Saturday, June 4, 2016

Cold War Calculus vs. 21st Century Linear Algebra

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.

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. 
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?

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:


Auntie Ann said...

These arguments seem to me to be ways to mitigate the lack of mathematics rigor in K-12 curricula and mathematics achievement by our young adults.

They are coming up with excuses and work-arounds for the fact that too many college kids can't manage the same math today that their parents were required to take 20 years ago. Instead of dealing directly with the failure of K-12, they have to find a way to say that the crappy educational outcomes are actually not a bad thing, and might be a good thing.

Niels Henrik Abel said...

Indeed. I have wondered about the apparent fascination that these people seem to have with statistics. It's often included in "quantitative literacy" type courses. Why, I don't know, because you really need a decent grasp of the calculus to understand the how and why of statistics. Otherwise, you just get a bunch of kids who learn to plug and chug with z-scores and t-tables - you know, that "rote learning" stuff they're always bellyaching about.

The applied linear algebra class (at our school, it was called "Linear Programming") that I had was dual listed with the comp sci department (although it was taught by one of the math teachers), which tells you something right there.

This fits hand-in-glove with what one of the drafters of Common Core (Jason Zimba) admitted: the Common Core sequence will not prepare students for a STEM college career, nor will they prepare students for admission into a selective college.

"I'm really awfully glad I'm a Beta, because I don't work so hard." Common Core is designed to produce not Alphas, but Betas (at best), Gammas, Deltas, and Epsilons....

Anonymous said...

"Naturally, few pieces about math education can resist claiming that part of the problem is rote learning"

What about the millions of people who watch NFL games on TV by rote? Couldn't a computer do a much better job?

gasstationwithoutpumps said...

I would not make a strong argument for linear algebra over calculus, but I would for teaching statistics—there is so much abuse and misunderstanding of basic statistics in the popular press that it is clear the subject is not being taught to all who need it.

Yes, higher-level statistics requires integral calculus, but one can do a fair amount with just good algebra. There is no evidence that calculus classes are any less rote than statistics classes—certainly most of the engineering students I see have treated all their math classes as rote learning (except for a few of the top students). Note: it is the student more than the teacher who determines whether the learning in active or rote—the teachers can nudge things one way or the other, but students can always refuse to think.

Linear algebra and linear programming are not the same subject—linear programming is an optimization technique for systems of linear inequalities and quite suitable for a high-school level CS/business course, though most college classes on optimization would only spend a week or two on it.

I don't buy the "things were better 20 years ago" argument. I've taught a lot of engineering students over the past few decades, and the math levels I'm seeing haven't changed much over 35 years—they're much lower than I'd like, but they have always been so.

Writing skills are probably a bigger problem for engineering students, but that has also not changed all that much in 35 years.