Math Is Fun Forum

I am a computer science and math student right now. I am aiming to do a lot of the learning on my own, (not in a college/school).

I plan on taking the CLEP Exam, info in the link. It is essentially the same as an AP Exam or a Credit by Exam for CALC1 in college.

http://www.collegeboard.com/student/testing/clep/ex_calc.html 

I want to learn math to see if I enjoy it for its own sake, and for being generally more logical/smarter (mental exercise), and for complimenting work I may do in the sciences like physics and computer engineering/programming.

So, from what I understand, a lot of lower level Math, in classes like CALC1,2,3, Linear Algebra, and Differential Equations, is all about algorithms.

Anyway, what does it mean to learn by doing? How much of the learning is conceptual? How much is of it, is pretty much robotic rule following?

My main problem in learning Math, is getting stuck in cognitive loops trying to make sense of the conceptual/theoretical discussion presented prior to the exercises in every section.

But then there is this approach called "Learn by Doing", which says you basically do robotic exercises and drills and memorization, and then you learn math that way. Look at the examples before the exercises, and then just apply those algorithms to the exercises enough times so that you know the rules by heart.

I read a book on the topic,
http://www.amazon.com/Equation-Excellence-Make-Child-Excel/dp/0980144604
. It basically argued that drilling, memorization, and other "cranking"/"grinding" learning behaviors are critical to developing math skill.

He also said struggling through problems, (preferably with a mentor/tutor), is the best way to learn deeply, and develop your mind.

More importantly, and to the point, what is the best way to prepare for this CLEP CALC1 exam?

Also, the other notion I've been getting, is that only through struggling through problems do you really gain a deeper understanding. The memorization and drilling is important, but they are useful as supports for the problem solving which is kind of where the light bulb goes off, and the understanding clicks in your mind.

The critical subtext to this, which has really been bothering me, is that you also need someone who has seen the problem type you are working on, and can interface with you and let you struggle, make sure you are adequately struggling, and offer as limited assistance as is necessary for you to discover the solution on  your own.

Now, I don't know if this subtext was intended, but the analogy of doing math to doing a bench press lift in the weight room, really solidified this problem for me. And this is a problem, because I don't have access to someone who knows Math really well who can work with me one on one, and who knows me well enough to interface with my psychology to kind of train me and make sure I am working hard.

If the analogy didn't make sense, it goes like this. Most people don't use math in their work. However doing math makes you smarter and more generally intelligent. Similarly, few people become serious body builders, but many people play sports. Just as you do math to enhance your general intelligence for various non-explicitly mathematical applications, you do the bench press (or running (which doesn't need a spotter, weakness in the analogy)) to help your general athleticism in various sports, although in no sport, are you expected to actually lay down and do the exact bench press behavior.

The critical part of this analogy which is making me nervous is that, as Arnold Schwarzenger said in his movie Pumping Iron, paraphrased, "It is those last few reptitions that separate the great from the champions". And those last few reps are almost always ones which require spotting. Spotting means you need someone there, who can interface with you, and make you sweat.

Similarly, in math, you need someone who knows how to do the problems, and who is willing to interface with you, and make sure you work your hardest, because as the analogy/learning-theory goes, you learn the best by struggling through the problems.

I hope this hasn't been too much to read, but it is driving me nuts.