Dec. 4, 2018 — At 8 am, students in Seth Montgomery’s Geometry class are proving themselves. Actually, they’re proving theorems at the white boards, which are soon covered with triangles, formulas, and numbers.
Montgomery isn’t where you might expect him to be. He’s at the back of the room, giving students the floor to present their ideas, answer questions, and discuss one another’s work.
“You’ll never see me at the front of the room presenting material,” he says. He uses a method developed by mathematician R.L. Moore, in which students work through a series of sequential problems to “discover” the subject. “In a sense, students go through steps similar to those Euclid took when he first compiled geometry in 300 BC,” Montgomery explains.
The class works on problems in teams of three, with each team documenting their assumptions, showing their work, and using deductive reasoning (and lessons learned from previous problems) to reach the final proof.
After the first group to present has found the tangent, sine, and cosine for their 90-degree right triangle, Montgomery asks the others, “Is their work complete, clear, and correct? Does everyone understand how they got that answer?” Nods all around, so the class moves on.
Problem two asks students to find the sine, cosine, and tangent for a triangle with an angle with a degree measure of 30. When a presenter needs a square root to several decimal points, smartphone apps provide instant calculations. The group gets their proof right, but not in the way Montgomery expects. “That’s clever; I like that,” he comments appreciatively.
Problem three involves proving that an exterior angle of a triangle is equal to the sum of the opposite interior angles. Linh Dinh ’21 explains each step her group took to complete the proof, confidently drawing diagrams, documenting assumptions, and showing her work on the board. This time, several hands are raised with questions, which Dinh handles with aplomb. Julia Rubright ’22 had taken a different tack for this problem, and comes to the board to demonstrate. The class debates whether Rubright’s proof is complete or only partial.
Montgomery again steps in after students have had a chance to consider the options. After tweaking both students’ approaches, he asks, “Are both methods valid?” Yes, students decide; he agrees, and the class continues its exploration of this ancient science.
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