Tuesday, May 12, 2009

Thoughts on Arithmetic-Part II

Arithmetic-Part II

4•10 means to add four tens, n•10 means to add up n tens. If I have (4 tens + 5) sheep I can write 10+10+10+10+5 or 4•10+5 sheep. I will then shorten 4x10 to 40. Finally I will say that 45 stands for 4x10+5. If my neighbor tells me he has 67 sheep, I know that he has 6 tens plus seven sheep.
It is my opinion that it is better to introduce topics as they arise rather than introduce a bunch of mathematics linearly and then see what problems you can solve.
A psychology student told me of an experiment using little octopi. The experimenters wanted to keep all the octopi at one end of the tank. The first attempt was to shock the little critters every time they went to the forbidden end of the tank. The second attempt was to shock the octopi randomly when they went to the forbidden end. It turned out that when the octopi were shocked every time they learned faster to stay away from the shocking end of the tank but they also forgot quickly and were soon wandering back. It took longer for the randomly shocked octopi to learn to stay away but once they had learned their lesson, they didn’t go back.
I was teaching for the long haul and addressed topics as they came up, which, while not completely random, was not linear either. Over time the student, with the teacher’s help can put the pieces together.
I had a three Honda Scramblers. Having work done on them in a shop was financially prohibitive so I bought a shop manual and some wrenches. I wanted to work on the transmission of the motorcycle I called “Flower Power”. So I opened my trusty shop manual and went to work. I made a lot of mistakes but I fixed the transmission.
When something broke on a motorcycle, I would go to the shop manual and fix it. Eventually, I could do everything to my motorcycles. And I understood how the damn things worked.


I am now going to suppose that we have the counting numbers. I have written notes where I go into the introduction of numbers more completely, if not completely, and would e-mail them to anyone who is interested.
I introduce a negative integer as the opposite of a positive integer. I don’t introduce a number line and don’t go into a geometric interpretation of an integer right away. When I do, negative integers are on the opposite side of zero from the corresponding positive integers.
I use “opposite” as the basic idea behind “negative” and try to avoid the philosophical problem of “less than nothing”. A negative velocity is in the opposite direction of a positive velocity. A negative time is before the clock has started, a positive time is after the clock has started. Owing money is the opposite of having money. A-B is the net result of combining A and the opposite of B.
I tend to base mathematics in conceptual thought and then use geometry as a picture of the ideas. When I think about adding 2+5, I think about 2 objects being combined with 5 objects. I don’t think about the addition being done on the number line. I think of the number line as a crutch.
So I have addition and subtraction.
If I’m selling eggs, I sell them in groups of a convenient size for my customers, say in groups of twelve. A cafe down the street orders 5 groups of twelve. They could express the order as 12+12+12+12+12 and if they only did it once that expression would work fine. But a species in love with acronyms would find a more compact way to express it. 5•12 denotes the total number of objects in 5 groups of objects with 12 objects in each group. (We call 12 eggs a dozen eggs so five dozen eggs, 5 groups of 12 eggs each, would be represented by 5•12 eggs.
I call multiplication “fast addition”. N•M is the number of objects you get if you combine N groups of objects where each group contains M objects.
I have not talked about computing 5•12 or more generally N•M. I don’t know if N•M is the same number of objects as M•N. And N and M both have to be positive. Actually, if M was a negative integer and N was a positive integer, I could compute N•M.

3•(-2)=(-2)+(-2)+(-2)=-6

I don’t know about (-2)•3. I don’t know how to add 3 to itself a -2 times.

At this point I have the concept of counting numbers and some schemes for representing them on paper. I have the start of a concept and the representation for multiplication of positive whole numbers.

