*Bits, ink, particles, and words.*

One of the things I like most about mathematics is its ability to generalize results to realms that one might not have previously thought of before. Historically, this is what happened with rational numbers, negative numbers, irrational numbers, complex numbers, and so on.

I’ve been thinking recently about what it takes to make a concept “stick” in a student’s mind. When first looking at a topic, it’s tempting to show a student easy examples that get them familiar with the mechanics, before moving on to more difficult problems. However, when this new concept is a new way to see an old idea, it can be difficult to sell the concept to the students if the old idea seems to be just as effective as the new one. After all, why should the student have to learn a new method if the old one still works?

If you’re in secondary school, chances are you’ve had to discuss a bunch of properties of functions. These include finding the maxima and minima, the roots (zeros), the intervals where the function is positive or negative, the intervals where the function is increasing or decreasing, and the domain of the function. While having to slog through page after page of this, you might be tempted to ask, “Why do I have to learn about this? Not only is it boring, it’s completely useless!”

When you first learned about algebra, chances are you learned about something called a function, typically one that looks like this:
\(\begin{equation}
f(x) = ax +b.
\end{equation}\)
This is nothing more than the equation of a straight line. You probably also learned how this could be represented as a graph (which is why you know it’s a straight line). This was simple enough, and you soon learned how to deal with different kinds of functions. These include quadratics (parabolas), exponentials, rationals, and a host of other functions. You learned what these looked like when graphed, and how to find various properties of these functions. This includes finding the roots of the equation (when $f(x)=0$), finding the domain and range, and characteristics of when the graph is increasing or decreasing.