Joe is going to paint his bedroom: walls and ceiling. The room is 11' by 13' with 8' ceilings. There are three windows that have dimensions 4' x 6'. How much area does Joe have to paint?This is typical of the sort of problem us math types come up with, and probably why students hate word problems. More fun than solving such problems is figuring out what's wrong with them. We had a good laugh because Joe doesn't have a door to his bedroom! I guess he goes in and out of the window.
It was interesting to watch the way Epsilon thinks about the problems. Clearly the teachers have a big job on their hands, and the simplest route to maximizing success on tests is to teach deductive processes through repetition. That is: there's a right way and a wrong way to do problems. Ironically, this can act like a rut on the test questions, which are multiple-choice. There were usually shortcuts we could employ that were quicker than using the deductive method. In case that dry description hasn't put you to sleep already, continuing the example above:
It's easy to see that the average wall length of Joe's bedroom is 12', so the perimeter of the room is 48'. This is quicker than the cookbook method of adding up 2x11+2x13 on the calculator. Then we have to multiply by 8 feet high walls, which is doubling three times. Round 48 up to 50, double once: 100, twice: 200, thrice: 400. Add in the approximately 12x12 ceiling to get 544, and subtract out the three 24 square foot windows (round up to 25, to get 75 square feet), subtract to get about 470 ft^2. This is close enough to find the right answer on the list of possible choices. And it passes the "smell test" because there are five surface to paint, each in the neighborhood of 10'x10'.
Part of my job was to let the kid know it's okay to take shortcuts to get the answer. She's at that stage I remember well, where her intuition tells her what to do, but she can't articulate it. Trusting that intuition will lead you astray occasionally, but in interesting ways. Not trusting it dooms you to reliance on formulas.
In terms of content, I was impressed by the number of topics addressed. Seventh grades here are learning far more than I did in seventh grade math. I don't have enough information to know if the standardization of the curriculum in math is ultimately good or bad. Math at this level is probably the best place to standardize because of the deductive nature of most of the work. Whether or not it works in English is another question.
*That's not really her name; it's a tip of the hat to Paul Erdős, who called children "epsilons" because in real analysis proofs that letter is often used to denote a very small quantity.