A fresh look at the new math days of the late 50s and early 60s

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?"
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."

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.

Levels of Looking at Learning: Cena's fragile knowledge about place value

Keith Devlin writes in his recent blog "How to design video games that support good math learning: Level 4":

"A major problem with video games, or more generally any mechanized educational delivery system, is that the system has no way of knowing what the player, or student, is learning. That a player who moves up a level in a video game has learned something is clear. Video games are all about learning. But all you can reliably conclude from a player’s leveling up is that she or he has leveled up. It could have been happenstance."

The student may not have learned anything significant. Appearances can be deceiving as Keith noted after he watched this video.

Keith continues:

"If you are like me the first time I saw this video, when you heard Cena’s answer in the class you concluded that she understood place value representation. She certainly gave the right answer. Moreover, to those of us who do understand place-value, her verbally articulated reasoning indicated she had conceptual understanding. But she had nothing of the kind, as the subsequent interview made clear." 

Let's review the video above. Here's what happens.

The teacher drew 49 stars and circled 4 groups of 10. When she got to the last 9 stars, she asked the class whether they should be grouped? 

"No," says Cena. "You got uhh one, two, three, four tens. You like put a four right there. And you have 9 stars left over so you put 9 right there." 

To which the teacher replies, "So does everyone understand?"

At that moment Marilyn Burns comes on and says, "Children's understanding is often fragile but what they know in one setting doesn't always transfer to another."

New scenario. Marilyn is now sitting next to Cena with a bunch of tiles on the table. Marilyn continues, "Put the tiles in groups of 10 and count out loud so I hear what you are doing." Cena counts out ten as she places them in a pile. "Can you make another pile?" Marilyn continues. "So how many groups of 10 do you have? "2," Cena responds. how many more do you have? "4". Do you know how many tiles you have all together? Cena responds with "Uh-uh (no)."

How is that possible? This is the same girl that so brilliantly knew to write that there were 49 stars by counting 4 groups of 10 with 9 left over. Keith didn't offer a possible explanation and neither did Marilyn. That, of course, was not their purpose; all they wanted was to make the reader/viewer aware that what you see is not necessarily what you get.  So that begs the question for me. What would you do: (1) as the teacher in the classroom to confirm Cena's understanding and (2) as a tutor ala Marilyn Burns?