| | Ivars Peterson's MathTrek |
June 15, 1998
Last month, North Bay's Nipissing University hosted the annual meeting of the Ontario Association for Mathematics Education, which attracted about 650 math teachers. As one of the featured speakers, I presented a talk on randomness. My examples included slot machines, Tilt-A-Whirls, random-number generators, dice, and tossed coins.
For my finale, I had set up about two dozen U.S. pennies so that they stood on edge on the surface of a table. During the course of my talk, most of the coins had fallen over. Those that remained upright toppled when I smacked the table. Nearly every coin ended up heads, a striking demonstration that the penny, with Abraham Lincoln's head on one side and the Lincoln Memorial on the other, has an uneven mass distribution (see A Penny Surprise).
The question that immediately arose was whether such a bias would show up with coins other than the U.S. penny. I didn't have an answer, but I suggested that it might be worthwhile performing some experiments to find out.
A little later, I heard that a group of teachers had tried the experiment with a roll of Canadian one-dollar coins, called loonies. The coin is polygonal rather than circular and features the head of Queen Elizabeth II on one side and a loon on the other. Based on their preliminary results, the teachers reported evidence of a definite bias. It's worth keeping such a potentially confounding factor in mind when doing coin (and dice) probability experiments in the classroom.
One striking feature of the meeting was the large number of talks and workshops that linked mathematics with the visual arts. Topics ranged from the golden mean, spirals, and tessellations to fractals, origami, and impressionist art.
Peter Taylor of Queen's University in Kingston, Ontario, teaches a course on mathematics and poetry and has written an applications-oriented high school calculus textbook called Water Earth Air Fire: The Elements of Calculus for Grade Twelve. In his presentation, he delved into the history of painting--the struggles of Paul Cézanne (1839-1906), Vincent Van Gogh (1853-90), Paul Gauguin (1848-1903), Pablo Picasso (1881-1973), and others to establish a new visual vocabulary--to say something about the present-day troubled state of math education.
Taylor noted how the introduction of the portable easel and tube paint in the nineteenth century allowed artists to get out of the studio and into the street and countryside. In the impressionist works facilitated by this technical innovation, painters endeavored to capture moments in everyday life. Paintings were no longer merely representations of static scenes. They evoked motion and change. This added vitality, in turn, was later transformed into vivid, disorienting expressions of a painterŐs encounters with reality, as seen in the works of post-impressionist artists.
Taylor argued that part of learning mathematics is learning what it's like to be a mathematician. He described the importance of capturing "mathematical moments" in the classroom, as students become involved in "rich" tasks that afford many opportunities for exploration.
Ken Stange of the psychology department at Nipissing University focused on current trends in the visual arts and mathematics. He argued that the combination of the personal computer, the Internet, and interest in fractals, chaos, and related mathematical topics has in recent years greatly facilitated the intertwining of art and math.
A mathematical image can readily become an art object, Stange said, and the Internet serves as a widely accessible art gallery and an ongoing, multithreaded symposium on topics of interest. Stange's Web site presents a gallery of his own fractal art and poetry (http://stange.simplenet.com/ken/personal.htm), and he provides links to other math and art sites (http://stange.simplenet.com/mathart).
Since 1992, artists interested in mathematics and mathematicians interested in art have been gathering and learning from each other at the annual Art-Math conferences organized by mathematician and sculptor Nat Friedman of the State University of New York in Albany. This year's edition takes place in August at the University of California, Berkeley (http://http.cs.berkeley.edu/~sequin/AM98/).
One of the best aspects of the math television series Life by the Numbers (see Math TV: Life by the Numbers) was its visual elements, ranging from colorful representations of fractals and four-dimensional objects to dramatic slow-motion photography of athletes in action. Unfortunately, because of the way PBS television stations chose to schedule the programs (or decided to ignore the series completely), few people got to see the shows.
The series was also intended for classroom use, and that aspect shows some promise. I met a teacher who is already using segments from a 2-hour videotape of program highlights to motivate the study of various mathematical topics and to illustrate the wide range of careers open to students interested in math.
My final visual memory of North Bay is of jogging early in the morning in brilliant sunlight along the mile-long lakefront park. Even at that hour, the bay was already dotted with the boats of fishermen casting their lines.
Copyright 1998 by Ivars Peterson
References:
Peterson, I. 1998. The Mathematical Tourist: New and Updated Snapshots of Modern Mathematics. New York: W.H. Freeman. ______. 1998. The Jungles of Randomness: A Mathematical Safari. New York: Wiley.
Comments are welcome. Please send messages to Ivars Peterson at ipeterson@maa.org.