The problem isn't that grade school teachers are sitting on a secret stash of cool theorems, and instead diabolically choose to have students solve quadratic equations year after year. It's that no one in the grade school pipeline has any experience with actual, modern, proof based mathematics. They've never even heard of analysis or group theory. Their knowledge does not extend beyond the latest meme textbook with MULTICOLORED (whoa!) text. There is no communication between modern mathematicians and grade school teachers in America; they may as well live on separate planets. Math classes in American grade school are little more than busy work meant to weed out people who hate busy work. Infer from that what you will.
So the way to get kids to like Math is by forcing them to learn... proofs? Not learning how it applies to the world and solves problems, but how it can be used to impress professors who care about rigor/pedantics?
Math is dirty. Math was created by businessmen who wanted to be monopolists. By gamblers who wanted to win, or at least to postpone the day when they would go broke. By scammers, and those who wanted defense against scammers. By people who wanted to blow up their enemies in war and who were moved by hatred and revenge. A recent mathematical construct born out of academia wants to destroy the US dollar and turn the world economy upside down. There are people trying to take over another planet with Mathematics. Others are trying to achieve geopolitical supremacy building artificial intelligence and robots armies.
Mathematics is blood. Mathematics is power.
There's a disconnect between academics and how real humans think if the modern solution to Math education is proofs.
I mean, that's mostly a nonsense sentiment, but I really enjoyed the allegory so I will not hold it against you. The widest ranging theorem in mathematics was created by a bunch of greek cultists who believed that the natural numbers were at the heart of existence. The most important mathematical textbook, one that was taught almost exclusively for over a millenia in the west, was steeped in the traditions of said cult. That tradition is still there and has, for the most part, always been there (although, as an extremely applied mathematician, I find it's practioners to be alien).
Plus, tell me: how do you learn proofs? They aren't statements in a textbook that you just learn and recite like poetry. You have to learn to see them, to understand how all those moving parts fit together. When done right, teaching proofs is an exercise in illumination. I find it is more enjoyable all around when performed as a creative, exploratory exercise: give the students the axioms, set them lose on a conjecture.
Of course, for that you need a) Teachers who can serve as guides in the mathematical worlds and b) Students who aren't culturally predisposed to dislike mathematics.
Proofs are only an example of what grade school teachers don't know. They are obviously missing all of the modern applications as well. Whether you want to call those applications "math" or something more specific ("engineering", "cryptography", etc) is a matter of taste I guess. The point still stands that kids are learning hardly anything of theory OR practice in their grade schools, and this is due to the teachers not knowing those things either.
We can talk about why the teachers don't know these things, but I think the real issue is American culture still sees math and science as something for pitiful, nerd-virgin dorks. You can try to sell math as this sexy, dangerous thing you use to win wars, but ultimately you will have to reckon with the image that most Americans have of mathematicians and scientists, which is illustrated in shows like the Big Bang Theory: they're ridiculous laughing stocks who use science as a kind of cope for not being socially successful.
> So the way to get kids to like Math is by forcing them to learn... proofs?
I mean, there are definitely issues with the whole "forcing them to learn" part, but assuming we're given the option to change the content of the math curriculum but not to fundamentally change the education system, then yes, absolutely. You don't need a crazy level of rigor, of course, but I think learning simple proof techniques could make students experience mathematics more like solving puzzles than just completing calculations.
Yes, yes, and absolutely yes! I'm a high school math and physics teacher (who majored in physics, not education, and is now doing self-study in more proof-based mathematics), and so much the issue is in grade school, in my opinion. The teachers there don't appreciate math. And, more than that, they don't understand math. Lots of them don't have a good well-developed number sense, so it's no wonder our kids don't either!
Then the problems compound when they just give kids calculators without the kids understanding why the calculator works, and by the time they get to high school hope is lost because they have no number sense and we're trying to explain variables and such...often divorced from real life. I've found my kids get a much more intuitive sense of slope when I explain it as the rate of change rather than "rise over run" that they're so often taught in middle school. Give them examples, like amusement park tickets (it cost $50 to get there, then $70 per person) and have them incorporate that into a slope-intercept form. It just makes things so much clearer that they often don't get because they don't have the number sense and because many teachers don't understand/appreciate math.
