Arithmetic-Part I

Arithmetic-Part I.



First, I am going to say why I think I have any standing in a discussion of teaching and learning. I went to school every fall from 1941 to 1995 as either a student or a teacher. During this time I was always learning outside of school, for example, starting with the 5th Grade, building model airplanes, sleight of hand, wood sculpture, motorcycle mechanic, classical and bluegrass guitar, banjo, writing, producing and directing video plays. I not only observed how my students learned, I observed how I learned...I observed how everybody learned.
One thing I have noticed is that there are at least two ways to approach learning something. One method is just fooling around with the topic. I have a friend who got a new computer and he would sit at it and “fool around” trying to do this and that. Eventually he got the hang of it.
I learn when I have a need. When asked how to use a PC I say, “Put it up in the closet until you have a problem that needs a PC’s help. Then take the PC down from the closet and read the instruction manual to see how to solve your problem.”
I am going make up a problem to act as a framework to hang the introduction of natural numbers on. My description, which follows, of how I use the problem is a framework I use to present my ideas of what should be included in the introduction to natural numbers.


I start with the problem of, say, telling Sam how many sheep I have. (I started my calculus book with the problem of trying to describe the motion of a falling rock.)
If my herd has ten sheep or less I can use my fingers to tell Sam how many sheep I have. If I’m too far away for Sam to see how many fingers I’m holding up, I’ll have to yell the information to him. This means that I’m going to have to have spoken words for 0 through 10. If I want to write my cousin in Boise how many sheep I have, I’m going to have to devise written symbols.
I’m going to suppose that words and symbols have been assigned to 0,1,2,3,4,5,6,7,8,9,10.

At this point there are philosophical problems. 4 is a symbol that represents certain groups of sheep. Which groups of sheep? Well, those groups of sheep that have...How do I tell you what four means? If I want you to put the same number of sheep in a pen as I have, how do I let you know what I want? I could tell you to put the same number of sheep in your pen as there are stars in ****. Then I could slide into defining cardinal numbers.
I could but I won’t.
What am I trying to do? I’m trying to make up a “just so” story that explains the evolution of numbers. I would encourage my students to make up their own creation story for counting numbers. Creation stories aren’t about deep philosophical questions, they are about a god creating everything and man in particular. Creation stories aren’t long; how long can it take God to create everything? The interesting story is what happened to man after he was created.



(The Genesis version of creation has the advantage of brevity, the evolution version drags on interminably.)
Some mathematician said that God created the natural numbers and the interesting story is what happens after they were created; or something like that. Landau didn’t number the pages in his book on number theory until after he had introduced the natural numbers.
Number is one of those concepts that seem self-evident but hard to define exactly what you mean, like “force”. If we were standing on a street corner talking and you asked me what four meant, I would probably hold up **** fingers and say, “This many things.”
Somehow I need to make you aware of the idea of a group of distinct objects. What if we lived someplace where everything was continuous? What if there weren’t any discrete objects? How would I tell you what four meant then? Would natural numbers even exist? If there is nothing to count, why would counting numbers arise?

I would and did give my students a rap like I have just written. I don’t think of the rap as wasting time. When I was learning how to ride a motorcycle across the desert, I also learned how to work on an engine, I learned that there is no place on an M/C engine that is hot enough to light a cigarette, I learned about the diminutive woman who slid her Harley 74 under the tractor trailer. I read motorcycle magazines and hung out at motorcycle shops. I learned the “folk lore of the tribe”.
I think “folk mathematics” should be taught in a mathematics course.

I assume the student already has the understanding that 4 denotes the number of stars in ****. But that is just an assumption I make. There are viable societies that take little interest in counting things. If a student lacks such understanding, teach him.
There was a movie where a high school English (I think. Maybe it was History.) teacher is in the lounge complaining that her students don’t know how to read. And another teacher says to her, “You’re a teacher. Teach them how to read.”
You have to teach the students in front of you. I don’t understand the point of insisting on teaching algebra to kids that don’t understand arithmetic. It’s like teaching to students in a language they don’t understand.

Back to the sheep.

