High school math does not look much like research math.
High school math is highly concerned with teaching 'algorithms' that compute answers, like long multiplication, long division, the quadratic formula, completing the square, synetic division, u-sub integration, ...
Most mathematicians don't work with calculating things. Rather, they're more interested in _generally_ characterizing how objects behave (and _proving_ that that characterization is actually correct)
For example, the definition of "continuous" you learn in pre-calc is probably something like: if you compute a function's limit, you get the same thing as computing the function.
The topologist's definition is that the function's inverse map preserves openness.
These are basically unrecognizable as being the same thing, but in fact they are. And the second doesn't actually reference calculation or even anything identifiable as numbers!
I believe this is a significant cause of not understanding "why" despite understanding "how". As an example, I remember in pre-calculus we spent at least an entire unit on partial-fraction-decomposition. Partial fraction decomposition is a strange algebraic manipulation that is basically only important to integrating rational functions -- something you don't do until at least a year later, and which doing by hand is basically pointless because computers are better at it than humans.
They also require very different kinds of intuition. Building intuition is hard, but important for people studying math. Working through many different examples and problems and applications can help you build intuition.
But I don't really have any advice for how to make math stick better, but just want to share that the mathematics you learn in high school is not representative of the entire field.
If you ever want to get a taste of "higher math" without having lots of prereqs, point-set topology and group theory are two (very different!) fields that are also fairly accessible.
Just for fun, an example of an application of partial fraction decomposition that doesn't involve taking integrals.
Consider the Fibonacci numbers 0, 1, 1, 2, 3, 5, 8, ... (with the convention that 0 and 1 are the 0th and 1st). Here's one way to get an explicit formula for them.
First, we package them up into what's called a "generating function": 0 x^0 + 1 x^1 + 1 x^2 + 2 x^3 + 3 x^4 + 5 x^5 + ... or, more concisely, the sum of F_n x^n over all nonnegative integers n. Call this thing F(x).
What happens if we add up F(x) and x F(x)? Well, x F(x) is just the same sum but with all the numbers shifted over by 1. It's the sum of x F_n x^n, or the sum of F_n x^(n+1), or the sum of F_(n-1) x^n, with the proviso that there isn't an x^0 term. So when we add this to the sum of F_n x^n, we get the sum of [F_(n-1) + F_n] x^n, with again the proviso that the x^0 term is just F_0 x^0 = 0.
But, aha!, F_(n-1)+F_n is just F_(n+1), at least when n >= 1. So this thing equals the sum of F_(n+1) x^n, except that once again we need to be careful about n=0; more precisely what we have is the sum of F_(n+1) x^n, minus 1. (Because F_1 is 1, but the actual x^0 term is 0.)
And the sum of F_(n+1) x^n equals the sum of F_n x^(n-1), or F(x)/x.
In other words, we have F(x) + x F(x) = F(x)/x - 1, or equivalently (1 - x - x^2) F(x) = x, or equivalently F(x) = x / (1 - x - x^2).
This is already pretty neat -- who would have thought that that big sum, involving the Fibonacci numbers which are defined in terms of a sort of recursive-adding process, would come out to something so simple?
Now comes the coup de grace, which is the bit that uses partial fractions. We can write that denominator as (1-ax) (1-bx) for some numbers a,b. We'll figure out what a,b actually are in a moment -- that's just solving a quadratic equation. For now, looking at the coefficients of x and x^2 we note that a+b=1 and ab=-1. And now we can do the partial-fractions thing. We know that we'll have x/(1-x-x^2) = p/(1-ax)+q/(1-bx) for some p,q. That means that x = p(1-bx)+q(1-ax), so p+q=0 and pb+aq=-1 or p=1/(a-b). That is, F(x) = 1/(a-b) [1/(1-ax) - 1/(1-bx)].
Why bother doing this? Because the fractions we now have are easy to expand in powers of x. We have 1/(1-ax) = 1 + ax + a^2x^2 + ... and 1/(1-bx) = 1 + bx + b^2x^2 + ... So F_n, the coefficient of x^n in F(x), is just 1/(a-b) [a^n - b^n].
Finally, we ought to work out what a,b actually are. It turns out that a = [1+sqrt(5)]/2 and b = [1-sqrt(5)]/2; that is, a is the famous "golden ratio" sometimes called phi, and b = 1-a = -1/a. So a-b is just sqrt(5).
So: the nth Fibonacci number is exactly (phi^n - (-1/phi)^n) / sqrt(5). Isn't that nice?
(Since |b|<1, the term b^n/sqrt(5) is always smaller than 1/2 in absolute value, which means that F_n is also always the closest integer to phi^n/sqrt(5).)
For the avoidance of doubt, none of this has anything at all to do with the disc embedding theorem. I just thought it was worth clarifying that partial fractions aren't just a one-trick pony used only for integrating rational functions; there are other situations in which representing something in terms of partial fractions gives useful insights into its behaviour.
Yup. I don't think the similarity is because I'm copying Knuth (I certainly wasn't deliberately doing so); this is a very well known argument and there isn't all that much scope for presenting it very differently.
(Oddly, I think the presentation in the later Concrete Mathematics is worse than in TAOCP; they're trying to make it more approachable but I think just make it a bit more roundabout.)
Oh, absolutely, I realize that. It’s just that I learned this from Knuth so I recognized the derivation, and I thought you’d like to see it, but I see you already know about it.
I fully agree that (high) school math is mostly calculus and very different from university/research math. I would recommend parent to start exploring what math actually is, which fields there are and where they come from, what some famous open problems are and what beauty means to mathematicians.
Good popular maths books are e.g. The Music of the Primes by Marcus du Sautoy [1] or Fermat's Last Theorem by Simon Singh [2].
It's also important to note that there are no shortcuts - it takes at least 1-2 years of studying maths to start to understand the why.
High school math is highly concerned with teaching 'algorithms' that compute answers, like long multiplication, long division, the quadratic formula, completing the square, synetic division, u-sub integration, ...
Most mathematicians don't work with calculating things. Rather, they're more interested in _generally_ characterizing how objects behave (and _proving_ that that characterization is actually correct)
For example, the definition of "continuous" you learn in pre-calc is probably something like: if you compute a function's limit, you get the same thing as computing the function.
The topologist's definition is that the function's inverse map preserves openness.
These are basically unrecognizable as being the same thing, but in fact they are. And the second doesn't actually reference calculation or even anything identifiable as numbers!
I believe this is a significant cause of not understanding "why" despite understanding "how". As an example, I remember in pre-calculus we spent at least an entire unit on partial-fraction-decomposition. Partial fraction decomposition is a strange algebraic manipulation that is basically only important to integrating rational functions -- something you don't do until at least a year later, and which doing by hand is basically pointless because computers are better at it than humans.
They also require very different kinds of intuition. Building intuition is hard, but important for people studying math. Working through many different examples and problems and applications can help you build intuition.
But I don't really have any advice for how to make math stick better, but just want to share that the mathematics you learn in high school is not representative of the entire field.
If you ever want to get a taste of "higher math" without having lots of prereqs, point-set topology and group theory are two (very different!) fields that are also fairly accessible.