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> I can't remember once being shown the physical problems that drove progress in math until I reached calculus.

Progress in math often (or usually) isn’t driven by physical problems, so there’s that. Just like progress in physics often (or usually) isn’t driven by “real world” problems. Discoveries are often made before applications are found.

As a concrete example, number theory is an incredibly profound and fruitful branch of mathematics. It is also largely “useless” to this day (outside of cryptography, which uses only a small subset of number theory, and isn’t really “physical” to begin with).

Another well-known example is non-Euclidean geometry, which came into being due to the purely intellectual pursuit of the Fifth Postulate, and almost magically found its place in the theory of gravitation long afterwards.

It takes a certain kind of personality to enjoy mathematics as is, even if it seems “useless”.

(Theoretical physicist who double majored in math and physics in college here. Definitely possess the “certain kind of personality”.)

>Progress in math often (or usually) isn’t driven by physical problems, so there’s that. Just like progress in physics often (or usually) isn’t driven by “real world” problems. Discoveries are often made before applications are found.

The use of math however is driven by physical problems for most people; teaching people how to use math (or physics, or chemistry) ought to be the point of most non-grad level courses? Anecdotally, I got more out of my math classes that were through the lens of either a solver e.g. matlab/mathmatica/R or a 'practical' subject like quantum chemistry.

That doesn't work for everyone though. I'm studying pure math and I find most applied math problems very boring. I also like solving puzzles, though. If a problem is little more than knowing which formula to use (with some manipulation) and then punching the values into my calculator then it just feels like I'm doing a computer's job.

So I think if we gear our curriculum too heavily to the applied side of things, we risk alienating some very gifted students who might otherwise go on to become brilliant mathematicians.

I would say 'plug and chug' applied problems don't work for anyone, past the first couple. At risk of anecdote: I like solving puzzles as much as anyone[1] but the purer side wasn't a puzzle to me because there is no purpose, no picture to be pieced together[2].

[1] Back to the original topic, chemistry interestingly inverts its approach with organic chem - a class in puzzle solving that often frustrated premeds trying to memorize their way through.

[2] I don't mean that the problem was open-ended, e.g. research but rather that it didn't exist. Most of my math education hasn't been trying to prove P=NP (or other easier puzzles) but demonstrating specific transformations (e.g. here are the rules for integration, after a trivial treatment of the proof). imo something like an intro to algorithms in highschool or early college would have do more for both the uninterested and the gifted than "integration, but with 1 more dimension!"

That’s disappointing. My math education so far has included the usual calculus and the basics of analysis up to and including multi-variable calculus as well as linear algebra on abstract vector spaces, inner product spaces, and complex vector spaces. Additionally, I’ve delved into combinatorics with enumeration, combinatorial proofs, and generating functions as well as graph theory, abstract algebra including group theory and ring theory, and mathematical logic.

I plan to take Euclidean geometry, real and complex analysis, fields and Galois theory, and eventually measure theory as well as differential geometry and maybe commutative algebra and representation theory.

The problems I’ve studied in all of these courses have been varied, interesting, and highly challenging. Far more entertaining than any problem I encountered in high school. But not all of the material was too complex for a high school student to understand. I could definitely teach a high school class in graph theory and I bet the students would have a lot of fun with it. Despite it being a beautiful and sometimes very abstract field, it can be applied to computer science as well as games and puzzles of all sorts. Yeah, it’s not useful to any of the hard engineering disciplines but it’s far more entertaining than grinding integrals all day.

Gerry Sussman has a really good talk touching on [2]. It was surprising to me that his real interests are in classical mechanics but he uses computation as a tool to convey the ideas.

> Progress in math often (or usually) isn’t driven by physical problems

This is true at the largest scale, but the math classes that most people take are directly applicable to (and often were developed partly to solve) physical problems. I, too, found the high school presentation of math dry and boring until I took a good physics class, and I was amazed by how much of the math I learned was directly useful for that subject.

Unfortunately, the dry treatment math gets in high schools also usually fails to highlight the beauty and subtlety of pure math. I didn't realize how much I loved pure math until I took my first proof-based analysis class after junior year of college.

I think we're seriously failing both the applied and pure camps of students with the way we currently present math. The fact that it took me till nearly the end of college to really like it, despite my proclivity towards the abstract and technical, really drove this point home for me.

Practical does not mean interesting. A lot of math is based around accounting and taxes.

How big is this field and how do I split it evenly into N pieces. That’s more or less why Geometry was considered foundational for so long. Carpenters are often used as examples, but they generally avoid anything mathematically complicated instead using simple rules.

> failing both the applied and pure camps

Yes. Neither its use nor truth. Just a bunch of stuff to learn.

Much less could be covered if driven by applications/problems or from axioms, and the rest of the math ecosystem is built around it. So, hard to change.

