Field of Science

Book review: A Divine Language: Learning Algebra, Geometry, and Calculus at the Edge of Old Age, by Alec Wilkinson

A beautifully written account of mathematics lost and found. The author got "estranged" from mathematics in school and now, at the age of 65 and after a distinguished writing career, has taken it upon himself to learn the fundamentals of algebra, geometry and calculus. The book is by turns funny and sad even as Wilkinson recounts his struggling attempts to master material that would be child's play for many bright teenagers. He is helped in his efforts by his niece Amie Wilkinson, an accomplished mathematician at the University of Chicago. I myself could empathize with the author since I too had an estrangement of sorts with the subject in high school because of a cruel, vindictive teacher, and it took me until college when, thanks to brilliant and empathetic teachers, I clawed myself back up to start appreciating it.

But while he may struggle even with high school mathematical skills (and he I share a particular loathing for word problems), Wilkinson brings a poetic, philosophical sensibility acquired through a long career to bear on the topic that no young 15-year-old whippersnapper genius in math could commit to paper. He ruminates on the platonic beauty of math and wonders whether and how some people's minds might be wired differently for it. He does not always understand how his brilliant mathematical niece Amie always "gets it" and she in turn doesn't always understand why her uncle has trouble with ideas that are second nature to her.

Often quoting from eloquent mathematicians and physicists like Bertrand Russell, G. H. Hardy and Roger Penrose, Wilkinson brings a fresh, beautiful perspective to the utility and beauty of mathematics; to the struggle inherent in mastering it and the rewards that await those who persevere. I would highly recommend the book to those who may have lost faith in mathematics in high school and want to pick up some of the concepts later, or even to young students of math who may be wizards at solving equations but who might want to acquire a broader, more philosophical perspective on this purest of human endeavors.

Temple Grandin vs algebra

There's a rather strange article by Temple Grandin in the Atlantic, parts of which had me vigorously nodding my head and parts of which had my eyebrows crawling straight up. It's a critique of how our school system tries a one-size-fits-all approach that does a lot of students disservice, but more specifically takes aim at algebra. 

First, let me say how much I admire Temple Grandin. A remarkable woman who had severe autism for most of her childhood (there's a very good profile of her in Oliver Sacks's "An Anthropologist On Mars"), she rose above her circumstances and channeled her unusual abilities into empathy for animals, becoming one of the world's leading experts in the design of humane housing and conditions for livestock. She has without a doubt demonstrated the value of what we can call 'non-standard' modes of thinking, teaching and learning that utilize visual and tactile ability. So she starts off strong enough here:

As a professor of animal science, I have ample opportunity to observe how young people emerge from our education system into further study and the work world. As a visual thinker who has autism, I often think about how education fails to meet the needs of our very diverse minds. We are shunting students into a one-size-fits-all curriculum instead of nurturing the budding builders, engineers, and inventors that our country needs.

So far so good. In fact let me digress a bit here. When I was in high school I was very good at geometry but terrible at algebra; I still remember this one midterm where I got an A and in fact the highest points-based grade in the class in geometry but almost flunked algebra. It took me a long time to claw back to a position where algebra made sense to me. In fact this appreciation of visual explanations was what drew me in part to chemistry, so I perfectly appreciate what Grandin is saying about being sympathetic to students who might have more of a visual capacity. 

But further down the pages she takes a detour into the evils of algebra that doesn't make sense to me. Again, some of what she says is spot on; for instance the fact that algebra (and math in general) can be taught much better if you can relate it to the real world. Too often it's presented simply as abstraction and symbol manipulation. But then there's this:

Cognitive skills may simply not be developed enough to handle abstract reasoning before late adolescence, which suggests that, at the very least, we’re teaching algebra too early and too fast. But abstract reasoning is also developed through experience, which is a good argument for keeping all those extracurriculars.

This part may make more of a case for tying algebra to specific real-world applications than doing away with the abstractions per se. The fact of the matter is that math is abstract; in fact it's precisely this abstraction that makes it a powerful general tool. And there are good and bad ways of teaching that abstraction, but the solution isn't to get rid of it or delay it. In fact, that kind of thinking feeds into the popular belief seen in some quarters these days that algebra and calculus both need to be optional classes.

It's when she gets to the end of the piece, however, that Grandin completely loses me:

"No two people have the same intelligence, not even identical twins. And yet we persist in testing—and teaching—people in the same way. We don’t need Americans to be better at algebra, per se. We need future generations that can build and repair infrastructure, overhaul energy and agriculture, develop robotics and AI. We need kids who grow up with the imagination to invent the solutions to pandemics and climate change. When school fails them, it fails all of us."

Say what? Building and repairing infrastructure, overhauling energy and agriculture and - especially - developing robotics and AI do not need algebra? In fact most of these professions involve a very solid grounding in abstract aspects of algebra and calculus. I think Grandin is treading very handily from saying that algebra should be taught better to saying that we should get rid of it or make it optional. Two very different things.

My concern based on this article and others I am reading these days is that, in our drive to reform the system, we want to consider it unnecessary. That is a grave mistake. Algebra and calculus and for that matter music and art are things that, even beyond the practical utility of the first two, help us appreciate our place in society and the cosmos better and in general teach us to be more human. Make them better we certainly should, but let's not burn the building down in our zeal.