If you teach at an institution that has a competitive sports team, you’ll probably notice that the coaches spend time on recruiting. It is easy to see why: though athletes train hard to improve their performances, their inherent athletic ability provides an upper bound of how well they will do.

I played sports in high school, but wasn’t within an AU of being able to compete at the college level, any division. I remember summer wrestling; those recruited to wrestle for our team basically had their way with me on the mat.

In my current sports, I always do poorly in competition. For example, in my best running marathon, the winner beat me by 74 minutes! (winning time was 2:19).

It wasn’t that I didn’t try or didn’t train: it was that because I am a poor natural athlete, training only “moves the needle” just a bit, and not nearly enough for me to be competitive.

A coach could give me this workout or that workout…and get angry with me. But I have athletic limitations.

The same principle applies in mathematics.

Right now I am teaching the second semester of calculus for non-technical majors.

One question was: find the maximum and minimum of . They were told that this function modeled daily temperature where was in days, on January 1.

Now I asked the class some questions. And, well, let’s just say that they didn’t just recognize what the various terms and factors meant.

Now we took the derivative to find the local maximum and local minimum values and most of them got . Now we set this equal to zero and all of them that we got zero when the argument was 0 or an integral multiple of .

But now, when we had I said “of course, this gives us the solution . And you guessed it…one of the students asked “why”. It took about a minute of explanation for her to see it. I kid you not.

So, I reminded myself of what it must have been like for my sports coaches in high school…..what it was like for them to work with me.