The pure mathematician is a relatively recent beast though, and most non-graduate students of math (i.e. almost all of them) study mathematics that almost exclusively was developed to solve a problem in the physical world.

Euclid was a pure mathematician.

Euclid used his geometry to solve problems... its practical utility is what made it last for so long.

Almost nothing is known about who Euclid was. Might not have even been a single person.

You might be other things as well, but if you write a textbook on pure mathematics, you are a pure mathematician.

I studied physics in undergrad many years ago, and it's been a long time since I used higher level math on a regular basis. I just picked up Mathematics: Its Content, Methods, and Meaning, on the recommendation of someone here. It's over 1000 pages, so it's going to be a lifetime reading project for me, but it's been wonderful to start reading. The first part of the book traces the earliest origins of math, and everything was grounded in real-world physical problems.

I've been a high school math teacher for most of my life, and I have deep frustrations with how removed from meaning math is presented to most students. Just because the teacher knows and states the possible relevance doesn't mean students should be expected to take the relevance at face value.

I was mostly focused on teaching algebra 1 classes, which is why I didn't use higher math all that often. But my understanding of higher math grounded my teaching of lower level concepts all the time, and I often spoke of higher level concepts with my students to help demystify math. My 8yo son loves math for now, and the moment school makes math meaningless to him I am planning to find some way to intervene.

https://www.amazon.com/gp/product/0486409163/

In my view, math suffers from getting pulled in multiple directions all at once. It is:

* A tool for solving practical problems.

* A sorting hat for the more lucrative college majors.

* An introduction to formal thinking and abstraction.

* A score in a tournament called high school.

* A source of entertainment, mystery, and beauty.

It's hard to convey all of these things at once, and there may be social or political pressures to prioritize certain aspects over others. I was a math / physics double major, and then went to gradual school in physics. I was propelled by different aspects of math at different times, perhaps as I passed from one mentor to the next, or was swayed by my many side interests (programming, electronics, music).

I've tried explaining to my kids that the curriculum du jour, whatever it is, will give them practice with the mechanics of math, so that they're not bogged down in it when they're ready to meet math on their own terms.

Being much older now, it's very apparent to me that whatever aptitudes I had as a younger student were driven by whatever good teachers I was fortunate to have at the time. In my case I had the good fortune of wonderful math teachers. As a younger student my Physics teachers did not engage me. It is only in my 30's that I appreciate Physics (largely due to personal effort, exposure to important historical problems, and understanding that my own learning style involves needing to empathize with the motivating problem)

I played with a lot of circuits from Radio Shack as a kid. I did not understand much of the math behind it till I got to college and took the electrical engineering courses. But I agree with you, it would have much more interesting to use electronics and computers to teach some of the math subjects.

As a counter point, I am put off by practical applications to mathematics and even avoid such physics based exercises in math books. I prefer pure mathematics. I find it interesting because of its abstract beauty and that motives me to want to learn more.

Boring is subjective. Many people find ”practical applications” boring but pure math fun. They would find your proposed curriculum deeply uninteresting.

If a teacher takes the practical applications angle, you can always ignore that part and just do the math.

The reverse is more difficult. My linear algebra teacher didn't ground the work in anything and as student I had no idea that the vectors and matrices could be used for, say, computer graphics.

you can always ignore that part and just do the math

No, not really at all. Most of applied math is raw calculations. A student interested in pure math is going to be bored to tears by that. They need challenging problems and proofs to work on.

It’s like putting a gifted programmer in a data entry class and telling them to “just focus on the typing” as though people like programming for the keyboard work.

For example, in linear algebra, I found Gaussian elimination boring as well. I was far more interested in learning how to prove things about vector spaces, subspaces, the fundamental theorem of linear algebra, the spectral theorems, and so on.

I would guess students that are more interested in pure math are a minority though. I definitely don't think this group should be ignored, but perhaps the core classes could be more "practical" while still offering advanced pure math electives (potentially through a magnet type school that services multiple "regular" school districts). I believe it is already pretty common for universities to have multiple offerings of the same general math subject to this end, although this might be limited to schools with larger STEM programs.

IME K-12 math wasn't taught in a very engaging way regardless of whether one was interested in theory or applications. The people I knew that were into math all did after school and/or summer courses, although I don't know whether the interest came before or after their parents enrolled them in these supplements. Personally I did not think math was interesting until undergraduate, where I first gained an appreciation of the applied work and subsequently found myself interested in some pure math areas as well. I wish I had realized how cool math can be earlier on in my education- I do recall enjoying very early math, but I think by 5th grade I was unfortunately jaded.

I guess I ultimately wish high schools had a bit more diversity in classes. I don't know if this is feasible, but I feel by grades 11 and 12 it would be greatly beneficial if students had more choice between a number of focused electives. Even at my high school which was generally considered high quality, the upper level courses were largely the standard AP curricula, which at least in the case of biology was still very much a survey course. I don't think this helped students decide what they wanted to pursue enough, and it seems that freshman year of college for most people was largely a repeat. Why not let students get some sense of a couple subfields before they have to actually commit to something?

