This was demonstrated to amazing effect in Shane’s Geometry I class. The group was winding up their current unit on Geometric Constructions, correcting the previous night’s homework and reviewing important concepts in preparation for an upcoming exam. The room thrummed with quiet activity as students asked for clarification or offered opinions on how the problems in last night’s homework might be tackled more efficiently.
Shane was just revealing to the class the algebraic expression that would have allowed them to discover a pattern in one particular problem when freshman Bethany Pelotte ‘14 raised her hand. The algebraic expression she formulated while working on the problem the night before was totally different from the one Shane demonstrated. And yet, it had given her the correct answer. This was of interest to both the class and Shane, and time was taken during and after class to discuss how Bethany had created expression and it was discovered that, while it wasn’t suggested in any text book, Bethany’s expression was perfectly designed to solve this problem. “Earlier this year,” said Shane, “we looked at patterns and talked about different ways of reasoning through and finding what those patterns could be. Bethany was doing inductive reasoning. She was seeing the pattern through trial and error and her knowledge of Algebra, and using that to come up with an expression to allow her to predict that.”
How long did Bethany spend on her expression? “Not long,” she said. “About fifteen minutes.”
After class, Shane confided, “What’s amazing about that – which is always amazing – is when students find their own ways of figuring something out. The fact that she used such a complicated algebraic expression to capture and make sense of her own reasoning. She started with the reasoning; the algebra just helped her to communicate it. I thought it was pretty impressive.”