Notions of Sameness

Notions of Sameness

By Bobby Neal Winters

Mathematicians use sets and structures on sets to model realities that they perceive only in their minds.  I dealt with this in the matter of topology in one of my recent essays.  These structures induce special properties that we want to preserve.  I am still interested in this mainly in the case of topological spaces, but I will work my way up to it via a series of examples from other areas of mathematics.

The first example is that of sets with no structure in which we can only discuss the concept of cardinality.  This is to say, how many elements does a set contain?

With finite sets we can say that a set is composed of five elements or seventeen elements or two trillion five hundred fifty four billion two hundred twenty seven million three hundred twelve thousand five hundred and two elements.  There is a number we can associate with them.  

For infinite sets, the above does not work; we have to do a flanking maneuver. We avoid the word number and use the term cardinality.  We say that two sets have the save cardinality if they can be put into one-to-one correspondence with each other.  

We use the language of functions to make such notions precise.  Functions are thought of as having a sending set (the domain) and a receiving set (the range).  The range is like a target.  When two sets have the same cardinality, there is a function from one set to the other that is one-to-one and onto.  A function is onto if every point in the receiving set is matched with a point in the sending set.  A function is one-to-one if no set in the receiving set is matched twice.

A function that has both of these properties at once is christened with the high-falutin’ label of bijection.  It’s technical and pretentious but remember it anyway because it’s important.

If two sets have a bijection between them, they have the same cardinality.  If one (and therefore the other) of those sets is finite, they will also have the same number of points.  By this mathematical slight of hand of using a particular type of function to create a new notion of “sameness” we’ve expanded the notion of number/size into the notion cardinality/size.

This is done in the realm of algebra as well.  And here I must be careful because I am using the word “algebra” differently than many use it.  If you had pre-algebra and algebra in middle and/or high school, or if you have had intermediate algebra or college algebra in college, then rest assured that I am talking about something that is related to that in the way that a Tomahawk Missile is related to, well, a tomahawk.

Algebraists, and I am over-simplifying, study sets with binary operations on them.  For example, they might study the real numbers with the operation of addition or the positive real numbers with the operation of multiplication.  (Notice I put the word “might” in there because they aren’t really interested in those two particular sets, but I will use them because I don’t want to talk about a lot of algebra now.)

The real numbers numbers with addition and the positive real numbers with multiplication are examples of groups.  Groups are sets with binary operations that obey certain rules.  Don’t worry about what those rules are; if you can’t help it, look it up on Wikipedia.

Those of you who have had a course in college algebra might remember the logarithm.  I put emphasis on the word might there because I’ve no doubt that you have seen the logarithm.  You might remember it; you might wake up in cold sweat screaming it at the top of your lungs; your lover might have attempted to comfort you afterwards asking, “What is a logarithm, Sweet heart?” only to have you deny ever having heard the word.

In any case, the logarithm is a function from the positive real numbers to the real numbers that is bijective, i.e. it is one of those important one-to-one correspondences.  The important thing about it is that it changes the operation of times to that of plus: log(a times b) is equal to log(a) plus log (b). In mathematical ages B.C.--before calculators--the logarithm was used as a means of doing multiplication by turning it into addition.  You may not have notice, but addition is a lot easier than multiplication.  

Such a function that is bijective and preserves the operations on the groups--in this case turning multiplication into addition--is called an isomorphism.  The groups are said to be isomorphic to each other and a mathematician in the capacity of algebraist can’t tell the difference between them.

I could go on listing the various areas of mathematics and the functions they use to preserve there important concepts, but instead of doing that let me focus on topology.

A topological space is a set along with certain special subsets that we call “open” sets.  Topological concepts are those concepts that can be defined in terms of open sets.  As a consequence of this, we are interested in functions that preserve open sets.  A bijective function between two topological spaces that takes an open set in the first to an open set in the second is called a homeomorphism.  A pair of spaces between which there is a homeomorphism are said to be homeomorphic.  Spaces that are homeomorphic are the same as far as a topologist is concerned.  

Most frequently, I see homeomorphisms defined differently.  We usually define a homeomorphism to be a continuous function which is invertible and whose inverse is continuous.  This is mathematically equivalent to the other definition I’ve just given, but it’s shorter.  

Continuous functions are interesting in their own right.  Indeed, one might argue that topology was created in order that we might better study continuous functions.  There is also quite a well-developed theory of continuous functions that allows us to check whether a particular function is continuous.  For example, it is quite easy to use calculus to check that the logarithm and its inverse the exponential function are both continuous; it might be less easy to show they preserve open sets.

But in our desire to be efficient with our definitions, we should take care not to obscure for the beginner the very thing that homeomorphisms do: they preserve open sets.  As they preserve, nay set up a one-to-one correspondence between the open sets of each space, any properties defined in terms of open sets will be preserved.  That is the point.

