Mathematics Teacher

2011-09-21T01:24:00.000-07:00

Mathematicsas a cultural artifact.

I have beenthinking about Martin Gardner’s critique of Reuben Hersh’s ideas in an articlehe wrote in Eureka. I am not sure Iunderstand Prof. Hersh’s ideas but Gardner’s comments were obviously wrong.
   I am going toassume that Hersh thinks that mathematics does not exist independent of people.Gardner says that Hersh asserts that mathematics is a “cultural artifact”.
In the firstplace, to think that humanity has discovered timeless, universal truths is anamazing act of human hubris.
   Since the righttriangles of mathematics don’t exist the question arises: What does it mean tosay there is something true about something that doesn’t exist? How do you saysomething true about unicorns?
   Of coursemathematics is a cultural artifact. How could one think otherwise?
In my viewmathematics can be likened to a computer game. The symbols of mathematics aregame pieces. The rules of mathematics are the rules of the game. The symbolsdon’t represent real objects. They are like the trolls, ogres and mages. The problemsolutions and theorems are like the gold coins, better guns and health.
   It turns outthat humans can make game pieces that seem to model the approximate “real”world and then make up game rules that when followed seem to lead to a gamesolution of a “real” world problem. Then humans perform experiments to see howthe game solution measures up to the humans’ approximation of what actuallyhappened.
   The game gives auseful answer often enough that mathematics game rules are taught in school.
   Different cultures come up with differentgames with different game pieces. When someone says that the PythagoreanTheorem is true everywhere, they are wrong. It’s only true where the Earthmathematics game can be played and the Pythagorean Theorem is a secret place inthe game.
    Inthe “real” world there are no right triangles, there are no line segments to
use as sides for triangles orpoints to use as vertices. A board 2 feet long doesn’t exist. These are allgame pieces but they were designed to model approximations to the “real” worldas humans saw the “real” world.
    The unit squareis no more “real” than a dragon. Saying things about the unit square is like sayingthings about a dragon. What does it mean to say things about imaginary objects?Assuming that “true” and “exist” have meaning, can you make true statementsabout objects that don’t exist?
    I feel like I’mswimming in molasses when trying to explain why mathematics isn’t real. I don’tsee why the idea is so resisted. It doesn’t require believing in theimpossible.
    It is not hardto think of an environment where Earth mathematics can’t be played. I don’tthink that the game can be played in the world of dolphins and whales.
    What need wouldwhales or dolphins have to count? They surely wouldn’t count plankton one byone. Humans tend to count because there are a lot of collections whose objectsare stationary enough to allow counting.
    I think thatcounting arose from a need. If there was no need to count, numbers would notarise. A culture that had no need of counting would have no need of numbers.
    The purpose ofnumbers was to convey a certain kind of information. It would be remarkable ifthere were only one way to convey that information and that it required humanphysiology.
    What does itmean to say that the Pythagorean Theorem is true everywhere? A culture may havean entirely different way to perceive the universe and have no way to thinkabout geometry but if you, from Earth, were there would you see the truth ofthe theorem?
    Well, how wouldyou check the proof of it? You would set up your game and see if you could playit. So, what if there is no way to set up the geometry game? Is the PythagoreanTheorem still true?
     On Earth thereare no right triangles that satisfy the conditions of the theorem. There are noreal triangles so the Pythagorean Theorem only makes sense in the world of thegame.
    I read somewherethat there are 50-100 billion neurons in the brain. (I have no idea why thereis that big a spread.) 10 billion of these neurons pass signals around thebrain over 100 trillion synaptic connections.
    So to have abrain you need 10 billion “things” that pass messages among themselves and to,say, 100 billion other “things”. You probably need other stuff too but the 100billion “things” is a good start.
    In the humanbrain the “things” are called neurons and the connections are hardwired. Idon’t see why a wireless connection wouldn’t work as well if not better. Thismeans that the “things” wouldn’t have to be localized in a skull. The size ofsuch a “brain” would be limited only by what speed of thought was satisfactoryto the “brain”. And who knows how the “brain” looks at time. The “things” couldbe light years apart.
    The “brain”could be 10 billion years old in which case a short interval of time would meansomething quite different to the “brain” than it does to me.
How would such a“brain” look at truth?
    Humans stillthink that they are the center of the universe. They think heaven is the nextstep in human development. Maybe the urge to give a mysterious reality to righttriangles is the desire for God.
     Why do humansinsist on the reality of ghosts and goblins? Why do they insist that theuniverse was made in six days? What’s the point?
     I can see wherebelief in God offers the possibility of salvation and good crops. What does ∞do for you if you believe it is real and not just a game piece?

And that’s allit is…a game piece.

And that is whatmathematics is…a cultural artifact.

Graphing

2009-05-12T14:06:00.001-07:00

Graphing Calculators
One of the major obstacles to teaching a topic in mathematics is that the student has little or no idea why they are learning it.
When I was learning the C Language, I never understood pointers because I never knew why they were there. What problem do they solve? Evidently the guy who invented them did it to make programming easier but I didn't see why.
I struggled with epsilon-delta process until we got into 20th Century mathematics; when I saw what the process was used for the concept became trivial.
As a rule, mathematics spring topics on students like a magician pulling a rabbit out of a hat. The rabbit is obscure because why would a person keep a rabbit in a hat.
The part of a method or a theorem that the student sees is the result of lots of thought with the thought removed. It is one thing to see that the steps of a proof lead to a correct result, it is quite another see where the steps came from and why you are proving the result in the first place.
And now we come to graphs.
What are graphs for? They allow a person to use their eyes to help in the understanding of functions. Graphs are of no use to a person blind from birth. Graphs were invented by people who could see.
There are several ways to look at functions visually. Some graphs use the magnitude of the distance between two points, as can be seen, to visually represent a numerical quantity. You see that points A and B are farther apart than points C and D. Another method is to represent a number as a color. This is the method used to tell when steel is hot enough to pour. Film comes in different color temperatures.
So, the point of the graph is to see a function and the basic idea is to represent numbers in the domain and in the range by the size of the separation between two points.
But to understand graphs, the student has to understand functions. If they don't understand functions they can't see a reason for graphs.
The next time you are trying explain graphs, see how many of your listeners can tell you what a function is.
Supposing that the student understands the function, then they are ready for graphs. I suggest Chapters 3,4 and 5 of The Calculus: An Opinion by J. Davis
I think that paper and pencil is the best way to start. I think that the student gets a feeling for drawing the picture of a function. If I want to really look at a face, I will sketch it, not take a Polaroid.
The graphing calculator is of use when you want a more accurate picture of the function like when you want a more exact number you use a calculator.

A suggested goal of teaching

2009-05-12T14:05:00.002-07:00

Teaching RevisitedThe following question came up in a conversation: Is learning to ride a bike the same kind of learning as learning the quadratic formula. A distinction made was that learning to ride a bike is learning a motor skill while learning the quadratic formula is learning a mental skill. But motor skills are handled in the brain so they are mental skills as well. I find much in common between learning music, learning how to shoot a basket, learning calculus, learning anything. Learning is learning.>And once you learn something, you don't forget it, like when you've learned to ride a bicycle. That's how you can tell if you have really learned something; you don't forget itThe brain directs the muscles in performing a motor skill. When a person first learns a skill they memorize how to do it and the person is mentally and consciously involved in performing the skill. After the skill has been learned the brain skips the consciousness and communicates with the muscles directly. Sometimes the brain communicates with consciousness when it solves a problem. Sometimes the brain tells the finger what note to play and the brain listens via the ears.A musician learns music and can pick up any instrument and play it. Maybe not as a virtuoso but they can play it. The musician has consolidated music into one instance.
A mathematician learns mathematics and sees what mathematics is all about. Then she can learn any chunk of mathematics. The mathematician has consolidated mathematics into one instance.When I am teaching max-min problems I see it as working the same problem over and over. I have consolidated max-min problems into one instance. The student has consolidated nothing and everything I put on the blackboard is new. They don't see one instance, they see a jillion instances. They are flooded with instances. A teacher should remember this.I don't remember how to do max-min problems, I just know how to do them. Like a musician doesn't memorize a song, he just knows it.I watched Sa Chen, the young Chinese pianist, play the piano on TV and I thought, "There is no way to memorize what she is doing." The whole sonata was one instance to her. It's more than that because she knows a lot of sonatas and other stuff too. The piano is part of one instance of music.I am considering extreme cases of consolidation and why extreme cases occur is a mystery to me. I don't know where Mozart and Gauss came from.But most people can consolidate something to some degree. A guy drove me right to the Harley-Davidson shop in Kansas City, Kansas and he had only been there once two years previous. He told me it was just something he could do. He could find his way back to any place he had ever been.
I think people have consolidations that they can only realize every so often. Every once in a while a song is consolidated and is played without thinking. You shoot a basketball and you just know you are doing everything right and it's going in the hoop. And it does.It isn't that consolidations aren't there somewhere inside us; the problem is in the extraction.A friend, Chris Barrett, pointed out this idea of consolidation to me. He defined God as that entity that reduced the universe to one instance.Anyway, we need to teach consolidating.I think, however, that changing the way things are taught would be a Harry Sheldon project. (Foundation and Second Foundation by Azimov)

Remarks About Numbers

2009-05-12T14:05:00.001-07:00

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.

Teach How To Learn

2009-05-12T14:04:00.000-07:00

Teach How To Learn

I was troubled by what I taught in elementary courses. It seemed like I was teaching stuff and I didn’t know why I was teaching it.
A hundred years ago students could put the mathematics they learned to use right away. Now that is not so.
It used to be that the student was looking for a job for life. It took generations to build a cathedral. Now technology moves so quickly that a student can look forward to many different jobs and each one has to be learned.
Consider a radio repairman. I doubt if there are any now, they had to learn new skills. From tube radios to transistor radios and TVs to VCRs to DVDs to Blueray. Every few years there is a new technology to learn.
The appropriate skill to teach is not how to, say, differentiate but how to learn how to differentiate. This should be started in pre-school.
And it doesn’t make any difference what the skill is. They should start learning something that appeals to them. The goal is for the student to have confidence in their ability to learn, in their ability to face an unknown skill without fear.
Memorizing is not learning. The confusion between the two is probably the biggest obstacle to learning. It turns out that it is easier to test memorization than learning and since the student has been memorizing from the beginning it is a preferred way to take classes. In an unholy alliance the students and the teachers take the easy way out.
I would have my students take an oral in my office where they would have to prove a significant theorem, say, that the sum of the angles in a triangle is less than or equal to 180 degrees in hyperbolic geometry. It was too long to memorize, they had to see the flow of the theorem and see how the parts fit together naturally. And I gave them multiple tries.
I found that students didn’t know if they knew something or not. A student came in for help on implicit differentiation. It turned out that he didn’t understand what the derivative of a function was. And then it turned out that he didn’t understand what a function was. He wasn’t trying to con me; he thought he knew these things and was surprised when he didn’t.
I have many stories like this. A student sat through an entire semester of business calculus and through the semester he thought it was beginning statistics. And he thought he was doing ok in the calculus although his highest grade had been a 25. A student thought he understood max-min problems and didn’t.
Why is it that students believe they know things when they don’t?
For one thing, when you learn something, barring serious head injury, you don’t forget it. You forget things you’ve memorized. You don’t memorize how to ride a bike, you learn how to ride a bike”¦and you never forget.
Some people learn things easier than others. I was with a guy as he drove right to the Harley shop in Kansas City, Kansas and he had been there once three years previous. Arlo Guthrie never learned how to read music. When he started piano lessons, his teacher always played the piece he was supposed to learn from the sheet music. Once he heard the tune, he didn’t need the sheet music and never looked at it. Some people don’t forget tunes they’ve learned.
I was playing the guitar chords to “Don’t Think Twice, It’s Alright” while a friend played lead. At the point where it goes, “When the rooster crows at the break of dawn” he stopped us.
I had played an A instead of an A minor or something like that. He said, as if he was pointing out that the sky was blue, “If you play the A there it ruins the tune.”
I realized that he was hearing something that I wasn’t. I had memorized the chords, he understood the sequence of chords. He would always play the right chord.
I have heard that the piano prodigy and the piano non-prodigy have about the same proficiency when they are in their twenties so if a person isn’t a prodigy they can still learn to play the piano.
So how do you teach a student to know what learning is?
First they have to understand what “knowing” is. They have to have some idea of when they have learned something.
I think you have to start when they are young. Really young kids seem to be into acquiring skills and knowledge.
I t

The wrong stuff is being taught; not even in the ballpark.

2009-05-12T14:03:00.002-07:00

What to teach? What to teach?

Every time I think about how to teach undergraduate mathematics, say, algebra through calculus, I can think of ways to teach the standard stuff but I always get hung up on the question: Why am I teaching this shit?
The first topic that should be taught in school is “learning technique”. If a student knows how to learn, the life of the teacher is much more interesting. There is a big difference between talking to someone who knows how to learn and someone who doesn’t. It’s more interesting for the student too.
So the topic that should begin in pre-school is learning how to learn. It doesn’t make any difference what they start learning because the point of the exercise is the act, the process, of learning. The child should start learning something that that isn’t unpleasant for them.
When the child is young they will probably want to learn something that most adults know how to do and are quite able to show the child how to learn it. The first things I learned were taught to me by adults and their teaching technique was quite satisfactory.
I went to Junior High in Cheyenne, Wyoming and we lived on Warren Air Force Base. As soon as we moved in my dad gave a plane geometry book, an algebra book and a college algebra book; I was instructed to learn them.
This was OK with me. The winters in Cheyenne were long and cold and learning mathematics while listening to radio plays was as good a way to spend winter evenings as any. Well, there were also the Friday night fights to go along with factoring polynomials.
But the mathematics that I learned wasn’t as important as the fact that I learned how to learn.
From that point on, school was no problem. I didn’t necessarily want to learn everything but if I wanted to, I did.
People who know how to learn do it at different rates. Actually people who don’t know how to learn do whatever it is they do at different rates.
As I think back on my teaching career, the time restraint always bothered me but I didn’t stop to examine time more closely.
The time that it takes to learn something is a statistical distribution. There are some students that can learn the next thing in an instant, there are others that take a long time before they are ready to learn something else.
Because most of the students don’t learn the mathematics, we have tests to evaluate”¦ It is not really clear what is being evaluated. My class grades gave a bi-modal distribution, those that learned how to take tests and those that didn’t. Learning the mathematics was a sufficient but not a necessary condition to pass tests.
The more I think about giving a 50 minute test every 2 weeks the more bizarre it seems.
In my upper division courses the tests were all the same: “Write down what you know”.
If the class wasn’t too large, about 15, plus or minus, I would give an oral final. They would have to give an hour talk on, say, the proof of the Heine-Borel Theorem without notes. They would get as many tries as they wanted until the day before grades had to be in.
But in larger calculus classes there isn’t enough time.
I didn’t give numerical grades, just letter grades. I couldn’t tell the difference between a 71 and a 72. It’s pointless to make such distinctions. A teacher should be able to tell the difference between and A paper and a B paper.
It’s not that there weren’t students who could do well on tests without learning the material. A student who does well on tests but doesn’t know any mathematics, has learned a skill. The student has been given a problem to solve, get through school with good grades. The problem is not “to learn mathematics”.
I think there is a better appreciation of what learning how to make a guitar or to use a lathe (no, not the computer lathes), how to turn a cartwheel. There is an understanding that something more than memorizing instructions is requires. Learning how to play a guitar is more than putting your fingers on the right frets and hitting the right strings. I have heard a man play the piano and hit all the right keys at the right time and it wasn’t music. He hadn’t learned how to play the piano.
Perhaps it is music they haven’t learned. Memorizing which notes to play doesn’t make music and memorizing how to work selected max-min problems doesn’t make mathematics.
People say that music and mathematics are related and then start talking about octaves and fifths and Pythagoras and group theory. They miss the point.
You learn how to groove on both.
I think one problem with beginning mathematics is the teachers. I think most teachers like to read and their students see this. I think that a lot of elementary school teachers do not like mathematics and that shows.
I knew a grammar school teacher who told her students that she didn’t like mathematics either but that it is something you have to learn. And she really felt that they should. As distasteful as it might be, she thought that balancing a checkbook is important.
The conversation took place some years ago.
It is my opinion that once you know how to learn, you can learn anything. Well, anything within reason.
One of the problems is the amount of material that is shoe -horned into beginning courses. In the Calculus I that I took we spent three weeks on conic sections. In the last Calculus I syllabus I taught it was about three days. What I learned about conic sections has stayed with me for over fifty years. That’s what learning something does for you.
I liken the Calculus syllabus to driving down the freeway at night at 90 mph with your dims on. You read the green signs but you miss the off-ramp. You touch on everything and don’t get to really teach anything”; no time.
The “include everything” mentality leads to books that are too big, too heavy, too poorly written.
But if the student knows how to learn, all these problems dissolve in the mist. You teach the important basics, like what a function is and what its graph is, the pros and cons of continuity, what the derivative actually is and what an integral actually is.
A student who knows how to learn can learn the technique of differentiation in a few days. I had a friend who claimed that he could teach a parrot to take derivatives. After all, there are five basic functions and five ways to combine them; you have to know the derivatives of five functions and how the derivative deals with the five ways to combine them. End of story.
Doesn’t anyone ever wonder why more than a day is spent on the derivative of a product? Could it be because the wrong things are taught and that memorization is called learning?
We spend all this time thinking about ways to teach stuff that the students should be capable of learning on their own. The fact that they are not capable is the fault of education, not the student. If I were a conspiracy theorist, I might see a conspiracy to keep knowledge from the populace.
One might have thought that as the country progressed from being a manual labor economy to a more mechanized economy, the citizens would be educated to keep up. But there is a problem with people coming to this country, legally or illegally, and are taking manual labor jobs. Why haven’t we left manual labor jobs behind for people in countries on their way up?

How Do Different Species Arise?

2009-05-12T14:03:00.001-07:00

An Idea on Evolution and Species Differentiation

I will start with the primeval ocean full of stuff and at some moment a piece of replicating DNA appears.
The first question is: Why did the stuff in the ocean combine to make something that was self replicating? I don’t know and when looking for a book to read I never choose biochemistry. I take the fact that we are here as prima facie evidence that it did happen.
Here I am making the assumption that life started as a very simple organism. This is contrary to the creationist point of view that man was created fully made some thousands of years ago.
I don’t see why more than one piece of DNA would appear but I see no reason that they couldn’t appear in a volume of ocean or even throughout the ocean. This mass appearance seems very unlikely to me.
I will suppose that just one appears. It seems to me that there would be just one primordial piece of DNA but that is more for aesthetic reasons than logical reasons.
It doesn’t really make any difference. For my argument to hold, the pieces just had to be small. After a fairly “short time” there would be so many pieces of DNA that the original number of pieces would be seen as a point.
The piece of DNA had no predators because it hadn’t previously existed to develop any. But it could have happened that there were killer molecules. If a killer molecule accidently bumped into a piece of DNA they might have combined in such a way as to ruin the DNA’s reproductive ability. I suppose a piece of DNA could have been destroyed if it was hit by lightening. I suppose there were a lot of ways a piece of DNA could have met an unreproductive end.
The presence of life as we know it implies that the DNA population grows faster than the DNA is annihilated so after a certain amount of time, the doubling time, the DNA population effectively doubles.
Since the population is pretty homogeneous in the beginning I’m going to model the early growth process as starting with one piece of DNA and that an individual doubles after a length, T, of time; T is the doubling time. Early on T is fairly constant.
I assume that the time estimates of science are in the ballpark so I’ve got three billion years to fool around with.
How many times will the population double in a million years? If it takes a thousand years for a population to double, so for example it takes a thousand years for the first piece to double, then in a million years there will be 21000 individuals. This is on the order of 10300 individuals. If a grain of sand was 1/64 inch on a side, 1060 of them would fill the universe.
Clearly the population of pieces of DNA doesn’t double a thousand times in a million years but after a million years there must have been a lot of DNA floating around.
Now I come to my point which depends only on the fact that there is a growing population of DNA pieces and that, since the pieces are small, there are significant mutations that are replicated.
At this early time a mutation of a piece of DNA would involve a significant part of the piece and would thus change it quite a bit. One gene changing out of thousands is different than one gene changing out of a few.
My point is that the differentiation of species starts here. The mutations initiate changes that lead to different species.
When the organisms get larger, mutation is followed by natural selection. The mutations do not give rise to a new species, they change the phenotype; the feathers of Darwin’s finches get darker but they are still finches.
Different species can trace their ancestry back to one of the early mutations of DNA and hence back to the primordial piece of DNA.

I have always wondered about the origin of life. I had this picture of DNA appearing all over the primordial ocean and this seemed odd. But then I thought that, because of exponential growth and in particular doubling, you only need to start with one self-replicating piece of DNA.
I have also always been bothered by the appearance of different species. I can’t see how that could happen. Breeders can develop very different dogs but haven’t come up with a new species.
If you start with an ape, how does it mutate into a human? Does a whole bunch of apes become a whole bunch of humans? Does one ape change a little bit and some how start the path to humanity?
So apes with bigger heads are chosen for? So what?
Why can’t small heads think? Computer chips get smaller every day and “think” better. Not that I think you can model the human brain as a computer.
If I assume that species were differentiated soon after DNA appeared these problems go away.
It seems to me that a mutation of an early piece of DNA, a very small piece of DNA, could make a big difference in the end result of that piece of DNA. The end result is sensitive to initial conditions.
I don’t think that the random mutations to apes change the outcome of the succeeding apes much. The end result is a change in phenotype but not a change in species. The more complex the organism, the less sensitive it is to initial conditions. Mutation gives rise to a change in phenotype but not species.
Early DNA changes from chaotic evolution to non-chaotic as the DNA becomes more complex, like being an ape.

Teaching and order

2009-05-12T14:02:00.004-07:00

Students know what their interests are, not teachers

I recently looked at “Rank Your Teacher”, a webpage I found through Google. By my name there were 7 or 8 entries. There was one that essentially gave me a zero and had dropped my class in the first few days. The consensus of the rest seemed to be that I was a good teacher except when I was teaching and then I was boring.
The classes represented were trigonometry and (not quite) college algebra. In retrospect I agree with their assessment. I couldn’t see how to talk about trigonometry in a way that might be interesting. As they are presently conceived I still don’t see how to.
My solution is to not teach them as separate courses. Teach algebra and trigonometry as they arise in the context of other instruction.
It seems to me that the present philosophical underpinning of mathematics education is the there is an ideal, linear ordering of mathematics and it should be taught in that order. This “order” is the “philosopher’s stone” of mathematics education; if it could just be found we could turn base metal of mathematical deprivation into the gold of mathematical literacy.
Unfortunately I don’t think this order exists.
A hundred years ago 90% of the population was rural. High school was enough education to qualify for a job that would put a person in the financial middle class. College algebra was taught in college.
The mathematics taught in the first twelve grades was not to be learned now and used at some indeterminate later date. Weights and measures (how much milk you took to the creamery), geometry (how much land your dad put into wheat), how to make change (helping out at the store, taking eggs into town to sell) could be used everyday on the farm.
It is less clear to today’s student where they are going factor polynomials or solve trigonometric identities in their everyday life after high school. It isn’t clear to me either.
Word problems are called applied problems. When I was in K-12 I knew that this was bullshit thought up by adults that they expected me to believe. The problems are obviously contrived. The mathematics text books of today proudly say that they have applications but this is just hype for the authors to convince a publisher to publish their book and for the publishers to convince a school to buy it. What are called “applications” aren’t put in for students, they are put in for adults.
It isn’t as though word problems have no value. Part of mathematics is about the collection of data and its organization to solve problems. This is what word problems are about, not real world applications. They should be presented as such. The education establishment uses the real world application lie as a daily staple. The students know they are being lied to. I hope.
It should be kept in mind that the important thing being taught is how to develop mathematics to model a problem. The problems don’t have to be linearly ordered.

Instead of adults deciding on the student’s path to knowledge, the teacher helps the student follow their own path to knowledge. I think that all early learning should be done this way. I remember grinding through Vanity Fair as a junior in high school. It wasn’t until I was older that I had the interest and life experience to appreciate Becky Sharp.

Consider some examples:

I remember starting piano lessons when I was ten. First there were scales and then little melodies written by “Schaum” in Book I. I graduated to “The Happy Farmer.” My mother 26 years earlier had started piano the same way and learned “The Happy Farmer”.
Both my mother and I were conscientious students and could eventually play some fairly sophisticated tunes. My mother could always sight read the copy of “Poet and Peasant” that she had learned as a teenager.
But neither of us could really play the piano. We gained some appreciation of piano music but didn’t know how to play the piano.
All the people I know who can play a musical instrument started by playing tunes they liked and wanted to learn. My music teachers had me learn tunes they thought I should learn, many of which didn’t really want to learn.
The more tunes a person learns that they are interested in the broader their interest becomes and they eventually become interested in tunes they, when first learning, thought were uninteresting.

When I was six (1941) my dad told me about the Bohr model of the atom and the infinitude of natural numbers. He told me that a rectangle was an unstable structure because it can change shape without changing the length of its sides. On the other hand a triangle can’t change shape and keep its sides in tact.
So I found an interest in triangles, infinity and atoms at an early age and six years later that early interest was expanded by books on geometry, mathematics and physics.

The problem with mathematics courses as they are now constructed is that they are “learn this now because you will need it later” courses. These courses are in the ordered development of mathematics.
In point of fact very little of it is used later and the syllabus is padded with topics that won’t be used at all. There are topics in algebra courses that are used only by teachers teaching algebra courses.
There are topics in early mathematics that teach some thinking and problem solving skills but memorizing algebra stuff or trigonometry stuff is not one of them.
Suppose a Junior High School student likes to go to carnivals. They are aware of carnival rides and the types of forces their body is subjected to. By riding in a car or bus or subway or wagon or bike or tricycle very young children have felt the effects of acceleration and deceleration. They have felt the reality of objects needing a force to change direction when the vehicle they’re in makes a turn.
Young people are aware of what faster and slower mean, aware that things change and that some things change faster than others.
The ideas of differential calculus can be discussed without numbers. The idea of rates and rates of change can be introduced using only elementary arithmetic.
When more mathematics is needed, that is the time to present it. It is my belief that this approach will have positive results if one is interested in students having some facility with mathematics.

