I am an adult trying to rebuild a foundation in math. I had three semesters of calc in the 90's but used the memorize and regurgitate approach to math classes. On a recent project that involved BLDC motors I realized that I could manipulate symbols but had almost no idea of the math behind the symbols.
For the last six months I have been taking a two pronged approach to rebuilding a decent foundation. I started working through Khan Academy... from the beginning. When I got to algebra 1 I started watching the video from The Teaching Company.
I tend to watch a the videos in order. Every time there is a new diagram, graph, or algebraic expression, I pause the video and lie down on the floor and think about it. I trying to explain to myself how the various pieces fit together, what those relationship imply, and how the new concept fits in with other stuff I have learned. Sometimes this can take a few minutes per concept, sometimes it can take hours or even days before an idea gels.
For some reason the proofs of the law of sines and the law of cosines resonated with me. The way the rather simple ideas of geometry, algebra, and trig fit together was an ah-ha moment.
For me working problems seems to be significantly less effective than pausing to think deliberately and deeply about a topic.
After about 6 months of study I feel I have a deep enough understanding of the materials up to and including precalculus. In Aug. I plan an starting a calc sequence. Under the theory of "If it's not broke, don't fix it," I plan on continue using Khan Academy and the great courses videos. There have been some pretty harsh criticism that Khan Academy helps one learn to perform calculations, but is weak at helping someone learn to think mathematically.
I would appreciate it if anyone could help a self study student understand the weakness of my approach and what it implies in my future studies.
Edit-- Thanks @Andre, I have a little bit of a love hate relationship with exercises. Due to past old habits it is VERY easy for me mindlessly churn out exercises without understanding the math behind them. I was a CS major in the 90' math, physics, and chem were just requirements to get through as quickly as possible:(
Khan Academy has a series of very easy exercises to help the learner verify that they know the information. I try to wait at least a month between watching the material and doing those exercise under the assumption that if I can still work them 4 weeks later I have internalized the material.
This might be splitting hairs but personally find challenging problems effective capstones to a new topic. By this I mean a couple of problems that take several hours each to solve. Ideally they require the use of multiple prior concepts to complete. I like to chew on these problems when driving, mowing the lawn, etc. I can feel the synapses forming interconnections as I struggle through them.
It seems like you have helped me answer my own question. Over the next phase, 2-3 semesters of Calc, I will find a good text and increase the amount of time and effort I spend on those questions.
After that, I'll explore communication aspects you brought up:) thanks