The Mathematics Content Standards for California Public Schools, adopted by the Board of Education in 1997, exemplifies my ideal of what government's role in society should be. The standards define "

**a student must look like at each grade level, but leaves it up to the teachers, the parents, the Sylvan Learning Centers, the ACI Institutes, and the indie math tutors (locally grown and organic, like me!) of the world to figure out "***what*"*" to actually go about creating those students.***how**One of the larger categories within the standards is something called "number sense". I interpret number sense as being able to understand numbers abstractly: what numbers mean, what they symbolize, and how they are related. A primary requirement for number sense, (1.0), states:

Students understand the relationship between numbers, quantities, and place value in whole numbers up to 1,000.

The sub-requirements are:

Count, read, and write whole numbers to 1,000 and identify the place value for each digit. (1.1)

Use words, models, and expanded forms (e.g. 45 = 4 tens + 5) to represent numbers to 1,000. (1.2)

I've been struggling with

*how*to teach Billy the concept of place value. He can identify the place value of each digit in whole numbers up to 1,000, per (1.1) but I know he doesn't really "get" it, because he's still struggling with (1.2). It's great that he knows how to identify the ones, tens, and hundreds place of a number, but it doesn't do him any good if he can't use this knowledge to get a "sense" of what the number actually means, abstractly.Today, I tried introducing a game involving different colored poker chips. I would give him a number and ask him to create that number using a combination of various green (1), blue (10), and red (100) chips. He started getting very fidgety after about 1 minute of gameplay. I was losing him, and we were both getting frustrated.

I decided to give up on the game when I realized that, while it was a functioning game, it wasn't functioning to teach Billy what he needed to learn. The poker chip game was teaching him to express abstract quantities through

*color*, when it should have been teaching him to express them through*number*. Hippies would argue that both are equally valid forms of expression, but I say hippies be damned. The reason we have standards is to ensure that, at some basic level, we can all speak a common language.I think next week, I'll replace the poker chips with monopoly money. I figure, if the kid is gonna gamble, it's better he learn to gamble on something that the government is almost guaranteed to subsidize, to prop up, to perpetuate as a false promise--an opiate of the masses.