The power of having more than one right answer: Ambiguity in math class

  • The power of having more than one right answer: Ambiguity in math class

    By Christopher Danielson, posted September 26, 2016 —

    For 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.

    So, 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?”

    2016_09_26_Danielson1_fig1 After 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.”

    Each 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.

    But 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.

    A more complicated and nuanced Which One Doesn’t Belong? example is this one:

    2016_09_26_Danielson1_fig2Many 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.

    A 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.

    Your turn

    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 resources useful.

    •    Which One Doesn’t Belong? A Shapes Book and 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.

    Acknowledgements

    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).



    2016_09_26_Danielson_1auPicChristopher 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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