Finding Geometry and Algebra in Quilts

If you walk into Peg Clapham’s third grade class at the Healy school in Somerville, MA on Tuesday mornings, you will see children making geometric quilt designs and talking math. Clapham and her students are among the six classes participating in EDC’s Technology Enhanced Learning of Geometry (TELG) project, which is using a computer-based curriculum to study elementary students’ geometry learning.

“Geometry has been a neglected element of elementary math curricula” observes Principal Investigator and EDC Senior Scientist Dan Watt. “Often, students do little more than learn the names and properties of different shapes. We wanted to see what K–5 students knew about important geometry concepts like symmetry, transformation, similarity, scale, and visualizing in two- and three-dimensions, and how technology-enhanced curricula could help them learn.” The project is a collaboration among Watt at EDC, Richard Lehrer at Vanderbilt University, and Douglas Clements at SUNY Buffalo. Each investigator is studying a different angle on geometric thinking, and at slightly different age levels.

The EDC portion of the study uses quilt patterns to help students in grades 2-5 learn about geometric units, patterns and their repetition, symmetry, and geometric transformations.

Once they’ve created a core square (see diagram), students make larger patterns by putting together core squares in different orientations and positions. “As students work with their core squares, rotating and flipping them, they learn how to visualize different transformations,” observes project Research Associate Zuzka Blasi. “At first, students may need to manipulate the computer images or paper cut-outs of their core squares, but pretty soon even second graders are able to visualize the effect of a rotation or flip on the core square pattern.” Watt and Blasi were fascinated to discover that the students in the project not only gained a solid understanding of fundamental geometric concepts, they also abstracted their understanding to create rules describing systems of geometric transformations. Blasi notes that, “Pretty soon, students noticed that certain transformations had predictable effects, and that they could get to the same end-point in more than one way. For example, three turns to the right was the same as one to the left; an up-down flip followed by a sideways flip was the same as two turns to the right or left. After a while, kids didn’t need the concrete materials to figure out the transformations they needed to get a shape to look they way they wanted.”

Watt and Blasi encouraged students to use symbols and written instructions to record the steps they took to make their quilt patterns. For example, students illustrated equivalencies like those on the left.

Watt notes that some students were, in effect, developing algebraic ways of thinking about the geometric shapes. “Students were able to abstract general principles about transformations. This really surprised us. Even many of the second graders were able to think abstractly about the effects of different flips and turns. Students ‘discovered’ properties of the geometric transformation such as inverse, identity, and equivalence. These properties are identical to the properties of our number system that students usually don’t study until middle- or high school.

Blasi adds, “We were also surprised that there weren’t big differences between the second graders and the fifth graders in their understanding of symmetry and transformation. None of the kids we observed had much prior exposure to the geometry concepts we were investigating, so they were all starting from about the same place. Their learning grew naturally from the design process.”

She also notes that students learned in a non-didactic environment. “While the students felt like they were fooling around with the core squares to make interesting designs, they were actually learning a lot. Watt and Blasi see a lot of potential in research that uses design processes, like quilt making, to engage students in mathematical thinking. “We found that students were building their understanding of units, symmetry, and transformation as they were having fun making quilt patterns,” they observe. “And in this process, they were motivated to find shortcuts that represented very sophisticated geometric thinking. Students used technology—the computer-based ’quilt world‘—as a tool both for thinking about the transformations and for communicating their ideas with others. We think this study indicates that geometry could be pursued much more rigorously in elementary school, and it could be a context for introducing students to aspects of algebraic thinking, too.”