I gave a quiz this week to assess my students’ understanding of functions, finite and infinite, and functional notation, including the application of a function to a subset of its domain, and it didn’t go well. None of the students finished on time; they all thought it was too hard. As we push deeper into formally defined structures of math the students’ intuitive understanding of what is being asked by a question is beginning to fail them. We’re reaching that point where the logic of the vernacular and the comfort of counting numbers is impeding, rather than enabling, their interpretation of more abstract mathematical notations and structures.
This week we looked more closely at the notation and structure of relations. Every function is a relation, but not every relation is a function. Both are subsets of cross products, but there are rules that functions must follow that relations can ignore. (I sense that there might be an analogy here with life but I’ll leave that for another time.) What was a highlight of the week? A student asked me about the validity of using mathematical induction (counting) to prove that two general sets had the same cardinality. And after a discussion he understood that it wasn’t quite good enough since the sets were not constrained; they might not be finite and they might not be countably infinite, they might just be uncountable. What was a low-light of the week? I made an error in writing one of the questions on the quiz. No matter how many times I reread a quiz, errors slip through. I also find this true in writing emails, or proofs, or papers, or programs, or blog posts, or anything at all. I remember rewriting a program to substantially reduce the size of an internal data structure—that was at a time when computer memory was much less abundant than today—and at the same time breaking, unintentionally, with a last-minute change, the reporting function of the same program, causing it to write out all messages in duplicate. When I went to demonstrate the savings in memory to the quality control group it was hard to convince them of the validity of the savings when everything they asked the program answered twice.
One other reflection on teaching. Reflection on teaching is hard work. But now I am reflecting on reflecting on teaching. Is there no end to this? Countably or unaccountably so?