How valuable is it to learn math deeply?

JonahS

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

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:

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?

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. 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 on Video 2 of Stephen Chew's "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 from one domain to another is often very low. 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.

My experience learning math deeply

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.

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?

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 on Video 2 of Stephen Chew's "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 from one domain to another is often very low. 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 used a similar approach to learning chemistry at university level (undergraduate to PhD level, although my PhD drifted a bit from pure chemistry into computing and education). There were lots of situations where, to solve a problem, you needed the appropriate applied formula. Many (most?) students tried to memorise these formulae and the situations they applied in. I struggled to memorise them, so instead focused on how to derive the applied formula from a much smaller set of basic equations. Often there's a mental trick that makes it easier - e.g. to derive the equation for surface tension of a liquid, you think about what happens if you split a cylinder of liquid in half. I found it a lot easier to remember that sort of thing than an equation. (I've not used that particular equation since an exam in 1991 or 1992, but I can still vividly remember the mental model to derive it.)

I can't say that my approach was a better one in terms of getting good marks. When you have to answer specific questions - which you know will be drawn only from the set of situations you were taught - it's much easier and quicker to produce the appropriate equation from your memory and apply it. With my approach, you have to spend valuable exam time deriving the equation before you can apply it. (You rarely got marks for deriving the correct equation.)

However, it has been enormously useful for dealing with real world problems beyond the constrained world of exams. My basic approach - work out what I want to know, think of all the equations that might help, then do a bit of thinking and dimensional analysis to see how to get an equation to relate them - is applicable even when you've never seen the precise equation you need. Or, indeed, when nobody has ever seen it, which is often the situation when you're doing genesis research. It's also extremely helpful for understanding when your equation might not be appropriate, since you only get the equation when you've thought a fair bit about the situation you're trying to apply it to.

Also, in the real world these days, it's trivial to look up an equation. If you're doing any serious work that might require formulae, you'll almost always be sat at a computer with an internet connection. It might be marginally quicker to produce the equation from memory, but the generic skill of being able to look up the right equation is much more widely applicable. It's also, I suspect, less prone to trivial errors (e.g. misremembering a sign) for most people.

I've switched to this approach with maths over time. These days I wouldn't dream of doing any serious algebra or calculus by hand - much easier, quicker and less error-prone to stick it in to Mathematica, Wolfram Alpha or whatever. Those systems know more tricky integrals than (almost?) any human. But I don't do any research in maths, I just use maths in my research.

This ready access to information and symbolic mathematics certainly wasn't the case when the people who taught me learned their basics - it was only becoming the case for some people when I learned them (early 90s). Luckily I was one of the early ones. Sadly, it appears to be taking a long time for the world of formal education to catch on to this shift in how knowledge work happens.

My intuition is that my focus on turning things in to transferable knowledge makes it more transferable. My subjective experience is certainly that this generic approach is valuable across domains, and it feels that that's part of what's enabled me to work in many different domains. But I'm not sure I have convincing evidence that it's the case.

It definitely feels more personally satisfying to me, when I feel that I understand what I'm doing, rather than following a recipe, and have the skills to get to work on a totally-novel situation. But that's a personal preference that is not universally shared. Though my guess is that it'll be more widely shared among LWers than among the general population.

If you're in a position to choose between these approaches (learn deeply and how to apply vs memorising large amounts), I'd strongly recommend learning deeply and how to apply it. It may well not get you the best exam marks, but it'll set you up better for dealing with new situations, and that's a more valuable long-term skill.

33

How valuable is it to learn math deeply?

33

My experience learning math deeply

How generalizable is this?

33

33

How valuable is it to learn math deeply?

33

My experience learning math deeply

How generalizable is this?

33