# How valuable is it to learn math deeply?

I've been wondering how useful it is for the typical academically strong high schooler to learn math deeply. Here by "learn deeply" I mean "understanding the concepts and their interrelations" as opposed to learning narrow technical procedures exclusively.

My experience learning math deeply
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When I started high school, I wasn't interested in math and I wasn't good at my math coursework. I even got a D in high school geometry, and had to repeat a semester of math.

I subsequently became interested in chemistry, and I thought that I might become a chemist, and so figured that I should learn math better. During my junior year of high school, I supplemented the classes that I was taking by studying calculus on my own, and auditing a course on analytic geometry. I also took physics concurrently.

Through my studies, I started seeing the same concepts over and over again in different contexts, and I became versatile with them, capable of fluently applying them in conjunction with one another. This awakened a new sense of awareness in me, of the type that Bill Thurston described in his essay [Mathematics Education](http://arxiv.org/pdf/math/0503081.pdf): 

> Mathematics is like a flight of fancy, but one in which the fanciful turns out to be real and to have been present all along. Doing mathematics has the feel of fanciful invention, but it is really a process of sharpening our perception so that we discover patterns that are everywhere around.

I understood the physical world, the human world, and myself in a way that I had never before. Reality seemed full of limitless possibilities. Those months were the happiest of my life to date.

More prosaically, my academic performance improved a lot, and I found it much easier to understand technical content (physics, economics, statistics etc.) ever after.

So in my own case, learning math deeply had very high returns.

How generalizable is this?
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I have an intuition that many other people would benefit a great deal from learning math deeply, but I know that I'm unusual, and I'm aware of [the human tendency to implicitly assume that others are similar to us](/lw/dr/generalizing_from_one_example/). So I would like to test my beliefs by soliciting feedback from others.

Some ways in which learning math deeply can help are:

*   **Reduced need for memorization (while learning math).** When you understand math deeply, you see how many different mathematical problems are special cases of a single more general problem, so that in order to remember how to do all of the problems, it suffices to remember the solution to that more general problem. This reduces the cognitive load of doing math relative to what it would be if one was considering each individual problem in isolation. When I taught calculus to freshmen at University of Illinois, I got the impression that many of the students studied for tests by trying to memorize all of the homework problems individually. There were too many homework problems to memorize, so this didn't work very well. Had they learned the material on a deep level, they wouldn't have had this problem.
*   **Ability to apply knowledge in novel contexts (that require mathematical reasoning).** When you understand general mathematical principles, you can apply mathematical knowledge to tackle mathematical problems that you've never seen before. This contrasts with mathematical knowledge that's restricted to knowledge of how to solve specified problems.
*   **Higher retention of (mathematical) material.** Cognitive psychologists have found that students retain information better when they engage in "deep level processing" rather than "shallow level processing" (see the [notes](http://www.samford.edu/uploadedFiles/How_to_Study/Teaching_Resources.pdf) on Video 2 of [Stephen Chew's](http://howard.samford.edu/psychology/bio.aspx?id=2147485427) "How to Get the Most Out of Studying" video series). Developing deep understanding of math reduces need to review mathematical material when one needs to know it for future units and courses (whether within math or adjacent to math). This cuts down on the amount of study time necessary to master later material. 
*   **Developing better general reasoning skills (across domains).** Learning math deeply is closely connected with developing mathematical reasoning skills. Distilling general principles from special cases involves abstract reasoning. In the other direction, when you understand general principles, it makes mathematical reasoning feel a lot less cumbersome, which incentivizes one to do more of it (relative to the counterfactual). Mathematical reasoning ability may be transferable to reasoning ability in other contexts, so that learning math deeply builds general reasoning skills.

Some arguments against learning math deeply being useful are:

*   **It may be too hard.** Sometimes when I suggest that learning math deeply is helpful, people respond by saying that most people aren't capable of learning abstract concepts with enough ease so that it makes sense for them to try to learn math deeply rather than just memorizing how to do specific problems. This is an ill-defined claim, but it can be made precise by specifying a population and a given level of mathematical abstraction. 
*   **The span of the payoff may be too short.** For people who won't go on to take many math courses, the benefits of reduced future study time and higher retention might not be worth the upfront investment of learning math deeply.
*   **Mathematical reasoning may not be very transferable.** A counterpoint to the "developing better reasoning skills" point above: it's known that [transfer of learning](http://en.wikipedia.org/wiki/Transfer_of_learning) from one domain to another is [often very low](http://econlog.econlib.org/archives/2012/08/low_transfer_of.html). So learning mathematical reasoning skills may not be an efficient way of developing reasoning skills that can be used in the context of one's career or personal life.

I'd be grateful to anyone who's able to expand on these three considerations, or who offers additional considerations against the utility of learning math deeply. I would also be interested in any anecdotal evidence about benefits (or lack thereof) that readers have received from learning math deeply.