I can speak only for myself but in my 50’s, while trying to learn guitar, I realized that I didn’t listen to myself play. I had taped myself playing a tune and I thought I had done it pretty well but when I played it back it sounded terrible, really bad. I was so occupied with which fret and which string that I didn’t listen.
So I consciously tried to listen to myself play. The first thing that became evident was that I couldn’t do two things at once, namely put my fingers in the right spot and listen at the same time.
In a flash of insight I realized why the one vinyl record of my piano playing sounded so much worse than I thought it should. It had no feeling or dynamics, just the correct notes.
What comes to mind is the conscientious student of mathematics who memorizes and then is surprised when they don’t know any mathematics. I had a student in Calculus II who claimed that he had become a max-min maven in Calculus I. I let him pick a calculus book and pick a max-min problem from the book to work on the blackboard. An hour later he was completely lost; he had no idea how to work the problem.
He wasn’t trying to con me; he really thought that he understood max-min problems until he looked at what he had put on the blackboard. I really thought that I could play “Bumble Boogie” on the piano until I listened.
Learning mathematics linearly entices the student to memorize. You just have to memorize the order. If I was stopped in the middle of a tune, guitar, banjo or piano, I would have to start at the beginning. I had memorized the order and when the order was broken I had to start over.
Needless to say I have observed this when students would try to put a theorem that they had memorized on the blackboard.
This happened to me in music but why not in mathematics? I could be asked a question in the middle of a problem or theorem and pick it up without dropping a stitch. I would sometimes get off on a tangent and forget where I was in the demonstration but as soon as I was reminded, I was back on track.
My music teachers could start anyplace in a tune be it “The Happy Farmer”, Beethoven’s Sonata Pathetique or “Classical Gas”. Rock guitarists jump around and sing while they are going crazy on their instrument. I watch country singers play effortlessly as they sing. I watch banjoists nod to a friend in the crowd while in the middle of “Foggy Mountain Breakdown”.
I can start in the middle of the Heine-Borel Theorem.
Learning mathematics linearly is like trying to understand Oklahoma by driving through it on I-40. I hear that “Oklahoma City is mighty pretty”.
You learn about Oklahoma by going to Ponca City, Hugo and Enid. You learn about it on the two lane roads and by stopping at the only café in a small town for a bowl of chili.
The musicians that I have talked to who had an easy relationship with their instrument learned this and that, learning tunes that they liked. As they played more they became more musically sophisticated and their musical interest broadened to rock, jazz and Bach.
But people who know things look back at how they learned them and think that they did it in the wrong order. If they had just learned the 43 fundamentals of drumming first. This was told me by an accomplished drummer who had started out just fooling around and was teaching his girlfriend the 43 fundamentals to start with. She gave it up after the first few fundamentals.
Mathematics books are monuments to the order principle.

Topics should be chosen as interest dictates and every so often the separate pieces can be consolidated into a single instance of a concept.
The choices of topics look random from the outside but they aren’t. They follow the path of the learner’s interest.

Single Valued Functions and Dolphins

2009-05-12T14:02:00.003-07:00

Dolphins, Single Valued Functions and Intelligence
Mathematics is dominated by the single valued function. We have technology because physics on earth is essentially single valued. Since physics is single valued it is repeatable. We think of physics as deterministic.
When I tune my radio to 102.5 I always get KIOT. If I turn the wheel of my car a certain amount, the car always turns the same amount. Freeway driving would be impossible if cars drove like bumper cars. (At least bumper car steering always seemed almost random to me, no repeatability at all.)
All of our high tech engineering depends on being able to accurately predict how our airplanes, rockets and radar dishes react to given inputs. The engineers expect the same response to a given input.
Of course, neither the input nor the output is exactly the same each time since nothing is exact; since our measuring capability is limited we couldn't tell if either the input or the output was exactly the same each time. But they are, within our ability to measure, close enough to make the single valued function a useful tool in modeling physics.
This repeatability is so ingrained that it is hard for humans to imagine how it could be otherwise.
But dolphins live in a non-repeatable world. In the ocean nothing happens the same way twice. If you drop rocks in a swimming pool it takes neither the same path nor the same time for each of them to get to the bottom. The world of the dolphin is not deterministic, it is stochastic.
It isn't the lack an opposable thumb that keeps the dolphin from technology, it's the non-repeatable world they live in. The single valued function is of no use to dolphins so it is hardly surprising that they haven't developed them and the accompanying mathematics.
The dolphin, on the other hand, has some advantages that humans don't. They have an inexhaustible food supply. They have a free run of 2/3 of the planet and housing is not a problem for them. They have no natural predators; anything they can't outfight they can outrun.
Except humans of course. Humans are the predators of everything from ants to whales both of which humans eat. Humans are the universal predator and dolphins have not been given an exemption.
So while they don't have radios or fast cars, they don't have wars either. As far as I know dolphins don't have fights. What do they have to fight about?
Well, there's always women to fight over but dolphins seemed to have solved that problem, certainly better than humans have.
Humans think they are smarter than dolphins because dolphins don;t have guns and shopping malls. And humans kill dolphins but dolphins don't fight back, dolphins don't kill humans. In fact there are reported cases of dolphins saving the lives of humans. I mean, is that unintelligent or what? So often humans base intelligence on the sophistication of weapons used to kill.
On the other hand dolphins perform at Sea World and Disney Land so they can observe the human species up close and try to figure out what goes on with this weird species that is ruining their ocean. Is that intelligent or what?

Brief Overview of Functions

2009-05-12T14:02:00.001-07:00

After we have a set, the next thing we need is a function. For most of mathematics, this is all that is needed. The rest is just studying the relation between the sets and the functions.So what is a function?

A function is a rule and a set. The rule associates an element of the set to exactly one element of another set, which could be the same set.
In particular, I’m going to consider sets of numbers although the sets could be groups, rings, or any number of other kinds of sets.
A function is actually a pair, a rule and a set. The set is often not mentioned explicitly but it is a crucial part of the definition.
Suppose that the set is the set of numbers between 0 and 10 inclusive, that is [1,10]. The rule is to associate a number in the set with its square. I take a number out of [0,10] and square it. The rule associates 5 with 25. The set of numbers that the rule is applied to is called the domain of the function.
I can use algebra to express the rule. Associate a number x in the domain with the number x².As is usual we give the rule a letter name, say f. We denote the number that the rule of f, associates with the number, x, by f(x). So we can write
f(x) = x².
(f(x) is not a function, it is a number. The function is f where f stands for the rule and its domain. I take a number from the domain and associate it with its square.)
[0,10] could represent the points along a 10 cm. rod. The function, f, could associate a point on the rod to the temperature of the rod at that point in degrees C. Suppose that the temperature at x is f(x) = x² so the temperature at x = 5 is 25°C.
If I have a rod 20 cm. long, the rule that gives the temperature at x could be the same as the rule of f, the temperature at x is x². But the domain of this function is [0,20] and so it is a different function than f and must be given a new name, say g.
g(x) = x² because f and g have the same rules but they are different functions because they have different domains.
Next we examine the behavior of functions, in particular how do they act on the operations defined on the set. Since I am looking at a set of numbers, whose operations are +, -, • and ÷, I look at

f(x + y), f(x - y), f(x•y) and f(x/y).
It is quite possible that they are nothing special but, on the other hand, they could be special.
Maybe f(x + y) = f(x) + f(y). Maybe f(x•y) = f(x) f(y). Maybe not.
In trigonometry there are formulae for sin(x + y) and cos(x + y). These formulae are just examples of how the functions sin and cos, whose domain I take to be all numbers, behave with the operation of addition.
There are other properties that a set may have and functions relate to. For example there can be a distance between two numbers.
Here I’m thinking of the numbers on a number line and the distance between two numbers, x and y, is
|x-y|
This is the distance you would find with a tape measure.
If x and y are close together, are f(x) and f(y) close together? How does the distance between f(x) and f(y) relate to the distance between x and y? This gives rise the concept of continuity.If we examine this relationship a little closer, we look at the ratio
f(x) - f(y) / (x - y).
This gives rise to the concept of the derivative which essentially describes how fast f(x) changes as x changes.By introducing the idea of graphs, we get a picture of how f(x) changes as x changes.
My point here is not the details of the analysis of the relationship between x and f(x) but the kind of questions that are asked.
This type of overview is something that most courses in mathematics leave out. One of the most egregious examples is the introduction of the epsilon-delta process. The student is asked to show that that sin x is continuous using epsilons and deltas when it is obvious that it is continuous. Until the time of Weierstrass mathematicians assumed that all functions were piecewise continuous and they did just fine, particularly with the kind of mathematics the undergraduate encounters. Undergraduate mathematics barely gets out of the 18th century.
When I was in graduate school, at the beginning of each semester an advanced student would tell me what was going to happen in the courses I had signed up for. Then I wasn’t driving 90 mph down an unknown highway at night with my dims on.
I would suggest that a course in mathematics start by telling the students what the point of the course is, what the instructor wants to accomplish and why.

General Principles

2009-05-12T14:01:00.001-07:00

General Principles

It is my contention that mathematics is a collection of general principles.

Mathematics has to do with sets and I will start with the set of positive counting numbers. My development is typical of the development of sets used in an algebraic way in general. (See Arithmetic, March 2006)
I start with positive counting numbers because almost everybody counts and generally they start by counting on their fingers.
When a child is given a roll of Life Savers she will count them and see how many she has and thus become aware of positive counting numbers. When all the Life Savers are gone a child will add zero to his set of numbers. When a friend gives him some Life Savers, he adds. When he gives some of his Life Savers away he subtracts. Subtraction is "take away".
He sees that if Ed gives him 3 Life Savers and John gives him 5, he ends up with 8 Life Savers regardless of who gives him Life Savers first. Thus he becomes aware of operations on counting numbers and that they obey certain rules.
In a later mathematics class, the positive integers are introduced more formally and the rules stated more explicitly but the student has everyday experience and her fingers to fall back on.
Most children start adding counting numbers on their fingers and the rules are introduced to make the process more efficient. Since the Arabic symbols 2, 3, 4, 5, 6, 7, 8, 9 give no clue as to how many fingers they represent, they have to be memorized.
In an arithmetic class the numbers are kept small until the student believes in the rules. But when the numbers become large, the numbers lose intuition. A student can't use their fingers to compute 2348 + 4729. Intuitively there isn't any difference between 4728 and 4729, the difference is mathematical. So, the rules must be extended into the mathematical world. Adding 2348 + 4729 must be done formally.

Multiplication is introduced as fast addition, 4 x 5 = 5+5+5+5+5 and a little more memorization is a good idea. Division is introduced as multiple "take aways". 14÷3= how many times 3 can be taken away from 14, 4, and the remainder, 2.
The point is that I have a set, of counting numbers, and have put an arithmetic on them, that is, operations on them.

When he loses a pack of Life Savers he doesn’t have in a game of horse, he becomes aware of negative numbers to some degree.
The idea of something that is less than nothing is an intellectual and philosophical leap. I can recall in algebra class that negative roots of polynomials were dealt with gingerly.
I think that the teacher has to make some sort of intuition for negative numbers. I say that a negative number represents the opposite of whatever a positive number represents. If a positive number represents distance to the right, a negative number represents distance to the left. If a positive number represents time after the clock starts, a negative number represents time before the clock started.
This implies that the meaning of a negative number depends on context and the meaning of a positive number. Since opposites cancel each other it makes sense that 5 + (-5) = 0.
(It also makes sense that all these ideas were eventually abstracted by mathematicians and made independent of context.)

This is where we start, a set, in this case the positive counting numbers, with operations, in this case addition, subtraction (take away), multiplication (fast addition) and division (multiple take away), defined on it. Some rules hold, like 4 x 5 = 5 x 4, 4+5 = 5+4.

But there is a problem. 5 x (-4) = adding up 4 losses five times = -20 makes sense but (-4) x 5 = adding up 5 a minus 4 times doesn’t.
Here I do something that will be done many times when a set of numbers is enlarged and the rules don’t make sense when applied to the new numbers. (-4) x 5 is not defined and if something is undefined it is ok to define it. I define it so that the rules work.

(-4) x 5 = -(4 x 5) = -20 so that (-4) x 5 = 5 x (-4) = -20

This is where most mathematics that leads to computation starts. The sets may be other than the counting numbers and the operations may be other than the standard arithmetic operations but mathematics starts with a set and some operations on it where the operations satisfy some rules.

The next step after integers is fractions. Making intuitive sense of the ratios of integers is not easy. Elsewhere I have gone into the problems of fractions in some detail. (Fractions and Rational Numbers Revisited, January 2006) Here I will just say a few words about the rules.
We have addition, subtraction, multiplication and division of integers and we now have to define those operations on fractions.
The problem is complicated by the fact that every fraction has many, many, many representations. 1/3 = 2/6 = 3/9 = … for example.
Another problem is multiplication. 5 x 1/3 = add up 1/3 (of a pie perhaps) 5 times. But what does 1/3 x 5 mean? How can I add 5 to itself 1/3 of a time?
1/3 x 5 is not defined and we define it so that 1/3 x 5 = 5 x 1/3.
When we introduce new numbers we have to define the operations so that the rules applied to the new numbers is consistent with the rules applied to the old numbers.
It is my opinion that when arithmetic is taught, this process should be pointed out to the students. It is my opinion that the student should be made aware of what is going on and that this would dissipate a lot of mathematics anxiety. The way it is now, as of this writing, arithmetic in particular and mathematics in general is taught in the dark and the students are afraid of the dark.

Thought on learning mathematics

2009-05-12T14:00:00.002-07:00

Methods of Learning

My approach to learning how to use a computer was to ignore it until I had a need for it and then figure out how to make the computer fulfill that need. I used this technique to learn about motorcycles; when something broke, I learned enough so that I could fix it.
I know people who as soon as they got their computer started fooling around with it, doing this, doing that, and this was how they became familiar with the beast.
I know people who took a course of instruction on the use of the computer. They have left the decision of their possible needs to someone else. Those who like the very idea of computers and enjoy doing anything on them, find the lessons a pleasant way to spend their time.
I used the "fool around technique" to learn about photography. I started taking pictures, saw what kind of pictures I could take and then tried to improve them.
Many guitarists I know started when they heard a tune they liked and wanted to play it. They learn the tunes they like.
I know a guy whose dad had traded a motorcycle for a drum set. It was just sitting around the house so he started fooling around with it and became quite an accomplished drummer.
I started to learn classical guitar by taking lessons from a former student, letting him decide what exercises and tunes I should use to master the instrument. Since I liked the sound of all the exercises and tunes it didn't make much difference where I started and the lessons were partially successful.
What about learning mathematics?
If a student, mirable dictu, is interested in mathematics then any of the learning techniques, need, fooling around or lessons can have some success.
Unfortunately, most students are in a mathematics course because they are required to be there. They see no need satisfied by mathematics and they would rather spend their time doing anything else than fooling around with mathematics so mathematics education falls back on an extreme form of lessons to teach the students.
It is like being forced to take guitar lessons without ever having heard a guitar, without knowing what the end result is supposed to sound like, indeed, without knowing what music is.
"Just put your finger here and pluck this string, then put your finger here and pluck this string..." and before you know it, voila, "Classical Gas". I had a friend who was absolutely tone deaf and claimed that he was going to use this technique to learn the guitar; he wanted to be the life of the party and felt that the guitar was the path to this desired end. I never knew how it turned out.
"Just follow these rules of mathematics, memorize these formulas and before you know it, voila, differential equations."
I am always surprised, although I suppose I shouldn't be, that when I ask a student who has passed a course in differential equations if they know what a differential equation is, they have no reply. Some can even solve one without knowing what it is.
A guitarist pointed out to me that the end result of the fretting and plucking is sound and that's the ball you want to keep your eye on.

Algebra 2

2009-05-12T14:00:00.001-07:00

Algebra 2

I have a young friend taking Algebra 2 at a local, well regarded, high school and I asked the student about it. He was non-committal; it was a course he had to take and hence deal with. The instructor jumped around in the book and didn't hand homework or tests back promptly. I asked about the text for the course. He said that he didn't read it, just used it for the problems assigned.
I asked him to let me look at it.
The book is Algebra 2 by Larson, Boswell, Kanold and Stiff. It was published in 2001 and I presume it is the First Edition.
The first thing that struck me about the book was its weight. It can't be read while holding it; you have to put it on a desk.
It has 1000 pages. and there are 17 pages in the Table of Contents. It is my contention that a high school algebra text doesn't need 1000 pages. You should be able to lift it with one hand.
This book follows the recent practice in mathematics texts of having every topic that anyone ever suggested to the authors. Since the whole book can't be covered in a finite amount of time, an instructor has to jump around.
The general format was childish with cute pictures and shaded or boxed formulae. I don't know why the "real world" applications were put in. They weren't covered in sufficient detail to give the student any real information. They seemed to be stuck in so that the authors could say they had "real world" applications. This was true about most of the topics covered.

I started with Chapter 2. Quotation indicate material taken directly from the book.

"Chapter 2 is about linear equations and functions." Does the adjective "linear" modify "equations and functions" or just "equations"?
A further reading seems to indicate that linear modifies just equations. The chapter is about functions and linear equations. Why these two topics are paired is less clear.
"A relation is a mapping, or pairing, of input values with output values."
What is a mapping? What is a pairing? What is an "input value"? What is an "output value"? Input to what? Output from what?
"Relations (and functions) between two quantities can be represented in many ways, including mapping diagrams, tables, graphs, equations and verbal descriptions."
What is a "mapping diagram"? What is a graph?
In the next paragraph "graphing" is used as a verb. I never did see a definition of "graph" although on the following page "graph" is used as a noun. I couldn't find any place where graph was defined.
I read more of the book, jumping around as seemed to be the drill when using it.
Rules are given for the computation of the determinant of a 2X2 and 3X3 matrix. Rules are given to find the inverses of these matrices. Rules, rules everywhere nor any reason why.
It is beyond my comprehension why this book was chosen. Maybe straws were drawn. It is unbelievably bad. I would be interested in hearing someone defend it.

Attack on the Teaching of Mathematics

2009-05-12T13:59:00.004-07:00

An Attack on the Teaching of Mathematics

Why is it that the United States ranks so low internationally in mathematics?
The reason that occurs to me is that the United States ranks low in respect for intellectual activity. The United States ranks high in belief of creationism or intelligent design. The United States elected Bush as President twice and regardless of how one rates his policy, I don’t think anyone ever accused him of being intellectual.
I think that every country can have its stars. A country can put together ten basketball players to win the Olympics but that country can’t staff an NBA. I would guess that every school in America from middle school on up has a basketball team. Everybody the U.S. knows who Shaq is but how many know who Smale or Stein is.
The United States can produce great mathematicians but can’t produce the players to support an infrastructure. We are missing the intellectual minor leagues.
Mathematics is nothing if not an intellectual pursuit.

The majority of students in the United States seem to feel that school is a social institution, not an intellectual institution. When I was involved in home schooling kids I was told that I was depriving them of the socializing influence of going to school. (This was a lame excuse for going to school in my opinion, an opinion that I am willing to defend but not here.)
The United States, until well into the 20th Century, was 90% rural. Kids went to school when the farm work was done. There was no centuries old tradition of scholarship.
This does not say that there were no American intellectuals but America was born as a Hog Butcher and Steel Maker, not a scholar. Intellectuality in America has not reached the "tipping point" nor does it appear that it will soon.
Not all that long ago people could build their own house and fix their own car. In the country it is still valuable to have the skills that built this country and made it great. A hundred years ago people were needed who raised hogs and poured steel. Intellectuality wasn’t needed, know-how was. With a high school diploma a person was prepared for their future life.
Now steel is poured overseas and hogs are raised on a mega-farms. A high school diploma gets you a minimum wage job.
Unfortunately teachers have grown up in an environment of diminishing interest in the intellectual. In fact, my dad’s grammar school in Deer Lodge, Montana in 1910 was more intellectual than the grammar school my daughter went to in 1998. “School to work” is a catch phrase of education. As the number of graduating mathematicians decreases, the Business School grows.
The teaching establishment looks for new ways to teach the same old stuff apparently thinking that this time it will work but doing the same thing and expecting a different result is a definition of insanity.
Every calculus book I have ever taught from asks the student to maximize the volume of an open top box with a square base with a given area of sides and base. They had the same list, more or less, of functions preceded by “Take the derivative of the following.” or “Evaluate the Integrals” etc. The even numbered are assigned because the back of the book has the answers to the odd numbered problems.
I think it is insane to expect better test results this semester than last.
Another example of this kind of insanity is thinking that armed force will change the minds of an indigenous population.

Instead of making the school experience intellectual, teachers in the lower grades tell me they try to make it fun.
Now my definition of fun is meeting a challenge but I get the impression that the definition the teachers have in mind is riding a merry-go-round. I understand that a merry-go-round can be used to demonstrate the coriolis force but I don’t think that was the fun they had in mind.

The first calculus text I used was Sherwood and Taylor, a reasonable book of reasonable size. The first calculus book I taught from was the first edition of Thomas. It was a bit thicker than Sherwood and Taylor but I thought it was a good book.
Thomas went through a sequence of new editions which evolved into new editions of Thomas and Finney. Every edition was thicker than the last until the last edition I taught from was so heavy that I had to cut it into two pieces so I could carry it.
Each edition, and God only knows what edition they are on now, was more unteachable than the previous edition, continuing a monotone sequence of increasing obscurity.
When I read a section (a section a day was the drill) that I had assigned, I would see that in three hours I couldn’t read it and work the assigned problems. The literary style sucked. The section was not self-contained and referred to previous sections scattered throughout the book. The shaded formulae were unhelpful. (I would suggest to my students that they take a black magic marker and blot out shaded formulae.)
I am not going to give a section by section critique of Thomas’ legacy but I would point out that the evolution of Thomas is typical of the evolution of solutions to any serious problem in this country, maybe in all countries; by making things worse.
To solve traffic problems a city will add lanes to existing roads and put more curlicues in highway interchanges asymptotically approaching perpetual gridlock.
The tax code is made thicker and more indecipherable each year as are laws generally, asymptotically approaching the livelihood of an infinite number of lawyers. Simplification is an unknown art.

As in many areas of human endeavor a lot of dead ends are followed. The New Math, Piaget rods, the Harvard Program are a few in teaching mathematics come to mind. (Surely the students know, deep down, that they are being fed crap.)
This procedure has many names: the quick fix; too little, too late; a day late and a dollar short; looking under the street light for your lost keys because the light is better there.
The surge of troops in Iraq falls in this category

Personally I have no hope that the teaching of mathematics will change for the better in the next several decades and for the general student population it will get steadily more irrelevant.

One of the evolutionary trends that I noticed over thirty-seven years of teaching was the evolution of the normal distribution of grades to a bi-modal distribution; students got it or they didn’t. The class average became a meaningless statistic. It isn’t as though we have no good students in the sciences and mathematics but the gap between the haves and have not’s is widening. Where is the B Student of yesteryear?

Why Teach Mathematics Revisited

2009-05-12T13:59:00.003-07:00

Why is Mathematics Taught Revisited

Why is mathematics taught? I ask myself this question from time to time but have come up with no universal answer.
Since mathematics is the language of science, the scientist must know mathematics but it doesn’t seem universally agreed on how much and what kind. Statistics seems a necessity for a variety of disciplines from business to biology and evolutionary genetics.

One answer is that mathematics teaches students to think rationally and critically.
I have espoused this raison d’etre to my students who have asked me why they are having mathematics inflicted upon them. I pointed out that the difference between people and the two dogs that lived next door to me, whose total activity was eating, copulating and sleeping, was that people thought rationally and critically.
A young man in the class remarked that if the dogs had a fast car their lives would be perfect.
He didn’t seem to put a high value on the human capacity for original thinking.

When I was in junior high school the cold war had yet to become an obsession and Strom Thurmond was just starting the Dixiecrats, but even the school hoodlums discussed these things. They went to the library regularly and read books about Black Hawk and Red Grange; they read about egocentric sports stars who learned about team work.
Cheyenne, Wyoming was not the intellectual center of the country in the late 1940s, it was a cowboy-railroad-military town, but I never heard a classmate say they hated mathematics or any of their courses as far as that goes.
I haven’t been back to Cheyenne since that time and I don’t know what it’s like now. I do know that the junior high, which was new when I started school there, has been torn down, sic transit gloria mundi, an expression I use a lot these days.
But, quoting myself, “Of all sad words of tongue or pen, the saddest these, I wish it was the way it used to be.”
But it isn’t the way it used to be and education in general has to be rethought.

I think students in the early grades can see that basic arithmetic is useful. The use of fractions is less obvious as is the technique of computation, but, the student is told, “Just do it and you will see why later.”
And then comes algebra. While the student could at least see how fractions could be used in dividing up a pizza, the uses of algebra were truly obscure. But, the student is told, “Just do it and you will see why later.”
For many students this is not true. It will be true for those who become scientists but I don’t think it is true for those who don’t. I don’t think most people, even scientists, compute how long it takes Ed and Bill to paint a house together.
I have thought about it and I can’t think of a time when I have used algebra outside of my profession in mathematics. Well, one time when in college I did a mixture problem making gallons of Manhattans for a party.
Even as a mathematician I have never done one of those long cancellation problems with fractional exponents.
I don’t think that the teaching of mathematics has really changed much since…forever. Probably the teaching of history hasn’t changed all that much either.

Maybe mathematics could be taught as topics arise instead of in some pseudo linear way. Why not introduce calculus before algebra and then calculus would supply a reason for considering algebra? Why not try to describe a damped-spring-mass system which would give a reason for considering calculus?
Why not try a different way of developing mathematics?

