# Convince Me

## By Wendy Petti

Saying "Wrong answer!" pronounces judgment and kills debate. Saying "Convince me!" stimulates inquiry, clear mathematical thinking, and animated mathematical discourse.

"Convince me!" has become the new mantra in my math classes. It's a positive and powerful call to action. Anyone can make the call. Sometimes, I ask my students to convince me. Sometimes they ask me to convince them. And sometimes they try to convince one another.

"Convince me!" is what mathematicians have been attempting to do ever since Thales of ancient Greece introduced mathematical proofs through formal arguments. But "Convince me!" in the math classroom can take a variety of forms. We might convince one another with equations accurately depicting the problems we are trying to solve. We might convince one another with drawings or diagrams. We might convince one another with step-by-step narratives or an informal oral explanation.

## A GOOD ARGUMENT CAN CONVINCE EVERYONE

"Which is greater, 1/6 or 1/8?" I asked my fourth-graders in an early lesson on fractions. Many of them knew that 1/6 is greater.
"How do you know?" I asked.
"Because our third-grade teacher told us that when the number on the bottom is greater, the number is smaller."
"Ah, but that doesn't convince me!" I said.
Now the burden was on them to convince me and to convince the few skeptics in their midst. I could step back as they huddled in small groups, trying to think of a convincing explanation.
"Oh, I know how to convince you!" TJ exclaimed to his group. "Which would you rather eat, one third of a pizza or one hundredth of a pizza?"
His point was instantly made and his group was convinced... and so was I. The students liked the pizza argument so much that some of them went home and convinced their younger brothers and sisters.

## "CONVINCE ME" EMPOWERS ALL LEARNERS

"Convince me!" helps all learners take ownership of a problem. When I ask my students a question like "What is 1/3 + 1/2?" they need to be able to justify their answer. Students discuss the problem in small groups. Certain students who might slump in their seats at times when I'm doing the explaining will eagerly join a small-group huddle when they know I'm going to call on a student at random to convince me of an answer.

The students who need extra time and extra help to grasp new concepts are empowered, along with the students with quick insights. Instead of saying, "I don't get it!" they now say, "I'm not convinced yet." They're not giving up; they're assessing the logic of arguments. They're using higher-order thinking skills, and so are their classmates, who need to think a little harder and a little more creatively to arrive at a new, more rigorous argument. There is pride on both sides when the hesitant student finally is able to say, "Okay, you've convinced me!" and is able to turn around and convince me of their argument.

## "CONVINCE ME" CAN BECOME A WHOLE-CLASS ENDEAVOR

 Illustrations courtesy of Wendy Petti.
A low moment turned into a peak moment one Friday afternoon when -- at first -- every student in my class seemed to be missing a fundamental point. We had created a Venn diagram of the multiples of 4 and 6; the students could see that 4 and 6 shared some common multiples: 12, 24, 36.... But when I asked if they thought 45 and 71 shared any common multiples, the entire class said no. They seemed quite sure. I fought off the inclination to say, "Well, you're all wrong! What about 45 x 71?!" Instead, I said with a smile, "Well, you're going to have to convince me!" They seized the challenge. Some students tried showing me that multiples of 45 would always end in 5 or 0, but multiples of 7171, 142, 213, 284... did not. As they huddled together, I heard students saying, "Yes, but how are we going to convince her?" A minute later, one group excitedly reported that Sebastian had found multiples of 71 up to 710, so there actually was a multiple of 71 that ended in 0. (Again I held my tongue while wondering why they were surprised; why all our past lessons on multiples of 10 had not come to mind.) As the students wrestled with how to convince me, they began to realize that their arguments were breaking down.

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Finally, a student had the flash that maybe 45 times 71 would produce a common multiple. But that wasn't intuitively obvious to the group; they needed to convince themselves. At their desks, they multiplied 45 x 71 and found 3,195. But was it a common multiple? From where I stood, the obvious was staring us in the face; all year long, we'd been demonstrating fact families, so if a x b = c, then c b = a and c a = b. But it seemed they couldn't convince themselves unless they could make sure that 45 and 71 each would divide equally into 3,195 without a remainder. Pauline ran to the board to divide 3,195 by 45 and was thrilled to find that the quotient was 71. Now, she and her classmates had finally convinced themselves that they had found a common multiple.

"I wonder if there's another common multiple," Daphne offered. Most doubted it; again I was surprised. But after wrestling with it a bit, and consulting the first Venn diagram with multiples of 4 and 6, they had the idea that since 45 x 71 had produced one common multiple, maybe if they doubled that number they would find another common multiple. Once again, they needed to work the division to be convinced. They were practically ecstatic when they found out that 6,390 divided by 45 produced a quotient of 142, which they knew from their Venn diagram was the second multiple of 71. They were so excited by their shared involvement in the quest for understanding that some of them told me it was the best math class they had ever had. Days later, they were still talking about it. It seems so simple here in the retelling, but there was incredible group energy in the room as they tackled and extended the investigation.

All I had done was ask them to convince me that their intuition was correct. In the process of ultimately convincing themselves of quite the opposite, they began to convince themselves of several fundamental mathematical concepts I thought they had mastered months or years ago. But they never "owned" the concepts until they needed to grapple with them to make sense of something that seemed at first impenetrable, to make new meaning.

## "CONVINCE ME" PROMOTES INQUIRY AND DISCOURSE

"Convince me!" withholds judgment. "Convince me!" says, "I'm willing to be convinced, if your argument is strong enough." Saying "Wrong answer!" pronounces judgment and kills debate. Saying "Convince me!" stimulates inquiry, clear mathematical thinking, and animated mathematical discourse.

Most great innovations in mathematics and science have been born of skepticism, when a bold thinker dared to challenge prevailing beliefs or methodologies. Albert Einstein, when asked how he discovered the theory of relativity, replied, "I challenged an axiom."

When, instead of passively receiving and believing everything we tell them, students become hungry to convince and to be convinced, a wonderful transformation occurs. Our students become active learners. They aren't just doing a student's work. In seeking to convince and be convinced, our students are doing a mathematician's work.

I hope you are convinced!