WHAT IF
What if we used our imaginations to learn mathematics? Certainly there is a place for imagination when learning to read and write, or create artworks, but mathematics? Isn’t math just right or wrong? We wondered if we might not use our students’ imaginations to learn more about shapes. We asked our students what they thought might happen if we no longer had squares or rectangles. What would our world look like? What would change? What happened was that we learned more about our students mathematical thinking than you might imagine.
The Mathematics
Geometric thought in the primary and early junior grades in Ontario involves fairly lowlevel thinking. Students are to identify, describe and compare shapes that they encounter in the classroom. For the most part these are going to be in the form of blocks  attribute blocks, pattern blocks, 3D blocks, possibly tangrams. Drawings of shapes are posted all around, and we might hunt for shapes in the objects around us. There is no requirement to think deeply about shapes or to reason about why everyday objects are made with certain shapes. Certainly there is no thought given to how everyday objects might change or even improve if we changed their shape. It was into this arena that we marched with our students. Armed with our imaginations we were prepared to wrestle with our question.
The Students
The students were able to approach thinking analytically about shapes in this manner with flexibility and creativity. Initially, they thought that if an object was made with a rectangle or square it wouldn’t exist. “No rectangles? Does that mean there would be no TVs?” Amid the gasps of horror from the students, we reassured them that there would be TVs  it was only that TVs would have to change shape. We asked them to imagine what shape a TV might be if there were no rectangles. What shape might it be instead? How would having a TV in this shape be better or worse?
Once this initial block was overcome, the students opened up, and ideas for all sorts of everyday objects flowed. Different shapes of TVs surfaced using trapezoids, octagons, ovals, even hexagons. Octagons and ovals were able to give enough space to the screen so that it could still be as close as possible to being a rectangle. The trapezoids were useful because one on top can be a screen while the other underneath can be a stand.
Triangles were useful for redesigning rulers, dressers, and beds. When asked, the students were all able to give reasons for their choice of shape. Triangular dressers are useful because you can put small or short clothes in the top drawers and long clothes in the bottom drawers.
Another idea that came up was the tiling of floors with different shapes instead of squares. This not only works, but the possibilities for creating interesting patterns increases. Clearly, the students were not only thinking about the shapes and their geometric properties, but also their functional and aesthetic properties as well.
The Teachers
The teachers had an interesting role during this lesson. The first important role they took on was that of a scene setter. In one class the rationale for this banning of squares and rectangles was a fairy tale land with a crazy king who just banned rectilinear shapes. In the other a story was coconstructed with the students about a land of shapes where a disagreement breaks out between a rectangle and an oval leading to the rectangles and their cousins the squares leaving shape land. In both of these cases, setting the scene allowed the children to use their imaginations. “It’s a story, right? We can use our imaginations. It’s not like we’re really doing math is it?”
After setting the scene, the teacher’s role shifted to that of a facilitator, circulating to probe the students’ thinking and to prompt them towards thinking about the changes in shapes and how they would affect the different objects. One student really liked the idea of oval books, but knew that it would be difficult to have a big enough binding to keep the pages together.
Another had two hexagons for her TV. You could either have both hexagons for one big screen, or use one for the TV show and the other for subtitles, which she said would be useful for people new to the country who did not yet speak English. This is thinking beyond the mathematics curriculum.
Implications for Practice
This was a very interesting lesson to do. Like a sled at the top of a beautiful, snowy hill, it needed a bit of a push to get it going, but once on its way down the hill, speed and excitement picked up apace. It is certainly possible to allow, even encourage, students to use their imaginations in mathematics class. This set of students not only knows and understands that a rectangle has four straight sides, four right angles, and parallel opposite sides. They know that it is a useful shape in design allowing for a flexible area for viewing, filling up, stacking and so on. Even more they know that other shapes can fulfill the roles of rectangles and squares, and they can think about the practical and aesthetic advantages and disadvantages of these changes. Without engaging their imaginations this would not have happened. Imagination does have a role in the mathematics class. With further exploration, we hope to find other areas of mathematics as open to imagination as was this part of geometry.
The Math Performance Festival is funded by the Imperial Oil Foundation, the Fields Institute, Research Western, the Faculty of Education at UWO, and the Canadian Mathematical Society. A project by George Gadanidis (UWO), Marcelo Borba (UNESP, Brazil), Susan Gerofsky (UBC), and Rick Jardine (UWO).
