Motivation for abstractness

Question

I'm seeking examples of concepts or theorems in school mathematics that are better understood when we generalize (when we deal with a more abstract concept where the former concept is a special case of the latter) such that the more general concept is also elementary (can be understood by a student).

Answer 1

One nice example is solving quadratic equations. If you know only about natural numbers or integers then it is difficult to give a concise universal solution for quadratic equations. But if you abstract your notion of "number" to allow negative, rational (possibly even imaginary) numbers, then one can give a single universal formula - the well known quadratic formula.

Before these more general number systems were constructed, there was no universal quadratic formula. Instead, the solutions bifurcated into many cases because the solution had to avoid the unknown numbers (negative, (ir)rational, imaginary etc).

Answer 2

This is a "chain" that I discovered recently.

The first element in the chain is Fermat’s little theorem which can be proven by the combinatorial necklace argument.

However I believe that this idea can be generalized and better understood after learning it through Euler's Theorem, which can be proved by the argument that multiplying two distinct relative primes of n by another relative prime of n gives two different numbers which are also relatively prime.. Thus if x is a relative prime and $y_1,y_2...y_{\phi(n)}$ are all the numbers relatively prime to $n$ then $y_1*y_2*...y_{\phi(n)}\equiv xy_1*xy_2*...*xy_{\phi(n)}=x^{\phi (n)}(y_1*y_2*...y_{\phi(n)})\bmod n$

thus $x^{\phi(n)}\equiv1 \bmod n$.

However this argument becomes even clearer after understanding Lagrange’s theorem, it tells us the order $d$ of the group generated by a number relatively prime to n (call it $a$) must be a divisor of $\phi(n)$ . Suppose $\varphi(n)=dk$, then $a^{\phi(n)}=(a^d)^k\equiv1^k\equiv1\bmod n$.

Clearly the first proof may seem as a funny coincidence and the reader might find it strange that a combinatorial argument can be used in number thoery. However after seeing Euler's theorem one can start thinking about the fact that numbers relatively prime are closed under inverses. Finally Lagrange's provides a very clear and generalized explanation.