two to turn this insight into formal mathematics. But the visual thinking—the colors—came first, and my subconscious had to grapple with the problem before I was consciously aware of the answer. Only then did I start to reason symbolically.
There’s more to the tale. Once the formal system was sorted out, I noticed a deeper idea, which underlay thewhole thing. The similarities between colored cells formed a natural algebraic structure. In our previous work on symmetric systems we had put a similar structure in from the very start, because all mathematicians know how to formalize symmetries. The concept concerned is called a group. But Marcus’s network has no symmetry, so groups won’t help. The natural algebraic structure that replaces the symmetry group in my colored diagrams is something less well known, called a “groupoid.”
Pure mathematicians have been studying groupoids for years, for their own private reasons. Suddenly I realized that these esoteric structures are intimately connected with synchrony and phase shifts in networks of dynamical systems. It’s one of the best examples, among the topics that I’ve been involved with, of the mysterious process that turns pure math into applications.
Once you understand a problem, many aspects of it suddenly become much simpler. As mathematicians the world over say, everything is either impossible or trivial. We immediately found lots of simpler examples than Marcus’s. The simplest has just two nodes and two arrows.
Research is an ongoing activity, and I think we have to go further than Hadamard and Poincaré to understand the process of invention, or discovery, in math. Their three-stage description applies to a single “inventive step” or “advance in understanding.” Solving most research problems involves a whole series of such steps. In fact, any step may break down into a series of sub-steps, and those substeps may also break down in a similar manner. So instead of a single three-stage process, we get a complicated network of such processes. Hadamard and Poincaré described a basic tactic of mathematical thought, but research is more like a strategic battle. The mathematician’s strategy employs that tactic over and over again, on different levels and in different ways.
How do you learn to become a strategist? You take a leaf from the generals’ book. Study the tactics and strategies of the great practitioners of the past and present. Observe, analyze, learn, and internalize. And one day, Meg—closer than you might think—other mathematicians will be learning from you .
7
How to Learn Math
Dear Meg,
By now, you’ve surely noticed that the quality of teaching in a college setting varies widely. This is because, for the most part, your professors and their teaching assistants are not hired, kept on, or promoted based on their ability to teach. They are there to do research, whereas teaching, while necessary and important for any number of reasons, is decidedly secondary for many of them. Many of your professors will be thrilling lecturers and devoted mentors; others, you’ll find, will be considerably less thrilling and devoted. You’ll have to find a way to succeed even with teachers whose greatest talents are not necessarily on display in the classroom.
I once had a lecturer who, I was convinced, had discovered a way to make time stand still. My classmates disagreed with this thesis but felt that his sleep-inducing powers must surely have military uses.
The vast amounts that have been written about teaching math might give the impression that all of the difficulties encountered by math students are caused by teachers, and it is always the teacher’s responsibility to sort out the student’s problems. This is, of course, one of the things teachers are paid to do, but there is some onus on the student as well. You need to understand how to learn.
Like all teaching, math instruction is rather artificial. The world is complicated and
There’s more to the tale. Once the formal system was sorted out, I noticed a deeper idea, which underlay thewhole thing. The similarities between colored cells formed a natural algebraic structure. In our previous work on symmetric systems we had put a similar structure in from the very start, because all mathematicians know how to formalize symmetries. The concept concerned is called a group. But Marcus’s network has no symmetry, so groups won’t help. The natural algebraic structure that replaces the symmetry group in my colored diagrams is something less well known, called a “groupoid.”
Pure mathematicians have been studying groupoids for years, for their own private reasons. Suddenly I realized that these esoteric structures are intimately connected with synchrony and phase shifts in networks of dynamical systems. It’s one of the best examples, among the topics that I’ve been involved with, of the mysterious process that turns pure math into applications.
Once you understand a problem, many aspects of it suddenly become much simpler. As mathematicians the world over say, everything is either impossible or trivial. We immediately found lots of simpler examples than Marcus’s. The simplest has just two nodes and two arrows.
Research is an ongoing activity, and I think we have to go further than Hadamard and Poincaré to understand the process of invention, or discovery, in math. Their three-stage description applies to a single “inventive step” or “advance in understanding.” Solving most research problems involves a whole series of such steps. In fact, any step may break down into a series of sub-steps, and those substeps may also break down in a similar manner. So instead of a single three-stage process, we get a complicated network of such processes. Hadamard and Poincaré described a basic tactic of mathematical thought, but research is more like a strategic battle. The mathematician’s strategy employs that tactic over and over again, on different levels and in different ways.
How do you learn to become a strategist? You take a leaf from the generals’ book. Study the tactics and strategies of the great practitioners of the past and present. Observe, analyze, learn, and internalize. And one day, Meg—closer than you might think—other mathematicians will be learning from you .
7
How to Learn Math
Dear Meg,
By now, you’ve surely noticed that the quality of teaching in a college setting varies widely. This is because, for the most part, your professors and their teaching assistants are not hired, kept on, or promoted based on their ability to teach. They are there to do research, whereas teaching, while necessary and important for any number of reasons, is decidedly secondary for many of them. Many of your professors will be thrilling lecturers and devoted mentors; others, you’ll find, will be considerably less thrilling and devoted. You’ll have to find a way to succeed even with teachers whose greatest talents are not necessarily on display in the classroom.
I once had a lecturer who, I was convinced, had discovered a way to make time stand still. My classmates disagreed with this thesis but felt that his sleep-inducing powers must surely have military uses.
The vast amounts that have been written about teaching math might give the impression that all of the difficulties encountered by math students are caused by teachers, and it is always the teacher’s responsibility to sort out the student’s problems. This is, of course, one of the things teachers are paid to do, but there is some onus on the student as well. You need to understand how to learn.
Like all teaching, math instruction is rather artificial. The world is complicated and
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