In Robert Heinlein`s sci-fi novels, one of the recurrent features (besides nudism) is that the hero is typically a math prodigy. I was reminded of this when reading that the State of California, like most states in recent decades, has put a lot of effort into coming up with “academic content” standards to delineate precisely what each public school student will learn. In fact, teachers are supposed to write the Standards on the classroom whiteboards so that the students can make sure that the teachers aren`t slacking off and leaving out anything that is officially mandated.
Unfortunately, the mathematicians who made up the California Mathematics Content Standards seemed to assume that the young people of California are characters from Heinlein novels.
Here, for example, is the very first of the 25 items in California`s Algebra I content standard (to put that into perspective, LA public schools students must pass Algebra I, Geometry, and, beginning this fall, Algebra II to graduate from high school). This is what California 8th or 9th graders are supposed to learn on roughly the day after Labor Day when they first begin Algebra I. (Although in many cases, they are 10th, 11th, or 12th graders who are trying to pass Algebra I for up to the fourth time.)
1.0 Students identify and use the arithmetic properties of subsets of integers and rational, irrational, and real numbers, including closure properties for the four basic arithmetic operations where applicable:
Now, I`m sure most of you are saying, well, ho-hum, of course everybody knows the closure properties for the four basic arithmetic operations. And how can students move on to studying Abelian closure without being introduce as soon as possible to simple closure?
Unfortunately, I`m not a Heinlein character, so to be honest, my eyes glazed over when I read that standard. With some effort, I`ve finally managed to focus upon what the words are, so I`ve been able to move on to trying to find out through Google what they mean.
The idea of `closure` is actually very simple. If you add together two whole numbers, you will always get another whole number. If you multiply two whole numbers, you will get a whole number as a result. So we say that whole numbers (integers) are `closed` under the operations of addition and multiplication.
What about division? Well 12 divided by 2 is 6, which is a whole number, so in this case we get a whole number result. But 12 divided by 5 = 2 and 2/5, so now we have moved out of the field of whole numbers. If we divide two whole numbers we cannot guarantee that the result will still be a whole number. So the set of whole numbers is not closed under the operation of division.
Positive whole numbers are closed under addition – you always get a positive whole number in the result. But they are not closed under subtraction, since, for example, 4 – 9 = -5 and -5 is not a positive whole number.
To decide whether a set of numbers is closed under some operation or other, look for cases where the result is no longer in the set you started with.
In the case of real numbers, which include positive, negative, fractional, and irrational (like sqrt(2)) numbers, the operations of addition, multiplication, division and subtraction are all closed (apart from division by zero which is not defined). But taking square roots is not closed because if, for example, we try sqrt(-5), we no longer get a real number as a result. In fact, we have moved into the realm of complex numbers.
Well, that`s rather interesting … but is this level of abstraction appropriate for the first thing taught to public schools students? In the Los Angeles Unified School District, less than one out of ten students will score 500 or higher on the SAT math test. What about the other 90+%?
I suspect that the mathematicians who dreamed up these standards wish that they had been taught like this in high school. They wouldn`t have been so bored if their courses had been geared at a much higher level of abstraction.
So, this is how they get their revenge on the assistant football coach who bored them so badly when he taught them Algebra I — by making him try to explain, on a hot day in early September, the closure properties of the irrational numbers to high school freshmen who add and subtract on their fingers.
It`s just another little victory in the endless war the right half of the bell curve is waging so successfully on the left half.