For one week in June, a group of educators gathered at the Stony Point campus for the â€śConstructivist Classroomâ€ť graduate course offered through VCUâ€™s Division of Community Engagement. Instructors Pam Oken-Wright and Marty Gravett led participants in discussing and experiencing tools for co-constructing understanding through hands-on, collaborative research. Using the constructivist educational philosophy of Reggio Emilia as a springboard, class discourse ranged from adopting new practical approaches (e.g. documentation, action research) to deeper questions about the implications of constructivism within different cultural contexts and across the PreK-12 spectrum.
I left the class with a renewed enthusiasm for a constructivist approach to education. Only a few days later, I had an opportunity to apply this new understanding of how we learn, the importance of multiple means of representation, and of that critical â€śtossing of the ballâ€ť among individuals as understanding and meaning are co-constructed.
â€śShe just told us what pi was â€¦â€ť
A few days after the last meeting of our workshop, I was working with a rising sixth grader on geometry. A bright, conscientious student, she was eager to tell me what she knew about circles - and in particular about pi. â€śHow did you find out about pi?â€ť I asked.
â€śMy teacher told us it was 3.14.â€ť Her answer did not surprise me. â€śShe said it was 22 divided by 7.â€ť I decided to let this common misconception slide for the moment and continued to probe. â€śI wonder how she knew that. Why 22? Why 7?â€ť The student drifted a bit. I could tell she was losing interest quickly. She shrugged her shoulders and said, â€śShe just told us what pi was.â€ť
Grabbing hold of the drinking glass in front of me, I asked, â€śWhat if we measured the circumference of this glass?â€ť
â€śThen we would need to multiply it by pi,â€ť the student offered, her voice fading.
To her surprise, I gave her a tape measure and asked her to follow through with her plan. After a few awkward attempts, she measured the circumference of the glass: about 26.5 centimeters.
â€śYou said something about multiplying the circumference by pi. Is that what you want to do?â€ť
â€śI donâ€™t know. Is that what I should do?â€ť She scrutinized my face, looking for that teacherâ€™s hint of â€śrightâ€ť or â€śwrongâ€ť.
â€śI donâ€™t know,â€ť I said. My tone conveyed both neutrality and deep interest. â€śWhat would that do - - multiplying the circumference by pi?â€ť
â€śWell if the circumference is the perimeter, then thatâ€™s the perimeter. We donâ€™t need pi now.â€ť She furrowed her brow. She knew that circumference had something to do with a funny Greek letter. After all, she had memorized two formulas (both with pi in them) just a few months ago. I asked her what other information she could get about the circle she held in her hands.
â€śI could measure the diameter.â€ť She did so and recorded â€ś8.5 cmâ€ť next to the measurement of the circumference.
â€śWhat do you notice?â€ť This question was all that was needed. She saw it in an instant.
â€śThe diameter times pi gives you the circumference!â€ť I asked her write down her formula and then reconsider it using the idea of radius.
â€śWell, the radius is half of the diameter, so itâ€™s â€¦ pi times the radius times two!â€ť The excitement was contagious.
Our time was up, but I encouraged her to go around the house and measure circles of all sizes, recording the circumference and diameter each time. What might happen? Would she see the same relationship between diameter and circumference? Could she prove it worked every time? Her passivity seemed to turn into a sense of ownership and competence as she learned to trust her own mathematical instincts. She had figured it out for herself.
Her investigation took much longer than my saying â€śTo calculate the circumference of a circle, multiply the diameter by 3.14,â€ť but I feel the experience was so much richer, so much more meaningful. I thought about one of the central questions of the workshop: what possible understanding can we expect children to construct when we â€śjust tell themâ€ť?
My use of the word â€śconstructâ€ť is not accidental; a constructivist approach to teaching and learning is quite different from the traditional â€śtransmissionâ€ť model many of us experienced in our own early education.
But on that summer afternoon, I thought about the work we had done in our class. I resisted the urge to correct and â€śfill upâ€ť a young mathematician with the abstractions I had been given so many years ago. Why not take her ideas seriously? Why not multiply the circumference and pi? (She seemed somewhat startled that I allowed her to choose a path of inquiry!) I trusted in her ability to make sense of the world, of dimensionality and shape.
When I saw her again, she had clearly been thinking deeply about pi. â€śI think pi goes on forever because â€¦â€ť she fought to articulate her moment of brilliance, â€ślike â€¦ because â€¦ well itâ€™s a circle, not a hexagon or octagon or something easy like that. Itâ€™s a circle.â€ť
Archimedes said it best: Eureka!