Tuesday, May 12, 2009
Remarks About Numbers
Numbers, Sort of I would like to talk about numbers and suggest that they should be talked about in the classroom and at the dinner table. What about the number 1? It‘s a symbol that conveys information; let‘s say from Joe to Sam. The information to be passed is that Joe has a single child. Here I run into difficulty. The phrase ‘single child’ means the same as ‘one child’ so I‘m not really defining anything. How do I tell a person what ‘one’ means? If both have sight and they are close to each other, holding up a finger might work. For the two guys on sight, this works but I am faced with explaining what ‘holding up one finger’ means. Having sight and being there is a big help in two people coming to agreement on what ‘one’ means. Some kind of sensory communication is needed. (See Johnny Got His Gun by Dalton Trumbo.) Let‘s suppose that Joe and Sam have the usual five senses and that they agree that they can convey a certain kind of information by holding up their fingers. A thumb down might mean that they don‘t have any of the objects in question. They may also decide that a thumb, forefinger and pinkie convey the same information as a forefinger, middle finger and ring finger. They can work out sounds that will designate different collections of fingers. This would be useful if Sam couldn‘t see how many fingers Joe was holding up and had to yell the information across a meadow. If Sam and Joe live in different towns they can devise marks to put on a clay tablet that will stand for different collections of digits. The Romans developed a pretty good system to convey the kind of information that I‘ve been discussing. Let‘s call symbols that describe how many objects are in a collection of objects, numbers. Or maybe a number represents a property of a set. I remember 1st grade as the place where symbols were attached to small sets. Here memorization was necessary since I, at least, saw no reason why 4 denotes **** objects. Maybe it does to an Arab since it is an Arabian system. Most people, however, agree on the meaning of number to some extent. I can‘t think of any cultures that don‘t agree on what ‘1’ means. 0 works ok for a place holder. 2056 means that there are no hundreds. 0 works ok to say that there are none of something. I have zero cats means that I don‘t have any cats. These are not philosophical ideas. Nothing deep here, whatever nothing means. I think one should start with the everyday meanings of numbers and then go to more abstract concepts. When enough is known about numbers, abstraction is natural. There is an opinion that if a body of material is presented logically and consistently, a student should be able to understand it. The New Math was a result of that opinion. It is not that the New Math is inherently a bad idea but it requires people who can teach it, which requires that the teachers know it. A miss is as good as a mile. Zero, one and two are concrete but at a certain point we come to ‘many’. I take eight pills from one bottle. When I pour them into my hand, I look for four and four or five and three, not the whole eight. I can‘t glace at a class and tell if there are 17 or 18 students present; I am in the realm of many and I have to count. I was giving a test to over a hundred students and to pass the time I counted the number of students present. I counted three times and got three different answers. At the end of the test I knew there were more than a hundred. The number of students had become an abstract concept. When I grade tests I first alphabetize them, then count the number of students for each letter and then add. I am fairly confident of this number; but not absolutely confident. My point is that in The Real World most numbers are to some degree abstract. I remark that a number can be partially abstract. I can usually tell if a number is big or small and this description is concrete. When it is announced that the attendance at the Super Bowl is 104,368, this number is not correct. If it were possible to count the house, it would probably be more than 100,000. The number on a test is an example of a number that is mistakenly thought to be exact by both teacher and student. When I was a new teacher I would give points to each problem on a test and add them up for a final score. Then I would decide where to draw the A, B, C, D and F lines. I would end up with a 79 C and an 80 B. This is ridiculous. A C is a ridiculous grade. There is the C where the student works 7 out of 10 problems correctly and doesn‘t answer the other three. And there is the ‘Partial credit C’ where the student can‘t answer any questions but somehow scrapes up enough points for a C. It is more reasonable to give A or B to the student that gets it, D and F to the student who doesn‘t; no Cs. I finally stopped giving numerical grades on tests. I could tell a B paper from a C paper and so forth, so I just put the letter down. In upper division courses, which are usually of a reasonable size, I would only accept correct homework and the students could redo it until they got an A. Some of my colleagues thought I should take the number of tries into account when grading. They were wrong. The whole point of the course was to get the student to learn and giving them an A when they achieved that end makes sense to me. Prisons are punitive, schools are supposed to be educational in a classical sense. So, what‘s the point of the Sam and Joe story? Mathematics presents itself in a pristine form and the student asks, ‘Where does this stuff come from? Why do we study it?’ I tried reading the Richie/Kerrigan book, The C-Language, to teach myself, with a little help from my friends, how to program in the C-Language. It‘s quite a thin book but I‘m sure all the information needed is in it. However I was unable to extract it and got stuck on ‘pointers’. I would say what a pointer is but I am still not sure. I realized that I didn‘t know why they were introduced; I didn‘t know what problem pointers were supposed to address; evidently pointers made something that was hard to do with earlier languages, easier to do but I had no idea what that something was; and The C-Language wasn‘t telling. Another example is the epsilon-delta process. Mathematics got along without it just fine until well into the nineteenth century. What undergraduate calculus student doubts that sin x is continuous? That the limit of x2 as x approaches 2, is 4? What did the epsilon-delta process make easier? Where was it really needed? Where was it used in a practical way? The thing about the epsilon-delta process is that when you see the point of it, the concept is obvious. I tried to present the epsilon-delta process in a more reasonable way in The Calculus: An Opinion. Actually, I don ‘t really talk about the process but try to present 18th mathematics in a way that makes the epsilon-delta process a natural thing to introduce. The point of my remarks about Sam and Joe is that mathematics should be talked about from the very beginning of a student‘s introduction to mathematics. What gap in the life of early humans was filled by adding numbers to their daily experience? Sam didn‘t tell Joe, ‘Remember these symbols because sometime you might need to use them. If I have kids someday, I might want to tell you how many.’ The larger the context a new idea can be placed in, the easier it is to understand the new idea. The context of mathematics can be started in the womb. Well maybe not in the womb, but shortly after emergence from said womb. I think that adults thinking that they know the best way to grow a context is incorrect, dare I say it, stupid. Numbers arise naturally. If they didn‘t, why study them? You don‘t start with Latin when you are teaching English at the elementary level. When a language context is big enough, you fit Latin in.