I will often ask a high school graduate if they took any mathematics and they usually admit that they have taken algebra. In New Mexico it seems that some algebra is required for graduation.
And then I ask them to tell me something they learned in algebra. The quadratic formula is the popular answer to my question but when I ask what the quadratic formula is, the fact that there is a square root in it is all they really remember. They don’t remember what the quadratic formula is used for.
Why is it that a student can take Algebra 1, Algebra 2 and Pre-Calculus and remember almost none of it? Why is it that mathematical amnesia doesn’t seem to bother anybody? Why isn’t this talked about?
Why don’t mathematics teachers discuss the fact that they spend hours and hours teaching kids things that go in one ear and, at the end of the semester, go out the other?
Why is it that if I ask a person who has passed a course in differential equations what a differential equation is, I receive a blank stare. (The derivative and differential equations describe physical systems. The integral computes.)

Why is it that nine out of ten people I ask what they think about mathematics say they hate it and aren’t any good at it. They often blame a teacher, most often their 7th grade teacher. Why isn’t this general dislike of mathematics talked about?
I recall that when a new book was chosen for algebra or calculus the procedure involved going through a lot of books but with no discussion about what should be taught. In retrospect I suppose this was a topic we would rather not talk about.
Why don’t mathematics teachers discuss the fact that they spend hours and hours teaching kids things that go in one ear and at the end of the semester goes out the other?
Why is it that if I ask a person who has passed a course in differential equations what a differential equation is, I receive a blank stare. (The derivative and differential equations describe physical systems. The integral computes.)

Why is it that nine out of ten people I ask what they think about mathematics say they hate it and aren’t any good at it. They often blame a teacher, most often their 7th grade teacher. Why isn’t this general dislike of mathematics talked about?

Freely Chosen Thoughts on Will?

2009-05-12T13:59:00.001-07:00

Freely Chosen Thoughts on Will?

I saw an episode of Law and Order where the point of contention was whether the will to rape is inherited, that is, genetic, and not in control of the rapist, or that one chooses to rape, as an act of free will, and hence is responsible for the act.
“Not his fault”, from one side; “Responsibility”, from the other.
Does genetics affect freedom of the will and if so, how much? I guess this is a question where “free will” needs a good definition, which unfortunately, I am not going to provide.

Let me begin by saying that I don’t really know what is meant by free will. I had heard the phrase and thought I knew what it meant but when asked to explain it, I fell silent.
When I decided to put on a CD of The Dixie Chicks, was it an act of free will or was it predetermined? Is this even an appropriate use of the expression, “free will”?
I have heard free will discussed in the context of cause and effect. Each link in a cause/effect chain is determined by the previous link. Each instant of behavior gives the initial conditions that determine what happens at the next instant. Or maybe it is the entire history of the chain up to and including the instant of “˜now’ that determines what happens in the “next” instant.
Was my choice to put on a CD of “The Dixie Chicks” determined at birth? Maybe it was determined at the beginning of the universe. Or was my choice of The Dixie Chicks the result of choosing randomly among my CDs? Is there some element of probability in free will? Does the operation of will belong in macro-quantum mechanics?
Trying to remedy my state of ignorance, I checked out free will in Wikipedia. Apparently there are as many ways that the will can be free as there are philosophers. I was soon swamped with versions of free will and after reading a few, I decided that enough was enough. Well, something decided, by hook or by crook, that enough was enough...sometime.
In a kind of straw poll, all the people I’ve talked to lately about the freedom of the will had a different opinion but all of them said that they thought there was some kind of free will. Most of the discussion was about the possible ways that the will might be free.

It seems to me that any concept of a free will would incorporate a choice, and a choice would require a “something” to make the choice. But does “something” evaluate and freely decide or is “something” preprogrammed? Or is there “something else”? Does “something else” program “something”, deciding to tell “something” the instructions to pass on?
Is there a “first decider” and on what would a “first decider” base its decisions?

I have heard that a certain spot in the brain lights up about half a second before the subject of the experiment is aware that she is going to reach for a glass of water. I’m not implying that the glowing spot is the “something” but evidently some kind of activity was going on before the thought occurred to her to reach for the glass. At the time when she thought she was making a choice, it had already been made. Perhaps the thoughts that we are aware of are how “something” puts its decisions into action.

I remember watching news reels during The War and thinking that the bombs falling on German and Japanese cities were falling on kids just like me. I wondered about the fact that I was born in Boise instead of Berlin or Tokyo. I marveled at my luck.
In later years I looked back at that feeling and realized that as a kid I had automatically assumed that the “something” that was me could have been born anywhere and I wondered how that choice was made, if at all.
It is that “something” that could have been born in Mongolia that is in question. It is the place where the will resides. It is this “something” that many philosophies claim doesn’t exist. Some say the “something” is a meme. Some say it is the soul. Some say it is the unconscious, a cop out if there ever was one. But whatever the “something” is, pretty much everybody starts off life thinking that they have one; thinking they have a “something” that is them.
This feeling of self-identity is very strong. Buddhists go to considerable effort to understand the meaning of “There is no I”.
So the “something” is there. We may not understand it but it’s there.

I don’t think it’s possible for the brain to understand the brain. A brain can only understand something simpler that itself. We will never understand the “something” that makes us feel that there is an “I”. Not in this world anyway, at least not this month.
It is my hypothesis that the will makes decisions, determined or free, and that the aware mind is logically unable to understand that which controls it, that is, the will.

A lot of people try to understand the brain by likening it to a computer and I question this analogy. The computer is discrete and does not have a free will.

I think the difference between a computer and a brain lies in the difference between discrete and continuous. It is more or less clear what happens in a computer when decisions are made. Even though computers can get petty complicated, so complicated that one person finds it hard to grasp the entire system, they aren’t as complicated as the brain. I would think that the number of possible connections in a brain is beyond human comprehension.
When a number gets so large that it is completely beyond human understanding I will call it humanly infinite.
A discrete collection of dots looks like just a bunch of dots but as the density of dots gets larger and larger the discrete looks continuous and may even be a recognizable picture. The discrete passes to a state I call humanly continuous.
When we pass from the finite to the humanly infinite or from the discrete to the humanly continuous, I think we pick up unanswerable questions.

As I write these words, it is my perception that I have the feeling that I am writing these words. Sometimes words come to my mind and I write them down. Sometimes I write words that my fingers seem to write by themselves, skipping the mind step. I have a sense that I am writing about free will and the words that come to mind are appropriate. I choose between two or three different ways of expressing a”¦thought? and either choose a rendition or dump the idea entirely. I have made a decision. I have used my will freely “¦or have I?
A little exercise that piques my wonder is listening to myself talk as I do it. Where do the words come from? I don’t think out each sentence before I say it. I stand amazed as words pour out of my mouth, seemingly from out of nowhere. It is much like the feeling I have when I realize that the objects I think are “out there” when I see them are really inside my head.
The actors who say their lines poorly sound as if they are reciting words that have been memorized. Good actors sound like the words are the product of the instant.
We can’t understand that “something” because it is humanly infinite and humanly continuous with respect to what might be called the reasoning part of the brain which is discrete and finite.

Is there a God? This question is about something which, if God exists, is humanly continuous and as such has no answer in a discrete thinking brain.
Absolutes often involve opinions of concepts that our discrete, reasoning brain can’t understand. “Nothing” is an example. I find the Big Bang theory hard to understand because I have to deal with the concept of “nothing”. What does it mean for “nothing” to exist? How do I describe “nothing”?
If the universe goes back forever, then the universe has no beginning. I find it hard to bend my discrete, overtly thinking brain around that.

I have read that in ancient Greece, the highest virtue was moderation. If we look at Homer’s Iliad as a morality play, the reason Achilleus had to die was that he had sinned when he went on an extreme in killing spree after Hector killed Patroklos.
I tend to follow this Greek philosophy of moderation when I consider my opinion of the freedom of the will. I don’t think that the act of pushing my glasses back up on my nose is determined by fate. On the other hand, when a doc hits my knee with a rubber hammer I don’t seem to have much choice in whether to kick or not. A free will lies between those bounds which more or less implies that talking about free will is for late hour bull sessions in the dorm.

I’ve looked at will from both sides now
From bound to free but still somehow
It’s will’s illusion I recall
I really don’t know will at all.
(a la Joni Mitchell)

Sacred Text

2009-05-12T13:58:00.000-07:00

Sacred Text

What about the Word of God?
Evidently the angel Gabriel recited The Koran to Mohammad when he was on a mountain. When Mohammad came down from the mountain he recited what the angel had told him to a scribe. This may not be quite right but the idea is that the word went from God to Gabriel to Mohammad to scribe. The Koran is just three degrees removed from God, so one would suppose that The Koran says pretty close to what God wanted.
The same is true of the Book of Mormon. The Words went from God to an angel and tablets of gold, I’m not sure of the order, and finally to Joseph Smith who wrote them down; Joseph Smith doing double duty as both the reader of the tablets and the scribe. The Book of Mormon is also three degrees removed from God and should also be pretty close to what God wanted written.
The books are quite different but are they so different that they contradict each other? Is it possible that both The Koran and The Book of Mormon were dictated by the same god?
Divine texts share a common property. They are all written in such a way as to allow a wide variety of interpretations. This rather complements the human fascination with interpretation. Humans can interpret the Word of God in such a way that the different texts complement each other or in such a way that the texts call for the destruction of any other God; and shades of gray between these two extremes.
I’m not sure about the Vedas. I have been told that the Vedas were existent before the gods and that in the beginning humans were smart enough to remember them. But humans degraded to a point where they couldn’t remember the Vedas and had to invent writing so they could write them down. This seems to beg the question as to where the Vedas came from since they were always there. But this is just what I was told and there may be other versions. I don’t consider the Eastern versions of gods, not because they aren’t cool but because I am, at the moment, considering monotheism.
Except for the tablets on which The Ten Commandments were written, it seems pretty well agreed that the Torah and The New Testament were written by people. The degree of separation from God is more difficult to determine. If I suppose the author of a particular part of The Bible was divinely inspired then the author was, therefore, memorizing or writing exactly what God wanted. The degree of separation between the words we read today and the original words depends on the number of times they were passed on orally and written until the earliest example we have of the written word. I’ll suppose that from the printing press on there wasn’t much error in reproduction.
I suppose the copyists could be divinely inspired but then copy wouldn’t change from generation to generation or recitation to recitation” unless God had reasons for letting a certain number of both random and intentional errors into the copies. Actually, God must have realized that there would be differences between copies. Perhaps God made the meaning of the words remain even after copying errors. As language developed there would be more versions of The Bible. God must have anticipated this.
By the time God started on The Bible, God must have had a pretty good idea of what humanity was like. Cain had killed Abel and the flood had occurred. It was evidently important to leave the story of Noah to following generations, that the first thing Noah after the flood subsided was build an altar and the next thing was plant grapes so he could make wine and get drunk. Perhaps God is pointing out something that is an important universal characteristic of humanity. Perhaps the inner message of these stories is a set of survival instructions. Hidden in the words is the knowledge of how humans could get along. Perhaps Biblical scholars could try to find that message. But God wouldn’t have restricted the knowledge to scholars; the knowledge would be there for anybody. Since copying errors don't affect meaning, the meaning must be in the stories. But that meaning would have to be in all ways to interpret the stories from all over the world. The search would be for nuggets that were common to all stories...which would require knowing all the stories. That should keep some people off the streets and out of the pool halls for awhile.
While the writers of the Torah are often known only by the word they used for God, much more is known about the writers of the New Testament. There were conflicts because there were different memories of how events happened and what was said. But maybe that was part of the lesson.

Instead of starting with an estimate of the veracity of The Bible, I start with what I am reasonably sure is correct. It was written by people.
I can read The Bible and see that it was written by people who were articulate. In my opinion much of the writing shows insight and depth and wisdom. Inspired by God or not, The Bible is a worthwhile read.
So why isn’t that enough? If Jesus really is The Son of God, then the words attributed to Him, the Sermon on the Mount, for example, should be given serious attention, but if Jesus is just a guy like, say, Billy Graham, then His words can more or less be ignored depending on how one feels at the moment of consideration.

Suppose that in some way God created humans. God surely must have foreseen that humans were going to develop scientifically, that they would multiply and fill the earth. Surely God must have foreseen that increasing population and technology would make the way humans lived their lives be quite different after thousands of years. If God were going to give humans something to help them through thousands of years, God had to write a book that would be useful beyond the time of Moses.
When we read a prohibition against eating pigs, it was there because that was a helpful prohibition before refrigeration. But after refrigeration it isn’t so helpful and we see that there is a deeper meaning, it means don’t ingest things that are bad for you, like meth. The pig represents things that may be harmful if you eat them. If the hook worm problem is controlled, eating bacon is no longer prohibited because it is no longer harmful. That prohibition in The Bible should now mean, "Don’t smoke crack.”
God knows, we could use a little direction now. Surely God, who guided the hand of the authors, would make a book that would be useful until humans could survive without it. Come to think of it, he would guide the hands that wrote the Vedas and The Koran. I don’t suppose that every book is inspired but who knows?
The progress of humanity is measured by how well it sees that the books were all inspired by the same god. Or by how well humanity finds the deep insight to find the pigs in their own lives. And maybe there is meaning behind the act of caring about trichinosis.
The stories grow in meaning as humanity ages and human minds expand to see an ever growing meaning in the stories. Would God write a book whose meaning was static? Would God write a book with just one meaning or would God write a book whose meaning grew with the growing experience of successive generations? Maybe God planned for a new edition every so often.
Would God intend the story of Cain and Abel to have the same meaning ten thousand years ago as it does today? Would God intend the meaning of “brother” to mean the same now as it did ten thousand years ago? Would God be so ordinary? Maybe there is lesson in the fact that there are four Gospels and not one.

The mystery of The Bible is how it is ever changing, how each reader reads their own Bible. Surely God meant it to be that way. God knows what humanity is like and wrote The Book for them. Is God’s point in inspiring The Bible to lead to conflict or to accommodation.

There is a concept of consolidation of information which was told to me by a student. The idea is that when you first look at bits of information, each bit is a separate instance. When I start teaching max/min problems the student sees a separate instance in each problem that I put on the board. Each step in working the problem is a separate instance. But I see max/min problems as a single instance. From my point of view, I’m working the same problem over and over. The student sees ideas flying at him thick and fast, I see throwing only one idea.
After continued study calculus becomes a single instance of one idea. The single idea carries all the information.
My friend said that God was that entity that had consolidated everything, the universe into one instance.
It is possible that the consolidation of The Bible into one instance is the path to understanding life. It is possible that that “instance” holds the meaning of it all.
With this possibility at hand, I wonder why sacred writings are read to find differences and not commonalities. Why are the scriptures read in such a way as to ensure perpetual conflict?
Well, The Bible points out that humans are an unruly bunch and not all that quick to learn.

Rambling Comments on God

2009-05-12T13:57:00.002-07:00

Rambling Comments on God

I do not know if there is one god, many gods or no god at all. After thinking long and hard on the matter, I come to no conclusions.
It is my opinion that no one knows for sure that a god exists. I say this while well-aware that many people claim to know; some say they know there is a god and others say they know there isn’t. But “knowing” that god or gods exist, that they are real in the sense that they materially affect material things, doesn’t imply that they do exist. On the other hand, “knowing” that there is no God in any, shape or form does not imply they do not exist.
“Knowing” that God exists is the same as having faith that God exists.

Having had some little but surprising experience with hallucination, I am amazed at what the human mind can convince the owner of that mind is there but isn’t really there. Thus eye witness accounts of appearances of the Virgin Mary, angels and various saints do not convince me that these manifestations of God were actually there. Magicians earn their livings by convincing the eye that something occurred that didn’t really occur.
I think it is all about faith that a particular god exists, which for those with faith, is the same as “knowing” that a particular version of God exists. If God were real in some obvious way, then God wouldn’t be a god anymore. If there is a God then It must necessarily be forever unproved; for if its existence were proved, It wouldn’t be a god anymore. God should be beyond human understanding else It is just another comic book super hero.

Given that the concept of God is beyond human understanding, how does one talk about God? What is God? What is meant by the word “God”? What is God made of? If God exists then It must be made of something. What does it mean for God to exist if It is made of nothing? Is God everywhere? Is God internal or external to the universe?
God, The Spirit and The Word are undefined terms. As far as I know there are not a set of axioms that God satisfies; certainly not a set that any sizable number of people agree on.
A God starts out as The One True God. A True God satisfies an amorphous set of axioms that arise from convenience and are sometimes added after the fact.
I don’t know how many One True Gods there are but it seems to me that asserting the existence of One True God is similar to asserting that there is One True Point; and you know which one it is, except that few are willing to kill and die for their “One True Point”.
The idea is that finding The Unique One True God is like finding The Unique One True Point, it’s a meaningless task.
There can’t be two “One True Gods” that are very different. I suppose that The One True God of a Baptist Church in Atlanta may be a little different than The One True God of a Baptist Church in Los Angles but nothing to get upset about. When the difference reaches some critical point the 2nd Baptist Church is born.
The Hebrews and Arabs apparently both started out in the Tribe of Abraham and they eventually came up with very different “One True Gods”. They were so different that acceptance of one implied rejection of the other. This has led to animosity.
Then the Northern Tribes came on the scene with their own “One True God” and put Him into the fray.

Philosophers and theologians try to define God but if there is more than one definition, then for all practical purposes you don’t have a definition of God, you have an argument.
According to Karen Armstrong in “Battle for God” an early Muslim scholar said that all religions are correct; different religions are different faces of God. This more or less makes sense but so what? This idea didn’t stay the hand holding a sword on its way to erase one of those faces.
I think that in a practical sense the existence of God is not a philosophical question but a psychological question. People who have a “One True God” aren’t interested in logically defining that Being. Their Being satisfies their psychological needs without befit of axioms very well, thank you very much.
I don’t think that logic convinces that God exists nor does logic convince that God doesn’t exist. How much a person believes or disbelieves in God does not depend on logic, it depends on the psychology of the person, depends on how the person’s brain is wired. The actual existence or non-existence of God seems to be logically independent of how many people believe God exists.

Evolution

2009-05-12T13:57:00.001-07:00

Evolution, truth or fiction

I have read that the dinosaurs were around for about 120 million years. It doesn’t appear as though the dinosaurs did all that much evolving in their 120 million years, certainly their table manners didn’t evolve to any great degree in their allotted time. In the next 65 million years the rat evolved into us. But mammals are fast evolvers, I guess.
But maybe the minds of dinosaurs evolved. Surely the ability to think gives an animal a survival advantage. Among the random mutations that evolutionists are so fond of appealing to, surely in 120 million years there must have been a few that gave rise to a better brain. Inside those thick skulls their supposedly tiny brains may have evolved into sensitive, very intelligent minds. Inside, the minds of these beasts may have been constructing poems, symphonies and deep philosophical essays.
The extinction of the dinosaurs could have been a mass suicide of animals that had become depressed that at the end of 120 million years they were still unable to express the sublime thoughts that were trapped inside bodies that couldn’t give expression to those thoughts.
Perhaps they philosophically viewed the impact of an asteroid as a cosmic end to their untenable life on earth.
Since we have no idea of what went on in these early brains it seems to me that one guess is as good as another. Humanity compares all brains and behavior to own, considering its own as the epitome of brains and behavior of all sentient life. Humanity takes as a given that humans have brains of perfect size and operation.

*****

A possibility, besides the rat to human hypothesis, for the appearance of the human species is the alien possibility.
Ancient Chinese used to cross breed the mutations of carp to make truly bizarre fish. But if you toss the mutations into a pond and let them procreate naturally they are back to carp in a few generations. This is true for hybrids generally and is why kernels of hybrid corn won’t grow the hybrid variety but will give the original corn.
So some aliens came to Earth, bred mutant apes and developed a hybrid that we call humans. But as hybrids do, the humans are returning to the animals from whence they came. We did not evolve from apes; we are devolving back to apes.
Apparently evolutionists claim that mutations can become permanent. I am told that the increase in cranial size, which gave room for bigger brains, was one such mutation. But I don’t see how this happened.
Was some woman born with a bigger head and the local studs thought that she was a hottie? Was some stud born with a bigger head and the ladies thought that he would be a good father for their children?
How many of these early proto-humans were so mutated? The death rate in those early years was very high so the likelihood of one mutation spreading would not be large. Did the mutation happen to a lot of beings at the same time? Was this a miracle?
The mutation must have been a dominant trait otherwise I would suppose that the big head mutation would revert back to the small head as the lion fish reverts back to a carp.
I suppose that evolutionists have an answer to these questions but it seems to me that it is not known from whence humans arose just as it is not known from whence the universe arose.
I have an opinion about how Homo sapiens came about. I think that maybe it happened like this:
Life started with one replicating strand of DNA. Since it had no predator it could replicate freely in the primordial soup until it was everywhere in the soup. At this time the radiation hitting our planet was more than at any other time and the likelihood of mutation was great. Further, the mutations would not have predators and would survive on the basis of being fittest; they would multiply freely. My idea is that the different species were started at this point. The big head was inherent in the mutation of a strand of DNA in the primordial soup.
But what do I know? Perhaps some energetic evolutionist will tell me why my opinion is outrageous.

*****
Creationism seems to dump evolution altogether. This would seem to imply that humans are as good as they get. I don’t really see why humans would be created with the urethra running through the prostate gland, a gland, my doctor tells me, enlarges with age which causes some problems for us of advanced age.
I have read that there are something like 400 species of ant in a single tree in the Amazon Basin. That’s a lot of creating in a single day.
As I understand it, Intelligent Design seems to accept evolution but an evolution that is directed by an intelligent designer.
The eye is given as an example of an organ that couldn’t have developed by chance but only with some help from a designer. I don’t see why an intelligent designer wouldn’t have given the eye a little infra-red capability, given it an extra translucent lid to keep dust out on windy days and made it a little more impervious to disease and river worms. The designer gave these properties to some animals, why not us? As a design project I’m not sure I would have given it an A.
But perhaps I expect too much from the designer. Maybe humans are still a work in progress and these little problems will be ironed out in the next million years.
I freely admit that I don’t know how our species came to be on Earth. It is one of many things I don’t know.
Many people claim to know that organically grown vegetables are nutritionally better than vegetables grown using chemicals. Richard Feynman was asked what he thought about this. Feynmann said that maybe they were but he didn’t know and neither did the people who said they did. What he did know was how hard it was to “know” something.

Existence Revisited

2009-05-12T13:56:00.002-07:00

Existence Revisited
But Existence Wasn’t Home

I think that it is remarkable that we live in the middle of an incredibly deep, insoluble puzzle. The puzzle I have in mind is the puzzle of existence. If we consider the universe we perceive around us, we don’t how it began and we don’t know how it will end. We don’t know why we’re here and we don’t know where “here” is.
I suppose it isn’t so remarkable when you think of it. In size, humans are somewhere between the universe and atoms, too small on the one hand, too big on the other. Being a poorly sized human, who am I to give meaning to existence?
None the less

“Existence” reared its ugly head in my investigation of numbers. I considered two worlds, the Real World where things that exist, like my hat, are, and the Ideal World, where things that don’t exist, like an ideal square, are.
Some symbols represent things in the Real World, things that exist. The symbol “3" is such a symbol.
Some symbols represent things in the Ideal World, things that don’t exist. The symbol √2 is such a symbol.
1/3 is an example of a symbol that represents both something real and something imaginary.
The first use of 1/3 is to be a written symbol for the real operation of separating my pile of rocks into *** piles, where each pile is equipotent with the others, and taking * of them.
By *** piles I mean that there is a one-to-one correspondence between the stars and the piles.
(I conjecture that “number” began when our species began to distinguish bigger piles of nuts from smaller piles of nuts. From this point on it was just a matter of refining the concept and developing grunts, hand signals and symbols that can be physically engraved on rocks, clay tablets and paper to express the forms of this concept.)
A “counting number” is a symbol that stands for a concept that originated in a brain, that concept being how to describe how many sheep were in the pasture.
”Counting number” arises from the desire of brains to communicate a commonly shared concept in the world around them. The concept that brains are trying to describe is about something, say, piles of nuts.
The second task of 1/3 is to be a written symbol that can be used in lieu of the symbol 0.333… to stand for an Ideal World concept. The symbol 0.333… stands for a concept that isn’t about anything. It doesn’t stand for a Real process, you can’t measure 0.333… inches.
So in the first use, 1/3 stands for a process that exists and in the second use it stands for a game piece that doesn’t exist.

But I’m throwing existence around as if I knew what it meant. What do I mean by “exists” and “doesn’t exist”?

I seem to be taking existence to be material existence. This is the “kicking Dr. Johnson’s rock” kind of existence.
There are things that I don’t think exist on Earth, or any place else, like dragons and unicorns but that is just my opinion. The things that people seem to truly believe in seem almost all-inclusive. But there are things that most people, indeed most living beings, believe exist; food, for example, and themselves.
As far as I can determine, the word “existence” is a property that people assign to nouns. As is the case with all words, they only have meaning with respect to people. People invented a word to express a commonly shared experience and this word was “existence”.
I remember reading a book by Alfred North Whitehead where he talked about “the unthinkable night” where all the stars were blazing away but there was no one to look at them. What did it mean for Orion to be there if there wasn’t anyone to observe it?
At the time Whitehead wrote the book he claimed to believe that the only life in the universe was on Earth and that eventually Earth wouldn’t be here. There would be no life in the universe. How would existence be defined in a universe that, perhaps, didn’t question existence because there was no non-existence?
What about the first seconds of the Big Bang? Surely there was no life in those first few seconds. In this primeval state there was no concept “existence”, because there was no “non-existence”. But who knows if there wasn’t any life at the being? As my mother would say, “Was you dar, Charlie?”
The description of The Big Bang that I read was a description of how it would appear to humans if humans had been there? But humans weren’t there. (You know, that’s too bad. It would have been cool being there from the beginning and seeing what went on. It would have been like being at Woodstock.)

