a fourth meditation on being a teacher, part 1

•30 January 2016 • 5 Comments

I spend a lot of my time thinking about teaching.  I’ve been teaching mathematics for 30 years, since I started graduate school in 1986, and aikido for 12 years.  But it’s the math teaching that most on my mind at the moment, having just finished teaching Graph Theory for the 4th time.

The more I teach, the more I become dissatisfied with how I teach.  I don’t think I’m a poor teacher; in fact, I think I teach well.  My own impression, and the evidence from the students’ end of semester questionnaire results, is that I’m entertaining at the front of the room (though my jokes are not particularly funny, but math jokes, what can you do, and one should never judge one’s own perceived level of being entertaining), that I structure and present the material clearly and explain things well, and I have developed the reputation that my examinations tend to be on the hard side.

But my overall impression is that my lecturing is standard.  I am a mathematician, somewhat old school, and I lecture like a mathematician, somewhat old school.  I stand at the board, go through the topics I need to go through on the day, answer questions from the floor, and do what things I can do to encourage participation.  For the past few years, for instance, I’ve used a hash tag and let the students tweet questions during the lectures.  For the relatively small class I’m teaching now, the students are willing to ask questions in lecture and so the volume of tweets is low, but that’s a story for another day.

So, the question that’s occupying me at the moment is, what can I do differently?  What can I do that’s unusual but nonetheless effective?  I’m already starting to think about things I can do to restructure the class for next year, and so I’m going to conduct an experiment.  Perhaps it’s an experiment that will end in abject failure, but let’s give it a try anyway.

I’m going to keep a record, at least in general terms, of what I’m doing to get Graph Theory 2016/17 ready for the students.  (And yes, I do have a reasonable level of certainty that I’ll be teaching Graph Theory again in 2016/17.)

One thing that I’ll be doing is to ensure that I have a complete set of printed notes available for the students on the first day of teaching.  I was writing the notes as I went along this year, but that didn’t go down well with the students, and so that I’ll change.

Some students seem to give printed notes an almost talismanic status, and I’ve been pondering why.  I’ve talked to a few students about this but those conversations haven’t given me any clarity.  Perhaps students think that having printed notes somehow delineates the scope of what’s being taught, that only the material inside the notes might then appear on the exam.  Perhaps.

 

 

 

Posted in Uncategorized

An apocryphal story about Michaelangelo and his horse

•18 October 2015 • Leave a Comment

I like apocryphal stories, particularly those apocryphal stories that might not be true.  That can’t be true. Indeed, I think I prefer those that can’t possibly be true.  I suspect that this story, involving Michelangelo, is squarely in this latter category.

The short version has someone asking Michelangelo, how do you, the great Michelangelo, carve a statue of a horse?  He answers, it’s quite straightforward.  One starts with a block of marble, and then removes everything that isn’t a horse.

This story shapes a lot of my teaching, particularly my aikido teaching, as well as my aikido practice.  Part of aikido is doing as little unnecessary as possible, actively removing the unnecessary, and Michelangelo’s horse always helps to remind me of this, that I need to be efficient in what I do.

This efficiency, a principle of least action of sorts, holds in mathematics as it does in aikido.  When we develop proofs of lemmas, propositions, theorems, we look not only for a correct argument, but we look for an argument that is beautiful in its efficiency. We work to polish our answers, to remove all the rough edges, to make them shine with the reflected glory of mathematical truth.

In both aikido and mathematics, we have models of this beauty and efficiency that we can work from. In aikido, we have recordings of O’Sensei and his students, as well as our own teachers. In mathematics, we have canonical examples of proofs that cannot be improved, such as Euclid’s proof that there are infinitely many prime numbers.

So how do get to there from here? My current struggle is to develop that efficiency myself, but as I’ve commented elsewhere in these pages, that for teachers, there is a seductive danger to this beauty and clarity. Understanding this clarity and efficiency takes a depth of knowledge and experience and lots and lots of time, and when we develop the ability to see this beauty and clarity, it is sometimes easy to forget that not everybody does. And therein lies the danger.

Posted in Uncategorized

a third meditation on being a teacher

•26 September 2015 • 1 Comment

On Monday, the day after tomorrow, a new academic year begins.  Actually, one could argue that the new academic year has already begun, with the campus buzzing with students new and returning, but my teaching begins on Monday morning.

I’m teaching this semester the same class that I taught last year and the 2 years before that.  I like teaching the same class for a number of years, because it gives me the opportunity to think deeply about how I approach the subject myself, whatever the subject may be, and how I structure it for my students.

