### In Defense of (Some) High School Calculus

I'm usually one of the first to express skepticism about high school calculus. This is a result of my transition from having a default assumption that students' backgrounds are mostly like mine (which is what I thought when I started teaching) to assuming that they don't know anything unless I have evidence to the contrary.

But my own high school calculus class was a really good one. It was a BC Calc class. We used whichever edition of Thomas-Finney was current in the late 1980s. Covered everything from epsilon and delta through all the derivative stuff (including logarithmic differentiation), all the integral stuff (including trig subs and very messy partial fractions), Taylor series, and a little bit of differential equations (first order with the integrating factor, second order homogenous and non-homogeneous).

Since I grew up in New York State, we didn't get started until after Labor Day. Since it was an AP class, we had to be done by the beginning of May (when the AP exam was given). Once you take out Christmas Break, Winter Regents (a week in January where high school students in NYS don't have to go to school just in case anyone is taking the Regents Exams), Winter Break (a week off in February), Spring Break (a week off in April), and all of the assorted other civil, Jewish, and other holidays that interrupted the school year, the number of classroom minutes that I spent learning BC Calculus was within an hour of the number of classroom minutes that we devote to our Calc 1- Calc 2 sequence.

The high school calculus class I took had higher expectations than the Calc 1 - Calc 2 we teach here. I was expected to be able to do epsilon-delta limit proofs for rational functions (linear over linear). Here we "expose" students to epsilon and delta. We proved the Mean Value Theorem! (The Stewart that we use here doesn't even HAVE the proof of the MVT. I prove it anyway)

The other thing I liked about my high school calculus class was that there was no assigned homework. He taught the material. Every now and then he gave a test. It was our jobs to learn the material and be ready for the tests. He didn't care how we did it. No one told me to do the page of over 100 integrals at the end of the chapter on techniques of integration. I just knew that if I wanted to learn how to integrate well enough to get the grade I wanted and to score well on the AP exam that I was going to have to evaluate a lot of integrals.

Oh, and we loved this: After the AP exam, the calculus teacher made a deal with us. We didn't have to come to class for the rest of the year as long as we did some worksheets on hyperbolic trig functions, came in and took two exams on Taylor Series, and didn't get in trouble for not being where we were supposed to be. Since Calculus was right before lunch, I had a 90-minute lunch for the rest of the year! Since I lived a short walk from school, me and one of my friends from the class would go to my house for lunch.

Based on that calculus class, I was prepared for my next math class. The following year I took Calc 3 at Union College and was near the top of the class. I got an A without much effort. In fact, I was secretly annoyed to discover how little new content was in Calc 3. Since my pre-calc class had done all that vector crap with the equations of lines in 3-space, equation of the plane, all that, I wished that my calculus class had done the whole thing in three (and more) dimensions.

But my own high school calculus class was a really good one. It was a BC Calc class. We used whichever edition of Thomas-Finney was current in the late 1980s. Covered everything from epsilon and delta through all the derivative stuff (including logarithmic differentiation), all the integral stuff (including trig subs and very messy partial fractions), Taylor series, and a little bit of differential equations (first order with the integrating factor, second order homogenous and non-homogeneous).

Since I grew up in New York State, we didn't get started until after Labor Day. Since it was an AP class, we had to be done by the beginning of May (when the AP exam was given). Once you take out Christmas Break, Winter Regents (a week in January where high school students in NYS don't have to go to school just in case anyone is taking the Regents Exams), Winter Break (a week off in February), Spring Break (a week off in April), and all of the assorted other civil, Jewish, and other holidays that interrupted the school year, the number of classroom minutes that I spent learning BC Calculus was within an hour of the number of classroom minutes that we devote to our Calc 1- Calc 2 sequence.

The high school calculus class I took had higher expectations than the Calc 1 - Calc 2 we teach here. I was expected to be able to do epsilon-delta limit proofs for rational functions (linear over linear). Here we "expose" students to epsilon and delta. We proved the Mean Value Theorem! (The Stewart that we use here doesn't even HAVE the proof of the MVT. I prove it anyway)

The other thing I liked about my high school calculus class was that there was no assigned homework. He taught the material. Every now and then he gave a test. It was our jobs to learn the material and be ready for the tests. He didn't care how we did it. No one told me to do the page of over 100 integrals at the end of the chapter on techniques of integration. I just knew that if I wanted to learn how to integrate well enough to get the grade I wanted and to score well on the AP exam that I was going to have to evaluate a lot of integrals.

Oh, and we loved this: After the AP exam, the calculus teacher made a deal with us. We didn't have to come to class for the rest of the year as long as we did some worksheets on hyperbolic trig functions, came in and took two exams on Taylor Series, and didn't get in trouble for not being where we were supposed to be. Since Calculus was right before lunch, I had a 90-minute lunch for the rest of the year! Since I lived a short walk from school, me and one of my friends from the class would go to my house for lunch.

Based on that calculus class, I was prepared for my next math class. The following year I took Calc 3 at Union College and was near the top of the class. I got an A without much effort. In fact, I was secretly annoyed to discover how little new content was in Calc 3. Since my pre-calc class had done all that vector crap with the equations of lines in 3-space, equation of the plane, all that, I wished that my calculus class had done the whole thing in three (and more) dimensions.