I see my approach as being basic. I have symbols that model a part of the real world, namely the size of herds of sheep. What does Mathematics do with symbols? It imposes an algebra on them and I have a hint of this from buying and selling sheep.
When I introduced numbers it was for passing information between sheepherders. + and • were not operations on numbers, they were operations on sheep and convenient in expressing the number of sheep involved in a transaction between sheepherders. 2+3 is about putting a group of two sheep and a group of three sheep into the same pen.
When I was a sheepherder, I spake as a sheepherder.
As a mathematician I see that 5 can stand for 5 cats as well as 5 sheep. 5 can stand for 5 of anything.
I now make the jump to putting an algebra on the integers. First, this is a step up in abstraction. 4 isn’t four sheep anymore, it’s...well, it can stand for four of anything. Well, it doesn’t stand for anything. It is a symbol. I am manipulating symbols that don’t represent anything. Well, that’s not quite right. I’m not going to manipulate squiggly line segments. There is a concept somewhere behind each squiggly line segment. And that concept is...
The fact is, I am wading in deep waters. When I think of the integers, I think of 1, 2, 3... and then I think of some of their properties. Sometimes I toss “infinite, cyclic commutative ring” into the mix. I do this because it gives me a solid starting point that I don’t have to get too philosophical over. But that is just how I look at integers and my point of view developed over about 15 years.
There is this thing where mathematics people finally figure out how to look at a concept so that the concept is completely trivial. Take limits for example. The first time I was introduced to limits, in Calculus I, I struggled. I couldn’t see why we were going through this bizarre process. The behavior of tangent lines seemed clear to me. I had no doubt that polynomials were continuous.
And then one day I saw it, “My God”, I thought, “limits are trivial if I look at it from this angle, and, wow, now I see how it works from all angles. I’ve got to tell my students of this remarkable point of view.”
But of course my students haven’t spent 15 years thinking about it.
I was teaching determinants for the first time and to augment the text, I read about determinants in van der Waerden. It was so much clearer than the text that I presented my class determinants out of van der Waerden. Should any of those students read this, I apologize.
What is very obvious to me, the students may just not have the solo time for it to be obvious to them.
I was being flown from Albuquerque to Alamogordo in a four seat, two engine airplane; a Beechcraft Bonanza. The pilot was an older gentleman who had thousands of flying hours and because of this experience didn’t bother trimming the airplane. He crossed his feet under his seat and ignored the rudder. The plane bounced and yawed across the sky in a manner that I suppose was safe but which I found quite unpleasant and occasionally terrifying.
I saw that I was the casual pilot in class and my students were terrified passengers. I saw that the student saw the landscape zipping by where as I saw the landscape as barely moving.

I’m not sure how to explain what “five” means in an introduction to integers so I stay with sheep. Number still represents that many sheep in a group. I soon stop writing “sheep” and let the student supply the “sheep” which they don’t do because it’s easier to drop it. That way, the abstract symbols are used as the algebra develops. Here I use the word algebra in its general sense that algebra is about addition, subtraction, multiplication and division.
It also gives the student a base to go back to. If he gets lost, he can always go back to thinking of sheep.
The fact of the matter is that if a student is going to go very far in mathematics, she is going to have to eventually leave “sheep” behind and form an intuition about using the cardinal number idea of 5.

So without going into Peano’s axioms I show properties of integers. Let’s try to “prove” that N•M = M•N.
Well, 2•3 = 3+3 = 6. I get this by adding on my fingers. Using the same sophisticated technique I see that 3•2 = 6. By golly, 3•2 = 2•3. What about 4•3?
Son of a gun, 4•3 =3•4. And 2•4 =4•2. It must be that N•M = M•N for all integers.

I use the principle that if something is true for a reasonable number of small integers then it is true for all integers. Now I know that there are examples where something is true for small integers and not true for some large integer, but these examples are not easy to come by. If I count on my fingers or draw a diagram to convince myself something is true for small integers it usually gives me insight why it is true for all integers. If I look at a 2 by 3 rectangle and a 3 by 2 rectangle, I can see why 2•3 = 3•2 and why N•M = M•N.
I don’t think it is intuitively obvious that 7539•4863 = 4863•7539. In the first place, 4863 and 7539 don’t mean much to me. As far as I am concerned, they are abstract symbols. 4863 nails is a lot of nails. I don’t know if there are 7539 nails in the bucket just by looking. I would have to count them...at least two times. Never being sure which count was the correct count, the number of nails in the bucket becomes a statistic. How would I verify that 7539 buckets, each holding 4863 nails would give the same total number of nails as 4863 buckets, each holding 7539 nails? I believe that I would have the same number of nails in each case because I believe that 2•3 = 3•2 and I believe that 2•3 = 3•2 because I have verified this by counting on my fingers and by drawing a picture.
In an introductory situation I want my students to believe things. A formal proof may show step by step that a result is true but it doesn’t make me believe it. To beginning students a proof doesn’t seem needed to show something as obvious as N•M = M•N.
If the student continues in mathematics they will become more sophisticated and will see problems with “it’s obvious” proofs. (I recommend “Proofs and Refutations” by Lakatos.)
In the 18th century they didn’t use ε,δ proofs to define and prove continuity because they assumed all functions were piecewise continuous. Then a function that was discontinuous everywhere was found. More and more bizarre examples came to light. Now mathematicians had to worry if their functions were continuous.
According to Morris Kline the 18th century mathematicians didn’t come up with wrong results; their intuition worked fine. Newton and Leibniz didn’t use εs and δs.
As a TA I taught calculus the “New Math” way. I already had a Master’s so I really enjoyed the courses, the students enjoyed them less so.
I had fallen into the trap of organizing a course in a clear, logical way that I felt even the slowest student would get. Alas, that’s too austere. Learning needs to be full bodied.

