What convinced me was the book Goedel, Escher, Bach, by Douglas Hofstadter. It was my first exposure to the more abstract and playful side of math, and motivated me to become a math major in college.
I was "good at math" in high school, to the extent that I could crunch through any problem in the math or physics textbooks. School math existed mainly to serve the science and engineering students. Now I was happy to use math, for instance in my electronics and programming hobbies, but that wasn't math as an end unto itself.
But proofs are what really made math come alive for me. I don't think this makes me a freak. A lot of people I know from my generation said that their favorite high school math course was geometry, which was heavily proof oriented at the time.
Sadly, the contemporary K-12 math curriculum is sorely lacking in proofs.
Now, what about the students who would be more motivated by usefulness than by abstract theory? For those students, I suggest looking at how people in the so called "real world" actually do math, and work backwards from there. My days of abstract math are behind me, but I am still one of the "math people" at my workplace, in an industrial R&D department. When I do math, I'm never far from a computer. Yet school math is still taught by mainly pencil-and-paper methods, even if those have been translated to online forms.
For many of my math tasks, I start by playing with numbers, e.g., computing a function and graphing it in a Jupyter notebook. It brings tears to my eyes that K-12 students are not exposed to this. For one thing, it's fun, and it's an honest portrayal of how people actually work. For another, you can brute force your way through a problem even if you've forgotten some particular formula, making it more likely for someone to use it later on. Graphing calculators are of course a thing, but they hardly go far enough.
Interestingly (I'm one of ther guys self-teaching as an adult), this is exactly how I'm learning. I'm doing a course that uses Jupyter to cover all of the major areas up to linear algebra. I also aced my Geometry class in high school exactly because of what you said: it gave me some of the flavor of proofs and "real math," which was fun and made me motivated. I remember at the time being sorely disappointed when no other math classes were like that again and figured it was some kind of lucky fluke.
I think you're onto something, because teaching myself programming was precisely my backdoor into math. I remember a line in the Clause Shannon biography "A Mind at Play" that said he was essentially the same way: he wanted to apply his mathematics and understand the proofs/structure behind it, but essentially (as the title suggests) it was a form of play.
I was "good at math" in high school, to the extent that I could crunch through any problem in the math or physics textbooks. School math existed mainly to serve the science and engineering students. Now I was happy to use math, for instance in my electronics and programming hobbies, but that wasn't math as an end unto itself.
But proofs are what really made math come alive for me. I don't think this makes me a freak. A lot of people I know from my generation said that their favorite high school math course was geometry, which was heavily proof oriented at the time.
Sadly, the contemporary K-12 math curriculum is sorely lacking in proofs.
Now, what about the students who would be more motivated by usefulness than by abstract theory? For those students, I suggest looking at how people in the so called "real world" actually do math, and work backwards from there. My days of abstract math are behind me, but I am still one of the "math people" at my workplace, in an industrial R&D department. When I do math, I'm never far from a computer. Yet school math is still taught by mainly pencil-and-paper methods, even if those have been translated to online forms.
For many of my math tasks, I start by playing with numbers, e.g., computing a function and graphing it in a Jupyter notebook. It brings tears to my eyes that K-12 students are not exposed to this. For one thing, it's fun, and it's an honest portrayal of how people actually work. For another, you can brute force your way through a problem even if you've forgotten some particular formula, making it more likely for someone to use it later on. Graphing calculators are of course a thing, but they hardly go far enough.