Words are imperfect attempts to communicate concepts. A collection of words, no matter how cleverly arranged, is never equivalent to the concept in the brain. (Music and art are also attempts to communicate concepts between brains.)
The common experience of self-awareness is expressed by the words, “I exist.” We extend the concept to, “We exist.” and “That exists.”
Descartes was right in the sense that thinking implies self-awareness and if you weren’t self-aware, your existence would be moot.
While this idea is intriguing, it is too centered on life as we know it.

Suppose I base existence on observation. I will say that B observes A if B is altered in some way by A. In this case I will say that A exists for B.
This definition is pretty broad. It says that one electron observes another because the force on it has changed and so electrons exist for each other.
Some people feel that God has altered them, so God exists for these people. Since B observing A does not imply that A observes B, it is an open question as to whether God observes the person who observes God.

If an object is not observed by either itself or anything else, the question of the object’s existence is otiose.
Existence occurs in pairs, the observed and the observer, and an object may or may not observe itself. An object may be sui generis and exists only as a singularity.
An object A exists for B if and only if an object B observes A. (And B may be A.)

A exists for B because B observes A; B knows that A is there. But A may not know that B exists, A may not observe B.

If I were so moved I could formally define God using this definition of existence. God is the universal observer whose observation makes the universe “exist” for God. God by definition observes everything in the universe and God “observes” everything means that God is altered in some way by everything.
This wouldn’t necessarily imply that God exists because the definition doesn’t imply that God is observed.
If God is part of the universe then by definition God observes God and God is self-aware.
But it seems to me that to be self-ware you have to recognize that you are something as opposed to nothing. But if you are everything, where is “nothing”? There is no “nothing”. “Nothing” doesn’t exist. “Nothing” doesn’t alter anything and so isn’t observed. Everything is something. I’m not sure how self awareness would work in this case or even if it does.
But if God is not part of the universe, where is God? Where ever God is, is God observed? What does it mean to not be part of everything?
God only knows.

I’ve looked at “real” from both sides now
From yes and no but still somehow,
It’s just illusions I recall
I really don’t know “real” at all.

(a la Joni Mitchell)

Yeah, yeah, yeah, so what?
It doesn’t make any day to day difference to me whether √2 represents something that exists or not, although I might remark in passing that I don’t think that the object represented by √2 is observed by anything and hence doesn’t exist. The existence I observe is the amount of propane in the tank, the temperature inside my cabin, and the dent in my truck.

What about God? Does God’s existence make a difference in my life if God exists?
First off, I understand that for reasons known only to God, God may require faith. It is less clear to me why God would require faith for physics. I don’t see anything contradictory to God in a straight forward physics that people have minds capable of understanding.
What’s wrong with a person being able to say, “Well, that wraps up the way the physical world works. I’m glad we’ve finally got that done and we can get to something important.”
I don’t think that will ever happen. I suppose there are scientists who think that it will but they are wrong. I don’t think that it is possible to have a universe that can be understood by a necessarily finite “brain” in it. I conjecture this as a theorem.

If I think God makes a difference in my life, that God alters me, then I am observing God and God exists for me. If I think that I’m not altered by God, then I’m not observing God and God doesn’t exist for me.
So I have come to the end of existence, lost in a Sargasso Sea of ifs and contradictions and no conclusions in sight. This is the way I always end my search for existence, open ended. I am awash with questions for which I find no answers, answers for which I find no questions. I begin a search for “meaning” thinking that this time I’ll find it, but all I find is a torrent of mysteries, some of which I haven’t thought of yet.
So I stop at the bottom of the hill and take a rest before I start pushing the boulder up again.
Pushing this boulder up a hill isn’t so bad; there are much heavier boulders that I could be dealing with. It’s kind of fun actually. I look for new approaches to and ways up the hill. Sometimes find level areas where the boulder will sit without rolling back down the hill and I can stop and look at the view. And the weather’s usually pretty nice. And pushing the boulder up the hill keeps me off the streets and out of the pool halls.
Surely I must exist because I wrote this, unless I’m just imagining it. Nah.

Send your kid to college...ha ha

2009-05-12T13:56:00.001-07:00

School days, school days.¦

I was going through some boxes looking for old photographs when I came across the letter that said I was accepted to Rensselaer Polytechnic Institute (RPI) and that I was the recipient of a full four year tuition scholarship. RPI was one of the pricier schools and was ranked with M.I.T. in the engineering world. The builder of the Brooklyn Bridge was a graduate of RPI and the first steel frame building that used welding instead of rivets was part of the campus.
The tuition was $800 a year. My dad came up with $100 a month for board, room and entertainment. The drinking age in New York State was 18 and beer was 5 cents or 10 cents a glass, depending on how classy the bar was.
I graduated as an electrical engineer with no debt and went right into graduate school.
In 1986, one of my students graduated from a state university, supposedly a less expensive option for higher education, with a $40,000 debt. Students now graduate from college directly into bond slavery.
I don't really see how this either encourages higher education or makes it better.
When I began graduate school my professors had leather patches on the elbows of their jackets. They lived in modest houses in more or less genteel poverty. They were excited about mathematics. They were my role models.
In the late 80s, by a trick of fate, I was on the Ph.D. committee of a student in electrical engineering. There was a get-together at the house of the student's dissertation advisor when all the student's papers were signed, sealed and delivered.
I was blown away by the house. You walked in and were confronted by a long room with a with a fire place at the far end which turned out to actually be in the center of a really long room and burned brightly on both sides. There was a balcony on the right side of the long room and doors that, I was told, led to bedrooms.
I recalled the houses that the Professors of Electrical Engineering had at RPI.
I constantly hear how the cost of higher education is rising. Well, times change and Professors of Electrical Engineering who get lots of grants cost more than they used to. Oh, I forgot; there didn't used to be grants.
And I suppose it does cost more to print all those multicolored pleas to the alumni on slick paper. As a matter of fact, since every organization on campus has to send its flyers on slick paper, there is a bit of an increase in printing costs. In the course of a month I could fill up an empty copier paper box with very important communications which I never got around to reading.
And then new buildings have to be built; I'm sure why the old buildings don't work anymore. And professors and administrators deserve offices that live up to their houses.
Perhaps it is worth bond slavery to graduate from a university that has new buildings and a well housed faculty and administration.

The Constitution

2009-05-12T13:44:00.002-07:00

The Constitution and Education

I have been reading the newsletter of the Constitution Project (www.ConstitutionProject.org) which is a group that seems to believe in The Constitution. One of the issues was about the recent attack on habeas corpus, an attack that I oppose, and I wondered about the mindset of the attackers and how that mind set was engendered.
And then I wondered from what seed my opinion of The Constitution grew.

I am a pro Constitution guy. When I think of the Constitution I hear "F R E E D O M" ringing in my mind. You really should hear me sing it to get the full impact.
My concept of the freedom of which I sing was formed during the fifty years between 1935 and 1985 with a heavy emphasis on the first fifteen.

I grew up in small towns in Idaho, Montana, Wyoming, Oregon, Washington, Nevada, Utah and Colorado; I guess all the towns in the west were small then. In 1945 even Los Angeles had clean air; San Clemente, Dana Point and Laguna Beach were separate towns surrounded by walnut groves. Berries were grown on Knott’s Berry Farm.
I rode in the back of a pick-up with no thought of it being dangerous. Cars didn’t have seat belts but they did have running boards. On long trips my mom would make sandwiches to eat along the way and around noon my dad would stop at a roadside bar, of which there were many, and pick up a few bottles of beer and a Nehi to have with the sandwiches. One of the bonding moments between my dad and me was when he taught me how to throw a beer bottle out of the window of a moving car.
When it was time for my dad to go overseas my mom and I spent a couple of weeks at the town near the airbase he was to leave from. One afternoon I was left in the Officer’s Club while my mom and dad went for a ride in the nose of a B-17. This was against the rules but my dad felt that sometimes you had to rise above the rules.
The officers sitting at the bar saw that I had nothing to do and decided that they would relieve my boredom by giving me a chance to shoot a submachine gun. We all went out to the shooting range and a bunch of drunken Army Air Force officers handed a loaded submachine gun to an eight year old kid. I couldn’t control the gun and I couldn’t stop shooting. The gun barrel rose up and turned to the right; the officers, suddenly sober, hit the dirt. Fortunately I ran out of bullets before I caused serious damage.

My grandfather, Bill, was a semi-pro baseball player, an avid fisherman, a guy who knew how to do a lot of neat stuff and the owner of a saloon in Spokane. When prohibition came he sold the saloon and bought a ranch up around Deer Lodge, Montana. My dad and his brother, Bob, were born on the ranch.
My grandmother, Rose, was French-Canadian and could play the “Maple Leaf Rag” on the piano. She prayed a lot. When Bill died she decided to stay in Butte which indicates that she was a very strong woman.
One winter my dad’s horse fell on his foot while riding to grammar school and broke the foot underneath the horse quite badly. Bill laid him on his bed and tied rocks to his toes to set his foot. A high school girl from a nearby ranch stopped by a couple of times a week to teach him the three R’s and an old man brought him books to read; Sabitini, Dumas, Kenneth Roberts.
But Bill spent more time fishing than ranching and the family lost the ranch. They moved to Butte where Bill and the two boys went to work in the Anaconda copper mine.
My dad put himself through college and then took a commission in the Army. His first assignment was running a C.C.C. camp. It happened that there was no athletic equipment in his camp so he bought some and charged it to the purchase of cabbages. This was discovered and he lost his commission but through the good offices of a friend he was reinstated in time for the war.
By the way, he never had any trouble with the foot and it passed the army physical with flying colors.
My grandfather, Papa Jess, was my male role model during the war. He had been too young for the Spanish American War and too old for World War I.
Papa or Papa Jess, as all the kids in town called him, had grown up on a ranch in western Washington along with four brothers and four sisters. All the brothers wore guns and all rode black horses. While still a teenager he rode his black horse alone from Goldendale, Washington to Missoula, Montana. His precipitous departure had something to do with a woman.
He next went with his brother-in-law and sister from Missoula to Long Valley in the mountains of Idaho about 80 miles north of Boise. There he worked on the railroad, homesteaded, married a blond beauty, Rosie, from Wisconsin and raised a family.
There is a story my mother told me about Papa Jess playing cards with two other men to see who was going to have the honor of shooting the guy who had burned down the local hotel. She didn’t tell me if my grandfather shot the arsonist; probably not since the man was only wounded and, quoting my mother, “Old Doc Noggle saved his life”.
There were no social security numbers before Roosevelt and no way to keep track of people. Workers on the railroad were known by their first name only and they were paid at the end of every week in real money by their section boss. There is a grave marker in the local cemetery with the name Tim Two; there had been at least two Tim’s.
People came to Long Valley to work in the mine, to work on a ranch, to work at the sawmill, to work as a lumberjack, to grow wheat and to disappear from the outside world if they wanted to.
Basques herded their sheep on West Mountain. A Japanese family had a ranch and a store in town. Papa Jess always took his cows to Takiuchi’s bull even during the war. Finn Hall was a few miles up the valley.
There were no African-Americans in Long Valley. There were no Jews; in fact I thought that there weren’t any Jews around anymore like there weren’t any Philistines around anymore.
In all of my many grammar schools I was told that all men were created equal and I believed my teachers. The twig was bent.

When my dad went overseas, my mother, my sister and I went to Long Valley to spend the rest of the war. It never occurred to me that we might lose the war. Men like my dad and the men in his squadron in the South Pacific, men like my uncle Raymond flying in North Africa or my cousin Darrel in New Guinea, men like this didn’t lose wars. Even in retrospect, I feel that way.
The United States seemed to have such raw power; the black smoke pouring from the stacks of U.S. Steel, the logging trucks coming nose to tail out of the mountains, the clouds of B-17’s flying over Long Valley heading east, the endless trains passing through town and a citizenry that would not be denied.
When the war started the men who worked at the sawmill joined up and a group of Japanese-Americans came to replace them. I never really knew what their status was. They lived together somewhere out of town and while they weren’t prisoners neither did they come into town except on Sunday morning in the summer when Papa Jess would open up Rosie’s swimming pool, she had a tavern and a swimming pool, for the Japanese. I saw that these young men did not look like the Japanese who were portrayed on the posters in the post office. I saw that even though Papa Jess told them not to, they ran on the cement around the pool and dove off the roof of a building that was pretty close to the pool, just like the local young men used to.

Medically I grew up with mercurochrome, iodine and little packets of sulfa powder and Cuita-Cura Ointment, Papa’s universal nostrum. I never cut myself so badly that gauze, Cuita-Cura and adhesive tape wasn’t sufficient to stop the flow of blood. I didn’t have a stitch until I was a freshman college.
We worried about polio, Rocky Mountain spotted fever and Typhoid Mary. About cuts, hardly at all. When the head flew off the axe and hit Dale in the head, Cuita-Cura and adhesive tape. When the chain that a truck was using to pull a freight car broke and hit Danny across the face, Cuita-Cura and adhesive tape. I admired Danny’s scar, kind of like a German dueling scar, but his high school sweetheart, whom he married, had him have it removed after he got out of the Navy.

This is a sketch of the world I grew up in and the people I grew up around.
This world view is my benchmark for life, liberty and the pursuit of happiness. My opinions broadened over the next sixty years but they are always compared, for better or worse, with my first fifteen years.
This is not to say that the opinions I formed in those years are good or bad or that my formative years were ideal. My early years were just what they were and that’s all. Everybody has a unique first fifteen years that is the basis of their unique view of life, liberty and the pursuit of happiness. This gives everybody a relationship with the Constitution; even if they don’t really know what it says; they think they know. And they think they know what is right and what is wrong, what is fair and what is not because they know how to judge those qualities in the world that they, personally live in.

My opinion on gun laws was formed as much from being around guns all my life as it was from careful rational thought.
My grandmother, Rose, was the only person in my family that ever went to church. My mother was indifferent to religion and my dad was hostile. I excise religious freedom by not participating. I don’t have an opinion one way or the other about putting The Ten Commandments on a Court House lawn.
A part of religion that I did like was Christmas Carols, which I never hear on the radio or in school programs anymore. I don’t see what the fuss is about school Christmas Programs with Christmas carols. While I didn’t believe the stories, I didn’t feel offended by them and I enjoyed singing the songs and reciting the verses from the Bible.
Papa Jess was supposed to have been named Jesse James Lefever but the preacher wouldn’t baptize him with that name so he was baptized Jesse Cleveland.
Racism is incomprehensible to me. When we moved to Cheyenne I attended an integrated junior high school but since I believed that all men were created equal I didn’t really think much about it. The cultural division in school was between the jocks and the rest of us. The jocks and their girlfriends, a very multicultural group, sat together at lunch.

So, when a Constitutional question arises I judge it in the world I live in as I have experienced it. My life in my world formed my take on what’s ok and what isn’t.

In 1790, around when the Constitution was written, the population of the United States was about 4 million. In 1950, when I turned 15, the population was 131 million and 65 thousand people lived in Phoenix. As of the last census the U.S. population is 300 million and there are as many people living in The Valley of the Sun now as lived in the entire country when the Constitution was written. (I remark in passing that China has a billion more people than The United States.)
It is a tribute to the men who wrote The Constitution for a country hugging the eastern seaboard and populated by 4 million people, that it works at all in a country of 300 million that stretches from sea to shining sea. The fact that the interpretation of the Constitution has changed over time is hardly surprising.

Any constitution requires reasonable, honorable men and women to make it work. Any constitution can be subverted by determined, dishonorable men and women.
When the Constitution was written it was not unreasonable to assume that reasonable, honorable men would be running the country. Honor was still a big deal and the country was small enough that it was hard to hide dishonor.
The men of the Constitution fought beside George Washington and the women kept the home fires burning. Their grandchildren heard Lincoln debate Douglas in person. When the Constitution was written the owners of businesses lived in the same neighborhood as the people who worked for them.
The people lived in the same world and the same country as the early presidents.
The last president I saw in person was Eisenhower at his inauguration parade. I saw Truman once when he passed through Cheyenne. The presidents since then have been images on a TV screen. I don’t live in the same country as President Bush, or the same country as John Kerry for that matter. I can’t imagine what living in the world of Rove or Chaney or Scalia would be like; I see that world but through a glass darkly. But their worlds formed their understanding of The Constitution and as we live in different worlds our understanding of The Constitution is different.

Why do people want to chip away at The Constitution? Why do they want to weaken habeas corpus, a legal concept that goes back to the Magna Charta? Why do they want to allow information obtained under duress in trials? Why do they want to allow the duress?

The ordinary citizen is propagandized by Pro-life, by Pro-choice, by national security and threats of terrorism, by Pro-gay marriage, by Pro-one man to one woman marriage, by Pro-gun, by Anti-gun and by Pro torture to name a few of those who propagandize to have The Constitution interpreted or changed to the benefit of their ends.
Unfortunately education has failed the citizens by not teaching them rational thought.

More on habeas Corpus

2009-05-12T13:44:00.001-07:00

More on Habeas Corpus

I listen to debates in the Senate and who is to say what the motives are behind the desire to weaken or erase habeas corpus. What is the motive behind the reluctance to outlaw torture? What is the motive behind attacking Iraq without a declaration of war from the Senate?
Like most motives I suppose they are mixed and run the gamut from heart felt conviction to lust for power and these motives have been with us for a long time.
The ingredient missing in the mix we have today is a reasonable populace.

Problems can't be solved in a vacuum. Education can't be a successful enterprise in an atmosphere that gives only lip service to the concept. Education can't be fixed with patches, you have to start with convincing people it is a good idea.
In the 1960's I learned that "working within the system" meant, "Go away, kid, and talk about it to your committee. Don't bother me. I've already made the decision."
The problem with committees and commissions is that they are ignored. Take, for example, the 9/11 Commission.
The way to protect The Constitution is to start with education and by education I mean teaching rational thought and critical thinking. For a populace to be reasonable it has to know how to reason.
If a reasonable population decides an issue in a way that may be contrary to the way I would have decided it, so be it. The population thought about it, I can ask for no more. If a reasonable population decides to change The Constitution, so be it. That is the will of the people because they thought about it; thinking about it instead of believing what someone tells them to think.

A Mathematical Look at the Theory of Destroying Values

Senator Frist (R-TN) says that the terrorists are trying to destroy our values. This lives up to the usual absence of information in the Senator's comments.
What values are we talking about? Honesty in business? Honesty in politics? Honesty in personal relationships? Did the terrorists make Abramoff pull his scams?
I don't know what values the good Senator has but I have a sense of honor, a sense of responsibility to my children, indeed, to humankind in general and a sense of the value of human life. I think I should pay my debts and not cheat people.
I don't see how the terrorists can destroy my values. How does someone blowing up a bomb change my values? I didn't start robbing stores and beating my kids on 9/11.
Maybe the only value at stake is the value of The Constitution although I don't feel that my appreciation of The Constitution has diminished since the war on terror began. I value my freedom more, not less, since the war started.
The only people who can destroy our values are ourselves. Only we can we say it's OK to torture, that habeas corpus is unnecessary, that we can start wars with impunity.
The irony is that in fighting the terrorist war we could well destroy our constitutional values in the attempt to win it, only realizing too late that the terrorists had won.
Funny how that works out.

Mathematics in the Real World and Habeas Corpus

2009-05-12T13:43:00.002-07:00

Applied Mathematics

First I’m going to present a basic principle. It goes by many names, for instance,
The Check the Air Pressure in the Tires Principle or The Look under the Street Light for
Your Keys Principle.
The idea is that you try to solve a problem by using an easy technique that has no chance of success instead of using a difficult technique that has a chance to succeed. It is easy to ignore and give your kvetching neighbor bad glances, it is hard to love her.
This is a Principle that is often applied by students of mathematics. It seems so easy to memorize some formulae and then try to see which one works. To be successful they eventually have to realize that they are going to have to look for their keys where they dropped them, so to speak.

I was on the Undergraduate Committee and one of our tasks was to select text books. Every three or for years, depending on the greed of the publisher I suppose, publishers changed the edition of their calculus books. A book rep told me that at most universities they just rolled over to the new edition but at The University New Mexico we never rolled over to the new edition; at least not in the thirty years I was there. I give us credit for seeing that all the books sucked. After three or four years everybody realized the current book wasn’t working and when the edition change approached we started a search.
Finding a new text was difficult because the books were all so similar. It was like choosing a tract house to live in or choosing the prettiest snow flake.
At one text selection meeting I suggested that, since picking a book and then deciding what to teach didn’t seem to work, why didn’t we figure out what we wanted to teach and how to teach it and then pick a text. If we couldn’t find a satisfactory text, we could write one.
I was told that my suggestion was too hard, that nobody had the time, that we could never agree on what to teach or on who would be the lead author. On and on.
Since we were a democratic committee we chose another faceless book.
I have heard that, by one definition on insanity, an insane person keeps doing the same thing over and over and each time expects a different result.

There appears to be a movement to abolish habeas corpus and the problem is to restore that principle of law to high esteem.
Since legislative acts to curtail habeas corpus are being presented and enacted, there must be a motive force. If one wishes to protect habeas corpus a counter force must be applied. But where and how?
Not the right question. The question is, “Why do people attack habeas corpus?” Once you know that, you can make a better effort.

Let’s look at an egregious example, the war in Iraq.
The problem was to prevent the country going to war. There were those who were against the war in Iraq and they tried to stop it. They demonstrated, they wrote letters, they had meetings, formed committees and lobbied Congress. Evidently these efforts were unsuccessful.
Did the movement to go to war have infinite mass and hence was unstoppable or did the anti-war group fail to find the point where a counter force should be applied?
Since the murmurings of a war with Iran seem to be raising less opposition, it could be that the protesters are disheartened. It seemed as though they did everything they could think of but to no avail.

This discouragement is found in calculus students. They get tutors and think they are studying hard and after three successive failures give up and switch major.
My freshman year roommate drank a lot of beer and was failing calculus. He decided to spend a night in and study.
He opened up his book, got his desk cleared off, had paper and pencil at the ready. He started a problem and then decided he could think better after a shower. After his shower he looked at the problem for a while and then remembered that he hadn’t called his girlfriend. So the evening went.
Later that night his drinking buddies stopped by; they had missed him down at Gainors’ Bar and asked where he had been.
“Oh, I stayed in and studied tonight.”

Protests, unless they involve significant portion of the population, don’t apply much pressure at all, sound and fury signifying nothing. Probably most letters end up in the shredder.
It is true that the numbers of protesters did increase to a critical point when the sons from middle class were drafted and killed or injured in the Viet Nam War but the war in Iraq has no draftees. The avowed reasons for the war are so muddled and the metaphors so mixed that it is hard to find a point to protest against.
What do you protest against? The war is a fiat accompli. There are not a lot options let alone good options for policy in Iraq.
For example, we could pull out and let the Iraqis deal with the problem.
During the week or so before classes started at the beginning of my first year in college (RPI-1953) the college freshmen would riot with the local high school kids. The nightly riot took place at a spot downtown where two streets merged into one and formed a large enough area to accommodate a pretty fair sized riot. The police formed a cordon around the area and once you entered the riot you were committed; if you tried to leave the police pushed you back in.
It was an easy way to control the riot and nobody was seriously hurt. But Iraq is not a school boy’s rite of passage. Cordoning off Iraq and letting whoever is there fight it out has some ethical problems, at least I think so.
Once the war began I couldn’t really see any totally ethical policy changes nor did leaving the policy unchanged seem any better. Talk about being between a rock and a hard place. I guess if ethics is dropped from the equation, more options are available.

Here we see a case of an over determined system similar to trying to satisfy four kids with three televisions. Mathematics is everywhere.

If all actions can be protested it isn’t clear what a protest means. If there is no way to get from Damascus to Shangri-La, it hardly makes any difference which road you take out of Damascus. Protest needs a sharply defined issue.

If a student didn’t know how to construct the correct polynomial they just made one up. Since they had no idea where I got the polynomial when I worked a similar problem on the board, it was easiest to assume I made it up and that’s what they did.

What about the attack on habeas corpus?
The reason seems to have something to do with national security. National security seems to be the reason for everything, indeed it was one of the major reasons given for the Iraq adventure. I can’t think of a bill in the Senate that didn’t involve national security.
This is particularly true in the case of habeas corpus because the recent legislation takes habeas corpus away from detainees acquired in Iraq and Afghanistan; these detainees seem to fall in the terrorist category. And you don’t even have to be a terrorist to have your Constitutional right of habeas corpus in jeopardy.

In debate I hear that terrorists don’t deserve habeas corpus. There seems to be a sizable group that finds it acceptable to put them in jail and forget about them. This is a little disquieting because the definition of “terrorist” seems to be in the eye of the beholder. It is also mean spirited and “chicken shit”.
I have heard it said that it is too hard to give habeas corpus to all detainees, there are just too many of them. The logic behind this point of view seems to depend on the “Take the Easy Way Principle”.
The idea to remove habeas corpus is at least as old as King John. In the King John case the people who supported habeas corpus had more power than the king. In the current situation it looks like King John has the edge.

The people who want to restrict habeas corpus want to put people away and forget about them because that seems to be the only thing that habeas corpus deals with. As far as I can see it is more or less clear why the original King John wanted to do that but why now.
I can only speculate why our King John takes this attitude and even if I did know, I wouldn’t be able to change his mind. If someone already knows that their actions are unethical and do it anyway because they think they can, I can’t think of a convincing argument to dissuade them.
And tyranny is so much easier than democracy and habeas corpus and all that stuff.

But the populace is still in the mix and a bigger question is, “Why do the citizens allow it?”
My opinion is that fear and a desire for revenge have left the citizens in a quandary and they don’t know which way to turn. If they display compassion, at one time considered a virtue, and a respect for the Constitution, they are being soft on terrorism and condoning the people who are trying to destroy our values, whatever that means, and plant bombs.
Not only could a freed “terrorist” continue terrorizing if released (fear) but he would be getting away with being a “terrorist” (revenge). It is often said that justice is involved but I think that in common parlance, justice, a malleable term at best, means revenge.
To get the support of the citizens you have get rid of their fear and their thirst for revenge. People in a state of fear and seeking revenge for being made afraid, tend to be irrational. In the presence of fear people don’t think rationally and there is no point talking to them; your words fall on deaf ears. When the fear is gone, they can be talked to.

