This year’s Brizendine Scholar is Glen Van Brummelen, an expert on the history of astronomy and trigonometry. Van Brummelen is past president of the Canadian Society for History and Philosophy of Mathematics, senior fellow at the Dibner Institute for History of Science at MIT, and a founding faculty member of Quest University Canada. In addition to authoring 30 scholarly and 10 encyclopedia articles, he is co-editor of Mathematics and the Historian’s Craft and The Mathematics of the Heavens and the Earth: The Early History of Trigonometry and author of the forthcoming Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry. Van Brummelen has taught has taught more than 30 different mathematics courses, including most traditional topics but also mathematics and music, democracy, computer graphics, spherical trigonometry (using a 19th-century textbook), and how to be an ancient astronomer. He presented the history of mathematics from Hellenistic and Islamic civilizations and worked with geometry, pre-calculus, and calculus students in their classes.
1. Why is the study of math in high school so important?
Math is the ultimate liberal art. It isn’t portrayed that way in curriculum these days, but it is. So much of how we think today relies on how the ancient Greeks first started thinking about mathematics. When the Greek miracle happened in the 4th or 5th century—the emergence of logic, rationalism, inquiry, and proof—early mathematicians found themselves in trouble, especially with the concept of infinity. Logic and infinity don’t cooperate with each other very well.
An example is Zeno’s paradox: Achilles and the tortoise are running at constant speeds, with the tortoise in the lead. At each stage Achilles reaches the tortoise’s previous starting point, constantly getting closer and closer, but never actually catches up to the tortoise—which is ridiculous. The natural reaction is to be cautious—to set up axioms and proceed logically one step at a time. This is how we think today. If you look at an income tax form (as you will be doing in a few weeks), it was actually designed in much the same way that the Greeks studied mathematics: one section depends upon what has come before. The legal system is also designed this way. Much of modern discourse is modeled on how the Greeks were studying math.
2. If you think about it that way, what is the partner of logic?
Creativity. Thinking of math as exclusively logical, or learning in general as exclusively logical, is dangerous. Logic is a way of grounding exploration, to understand bounds and limits and the extent to which you can trust your conclusion. But original exploration doesn’t happen through logic. The Greeks didn’t arrive at concepts like prime numbers through logic—they came up with a big idea, and the logic came later to make sure it was reliable. Creativity was the first step. Mathematical creativity is almost impossible to define; by defining it, we’d probably lose its essence.
3. Who do you read? Who is your inspiration?
When you are historian of any kind you are constantly reading other people—different cultures, different mindsets. Euclid is a major author in mathematics—he developed the logical structure for geometry to follow. But, even with the logical structure, his books are almost like a novel—they build towards a climax, and all is revealed at end. This sense of play within a logical context is inspiring to me. He is able to explore both the logical and the creative sides without compromising either.
Neal Stephenson is one of my favorite authors right now. I’m reading The Baroque Cycle—the story of the controversy between Newton and Leibniz on discovery of calculus. But this is a small part of the book. He represents all sorts of aspects of 17th century life: Newton’s role as master of the mint and his motivations as a scientist and theologian; political and social views in Europe in 17th and 18th centuries; and interactions with Arab and other cultures along the Mediterranean Sea. The clash of ideas and different points of view all coming together at once is a rich and glorious mixture. Calculus was born at this time, and it established the scientific trajectory that we are still on today. The inventors of calculus were not just thinking mathematics and science, but also philosophy. Leibniz was attempting to come up with logical, computational machines to study everything in the universe, and calculus was part of that. If Leibniz’s machine had been realized, you could compute everything: weather, which restaurant to go to, et cetera.
4. What has your experience at MA been like?
I feel right at home here. I teach at Canada’s first private, secular, liberal arts and science university. Students at Quest are similar to students here at MA. They are interactive, willing to try anything. They really want to take possession of material and give it a good shake. I have a harder time visiting traditional high schools where students just sit quietly.
5. In education we hear about the words “innovation,” “21st century skills,” “design thinking” constantly. As a teacher and thinker, what does that mean to you?
I don’t think of “21st century skills” in the obvious way (technological training, technical ability). When you’re passionate about any issue—ending child poverty, global warming, et cetera—you’re not going to get very far unless you understand it from many different points of view at once. To understand climate change, you must understand climate science and physics, but also social realities as well as the cultural, political, and economic effects of change. If you want to make a difference, you have to link fields like economics and philosophy. This is where innovation will happen over next 50 years.
If you have a conventional education, you are trained in the here and now—how we view the world in 2012. By the time you get to 2015 the world will be a different place. You might be able to exist in current system for a few years, but what will get you far is flexibility in thinking.
I want to have students who are prepared and willing to jump in—with both feet—somewhere they’ve never been before. The students who are going to succeed will be the ones that adapt to change.
At Quest, the campus was constructed so that the largest classroom holds 20 students. All courses have seminar-style discussions—there is no back of the room. Everyone is involved in every stage. No lectures—if I talk for more than 15 minutes, my students get restless and want to start experimenting. That’s not unusual in high schools, but completely foreign at many universities. Our students excel at discussion and reasoned arguments. They are very good at presentation skills because they are doing it all time. They are very good at doing research and can find the right sources to get at something new. They know how to interact with scholars and researchers. Compared to students who are educated conventionally, our students don’t have as much technical knowledge in a particular area. I don’t see that as a huge loss. Research shows that within three weeks, most students forget 90% of the detailed technical knowledge they learned.