Coming up with math explanations for students that don’t always “get it” can be tough. You have to be thinking on your toes, and you have to make your explanations understandable. It’s not always easy to succeed.
Two Recent Examples:
How would you explain to a student in a non-accelerated Algebra II (or even non-AP Calculus class) why it is that when you solve in the standard algebraic way, you get and ? Why is it that you generate the extraneous solution, that you then have to eliminate, because it can’t “plug into” the original equation? Where does that extra solution come from?
How would you explain to a student — without using the formal definition of a limit — what a limit is? So that they understand it intuitively. Now ask yourself: would your explanation work for the constant equation ? What would you say to a student who says “Why is the limit of f(x) as x approaches 0 equal to 6? The function has already reached 6. It isn’t approaching 6.”
And for those of you who want to prove your mathematical mettle, here’s a question that recently circulated through our math department. What’s the answer, and how would you explain it to a stuck student?:
Suppose that a function is differentiable at and . Find and .