Again I look at a first semester calculus class although the fear and thirst for revenge may start much earlier; fear of mathematics and revenge on the people who inflicted mathematics upon them.

The citizens must be told that freedom and democracy are neither safe nor easy. If the citizens can’t accept that, habeas corpus questions are beside the point.
Sorry, guys, that’s just the way it is. It all goes back to the Second Law of Thermodynamics and you can’t get around that.

Learning mathematics neither safe, I have heard of people becoming addicted, nor easy.

Jeffrey R. Davis
Assoc. Prof. Emeritus
Department of Mathematics and Statistics
University of New Mexico

He who steals my purse, steals trash
'twas mine, 'tis his, and has been slave to thousands.
But he who takes from me my good name
Robs me of that which not enriches him
Yet leaves me poor indeed.(The Bard)

More Dispatches from the Front

2009-05-12T13:43:00.001-07:00

More dispatches

When I taught Calculus I in 1957, my class had twenty students and all had taken college algebra, trig and plane geometry in high school. Their parents were paying a lot of money for their kids to go to this school.
The students in the Calculus I class I taught in 1995 had no coherent preparation. I was asked if ½ x was the same as x/2 by a student who picked up the idea of the derivative as a rate of change with no problem.
Well, you teach the class in front of you, you don’t wish they were some other class. There are a hundred students instead twenty; they didn’t get a good preparation in algebra, trig and geometry in high school and it doesn’t cost $40,000 a year to go to school.
I think that Communism doesn’t work because Marx designed his system for a race other than the human race. It was an intellectual design, no pun intended, based on how he wanted people to be.
Mathematics education seems to have the same design flaw. The “New Math” is an egregious example of other worldly educational design.
I must confess that when the “New Math” first appeared it seemed like a neat idea but I was a young and foolish graduate student. At that time in my career I thought that a topic that was clear to me was teachable.
I taught determinants and Cramer’s Rule in their full generality to a freshman calculus course. I taught Fourier Series from the point of view of Banach Algebras. It was so obvious; how could the students fail to understand. I eventually had the epiphany that what was obvious to me was not obvious to my students, probably because they didn’t have five years of thinking about mathematics behind them.

But I did finally realize that my students were not me and put some effort into seeing who they actually were.
I am not saying that students need to be talked down to; I’m saying that mathematics education has to be revamped so as to release the potential of our students.

I don’t think that traditionally mathematics was taught to 100 students at a time. Populations weren’t large enough to need 100 student classes. Throughout most of the educational history of the U.S. the country was agrarian; school houses and the classes held in them were small.
It is my contention that a conscientious teacher in a small class will turn out an educated student.
That’s why people say, “Why, I went to a one room school house and got a good education. I don’t see why these kids today can’t get a good education. Maybe they need standardized tests.” The reason the Senators from North Dakota got good educations was that the classes they attended were small.
My dad lived on a ranch in northern Montana when he was in grammar school. One winter on his way to school his horse fell on his foot and broke it badly. A high school girl from a nearby ranch was his teacher and an old man would bring over books for him to read; Dumas, Sabatini, Kenneth Roberts. He was a well educated man.
My grandmother, Rosie, had a high school education and taught school. She was at least as well read as many school teachers are today and could teach the three R’s with the best of them.

When towns grew into cities and school populations got bigger there were two ways to go; get more teachers and keep class size small or have the same number of teachers and let the class size grow. That part of government that finances education has generally taken the penny wise, pound foolish path. For reasons that are obscure to me it doesn’t seem to be universally recognized that viable economic and political systems are rooted in an effective education.

When I went to high school, the trades were a viable career option. One of my classmates in college figured that engineering and bricklaying had about the same financial future. I remember our family driving across the country in 1952 and passing by the U.S. Steel plant, its smokestacks belching smoke and declaring the health of American industry.
The copper pit in Bisbee, AZ and the smelter in nearby Douglas were going 24/7.
One of the effects on education was that high school became a reasonable terminus. A kid could finish high school and expect to get a good job at the mill or plant or factory or mine. I guess that in 1950 the girls looked forward to marrying the guys who went to work in the mill and raising their kids. Reading, writing and arithmetic were all that were necessary for a good life, sort of.
College Algebra was a college course and the elite course in high schools. In 1950 a College Algebra book covered Descartes’ Rule of Signs, Horner’s method of Extracting Square Roots and the solutions of the general cubic and quartic equations.
But that is the way it used to be. The jobs of society with an industrial base aren’t there anymore. An education that worked in the past doesn’t work in today’s America but we are trying to shoehorn the teaching techniques and educational philosophy of yesteryear into our brave new world.

And whatever changes are made, the most important is a “small teacher/student ratio”, the smaller the better.
I say “small teacher/student ratio” rather than “small class size” because classes may be one the artifacts that have to go. But I will talk about class size because I think that is where change has to start.

I asked a high school mathematics teacher in Tucson what the limit on class size was. She said that it was 46. I opined that I thought that was little large and she said, “I can handle a class of 46.” She didn’t mention teaching a class of 46.
A colleague told me that he could teach calculus to at least 60 students at a time. Yeah, right.
The idea seems to be that you lecture clearly and maybe work some problems; the student takes notes and goes home and studies them. If they have any questions they come to office hours and get them answered. There should be no limit to the number of students you can teach.
This sounds so reasonable but unfortunately it doesn’t work. Well, it worked for the student I was but that student isn’t in my class; most of my students aren’t planning for a career in higher mathematics.

So, what’s wrong with large classes?

If I am talking to one person, they pay attention to me; they try to understand what I’m saying. Say we’re sitting at a table in a bar. Somebody else joins us. Two people will now pay attention to what one of us is saying. If there are five people at the table, the attention of one of them might wander unless the speaker is quite forceful. The speaker has to hold their attention. As the crowd grows the conversation will split into two conversations; or maybe someone will pick up a newspaper left on a nearby table and read while the rest converse; or maybe one will take out her laptop.
In my calculus classes of 100 students, some of the students knew how to work the problem I was demonstrating and were bored, some were lost and had no idea what I was talking about, some were trying to pay attention but couldn’t stay awake. Some read the school newspaper, some took out their laptops.
When two people are talking to each other, I’ve never seen one of them fall asleep. I’ve never had a student come to office hours and start playing a game on their laptop in the middle of working a problem.
If I’m talking to two people, they both feel I’m talking to them. It is hard to make 100 students feel you are talking directly to them, especially if many of them are disinterested.

A bad teacher will get better if they teach small classes. In a small class teaching gets personal and you don’t want to look like a jerk in front of a bunch of kids you are getting to know.
In a large class, not only do the egos of the students get lost but so does the ego of the teacher.

Perhaps small classes would not solve all the problems but that change would be a giant step for education.

Dispatches from the Front

2009-05-12T13:42:00.002-07:00

Dispatches from the Front

I taught my first calculus course in the fall of 1957 at RPI. Calculus was the lowest level mathematics taught, so first year TAs were given calculus to teach. Calculus I met five days a week and I had two sections. The previous spring they had given me a copy of the first edition of Thomas and told me that this was the book I would use. During that summer I worked all the problems in Thomas; it was the only thing I could think of to do to prepare for my first teaching experience.
On the first day I realized that I would have to teach without notes because I couldn’t hold them with a steady hand; never let ‘em see your notes shake.
The classes had twenty or fewer students. I taught in the same way I had been taught at RPI. I taught the same material and the tests I made up were based on the tests I had taken. I wrote my tests on the blackboard; the only tests that were mimeographed were the three hour finals which were given in the gym en mass.
I found out at the end of the semester that the finals were graded during a marathon session that started right after the final was collected.
My biggest anxiety was making a test that was too easy and yet,not too hard. I ended up with tests whose problems were trivial if you saw how to work them and impossible it you didn’t. Of course the students hadn’t worked on making up the problems all night and so didn’t see how they were obvious. The grades were dismal. I would then add insult to injury by going over the test with the class and showing how easy it was.
This made the class feel bad; it made me feel bad.
“Why didn’t I see that?” they would ask themselves.
“Why didn’t they see that?” I would ask myself.
I would recall that in my Statics and Strength of Materials class there was an A, a B, a C, a D and 16 Fs so giving too many Fs wasn’t a big concern.

I don’t remember what my early grade distributions were. I do remember that a regular faculty member looked over our grades for irregularities before we handed them in. I suppose my grades must have fit some predetermined norm because I don’t recall any bad glances directed my way.
At my next graduate school calculus was taught in a lecture hall to a lot of students and tests were given in the evening in the lecture hall. Two versions of a mimeographed test were made up by the TA and approved by a faculty person. The tests were multiple choice and machine graded. I had nothing to do with assigning letter grades to scores.

I had a student in College Algebra who passed the bi-weekly tests and failed the final dismally. He told me that this had happened all through high school. He’d have an A average going into a final, fail it badly, and get a B in the course.
I told him that he could memorize two week’s worth of material but not a semester’s worth.
But I have had students who could memorize a semester of problems and formulae. And I have seen instructors who would encourage memorization by putting the page and number of the problem whose numbers they had changed to make the test problem.
At the risk of appearing grumpy I think this method of teaching and testing is insane.
One thing that seems clear to me is that the test should be made to fit the course and not the course designed to fit the test.
When I was a graduate student I made some extra money tutoring a high school kid in plane geometry. In New York State one of the requirements to pass a high school class was, and maybe still is, to pass the Regent’s examination in that subject. The previous 15 or so years of Regent’s exams in plane geometry had been compiled into a book and this was the text that the student was using.
The whole point of the student’s course in plane geometry was to pass the Regent’s exam. His course was designed around the exam and not the exam designed around the course.

A Few Thoughts on Teaching

2009-05-12T13:42:00.001-07:00

A Few Thoughts on Tests and Testing

I looked at tests as a student for 22 years and I looked at tests as a teacher for 37 years. I’ve looked at tests from both sides now, from years in school and still somehow, it’s test’s illusions I recall, I really don’t know tests all. (Judy Collins, sort of)
I have sensed and heard the frustration of teachers over testing, particularly if they really wanted their students to learn something. Teachers are trapped into playing a game they can’t win. If they give a lot of bad grades, then they are too hard and unfair to their students. If they give a lot of good grades, then they are too easy and aren’t keeping standards high and making it even more unfair to the students who drew the ‘hard’ teacher.
There are no bowls of porridge that are just right.

**************

As I was walking across campus one day I stepped aside to let a man blowing leaves off the sidewalk pass. It occurred to me that the university hired a lot of people to clean stuff up.
And then it occurred to me that the university hired people to do a lot of work that the faculty and students could well do themselves; for example blowing leaves off the sidewalk (at least a work study job) and cleaning our own offices (at least emptying ashtrays and waste baskets).
And then I wondered where the leaf blowers and janitors came from and what their lives had in common. There was one thing they probably all had in common; they did poorly on tests and left the educational system after high school if not before.
And then I started thinking of all the uneducated people I knew who were really quite intelligent. The character in “Driving Miss Daisy” who drove Miss Daisy was uneducated but fixed the elevator for Dan Akroyd; that is the kind of intelligence I’m talking about.
A guy who lived across the street from me went to a welding school after high school but he could do an amazing number of things besides welding very well. This would seem to imply that he could learn.
On the other hand I know of a lot of students who do well on tests but don’t know how to do hardly anything. I know of a student who learned chair forms in organic chemistry by making up a poem for each form that had nothing to do with organic chemistry, a straight A student in fact.
The correlation between grades and smarts didn’t seem to be real high. I know there are smart kids who get good grades but there are smart kids who don’t.

**************

I think that the rationales for testing are many and most of them are obscure. The consequences of testing are many and many are unintended. A test is a selection process but what are they selecting for?
One of the consequences of testing is that it gives society a constant supply of unskilled labor. Because of the uncertainty in testing, a lot of people who are quite intelligent get put in that labor pool. Some become the person you go to when you have a problem; some become the guys who rob the corner liquor store.
Another consequence is that some incompetent people get jobs that require skill and intelligence. This circumstance results in disasters of various magnitudes.

*************

Tests are constructed in such a way that they automatically give a certain percentage of bad grades. This is possible because of the general acceptance of the bell curve.
Usually the tests are about how well the students can memorize, about how well they can work standard problems, about how fast they can work standard problems, or about how well they can write the response the teacher is looking for; all are easily fit into a bell curve.
The bell curve requires right or wrong answers so questions on tests have to have right or wrong answers. I think this is a serious drawback.

Some memorization is needed. A chemistry professor told his class that there was some information that was reasonable to look up but there was also some that you were expected to know, like the chemical formula for water and hydrochloric acid; or 9 x 7.

Standard problems are standard and really fall into the memorization category.

The question of speed is a little trickier.
My school started considering learning disabled students although I never fully understood the criterion a student had to meet in order to fit in that category. (It was interesting how the office that handled learning disabled students moved from the basement and staffed by a couple of people to a full fledged power bloc.)
The idea was that if a student was learning disabled they were to get extra time, 15 minutes if I recall correctly, on a test. One young man’s hands were badly crippled by youthful arthritis and it seemed reasonable to give him extra time. Another student in the class brought me the required notification slip and sat the extra time with the kid with arthritis. Her test papers were close to perfect.
Since the young woman could evidently comprehend the material and seemed to have no physical impairment, I concluded that her disability had to do with how fast she could work a test. It made me wonder how many of the students in the class would improve their grade with an extra 15 minutes.
By setting the time for the test to be 50 minutes a bell curve was introduced. If everybody in the class knew how to work all the problems the scores would still fall on a bell curve. Students who knew the material could still get a bad grade.
The bell curve is one of those commonly accepted facts of life that I think is, to say the least, wrong. In practice I found that my tests scores were bi-modal. There were those who got it and those who didn’t. I had very few C’s.

*************
A few anecdotes:
Instructors, often unwittingly, lead their students into traps. I was sharing a cab going to the Boston airport with some colleagues when one of them said that she sometimes gave students a break by giving them a C instead of the F they deserved. I made an enemy that day when I pointed out that besides lying to the student and leaving them thinking that they knew more than they actually did, she was putting them in a hole. The student didn’t know enough to pass the next course in the sequence but didn’t want to retake a course they had already passed. I used to see this particularly in College Algebra to Calculus I and Calculus I to Calculus II.
The student would try, say, Calculus I two or three times and then either change major or drop out of school.
I made, after much soul searching, the decision to pass a student if I thought they knew enough to have a reasonable chance to pass the next course. When I didn’t pass a student I always gave them an opportunity to discuss my decision and I would tell him the reason I thought he would fail the next course and would explain the trap.
I did this with a student in Calculus II and explained to him how he had improved but that he wasn’t ready for Calculus III. If he repeated Calculus II, he would solidify his improvement and do well in Calculus III. He came to my office a week later and said that because of the D his average had fallen too low and Electrical Engineering had disenrolled him. So I called Electrical Engineering and pointed out that besides voiding all the good work the student had done in Calculus II, they were missing the whole point of education which was, ultimately, being a facilitator in a student’s acquisition of knowledge.
The student came back the next day and said that E.E. claimed to have made an error in the calculation of his average and he was readmitted. He got a B in both Calculus II and Calculus III.
*********

I was teaching Math 180 which was the first semester of a two semester sequence in calculus for the biological and social sciences. At the end of the semester I gave my little talk that that I passed people in Math 180 who I thought could pass Math 181.
A student who had attended all the classes but hadn’t gotten a grade above 25 raised his hand and asked, “What’s Math 180?”
“It’s the class you’ve been sitting in all semester”, I replied.
He was stunned. “This isn’t Math 102?”
Math 102 was a very elementary course in statistics and the student was a physical therapy major.
After class I walked the student over to the Math 102 instructor and introduced him to the student who hadn’t been in class all semester.
On the way the student said, “You know, I was really starting to get the hang of your course.”
************

Walking back to my office from class, a T.A. stopped me to ask me about one of his students. He said that the student was somehow getting a C but didn’t seem to know anything. He said that the student had the best penmanship that he ever seen and when he graded the student’s tests he seemed to give him a few points on all the problems and they added up to a C.
As it happened I had graded this student’s paper in his two attempts to take College Algebra by correspondence. I failed him both times; beautiful handwriting and absolutely no understanding of mathematics. He had apparently made it through College Algebra and Calculus I based on his penmanship.
There is a short story by Lionel Trilling, “Of This Time, of That Place” about an English teacher trying to grade the paper of a ‘different’ student, who was a main character in the story. The teacher sweated over the grade and at one point wishes that he could give the student “M for mad”.

*************

A C is probably the most ambivalent of all the grades but it is not clear what any grade actually means.
I had a friend whose daughter had gotten a 50 on a high school final exam in history and flunked the course. My friend complained that 50 should have been a passing grade. I told her that it didn’t make any difference, in a cosmic sense, what number her daughter got on the final; her daughter didn’t know any history and should retake the course.
I told the irate mother that her daughter’s knowledge and the placing of the passing line were independent of each other; putting the passing grade at 10 wouldn’t make her daughter know any more history.

***********

Second Foundation

2009-05-12T13:41:00.002-07:00

Second Foundation

A Picture is Worth a Thousand Words

Functions are most often given by formulae, or at least by some method of giving numbers. Functions at this point are symbolic and they don’t have to get very complicated before they become a blur of symbols that can yield a number but very little insight as to what is going on with the function. I want to see what’s “going on”; I want to see what the function is “doing”. I use the word “doing” because it is short and gives an expression of what I mean; how the graph goes up and down as you follow the graph to the right. I suppose I should talk about what I mean by using a lot more words but I’m not going to; I’m going to say that the function or its graph is “doing” something, like increasing or decreasing or neither.

I can spend time describing a dog biologically or I can draw a picture. (I think about Fonzie’s picture of his lost dog.) Graphs were developed because people needed to see functions; graphs were developed to satisfy a need.
I introduce graphs as pictures that give some insight into what the function is “doing”. The graph is secondary to the function and is just one of several ways to give a visual representation of a function.
I point out the obvious fact that graphs were developed by the sighted.

If I have a function given by a formula and I draw a graph of the function by plotting points, I have missed the point of making a graph, so to speak. Plotting points is a static process; my pencil is still as it plots a point. I am just thinking about what is happening at that point. I want to know what the function is “doing”. I want to know how its graph passes through a point.
The robotic plotting of points is not thinking about what might be happening between the points. I had a student who swore his graph was correct until I showed him that his picture didn’t show what the function was doing, which was going to infinity and back between two of the points he had plotted.
Graphing by plotting points (and later by finding where the derivative is zero etc.) is like getting information through torture, you get answers but they are not reliable. Instructors like the plotting points technique because it is easy to teach and they don’t have to understand graphing all that well themselves. Instructors like rote methods, they like “step one, step two”, techniques.
There should be a Teaching Mathematics Oath: Do no harm.
Graphing by plotting points and not by trying to see what the function is “doing” is harmful.

There aren’t a whole lot of things a function can do. It can increase, decrease, neither increase nor decrease or it can change from increasing to decreasing or vice-versa. And there are only three ways a function can increase and three ways it can decrease. Each of these behavior options is clear from the function’s graph.
You sketch the graph of the function before you take derivatives of the function. From a sketch of the graph it is clear where the derivative is positive, where it’s zero and so on. If you want to find the exact point where something happens, say attain a local maximum, then you set things equal to zero. A graph has a “what it’s doing” aspect and a “computational” aspect.

Can Graphs be Skipped?

I went to a lot of grammar schools and hit the third grade knowing only how to print. My school required cursive. I told the teacher that I could communicate quite well printing and could I skip cursive. I was told that I couldn’t skip cursive.
I suppose that graphs could be skipped. I know a lot of students try to get by on cranking out numbers but they don’t make it, well, they shouldn’t make it.
On the other hand there are blind mathematicians and I have heard of a chess master who played 50 games simultaneously while blindfolded.
But unless the student is exceptional, she shouldn’t skip graphs, nor should the instructor.

Testimonials for Graphs
I was reading the proof of a theorem in complex function theory and there was a point that the author claimed was clear. Being a conscientious student, I thought I’d better go through the proof so that it was, indeed, clear. Several hours later the fog had yet to lift.
The author evidently felt that pictures were for wimps and my attempts at leaping my hurdle were technical. At last I drew a graph and the author was right, the proof was clear. Not only was it clear why the theorem was true but the graph pointed the way to a formal proof.

I was reading the famous book on functional analysis by Riesz and Nagy as a way to practice my mathematical French. I came upon a lemma that I could go through and see that it was step by step true but I had no idea why a person would want to know this arcane result.
A few years later I was taking a course in Harmonic Analysis and the instructor proved the Sun Rising in the Mountains Lemma. Lo and behold it was the lemma in Riesz-Nagy. But now the instructor drew a picture of the sun rising in the mountains and not only was the meaning of the lemma clear but so was the proof.

Teaching graphs
Perhaps functions and their graphs should be taught more in parallel than I do and I have considered this. I chose the more linear approach of first functions and then graphs for several reasons. It seems more natural to do functions first. While representative art may have appeared earlier than numerical descriptions of nature, I am trying to describe nature using numbers. As this description becomes more complex it becomes evident that a picture would be helpful.
I am a big believer in presenting things when there is a perceived need.

Once both concepts are on the table, they can be viewed in parallel, a stereopticon effect. The advantage of sight in the communication of information is that sight can receive information in parallel. The expression, “I see”, implies understanding something all at once, understanding a totality.
A blackboard can display several ideas simultaneously. Books are traditionally linear but can have pictures, for example, graphs or photographs. Speech is linear; I can’t envisage saying two different words at the same time. Movies give information linearly and the flash-back is a device used to overcome this limitation.
Since most of the information that the brain processes comes to it linearly, through the ears as speech, much of what comes through the eyes including the written word and television, it isn’t practiced in parallel processing. I would expect that how well the brain can process two streams of information simultaneously is minimal at best.

I put the expression that gives the rule of the function on the blackboard and a graph of the function next to it. The student now has two concepts to process and it is important that the student get a strong connection between the two. When the student sees an algebraic or trigonometric expression, its graph should immediately come to mind; well, maybe not immediately. And if she sees a drawing that looks like a graph she should wonder what kind of function would have a graph that looks like that. If I see the word ‘dog’ I immediately have a mental picture of a dog. If I see y=2sint, I immediately have a mental picture of a wavy curve moving up and down along the x-axis. I also have a mental picture of a rock bouncing up and down on the end of a spring. The more mental pictures of a function I have, the better.
In my calculus book (mathematicsteacher.org) I give a chapter to functions and then a chapter to graphs. Then I have functions and at least one visualization of them.

Foundation (Calculus)

2009-05-12T13:41:00.001-07:00

Foundation
From Ideas Presented in The Calculus: An Opinion
(mathematicsteacher.org)

If I am faced with a Calculus I class on the first day of the semester, I have to decide what I’m going to base my presentation of calculus on and I have to decide what cohesive block of calculus I want the students to leave with.
It is important that the calculus is coherent so that the student has a solid foundation to build on, not the sand of unrelated techniques scattered here and there.
In what follows I present some of my ideas on teaching beginning calculus. Since my approach doesn’t get to the stuff engineering wants, or even the stuff that mathematicians want, I doubt that it will be tried in my lifetime.
None-the-less I present them. I think that a foundation that gives the student confidence in what she knows early on and then she could learn what engineering or mathematics or physics wanted her to know. It is my belief that without a solid understanding at the beginning, the rest of the calculus sequence is a waste of time.
What is this fascination that mathematics has with giving information to people who haven’t the faintest idea what you are talking about?
In an effort to get my first job I was giving talks at various universities and hoping my dissertation, which was the topic I chose, would dazzle my listeners. After one such talk a professor at the university I was visiting came up to me and said, “Your talk was great. I didn’t understand a word of it.”
I might say in passing that it is interesting to see what famous names did in their dissertations.

So, my goal is to provide the student with a coherent framework for mathematics. I want the student to feel comfortable with the ideas. More to the point, I want to erase their fear.
In one of the Castaneda books Carlos asks Don Juan how you become a man of knowledge. Don Juan replied that there are four enemies you must defeat. When you start to learn something, really learn something, it is totally different than you thought it was going to be. The familiar landmarks and toeholds are gone and you meet your first enemy, which is fear.
A student comes from high school thinking they have taken some mathematics, perhaps even some calculus. They think they know what mathematics is all about. They think that geometry is an answer on a multiple choice test and that algebra and calculus are formulas and rules to memorize.
And then they get me for Calculus I. It is nothing like they thought it was going be. If the student understood the ideas, my tests were trivial; if he tried to memorize his way to success, my tests were impossible. The students met their first enemy, fear.
Don Juan tells Carlos that you must take your first step into the face of this enemy if you want to eventually defeat it. Some students take this step, others drop the class.

Genesis

The first brick that I put in the foundation is ‘function’. The grist for the calculus mill is function; calculus is about functions; no functions, no calculus. Any understanding of calculus must be preceded by an understanding of ‘function’. And ’function’ must be understood well enough that the idea seems worth emblazoning on one’s very soul.
I say “worth” on purpose. If “functions” have no value to the student, then I can’t suggest a humane way to teach them. The first step is to convince the student that functions are worth understanding.
It could well be that this step is never achieved. I’m sure that many people live satisfying, productive lives without any knowledge of functions. But the students in front of me have voluntarily signed up for the course, usually to satisfy a degree requirement and their degree program must have some rationale for requiring calculus. This rationale provides a toe hold and if there is no rationale, I invent one.

