Last week, in Fundamental Understanding of Mathematics XXVIII, we talked some more about fractions, but I really felt I was just scratching around at the surface of the ideas involved. What to do to get organized?
Have you ever had a speaker come up to the podium, start talking, and it seems like they've been watching you work for the past year? They know, to the letter, every single issue you've faced, which are important, and which are sidelines? A couple of years ago, I went to a Math Seminar at UCLA, and in walked Professor Hung-Hsi Wu, and he did exactly that.
So, recalling that Dr. Wu had a very good handle on what goes on in K-12 mathematics classrooms, and that he writes a great deal about what the problems are, and proposes possible ways to fix those problems, I decided to visit his page at Berkeley, where he teaches, to get some insight into what I could talk about here.
One of Dr. Wu ideas is the concept of math teachers being mathematical engineers. That is, we don't actually do professional mathematics the way professional mathematicians think about it, we don't develop new mathematical ideas, prove new theorems, solve hitherto unsolved problems, rather we take mathematical knowledge and customize it and make it useful to non-mathematicians, namely, math students. He calls this mathematical engineering, an analogy to mechanical engineers who take theoretical mechanics and material science and make tools or machines, or chemical engineers who take chemistry and apply it to develop plastics or dyes or window cleaning sprays.
When I was in seventh grade, the University of Chicago loosed the School Mathematics Study Group's work upon the nation: the New Math. One of the things I had the most fun playing with was number bases. We took a look at base eight, if I recall correctly, and got a handout about the duodecimal system, or base 12. I remember making addition and multiplication tables in base 4, to try to memorize them and become fluent in base 4 arithmetic. Dr. Wu points out that working with number bases is a good way to understand place value.
Let's suppose, then, that we are a robot with one of those robot hands: one finger and an opposable thumb. Like a lobster claw. Having only two fingers, we only invent two symbols for numbers: 1 and 2. Being clever robots, or lobsters, we also invent a number for nothing, so we have three symbols to use: 0, 1, 2.
Now we have a problem if we have a large pile of things to count. Initially, we try to solve that problem by simply reusing the symbols:
0,1,2,
0,1,2,
0,1,2,
but if the pile is big enough, we lose track of how many times we have repeated our three numbers. To remind ourselves, we can put a second symbol off to the left of our count, to keep track of the repetitions:
00, 01, 02,
10, 11, 12,
20, 21, 22,
Now we are sure of where we are in the counting process, and if the pile of things is still larger, we simply repeat our earlier strategy: reuse what we've got
00, 01, 02, 10, 11, 12, 20, 21, 22,
00, 01, 02, 10, 11, 12, 20, 21, 22,
00, 01, 02, 10, 11, 12, 20, 21, 22,
but the 12 in the first row looks too much like the 12 in the second row. To tell them apart, we again reuse a familiar strategy: mark the repetitions with a digit to the left:
000, 001, 002, 010, 011, 012, 020, 021, 022,
100, 101, 102, 110, 111, 112, 120, 121, 122,
200, 201, 202, 210, 211, 212, 220, 221, 222,
Now we can count all the way up to 222. If we were going to count past 222, our next round would start with the symbol 1 off to the left: 1000, 1001, 1002, 1010, and so on.
We notice that, when we start counting a large pile, we don't know in advance how many repetitions we will have, so we don't know how many zeros to put in our very first number (notice how that first number grew from 0 to 00 to 000 as we kept counting.) But we also discover that we can simply not write those zeros (the zeros on the left) without losing track of where we are in the count: 12 can't be confused with 112 or 212 or any other number ending in the digits "12."
We check to see whether that time saving convenience of not writing zeros on the left side works with the zeros on the right side... but it doesn't. If we take 120 and leave off the right hand zero, it looks too much like 12. Since those are different numbers, we need to be able to tell them apart, so the right hand zeros must stay.
How do we know they are different numbers? Well, we are counting, after all, and we can represent counting by using a number line.
We can see that 12 and 120 are in different places on the number line, so they represent different numbers. But what are those numbers? Well, since we are using base three, they are 12 and 120. Who counted along the number line and thinks, "Aha! they are 5 and 15!"? We can do that, because, unlike our robot lobster, we have more than three symbols to use in our counting system. We have ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
Have fun in the comments.