The Calculus is finally having its moment in the sun, or more accurately, its moment in the spotlight.
I’m not sure why the calculus issue chose this moment to bubble to the surface, but consistent with the zeitgeist of public education at the quarter-pole of the 21st century, the framework for the discussion is equity and the perceived, actual, and/or changing role of calculus as a gatekeeper to college admission, at least admission to highly-selective colleges.
Whatever lies behind the current scrutiny, the question that I’ve been asking myself for well over a decade now is why calculus in the first place.
How do we explain a 2500% increase in enrollment in calculus in high school since I last cracked open a calculus book back in the early 1980s.
Could it be the result of masterful marketing of AP Calculus by the College Board? Given their recent track record with the SAT that hardly seems likely.
The surge in enrollment as a byproduct of Stand and Deliver seems a more plausible explanation. Sure, the dark-suited psychometricians came off as the bad guys, but overall the film was a pretty strong advertisement for AP Calc.
As enrollment in AP Calculus soared, as described in the article, its presence on a high school student’s transcript became a signal of rigor, resilience, academic seriousness, and all of those other things that college admissions officers look for in potential students. So, even high school students interested in majoring in the arts, humanities, or even, good gracious, English took AP Calc because that’s what you do to get accepted at a good college. That’s just the kind of warped thinking that makes perfect sense to 18-year-old-seniors and their college counselors. And before you know it, the rigorous academic record phenomenon becomes a self-fulfilling and self-perpetuating philosophy.
But what if it’s the students who made the course and not the course that made the students? You know, as a people we’ve never been really good at the whole cause and effect thing. Correlation does not imply causation. If top high school students self-select into AP Calc courses in high numbers then AP Calc becomes a pretty good predictor of students’ success in college – whatever their major.
However, if we put out the word that admissions officers at highly-selective colleges looked favorably on students able to make a tasty croissant, you can bet dollars to donuts that by 2026, the College Board would be offering AP Baking and enrollment in it would rise like a well-proved dough.
But even if that’s true, Charlie, you cannot question the rigor of Calculus.
Hold my spatula.
Calculus I: Smoke, Mirrors, and Rigor
It’s been a pretty poorly kept secret for a very long time that there is not a whole lot of rigor in a first-year course, introductory calculus course, even, or especially, AP Calc.
First-year calculus when I took it a couple of times back in the late 1970s and early 1980s was all about the derivative and learning new rules and procedures. It was a starting point to a new kind of mathematics and mathematical thinking, a cognitive complexity reset, if you will.
When I took Calculus as a senior in high school, it was a welcome respite after three years of mathematics courses that had increased in cognitive complexity and demand each year. Silver-colored books still trigger flashbacks to Herberg and Bristol’s Elementary Mathematical Analyses, which I encountered in the eleventh grade. In sharp contrast is the calming tan of Loomis’ Calculus, 2nd edition.
The same was true when I took the course again as a freshman in college. (Take the course again they told us because they teach things differently in college. They didn’t tell us that meant that in college we would be taught by a grad student with little to no teaching experience or interest in teaching rather than by the highly-trained veteran mathematics specialist we had in high school.)
I’ll admit that I did hit the mathematics wall hard when I tried to go beyond first-year calculus, earning the same middling grades that I received as a music major in my theory and composition courses, but fortunately I had the social sciences to fall back on.
Now, the 1980s were a long time ago and there was a major reform of the calculus curriculum in the 1990s, but to a large extent that effort can be summed up by the following quote:
In 1997, The Chronicle of Higher Education published its post-mortem of the Calculus Reform movement. The article concluded with a discouraging comment from Ed Dubinsky, one of the fathers of this effort, “Except for a small number of isolated pockets, it will be hard to tell that there was a calculus reform. [In a few years] we’ll become upset that very few people are really learning calculus and we’ll have another round of reforms. I hope that round survives.”
As evidence of the prescience of Ed Dubinsky, the word that my daughter and her cousins used when I asked them to describe AP Calc courses they took between 2008 and 2012 was procedural.
Of course, the teaching of calculus, like all subjects, is constantly changing, and the next round of reform might be different.
That still leaves us with the question of who, if anyone, should be taking calculus in high school or who needs to be taking calculus in high school.
Calculus For Whom
I remain fully committed to Algebra 1 for all. I was willing to carefully consider for a time the well-crafted arguments of those who promoted Algebra 2 for all in the early part of the 21st century, and I thoroughly enjoyed the heated debate between a Stanford mathematics professor and representatives from community colleges in New Jersey over what exactly Algebra 2 comprised. But calculus is where I draw the line.
Reading the arguments of proponents of calculus as the peak of high school mathematics one would think that there is no limit to the function of calculus in high school and to the number and type of students whom will benefit from a foundation in calculus. The list begins with future physicists and, of course, engineers. Economists are usually the next group mentioned before we move on to meteorologists, finance specialists, data scientists, and pretty much anyone else who deals with mathematical models from animators to actuaries. And for good measure throw in software developers and anyone involved with computer science. Strangely mathematicians are rarely mentioned, perhaps because we tend to focus on applied fields.
After a bit of close reading, however, the benefits of calculus to most of those professions and professionals start to sound fairly vague, general, and very much indirect like the benefits I was supposed to reap from five years of studying Latin.
I’ve been involved to some degree in educational measurement, psychometrics, and assessment for more than three decades and have never once come close to using calculus. The same is true of highly successful professionals, young and old, whom I know in several of the field listed above. There are others in these fields using calculus and other types of statistics and mathematics I’m not well-versed in to develop new models, but they are a different breed.
My bottom line is that Calculus 1 should never be someone’s terminal course in mathematics. It is a starting point, not an end point. For those preparing for a field that requires calculus, perhaps already participating a dual-enrollment program, have at it. For the rest of us, we need something different.
What then do I suggest as an alternative for a fourth-year mathematics course for the rest of us?
The popular and logical choice is a course that introduces students to key aspects of data analysis, statistics, and probability, the redheaded stepchild of most state’s mathematics standards and mathematics curricula. There is certainly a pressing need there, and I would support such a course offering as a replacement to AP Calculus.
However, as the last chance to answer questions like ‘Why am I taking this?’, and ‘When am I ever going to use this?’, I’m envisioning a different type of course for high achieving students in high school. If this is the last serious mathematics (and possibly science course) that a student will take, then I’m looking for a course centered around discussing readings such as
Infinite Powers: How Calculus Reveals the Secrets of the Universe, by Steven Strogatz
Gödel, Escher Bach and Eternal Golden Braid, by Douglas Hofstadter
A Brief History of Time by Stephen Hawking and
A Whole New Mind: Why Right-Brainers Will Rule The Future, by Daniel Pink
And for good measure, throw in Leonard Bernstein’s The Unanswered Question and Richard Feynman’s The Meaning of It All. (Feel free to diversity my list of authors but don’t deviate from the main idea.)
And if that course doesn’t work for you, my other alternative, to paraphrase Marie Antoinette, is Let Them Bake Cake.
Really good cakes. From scratch.
There’s enough mathematics, science, psychology, and sociology (not to mention cultural relevance) in baking a cake to make everyone happy.
Embrace the Absurd 