I introduce functions by associating how far an object has fallen with the time it took to fall that far. I try to convince the student that it is reasonable to want to associate the numbers that describe one physical quantity with the numbers that describe another physical quantity. The problem is that I don’t know of any earthly reason why a person would want to make such an association. I can say why this process interests me and many of the people I know and hope that one of these reasons clicks with the student.
In any event, after having made the association of time and distance I slowly generalize the idea. I point out that Western Civilization is into single valued functions because the world we live in appears to our sense organs to be single valued. Dolphins don’t live in a single valued world and have only an academic interest in single valued functions; probability may arise more naturally in their environment.
If I watch a leaf fall from a tree, at every instant of time the center of gravity of the leaf is some distance from the ground. Thus I have a function that associates the numbers that give instants of time to the numbers that give the corresponding distances from the ground.
But while that function exists, its rule can at best be tabulated using a finite number of approximate distances at corresponding approximate times. Further, since every leaf falls differently, the act of tabulation is the only time that the function applies to anything.
Students get so used to seeing functions given by formulae that, if they aren’t careful, they can possibly come to think that formulae are the only functions there are. Through out the course I interject weird functions that can arise from experimentation.

I start out with functions defined by rules that arise from experimentation and the notation is all symbolic, either numerical tables or rules that use the symbols of mathematics, t, sint, cost and the like. (Beware the deadly symbol. I think that the confusion between a symbol and the concept, real or imagined, that it is supposed to represent is harmful.)
I spend a lot of time on the symbolic aspects of functions and I point out that functions whose rules can be expressed using the symbols of algebra and trigonometry, the standard symbols of mathematics, are relatively rare.

Teaching and the Philosophy of Mathematics

2009-05-12T13:40:00.002-07:00

Teaching and the Philosophy of Mathematics

I will first assume that I am talking about students who are being introduced to mathematics (which doesn’t mean that they haven’t had previous courses that were listed in the catalog under mathematics) and want gain facility with mathematics as a tool.
I think that students who feel a calling for the Ideal world early in life should be aided in their studies in ways that are appropriate for them.
Since ‘continuity’ is an Ideal World concept, beginning calculus is a course in elementary-calculus-game-playing. Continuous functions are probably the most popular game pieces although their use in the game is often poorly taught.
But I am getting ahead of myself. There are those who don’t consider calculus to be in any “Ideal World”. They think of it as the stuff Reality is made of.
Now, I do not judge lest I be judged. While I don’t believe that f(x) = x defines a function in the Real World, I would defend to the very death a person’s right to have such a belief.
On the other hand I would judge a teacher whose students couldn’t model a simple physical system using calculus.
I don’t have a problem with a teacher that bases the use of the calculus tool, or even as far back as the arithmetic tool, on magic, mystic runes and spells. Well, yes, I do have a problem with that. Indeed, that’s the way mathematics is often taught; consider the magic word, “FOIL” and the mystic incantation, “Invert and Multiply”. (This is a little known passage in the Old Testament that has been much over shadowed by the more familiar, “Be Fruitful and Multiply”. “Being Fruitful” was a not unpleasant activity and quickly caught on; people have been following that exhortation with great energy ever since. “Inversion” was a short-lived fad and faded quickly from the scene.)

Everybody seems to have a different idea about how to teach mathematics, indeed, a different idea about what the tool even looks like or what it’s used for.
My stab at teaching beginning calculus has a link on this site. I think I would make modifications if I were to write it again. In 37 years of teaching calculus, I don’t think I ever taught it the same way twice.
My general idea, to which I remain faithful, was to motivate the ideas behind the making of a mathematics-game by looking at a specific example; in the book I study the motion of a falling rock.
The method in my madness derives from watching people work on automotive engines. For example, if you can work on one motorcycle engine, you can work on them all. I heard this some years ago when motorcycle engines, even car engines, could be worked on in your garage.
But the mathematics in Calculus I was known by Newton. Calculus is not a new tool on the cutting edge, so to speak. Teaching Calculus is analogous to teaching backyard mechanics to repair lawnmower engines.
Calculus is an Ideal World game but in many cases models the Real World very closely. In fact the calculus-game was designed with modeling the Real World in mind. You play the game and see where an Ideal cannonball lands in the Ideal World. Then you shoot a real cannonball and see how good a prediction the game made. Well, the calculus-game makes really good predictions; so good that classes in major universities are given in playing calculus. There are even advanced courses in the calculus-game.
A course in calculus is a course on “The Construction of Simple Ideal World Games for Fun and Profit”.
My approach in the calculus book was to construct a game in the Ideal World that I could use to predict the motion of the rock when it was dropped. Proofs were aimed at making the results seem reasonable, both as to why they were true in the Ideal World, using the close relationship between the Real and Ideal World physics, and why anybody really cared.
In the book I stress the difference between the Real World and the Ideal World. I do this because of a personal opinion that it is important to distinguish between what is Real, what is Earnest, and what is sublime, what is a game. I think that this distinction should be made generally and not only in mathematics; and I believe in “teach the whole child”.
So, what I teach is affected by how I look at the world generally and I would suppose that this is the case with teachers generally. I don’t see how it could be otherwise.
I couldn’t keep a personal philosophy out of the calculus courses I taught, I can hardly expect others to rarify their courses.

The whole point of calculus instruction is to, at the very least, turn out good mechanics. Metal Shop 101 doesn’t start with metallurgy; it starts with making a chisel. Starting calculus with epsilons and deltas for the general run of people, like myself as a student, is so unrealistic as to be stupid. I can remember going to an office hour in an effort to get some inkling as to what this ε, δ stuff was all about. I left the office of my teacher no wiser than when I entered. I couldn’t see the problem that ε and δ were supposed to solve. I had no doubt that the function, f, given by f(x) = sin x was continuous, any damn fool could see that by looking at the graph. Archimedes found the area under a parabola without the help of ε and δ.
During a time of mental illness I tried to teach myself the C-Language using the pamphlet written by the guy who invented it. After many frustrating hours I saw why I was having trouble. I didn’t know what problem the inventor was trying to solve. ‘Pointers’ seem to come out of nowhere but I supposed that there were people who saw ‘pointers’ as a clever solution to a problem that other programming languages had.
I was in the same quandary with ε and δ. The people who developed the calculus I was learning didn’t use them, so they clearly weren’t necessary.
When I started playing advanced Ideal World games, it became obvious why you needed them to get to the next level. I had expected that insight into ε and δ would be an epiphany but instead it was obvious and not all that exciting. “Was it for this I screamed and cried and kicked the stairs?”
It was so obvious that I couldn’t understand why I hadn’t seen it before. Now I do understand. It was because ε and δ had no place in the Calculus I game.
I had the same problem with the exercises at the beginning of Lara Croft: Tomb Raider.

Newton was a natural scientist, not a pure mathematician. Newton constructed his game to be as close to the Real World as he could make it. Calculus instruction should start in the Real World and with Real World intuition. If the first-game in the series, which is Real World based, is mastered, then the advanced versions of the game are easier to learn. Some of the advanced games, while enjoyable to play, give little or no insight into the Real World nor does the Real World help much in playing the game.
(I have found that mechanics are quite interested in the 2nd law of thermodynamics and pressure/volume graphs showing how the various internal combustion engines work. I conclude from this that mechanics had some interest in advanced game theory.)

If a course talks about the general structure of the games of mathematics and then studies one of the games in detail, it doesn’t really make a difference which one, then the student is being taught how to learn any mathematics-game with Real undertones.
It seems strange that students, for example engineering students, who will spend much of their lives learning some new mathematical technique, are not taught how to “learn” mathematics on their own.
So often I hear, “I want to learn about something, I think I’ll take a class.” It doesn’t seem to occur to them that it is possible to learn things on their own. The musicians that I have talked to started out by trying to play a tune on a piano or a guitar, whatever. I read “Audel’s Handbook of Auto Mechanics” but I learned about engines from taking one apart and putting it back together.
When Pete, a friend of mine was twelve, his brother got him an old engine from a junk yard and put it in the front yard. Pete spent the summer taking it apart and putting it back together again, over and over; kind of like Zen archery.
The same thing occurred the next summer at the end of which an Oldsmobile engine could have been seen sitting on the grass in Pete’s front yard and running cheerily.
I think these examples have relevance to learning mathematics, actually, to the learning of anything. Well, maybe not everything, but I can’t think of a counter-example at the moment. You teach how to learn. What’s so hard to understand about that concept?

So what is my opinion about the appropriate place of Philosophy in the teaching of mathematics game-play? (I don’t think cheat codes, like FOIL, are useful. I think that authors who put shaded formulae in their game-play-books should be subject to criminal prosecution.)
I don’t think that philosophical biases should displace the study of good, solid game-play. But if someone converts a student to a particular ism as well as facilitating the student to become a competent game-player, so be it. I suppose that an excellent teacher of game-play could be as crazy as a March Hare and that some of his students might join the tea party; intellectual growth does have its risks.
As long as good gamesmanship is taught, I would not limit the free speech of the teacher. Personally, I try to put forward the idea that mathematics games are good examples of using critical thinking and reason. Oh yeah, and that there is a difference between the Real World, where things exist, and the Ideal World where they don’t.

Remarks on The Philosophy of Mathematics

2009-05-12T13:40:00.001-07:00

Remarks on The Philosophy of Mathematics

I have been reading “What Is Mathematics, Really?” by Reuben Hersh. This book is about The Philosophy of Mathematics”, I guess. I’m not really sure what The Philosophy of Mathematics is; but I know it when I see it.
My answer to his question is that Real World Mathematics is a tool; Ideal World Mathematics is a game.

After reading Prof. Hersh’s book and thinking about it for a while, I realized that The Philosophy of Mathematics wasn’t much different than Theology, The Philosophy of Religion, and they both study Ideal World Games.
I liken The Philosophy of Mathematics to The Philosophy of Religion and mathematics to religion. There are religionists who work in the trenches trying to make the lives of their flocks make sense on a day to day basis. There are mathematicians who work in the trenches of applied mathematics and trying to understand the Real World and teachers of mathematics trying to pass that knowledge along to the next generation.
And then there are the elites who count angels on the heads of pins and talk about The Trinity; who untie knots in six dimensions and try to well-order the real numbers.

Talking about a god for whom everything is possible is like talking about the set of all sets.
They both claim to seek truth without knowing what truth really means in the Real World and then defining truth in the Ideal World by fiat. They both ask questions that can’t be answered in the Ideal World and have no relevance in the Real World. Paradoxes are in the Ideal Worlds of both religion and mathematics.
Both disciplines build very complicated edifices out of Ideal World bricks, so complicated, indeed, that only the very few, the very elite, can understand the marvelous constructions. Both disciplines develop a shaman class to tell the ignorant masses what they need to know about janitorial jobs.
As a child I took what I heard in Sunday School as truth and was amazed. As a beginning graduate student I took the theorems I studied as truth and was amazed. Disillusion has followed in both cases.

Real World Mathematics is a tool that humanity has developed to help solve the problems of survival, like a scraper or a club. Societies developed the tools that were useful for their particular society and didn’t make tools that were not utile. Tools that are not useful are called objects of art. The wheel was not found useful by the Native American so they didn’t invent it; although I have seen circles in Pre-Columbian art.
Some societies were interested in counting things like money, wives and sheep and numbers were developed to make this sublime goal a reality. They needed to make buildings that didn’t fall down so geometry and a way to measure was developed.
Some societies felt that trees had to be chopped down so they sharpened stones and made axes. In order to chop the trees down faster they developed bronze, then iron, then steel, then chainsaws.
But as swords were developed to be more deadly, they were also developed as an art to be worn in dress parade and not on the battle field.
As axes were developed for a variety of uses, the single bitted axe, the double bitted axe, the hatchet, the machete, the meat cleaver, so was Real World Mathematics.
A machinist works within an error tolerance and the dimensions on blueprints are given ‘plus or minus’. The blueprint is a Real World document that tells how make a Real World object on a Real World lathe. If error tolerances are not given in the blueprint, then it is a Real World document that is a picture of an Ideal World object, like the ceiling of The Sistine Chapel.
A differential equation is a Real World picture of an Ideal World object. A harmonic oscillator is in the Ideal World. x``+x = 0 is a Real World picture that tries to express an Ideal World rock bouncing on an Ideal World spring in an Ideal World vacuum. (Like the ceiling tries to express an Ideal World God giving Ideal World life to an Ideal World Adam.) Does the picture of the Ideal rock and spring give any insight into a Real rock and spring? Experimentation has shown that for some Real rocks and Real springs, it does. Experimentation is still being done to see if the ceiling gives any insight into the origin of life.

Real World Mathematics is a tool. Ideal World Mathematics is a game. It turns out that parts of the game of mathematics model parts of the Real World; not model it exactly but close enough to be useful. How well the model fits the Real World is determined through experimentation. End of story; what’s to get philosophical about?

For example, consider the Banach-Tarski Paradox game. I, the gullible graduate student, was told that Banach and Tarski had proved that you can take a pea, cut it up into a finite number of pieces and reassemble the pieces into a pea the size of a basketball, or the sun for all that matter. Well, that seemed pretty cool.
Now, the way the Greeks found the area of a circle was to approximate that area by the areas of inscribed regular polygons and then approximate the circle’s area by the areas of circumscribed regular polygons. Since regular polygons were made up of triangles, any damn fool could find the area of a triangle, the areas of the triangles could be added together to find the areas of the polygons and get a lower and upper estimation of the circle’s area. If the outer approximations by circumscribed polygons get close to the inner approximations by inscribed polygons as the number of sides of the polygons increases, the Greeks figured that the common value of that the approximations were approaching ought to be the area of the circle. This is a little down and dirty but most high school sophomores can straighten out the demonstration.
The idea is that you approximate the area of a Region by the areas, which you know, of regions that contain the Region; call the number these approximations get close to the outer measure of the Region. Then you approximate the area of the Region by the areas, which you know, of regions contained by the Region; call the number these approximations get close to the inner measure of the Region.
If the inner measure of the Region equals the outer measure of the Region, then the Region has an area equal to the common value. This was the case with a circle.
Does there exist a region whose inner measure doesn’t equal its outer measure? Such a set, if it existed would be said to be unmeasurable. Well, you won’t find one at Wal-Mart but not to worry, mathematicians claim to be able to make them.
To get an unmeasurable region you need to use The Axiom of Choice. Being young and naive, I accepted The Axiom of Choice. What the hell, wasn’t it called an axiom?
As it turns out, The Axiom of Choice is obviously true in the Real World. When it was idealized and made a game piece it was like a football. (The genius of the ovoid shape of a football is that it bounces erratically and not true as a round ball would. This was pointed out to me by Sue Heim nee Smith. In support of my hypothesis that every number is really equal to 12, it was she who pointed out that spiders have 12 legs.)

My examples have been in two dimensions but roughly the same ideas work in three dimensions, where peas and basketballs live.
But if I break the pea up into unmeasurable sets, I have lost control over the size of the unmeasurable component parts so it hardly seems surprising to me, being no longer young nor naive, that I can put the parts together in such a way as to be any size I want. Duh.
I have never gone through a proof of the Banach-Tarski Paradox and my remarks don’t prove anything, but my point is that I am no longer in awe; it seems quite possible, even plausible.
It seems plausible in Wonderland, the world of unmeasurable sets and the Axiom of Choice, because it seems to fit within the rules of Wonderland; even Wonderland has laws which must be obeyed.

Because there are mushrooms and cats in my world, I can see where I might make up a game where the mushrooms were magic and the cats were Cheshires.
Because I can cut up peas and oranges in my world, I can see where I might make up a game where the parts of the peas and oranges were unmeasurable.
We invent a world and its natural laws, objects in the world and rules that govern how the objects interact with the world and each other and voila, we have a game.
Some games are refinements of other games. I think of the progression of games from Wolfenstein to Doom to Doom II to Quake; or the progression of games from the line to the plane to 3-space to finite vector spaces to Hilbert spaces.
Some games are made up and have little or no contact with the Real World, like Tetrus or Banach Algebras.

Another example of a game is Goedel’s Incompleteness Theorem. This theorem says that there are true statements that can’t be proved. On the face of it this seems like a surprising result until you realize that it is part of a game whose rules preclude planting the “provable flag” on some “mountains”. You can never plant the “Nineteen Point” flag on a cribbage hand.
The objects and rules in Goedel’s game are those of a carefully prescribed logic and the natural numbers. Some of these objects are called statements and some of the rules allow connecting statements to make chains. Any statement that is the tail of such a chain is said to be provable.
Some statements that are marked red (false) and some are marked blue (true). The object of the game is to start on a blue statement and build a chain of statements to any other blue statement.
Considering the huge number of blue statements in the version of the game Goedel was playing, it really doesn’t surprise me that there was a blue statement that wasn’t the end of any chain.
I readily admit that Goedel’s accomplishment was a tour de force of gamesmanship; certainly the equal of DiMaggio’s consecutive game hitting streak. I have gone through Goedel’s demonstration and I was impressed at his ingenuity.

I feel that The Philosophy of Mathematics is akin to studying the computer game Civilization and thinking that the game is reality. So why am I talking about it. Am I joining the ranks of those who count angels?
I have a real problem with the idea that Ideal World constructs are real and should be addressed as such. It is true that the Ideal World of Mathematics hasn’t been the justification something as high in body count as The Crusades, or the St. Bartholomew’s Day Massacre, it was used to justify “The New Math” in the 1960’s and early 1970’s.
There is nothing inherently wrong about constructing new games or refining old ones in the Ideal World. I do it often myself. I have several Ideal World theories; that we are devolving back to apes; that existence needs an observer and God was the primal observer; that the extinction of the dinosaurs was a mass suicide due to 100 million years of boredom; that God arose from a stable configuration of fields that occurred in The Big Bang. I have already referred to my theory that every number is really twelve. (See The Calculus: An Opinion; a link can be found at mathematicsteacher.org. Of course, if you are reading this, you know about this link, so I’m telling you something you already know. Is this Real World or what?)
But I don’t wish harm to come to anyone to thinks that the dinosaurs died because of the consequences of a celestial body crashing into the earth.

Infinity Revisited

2009-05-12T13:39:00.002-07:00

Infinity Revisited

As a child I seldom talked to adults in a conversational way but almost always in an information gathering mode. I had the naive belief that all the information I gathered from adults was accurate and the fact that it was not was often a reflection of the ignorance shared by the planet at large. So, when I learned about infinity it was much like learning about God in Sunday School. They both did things that defied my understanding but must be true because adults said they were.
I went through graduate school thinking that the infinite number of infinities were real in some way. I didn’t question their existence; I concentrated on how infinity behaved. Zorn’s Lemma was as real as a bacterium.
I will not dwell on the process that made me realize that the bacterium was Real and that infinity was not but I did make that realization. I am an infinity skeptic.

I suppose that the idea of infinity is an abstraction of “a whole lot".
I think of abstraction as a process that humans participate in and it is not necessarily harmful. I think that the abstraction from cow pie to manure to that which "happens" is useful.
That which is abstract to one person will be concrete to another. The concept associated with “one" is concrete to most animals if only because an individual animal forms a set exemplifying that concept. Most human animals find 1,2,3,4,5,6,7,8,9, and10 concrete. I suppose 1,2,3,4,5,6,7, and 8 are concrete to an octopus. I have read that crows count one, two, many. Homer says that the Greek troops at Troy were as many as the leaves on the trees or as many as the grains of sand on the beach; which are poetic ways of saying, “There were many Greeks.”
Natural numbers have varying degrees of abstraction and the taint of abstraction does not necessarily bar a number from the Real World. Four trillion is pretty abstract. I can’t picture 4 trillion bricks but I can conceive of 4 trillion bricks. My inability to count to 4 trillion lies in design flaws in the construction of my brain; the fault lies in me, dear Brutus, not in the 4 trillion.
By Arithmetic I mean classical arithmetic; I mean the arithmetic that uses the Natural Numbers of the Peano Axioms, the arithmetic that allows an infinite (as defined in the Ideal World) collection of Natural Numbers. If I say that the Natural Numbers form an ‘infinite set’ I mean that there is an unlimited supply of them; no matter how many I have, I can always find more.
Arithmetic doesn’t exist in the same world as stock markets, wars, starvation and Miami Beach but stock markets, wars, starvation and Miami Beach find that a finite part of it useful.

I think that there is no infinity so that the number of things in the universe can be counted and there is a largest counting number. Natural numbers larger than the largest counting number are in the Ideal World.The real numbers are all infinite decimals and hence in the Ideal World. “What about 2.0?” you say.2.0 is really 2.000… . Every one of those zeros is necessary and should be included with the ….There are those who consider ½ and 1/3 as Real Numbers in which case they are not in the Real World. If I think of a half and a third as real numbers then I think of 0.500….and 0.333… Rational numbers are real numbers whose infinite decimals repeat. ½ and 1/3 are fractions, that is, ratios of Counting Numbers or proportions; they are in the Real World.
The Natural Numbers are used to count, the fractions give proportions and the Real Numbers measure. We picture the Real Numbers as the real line which has an uncountable infinity of points. None of the Real Numbers, however you define them, say Cauchy sequences, are in the Real World.
1.000… = 0.999… is a real number in the Ideal World and not in the Real World. 1 is a Counting Number and in the Real World.
These are two different 1’s, and the same symbol is often used for both; just as ½ is used for the proportion and the Real Number 0.50...
(Don’t blame me for the confusion; I didn’t design the number system.)
There is no board that is exactly 1 unit long. This ‘1’ is not counting anything. The standard meter in Paris is not exact. There is no concrete realization of any Real Number in the Real World and this is because of the stochastic nature of the Real World. No concrete realization, not in the Real World.

Remark
There are some interesting attempts at conceptualizing very large integers. In India there is the length of time it takes to make a cubic mile of wool if one fiber is added every century.
The one I like best goes as follows:
I define an ‘a’ inside a triangle to equal ‘a’ raised to the power ‘a’. So a 2 inside a triangle equals 2*2 = 4.
I define an ‘a’ inside of a square to equal an ‘a’ inside of ‘a’ triangles. So a 2 inside of a square would equal a 2 inside of 2 triangles. The 2 inside the inner triangle equals 4 and we are left with a 4 inside of a triangle = 4*4 = 256.
I define an ‘a’ inside of a pentagon to be an ‘a’ inside of ‘a’ squares. So, a 2 inside of a pentagon equals a two inside of 2 squares. The ‘a’ inside the inner square equals 256 so we have 256 inside a square which equals 256 inside of 256 triangles. At this point I stop. This number is so large that I can’t comprehend it. The only way I can write it is 256 inside a square.
When I first read this scheme (The Mathematical Experience by Phillip Davis and Reuben Hersh) I took out a piece of paper, drew a pentagon and wrote a 2 inside of it. I stared at the marks I had put on the paper, my mind a blank. I drew a hexagon and wrote a two inside of it. It was like looking at an unexploded bomb. I turned the paper over and left the room.
I think that a 2 inside a hexagon does not exist in the Real World because it is too big. It exists in the made up world of mathematics. Putting a 2 inside of a hexagon doesn’t mean that it represents a number in the Real World any more than a picture of a unicorn represents a Real World animal.

Remark
Why can’t I add one to the largest Natural Number in the Real World and get a larger Natural Number in the Real World? The old “you can always add 1” ploy. There may be no largest Natural Number in the Ideal World but there is in the world I live in, the Real World. I’m going to shorten the symbol, 2 inside a hexagon, to 2-H. I think that 2-H is an upper bound of the counting numbers considered as a subset of the Natural Numbers.
All Natural Numbers (which are in the Ideal World) that are bigger that 2-H are not in the Real World. I hypothesize that the largest Counting Number is less than 2-H. For good measure make it 2-M, a 2 inside a megagon.
The denominator of a proportion can’t be larger than the Counting Number associated with the set of all subsets of the elementary particles. The denominator of a proportion is a Counting Number that counts the sample space which must be less than the Counting Number that counts EVERYTHING.
This idea of being able to add 1 and get a bigger Natural Number comes from the standard ways of defining the Natural Numbers, say the Peano axioms, which implicitly assume that the Natural Numbers are unbounded. I can always add ‘1’ in the Ideal World but not in the Real World. If I add one to the largest Natural Number in the Real World, I get a Natural Number that doesn’t count anything and is therefore not in the Real World.
I have been considering large Natural Numbers that have left their good homes in the Real World. What about numbers, in particular Fractions, that are very small? How small can they get and still be in the Real World? I haven’t used “zero” because I don’t know there is a “zero”.

I always cringed when I let Δ x get arbitrarily small in a calculus class. It was especially inappropriate if x represented a number of light bulbs. In the Ideal World I can let Δ x get arbitrarily small without a qualm. But in the Real World there is a smallest number. The reciprocal of 2-M is smaller than anything in the Real World (This needs a proof.).

If Achilles is trying to catch a hare in the Real World by cutting the distance between them into approximately two equal pieces, there comes a point where the distance can’t be cut in half. There is no ‘in between’ the hare and Achilles.

Zero is a little tricky. Being broke is a Real World concept and if I say I have 0 dollars, I’m saying that I have nothing to count. I see 0 as short hand for “I got plenty of nothin’”.
But zero as a distance between two objects doesn’t exist in the Real World. There are no instants of time whose duration is zero. It may seem that the distance between the sole of your shoe and the floor is 0.00..., but it’s not. The distance is small but greater than zero. 0.00... considered as a Real Number isn’t in the Real World because in the Real World there is a smallest positive rational number. In the Real World there is no ‘nothing’. There is no concrete realization of the Real Number 0.00... in the Real World. There is a Rational Number, considered as a Real Number, that is smaller than the mass of an elementary particle, that is smaller than a quantum of distance, that is smaller than a quantum of time and so there is nothing small enough for this or any smaller Rational Number (considered as a Real Number) to measure, nothing small enough to be a concrete realization of it in the Real World.
The symbol 0 in the Mathematical World has the same relation to small Rational Numbers in the Ideal World as the symbol ∞has to large Natural Numbers.

Remark
Since I have cast ∞and 0 out of the real world I have effectively tossed out continuity as well.
Since the eye was the premier measuring device for much of the development of the basic physical ideas and motion ‘looks’ continuous it is not surprising that continuity was assumed. But movies ‘look’ continuous even though movie time passes in 1/32 second quanta. I can be on one side of a fence in one frame, on the other side in the next frame which violates continuity.