Sets with Structure

By Bobby Neal Winters

This semester I am teaching a course in topology after a hiatus of six years.  I am using a classic text by James R. Munkres with the title, fitting enough, Topology. This is the text I had my graduate level course from.  It’s been like meeting an old friend again after an extended separation.  Only someone who’s done that can appreciate all of the levels of meaning.

Topology is a word-like every word now that I think of it--that carries a bundle of meanings.  On the level that is most accessible to a popular audience, it is understood to mean that branch of mathematics in which a coffee cup is no different than a phonograph record. (That’s a CD to you, you young whipper-snappers!) For the sake of precision, we could make a distinction by saying geometric topology or even low-dimensional topology, but in practice clarifying adjectives or adjectival phrases get stripped off and we are left with topology left alone, forced to hide the other meanings it carries.

Today, I would like to venture into one of those areas where angels fear to tread to talk about the subject that mathematicians (especially geometric and low-dimensional topologists) refer to as general or point-set topology.

One can could say that low-dimensional topology is a sub-speciality of general topology, and I will justify the sense in which that is true in the sequel, but such a statement blurs over differences of mindset among the various practitioners.

Let me say I was drawn to my first topology course having seen the pictures of coffee cups being blithely changed into phonograph records, donuts, etc, only to find something entirely different.

A course in general topology begins with a topological space.  A topological space is about as abstract a concept as the math major will meet as an advance undergraduate or beginning graduate student.  It is a set which is paired with a special collection of its own subsets, and this special collection of subsets have a set of laws they must obey.  I won’t tell you now no matter how much you beg me.  We give a name to that special collection of subsets and call it a topology.  I told you the word carried a bundle of meanings.

The most common example of a topological space is the set of real numbers.  Topologists who’ve just read that sentence are now picking up pencils from their desks to write in “with the usual topology” between the “s” in the word numbers and the period that follows it.  I left it out on purpose just to annoy them because it is the usual topology. It is based on the open intervals that students learn about as early as middle school.  The open intervals are used to construct open sets and the set of all of the open sets of the real numbers is the usual topology on the real numbers.

The usual topology on the real numbers is such a natural thing to us--and my “us” I mean “geeky math types”--we don’t even notice that it’s there.  We use the real numbers with the usual topology first in calculus and later in analysis, and I have talk these courses without ever uttering the word topology.  Most of the basic results in those areas can be reached without naming the topological concepts explicitly.

Perhaps the concept of a topological space would never have been created had mathematicians not ventured beyond the real numbers, but--you know those scamps--they did.  They ventured into the plane, into 3-space, into sets of functions, and so forth, and they discovered sets of subsets in each of those areas that behaved like the open subsets of the real numbers behaved.

If I knew more of the history of the subject, this would be an opportunity to segue into a case study in abstraction.  Those three examples I listed above have quite a bit of structure on them.  They have ways of doing arithmetic, they have ways of measuring angles, and they have ways of measuring distance. They are groups; they are vector spaces.

When we push out to the level of abstraction required by the topological space, we forget about all of that other structure.  You can’t do arithmetic; you can’t measure distances.  You think about only the set and its topology.  You only define properties that can be discussed in terms of the members of the topology.  You only discuss functions which respect the members of the topology.

In some sense, learning general topology first requires that you forget everything else you know about anything. You become a slow thinker; you become a deliberate thinker; you always must be careful that your intuition--raised as it was in the fertile fields (nerdy pun fully intended) of the real numbers--does not lead you astray.

This sort of abstraction allows us to prove theorems that apply to a wide range of areas. It allows us to create language to see an underlying unity in diverse areas of knowledge.  It also provides a trap-door into what has been referred to as centipede mathematics, as in “How many legs can I pull off the centipede before it can’t walk any more?”

I called it a trap door, but I am not sure that metaphor works.  It makes what happens sound like an accident.  The truth is more complex.  Many--most--who are drawn into mathematics find this sort of abstraction attractive, not to say intoxicating.  Going deeper and deeper into abstraction leads us into what our appetite desires.  It is like the wind buoying up our wings, lifting us farther and farther from the ground.  Here the story of Icarus is attractive, but also inaccurate.  We don’t go so high that the sun melts our wings; we are lifted so high we are never seen again.

There is a quote I’ve heard attributed to RH Bing, a Texas mathematician who is a personal hero of mine.  When asked about a visiting topologist, he is said to have replied, “He studies spaces of which there are only one example and only in England.”

Mathematics, especially abstract mathematics, is best when it is equipped with numerous examples. Examples give breadth and richness.  Examples guarantee you aren’t just proving theorems about the empty set.  But I digress.

General topology is alive with examples.  It is wide and it is deep.

