How high-schooler discovered new math theorem
He's not in the honors program. His papers, one teacher says, are a mess. In a school where the best and brightest are well dressed, he wears tatty running shoes, jeans, and a brightly colored T-shirt. But Rob Stringer, a high school geometry student in this upscale Boston suburb, has come up with something unknown since Euclid first discovered geometry. With the aid of a computer program developed by researchers at the nearby Massachusetts Institute of Technology (MIT) and Harvard University, he has hit upon a previously unknown geometrical theorem for dividing a triangle into smaller triangles of equal size.
Nor is he alone. Two other students at nearby schools, working with the same program, have also reportedly come upon undiscovered theorems.
Along the way, they have proved once again the potential of computer-aided instruction in provoking surprising and unexpected educational results.
``It was just an accident,'' says Rob Stringer -- unconscious, perhaps, that ``accident'' and serendipity have formed the basis for many of the world's scientific advances. ``I didn't try to do it,'' he explains with a grin, adding that it was ``part of a day's work.''
The ``day's work'' was, in fact, an assignment from math department chairman Richard Houde. ``He wanted us to divide a triangle into five equal parts,'' Rob recalls.
When Rob turned in his paper, says Mr. Houde, ``I had never seen [his proof] before.'' Neither had the writer of the particular piece of computer software, Harvard graduate student Michal Yerushalmy. Nor had Judah Schwartz, professor of engineering science and education at MIT and co-director of Harvard University's Educational Technology Center, who masterminded the computer program, nor any of his mathematical colleagues.
``We believe this is an original theorem,'' says Houde, a veteran math teacher whose particular interest is geometry.
``It appears to me to be new -- I've never seen anything like it,'' says Prof. Nicholas Kazarinoff, a mathematician at the State University of New York in Buffalo.
The Weston High School student's ``hitherto undistinguished mathematical career'' (in Houde's words) took a sudden turn when what Stringer describes as a ``guess'' turned into ``Stringer's Conjecture,'' as his theorem is affectionately known here (see sidebar).
Always interested in math (he is now taking two math courses), Stringer was not among the school's recognized intellectual leaders. ``What no one recognized up to this time,'' says Houde, ``is that he has a superior mind.'' Now, he adds, ``the other kids know it -- and he knows it, more important.''
Indeed, the discovery has bounced Rob into unexpected prominence among his peers. Referring to the rise of quarterback Doug Flutie from Natick High School through Boston College and into national prominence (despite his short stature), Houde quips that nowadays ``Stringer is to Weston as Flutie is to Natick.'' Stringer has no particular plans to become a mathematician, although he plans to enroll in a Bennington College program this summer to study philosophy and writing.
Houde first got interested in the geometry software several years ago when the university team, needing a site to test its materials, approached him. Weston High School, which had recently gone 50-50 with Digital Equipment Company on a $250,000 investment in computer equipment, agreed to release Houde from some of his teaching responsibilities to allow him time to integrate the computer setup into his classroom.
``I didn't ever imagine that computers could be used to do this sort of thing,'' the veteran math teacher says. He got into it, he says, not because of the novelty but because he was looking for ``ways to involve kids in thinking about geometry.'' He began by determining exactly what theorems he wanted his students to know.
Then, he recalls, he asked himself, ``How could I lead them to these theorems -- without me telling them what the theorems were?''
Enter the software, which is designed to create and measure graphic images. The version Houde uses generates triangles -- at random, or according to measurements typed onto the keyboard by the user. It will then perform various tasks -- construct a median, for example -- without the student even knowing that a ``median'' is technically a line from one vertex to the midpoint of the opposite line. Which, says Professor Schwartz, is one of the program's primary virtues. It allows students to study numerous examples of similar constructions -- to discover for themselves, for example, how to define ``median.''
For Schwartz, the real purpose of the program is to help students ``expand their intuitions'' and their capacity to deal with abstractions.
The software also allows students to measure angles, areas, and lengths of line segments and compare several sets of measurements. For Houde, ``the measurement is the key,'' in that it performs these sometimes complicated calculations instantly, allowing students to sift quickly through a wide variety of examples.
``Kids began finding geometrical ideas on their own that were not included in my thinking and were not in the textbook,'' Houde says. ``They would make conjectures -- then the final step was proof.''
Early on, one of his students, asked to define what the program allowed him to do, said, ``I think I'm supposing about geometry.'' From that, the program's name was born. ``The Geometric Supposer,'' as it is now known, is published by Sunburst Communications for the Educational Development Center in Newton, Mass. Earlier this month a sister package became available, which works with quadrilaterals.
Schwartz is now testing the program in some 15 area schools, including private schools, middle schools, and a vocational-technical school. In the United States, he says, geometry is traditionally taught at the 10th-grade level. Next year, Schwartz hopes to ``see how far down [in grade level] this set of tools can be used.''
Schwartz cautions against premature enthusiasm. The program, he insists, is still in its formative stages. ``It's terribly important not to get deluded by the excitement of the greenhouse,'' he warns.
But he is clearly pleased not only by the impact on students but on teachers as well. He emphatically rejects the notion that the computer itself is or can ever be the teacher -- which is one reason he's pleased with the work Houde is doing.
``His soul is in the mathematics -- it's not in the Mickey Mouse of computers,'' says Schwartz. The program, he concludes, ``changed his teaching -- it changed his life.'' Chart: Stringer's Conjecture.
What exactly is Rob Stringer's previously undiscovered theorem?
Draw a random triangle labeled ABC. Construct a median from point C to the midpoint of line AB (see diagram).
Now suppose you wish to divide the triangle into five equal parts. Start by dividing the median (AD) into five equal segments by adding points E, F, G, and H.
Draw lines from every other point to the opposite vertices (FA, FB, HA, and HB). Finally, erase line HD (to produce triangle ABH).
The result: five triangles, all of which have identical areas -- a fact which can be verified by various standard geometrical proofs.
Stringer's Conjecture works when dividing any triangle into any odd number of smaller triangles. (A similar theorem, which does not require erasing any lines, works for even numbers of triangles.)