The power of having more than one right answer: Ambiguity in math class
Christopher Danielson, posted September 26, 2016 —
many students and teachers, math class is a place of great certainty. One plus
one equals two; six is greater than negative four; and the area of a triangle
is half the area of a related parallelogram. In this view, math is the place
where there are right answers, wrong answers, and no mistaking which is which.
it is maybe a bit unsettling the first time I work with a group of students,
show the image below and ask, “Which one doesn’t belong?”
asking students to think silently about which shape they would pick, and how
they would tell someone else why they picked it, I call on a few students. These
are some typical responses:
• “The triangle only has three sides; the
others have four.”
• “The one in the lower left isn’t colored in,
while the others all are.”
• “The one in the upper right is the only that
becomes a square when you turn it.”
• “The one in the lower right has a flat
bottom; the rest have pointy bottoms.”
of these is a true statement about the relationships among this collection of
shapes. That means each is a right answer. One question has four right answers!
That’s not certainty; that’s ambiguity.
ambiguity—a messy place—can be where important mathematics begins. What should
you pay attention to in a collection of shapes? If you count the number of
sides, you’re doing geometry. If you pay attention to shading, you may be
thinking about area (the shaded part) and perimeter (the length of the
boundary). If you pay attention to orientation, you may be working on your
spatial visualization skills.
more complicated and nuanced Which One Doesn’t Belong? example is this one:
wonderful mathematical investigations can begin by discussing the differences
and similarities among these shapes. One common point of discussion among
elementary students is how many angles each shape has, and in particular, how
many angles the heart has.
common claim is that the heart has no angles, because angles exist only when
two straight edges come together. Another common claim is that it has two
angles, because there are two places where curves come together. Which of these
claims is right depends on your definition of angle. Once you have defined this
term, the ambiguity disappears. A precise definition of angle that allows you to
measure the angle at the bottom of the heart requires calculus, but even very
young children have an intuitive sense that there really is an angle there, and
that its measure must be very small. Ambiguity allows students to discuss deep,
rich mathematical ideas. If you and your students are interested in exploring
this precise question about angles in the heart, you may find this zoomable
heart to be a useful tool.
What are some ways you can use Which One
Doesn't Belong? or other tasks to introduce ambiguity and open up spaces for
conversation in your mathematics classroom? Please share in the comments
section or reach out on Twitter (@trianglemancsd or use the hashtag #wodb).
Whether you join the public conversation or not, you’ll find the following
• Which One Doesn’t Belong? A Shapes
Teacher Guide, published by Stenhouse.
• The companion website for the book.
• The Which One Doesn’t Belong? website maintained and curated by Mary Bourassa.
For full-size, classroom-ready versions of these images used in this post, go to https://talkingmathwithkids.com/2015/01/07/building-a-better-shapes-book-2/.
In the next post, I’ll share my current
thinking and resources on extending ambiguity to new content—counting.
These resources would not exist without the
inspirational work of Megan Franke at UCLA and Terry Wyberg at the University
of Minnesota, nor without collaboration with teachers I know through Twitter
and blogs (known collectively and informally as the Math Twitterblogosphere).
Christopher Danielson is on the teaching faculty at Desmos, which offers a set of free digital math tools along with a growing library of activities developed by the community of users. He is the
author of two books—Common Core Math for Parents For Dummies, and Which
One Doesn't Belong? A Shapes Book. You can find more of his writing at
his website: Talking Math
with Your Kids.
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One of my FAVORITE activities to use with students! I did a "Which One Doesn't Belong" on Day 1 last year and gave the students time to debate with their table groups. We listened to several students defend their choices, and finally one student asked me, "So which one is it?" I polled the class. After getting them psyched about a "winning" selections, I told them they were all right, or all wrong.
"But I thought..." "Wait..." "What?!"
I don't think they had ever been exposed to mathematical ambiguity. For some students, this was eye-opening. The next time I posed a WODB, they remembered that more than one answer was possible and tried to justify more than one solution to the problem.
As the school year progressed, I would pose questions in other math contexts or even other subjects. Students would sometimes find other ambiguous answers. There was always one student who would connect it back to that Day 1 WODB activity: "It's both--you know, like the shapes."