There was a time in my career, and I will say this without shame, that I taught subjects simply because I wanted to learn them myself, without regard to the student.  I say it without shame because the students can still get a lot of value from that provided they are motivated themselves and their needs are being regarded other places.  Time has dealt with me in any case. I find myself singing along with Bob Seger:

Well those drifter's days are past me now

I've got so much more to think about

Deadlines and commitments

What to leave in, what to leave out

As I teach my courses now, I try to focus on what I think the student needs.  One great need that students have as they enter into graduate mathematics is to have their pre-assumptions stripped away. The abstractness of general topology is the best method I know.  That having been said, there is so much of it. What do I leave in?  What do I leave out?

In the end, my prejudice is to choose topics that will lead my students toward areas where mathematics is growing, places where many branches come together, places where there is structure--much structure. Then they will be able to choose.

The Tin Can Telephone

By Bobby Neal Winters

When I was a kid, there was no such thing as trash service in rural areas.  You burned your trash to minimize its total volume.  Then, when your burn barrel was full of things that would no longer burn, you hauled it off an dumped it in a isolated area where no one was looking.  I’m not proud of it, but that’s the way it was.

Sometimes we dug tin cans out of the trash and made phones out of them.  The idea is simple and I am sure many of you have done it--or something similar--yourselves.  You take two cans, put holes in the center of the bottom, and attach the cans with a light string.  You then holed the cans so the string is taut and talk into one can while someone listens in the other.

The model for communication theory is only a little more sophisticated. You have the equivalent of the two cans: call one the transmitter and the other the receiver.  And you have the string: call in the channel.

Instead of talking on one end and hearing on the other, you are sending symbols on one end and receiving them on the other.  When we say symbol, you can think what you want; the model is abstract enough to admit just about anything.  In practice, the folks who do this sort of thing will think of a symbol as being a string of ones and zeros.

The channel--the string, as it were--brings in an little more complication because it is a device through which we can add noise to the signal.  Those of us who have used the tin can telephone know that sometime the wind would whistle through the string.  This model will allow for that, but it will also allow for electromagnetic disturbances disrupting those ones and zeros being transmitted.

As an exercise, think about the following situation. Agents have captured an enemy operative.  She is a beautiful blond bombshell, a perfect exemplar of the “Bond Girl.”  You send a message, “Kill the prisoner.”  As you do, lightning strikes and your agents receive, “Ki** the prisoner.”  

There is ambiguity in the message.

While it can be reconstructed correctly, it can also be reconstructed as, “Kiss the prisoner.”  Depending upon the proclivities of your agents, they might find this message more attractive.

One value in creating a system to communicate effectively is to minimize the chance of this sort of ambiguity.  One way around it is to create a code wherein only certain things can be said.  This book, possibly, wouldn’t include the possibility of kissing an agent.  In practice, the symbols of ones and zeros are constructed so that only a few strings of ones and zeros are acceptable and corrupted ones are no longer in the alphabet, as it were.

The military does this with they so-called phonetic alphabet.  Interpreting strings of letters over a telephone line can be difficult.  The letters ess and eff  can sound the same, for example.  Instead of saying “Ess eff,” which could be heard either as “ess ess” or “eff eff,”  using the military phonetic alphabet you would say “Sierra Foxtrot.”  A set of symbols has been created so that, even when transmitted over a noisy channel, there is a reasonable chance of recovering the original symbols.  

So you could say “Kilo India Lima Lima” the prisoner and that wouldn’t be heard as “Kilo India Sierra Sierra” the prisoner.

What we’ve done here is to start talking about using a code.  The word code is often used to mean hiding the meaning of a message as when we say that people are talking in code to one another.  This is what mathematicians refer to as encryption, which is a different sort of thing. Encryption is about hiding meaning, but codes are about trying to transmit messages accurately.  I won’t chide you about blurring the distinction in casual speech, but in this context I will keep the distinction.

One practical issue that does occur in communication is whether the transmitter and receiver have the same code book.

I was watching a television show the other night where a young woman invited her date for the evening in for “a cup of coffee.”  His code book interpreted that phrase to mean an invitation for a hot, caffeine containing drink.  In her code book, it was intended to convey the possibility of insuring wakefulness by other means.

This is by no means an artificial example, nor is it unique. When adults are talking to children, the children have a different code book, as there vocabulary is smaller.  Communication is possible between parent and child though there is sometimes frustration in both direction.  There is also much comedy, as in the preceding paragraph, based on the characters having different code books.

It seems to me that an important element in basic communication is for the transmitter to know as well as possible what code book the receiver has and to craft the message accordingly.

The folks in marketing are masters at this.  They will tailor their messages to a particular demographic, folks with a particular code book and get their message through to that market.

