I was subbing in a 6th grade recently and the warm-up question for the math lesson (directly from the teacher's manual) was something like this: A certain state has chosen to use the following format for their license plates: a single letter, followed by 5 digits. How many different license plates could they make that start with the letter A? The students were supposed to write their answers on their individual white boards and then show them to me to verify their answers.
The first answer I got was 28. I was completely baffled. Then came the other answers 15, 59, 25, etc., etc. NONE of the answers was even above 100, let alone near the correct answer. When I told the class (of 29 students) that their answers were all way too low, they started guessing above 100. But their guesses were completely (to me) random.
I decided to give them a hint: if they could use only 1 number after the A, you would have license plates A0, A1, A2, A3, ..., A9 - for 10 license plates. With two digits, you would have A00, A01, A02, ... A99 - giving you 100 license plates. They still didn't get it.
Pedagogically, I was so baffled by their lack of understanding, that I missed a golden opportunity to ask them what their reasoning was. I wish I had asked. "My bad", as they say. But now I am left wondering how they could possibly have thought that 28, 15, 59, or 25 could be anywhere near reasonable. This was a charter school, where the kids had to be delivered to the building in cars by their parents every day and picked up at the close of the day. How could they possibly think that 28 license plates starting with A, 28 starting with B, etc., would be enough? There are nearly that many cars at that one school in one day.
I was left thinking that the thinking habits of those kids were pretty bad.
Then came the lesson. It was on the number of degrees in specific turns. Though this wasn't in the teacher's manual, I had them stand up and demonstrate turning clockwise and counterclockwise. That isn't nearly as intuitive was it was in the days of all analog clocks. Our digital kids nowadays don't seem to have quite as much familiarity with the rotation of the hands on clocks. So I had them practice. First we established that a full turn, either CW or CCW was 360 degrees. Then that a quarter turn was 90 degrees and a half turn 180.
As I expected, a lot of the students mixed up CW and CCW, until we had practiced quite a bit. What surprised me a little was that several students refused to participate at all. This is a fairly strict and structured charter school and the non-compliance was unexpected. I didn't make a big deal of it, however, since it was an unplanned part of the lesson. I was more interested in the fact that this was actually a fairly difficult exercise for 6th graders.
That was just about the whole lesson - that and a few word problems. I was not terribly impressed with the curriculum, but this is a curriculum that I am not terribly fond of, anyway, so I am not going to name it. I was more interested in the seeming lack of comprehension of the students. They could do the rote problems, but the applications seemed to baffle them.
I would love to blame it on the curriculum, but I am not so sure that that is the problem. I have seen similar things with other curricula. What seems to me to be more evident is that kids are not particularly interested in making sense of things. They are willing to learn the arithmetic procedures - essentially just memorizing "how to do the problems", but they have very little (no?) interest in understanding why things work as they do. How have we gotten such disinterested kids? Was it always this way?
I remember hundreds of years ago, when I was a child, that I wasn't particularly interested in math. I could do the problems reasonably well, but the mathematics behind the arithmetic wasn't compelling to me. I don't remember if it was taught. I just remember that I was good at math, but, to me, that meant that I was good at arithmetic.
Then came my own 6th grade. The math teacher taught us about number bases - the reason behind carrying and borrowing when you get to the number of the base. It was, for me, a whole new ball game. Math became much more interesting. But that kind of enlightenment was sporadic. I remember asking several times in calculus classes how calculus was used, but I usually got either completely useless answers (It is used in everything!) or vague answers (In physics it is used to derive the laws of motion).
So now, I am wondering: it is counterproductive to try to explain mathematical reasoning to young children? Perhaps they just need to learn to do arithmetic very well. Then, in middle school or high school, with a bit more mature brains, they should take a class called number theory - and learn the reasoning behind the algorithms.
It is scary, though, to think that kids have so little practical understanding of math that they can't see the unreasonableness of the answer 28 for the number of license plates starting with A and having 5 numbers following the A.