There seems to be a rush in teaching mathematics. Syllabi are harsh taskmasters. If time were taken at the beginning, in the introduction, so that the learner was given a solid base, then the student can learn quicker. Certainly mathematicians must realize that the more comfortable they were with mathematics, the faster they could learn new mathematics. I seem to remember reading somewhere that building a house on sand is an ineffective construction technique.

Finally I want to deal with (-2)•3, that is, multiplication by a negative integers. I have seen attempts at making up some kind of problem that would supposedly give insight into using negative integers. I have always found them contrived and not helpful.

I am going to go over what I have done so far.
I began with the positive integers, then I added 0 to get the non-negative integers, then I added the negative integers to get the complete set of integers. The idea being that I started with the counting numbers so I could count my sheep. 0 gets thrown in so I tell someone I don’t have any sheep. (Note that having no sheep is the same as having no oranges. 0 sheep = 0 oranges so sheep = oranges, a little known fact of animal husbandry.)
I need negative numbers so I can see how many sheep I lost in the big snow storm.
The first operation was addition of counting numbers. When 0 becomes a number, I have to see how to add 0 to a counting number (0+n)=n+0=n for all counting numbers, n.
Nothing is a funny concept. The phrase “nothing is better than a cold beer” seems ambiguous to me. It could mean that I would rather have nothing than have a cold beer. Or it could mean that the set of objects better than a cold beer is empty.
Negative numbers were a philosophical problem. DeMorgan of DeMorgan’s Laws fame said that the idea of “less than nothing” was foolish. The negative roots of polynomials were as much of an embarrassment as imaginary roots. When I was young, negative roots were called “extraneous” roots.
I’m going to say that -3 is a number such that 3+(-3)=0. I’m going to define subtraction by
M-N = M+(-N). 5-3=5+(-3)=2+3+(-3) =2 +0 = 2. In this situation -3 is the opposite of 3 and 3 turns out to be the opposite of -3.
I have just defined the operation of subtraction. Before this moment in time subtraction didn’t exist. Nota Bene: M-N does not equal N-M. Subtraction does not commute.

Having defined a negative integer, I have to see how I’m going to add negative integers to the rest of the numbers. (a+(-b))=a- b where (-b) is the opposite of b, i.e. (-b +b=0)
Although I haven’t talked about it much there is a rule that is obvious enough to use without thinking about it,
N+M=M+N.



Regardless of how obvious it is, it still has to be verified that it works for an enlarged set of numbers. Since subtraction does not commute, there exist operations that don’t commute. But for positive integers I can prove it by noting that it doesn’t make any difference which herd of sheep I put into the pen first, I end up with the same number of sheep in the pen. Good ol’ sheep.
I can use the same kind of reasoning to show that commutation works if one or both of the addends is negative.
Another rule is n•m=m•n but I only have it for positive integers. I want it for all integers so I define multiplication for all integers. Note that at this point, the multiplication of, say, a negative number by a negative number is not defined. Since it is undefined I can define it anyway I want to. I could define (-3•(4)=-3+4. I know of no law of God or man that would prohibit me from doing so. The problem is that

2•(-3)•(4)=2•(-3+4)=2•(-3)+(2•4)=2+(-3)+2+4=5

and

2•(-3)•(4)=2•(-3+4)= 2•1=2.

This is not good. I want multiplication to behave the same for all integers. Well, it does if I define it correctly.

I give some proofs, not because I expect a beginning student to understand them but it lets them know that they can be proved. Their teacher can prove them. This, I think, gives the student confidence that they are working with the real stuff. In my calculus book I proved things that I liked to prove and I put them in so that the student could see that proofs existed; so they could see what a proof in calculus looked like.
When mathematicians are doing research they use theorems that they haven’t proved. Sometimes they will read through a proof to see what kinds of techniques are used. I personally don’t know anyone who claims to know the proof of the Jordan Curve Theorem yet it is used.

What I have tried to do is get a flow. I start with positive integers and the operations of addition and multiplication which have a definite intuitive meaning. I enlarge my set of numbers to the integers and get a new operation, subtraction. I have to extend + and • to all the integers and make sure the new numbers and operations are compatible with previously defined numbers and operations.
The next step would be to extend the integers to the rational numbers. Then division can be defined and related to the previous operations of +, •, - .
Counting numbers are a subset of the integers which are subset of the rational numbers.
This idea runs throughout mathematics.

I have not tried for logical correctness or completeness. I have tried to give the flavor of my approach to arithmetic. I can see problems if my program were carried out with a vengeance. But I do think it is important to give the student a point of view, to give them an idea of what a definition is.

No comments:

Post a Comment