I can express counting numbers greater than 10 by using a hand motion to indicate the number of 10’s I have and using fingers to give the number remaining. So suppose I have three groups of 10 and seven left over.
If I have fewer than ten tens of sheep, I can use the symbols I already have to express the number of sheep in my herd.
I can say that I have three tens and seven more sheep. I introduce the symbol + to indicate that I’m combining three tens of sheep to the seven sheep; I have 3 tens + 7 sheep.
I can see that the expression “some number of tens” is going to appear a lot. Three tens would be 10+10+10 but 10+10+10 isn’t much simpler than saying “three tens”.
. Here we run into what I consider is a major problem in teaching mathematics. We think and speak linearly but mathematical ideas often come in parallel. Here multiplication is needed for three tens (3 x 10) before we have defined multiplication in general. When we get to whole numbers bigger than ten tens we will need exponents. I won’t want to keep writing 10 x 10.
Well, I introduce concepts as they are needed. There seems to be a tendency to invent things for some possible future use. But we don’t really invent, we use hindsight to know what is going to be needed later.
I hate, well, I don’t really hate, slick cleaned up proofs. The first time I read such a proof I can be heard muttering under my breath, “Where did that come from? Why did he try that? What in the world is he doing?”
Why do we learn long complicated proofs in graduate school? Why are they on our quals? The only time I used the proof of Fubini’s Theorem was when I taught it.
We learned those theorems in order to study the technique of proving theorems. In my theorem proving days I would try to present a proof where each step was a natural consequence of the previous step. The “slick proof” leads to the belief that mathematics is “a bag of tricks”. (Now there’s a phrase I really do hate.)
In classes that were small enough to allow it, I gave an oral final. The students would have to come to my office and put a proof on the board that needed previous results to be proved also. They couldn’t use notes and would have to answer any questions I might have.
In a geometry course for high school teachers the theorem was to prove, in hyperbolic geometry, that the sum of the angles of a triangle was less than or equal to 180 degrees. In Advanced Calculus it was the Heine-Borel Theorem. They each take about an hour to prove.
Since most students tried to memorize the proof, they were doomed to failure. I would let them get about 5 min. into the proof and then ask them a question. It was like when I stopped in the middle of a song I had memorized, I had to start from the beginning. And that’s why musicians don’t memorize songs, they grok songs.
Since I knew they were going to fail the first time, I gave them as many tries as they wanted with no grade penalty. I wanted them to get some kind of gut feeling that you didn’t use tricks to prove theorems, that there was method to the madness. I saw no point in punishing them for trying to learn what I wanted them to.
When I proved theorems in class I would start with the conclusion and ask, “Why is this true? What would make it true?” And answer, “Well, if this were true then the conclusion of the theorem would be true. It’s pretty easy to see that this implies the conclusion. What would make this true?” And then I would find that which easily shows this to be true. In rather natural steps, no tricks, I continue until I reach a first cause that everybody agrees is true. Since each step was proved, I have a proof of the theorem that is probably closer to the way the first person who proved it went at it. I haven’t shown them the only way to look at a theorem; I have shown them one way.
The interesting thing was that the kids did it. Some would take five or six tries before they did, but they did it. After an unsuccessful attempt I would talk to them about what they weren’t getting so each try took an hour and the process was time intensive. My teaching techniques tended to be time and labor intensive so they never caught on with my colleagues.
In the graduate school I attended, finals and quals seemed to consist of randomly chosen theorems and problems. It was clear to me that memorization was not a viable technique in preparing for these exams. Instead I asked myself, “Why is this theorem true; what makes it true? Why is the next step an obvious step to take? Where does this theorem fit in the scheme of things? Why are we proving it? What follows from it?”
If I was going to make it in graduate school I was going to have to see why theorems are true.
I say this in hind sight. I started looking at the learning process this way in junior high and I have no idea why I started, I just did.
I started Jr. High in Cheyenne and I started rewriting notes. I wouldn’t write anything down unless I could explain it and the picture in my mind was teaching it and having to answer any question I could think of. “But what about...? What does that mean?” I would then write the answers to my question in my notes.
I tried to pass this along to my students.
So I am going to introduce as much multiplication as I need. When I get more sheep, I’ll deal with bigger numbers.