In talking so much about the transmitter and receiver, let’s not forget about the channel.  There is only a certain amount of information that can be sent across a channel.  What is not sent can be as important as what is sent.

There have been time when I’ve met people in church.  They’ve got nice clothing on.  They are driving a late model car.  The overall impression, their image, is one of prosperity.  The truth is that they live in a modest home and that the car isn’t paid for and the clothing are saved for special occasions.  They can make themselves look rich by hiding their bank accounts and their homes.  It’s not only what is seen; it’s what’s not seen.

Celebrities make use of this as well.  They have their image, their public persona, but they have their private selves as well. For them, the image is a commodity that they sell just like a farmer sells produce.  They must master what is seen and what is unseen.

It is a mistake, though, to believe that only celebrities have images.  Each of us has an image as well.  We use different language for it; often we call it a reputation.  It doesn’t take long to get one, and once you’ve got a bad one it can take a while to improve it.

We build our images, our reputations, by the signals we send.  Some are masters of image creation.  It is relatively easy to convey an image of being prosperous; you just have to be sure to spend your money where people can see it.  It is relatively easy to appear to be intelligent; much of time it consists of keeping your mouth shut.

Beyond that you have to know your market and what code book they have.  It also helps to be rich, smart, or whatever you are trying to portray yourself to be, but because of the narrowness of communication channels, it’s not always necessary.

Numerous Numbers

By Bobby Neal Winters

What is man, that thou art mindful of him?
And the son of man, that thou visitest him?
--The Psalms

The Psalmist asked the timeless question “What is man?” thousands of years ago. The answers have come back in many forms.  Darwin said man is an animal; Freud said man is a sick animal. Others would say that man is an animal sick enough to care about math.

At least some of us.

Some of us care about mathematics.  Some of us care about numbers. The modern, mathematically-minded psalmist might ask: What is number that man art mindful of it?

For most people, that is a truly strange question.  Numbers are those things that are written on your bills.  You write them in your check register; you add them up at the end of the month; none of them has more than two decimals.  

Other people had encountered numbers in a somewhat more sophisticated way.  They’ve been in science classes and have encountered Avogadro’s Number, Pi, and the speed of light.  Still these numbers are, in most minds, yoked, nay, identified, with their decimal expansions.  Our teachers do tell us--and the sicker ones of us do care--that Pi can’t be completely captured by its finite decimal expansion, but for most the distinction is not made between the decimal expansion for the number and the number itself.

In a certain way of looking at the world, the failure of making that distinction is not a bad thing.  If you putty over the difference between the two, you can build the pyramids, create the hydrogen bomb, and work on cold fusion.  If make the distinction, you might not be worthy any activity besides mathematics.

Mathematicians are careful about making such distinctions and precise about language because they need to prove their assertions.  Mathematicians prove their assertions not only so that people will believe them but so their students will understand.

One means of laying the ground work for proof is setting up a system of axioms.  Those of you who’ve been through a course in geometry have experienced a system of axioms.  Axioms are statements about the objects in your system that allow you to do proofs.  What can be done for geometry can be done for the real numbers as well.  

This is called a synthetic description of the real numbers.  My aim is to stay as un-technical as possible so I won’t go too deeply into detail, but the axioms for the real numbers state the properties of the four arithmetic operations and how they deal with each other and with the order properties of the real numbers.  These axioms can be packed into the phrase that the real numbers are a complete, ordered field.

Dealing with mathematical objects synthetically, i.e. by listing properties in the form of axioms is clean.  It can be tricky because sometimes one must be rather clever.  It is much like trying to tie your shoes when you are too fat to see your feet: you have to be patient and have a good imagination.

There is also the danger that the object you are describing with your axioms might not actually exist.

There is a joke about a woman who went into a store to find a husband.  She came to two doors.  The one on the left said choose this door for men who are kind and the one on the right said choose this door for men who are kind and make a good salary.  She chose the one on the right.  

She then came into a small hall that again had two doors. The one one the left said choose this door for men who are kind and make a good salary and the one on the right said choose this door for men who are kind, make a good salary, and are handsome.  She again choose the one on the right and again she was in a room with two doors.  

This time the one on the left was one like she had just chosen but the one on the right said choose this door for men who are kind, make a good salary, are handsome, and are fantastic lovers.  Very excitedly, she chose the door on the right and found herself back out on the street.

Whatever point the one who made this joke had, mine is that sometimes you can put so many conditions upon an object, making them rarer and rarer, until they disappear entirely.

Mathematicians like to have at least one non-trivial example of whatever class of objects they are talking about.  These examples have to be described in terms of other well-understood mathematical objects and the language of set theory.  This is referred to as making a model.

One means of creating a model of a the real numbers is to begin with the rational numbers.  As I said earlier, the real numbers are a complete ordered field.  The rational numbers are simply an ordered field; this is to say they lack the property of completeness.