Remark
Suppose I drop a rock to the floor and want to describe, to model, the motion mathematically.
The picture I see is that of two strings of beads. One string has the time quanta beads and the other string has the path beads, i.e. the distance quanta beads on the path that leads from my hand to the floor. As the rock falls, the path beads light up indicating where the rock is and time beads light up indicating when it got there. Note that the path beads are distance quanta beads and are dependent on how the time beads are lighting up. The falling rock connects path beads to time beads to make velocity beads. The relation between time and distance is what I want to use in the description of the rock’s motion. I pick a time bead and when it lights up I’ll note the path bead that lights up and connect the string between them.
Could more than one path bead light up when the time bead lights up, leaving me in a quandary as to where to tie the string? But if more than one path bead lit up, the rock would be at more than one place at the same time. So I suppose that only one path bead lights up and in my model I can run a string from each time bead to exactly one path bead.
Could strings run from several time beads to one distance bead? The problem with this possibility is that if, say, four consecutive time beads lit and only one distance bead it would mean that the rock had stopped for those four time beads and I don’t want a model where the rock stops for four time beads on the way down to the floor..
So when I make my model, I’m going to tie a string from exactly one time bead to exactly one path bead. As the rock falls to the floor, all the time beads and all the path beads light up. There is a 1 to 1 relation between all time beads and all path beads. Velocity is the ratio of the length of a path bead to the length of a time bead.

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My point is that motion, like time and distance, comes in quanta.
This discrete model can be made to work, more or less. It has to be made to work because we live in the Real World which is discrete.
I read where an applied mathematician commented that in his universe the only numbers that existed were finite decimals that fit in a computer and I took his comment to heart.
When the number π is used, the author is leaving it for someone else to drag it kicking and screaming into the Real World. Nobody in the Real World can use π in their Real World computations. You don’t build an altar bowl using π, you use 3.
In a way, my Real World is the right one by default. The only numbers ever used in a computation are my Real World numbers. We all live in an everyday world that is finite.

Back to the Model of the Falling Rock

How can I keep my strongly felt intuition that the rock doesn’t stop and can’t be in two places at once? How can I make a model of the falling rock that keeps the 1 to 1 relation between time beads and path beads? .

I can bop over to the Ideal World and use infinite sets. I model a path where the path beads have reduced to points and the time beads have reduced to instants. I model in a world where paths have an infinite number of points; intervals of time have an infinite number of instants.
But I have lost velocity. I can't take the ratio of the length of a point to the length of an instant. I will leave this for the derivative to fix.

While this model may not be in my Real World, it comes close enough for government work and to my way of thinking it’s a whole lot easier than the discrete model. I can visually see the Ideal World model by drawing “unicorn” pictures often called graphs.
I don’t say that modeling in the Ideal World isn’t elegant or doesn’t give results that are useful in my Real World. Basic physics is modeled in the Ideal World and has given Real World numbers that have worked well enough to get to my Real World moon. But no matter how elegant the formulae, at the end of it all somebody has to set a dial to a number that is a finite decimal small enough to fit on the display.

Remark
I am forced to consider what difference it makes whether infinity is in the Real World or not. Infinity is used to solve Real World problems and the answers so obtained seem to give reasonable answers, so who cares if Infinity is Real or not?
I’m not sure what does make a difference. Does it make a difference to a farmer whether the earth turns or the sun rises?
Does it make a difference that √2 is not in the Real World? I guess it makes a difference in the sense that it would be a very different Real Universe if √2 were in it.

Infinity Part I

2009-05-12T13:39:00.001-07:00

Does anybody know what reality is? Does anybody really care?

Even though I speak as though my words represent truth, I mean them to only represent my opinions. For almost everything I assert, I can think of counter-arguments, and counter-counter-arguments.
I am rather in the position of Aquinas trying to prove the existence of God. I don’t know that the totality of natural numbers exists in the universe...or not, any more than I know that God exists...or not.
I do believe, which is a stronger verb than think, that there are only finite sets in the universe and that there is both a largest counting number and a smallest fraction number.
I’m an advocate of Lord Occam and infinity is too complicated and too far out for me to take seriously.
Evidently the latest concept of the universe is that it is a finite volume that has no outside. That’s pretty weird when you stop and think about it, but I don’t find it as weird as infinity.
I think of the Ideal World as a fantasy world like Alice’s Wonderland. Games are invented in Wonderland where the game pieces are Mad Hatters and disappearing cats. In the Ideal World games are played with derivatives and infinite series. In the Ideal World there are the ideal geometric figures of Plato. But these game pieces are not in the Real World.
Some the games played in the Ideal World seem to model the Real World well enough to use in the description of the Real World. The Pythagorean Theorem is an example. Even a fairly crude 3-4-5 triangle gives a right angle close enough to build a square based burial tomb.
I think the world is finite and stochastic. There is no such thing as an exact measurement beyond the simple counting of small sets. Any number given for the number of baseballs in a 100 yard cube is part of a probability distribution. We can conceive of an exact number of baseballs, so it’s a Real World concept, but it can’t be known for sure.
I think that because of the nature of space and matter at atomic sizes there is no exact distance in the Real World, only a statistical distance. In the mythical, Ideal Mathematical World boards are exactly two feet long.
There is no number in either world, except a small counting number, that is realized exactly in the Real World.
In the argument that follows my universe is finite because I make assumptions that imply that it is finite. I say what my assumptions are and try to explain why I make them. I’m not trying to convert anyone but I do think that spreading the Mathematical Ideal World into the Real World is mysticism.

A “concept” is a mental construct. This definition ties the idea of concept to humans; well, to intelligent life. “Unicorn” is a mental construct that is a concept of something in Wonderland, not the Real World. If there was no brain to think of a unicorn then a unicorn wouldn’t exist in any world.
I say that a concept “exists” in the Real World if I can conceive of it being realized by a concrete example.
I think that courage is a Real World concept. I can think of occasions when I have observed courage so the concept of courage is realized in my Real World. Courage may not be fully understood but I claim that the concept exists in my Real World.
Concepts like ‘Courage’ and ‘Mother Love’ are tricky and I will leave this kind of stuff for another day.
I say that a counting number “exists” if there is something in the Real World for it to count, that is, if I can conceive of it being realized by a concrete example. The counting number 5 exists in my Real World as a counting number because it is realized by the fingers on my right hand. I can conceive of the counting number four trillion. I can’t picture our national debt but I can conceive of it.
I will say that a set of objects is finite if it is the realization of a counting number.
When is a natural number a counting number? A natural number is a counting number when the natural number can be realized by a finite set of objects in the Real World. That is, when it purports to count something in the physical world.
Are there arbitrarily large natural numbers that are also counting numbers? The answer is, “No”.

My definition of Real World implies that every living organism has its own personal Real World so there is no general consensus on what the Real World includes. This is what happens when you start talking about mysticism. I thank God that I’m not deciding which world She lives in.

Assumption 1. The universe began with a set of identical elementary particles each of which has the same finite, non-zero mass. The universe has to make do with a fixed set of elementary particles; after the beginning it doesn’t get any more elementary particles and it doesn’t lose any.
A particle has to have something for it to exist. I take it that mass and energy are the same thing and if the particle has no mass, in what sense does it exist? I can’t think of any so an elementary particle has mass. I can’t conceive of an elementary particle having ‘infinite mass’; I don’t have a glimmer of an idea of what ‘infinite mass’ could even mean.

(I don’t know that cosmologists think there was just one elementary particle, they think there are quarks at the bottom, but it makes sense to me. If God made the primordial ball, why would She start with two elementary particles when one would do. Certainly we wouldn’t want one of the characteristics of God to be sloppiness. And if She didn’t make the primordial ball, if the ball was “just there”, why would the ball be made of a variety of different particles that just happened to be there?
The beginning of the universe is also the beginning of nature and Mother Nature is well known for her parsimony.)

Assumption 2. All of the objects in the universe are finite sets of elementary particles.
Definition 1. I will say that an object in the universe is finite if the set of particles it contains is the realization of a counting number.

There are several equivalent assumptions I could make at this point
Assumption 3. I assume that the universe began with a finite mass.

Since each particle has an identical, positive mass and the total mass is finite, I conclude that initially there was a counting number of elementary particles. I will denote this number by P which is both a natural number and a counting number.
The natural number associated with the set of all subsets of a finite set (of elementary particles in this case) can be computed. I’m going to denote this natural number by E. All of the objects in the universe are subsets of the elementary particles; the “totality of all the objects in the universe” has an associated counting number, which is also a natural number, less than or equal to E.
Every natural number greater than E is not a counting number; everything has already been counted. Since natural numbers greater than E don’t count anything, they have no concrete realization in the Real World and hence they are in the Mathematical Ideal World but don’t exist in the Real World. So the set of natural numbers that are also counting numbers is bounded. A bounded set of natural numbers has a maximum. This is the largest natural number in the Real World.
Note that since the building blocks of objects are finite in number there can’t be an unbounded set of physical objects.

(I am often asked if a thought is an object since it doesn’t appear to have mass. But a thought is a set of arrangements of particles in the brain so that a thought is a finite collection of finite sets of elementary particles.
Thoughts seem to flow continuously and continuity would seem to imply the existence of infinity. But movies seem to flow continuously and the images on the screen change every 1/32 second.
I think of time as coming in positive quanta so that in the time it takes for a thought, there only a finite number of sets of elementary particles. Over a fixed span of time, humanity has a finite number of thoughts and since everybody seems to agree that humans have only existed for a finite amount of time, humanity has had a finite number of thoughts, period. Eventually humans will all be dead or they will have changed to something else so the span of time that humanity is on the stage is finite. The totality of all human thoughts is finite. Thus the set of things humans can dream up is finite.)

When I first encountered the concept of the natural numbers and infinity, it seemed I was being asked to believe in God. How else could I lose every hand of poker but end up with all the money?
I finally decided that I was playing poker in Wonderland where poker games could go on forever. Unicorns and Medusa and Isis live in Wonderland, not in the Real World. If I can deal with talking caterpillars in Wonderland, I can deal with an unbounded set of natural numbers in the Mathematical World.
My Wonderland seems akin to Plato’s ideal world where the ideal geometric figures reside. The difference seems to be that Platonists believe their ideal squares and what not, actually exist; I don’t know what Plato’s definition of exist is so I’m not sure in what sense they are supposed to exist. Ideal squares are invisible to start with. Its sides are line segments that consist of an infinite number of points. I can’t conceive of a concrete realization of an ideal square. The square I draw is in the Real World, but I can’t conceive of a concrete realization of an ideal square. The jump between the Real World square and the ideal square is too great.
The square I draw is a picture of an ideal square and is like the picture I draw of a unicorn. A picture doesn’t imply the existence of either an ideal square or a unicorn.
Infinity is not in my Real World because I can’t conceive of a concrete realization for infinity.
I don’t allow arbitrarily small numbers in the Real World. For small numbers in the Ideal World I’ll use the reciprocals of natural numbers. I have absolutely no conception of what distance means at the size of Plank’s constant. But I do think that a journey from A to B requires at most a finite number of legs. I think time is like my old grammar school clock, it was 3:29 or 3:30, nothing in between, only the time quanta are much smaller than a minute.
I realize that relativity raises its head when considering distance and time and I have more or less ignored that aspect of space-time. But I don’t see how the properties of relativity would give birth to more elementary particles.

Thoughts on Arithmetic-III

2009-05-12T13:38:00.001-07:00

Arithmetic Part III

Counting numbers are a subset of the integers which are a subset of the rational numbers.
The Rationals are a subset of the Reals are a subset of the Complex Numbers. The algebraic operations are extended at each step to the new numbers. While the algebraic operations were originally defined to fill a real, everyday sort of need, as they are extended to these larger sets of numbers they lose a real world intuition which is replaced by mathematical rules intuition. It is the laws of exponents that have to be kept intact, not intuition using herds of sheep. “It’s the laws, Baby.”
2*3 means to multiply 2 times itself 3 times = 2 x 2 x 2. The intuition is intact. But 2*-3 doesn’t mean to multiply 2 times itself a minus three times. Physical intuition is lost. 2*-3 is defined as 1 / 2*3. 2*-3 is defined in terms of previously defined things, 2*3 and division.
The basic rule in exponents is that a*n x a*m = a*(n+m). This means that I want
2*3x2*-4 = 2*(-4 + 3) = 2*-1 = 1 / 2
and so it is for this reason that I define a*-n = 1/a*n.

Now I have exponents defined for all the integers...sort of.

I have not tried for logical correctness or completeness and perhaps I should have said more. As a matter of fact it is a decision for the teacher to make: How much should I say?
But something has to be said. Magician manuals say that when presenting a trick, you tell the audience what you are going to do. (I’m going to make a coin disappear), do it (I make the coin disappear) and then tell the audience what you have done. (“See, I’ve made the coin disappear.”)
The same technique might be used in class.
Dick Askey was an instructor at the same university where I was going to graduate school. At the beginning of each semester Dick would ask me what courses I was taking and then he would tell me what was going to be covered in those courses. He would tell me the flow of the theorems and how one followed from another, he would tell me the ideas behind the proofs. I would then know what the course was about and what the point was; all that was left was seeing the actual, formal, complete proofs. It didn’t really take him all that long, I usually took three courses, and it was very, very helpful. Topics didn’t come out of nowhere to whiz by before I knew what the point was. I liken it to driving down the freeway at 85 mph at night with the dims on. Off ramps come and go before you can read the green sign.
Of course I was prepared to assimilate the information Askey passed onto me. And a teacher must prepare his students to be able to accept general mathematical ideas.

I have tried to give the flow of my approach to arithmetic. I am a big believer in “flow”. I see the counting numbers, N, as the font from which all good things flow. N spreads its glory to the integers, I; + , x and exponents are carried along.
The integers pass the glory on to the fractions, Q; +,÷, x and exponents are carried along; division is born.
By “carried along” I mean that, say, exponents have to be defined for rational numbers in terms of how they are defined on the integers. That is, a*1/n = n th root of ‘a’ if a>0.

The idea is that the objects that are manipulated need a context; I try to make the original context the more or less observable world for both the objects and the manipulations.
But the context has to change to a mathematical context. If you keep everything in the Real World you lose the power of mathematics. When you move into the Ideal World of mathematics it is to some degree like moving to Point Barrow. It is a cold, unforgiving world. Life is not lived sloppily there.
The context here is the obedience to rules. The rules of exponents still have real world significance in special cases, say compound interest, but that is a small part of the mathematical significance of exponentials; and the real world becomes a third world. (e * iπ = -1, give me a break.)
I think the difficulty in making this change of context is considerable. It is jumping off a cliff into the sea of abstraction...with your eyes closed. Don Juan (per Castaneda) says that to become a man of knowledge there are four enemies you must defeat and the first enemy is fear. When you first start to learn something, I mean really learn it, to understand it, it is nothing like you had imagined. The familiar hand holds and crutches are gone; this doesn’t seem to be what you wanted to learn. And you meet your first enemy, fear. Carlos asks Don Juan how to defeat this enemy and Don Juan says you don’t really defeat it. You take one step after another in the face of your enemy until that first step when your enemy retreats. Then you see that you can learn anything. Some people retreat from the enemy. (And drop the class, I thought.)
I thought this was an excellent metaphor for the student’s first meeting with mathematical abstraction.
So, you might say, some people are just born with the ability to abstract. I saw a 14 year old Chinese kid play a difficult piano concerto with the Julliard Symphony. I guess it lasted about twenty minutes. How did he memorize it let alone get his fingers to operate correctly? Mozart wrote a symphony when he was seven; Daniel Boone killed a bear when he was only three.
In my teaching career I met some really bright kids, maybe not in the Norbert Weiner class but they had no problem with abstraction. But they are a small minority. The teaching problem is how to get through to kids who have difficulty with the transition

Thoughts on Arithmetic-Part II

2009-05-12T13:37:00.002-07:00

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.

Arithmetic-Part I

2009-05-12T13:37:00.001-07:00

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.

Let's throw the bathwater out with the baby

2009-05-12T13:36:00.002-07:00

Revamp of Mathematics?

Should fractions be taught?

That’s jumping the gun. Should counting numbers be taught?

Your question is ill defined because what does “being taught” entail? But, yes, counting numbers should be introduced.
In the first place, like it or not, our society counts things. It is my understanding that Native Americans weren’t all that into counting numbers. There are expressions like “a lot”, “a few”, “as many as there are leaves on the trees”, and so on that give numerical information without getting as fussy as our society does.
I remember that as a TA I was involved in the mass teaching of Calculus I. The four (I think.) tests were given at night, the answers were marked on punch cards and machine graded. I always thought it was interesting that the course started off with analytic geometry based on two and three dimensional inner product spaces; it defined the angle, x, between two vectors A and B in terms of the inner product. And then we gave a multiple choice test.
The scores would eventually be posted and lines were drawn across the printout. Above the first line was an A, below it was a B. The line’s placement wasn’t so much a function of the score as the number of lines above it. One point would be the difference between an A and a B.
I could look at a worked out problem and decide if it was worth an A, B, C, D or F. And the student could too. I would say, “This is a B solution to the problem.”or “This is a C test paper.” They would look at it and know I was right. If they argued, they usually had a good case and I could see that they were right.
On the other hand when you give points you have to decide between 6 and 7 on a 10 point problem. I think this is foolish. When I started, and a long way into, teaching I used to give points until I realized that it was crazy. It is an example where counting numbers (God forbid fractional credit) are not useful.
Our particular culture seems fascinated with whole numbers. (I have discussed this in my Calculus Book that is available on this blog.

Why is it hard for kids to remember 9 x 7?

Because they can’t visualize 67 things. (Just kidding.)
I can think of three things and I have a gut feeling for what 3 means. 6,872,970 means about 7 million which means more than I can count. Each expression has its use.
I accept the fact that there are times when accuracy is needed and the people who need it will learn how to carry more significant figures. But the very fact that we round off says that we have to handle something less than exactness. Going from 6,872,970 to 6.8 x 10*6 is just a step on the road to “as many as there are blades of grass in the park”.

Aren’t whole numbers of any use to the non-mathematician?

Of course they are of some use. They have to know which concepts correspond to which symbols. (Five corresponds to the symbol 5 and both correspond to ***** stars) I would present the basic ideas of practical arithmetic and would limit the numbers to three digits. If the student understands and has some facility with three digits they can use more digits if the need arises.
I would also introduce exponents and logs. I’m not sure when but certainly after the first grade. (Most kids aren’t as precocious as I was.) I have heard that the brain doesn’t stop growing until a person is in their twenties. Apparently the front part is where reason and cognitive functions happen and that’s the last part to develop. I suppose all this is well known. Anyway, if that is true I don’t see any purpose in trying to introduce abstraction before the brain of a student is ready to process abstract ideas.
There was an old algebra book on my grandmother’s bookshelves and when I was in the 4th grade I tried reading it. The x’s made no sense to me at all and trying to make sense out of x + 2 = 4, x = 2 frustrated me. By the time I was in the 7th grade and tried again, it seemed obvious.
So, just for the sake of argument, I would start logs and exponential when the student was mature enough, about 12 years.
The logarithm is an example of a transform like the Fourier transform or LaPlace transform. The LaPlace transform changes a differential equation into an algebraic equation, in some way the transform changes the differential equation into an easier equation to solve. The logarithm changes a multiplication problem into a simpler addition problem.
So the logarithm is an example of a much more general concept, changing a given problem into a simpler problem.
There aren’t all that many basic principles and because they are basic they can be introduced fairly early. As one progresses, the student sees that new topics are really just variations on themes.
So what are counting numbers good for?

Well, they are an introduction to unending sequences. I remember my dad telling me about the whole numbers and how there was no largest when I was 6. Knowing nothing else about whole numbers, I could see that whole numbers were a heavy concept.

I don’t know for sure what mathematics should be taught but I do know that the mathematics curriculum should be given a serious reorganization. As far as I can see, K-12 is taught a whole lot different now than it was a hundred years ago. I guess they don’t teach Hooke’s method of extracting square roots any more.
I hate the hand held calculator but I have to realize that they exist and that they are a part of society. After teaching that A divided by B is the number of times I can subtract B from A plus a remainder, I would then give three digit practice and how to approximate higher digit divisions. Then let the long tedious divisions be done on a calculator.
The most important thing is to give the student a way of looking at mathematics that gives them a facility in learning mathematics.
Another idea that has occurred to me is to put all the money into more and smaller classes, say (one teacher /ten students) let the teachers teach how they want to. I have heard that an NC in the number on a small airplane means that if you go into a spin, you can take your hands and feet off the controls and the plane will right itself. Maybe if we took people who had some knowledge of a subject, told them to teach it and then let them teach, the system might right itself.

rational numbers revisited

2009-05-12T13:36:00.001-07:00

Rational Numbers Revisited

Numbers like 1/3 seem to be on the edge. It is a fraction that is also an infinite decimal. If I am considering eating one of three apples, a third of the apples, then 1/3 would seem to be a real world faction. But if I want a third of a pie, I have to use decimals and
1/3 = 0.333..., which is in the Ideal World. A third of a pie is an Ideal World piece of pie.

Should these more or less controversial distinctions be made to students? I think, “Yes.” I think that it is good for students to think about the meaning of numbers. Whether they agree with me or not, they have thought about the meaning of numbers and hence made them less fearsome.

Every computation is made with rational numbers, in particular decimals. We see √2 and π in formulae so often that it is easy to think that they are numbers that exist in the real world but they do not. I know people that disagree with me on this but they are wrong. Before √2 was invented, it didn’t exist. It was invented as a convenience in constructing a model of a line. A model is not the reality of that which it models. Lines do not exist. Lines are part of the model that was made as an aid in thinking about the mark made on a paper with a pencil and straight edge. √2 was added to the Ideal World to fill in a hole in the Ideal World line.

When geometry appeared on the scene, the eye was the best measuring instrument available. This makes it easy to think that real world lines didn’t have holes. The proof that the square root of two isn’t rational just proves that √2 doesn’t represent something in the Real World but is a symbol for something in the Ideal World.

Evidently I believe that if I construct an eight-legged, three-headed, seven-horned character to fill in a gap in the plot of my Sci-Fi book, I have not added to the population of the Real World.

These last remarks may appear to be a digression but they aren’t really. I think that the classroom discussion of the philosophy of mathematics would be helpful. The mystery of mathematics isn’t in using the rules and symbols, it’s in wondering what the existential meanings of the rules and symbols are. And if the student sees where the real mystery lies, using the rules and symbols may lose its mystery. Anyway...

If a person wants to do something in their life that involves computation, they going to have to deal with rational numbers. I think it was Feynman who said that to understand physics, you had to know calculus. In the same way, if you want to compute you have to know decimals.

I have seen old arithmetic books that introduced decimals before fractions. Decimals were more useful and the author felt that they were a simpler approach to rational numbers than fractions. I think he had a point. It’s surprising how hard it is to get a kid facile with a carpenter’s tape measure.

I have given thought to how to teach fractions and sketched a possible approach but even as I sketched I could see problems teaching it. I kept asking myself, “What if a kid just doesn’t care about fractions?” I was saying things that would only be heard by students who cared.

I think that consideration should be given to making fractions an elective. It seems to me that the early years in school should be devoted to teaching things that the kids want to learn. I would make the point of the early schooling to give the student the experience of learning something. I found that university students didn’t know what it felt like to learn something. They didn’t know what it felt like to ‘know’ something. Giving the student this experience gives them the ability judge for themselves if they know something or not.

A student came in for help on implicit differentiation. I often tried to find the first place where the student got stuck. This particular student felt that he had everything under control up to implicit differentiation. But it turned out that he didn’t know what ‘differentiation’ meant. And when we looked at ‘differentiation’ it turned out he didn’t know what a function was, much less what an implicit function was. He really thought that he knew these concepts; he wasn’t trying to con me. He didn’t know what it felt like to ‘know’ something.

I have lived long enough to see kids grow up, mine and other people’s, and then see their kids grow up. Kids seem to like 1st grade and somewhere along the way their interest wanes. Not every kid, but a lot. I, myself, always liked school. I went back to school in one form or another every September from 1941 to 1995. I didn’t want to get out of graduate school; I loved graduate school. So I don’t know what it feels like firsthand what it feels like to ‘not like school’, but I believe them when students tell me they don’t like school.

When I got involved in home schooling I was surprised at how little time it took to cover the daily lesson plan. Since these lesson plan books were apparently covering the same material that regular school was covering, I assumed that a school room was an inefficient way to teach regular stuff.

It seems obvious to me that when you take a bunch of people and decide that they all need to know something, your success in teaching that ‘something’ depends on the size of the bunch and to what degree the bunch of students agrees that they need to know what’s being taught. The student is forced into classes that someone, often in the distant past, decided they should take.

But what about trying a different way to group students? Instead of grouping students by courses that are decided by forces external to the students, group them by common interests that they have. At the beginning of the school year, there would be a period of time when the students would form groups of common interest and then teachers would be assigned to the groups where they were competent in the interest of the group. A group can’t be an empty set of students but it can have a single element. A group could be quite large; the students may have a group that wants to be a drum and bugle corps or a symphony orchestra or a football team or a chess team...whatever.

If a person knows how to learn and knows how to recognize the difference between knowledge and ignorance, they have the world open to them. The basic elements of mathematics are not many and the different fields of mathematics just put them together in different ways and give them different names. A particular question may be difficult to answer but not hard to state.

I think that if you know how to learn one subject, you can learn any subject should the spirit so move you.

Now in my utopian school, when it comes to teaching fractions, the teacher would start with students who more or less want to be there. There is some ground floor to stand on.

I also think that the educational process is hindered not only by the counter-productive grouping of students into fixed courses but by the age restriction. When a kid is ready to learn, teach him or her. The mixture of ages in a classroom would lend stability. The younger kids would see an older person who feels that learning is worthwhile. I do not understand why there is segregation by age in schools.

