Earlier this month I participated in a fascinating two-day seminar on The Big Ideas of Algebra, taught by Deborah Hughes-Hallett and sponsored by Teachers as Scholars. Although I undoubtedly talked too much, I figure that that was because I had a lot to contribute. Nevertheless, I learned a number of valuable things from the seminar, and they will be as useful both at Weston High School and at the Crimson Summer Academy. Mostly this was a matter of focusing my attention on things I already knew and believed but wasn’t thinking much about. In particular, I am now convinced than ever that we need to pay a lot more attention to symbolic literacy (understanding the symbols and combinations of symbols used in algebraic expressions) and to the distinctions among expressions, equations, and functions. Because of my linguistics background I have always believed that one important way to look at mathematics is that it is a language — in fact, that’s probably what got me to make a smooth transition from linguistics to math — but I hadn’t thought enough about the implications of that point of view when teaching math.
In addition, among the other matters we talked about in the seminar, we discussed the assignment of partial credit for work in solving a problem — more on this later, but it definitely reflects one’s views on what the big ideas are — and whether the study of algebra is distinct from (and prior to) the study of functions. More on that later as well.