Why does today's new math look like the old math repainted? Back in 1969 Morris Kline had a problem with what was called the new math at that time. Here's what he wrote in chapter 1 of his seminal work "Why Johnny can't Add."
"Evidently the class is not doing too well and so the teacher tries a simpler question. "Is 7 a number?" The students, taken aback by the simplicity of the question, hardly deem it necessary to answer; but the sheer habit of obedience causes them to reply affirmatively. The teacher is aghast. "If I asked you who you are, what would you say?"Technically the teacher is right. 7 IS a numeral that represents the idea of "7ness" meaning 7 of something: an abstraction. Though accurate it was way beyond the call of a beginning student's duty to need to contemplate such subtlety. Though a bit of a stretch sometimes this kind of distinction gets play time when discussing traditional pure math and the necessity for rigor like in the numeral/number distinction. Rigor has its time and place. But not when you want students to have creative adventures in discovering powerful ideas about math or even having students invent their own math. There is a big distinction between what a professional mathematician would consider "inventing new math" and what an average student might do to behave like a mathematician. I discovered that distinction when I personally discovered Pick's Law. Of course, I didn't invent it; it was already a part of mathematical lore thanks to George Pick and Hugo Steinhaus. What I did was recreate it from scratch and that adventure was very meaningful for me. I was now a "mathematician" actually doing math (albeit not inventing it) and certainly not just practicing ideas that the teacher insisted I know whether it was currently relevant or not. I was on a mathematical journey where I made up the path as I went along. The endpoint was defined, but I had to determine the details of the journey following clues along the way. This to me is the new "new math" not an extension with a fresh coat of paint but something all together different. Something that is lacking in most classroom enviroments. I will be sharing some adventures (examples of what I mean) that I've been thinking about in future blog entries.
The students are now wary of replying, but one more courageous youngster does do so: "I am Robert Sinith."
The teacher looks incredulous and says chidingly, "You mean that you are the name Robert Smith? Of course not. You are a person and your name is Robert Smith. Now let us get back to my original question: Is 7 a number? 0f course notl It is the name of a number. 5 + 2, 6 + 1, and 8 - 1 are names for the same number. The symbol 7 is a numeral for the number."