Dennis and I were talking about multiple representations. Multiple representations are one of the Big Ideas that wend their way through all our math courses. A table and a graph and a function machine and a mapping diagram are all multiple representations of the same thing.
But what’s that thing?
A function per se is an abstraction, a possibly infinite set of ordered pairs. The graph doesn’t represent the table; both of them represent something else.
It reminded me of a conversation with a Lincoln-Sudbury parent back in the mid-’70s. He had noticed a disparate array of interests of mine that showed up in the two courses in which I taught his son. These interests ranged from aspects of mathematics to linguistics to cryptography to cartography to musical notation and even to model railroading. “What did all these have in common?” he asked himself — and me.
“I don’t know,” I replied. “What do they all have in common? I’ve never thought about it.”
“Well,” he said, “after pondering it for quite awhile, I realized that they all have something to do with representations. What you’re really interested in is the different ways that abstractions can be represented.”
So a parent of a student helped me understand myself better. Maybe he’s right.