Completeness, in the way of mathematical words, has a very precise, very technical definition. One can discern from the meaning of the ordinary English word completeness that a complete ordered field, such as the real numbers, has something that an ordered field that is not complete, such as the rational numbers, lacks.  What is this?  

A quick and--to the cognescenti--smart-alecky answer to this is the irrational numbers such as the square root of two and Pi.  This is smart-alecky because it ignores a the very real need that the rational numbers have for those irrational numbers.  The incompleteness of the rational numbers--again in the English sense of the word--signifies a lack, a deficiency.  This lack can be described in two different ways.

The least technical of these two ways involves the existence of least upper bounds.  The set of positive rational numbers whose square is no more than two does not have a least upper bound that is a rational number.  This fact--in different language--was discovered by the Pythagoreans in ancient Greece some time in the Sixth Century B.C.   

The more technical of these two ways involves certain sequences of numbers.  You may have heard of infinite sequences of numbers such as ½, ¼, ⅛, and so forth.  This sequence of numbers converges to zero.  There is a certain type of sequences that are referred to as being Cauchy.  All sequences that converge are Cauchy, but not all Cauchy sequences of rational numbers converge to rational numbers.  Again, one can easily find Cauchy sequences of rational numbers than converge to Pi and to the square root of two.

What mathematicians do in these two cases is to construct models based on the rational numbers.  In the first case, special sets of rational are created and the arithmetic functions are extended to those sets.  The objects in this model are no longer rational numbers but sets of rational numbers. In the second case, the objects in the model are sets of sequences of rational numbers.

These two different models of the real numbers are clearly different from each other in terms of what they are, but both of them satisfy the axioms.  Each as a complete, ordered field.  We call that field the real numbers, and there is a very precise mathematical sense in which that definite article is justified.

But I am becoming a mystic.  There are more numbers than we can know.  Our need for numbers springs from the world around us in numerous ways and I wonder if by drilling down to one idea of the real numbers if we are missing other things.

But the timeis late, and I want to go home.

The Historical Approach

By Bobby Neal Winters

Introduction: The Historical Approach

As with so many things, it came into my hands through the recommendation by a colleague of a book she had not actually read.  The book was A Radical Approach to Real Analysis by David Bressoud. The thing contained therein was the historical approach to teaching mathematics.  

Mathematicians are by their natures optimizers.  Perhaps the greatest mathematician of all time, Karl Friedrich Gauss (1777-1855), said, “A cathedral isn’t a cathedral until the last piece of scaffolding is removed.”  His personal motto was “Pauca et Matura,” few but ripe.  This is his testimony to the fact that after mathematicians have scaled to the top of the mountain with ropes and spikes and what not, they try to build trails, roads, or even rails roads up for the rest of us.

This is a great service for the rest of us, but in looking at the road, there is a tendency to forget that this is simply the endpoint of a process which as included numerous researchers, teachers, and students.

Claude Shannon (1916-2001) is known as the father of information theory.  In his theory, he imagined information going from one location to another via symbols.  The teacher/learner relationship is a system of communication, but not necessarily in the way one might naively imagine it following on the heels of that sentence. A teacher is a facilitator of the communication process. It is here that I want to thread carefully, because I dwell among those who kill upon hearing the phrase “Not a sage on the stage but a guide on the side.”

In communicating from a transmitter to a receiver, each has his (or her) on set of symbols.  Practical communication theory tells us there will probably be information lost even if both sets of symbols are the same.   In the teaching/learning process, the symbol sets are probably not the same.  The transmitter may be a German scholar who was doing his best work when Napoleon was marching across Europe and the learner might be a wannabe scholar growing up in the Oklahoma oil fields when Jimmy Carter was in the White House.

My point is there is a living system in place that transmits between those two points and more.  The Gaussian attitude of removing the scaffolding is an important part of presenting that beautiful cathedral, but we as teachers must keep in mind that the scaffolding is still out back in the shed.  Knowing that the beautification of mathematical results (or indeed the corresponding acts in any discipline) is an important part of pedagogy, should empower us as teacher-scholars.

Origins in Applications

One piece of scaffolding that often remains hidden is the Primum Movens in mathematics is physics.  I use the word physics here to be broad enough to include engineering.  This Prime Mover has interfered with mathematics multiple times in history.

The Greeks did pursue geometry for its own sake as an intellectual game, but Euclid (circa 300 BC) was known as Euclid of Alexandria.  Alexandria was the city of Alexander the Great who, while not a Greek’s Greek, did spread Greek culture to lands the old fashioned way: by conquering them.  It is not difficult to imagine Euclid and his like codifying the discoveries of those who’d been involved the many construction activities of the Egyptians. In effect, they were creating an orderly way for students to learn the geometry without to actually build the pyramid first.

