Rewiring my Brain: (gentle) Help Appreciated

There seems to be a mismatch between your description of the problem and your description of the solution. If you are already able to grasp concepts, but not to apply them rapidly, then the solution ought to be on the application side, not on the comprehension side as implied by "turbocharge [...] ability to absorb and deeply comprehend".

This being so, I suggest you practice doing problems. Lots of them. Go through your physics textbook (or calculus, or whatever) problem by problem, and do them all. Practice makes perfect.

In addition, it is my experience that people who say they grasp the concepts but can't apply them haven't actually grasped the concepts at all. I have yet to encounter anyone (including myself) who really fundamentally gets it (where 'it' may be physics, calculus, or even algebra) without doing a lot of dang problems.

Finally: It may be the case that your current learning effort is a bit misdirected. You say you want a career in a software-ish direction; you don't need to know a whole lot of either math or physics for that. You mention pre-requisites, so perhaps it is unavoidable, but any additional effort besides coursework may be better spent on programming. Which, again, you should learn by doing exercises, not reading books. Fire up emacs, write "Hello World", compile from the command line, run ditto. Iterate from there. For the sake of the absent gods do not start with an IDE or anything that hides the compilation step.

As a philosophy major who picked up CS after graduating, I have a recommendation:

If you want to start programming, start programming. The gaps in your knowledge will become apparent when you get to a problem you can't solve BECAUSE you don't understand the math behind it, and then you can backfill. Chances are you can develop a pretty solid grasp on the basics of programming before you get to that point.

This is not to say that taking math and science beforehand is a bad thing, but often people get hung up on prerequisites to the things they want to do and never get started.

My first course on programming was an earlier version of this course on edX: https://www.edx.org/course/mit/6-00-1x/introduction-computer-science/1122

It was far and away the best online class I've ever taken. Extremely high quality, designed specifically for the MOOC format, difficult enough that I had to really work hard, but doable with enough effort.

They're actually making it a part of a CS core series that looks like it will be very comprehensive.

Good luck!

Seconding RolfAndreassen's comment. Quit learning concepts for a while, instead take a big pile of problems that should be solvable way below your current level of knowledge, and solve them. For example, you could Google for some 8th grade math olympiad problems, and start going through them without skipping any. If the first ten yield too easily, go up a grade; if they're too hard, go down a grade. Don't be afraid to go down as far as you need to be comfortable. After you go through a couple hundred problems, some habits should start to form.

My problem is that I can grasp concepts and ideas, but when it comes to solving specific problems with actual numbers, I seem to shut down. Specifically, it takes me much more time (read "hours") to solve problems that ought to take 10 minutes. This is a particularly bad thing on tests and exams.

I believe that the difficulty I am having stems from so little exposure to symbolic reasoning in the past 5-6 years. I am looking for resources, techniques and advice to "turbocharge" (to use CFAR terminology) my ability to absorb and deeply comprehend technical material, so that solving problems becomes second nature.

So, typically when you look at a math problem there's a process you can go through that reliably gets from the problem to the solution. Do you have a good idea what all of the steps in this algorithm are? Can you identify where you're getting stuck?

My guess is that learning how to do that will be the largest speed boost.

I suspect that watching people solve problems, interleaved with solving them yourself, is the best way to practice. It might be possible to identify the step you're bad at, and set up problems which only go through that step- but this may require cooperation from someone else to be effective.

If you get the concepts and can slowly do the problems, it likely means you need to work on "internalizing" what you know, or teaching your type 1 systems (instincts) to direct your type 2 systems (thoughtful logic) to look in more fruitful directions earlier. The only way I know to really get good at this is practice, preferably with the involvement of other people who you can race with and against (and who can show you where you're going wrong when you spend time on a bad path).

Finding people doing topcoder or other programming/puzzle contests would likely make short work of this.

Specifically, it takes me much more time (read "hours") to solve problems that ought to take 10 minutes.

How long can you concentrate on a single problem? Can you stay with a problem for an hour or does your mind drift?

It is always going to seem like it is too slow. You ARE learning. I got a PhD in Applied Physics from Caltech. Working together, first year graduate students would spend about 15-20 hours a week doing the homework assignments in "mathematical methods of physics" course. I worked mostly alone and drew the line at 12 hours. Whatever I had done in 12 hours I turned in. I got a C+ 1st term and I think low B's the next 2 terms.

