What is the value of The Great Courses for general math?

Question

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

Answer 1

Unfortunately, a very essential part of way to master mathematics is through exercises. As someone once said, you cannot learn mathematics from watching someone else do it, you need to do it yourself. It is similar to the way you cannot learn to properly do sports by watching other people do it. You need to get your hands dirty. Try different things. Fail. Try other things. Fail again. Until you have found a way to solve the problem.

Every mathematician has gone through this, it is part of what it means to do mathematics. Even though you might not want to become a professional mathematician, exercises are important nevertheless. So if I were to give you one piece of advice only, it would be this: exercise! It is important to ponder the concepts that you encounter, no question. Your approach seems very good in this respect. Take your time to digest the arguments one step at a time. If you want to build a watertight argument, you need to be sure that (and more importantly, why) every single step works.

From my own experience I know that proceeding too hastily without proper exercising can lead to you losing touch with what is going on really. You think you understand the symbols and the calculations, but sometimes one just needs to do things with his own bare hands to see how they work.

When I was younger, I sniffed at examples. The more abstract the better, but around 1 1/2 years into my degree this led to serious problems. A way that solved the problem eventually was to teach other people what I knew by leading tutorials for junior students. This is my final piece of advice: you have only fully understood something after explaining it to someone else in your own words.

I haven't checked Khan Academy yet, so I cannot help you with any specific things regarding this. Does it provide exercises and maybe also a forum-like way to discuss them and to receive feedback? This would be nice, so once you are confident in a subject you can try to help other people with it, enhancing your confidence even more, thus providing a better foundation to move on.

The only thing in this direction that comes to my mind is this great homepage by physics Nobel laureate Gerard t'Hooft. His aim is to become a physicist but the mathematics scripts offered there might help you as well.

I wish you great luck in your quest to master mathematics!