical applications whereas they are more cerebral and want an abstract mathematical understanding of what they're doing and why it's valid. But you have to reverse that completely if the problem is the complete opposite, saying "here's the equations that I used in my Master's degree on an everyday basis, here's how they work and what they mean, oh, and here's the convection-diffusion equation that describes just about every physical process you experience in day-to-day life, and here's what you can learn from it."

I find that looking over someone's shoulder as they're working out a problem helps, but it has the nasty side effect that some students will start to lean on your corrections if you make them too dynamically. A good question that I've used sometimes is "how can you check whether that's right?" or "does that fit within your expectations?"

I have a perverse belief on this: I firmly believe that confidence is about 80% aggregated suffering; once you know all of the ways that you've been wrong you also get to the point of "I've accounted for all of these ways that I've seen myself fail before, unless I'm missing some new avenue of failure this should be robust against all of that stuff." So the important point is just to do a lot of work, fail a lot -- with empathy! I try to always emphasize that I've failed even more than they have. Only then can you get the feeling for when you can trust yourself to have not failed. However, every once in a while you can walk into some physics argument with a mathematical proof of your own correctness, and there is just no substitute for the confidence that this sort of proof can give you. It's very much an "I know what you're saying can't be true" confidence that gives you a tremendous perch of clarity to stand on. So when that's possible, it is powerful. But mostly it's about having hit enough stumbling blocks to know when you're in the clear.