EVERYBODY in a Caltech Physics PhD program are the kind of undergrads who ripped through Calculus including multidimensional and dfferential equations and complex analysis, in GOOD undergrad schools, at a trot. WE were now banging our heads against problems.

It was just a scale change. And we learned how to do harder problems. We learned by doing them.

I am currently coding Android Apps, sort of for work, sort of for fun. I work as an R&D engineer on the "systems" and "rf" sides, and am definitely NOT one of the Software Engineers who make up the majority of my employer's engineers. But it is hard to get a software guy to do coding for you, and coding Android is almost as compelling as websudoku. So I am hacking away at it. And I would estimate I am about 1/7 as productive as a software guy, I think what they can do in a day takes me a week. And I've been at it for pushing a year (not full time though). I have little doubt that I am as smart as these software guys. They just have YEARS of built up experience at solving software problems. It took me months to realize the code in an object, some of it could be running in one thread and some in another: the object was not what was in the thread. But now 100 things are easier with my enhanced understanding. But there are thousands of things for me still to learn. And don't get me started on UI's!

My point being, just keep doing it. The one piece of advice, spend one or two hours on a problem, and then seek help. I don't think sitting their clueless and then finally getting it after 10 hours is the most efficient way to learn, and I think this is why people at caltech did problem sets in groups. Spending SOME time searching is needed, but you learn something when you see somebody else's solution AFTER that searching hour. never just copy solutions, though, understand them as you write them down. Ideally, you would be able to do the problem a 2nd time without referring to the other person solution.

So yes, that is why we make the big bucks. A hell of a barrier to entry. A hell of a lot of intellectual capital constructed in our crania.

Enjoy!

Unless you lack aptitude for math (did you do well in high school?), it looks like you just need to fill the gaps in your basic math education and practice what you learn enough to achieve fluency. Consider walking into your nearest Kumon center and getting an evaluation, then taking your time working through their worksheets, starting from the assessed level. Their program is very instrumental, it is designed to fill the gaps and to build fluency.

As for physics, in my experience most of the trouble people have with freshman physics is due to insufficient math training, though there is an occasional person who just "doesn't get" physics.

Is it possible you have a mental disorder or anything of that nature getting in your way? I ask because this sounds a lot like my own experiences with math classes. (I have a visual/spatial problem rather than a math problem, but having a brain that's bad at figuring out how the positions of things relate to other things makes it harder to read math problems without getting confused.)

Depending on what you want to program, it's also possible to get pretty far without a lot of math training. (Although a traditional computer science program probably would require math and physics classes.) This... might be a less thorough route, but it looks like someone can go pretty far as a developer without learning math other than what they pick up in the course of programming. (This coming from a junior undergrad programmer with two internships. So I'm not a very reliable source here. But from what I've done so far, the only things from math classes that I've found myself using for work and class projects are really basic discrete math, mostly formal logic and base 2 numbers.)

In addition to just practicing a lot, I strongly recommend learning discrete stuff: combinatorics, basic number theory, basic discrete math, graph theory, linear algebra, algorithms, theory of computation, probability (not the theory, I mean get a book of problems), even group theory. If you really are philosophically minded, set theory and logic are good ideas too.

Some of these might sound scary, but learning the basics is not harder than calculus - get a few introductory books, and choose one you like. And pretty much everything I listed will be more useful to you in computer science / programming. I'm not saying you don't need to know calculus, it's just that studying the above will more directly teach you mathematical reasoning, as well as being of direct use in CS.

Try chalkboard challenge on the brain fitness website lumosity. You have to quickly decide which of two expressions is bigger.

You could also try increasing your short-term memory (so you can keep track of what all the variable stand for) with memory exercises on http://cognitivefun.net/

It sounds like you have a grasp of the abstract formal math, but have an aversion to raw remedial arithmetic. In that case I would recommend diagnosing why you have that aversion. Did you take longer on your multiplication tables than you thought was acceptable? Do you feel a similar aversion to grade-school knowledge and practices? Perhaps the teacher was bad/demotivated you? Once you know what the cause is, a solution should be obvious.

I could be wrong, though. Not 100% sure what you meant by, "specific problems with actual numbers."

Check out Tricki.

It's a repository of useful mathematical techniques. From my experience, many skills can be developed through practice.

For calculus I strongly recommend Rudin. Reading the book (~ half of it) line by line and doing the great exercises was very difficult but gave me a real insight into calculus and mathematical thinking in general.

I am graduating as a philosophy student shortly

Why???