Fourier Series were developed by Joseph Fourier (1768-1830) for problems connected with the theory of heat. Fourier, while getting great results, didn’t have a theoretical foundation for his mathematics.  Providing those foundations gave birth to new fields and new directions to old.

We like to teach the pursuit of knowledge for its own sake, but knowledge has an end, a purpose.  Mathematics need not teach mental masturbation. It should, in fact, teach the opposite: the delay of gratification.  The delay of gratification, I am convinced, is one of the cornerstones of civilization.

While in the process of learning to delay gratification, it is helpful to the student to know there is some gratification to be had at the end of the process.  The engineering student will endure Heat and Thermodynamics because he knows he must have it to attain his engineering credential.  He will endure Engineering Mathematics because he knows he must have it to pass Heat and Thermo.  He will endure Fourier series because he has to understand them to pass Engineering math.

As a mathematics teacher, I can help him to build that narrative. That process will be aided if I know the story myself in broad terms.  Whereas the details of the story are not a necessary part of my intellectual equipment as a mathematician, per se, they are an important part of my tool box of teacher as scholar.

Bridges

Another value to the use of the historical approach is in helping the student cross the bridge from where he is now to where he needs to be.  One commonality shared by a modern student of mathematics before taking up Fourier series and a historical figure like Fourier he began his work is an ignorance of Fourier series.  We can begin at the same place Fourier did and go from there.

Fourier didn’t spend much time on the theory.  The results he got worked very well for him when they worked. When they didn’t work, well, that could be taken care of later.  

And it was.

The first part of the mathematical program consists of calculus and other courses that are of use to engineers, physicists, chemists, biologists, and economists who are interested in mathematics as a tool rather than as an end in itself.  Mathematicians go through these ourselves because we do value the utility of the subject and want to equip ourselves in its more practical aspects; this is true, but there is more.  In addition, the number mathematics majors is typically too small to pay for a separate track for our majors.

This having been said, our majors get a somewhat skewed view of the subject.  Many absorb the view of a subject that consists of methods that have already been worked-out, methods that they need only memorize and master.  Mathematics is a living subject and our students need a bridge from the civilized, cultural center of the subject across the river of uncertainty to the frontier.

The historical approach supplies such a bridge.

As someone who has lived most of his life in the center of the country, first Oklahoma and then Kansas, I will claim some familiarity in the frontier.  Just like in the movies, there are times on the frontier where the rules are temporarily...um...ignored.  We can see Newton, Euler, and Cauchy doing things we would rap our student’s knuckles for.

Isaac Newton (1642-1727) developed series presentations for sine and cosine.  His work is absolutely brilliant, but it’s also stupid.  This is an exercise today for a freshman using Taylor’s Series.  Seeing the insight behind Taylor’s Series takes half a second, but Newton, as brilliant as he was, didn’t have it.  This didn’t stop Newton, however, as he experimented, found patterns, and verified the patterns to his satisfaction.

Leonhard Euler (1707-1783) played--and I will stick by that verb--with infinite series and obtained tremendous results.  Toward the end of his career, some of his contemporaries thought he was going off the deep end because some of his series didn’t converge.  It was more than one hundred years after his death before the foundations were laid that justified these wild calculations.

Augustin Cauchy (1789-1857) began to put analysis such as was done by Newton and Euler on firmer foundations, but in doing so he was himself faced with the dangers of virgin territory.  In Calculus I, we teach the Mean Value Theorem whose formulation is due to Cauchy.  When we prove it in the introductory analysis course, its prove can be contained in a single paragraph because the modern theory is so well developed.  Yet Cauchy’s proof is much longer and contains some jumps within it that would cause us to paint a student’s paper red: but it’s brilliant!

It is good for students to see that our subject is an adventure.  While as undergraduates, they might not have received the preparation to go into the subjects of current research interests, they have had enough to appreciate the research of the eighteenth century.  Seeing the spirit of that frontier might inspire them to explore current frontiers.

The Lasting Effect of History

As scholars who teach, we should be aware of the effects of history upon us.  We are part of a larger world that has been growing and changing for many centuries.  In many cases we bemoan the publish or perish culture that has grown up around us in modern academe, but, in doing so, we forget that publishing results is our gift to civilization.  Great minds like Archimedes (c. 287 BC – c. 212 BC) sent letters to others whom they thought might understand them.  Many times they played tricks upon their rivals to confuse them.

Others like Niccolò Fontana Tartaglia (1499/1500–1557) solved important problems like the general cubic equation, but kept the results to themselves so that they could pose challenges to others and build up their own reputations.