You aren't wrong. Until extremely recently, algebraic geometry was a field by and for the purest of the pure mathematicians. These days algebraic geometry has made its way into string theory.
1. A far reaching generalization of the Riemann-Roch theorem, which is now called the Grothendieck-Riemann-Roch theorem. As with a great deal of his work, it's about drawing conclusions about global structure from local data:
His work wasn't focused on solving particular problems so much as it was on finding the right language with which to describe problems. The philosophy is that, with the right language, your proofs should become obvious. This is somewhat in the same spirit as Leibniz's quest for the 'Universal Characteristic'.
The phenomenon you are talking about is essentially that of a homomorphism, or homomorphic structures. That is, structures that appear superficially different but share an underlying common structure.
The concept of a 'functor' was invented to describe a higher order 'homomorphism of homomorphisms'. An example most people miss is the total derivative in multivariable calculus: the chain rule implies that the total derivative is a functor that maps the composition of differentiable functions on a manifold, to matrix multiplication (of matrices acting on the tangent space).
You might also be interested in various 'dual' concepts, like that between tangent spaces and cotangent spaces in differential geometry.
For algebra, I'd recommend Pinter's Book of Abstract Algebra.
The first example there is: given a base point X and two vectors V,W based at X, the 2-form gives the "signed" area of the parallelogram spanned by V and W. Determinants (which measure n-dimensional parallelograms), when viewed as functions of their column vectors, have all the properties of differential forms.
Differential forms are a bit like generalized determinants and in a sense specify a way to measure something like an abstract volume in the neighborhood of a point of a manifold, in such a way that the Jacobian needed for changing coordinates is "built in".
The point of differential forms is that they give a way to express geometric theorems in a coordinate free way. Coordinates are seen as obscuring the pure geometric content of theorems. They are sometimes necessary artifacts of doing concrete calculations, but the idea is that geometry shouldn't depend on a choice of coordinates.
The important ideas can be found in pages 9-10 in this link:
> [...] but the idea is that geometry shouldn't depend on a choice of coordinates
Indeed, in "Tensor Geometry" (Dodson & Poston), they note:
> Most modern "differential geometry" texts use a coordinate-free notation almost throughout. This is excellent for a coherent understanding, but leaves the physics student quite unequipped for the physical literature, or for the specific physical computations in which coordinates are unavoidable. Even when the relation to classical notation is explained, as in the magnificent [Spivak], pseudo-Riemannian geometry is barely touched on. This is crippling to the physicist, for whom spacetime is the most important example, and perverse even for the geometer. Indefinite metrics arise as easily within pure
mathematics (for instance in Lie group theory) as in applications, and the mathematician should know the differences between such geometries and the positive definite type. In this book therefore we treat both cases equally, and describe both relativity theory and (in Ch. IX, §6) an important "abstract"
pseudo Riemannian space, SL(2;R).
Background in pure math here. Generally researchers learn about what problems are fashionable as they talk to others in the field. As a PhD student, your advisor should give you ideas for problems to work on. Solutions to old problems tend to open up new lines of inquiry. It's a potentially infinite process, with the only limit being the capacity of the human mind.
The odd perfect number problem has been unsolved for thousands of years:
Why would anyone want to know the answer to this problem? The truth is that the motivation for most pure math research is purely aesthetic. You could also ask arts or english departments "How can you apply your paintings or novels?!?". Some mathematicians (mostly geometers and topologists) are inspired by problems in physics, but most aren't. In truth you never know what structure or theorem might have some future application; number theory was totally "useless" until modern cryptography made (some of) it useful.
For just a tiny taste of one small area of modern math, I dare you to click on any of the links here:
We are flush with structures to investigate.
The idea that "most discoveries have been made" is nonsensical in a domain where the discoveries to be made are literally infinite. To give a hint as to the infinite nature of mathematical inquiry, you probably are familiar with the idea of a function mapping a number to another number. A good deal of modern math is involved with much higher order functions; we can have functions that map functions to numbers, functions to functions, functions to spaces, and so on and so on. And then we can consider functions between those functions (and so on). Category theory is an attempt to give a framework to some of these "meta" relations. It should be obvious there is no limit to these structures and no limit to the number of problems one could pose about them.
