So, after a wild foray into differentiation and applications of differentiation, we just started to learn how to deal with exponential and logarithmic functions, before starting our equally wild foray into integration. I gave a pre-test to see what they remembered about exponential and logarithmic functions from the past.

It was less than encouraging when I think about the quality of math education we give to high school students.

I gave them a little more than 10 minutes to fill in what they knew on a two page pretest. Most were just about as blank as the one below:

Most students could not sketch anything that even looked like an exponential function (these are four different student’s answers):

Which might have also been because not even had any idea of what e was:

Okay, so I have fully embraced the whole idea of “teach the where they are.” I enjoy teaching this class because I have the liberty to slow down and fill in gaps in their math backgrounds and then can teach the Calculus material with the depth in conceptual understanding but without the depth in mechanical skills (if that makes any sense). Sure it’s harder to explain the idea of a limit as x goes to negative infinity for e^x when they don’t fully even understand what a negative exponent is, but I wouldn’t be in teaching if it was an easy job. So, I’m not complaining that I have to change my sacrosanct yearly plan and “lose time” – I’m just wondering if this is all they can remember about a major family of functions that I’m sure they learned about in both Algebra II and Precalculus, what did we accomplish in teaching them this in the first place? What do we expect our students to know coming out of high school? How big is that gap between what we teach and what they actually learn? 

I’m not blaming their past teachers, and I’m especially not blaming my students (though my non-AP Calculus students are labeled “weak,” many of them are really quite talented but have yet to find that spark of loving mathematics or need a bit more time). Worse, I have no ideas for solutions, just those questions. I guess I can find comfort in knowing that other people are asking the same thing, and even better, that those are the questions that are driving (some of the) reform in mathematics education right now.