The Big Ideas of Algebra, Part Three

Four or five years ago, I wrote a couple of posts on the Big Ideas of algebra: Part One was published on November 30, 2008; Part Two on January 7, 2009.

It’s time for Part Three, isn’t it?

Fortunately, the inestimable Grant Wiggins has done my work for me. In his post entitled “4  big ideas of algebra — a reply to my challenge, and my response,” he discusses Patrick Honner’s version of the Big Ideas of algebra:

  1. algebraic structure
  2. binary relations
  3. the Cartesian plane
  4. function

Are we to object that these are noun phrases, not sentences? In the words of one commenter, an English teacher would say that these are topics, not theses. Shouldn’t a big idea be a thesis statement? Well, yes…and Wiggins brings up that very point. Nevertheless, each item could be considered shorthand for a full statement: as Wiggins says, “you may mean all of my phrasings to be implied.” The trouble is that you and I are likely to come up with quite different statements that each of us thinks is implied by the noun phrase.

Wiggins adds a new requirement: “I am looking for those ideas that are big – powerful and fecund – for both novice and expert.” It’s hard to disagree with that requirement.

I rather like the list of Big Ideas above, though I’m a bit unhappy about the second one. Wiggins suggests expanding it to “binary relations provide useful equivalences, based on complete reciprocity,” a rather obscure formulation that unfortunately doesn’t add much to the idea. Furthermore, what is special about binary relations, as opposed to any other kind? Unary? Ternary?

In any case, this is a start. Not the last word — just the start.

Categories: Math, Teaching & Learning