...taught by James Marshall of Sarah Lawrence College in New York: 'Is the Singularity Near?' His motivations for the course are outlined here. He got his Ph.D. studying under Hofstadter.

Khan Academy now also features a Computer Science category. There are not many lessons yet but about 3 new videos are being added each day. They are going to add CS exercises soon too.

If you don't want to wait for the exercises, there is always the incredible Project Euler that you can use to hone your math and programming skills.

If you're a smart young teenager who plans on attending college, consider taking lots of AP Exams.

I program, and am also presently working my way through some math books. I find that I often have to backtrack to look up pieces of notation like variables and operators. Unfortunately, this is very problematic. Greek and Latin letters give no indication of where they came from and are not usable search terms, even knowing the full context in which they appeared. Many authors have their own bits of idiosyncratic notation, often combinations of subscripting and line art generated by TeX macros. Since expanding equations out to their definitions is so difficult, I sometimes don't bother to investigate when one looks odd, which as you might expect leads to big trouble later when errors in understanding creep by.

This is not nearly as big a problem for me when reading code, however, because there every variable and nonstandard operator has a descriptive name wherever it's used, and documentation is never more than a few hotkeys away. The same thing could be done for math. Suppose you took a typical higher math book, and replaced every single-letter variable and operator with an appropriate identifier. For me, this would make it much more readable; I would gain a better understanding in less time. However, I don't know the effect size or how broadly this generalizes.

Do other people have this problem? Might this issue deter some people from studying math entirely? Has anyone tried the obvious controlled experiment? How about with an experimental group specifically of programmers?

A Slate article by psychologist Alison Gopnik about how preschoolers have already learned to accept what the teacher says rather than exploring things to develop their own understanding:

[...] Daphna ran through the same nine sequences with all the children, but with one group, she acted as if she were clueless about the toy. ("Wow, look at this toy. I wonder how it works? Let's try this," she said.) With the other group, she acted like a teacher. ("Here's how my toy works.") When she acted clueless, many of the children figured out the most intelligent way of getting the toy to play music (performing just the two key actions, something Daphna had not demonstrated). But when Daphna acted like a teacher, the children imitated her exactly, rather than discovering the more intelligent and more novel two-action solution.

[...]

These experts in machine learning argue that learning from teachers first requires you to learn about teachers. For example, if you know how teachers work, you tend to assume that they are trying to be informative. When the teacher in the tube-toy experiment doesn't go looking for hidden features inside the tubes, the learner unconsciously thinks: "She's a teacher. If there were something interesting in there, she would have showed it to me." These assumptions lead children to narrow in, and to consider just the specific information a teacher provides. Without a teacher present, children look for a much wider range of information and consider a greater range of options.

This experiment is from:

D. Buchsbaum, A. Gopnik, T.L. Griffiths, and P. Shafto (2011). Children's imitation of causal action sequences is influenced by statistical and pedagogical evidence. Cognition (in press). pdf

The other paper cited in the Slate article is:

E. Bonawitz, P. Shafto, H. Gweon, N.D. Goodman, E. Spelke, and L. Schulz (2011). The double-edged sword of pedagogy: Instruction limits spontaneous exploration and discovery. Cognition (in press). pdf

The signaling view of college holds that graduates of elite colleges earn high average salaries not because of what they learned in school  but rather because top colleges select for students who have highly valued traits, the two most important probably being high IQ and strong work ethic.  Since in rich countries almost every smart, hard working person attends college not going to college sends a loud negative signal to potential employers.  Elite colleges, of course, are fantastically expensive signaling devices.

Although I teach at an elite college I have a proposal for an alternate much less expensive and probably even more accurate signaling mechanism.  An organization could have a one month program which only admits those who get a high score on the SATs or some other intelligence test.  Then the entire program would consist of spending sixteen hours a day solving by hand simple addition and subtraction problems.  The point of the program would be to show that its graduates can spend a huge amount of time doing extremely boring tasks with high accuracy.   Graduating from the program would signal that you had both a high IQ and strong work ethic.

If the program had a reputation for graduating valuable employees then I suspect it would become desirable to many recent high school graduates.  The challenge would be for the program to initially earn its reputation.  Perhaps it could accomplish this by having some well-known backers, by giving big cash grants to its first few graduates or by promising the first few graduates attractive jobs such as at the SIAI.