When I was 30 years old I took up the motorcycle. I would ride on a desert area a couple of blocks from home in the company of teenagers. These boys could ride better than I and knew more about motorcycles generally. The experience of being taught by these kids was thought provoking. They were surprisingly adult in their attitude toward motorcycles and telling me what they knew. Learn is learning and these kids taught me a lot.

Fractions

2009-05-12T13:35:00.000-07:00

Rational Numbers

I taught a course in arithmetic for elementary school teachers and I did a terrible job. Among the many reasons that it was not a good course was how I dealt with fractions.
After all, I thought, the fractions are just the field of rational numbers and there aren’t a lot of field axioms to memorize. You don’t care where the axioms come from; they’re axioms for God’s sake. You don’t have to know where they come from, you have to accept them.
Mea culpa, mea culpa, mea culpa.
At a later time I thought about how I would teach fractions if I were given a second chance...which I was, probably wisely, not given.
I started with integers as counting numbers. I realized that some memorization was needed. Not being familiar with Arabic, I get no clue from the symbol 2 that it stands for ** that many stars. I must have memorized a meaning in terms of how many things I was looking at and associated one of the symbols 1, 2, 3, 4, 5, 6, 7, 8, 9 or 0 with that meaning.
I picture the first attempt at transferring numerical information from one person to another as holding up fingers and some agreed upon hand signal to indicate repetition. Next I see a need to give name to the number of fingers I hold up.
I then go into a story that sees the need for arithmetic and a reason for developing addition etc. I make no claim to historical accuracy which is irrelevant. What I want is a history that makes some sense and gives a cohesion, a framework for further ideas.
I also realized, as I thought about it, that small integers were different than big integers. Homer talks of as many Greeks before Troy as there were leaves on the trees of Agamemnon’s olive orchard. Something like that. We hear on the TV that 102,000 people attended the USC-Notre Dame game. How much more does 102,000 convey than ‘the number of leaves on the trees’?
If I had a class of 20 students, I would have to count them to see if one was missing. If you were a crow and watched n hunters go into a barn and (n-1) hunters come out, how big would n have to be before you couldn’t tell the difference between n and (n-1)?
I define multiplication as fast addition. n x m means ‘add m to itself n times’. An obvious question: Is ‘adding m to itself n times’ the same as ‘adding n to itself m times’?
I am driving down a road and on the right side are 734 piles of bricks and in each pile are 2147 bricks. On the left side of the road are 2147 piles and 734 bricks in each pile. Which side of the road has the most bricks? And why do you believe your answer is correct?
“Well”, you might say, “The number of bricks on the right side is 734•2147 and the number of bricks on the left is 2147•734. Since 734•2147=2147•734, there is the same number of bricks on both sides.”
That last equality isn’t obvious to me. I can’t realistically count the bricks to verify it. I don’t have a have an intuitive sense of 734 piles or 2147 bricks much less of adding 2147 to itself 734 times. Why should I believe your answer?
But I do have an intuitive sense of adding 2 to itself 3 times and adding 3 to itself 2 times. I get the same number of fingers either way I do it. For integers that are small enough for computation on my fingers, I can see that m•n=n•m. Not only that, if I look at it geometrically and compute the area covered by square unit tiles, I can see why it works and see no reason why it wouldn’t work for any integers, no matter how big they were. I believe that multiplication commutes for large numbers because I believe it for small numbers. Thus I believe your answer.

I should explain again that I am after belief. I have gone through proofs that I could see were logically correct but I couldn’t see why they worked so I couldn’t really believe their conclusions.
While pictures are not acceptable in a formal proof they do wonders for belief and understanding. There are no rules in a knife fight.
Principle of Induction (modified): If something is true for a few, consecutive small integers starting at one, then it’s true for all integers.
I seldom use more than the first ten.

At last I come to finding a numerical way to describe parts of things. I have an estate valued at $21 and I want to divide it equally between my three children. I want to divide 21 into three equal parts. I’m going to define 1/n•A to mean breaking A up into n equal parts and 1/n•A stands for the size of any one of them.
I’m going to break my $21 into three equal parts and give each of the three children one of the three equal parts. I will give each child (1/3)•21=7 dollars and end up with three piles of dollars with 7 dollars in each pile. 3•7=21
New problem: I’m going to give a kid $3 to help sweep out my bowling alley. If I have $21, how many kids can I hire? The question here is how many times can I subtract 3 from 21? But this concept already has a name, division, and a symbol 21÷3. I can subtract 3 from 21 seven times and 21÷3=7. Now I have seven piles of dollars with three dollars in each pile. 7•3=21.
New problem: I cut a whole bunch of pizzas into thirds. (I am in the Ideal World. You can’t cut a real world pizza exactly into thirds. Actually, cutting something in half needs an “I cut, you choose”procedure. I passed the pieces out to the multitudes and after everyone had eaten their fill there were 21 pieces left. How many whole pizzas would that make?
This is the problem of adding up 21 pieces of pizza, that is, adding 21 thirds.
I can write this as 21•(1/3)=7.

The last example takes some short cuts but the idea is that there is a separate problem with a separate intuition for each of the three, numerically equal, expressions.

7=21•1/3 = 21÷3=(1/3)•21

They correspond to the word problem types: 7•?=21,?•3=21 and 7•3=?

The rules tell you that (1/3)•21=21÷3=21•(1/3) but the rules don’t help in deciding which form of the rule to use in a particular problem.

I am not necessarily saying that my ideas should be used verbatim if at all, but I don’t see how a person, say, a student, can deal with these kinds of word problems and not at least be aware that there are three kinds of problems. How can a person teach these kinds of word problems and not understand the basic ideas?

I don’t know how much stuff should be put into the introduction of a kid to fractions. I have not had a whole lot of success in teaching fractions to kids. One problem was getting the kid to realize that fractions have a relevance to their lives.

How do you design a course for people who are sure that they will never think of the subject material again? How do you teach ‘the null course’? What about teaching to students who will, indeed, never think about it again?

I think that one of the basic questions is what if anything should be taught. When I think about how I’m going to teach something, I assume at least some minimal interest in learning what I trying to teach. I make this assumption because otherwise anything I might say about fractions has the same teaching value...none. So, if knowledge of fractions is judged to be an important part of knowledge, you have to start somewhere other than fractions.

Any damn fool can teach students with a hunger for fractions. Hungry students force the person in front of the class to at least do no harm.

But back to fractions:

I thought about how to bring intuition to the rules used in computing with fractions and started running into trouble with division by a fraction. Trying to give a useful intuition to (1/2+1/3)÷(1/5+1/7) was beyond me and I haven’t yet come to using large integers in the fractions. And the “simplify the following fractions” problems seemed to have no other way to solve them than mindlessly following rules. The application of the rules is not entirely straight forward and there are often many ways to simplify a fraction. Why is one form simpler than another?

It seemed to me that a ‘rule intuition’ had to be developed. Simplifying fractions is more akin to chess than mathematics. In both endeavors, if you learn a few basic rules you can play the game but it takes practice to gain intuition for the rules and be good at it.

There is a segment of technology that is devoted to making sure that simplifying fractions is an obsolete skill. I think that part of fractions could be done away with, or at least that possibility considered. Decimals are the big rage now, decimals.

It would seem that educators would spend time discussing what to teach and then discuss how to teach it. And there are good cogent reasons to teach fractions...but maybe not for everybody. For my language in Jr. Hi, I chose Latin. And I probably would have chosen fractions as well. But I am not the only person in the world.

My point is that the understanding of fractions is non-trivial and it isn’t clear that it is worth the effort to force it down the student’s throat. I think some more thought could be put into arithmetic courses, in particular what should a course in arithmetic accomplish?

Does the square root of 2 exist?

2009-05-12T13:34:00.000-07:00

I recall a student asking me what the “square root of two” (hereafter called √ 2) meant. I was about to say that it was a number that when squared you got 2 but that seemed a little too glib so I held my tongue and told her that I would think about it and tell her tomorrow.
The more I thought about it the more I realized that her question was not trivial. √2 is an infinite, non-repeating decimal but what does this mean. To make a long story short, I decided that √2 did not exist in the Real World.
In my state of ‘eureka’ I told a colleague that √2 didn’t exist in the Real World. He said that of course it did. It was the length of the diagonal of the unit square. I then asked him, When was the last time you saw a unit square.” Unit squares aren’t in the Real World either.
When teaching about the graphs of functions I would take a function, say f(x)= x+3, and plot a few points, talk about slopes and intercepts and points and eventually draw a line on the blackboard. Then I would say, “This is the graph of the line f(x) = x+3.”
But it wasn’t the graph of f(x) = x+3. It was a picture of the graph of f(x) = x+3. The graph of a function is a set of points and points aren’t in the Real World. I can’t say that a dot on a blackboard is a point. It is a picture of a point.
I liken the picture of a graph to the picture of a unicorn. The picture of a unicorn is not a unicorn. It is the picture of something not in the Real World.
And here is one of my main points. The symbol is not the object. The symbol is a representation, a picture if you will, of the object.
The proof that √2 is not a rational number is not a proof that √2 exists, just that if √2 did exist it wouldn’t be a rational number.
Here I think the Greeks went astray. Previous to the proof that √2 wasn’t rational the Greeks had assumed all numbers were rational. Since √2 wasn’t rational they concluded that there were other kinds of numbers that weren’t rational instead of concluding that √2 didn’t exist.
In my calculus book I introduce the “Ideal World” and put √2 in the Ideal World along with infinity. Unit squares are in the Ideal World. Graphs are in the Ideal World. Unicorns are in the Ideal World. Real World problems are modeled in the Ideal World and solved in the Ideal World. The answer is interpreted in the Real World and then experiments are done to see if the answer makes Real World sense. The pure mathematician stays pretty much in the Ideal World and if the Ideal World laws are obeyed she thinks the problem is solved. It is left for the applied mathematicians and scientists to see if her answer makes sense.
So what is in the Real World and what is in the Ideal World? I seem to be edging perilously close to Platonism except that I don’t think the Ideal World exists. I think the Ideal World is a construction of humanity. I wonder at my colleagues who are atheists but believe in √2.
What about functions? I think functions as we know them are human constructs made in the Ideal World. Our concept of function is that of a single valued function. This arose because to a first approximation our world is single valued. A rock always falls 16 feet in the first second. Note that I am already in the Ideal World. When we teach mathematics, we don’t say that the rock falls about 16 feet in about 1 second. In the Real World nothing is exactly 16 feet in length and no interval of time is exactly 1 second.
None the less, when I tune my radio to 89.9 I get KUNM every time. We believe in the repeatability of physical laws and hence single valued functions.
But in the world of the dolphin there is no repeatability. If you drop a rock repeatedly in a swimming pool it doesn't fall to the bottom in a single time; not even close. So the dolphin wouldn't develop single valued functions. Functions are a human construct.
I read where an applied mathematician said that the only numbers in the Real World are finite decimals. But a decimal point followed by a billion zeros followed by a 1 is a finite decimal. To a mathematician it is a perfectly good positive number. In the Real World it equals 0. Mathematicians let Δx go to zero without a qualm, even in calculus courses for business majors, even if x represents light bulbs. What does a billionth of a light bulb look like?
Personally I put very, very, very small numbers in the Ideal World. Likewise with very big numbers, I put 10 raised to the billionth power in the Ideal World. I am using extreme values and a number slips into the Ideal World a long time before 10 raised to the billionth power.
I think the student wished she hadn't asked about √2.

Why mathematics should be taught.

2009-05-12T13:33:00.002-07:00

My students would often ask me why mathematics was inflicted upon them. If the student was in the sciences, the answer was more or less obvious but seldom expressed: You have a problem. You collect data, organize it and use the organized data to solve the problem. That is what mathematics is about.

Mathematics also provides techniques to solve particular kinds of problems and unfortunately courses in mathematics give a lot of time to techniques and very little to understanding.

Pre-med students were particularly vocal in their distain of, say, algebra. But it seems to me that diagnosis is the collection of data, organizing the data, and using the data to deduce the cause of a patient's discomfort. Not too much different that finding out how long it takes Joe and Harry to paint the house together.

Of course the budding doctors do have a point in that the algebra course (and prehaps the required calculus course) was probably more about memorization than investigating the power of rational thought in solving problems.

I must admit that the ability to memorize is a valuable skill for a doctor, all those bones and blood vessels have names. I wish I would have thought of that point when I was teaching the course. Or maybe not. Why would I think of ways to justify a teaching technique that I think is counter productive to rational thought.

I think that rational thought should be an element in all teaching. It can't hurt. An artistic person told Richard Feynman that knowing the chemical composition of a rose didn't give one an appreciation of the rose. Feynman replied that he may not see everything in a rose that an artist does but that it can't it can't hurt to know more about a rose.

Mathematics is a model of rational thought. If a person doesn't want to think rationally I guess that's their business. I had a person tell me that they didn't believe in rational thought. They made their decisions intuitively. Well, it's a free country and I suppose if a person wants to intuit their way through life, then so be it. But even if a person always decided to follow their gut feeling, how much could it hurt to put a little rational thought into the hopper.

I guess how much it could hurt is another topic for another day.

Where is mathematics going and what is it doing in that hand basket?

2009-05-12T13:33:00.001-07:00

I was browsing through my book shelves and came across an old arithmetic book. Samuel G. Kimball of Thompson, Connecticut had written his name several times on the fly page, as is the wont of young men, in 1836. He could well have fought in the Civil War.

The book has a little over 200 pages and is about 5'' by 7''. The author has given some thought as to the order topics are presented and in the preface explains his choice.

A topic is presented, say, how to change the currency of New England or Virginia into Federal Currency. Some examples are given which are followed by about a dozen problems.

It seems to me that the teaching of mathematics hasn't changed much since 1836. The books have gotten too heavy to lift, too many topics to cover, too many examples to go over, and too many problems for a student to work before the instructor beserks to the next section, but other than these differences, mathematics books of today are pretty much the same.

The main difference between then and now is an extreme application of the 'more is better' priciple. I think that the general teaching of mathematics hasn't changed except for the worse.

I hear of conferences held by educationists, I read the papers given in these conferences but I don't see any of these ideas reflected in the classroom.

I don't see the question: Why do we teach mathematics? addressed in any conferences.

There are students who will learn and understand mathematics no matter how bad the teacher or the book. There are students who will never understand mathematics no matter how good the teacher or the book. There are a multitude of reasons for this: hormones, parents and sports, for example.

I could go into detail on the problems students face but I will first give my ideas on why mathematics should be taught and on how it should be taught.

First, "How".

It seems to me that almost all mathematics, certainly elementary mathematics, deals with the same concepts.

There are a collection of objects, for example, counting numbers, or fractions, or real numbers, or vectors. A collection of objects is given a name, for example, the integers, or the rational numbers, or vector space. Some of these collections are called groups, or rings or fields.

At this point we have a set of objects. Now a way is given to combine the objects in a collection. Take for example, the integers. Being a greedy species we want to get a number that measures our existing herd of cows and the cows we rustled last night. We add integers. We want to know how many cows we have left after the coyotes killed some last night. We subtract integers. If I rustle 15 cows every night this week, I want an easier way to represent my ill gotten gains than 15+15+15+15+15+15+15 so I invent fast addition, also called multiplication and write 7x15.

We then study the methods of combination. Is 4+7=7+4 ? Is 4x7=7x4 ? Regardless of what the collection of objects is, we look for general properties of the combination methods for that collection.

Finally, there are functions. We want to associate a cow to a price it will bring at the slaughter house. We want to associate the time it is to where we are. We want to associate the current through a resistor with the voltage across the resistor.

It is my contention that if a collection of objects, the ways of combining the objects and the functions defined on these objects are studied deeply and understood, then the students knows how to approach any collection of objects with its own rules of combination and functions.

Instead of touching on a lot of subjects, I suggest going deeply into one subject. I think that if a student knew one collection of objects well, he would be prepared to learn any collection of objects with their rules of combination and functions.

Indeed, they would see that functions are just another collection of objects and that we can combine them. We can add functions, compose them, multiply them-these are just ways to combine objects called functions.

Instead of having the students memorize a bunch of stuff, we should be teaching students how to learn mathematics.

I have talked to graduate students in physics or engineering that didn't realize that a linear differential equation is essentially a problem in functions and that it is essentially no different than solving ax=b.

Mathematics studies a set, the rules of combination for the elements of the set and functions defined on the set. End of story.

Why mathematics?

2009-05-12T13:32:00.000-07:00

When I was in grammar school, 1941 to 1947, I was told that I should learn arithmetic so that I could make change and balance my check book. I already knew how to make change and I couldn't balance my check book until the bank did it for me and I read it on line. Fractions and long division were fun and I can still do them but the hand held calculator, which I don't use as a matter of principle, have made my knowledge obsolete.

In Jr. High School I learned about logarithms so that I would understand the slide rule and compute with trig functions. Alas, students of today have never heard of a slide rule.

In High School I learned how to take square roots by hand. I learned about Des Cartes rule of signs and arcane methods of factoring polynomials. I don't think they cover these topics in high school anymore.

I do however remember these topics and techniques. Well, I don't remember how to take square roots with pencil and paper.

Recently I asked a student who had graduated from high school a couple years ago with A's in algebra what he had learned. He couldn't think of anything he had learned. I asked another student what he had learned in algebra and he said he knew the quadratic formula. I asked him to tell it to me. He finally said that was a square root in it.

Both of these young men said they had a good teacher. What does it mean to be a good teacher if your students don't carry anything away from your course? What good does it do to pass tests and get a good grade and then have what you learned evaporate?

What do all the tests that students take today mean?

I helped a friend grade tests from a high school geometry class. The questions were by and large multiple choice and fill in the blanks. They had to know volume and area formulae, the volume of the frustrum of a pyramid comes to mind. Those students that remembered the formulae couldn't do the arithmetic. As I marked problem after problem incorrect I wondered if the few students who correctly found the volume of the frustrum would know any more after the final exam than the students who missed the problem. There were no proofs required although there was a two column proof on the test with one of the reasons missing; fill in the blank.

When I took geometry in 1950 I was told the study of geometry would help my ability to think clearly and precisely. I was told that arguments should be based on axioms and the rules of logic. I was told that the terms used in discourse should have clear definitions. I was told that these ideas could be used in any area of human endeavor.

Now, this made sense to me. It was not a particular proof that was important, it was what constituted a proof, what did it mean to prove something. It was not the conclusion but the means by which the conclusion was reached that was important because it was the means that gave the conclusion validity.

It is true that my sample size is small but in my small sample logical thinking has been replaced by memorizing and multiple choice tests.

What is the point in teaching things that are forgotten as soon as they are learned...if they are learned at all? And if they are remembered, seldom if ever used?

What is the point of covering material so quickly that by the end of the course the students are lost? In college calculus courses the rule seems to be a section a day. At the conclusion of these courses the passing score is set so that half of the class passes.

My viewpoint:

Teach fewer topics and cover them deeply so that the students understand concepts and how to approach learning mathematical concepts. It is my opinion that learning high school algebra is not particularly different than learning abstract algebra; or learning any mathematics. How to learn mathematics is the skill that should be taught and it doesn't really matter which mathematics course is used. If the student knows what the point of mathematics is and how to learn it, the student doesn't need a teacher they just need a book.

Of course, teachers have to know how to learn mathematics.

Why algebra?

2009-05-12T13:31:00.002-07:00

A high school student asked me why they had to take algebra. They were planning to into the arts and didn't see any need for algebra.

This has come up before and in point of fact I think making algebra a universal requirement is stupid. Yes, I said it. Stupid.

I suppose requiring algebra of everyone in high school is some residue of the 'new math' insanity.

I know a guy who started working on cars when he was twelve. His brother got an engine from a junk yard and he spent the summer taking it apart and putting it back together. In the summer of his 13th year his brother got him another junk yard engine and by the end of the summer it was sitting in the front yard running.

The mathematics he needed after HS was knowledge of angles involved in crank shafts and cam shafts, the mathematics of displacements and piston ring gaps; no algebra. But his HS required algebra and so I had to teach him about angles and displacements in his thirties. He never saw an 'x' after HS.

If you are in science or mathematics algbra is used all the time although the only time I have personally used HS algebra was in altering the bourbon-vermouth proportions of four galleons of manhattans.

And it isn't that an English major couldn't use some mathematics but that mathematics is not algebra.

Mathematics is said to enhance thinking skills and I believe that. But a HS algebra course, or a university algebra course for that matter, may help memory skills but not thinking skills.

I know of a HS that gave up on algebra word problems and stopped teaching them. Any possible value of algebra was taken out. But algebra was still required.

A kid that lived next door to me took all the mathematics courses in HS, including calculus, and tested into algbra when he entered university. He aced the university algbra because it was pretty much the same as the HS algebra that he had aced.

There is the concept of "core courses" that all students are required to take and of course mathematics gets a share of the pie. Since all students are supposed to take a core course there are a lot of huge sections, hence more teachers, hence more positions to fill.

I think the "core course" is a stupid idea. Yes, I said it. Stupid.
To make algebra a "core course" is stupid supersized.

If mathematics understanding isn't taught, why teach it at all? If memoriztion is the object, save money and use the phone book for a text.

I had a freshman in a calculus I class who seemed to get the calculus pretty well but asked me why x/a = (1/a) x.

Why teach algebra to your students when you haven't taught them arithmetic?

Because in the scheme of things maybe it doesn't matter?

Selling knowledge and selling sex are both forms of prostitution

2009-05-12T13:31:00.001-07:00

Where do I start?
I talked to a HS teacher and he told me that he told his students that mathematics was just a bag of tricks. I would tell my students that there are no tricks in mathematics, just understanding.

Perhaps this is one of the great divides in mathematics education.

I suppose that algorithms are useful but as Stein says, "Algorithm stops thought."

Algorithms also make students think they know something that they don't.

I was teaching calculus II and a student said he didn't understand why he was failing. He told me that he did well in calculus I, in particular max min problems. I handed him a calculus book and told him to find a max min problem, his choice, and work it on the blackboard. This occurred during office hours and while I answered questions he worked at the board. After about an hour I had answered the last question of the last student and I looked at how the max min problem was going.

He had chosen to find the dimensions of the biggest open topbox with with a square base that could be made with a given amount of cardboard.

He had made no progress. He had made a slice through the box as if he had some kind of integral in mind. He had no idea how to work the problem.

The student wasn't trying to con me. He really thought he knew how how to work the problem and was dismayed that he didn't.

A High School student told me that he had done well in algebra. I asked him to tell me something he knew. He said he knew the quadratic formula. So I asked him what it was. He knew that there was a square root and that there was a denominator. He was surprised that he didn't know it.

Another High School student complained to me that he could pass the weekly tests with A's but couldn't pass the monthly tests. I told him he could remember a week's worth of algorithms but not a month's worth.

Students like the idea of having to memorize. It gives them something concrete to do. And algorithms are easier than understanding to teach.

I am going out on a limb but most High School, Middle School and Elementary School teachers have little if any understanding of mathematics. Many tell their students how hard mathematics is and how they didn't like it when they were in school, remarks that I have no doubt are true.

But these teachers were taught by people that didn't understand mathematics. I was told by a woman in the Dept. of Education that she didn't really know mathematics but felt qualified to teach students in mathematics education how to teach mathematics. Go figure.

I don't think that these teachers are bad people. They aren't giving their students a bad course out of spite of meanness. They think they are teaching the best course they can and in the absense of understanding mathematics themselves, they are. In general they like their students and wish they could find a way to help their students memorize better. (Although some teachers seem to feel that students fail just to spite them.)

People who can memorize algorithms well become mathematics teachers. They work out tricks to help them memorize and pass these tricks on to their students, telling them that mathematics is just a bag of tricks.

Mathematics as algorithm is passed along from teacher to student, and the student who excells at the algorithm game becomes a teacher and continues the tradition.

Students learn the algorithms and pass courses thinking they know some mathematics and then don't understand why they don't.

Passing a student who doesn't know the course well enough to work a basic problem at the end of it is being lied to. A teacher told me that she sometimes gave a student a break and give them a C when they really deserved an F. She wasn't giving them a break if they didn't know enough to pass the next course. She wasn't giving them a break, she was condemning them to the limbo between a course they didn't want to repeat because they passed it and a course they weren't prepared for. Some break.

If the student payed good money for the course, she is being cheated.

But there aren't good guys and bad guys, only people doing the best they can without understanding.

What's Wrong

2009-05-12T13:30:00.001-07:00

I should perhaps talk about problems in teaching rather than anything being wrong. I think 'what's wrong' would lead too far into the philosophy of right and wrong.

There several fundemental problems:

The ambient attitude toward mathematics that we live in. When I inform people that I am a mathematician, there follows:

"I always hated math."
"I was never any good at math."
"Just can't do math."
"I was good at math until I had a bad teacher in 7th Grade and got turned off."

This isn't everybody, of course. I try to avoid the universal quantifier. I remember going to an Urgent Care and the doctor was reading Strang's "Linear Algebra".

None the less, such instances are rare.

I listened to The Senate when they gave bigger grants to students that were going into mathematics and science or take a crucial foreign languages. (I wonder what those are.)

The Senate seems to believe that money is the universal nostrum.

I began graduate school before Sputnik went up. I recall that someone in the teacher's lounge remarked "Sic Transit Gloria Mundi." when a Vanguard blew up on the launch pad. I remember the day when the Russians were successful at orbiting their first spacecraft.

There followed a torrent of money into mathematics and science. Grant money was there for the asking. In universities mathematicians had two course loads. Before Sputnik the Mathematics Professors had leather patches on their elbows.

After Sputnik the number of mathematical journels grew as without bound. If you had a Ph.D. in mathematics, you had a good job. You could drop out for 5 or 10 years and come back to a job.

And the "New Math" was born. An injury from which mathematics education has never recovered.

(To be cont.)

What's Wrong?

2009-05-12T13:29:00.000-07:00

I want to take a look at what is wrong, yes wrong, with the teaching of mathematics.

Welcome

2009-05-12T12:33:00.000-07:00

This category is all about teaching, learning and education