Which One Doesn’t Belong? A Shapes Book and Teacher’s Guide, by Christopher Danielson. Portland, ME: Stenhouse Publishers, 2016. 111 pp. $33.33
Earlier this year, at an annual meeting for math teachers, Dan Meyer gave a fascinating talk on how to get children to love and succeed at mathematics.1 The talk was fascinating in part because it challenged some popular pedagogical views. Meyer observed, for example, the futility of trying to motivate students by merely identifying mathematical relationships in nature, or saying that math will be useful in their future careers, or formulating word problems with objects to which students might relate. By themselves, said Meyer, such tactics fail to interest many students in math and can even turn them off to the subject.
Most of all, however, the talk was fascinating because it offered positive ways to engage children in an immensely useful subject. Among other things, it showed the importance of not starting formally, with precise definitions about something very abstract, but rather of starting informally, with something perceptual that raises interesting questions and gets students wondering and debating about what the answers might be.
The process Meyer advocated does not ignore the more careful measurements or more precise definitions that help ultimately to answer questions raised. Rather, it saves those for later, focusing initially on ensuring that students become interested not only in the answers but also in the journey toward their discovery.
Since hearing this talk, I’ve been implementing the ideas with my son, and I’ve been on the lookout for resources that might be of assistance. One of the best resources I have found is Which One Doesn’t Belong? A Shapes Book and Teacher’s Guide by Christopher Danielson.
As its title suggests, the book presents sets of shapes and asks of each, which shape is different from the others in the set. There are other shape-comparison books on the market, but this one involves an important difference. As Danielson explains:
In this book, the question “Which one doesn’t belong?” is ambiguous: there isn’t just one answer. In fact, any one of the shapes on any one of the pages can be the one that doesn’t belong. All choices are correct, which shifts the focus to justification. Which One Doesn’t Belong? isn’t about guessing the right answer; it’s about expressing mathematical relationships precisely in order to communicate with others. (p. 3)
One of the pages, for example, has a small blue square in the upper left, a large red square in the upper right, a small red square sitting on one of its corners in the lower left, and a small red rectangle in the lower right. Which one doesn’t belong? Well, that depends on whom you ask—and what he focuses on.
For example, my son initially noticed that three of the shapes were red and thus pointed out that the blue square was obviously different. But no sooner did he say that then he noticed that three of the shapes were roughly the same size as opposed to the square in the upper right that was much larger. For fun, then, I observed that the square in the lower left was oriented differently—it was the only shape that seemed to stand on one of its corners. That said, the shape in the lower right was practically begging to be noticed, which happened soon enough when my son observed that it was the only rectangle. Every page of this book sparks such conversations.
Another page, for example, has a triangle in the upper left, a rectangle in the upper right, a shape that is almost a triangle—but whose sides don’t touch—in the lower left, and a shape made of inwardly-curved sides that resembles a triangle in the lower right. Again, these shapes are all distinct in a way that is easy to grasp visually, and they all involve attributes that help to clarify what a triangle is and is not.
Children, of course, do not always have the words to describe what they see. And this book helps in this regard as well, as it motivates them to inquire about the words they need. For instance, if a child does not yet know the word “concave” but sees a shape with concave elements, he knows that he lacks the word to identify this characteristic—and he wants to learn what it is so he can answer the question. Such moments enable the teacher or parent to offer the word and a contextually appropriate definition. This also enables the child to see that his observation is important—so important, in fact, that there is a special word to describe what he sees. The child obtains and integrates a new concept as necessitated by his own observations and curiosity.
Unfortunately, such moments are rare in today’s educational environments. As Danielson observes in the teacher’s guide (which can be purchased with the book):
This is the precise opposite of how a lot of instruction in mathematics proceeds. Typically, students are given a definition and asked to apply it. “A triangle is a polygon with three sides. Circle all the triangles on this page.” After enough examples of this sequence, students come to believe that this is how math works. (p. 38)
Parents or teachers who have had enough of that kind of instruction and who want better for their children or students will find great value in Which One Doesn’t Belong?. Here they will find an approach that enables children to experience math as an engrossing challenge and a mental adventure.
And, in the accompanying teacher’s guide, educators will find tips on how to use the shapes book, an answer key that shares different ways to think about the shapes, and ideas for creating additional sets of shapes in this vein. This last perk in the teacher’s guide is a crucial value because, if Which One Doesn’t Belong? has a flaw, it is that the book ends too soon.
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