Online courses can maybe help but I feel they will have to be grounded somehow in the high schools to be fully effective. Anyways this got a bit off topic, but if anyone has any related information/thoughts I'd love to hear it- I am certainly interested in education but don't have much practical knowledge/experience outside of my own bubble at this time.

I told my kid that math might be boring perhaps through the end of high school because the time was taken up learning the “alphabet” of math, and see what the boring process learning the actual alphabet resulted in? I guess he believed me as he became a math major.

race car dynamics is a great math application of interest. and good for algebra through diff eq

I concur. I hated math until calc, when it explained physics. Now I love math. But most folks never get there - it’s taught too removed from both the application and the beauty. Looking at my children’s teachers, some make it come alive, some struggle with large classes, and some just don’t know the material well themselves.

My interest in math and all things STEMy stemmed from computer games. Then I realized basic solutions could be generalized, and that got me interested in more abstract aspects of mathematics.

I think a curriculum based around game programming could provide a concrete context for many STEM disciplines.

Then you get into the story, consistent gameworld, art and sound side of computer games and you can introduce humanities subjects such as the classics, philosophy, art, and music.

So, in short, we could base an entire Uni curriculum, spanning the sciences and arts, all centered around creating a computer game.

And best of all, the student's capstone will be something any person can play, instead of dragging themselves through pages and pages of hard to read writing.

What I hated was having to memorize orbitals in freshman chemistry, then after taking modern physics as a junior, the orbitals were deduced easily.

Anyway, I hated chemistry, as it seemed all rote memorization. Physics was so much better. It’s also more applicable to RF.

Totally agree. The idea that chemistry is separate from physics is pretty dumb anyway. I'm kinda talking out of my blowhole here, but it seems the reason chemistry and biology are distinguished from physics is that there are a lot of emergent phenomenon that we didn't and in many cases still can't predict from the fundamental physics. But, it's still there. Maybe chemistry should be called molecular dynamics or something like that. Jazz it up a bit.

Also, more explosions and experiments with fart gas.

This old adage is called ‘reductionism.’

TIL! Thanks

And the obligatory XKCD: https://xkcd.com/435/

Nice!

But some sciences really do require a lot of memorization, and even if we could understand the entire world from first principles, biology would still remain a separate science from physics, especially in the area of anatomy and ecology. Perhaps one day we will be able to account for each biological process starting from physics, but that still won't help us find new species of orchids and which bees pollinate them.

Even for chemistry, sure, you can use quantum mechanics to predict what combining hydrochloric acid with caustic soda produce, but that is far, far more difficult than remembering some valence for the few common elements.

> The idea that chemistry is separate from physics is pretty dumb anyway.

Applying reduction to say "everyone young person should know everything" is hubris.

It's a practical division, at multiple levels. eg Specialization, profession, etc. Renaming (Molecular Dynamics) is just adding to the soup. Yes there is some overlap, but it's not worth trying to jam into the minds of students who dont have the drive or aptitude. re: Trying to teach everyone software development or CS

An example of reductionism in CS: one of the great minds of the field[0] adamantly believed that compiled programming languages were literally a waste of time, since the machine code they spit out was not as optimal as handwritten assembly and the compiler used extra computer time to build it. Can you imagine the state of CS today if that view had prevailed?

Chemistry is a useful abstraction over physics in the same way. Hell, so is Newtonian mechanics over quantum mechanics, for that matter. The difference between these and my CS example is of course that the amount of stuff in physics itself is much more vast and undiscovered. The same is certainly true for biology, psychology, and sociology; and the exploration of their respective spaces are only hampered by focusing too much on the fields they rest upon.

[0] I wish I could find the reference. I think it was Von Neumann.

One of the challenges with curricula generally is that there are a lot of dependencies and you sort of have to use a certain amount of rote memorization to jumpstart things. Organic chemistry in particular, which I did not get along well with, I recall as requiring a lot of memorization because at that level you just don't have the tools to figure out how reactions will progress from first principles.

Even physics, if you haven't had calculus yet, you end up having to memorize a lot of formulas because you don't know how to derive them.

In high school chemistry, we had to memorize the periodic table in the sense of symbol <=> name. I hated memorizing stuff, but I found a way to make it enjoyable - I wrote an adaptive flashcard program to use.

My highschool chemistry class required us to memorize that stuff, but they also let us use graphing calculators on every test... Memorizing it still pays off because it saves time, but I think a load of students didn't bother and just 'cheated.'

(I'm not really convinced this is truly a form of cheating since if you look up the answer frequently enough you'll end up memorizing it by accident anyway. You can only look up Pb so many times before you start to remember it means lead.)