*Bits, ink, particles, and words.*

I pride myself on being honest with the students I tutor. I don’t like the idea of telling a student that they’ve made a “good” attempt when really there answer is incorrect. My goal with them is to help them learn whatever material they are struggling with, not to give them what some people call “compliment sandwiches”. These are the result of giving someone a piece of criticism in between two compliments in order to make the criticism easier to digest. Perhaps that works for some people, but for myself, I find it more efficient to simply tell the student what’s right and what isn’t.

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!”