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.

Reason to learn math deeply: by forcing you to master alternating quantifiers, it expands your ability to understand and handle complex arguments.

This falls, possibly, under your "developing better general reasoning skills", but I would stress it separately, because I think it's an especially transferrable skill that you get from learning rigorous math. Humans find chains of alternating quantifiers (statements like "for every x, there exists y, such that for every z...") very difficult to process. Even at length 2, people without training often confuse the meanings of forall-exist and exist-forall. To get anywhere in rigorous math, a student needs to confidently handle chains of length 4-5 without confusion or undue mental strain. This is drilled into the student during the first 1-2 years of undergraduate rigorous math, starting most notably with the epsilon-delta formalism in analysis. The reason this formalism is notoriously difficult for many students to master is precisely that it trains and drills larger chains of quantifiers than the students have hithertoo been exposed to.

(other math-y subjects have their own analogues; for example, I think the chief rea... (read more)

Random thoughts:

The decision that smart high school students should take calculus rather than statistics (in the U.S.) strikes me as pretty seriously misguided. Statistics has broader uses.

I got through four semesters of engineering calculus; that was the clear limit of my abilities without engaging in the troublesome activity of "trying." I use virtually no calculus now, and would be fine if I forgot it all (and I'm nearly there). I think it gave me no or almost no advantages. One readthrough of Scarne on Gambling (as a 12-year-old) gave me more benefit than the entirety of my calculus education.

I ended up as the mathiest guy around in a non-math job. But it's really my facility with numbers that makes it; my wife (who has a master's degree in math) says what I am doing is arithmetic and not math, but very fast and accurate arithmetic skills strike me as very handy. (As a prosecutor, my facility with numbers comes as a surprise to expert witnesses. Sometimes, they are sad afterward.)

Anecdotally, math education may make people crazy or attract crazy people disproportionately. I think that pursuit of any topic aligns your brain to think in a way conducive to that

Some lessons that I've learned from attempting to solve hard and tricky math problems, which I've found can be applied to problem-solving in general: (a) Focus hard and listen to confusions; (b) Your tendency to give up occurs much before the point at which you should give up; (c) Don't get stuck on one approach, keep trying many different approaches and ideas; (d) Find simpler versions of your problem; (e) Don't beat yourself up over stupid mistakes; (f) Don't be embarrassed to get help.

But of course I don't mean to say that learning math is the only way or the best way to learn these techniques.

I agree that math can teach all these lessons. It's best if math is taught in a way that encourages effort and persistence.

One problem with putting

toomuch time into learning math deeply is that math is much more precise than most things in life. When you're good at math, with work you can usually become completely clear about what a question is asking and when you've got the right answer. In the rest of life this isn't true.So, I've found that many mathematicians avoid thinking hard about ordinary life: the questions are imprecise and the answers may not be right. To them, mathematics serves as a

refugefrom real life.I became very aware of this when I tried getting mathematicians interested in the Azimuth Project. They are often sympathetic but feel unable to handle the problems involved.

So, I'd say math should be done in conjunction with other 'vaguer' activities.

In my anecdotal experience, math is the most transferable of all skills I've learnt.

I'd start with an anecdote from the local practice here, with regards to learning math shallowly vs with an understanding from the grounds up:

It is fairly common to derive supposed ultra low prevalences of geniuses in populations with lower mean IQs.

For example, an IQ of 160 or more is 5 SDs from 85 , but 4SDs from the 100 , so the rarity is 1/3,483,046 vs 1/31,560 , for a huge ratio of 110 times the prevalence of genius in the population with the mean IQ of 100.

This is not how it works; the higher means are a result of decreased prevalence of negative co... (read more)

I was very successful in my early mathematical education. I'd get As with ease, take exams early, enter mathematics competitions, etc. I had a deep understanding despite doing very little work because all the concepts seemed obvious.

I continued in the exact same way and my performance declined to the point where I was struggling to get Cs. I was now meeting concepts that were not intuitively obvious (eg. limits, proofs, complex numbers), and because of my previous success I had not developed any techniques to gain deep understanding of them. I lost all sen... (read more)

(This discussion doesn't distinguish what could be called the rigorous and post-rigorous levels of skill, and so feels a little off (at least terminologically). At the rigorous level, which seems like what you are talking about, you know how the tools work, and can reassemble them to attack novel problems. At post-rigorous level, which seems like a better referent for "learning math deeply", you've sufficiently exercised intuitive mental models to offload most routine observations to System 1, freeing up conscious attention and allowing more ambitious intuitive inferences. Fluency as opposed to competence.)

It's interesting that the comments on this post are split in terms of whether they interpret the focus to be on

mathor ondeeply. It's also worth noting that the term "deeply" has many different connotations. Stephen Chew, whom you link to, is using deep learning in the sense of learning something by pondering its meaning and associations. But it's very much possible for an unsophisticated to learn something deeply in the Chewish sense without acquiring a conceptual understanding of it that has transferable value. For instance, one might "d... (read more)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 de... (read more)

Up until University where I am now I have never actually had to think hard about math problems I was presented with.

Last summer I had an epiphany in abstract algebra, and it has been hugely beneficial to see these structures everywhere in computer science. That is a lot of handy theorems you get for free.

I think that high-level pattern maching strategies are very valuable. Category theory, abstract algebra, etc.

I don't remember a period of my life where I

didn'tfeel like I had a deep understanding of math, and so it's hard for me to separate out mathematical ability and cognitive ability.I've also seen advice from a handful of places I respect to learn as much math as you can stand, because there often

istransfer from mathematical topics to practical applications. This is much more true for engineers, physicists, and software developers than it is for people in other professions, but still suggests that the first negative consideration you raise is strong (unle... (read more)Deduction and analogy seem like largely different reasoning processes. I suspect that what you're describing is that by learning the notation and doing enough deductive arguments, the tasks begin to become intuitive, that is, they begin to become analogical and not deductive.

Deductive thinking is conscious, deliberative, and "slow." Analogical and intuitive thinking is unconscious, nondeliberative, and "fast." So you're probably right that by learning to relegate many mathematical tasks to analogical thinking, one increases their ef... (read more)

Perhaps this comes under "Reduced need for memorization" but when someone says "deeply" I assume they mean understanding the underlying principles - specifically understanding the limitations of the tools being used:

An extremely trivial example might be how often people in businesses communicate using measures of central tendency (mean) but almost never talk about spread (standard deviation). Yet the SD is as important as the mean.

Perhaps less trivial might be that the analysis of small samples (N < 50) often use T-Statistics. This... (read more)

Learning math deeply is better than not taking math courses at all, which is better than memorizing some formulas for the exam and forgetting them afterwards.

The analogous statements are true of every other field.