The issue is the scope of the course. I know plenty of professional mathematicians. None of them can calculate a tip. The way to excite their mathematical minds is to drag out the Cosmic Wimpout dice, play it with them until they understand the rules, and then ask them, "so, it's obvious that strategy doesn't matter much, but what's the optimal strategy?" Bam, they're writing Markov diagrams and fussing over the fact that one of the dice has a wildcard which you can technically choose to be anything in certain rolls, so which one should it be, and so forth.
The most accessible and iconic piece of plain mathematics that I know is the Fundamental Theorem of Calculus. In Math class in high school, you may have been taught this by saying "it's very complicated and rests on some very difficult theorems and trust us, the mathematicians have proved it. But no. That's not mathematics.
I occasionally get the opportunity to talk to high school students and college freshmen about calculus, not in any formal setting, but just one-to-one. I tell them that the fundamental theorem of calculus is about + and −. The − is confusingly called a "derivative" and the + is confusingly called an "integral", but the fundamental theorem just says: + undoes −, and − undoes +, and that's it.
They look at me funny. Suppose we start defining sequences (F₀ F₁ F₂ ...), and suppose we start defining ΔF as a new sequence, Fᵢ − Fᵢ₋₁ . And let ΣF be a new sequence F₀ + F₁ + ... + Fᵢ .
Then it's not too hard to see that Δ(ΣF) = F, in other words, − undoes +. And it's not too hard to see also that Σ(ΔF) = F, because if you look at it:
[Σ(ΔF)]ᵢ = Fᵢ − Fᵢ₋₁ + Fᵢ₋₁ − Fᵢ₋₂ + ... + F₁ − F₀ + F₀.
And those cancel out by pairs into just Fᵢ.
Σ of course forms the basis for a Riemann sum and Δ forms the basis for a difference quotient, and we take limits because the product rule says:
δ(fg) = (f + δf)(g + δg) − fg = g δf + f δg + δf δg
...and we want to say δf δg → 0 to keep the algebra pretty, because if δf and δg are one in a million, g δf and f δg are a million times larger than δf δg, and so on until δf δg is negligible by comparison.
I kind of lost my point, so let me return to it and say that being able to see ∫dx as + and d/dx as −, being able to feel these patterns, that is actual mathematics. Being able to write out the matrix transform of complex multiplication and just seeing, "oh wow, that's a scaled rotation matrix!" and the implication -- "wow, complex-differentiable functions must preserve angles!" -- that sort of thing is mathematics.
I think it would be nice if we taught this sort of thing in a mathematics class, but in the meantime, I'd prefer if we just insisted that "math class" doesn't teach mathematics until you get toward college.