When Isaac Newton wrote Principia Mathematica he invented the language to describe his new system of the world, and while there might be a jot or a tittle moved here or there, we still use basically the same language today.  I am not sure he was thinking about this three hundred years ago, but the effect of his work, his words still stand.  Without learning history, however, we might not understand that he had to be pushed, bullied, and cajoled into writing his book and he made Edmund Halley pay to have it published for him.

When groups such as the Royal Society and the various journals arose as means of disseminating research results and establishing priority, progress hastened. It is difficult to over-emphasize the value of these steps toward openness and publication.  Yet, history should also show us that we are continually changing.

I believe we need to broaden our ideas of scholarship to include activies which are, in fact, necessary to teaching, learning, and research continuing in our universities.  Ernest Boyer explored this generally in his book Scholarship Reconsidered: Priorities of the Professoriate.  I believe that an exploration of the history of my subject, mathematics, vindicates this idea.  

Research and teaching are inextricably linked. Research discovers the mysteries of creation, but teaching, in a continuum of forms, transmits those discoveries to the ages.  I can have conversations with Archimedes, Newton, Euler, and Gauss because teachers who were themselves scholars have worked to make that happen.  They have learned the language of the countries of the past and have shaped the language of a boy from the Oklahoma oil field so that he may understand, if only just a little.

That is what teachers can do and that is little short of magic.

Elementary Statistics

By Bobby Neal Winters

The Learning Phase

Mathematics and Statistics are separate disciplines. This is something that both mathematicians and statisticians are insistent upon. Statisticians use mathematical tools much in the same way physicists and chemists do, and many mathematicians take statistic for the same reasons they take physics: as an application of their mathematics and to make themselves more marketable when it comes time to look for a job.

My undergraduate degree included a course in Probability and Statistics, but it was very theoretical in nature without much indication of how one might actually apply this in the real world. This is common among undergraduate math degrees. Also commonly offered is a class in elementary statistics that is rather short on theory but long on practical applications.

Fifteen or twenty years ago, as a part of my responsibilities for teaching service courses in the math department, I was assigned elementary statistics. It was a course I’d never had. The first time I taught the course, being a team player at heart, I used the text that had been adopted by the department and, as I taught, staying a few sections ahead of the students, following the syllabus that had been determined by the department.

It was in this way I learned statistics. As I learned it from the book the department was using, that book must have done something correctly, but--and you saw this coming--I hated the book.

Why?

You may be familiar with the French post-Impressionist painter Georges Pierre Seurat. If you are, then you probably know more about him than I do. What I know is that he painted pictures using dots. If you stand with your eye just an inch or two away all you see are dots; if you stand back there is a park or dogs playing poker or something else.

This text was written like that. When I was done teaching the course, I took a step back and saw the whole picture at once, but I was not convinced that any of the students would do that. For them, it was all dots. Or, if I my phrase it another way, it was just one damn thing after another without any connection between them. Nevertheless, being a team player, I continued in this way.

The Synthesizing Phase

In thinking about the course, it occurred to me there were four parts: descriptive statistics, basic probability theory, applications of the Central Limit Theorem, and advanced applications.

The descriptive statistics consisted of the topics that one, for the most part, could cover in a high school or even middle school class: Make a frequency distribution; list numbers from largest to smallest; find the average; find the standard deviation; draw a bar graph; draw a pie chart. Students, even the ones who were ultimately going to fail the class, rocked on this. Among those who showed up for the exam, it was rare to see anything less that a B. This made the second part of the class even more tragic.

Students find what I refer to as basic probability theory to be difficult. Topics in this section can be anything from if you have a can that has 300 red beads and 700 blue beads, then what is the probability of drawing a single blue bead (Yes, it’s 0.7.) to the hypergeometric distribution. This section cleaned the students clocks, which is to say they uniformly found it to be quite difficult.

The portion of the course I describe as “applications of the Central Limit Theorem” is actually the meat of the course. This is the part of the course that our clients in the university, i.e. those who use statistics in their classes, want their students to know. This includes estimation and hypothesis testing.

The final portion of the course consisted of the sample proportion, which is used to estimate the percentage of the vote a politician will get, and the chi-square test, where the question of whether particular models fit is discussed.

Teaching as Literature

After I’d taught the course a number of times, it was clear that it all did fit together like one of Seurat’s paintings, but that the students might be better served if it were taught like a classic work of literature. By this I mean, I thought I should take advantage of certain opportunities to foreshadow concepts.

For example, there is something called Chebyshev’s Rule. It is the most difficult topic covered in the first section. To use layman’s terms, it says that only a small proportion of the data can be very far away from the average. This opens up the opportunity to mention proportion, which has already come up with relative frequency, and foreshadows the central means of hypothesis testing. In terms of literature, it helps to tie the course together.

Spending more time on Chebyshev’s Rule also serves to make that first section a bit more challenging. If it is de-emphasized, as I tried a few times, there is little to give the students that they are now in a college course because so much of the material in that first part of the course is at a middle school level. Chebyshelv’s Rule, by way of contrast, can be challenging in a number of ways.

