Getting to Know a Triangle

One of the first experiences that convinced me the use of technology in mathematics education had enormous potential was interacting with dynamic geometry software. With programs like Geometer’s Sketchpad and Cabri, a user can create a 2- or 3-dimensional shape and then manipulate it. The very idea of having some control over a shape, rather than being restricted to looking at a static representation on a printed page, was intriguing to me. But what really blew my mind was not the freedom, but the restrictions on the manipulations. I could drag the corners and the sides around, but the way the shape responded would always maintain the defining properties of the figure.

Take a square, for example. If I used the software’s square tool to draw a shape, I would get a square, for sure. And then if I made a side longer, all the sides would get longer too. If I dragged a vertex, the sides might change in length and the orientation of the shape might change, but all four angles stayed right angles. It was amazing. Even though I was a 20-something holder of a B.A. in mathematics who felt she knew what there was to know about squares, the experience pushed my understanding of a square.

The whole idea became even more interesting when I learned about research showing that young children often struggle to apply the names of shapes beyond prototypical examples (e.g., Verdine, Lucca, Golinkoff, Hirsh-Pasek, & Newcombe, 2016). Young kids tend to think the triangle on the left — the one they see most often in the world — is indeed a triangle, but the shapes on the right are not.

triangles
When it came to 2-dimensional shapes, exposing kids to more and different examples of non-prototypical shapes seemed feasible to me without the use of dynamic geometry software. All sorts of triangles and rectangles, for examples, are embedded in everyday objects. Still, there seemed to be something special about manipulating a single example as opposed to seeing multiple examples.

Two pieces I’ve read in the last few weeks have helped me to reflect on why the manipulation of a dynamic triangle might be different and more powerful than examination of many different triangles. The first is a book that I’ve been meaning to read for some time: Seymour Papert’s Mindstorms. In it, Papert describes the LOGO programming environment and the ways in which it could be used to support the development of new ways of thinking and learning. This particular passage really struck me:

Working in Turtle microworlds is a model for what it is to get to know an idea the way you get to know a person. Students who work in these environments certainly do discover facts, make propositional generalizations, and learn skills. But the primary learning experience is not one of memorizing facts or of practicing skills. Rather, it is getting to know the Turtle, exploring what a Turtle can and cannot do. (Papert, 1980, p. 136)

The notion of getting to know an idea the way one gets to know a person brought me back to the dynamic triangle. The reason that manipulating a single triangle, rather than creating or examining multiple examples, feels like a more powerful experience is because the manipulation allows for a student to get to know what it means to be a triangle. They get a feel for what the entirety of the triangle category looks like, rather than deciding whether particular examples fall into it. I really love Papert’s emphasis on qualitative thinking, and thinking about it in relation to dynamic shapes was powerful for me.

The second reading that pushed my thinking on this issue was Stephen Kosslyn’s response to The Edge’s Annual Question in 2010: How is the Internet changing the way you think?” In his response, Kosslyn (2010) says, “The Internet has made me smarter, in matters small and large. For example, when writing a textbook it’s become second nature to check a dozen definitions of a key term, which helps me to distill the essence of its meaning” (5th para.). As we discussed this passage in one of my courses, we came to the idea that the ability to quickly and easily see many instantiations of something leads to a softening of your definition of that thing. Children appear to have a rigid definition of triangle when they only consider horizontal-based, equilateral examples as triangles. Manipulating a dynamic triangle, however, allows them to explore all the ways in which they must soften this definition to come to what defines a triangle mathematically. Examining multiple individual examples can work toward this, of course, but it is the all-ness that seems most important. With dynamic triangles, children can directly confront the properties that they think define a triangle, whatever and however many of those things there may be.

That’s what makes dynamic manipulation so powerful. Cool.

As with most connections I made on this blog, I suspect that in both cases, I am not the first to realize this. If you know of pertinent papers to read on these ideas, please let me know in the comments!

References

Kosslyn, S. M. (2010). A small price to pay. Retrieved from https://www.edge.org/responses/how-is-the-internet-changing-the-way-you-think

Papert, S. (1980). Mindstorms: Children, computers, and powerful ideas. Basic Books, Inc..

Verdine, B. N., Lucca, K. R., Golinkoff, R. M., Hirsh-Pasek, K., & Newcombe, N. S. (2016). The shape of things: The origin of young children’s knowledge of the names and properties of geometric forms. Journal of Cognition and Development, 17(1), 142-161.

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