mathematics

"The Calculus of Friendship" by Steven Strogatz

In The Calculus of Friendship: What a Teacher and a Student Learned about Life While Corresponding about Math, Cornell University math Professor Steven Strogatz shares some of the correspondence he shared over several decades with his high school calculus teacher, Don Joffray.

The book is short, and a quick read if you don’t try to follow too closely the mathematics the two correspondents toss at each other. The two men reverse roles over time — originally Strogatz was the student, but in time he becomes the teacher. Along the way, Strogatz and “Joff” share the joy of being challenged by interesting math problems. But their interaction, while lively, is also frequently sporadic. Neither man (particularly Strogatz) seems comfortable in becoming a friend who shares more than just a love of math and the highlights of day-to-day life. It is only as the two men grow older that they bridge the gap and communicate their more personal feelings. Both men suffered the kinds of personal losses common to most of humankind, but they each failed to share their vulnerability and need for emotional support. One can appreciate the special bond between these two men, even as one can imagine what a deeper relationship might have brought them.

Yes, mathematics is key to this story. But the real theme is friendship, what brings us together, and what we may miss out on by not opening up on a deeper level.

"Birth of a Theorem" by Cedric Villani

"Beauty is in the eye of the beholder" is a well-known cliche'. And it can be very difficult, if not impossible, to explain to someone who has no knowledge or experience to fall back on why something as abstract and abstruse as mathematics can be said to be beautiful. It would be interesting to conduct an informal survey to test the hypothesis that those who succeed in mathematical studies are those most likely to recognize the beauty of mathematics.

Frenchman Cedric Villani won the Fields Medal (the mathematics equivalent of the Nobel Prize) in 2010 for his work, with former student and colleague Clement Mouhot, on "nonlinear Landau damping and convergence to equilibrium for the Boltzmann equation."

I have no idea what this is, except I do know who Ludwig Boltzmann was, and that this work addressed a problem in mathematical physics.

But I didn't need to understand what this was about in order to follow Birth of a Theorem: A Mathematical Adventure, Villani's story of his work in this area. Yes, there are plenty of equations about as comprehensible as hieroglyphics. But the reader does not need to comprehend the math in order to grasp what life as a mathematician is like, as Villani describes it. Villani describes the joy of discovery, the fear of making a mistake that could negate all his work, the struggle in wrestling a problem to the ground. As many wunderkinds do, he spends long hours absorbed in his work, trying to make sense of a small part of the universe. When he succeeds, the reader breathes a huge sigh of relief with him.

I grasped just the tiniest bit of the mathematics he describes, but it reminded me of the pleasure of seeing something unfold in mathematics. I only understood part of the esoterica presented on these pages, but it left me wanting to dig into old math books to decode and more deeply appreciate the mathematics. As a graduate student taking courses in theoretical statistics. I had glimpsed one example of mathematical beauty when I learned about the close relationship between the concept of moment in mathematics and the concept of moment in physics.

You don't have to be a mathematician to appreciate this book, but I must concede the obvious: only someone with an interest in mathematics may appreciate it.

"The Math Myth" by Andrew Hacker

They say it is a good idea to read things that challenge one's cherished beliefs or values, both to expand one's mind and to test one's beliefs. But, more often than not, this is uncomfortable to do. (And that's why the normal tendency is to avoid doing it.) As in reading The Math Myth: And Other STEM Delusions, by Andrew Hacker. Professor Hacker has a number of arguments against mathematics education in high school and college, not the least of which is that the requirement for taking algebra, let alone classes such as trigonometry or calculus, forces students who have no desire to enter technical fields to endure a painful winnowing as they pursue their education. The acronym "STEM" stands for Science, Technology, Engineering, and Mathematics, and is used in a number of education policy approaches aimed at improving what students know in these subjects, with the objective of improving the competitiveness of the U.S.

As I recall, I was taught the oft-maligned "New Math" in my elementary school. I had no trouble with it, so neither did my parents, because I didn't need to ask them for help with my homework. (Unlike the father in the movie, Incredibles 2, who struggles to help his son with his "new math" homework and exclaims, "I don't know that way! Why would they change Math? Math is Math! MATH IS MATH!")

By seventh grade I was being pulled aside to do more challenging math assignments than the rest of my class was doing. By the time of high school graduation I had taken two years of algebra,  as well as geometry, trigonometry, and calculus, with high scores on standardized tests. I pursued chemistry as a college undergraduate and took calculus and differential equations. I pursued biochemistry in graduate school before moving into a fledgling PhD program in biostatistics and computer modeling. There I continued with coursework in theoretical and applied statistics, operations research, signals and systems analysis, complex systems, and numerical analysis, among others. In my dissertation research I had to apply what I knew about differential equations and numerical analysis to hypotheses in the biochemistry of vision. My first job out of graduate school was in biostatistics for pharmaceutical research.

I have always valued my mathematics education. It was very much aligned to applied mathematics and not theoretical mathematics. I am matho-philic, not matho-phobic, so it is a real challenge for me to put myself in the shoes of those for whom math does not come as easy.

Then along comes Andrew Hacker, with The Math Myth. One by one, Hacker tackles the assumptions that guide everything surrounding math education. As I read this book, I often felt like the subject of Edvard Munch's "The Scream", because I resist his arguments even as I have to grudgingly admit he may be right. Allowing one's beliefs and values to be challenged by reading a book like this is no easy task, especially for someone for whom mathematics and science has dominated their education and career.

I won't attempt to summarize Hacker's arguments. (It would be interesting to read a counterpoint.) But I will touch on one argument, the question of taking coursework that will not likely have practical value for a student. I bristle at this, especially when someone I know tries to tell me that because a knowledge of history has no value in the workplace, history education is a waste. On the other hand, I have been known to argue with one of my friends that there is little or no value in taking classes in Latin. And I do not understand those who put the so-called "classical" education on a pedestal.

What should students be required to learn, and when? This is a really tough question. I could ask how physical education ("P.E." we used to call it) prepared me for supporting myself and my family when I finished school. I could easily grumble over my grades in P.E., grades that were not impressive because I didn't have the athletic "aptitude" that others did. But if I fall back on that argument, was it fair to my fellow students who didn't share my mathematics aptitude for them to be compared against me?

There are no easy answers to many questions in education policy. It may not be comfortable to admit that, but it is a good thing that people like Andrew Hacker challenge our thinking.