Nikhil Datta ’21 jumps right in. “I have a question on 8. I figured out that the two triangles are congruent to each other. After that, how do you prove that those two sides are parallel?”
Math teacher Kevin Bau’s band of eager ninth- and tenth-grade Honors Geometry students has been focusing on triangle properties as they learn how to compose proofs. “We’re working on using logic, on justifying assertions,” Bau says. “My hope is that this work is transferable to building an airtight argument about anything.”
Looking at the homework problem projected on the whiteboard—two side-by-side triangles joined at point M—James Wade ’21 suggests, “Can’t we say that angle DMA is equal to angle B, so the lines are parallel?”
Bau scans the class. “What do you guys think?
“James, what’s the reason for that?” asks Anoushka Mahendra-Rajah ’21.
“Because the angles are the same.”
Alluding to a postulate they’ve learned, Eli Jensen ’21 adds, “They’re corresponding angles.”
“Oh, I get it now,” says Sam Subramanian ’21.
In addition to developing students’ critical thinking skills, Bau says that he aims to create “a culture of respect for everyone’s voice in the class,” the benefits of which are not lost on the students. “My two main class rules,” says Bau, “are to work hard and be nice. I try to make sure I’m modeling those for the kids all the time, too.”
“We have a lot of friendly banter in class,” says Julia Shephard ’22.
Samantha (Sam B) Bernstein ’21 ascribes her enjoyment of the class to “the other students’ motivation and love of math, which is infectious—and of course, Mr. Bau’s love of geometry. It really is incredible what a good teacher like Mr. Bau can convey to his students.”
Bau relies not just on elegant proofs or the golden ratio, though, to inspire his students. He believes in the “power of stories” to engage students and get them to think about why they’re working in certain ways. With no more questions raised about the homework, he now projects an early twentieth-century newspaper photo of a small crowd surrounding a horse.
“This is the story of Clever Hans, a horse in early 1900s Germany. Apparently, he could do math.” Bau explains that Hans would stomp his hoof the correct number of times in answer to simple arithmetic questions. “His trainer could also ask him more complex questions, like, if today is Tuesday, the second of October, what day of the month will Friday be? And he’d tap out… six, is it?”
“Wouldn’t it be the fifth?” Sam B asks.
“Hold on,” says Bau. “Oh, yeah….” He breaks into a grin that invites everyone to laugh. “Clever Hans is possibly smarter than I am.” Bau goes on to say that the trainer was not out to make money; he just thought he had an amazing horse. The skeptical German government, though, sent in a team of investigators. “One psychologist discovered that Clever Hans didn’t actually know math.”
“Oh, this is so disappointing,” says Sophia Cohen ’21, making her classmates and teacher crack up.
“He figured out that Clever Hans could answer questions correctly only if he could see the trainer and if the trainer himself knew the answer.” In a nutshell, the trainer’s subconscious body language signaled to Hans when he should stop stomping. “I tell you this,” explains Bau, “because I want to avoid training you to be Clever Hanses. I shouldn’t always be the person telling you, even with my body language, if an answer is right or wrong. I want you guys to be the arbiters of deciding—together.”
For a moment the story’s lesson seeps silently into the students. Then Nikhil asks, “But what if we don’t know what the right or wrong answer is?”
Laughing along with his students, Bau says, “Good question. I’m not going to totally abandon you and ‘Peace out,’ but my hope is that I’m just another voice in this classroom and that we can decide collectively what we’re convinced by and what we’re not. With that in mind, in your teams, I’m going to have you work on a slightly harder question.” Bau turns to the whiteboard, his signature warm smile assuring support for students’ efforts.. “We’ve got a regular pentagon. You guys remember what ‘regular’ means?”
Sophia responds, “All sides and angles are the same.”
Inside the pentagon, Bau draws a regular (or equilateral) triangle. Pointing to the pentagon’s leftmost vertex (an angle’s point), then the triangle’s top vertex, then the rightmost vertex, he says, “Your task is to decide, if we connect these three vertices, do we have a straight line or not? You need to be able to explain your thinking convincingly to the class.”
Champing at the bit during the last of Bau’s directions, the students decamp to the whiteboards to collaborate in trios. “It always helps to have someone to bounce your ideas off of,” says Alex Wu ’21.
Visiting each group in turn, Bau listens and watches as the students, thinking out loud, mark up their diagrams.
Marie Quintanar ’21 points to the angle on the exterior of the equilateral triangle. “Is this 60, as well?”
Anoushka taps all the angles formed by lines radiating from the interior vertex to the pentagon’s five vertices and says, “If these were all 60, we’d have a hexagon, not a pentagon, right?” She has realized there’s an inconsistency among the angles, which James and his group discover by remembering that a pentagon’s internal angles add up to 540 degrees, making each angle 108 degrees. Subtracting the equilateral triangle’s 60-degree angle from 108, James states, “These two are not equilateral triangles.”
“Convince me,” says Bau.
“Because this angle’s 48.”
“Great!” says Bau, eliciting the students’ laughter. “That’s pretty convincing.”
In another group, Matt Hong ’21 turns to Bau and says, “Oh—this is why you said the other day that we couldn’t say the line necessarily divides the angle in half, right?”
When Bau shrugs, Sam B calls over, “By looking at the body language, I’d say yes.”
Laughing, Bau says, “I’m going out of my way not to Clever Hans you guys, but I am like the worst actor ever. I’m trying not to signal anything.”
Reading his own signals, Bau recognized that he’d become disenchanted after earning an M.B.A. and working for several years as a Wall Street investment banker. He did some volunteer work, running a tutoring program for kids in Harlem, and realized, “This is kind of hard, but it’s kind of fun, actually. It feels nice to have a kid come in totally confused and leave a little less confused.” With an Ed.M. from Harvard, he launched his new, more fulfilling career.
“Julia and Sam, why don’t you talk to us all about what you’ve figured out so far.”
“Okay, so if this is an equilateral triangle, then this angle is 60,” says Julia, pointing to an angle created at the pentagon’s rightmost vertex. “Sam thinks yes; I am an unconvinced.”
“It’s okay if you have different conclusions,” says Bau. “Sam, can you tell us what you figured out and why?”
Pausing, Sam readjusts his conclusion. “These two angles can’t be equal because we don’t know if these lines are parallel.”
“Nice! So, we can’t say for certain that that top-right angle is 60,” says Bau, nodding.
Sam B reports on the work she did with Sophia and Abby Chen ’22. “Using the isosceles triangle theorem, we knew that the angles opposite the congruent sides must be equal to each other. If this angle is 48, then 180 (the total degrees in a triangle) minus 48 equals 132, divided by 2 (angles) is 66.” Noting the three angles anchored at the pentagon’s central point, Sam concludes, “So, 66 + 60 + 66 does not equal 180. It is not a straight line.”
“How do people feel about those 66-degree angles?” asks Bau.
As others are nodding, Tori Huang ’21 says, “Nice job!”
It’s been a good class, Bau believes, when the students leave feeling challenged, “pushed a bit but not spent,” he says. “I’d like them to leave feeling ‘that was kind of cool,’ appreciating some of math’s beauty.”
Written by Sharon Krauss, Upper School English