20150915 THE CURSE OF NOTATION:

Making connections between representations.

In SHNUMBERS, I disclosed that I am an oversimplificator. I am simply not smart enough to understand and appreciate the subjects like reform math. To me this reform means that teachers got science off their backs, and happily allowed themselves not to teach. The school is now free, and nothing in this democracy can bring it under control.

I wish it was so simple. In reality, school keeps itself employed imitating teaching. It looks like they throw at the students everything in hope something may stick (guess, that's how they understand the multiple embodiment principle). Knowing arithmetic is no longer enough, and, actually, is not even required. The students got to learn the silly stuff, call it common core or whatever. Or else.

"Standards" (another name for silly stuff meant to keep teachers busy) can be good for bad students. They may not know how to multiply, but if they learned to look up the data in the linear tables or recognize the patterns like 13579, this would be something.

School operates under an assumption that a standard American student can not learn multiplication. I don't believe in this, but I have something else to care. What if my children can? School make them learn as if they could not, or punish. Trust an immigrant, who can't even speak decent English: teachers still think that they are surrounded with cavemen, and believe they are the ones who carry the torch.

Even for the teachers, picking and getting the silly stuff falling from above is hard. The best defense I can think of as a parent is: aim high. Teach representations that may knock the reformers' socks off.

Here is a recent example. I have a plastic jar with 1000 1 cm cubes, like the ones in Dienes sets. We don't open it often. The cubes are too small, poorly made, and the colors are ugly.

Suddenly, I realized that there were 10 colors in the set. I built the table linking them to numerals and suggested to perform computations in colors. This did not make TLG happy - she hates everything new - but she quickly got the idea. By the following lesson she remembered them all (I did not) and called the colors by their numeral names quite confidently.

I challenged TLG to build the chart of additions in color. She agreed, but I had to do the gluing. Otherwise she wold probably get mad at me because the idea became too simple to her to be exciting.

So TLG was telling me which color to glue next, and we ended up with what you see on the picture. We could not finish it because I ran out of glue dots, but the progress was convincing enough.

Teaching at home, I present new concepts from different angles or try to derive them from what TLG has learned before (not necessarily in mathematics), but it's not multiple embodiments. Even at it's finest, the idea of multiple embodiment contradict all my learning and teaching experience. I believe the abstract ideas must be acquired on some familiar substrate (I use pegboards a lot). TLG's intellectual life has been revolving around those awful pegboards since she was 2 years and 4 months old.

This said, there is a similar problem, which American school with it's dependence on formal testing greatly amplifies. Once kids learn notation, they quickly forget what it means. They may know how to use it, but in the core their knowledge is absurd. I am constantly trying to break the ice, offering new disguises for well studied concepts. In the language of reform math, I want them to make connections between concepts and representation.