Saw this nice essay from James Tanton today:

What especially caught my eye was the 2nd question – What is the difference between familiarity and understanding?

A version of that question has been in my mind for a few days because of this comment on my blog from Paula Beardell Kreig:

A comment from Paula Beardell Kreig that’s been on my mind for a while

Also, Amy Hogan made a nice point when I showed this comment to Tanton today:

I wish I had a good answer to Tanton’s question. I’m don’t, and I’m not even sure where I fall on the mythical “traditionalist” to “reformer” line sometimes used as a measuring stick in the math wars.

One piece I go back to again and again that is sort of an indirect answer to the question, though, is the interview that Wild About Math did with Julie Rehmeyer. What’s always stuck with me from this interview is the story that begins around 31:30 and in particular the part beginning around 34:40 about proving that 0 + 0 = 0.

Julie Rehmeyer’s “Inspired by Math” interview

When I think about familiarity and understanding my thoughts always drift to this interview. Though I show my kids lots and lots of different ideas with the idea of trying to get them familiar with all sorts of different mathematical ideas, the part about mathematical understanding that comes in the “0 + 0 = 0” story is a good description of my long term goal in terms of building their understanding.