Tuesday, May 12, 2009


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

Teaching Revisited The 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 it The 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

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

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.

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?

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

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

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

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
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

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

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

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

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

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?

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

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

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, 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.