Compare Helen Keller with Ildefonso who was deaf and mute, and didn't learn language until age 27:

" But the interesting thing that he said is that he can’t even think that way anymore. (Music.) He said he can't think the way he used to think and when I pushed him to ask about what it was like to be languageless, the closest he ever came to any kind of an answer was exactly that. I don't know, I don't remember. I think differently now. " - Radiolab episode on words

Wow, excellent post!

For the floating ball problem, I had the same intuition as Bucky: A floating object will displace its own weight in water, so we can replace the ball with an amount of water that exactly fills in the hole created by the submerged part of the ball. So the box is balanced no matter where you move the ball to.

"Thinking Physics" by Epstein is an excellent source for puzzles like this, where you identify the relevant physical principles to give qualitative answers as opposed to doing tedious calculations.

Thank you for the write up! Matthew Walker was also on The JRE Podcast and has a Talk at Google. Well worth checking out.

Some of my experiences tutoring math over Skype:

One-to-one tutoring is superior to one-to-many tutoring.

One-to-one tutoring can be structured as a conversation with lots of back and forth and keeps student participation high. It also means the teacher can adjust their teaching style to that particular student's level and personality.

One-to-many tutoring is more of a performance on part of the teacher and is quite different. The students are much more passive in this situation. I don't know how to do this well.

The most important thing is to get the student to talk out loud about their thoughts when they see a problem. If they're intimidated or confused, it's very useful to ask the student what particular bit of the problem causes the confusion. (Is it because there's an $x$ in the denominator in the fraction $\frac{2}{x}$ in an equation you need to solve? Or is it because you need to differentiate $\cos \left(x^2\right)$ and the student doesn't know how to handle the $x^2$? Etc.)

Being able to identify and name the bits of the problem that gives them difficulty is a skill in itself and requires practice. When they do name them, you have a great opportunity to develop empathy with the student by reflecting their thoughts back to them. ("Yes, when there's an $x^2$ inside the cosine it really is a bit more tricky to differentiate than we're used to!" Etc.)

Then you can remove the complicating factor and solve a simplified, related problem. (E.g. what if the equation had $2x$ instead of $\frac{2}{x}$? Or what if it was just $\cos \left(x\right)$? Would the student then be able to solve the problem?)

Then you add back the complicating factors incrementally until you get to the original problem, explaining how to deal with the complicating factors using a similar problem with different numbers/functions, etc.

Optimally, the student should be teaching me and telling me step-by-step what I should write down (and how) as we solve the problem, with me only playing a supporting role, giving suggestions when the student gets stuck. Or even better: with me prompting for suggestions from the student, which we then try out, whether or not they work.

This role-reversal, with me being just a robot-hand and the student being the controller, gives great insight into their thought process and helps debugging it. (E.g. are they able to recognize that we can use Pythagoras because we have a right-angled triangle? Do they know the correct rules for multiplying together two parentheses? Etc.) It probably also strengthens the student's memory.

Some students are closed up and will simply say "I don't know." I find it's important to encourage them to guess, sometimes wildly, and then receive that guess non-judgementally, and then try it out. If it doesn't work, you can nearly always learn something from *why* it doesn't work. Does it *almost* work? Does it get us *closer* to the right answer? Or further away? (E.g. if dividing by $x$ makes the equation more complicated, the student himself will often notice that the opposite approach, i.e. multiplying by $x$ works better. Etc.) This also works as a free-recall exercise, helping the student connect related bits of memory together.

Removing the student's fear of math and self-labeling defeatist attitudes, and increasing their self-confidence is more important than any theorem you can teach them. But you cannot attack these beliefs directly. They will fade away by themselves in proportion to how many problems they solve successfully and how they learn to deal productively with problems they don't immediately understand.

If I at any point during the session get frustrated or negative or judgemental, I lose. The student will not look forward to the sessions; they'll become cautious of guessing because you shoot them down; and they'll be reluctant to tell you their working out in fear of being judged.

Polya's /How To Solve It/ has a useful list of questions to ask yourself (and therefore also the student) as you go about solving problems.

E.g.

• "What information do we get from the problem statement?"
• "What is the unknown? / What quantity does the problem want you to find?"
• "Is it useful to draw a figure?"
• "Do we know any related theorems or rules that can help us?"
• "Have we solved a similar problem before?"
• etc.

I never looked at the notes I took [...] I did notice, however, that I remembered things better when I wrote them down, so for a time my plan was to simply take the notes and forget about them in my binder.

Beethoven did the same:

Beethoven left behind an enormous number of sketchbooks. Yet he himself said he never looked at a sketchbook when he actually wrote his compositions. When asked, "Why then, do you keep a sketchbook?" he is reported to answered, "If I don't write it down immediately I forget it right away. If I put it into a sketchbook I never forget it, and I never have to look it up again.

Source: https://books.google.no/books?id=1YN3kc31nqAC&pg=PA148&lpg=PA148&source=bl&ots=w5G9bts1uF&sig=0E-DqVfesnTLnCTwXaFN3FT1WI&hl=no&sa=X&ei=UIyxU ikEMS_ygPey4F4#v=onepage&q&f=false

I'll add another supporting quote. Mathematician Niels Henrik Abel says of Gauss : "He is like the fox, who effaces his tracks in the sand with his tail."

A counter-example to this sleight-of-hand behavior is Euler. As related by Polya here: https://archive.org/stream/InductionAnd AnalogyIn Mathematics1 #page/n109/mode/2up

(markdown bugs up the underscores in my link)