I've been mulling on and off over a two-month old Wall Street Journal Op-Ed entitled Calculus Is So Last Century. The Op-Ed, co-authored by a data science firm CEO and a computer science professor, argues that Calculus is mainly useful for Cold War-era careers in physics that are now eclipsed by 21st Century careers in bio- and information-technology:
Calculus is the handmaiden of physics; it was invented by Newton to explain planetary and projectile motion. While its place at the core of math education may have made sense for Cold War adversaries engaged in a missile and space race, Minute-Man and Apollo no longer occupy the same prominent role in national security and continued prosperity that they once did.
The future of 21st-century America lies in fields like biotechnology and information technology, and these fields require very different math—the kinds designed to handle the vast amounts of data we generate each day. Each individual’s genome contains more than three billion base pairs and a quarter of a million genomes are sequenced every year. In Silicon Valley, computers store over 100 GBs of data—more information than contained in the ancient library at Alexandria—for every man, woman and child on the planet.It had never occurred to me that the only useful applications of calculus were in missile technology and rocket science. So last weekend I checked with Uncle M, who is both a mathematician and an academic dean at a major university--and someone who has spent many hours engaged with issues relating to required courses and prerequisites. What did he think about this proposal?
Accompanying the proliferation of new data is noise, and a major job for data analysts and scientists is to tease out true signal from coincidence and noise. Knowing when a result is due to chance versus when it is statistically significant requires a firm grasp of probability and statistics and an advanced understanding of mathematics.
We no longer think of outcomes as being triggered by a single factor but multiple ones—possibly thousands. To understand these large and complex data sets, we need an educated workforce that is also equipped with a firm understanding of multivariate mathematics and linear algebra.
Not much. Economics, engineering, and mathematical statistics, he pointed out, also require knowledge of calculus. Indeed, my oldest son, who just received a bachelor's in mechanical engineering, reports that differential equations--and not linear algebra--were crucial to his course of study.
Plus, Uncle M points out, calculus is a lot more challenging than the kind of applied linear algebra favored by the authors. They aren't explicit about this, but there's linear algebra, and then there's linear algebra. There's an abstract variety, taught by algebraists, in which one operates in n-dimensional space with complex numbers and proves theorems about domains, ranges, kernels, vectors, and eigenvalues. And there's an applied variety, often taught outside mathematics departments, in which one works with matrices populated with actual numbers and applies these to real-life situations. The two varieties are really completely different subjects. Abstract linear algebra, in my experience, is really tough: riddled with mind-bending abstractions and impossible to visualize. In comparison, applied linear algebra is a piece of cake. It's also easy compared with calculus. And the same is true of applied statistics.
They're not explicit about this, but it's applied linear algebra and applied statistics that our data science CEO and computer science professor are advocating for.
Since calculus is harder, argues Uncle M, it makes sense to introduce it gradually, starting in high school. Not doing so burns bridges--bridges to math, physics (which still exists), mathematical statistics (as opposed to applied statistics), engineering, and economics. It's far less clear that continuing to mostly not teach linear algebra in high school burns any bridges.
Finally, Uncle M observes, there is currently a glut of unemployed biologists. Maybe there are tons of jobs in biotechnology, but not enough for the biology majors. Career-wise, engineering (supported by Diffy Qs) continues to be the more promising field.
Naturally, few pieces about math education can resist claiming that part of the problem is rote learning:
Computers and computation are ubiquitous and everyone—not just software engineers—needs to learn how to think algorithmically. Yet the typical calculus curriculum’s emphasis on differentiation and integration rules leaves U.S. students ill-equipped at posing the questions that lead to innovations in computation. Instead, it leaves them well-equipped at performing rote computations that can be easily done by a computer.Perhaps the authors mostly experienced calculus as a bunch of rules about differentiation and integration, but good textbooks and good teachers show otherwise. Indeed, even the authors say:
Calculus, like any rigorous technical discipline, is great mental training. We would love for everyone to take it.It's just that:
the singular drive toward calculus in high school and college displaces other topics more important for today’s economy and society. Statistics, linear algebra and algorithmic thinking are not just useful for data scientists in Silicon Valley or researchers for the Human Genome Project. They are becoming vital to the way we think about manufacturing, finance, public health, politics and even journalism.Even journalism? That seems like a bit of a stretch: