To the editors:
As a fellow European mathematician, I found the interview with Sergiu Klainerman to be a particularly interesting read. Even though I am now primarily a teacher rather than a research mathematician, there was still much of interest for me in the article.
Klainerman begins the interview by reflecting on his time as a student in Bucharest. He mentions a mantra that was often heard among the mathematics students: “Climb the mountain first and then the true meaning of the subject will come.” It may not seem that way at first, but this is good advice. When learning mathematics, it is important to be persistent and not get discouraged. Mantras like this can be helpful and Klainerman himself is proof that the approach works. After all, he became a fantastic mathematician.
I felt the same kind of frustration when I was a student at ETH Zürich. At first, we were fed a lot of information and I found myself missing the creativity I was able to express as a high school student. Looking back, I can now see that learning a lot to begin with had many benefits. The knowledge we acquired helped us to make connections and be more creative later on.
When learning a new language, or a musical instrument, we often begin by practicing with existing material; we read and play the masters for many years before we start to write poetry or compose on our own. In these early stages, we might enjoy a piano concert, for example, without really understanding how and why it works. Building that level of understanding can be a challenging process for beginners. If frustration builds up during these early stages, a student may ultimately decide to walk away. Mastering any new skill or subject requires the investment of an enormous amount of time and energy.
Understanding is valuable, but creativity is even more important. There may be psychological factors that play a role in the process, but the rush to force children to understand or to be creative can also be a turn-off. Creativity is hard. How hard? Try to be creative and come up with a new theorem in geometry or number theory on your own. It is far from easy. Nonetheless, we should try to practice being creative as a way to improve our skills. Even when I have some success at tasks like composing music using a computerized algebra system or building 3D objects, I’m constantly reminded that creativity remains difficult for many people. During my final, and mostly oral, examinations as an undergraduate I was asked the hardest question of all by Ernst Specker, who was testing me in logic. “Mr. Knill,” he inquired, “tell us something.” It was a really difficult question to answer and also one that illustrated just how challenging it can be to tackle open-ended queries.
True understanding can also be hard. One reason is that understanding often requires a lot of knowledge beyond the subject at hand. Teaching with this goal in mind is important, but also difficult. How difficult? Pick up a mathematics book on an unfamiliar subject and try to not only solve a task, but understand it properly. In such situations, a good approach is to start by solving the problems provided in the book. This is a good exercise because it evokes the feelings of a student experiencing a subject for the first time. To begin with, whether the topic involves algebra, geometry, calculus, or any other area of mathematics, the initial impressions are disorientating. First encounters can often be baffling. These are the same feelings Klainerman recalls when reading the work of Lars Hörmander or Fritz John. I have the same experience now when I read a paper by Klainerman. To understand the paper properly, I would need to brush up on harmonic and functional analysis, learn inequalities, and so on. As an educator, I feel that neglecting to teach substantial amounts of material is one of the most important mistakes currently being made in mathematics education. I can’t help but feel it is also the reason why other countries do so much better in tests. Teachers are inculcated that teaching content is bad and that conceptual understanding is the most important precursor to knowing how to do things. This approach is reflected in Bloom’s taxonomy. But for understanding to flourish, there must be some level of interest in a topic. It is not necessary to know precisely how an internal combustion engine or electric motor works before learning to drive a car. It may, in fact, be the act of driving itself that piques one’s curiosity in these topics. Similarly, we need to have written a few programs in a programming language before beginning to understand why the syntax is built in a particular way or how the compiler works.
Like Klainerman, I was also frustrated as a student not to have achieved a deeper understanding on a range of topics. It was a problem Klainerman overcame by staying motivated and curious. Having been provided with a lot of additional knowledge, in time, he was able to progress and make a name for himself as a mathematician. The most important objective of education is cultivating understanding. From more than three decades of teaching mathematics and over five decades of learning mathematics, my own experience is that understanding is difficult to achieve if it is not preceded by some preliminary “how-to” work. The very best teachers can attain understanding in their students in the early stages of introducing a subject, but not many teachers are so skilled. The enduring popularity of “how-to” guides on the internet are proof that this is indeed the case.
Learning also requires some degree of memorization and practicing. This approach would never be questioned in other fields, such as music. Playing a piano concert is made much easier by committing the piece to memory and having played it many times. No one would tell a pianist that they should avoid memorizing a piece or abstain from finger exercises. Similarly, a rock climber needs to memorize each move on a difficult route. An unrehearsed approach will inevitably lead to failure. The recent films Dawn Wall and Free Solo show just how much work and repetition are needed to master the hardest and most technical climbing routes. In mathematics, even seemingly boring things like the multiplication table are treasure troves. There are patterns waiting to be discovered. Nonetheless, some level of familiarity is a prerequisite to become interested in a subject. This is a crucial step because once interest has been established, the learning process becomes not only much easier, but also enjoyable.
I agree with Klainerman’s assessment that partial differential equations (PDEs) are an important topic in mathematics, and one that is connected with many other fields, such as geometry and probability theory. Although PDEs are usually thought of as part of the field of analysis, and especially functional analysis, there are also connections to combinatorics or algebraic geometry. The name Klainerman first attracted attention while he was taking a graduate course in nonlinear wave equations given by Michael Struve at ETH Zurich. Here, the local and global existence of initial value problems arise in a similar manner as they do in fluid dynamics or general relativity. Klainerman made important early contributions in this area. It is also a topic that I was curious about as a graduate student due to my interest in integrable partial differential type systems.
The topic of PDEs can become very technical, as can be seen from Klainerman’s own work. It is also an area where the mastery of basic skills, along with knowledge of many inequalities and analytic techniques are essential. The broader field of PDEs is not only huge, but also one of the richest and deepest in mathematics. PDEs are also pervasive in physics. Recent discoveries by the Event Horizon Telescope have reinvigorated interest about PDEs in relativity. And then there is the million-dollar prize on offer for the Millennium Problem concerning the Navier–Stokes equations. PDEs have even appeared in pop culture from time to time, such as in the movie Gifted.