There are other opportunities to create ties they run the length of the course. I use the jar of colored beads as a recurring example. It’s used when I introduce probability, when I talk about Bernoulli Trials, and when I talk about the sample mean. By this means, and others, the student has a better opportunity to see that the course is unified and that it’s not just one damn thing after another.

Embedding in Time

Shortly after I’d separated the Elementary Statistics course into four parts, I took a year of sabbatical a Brigham Young University. They had a testing center that was very popular with the students. Teachers left copies of the exam with the testing center for an period of time--I think as much as a week--and during that interval of time, students could come and take the exam when they felt as if they were ready. Staff at the center proctored the exam, so pressure to be a policeman was removed from the professor. As I said, the students loved it and I grew to love it, but there was a catch. In order to use it, you had to schedule your exams ahead of time.

Up until that point in my career, I had not. I had exams when I felt we’d covered enough material, and the idea that this could be a predictable thing was foreign to me. In order to use the testing center, I scheduled my exams and discovered that this was not such a difficult thing to do.

When I can back from sabbatical, I scheduled my exams for Elementary Statistics and discovered a number of things. One of them was that the students didn’t complain. Indeed, while my students have never said so directly to me, I’ve read studies that support the idea that students like structure.

More important to me, however, was the discovery of how easy it made everything. Instead of having to weigh the decision of whether to have a test on a particular day, there it was on my calendar! I could make out the damn test and not procrastinate. Once those dates are set, they serve a similar function for the course as the arbor does for a grape vine: the course just grows around them.

Quite frankly, in some very real sense, when the dates are set for the exams and other assignments there is nothing much else for me to do.

Assessing and Modifying

As you’ve seen, course design is a dynamic process. I’ve modified my course as I’ve learned new things. I have changed as I’ve observed that worked and what didn’t. I was doing course design before I new the name for it. I was doing assessment before I heard the word.

That having been said, those words and phrases have power. When we teach a course, there are certain things that we want students to learn or we wouldn’t be teaching the course. Those items are called student learning outcomes. When we’ve taught those things we want to determine whether the student has learned them. This is called assessment.

When we assess, we are assessing a communication channel. The channel has two ends, the transmitter and the receiver. We give a lot of attention to the receiving end of the channel in that we assign a grade of A, B, C, D, or F to the student, and this is right in that it is the student who is paying money to learn and those who hire that student will want some measure of how well the learning has taken place.

But.

But there are two ends of that channel. For my own sake as a teacher, I need to know how well I am doing and I need to change what I do in order to make it better. When I learn what I need to change, I should carry it out.

Technology

Over the last twenty-odd years there have been numerous changes in technology. When I began teaching, I had to walk two blocks across campus to check my e-mail. I did it once a week, and frequently I didn’t have any. Now, God help me if I skip a day of clearing out the junk mail.

Regardless of the downside, technology has provided more tools to reach out to the student. I would like to mention three of these: PowerPoint, lecture-capture, and the learning management system.

Much has been made of PowerPoint and the effect it has had on bullet-pointing the learning process. There is a danger than the medium will affect the message. That is a legitimate concern. PowerPoint is not a panacea. I still use the chalkboard in many classes as I believe that the students need to see the practitioner at his (or her!) craft so that they may be empowered themselves.

If I’ve gotten my hands dirty, then they know they might have to get their hands dirty.

Yet there is so much material that simply needs to be put in front of the student and talked about. This can be done badly with PowerPoint, yes, but it can also be done badly with acetate slides and with chalk on the board. What I put on PowerPoint, I used to put on sheets from a yellow-pad and just transfer it to the board. There is no loss of empowerment and there is a great gain in efficiency.

As far as the bullet-point-ization goes, this is taken care by talking and adding meaning while you lecture over the slides. The audio can be recorded and synchronized with the slides on your computer by what is called--straightforwardly enough--lecture capture. It can be done in an almost effortless fashion and uploaded to the Internet.

On the Internet, we has access to our university’s learning management system (LMS). We can upload the PowerPoints ahead of time for the students to print-off and take notes on. We can put the captured lectures there by them for the students to listen to.

I have my whole course online, organized in four parts. In each part there is a schedule of what I plan to do everyday. All of the deadlines are there. I’ve interspersed quizzes for the students to take online, where they are automatically graded and recorded.

For the students who want to learn, there is unparalleled opportunity. The students who don’t want to learn are forced to be more creative in their excuses. It’s win-win in other words.

The Human Touch

Yet with all of this, we cannot forget the human touch. If we don’t care, the students won’t care. If we aren’t excited, the students won’t be. If we don’t think the material is important, then the students won’t.

In other words, you still have to teach.