People might disagree strongly, but my recommendation for you is to look into lambda calculus for this.

There are two important concepts that can be generalized: - Substitution, the method of renaming and replacing variables - Abstraction, replacing a variable with another term (function) that can also serve as output

This should give you a more natural feel to concepts like parametrization or what it means that the solution to one function may be another function. Perhaps you might even look a bit into the simply typed lambda calculus which has given me yet more perspectives on the fundamentals of mathematics.