As I’ve written about before, I teach both mathematics, graph theory at present, and aikido.  In thinking about all of the teaching I’ll be doing, I’ve made an observation.  This observation arises from the fact that in both the mathematics and the aikido I teach, I have a deeper understanding than I did a year ago.

This depth of understanding is good for me, in the sense that part of what I find interesting about teaching is the opportunity to deepen my own understanding, and is good for my students, in the sense that the better the understanding of the teacher, the better the experience of the student.

But within this understanding there lies a trap.  I encountered this trap directly a number of years ago, when I found myself thinking, I’ve been explaining these things to the students for years.  Why aren’t they understanding them yet?  The answer came to mind almost immediately.  My understanding has been growing deeper year on year, but each year, I am teaching a group of students that is encountering this material, be it mathematics or aikido, for the first time.

So each year, I need to engage in time travel, and I need to engage in more each year.  Each year, I get farther and farther from the starting point of the material I’m teaching, and I need to actively acknowledge that distance.   Failure to do so means that I’m not teaching at the same level that I taught the material when I first started, and as I go through the years, I may drift more and more.

An uncomfortable question lies at the heart of this observation.  Is it possible for a teacher to develop such a deep and advanced understanding of their material that they are no longer capable of teaching beginners or even intermediate level students?

I am convinced the answer is No.  But this is a qualified No, and one that relies on the willingness of the teacher to reflect on their own teaching.   After all, good teaching can be learned, and is learned by many thousands of people each year.  Nonetheless, I believe the trap is real, and as I walk into the lecture theatre on Monday morning and face a fresh group of students, I will step carefully.

Posted in aikido, mathematics

on issues of scale

•28 June 2015 • 1 Comment

Though it is not widely recognized as a superpower, I am coming to the point of view that having a proper understanding of scale is a hard and rare talent indeed.  So what do I mean by an understanding of scale?

Most of the numbers we encounter in our daily lives are fairly small, in mathematical terms.  To be honest, any number is small, in mathematical terms, but that’s a discussion for another post and another day.  Groceries and train tickets, vacations and books, these prices are measured in the tens, hundreds, perhaps thousands of pounds.

However, when reading the news, we encounter much larger numbers.  Going back to an earlier post, we disguise the size of these numbers by using words such as million, billion and trillion.  These words all sound remarkably similar and that similarity disguises the differences in their sizes.

These numbers are important because these cover income and expenditure at the national and international level.  The annual budget deficit, the amount the government spends on welfare and defence, the amount the government collects each year in taxes, these are measured in the billions.

I wonder whether it is our lack of understanding of just how big 1 billion is, or 10 billion, or 100 billion, whether it be pounds or grains of sand or anything, that contributes to a lack of comprehension of the scale of the government deficit each year.

What does it mean, to my imagination, to the way that I view the world, to say that 1 billion is 1000 times as large as 1 million, when I don’t have a good conception of what 1 million means, and I barely understand what 1000 means.  I’m not sure it means much of anything, as simply stated as this.

I’m sure there are there’s been some psychological research done on how we perceive large, very large things, but I haven’t yet dug into this.  (If you have a suggestion of where to look, definitely let me know, because I’m curious.)  But I find it an interesting question, to what extent does a difficulty in imagining and understanding scale affect political debate, particularly in the modern world of economic crisis and deficit.

Posted in bizarre and random things, mathematics

aliens, oak trees and the difficulties of communication

•3 May 2015 • 2 Comments

One of the advantages of working at a university is that it allows us the possibility of participating beyond the normal limits of our chosen subjects.  Whereas my chosen subject is mathematics, I have also had the opportunity to do a small amount of teaching beyond the normal mathematics curriculum.

Several years ago, the University of Southampton set up a collection of modules spanning the standard discipline boundaries that govern many modules here and elsewhere.   (If you’re interested, you can find out more at http://www.southampton.ac.uk/cip/index.page)

As the result of a conversation here leading to a conversation there, I have had the opportunity to give a lecture in Intercultural Communication in a Global World in each of the past 3 years.  And it’s not, as you might immediately think, about mathematics as a culture in and of itself, or of the perceived difficulties that non-mathematicians might have in communicating with mathematicians.  Rather, my topic comes from a favorite topic of speculation from another part of my life, as struggling science fiction author.

Suppose we were to start ranking how difficult it might be for humans to communicate with non-humans.  This is an extreme form of intercultural communication, but reasonable communication with extra-terrestrials is a common theme of science fiction and has become mainstream.  But why start with extra-terrestrials when there are so many interesting speculative possibilities closer to home.

EASIEST: Regardless of what else we might think, humans communicating with humans must be at the easy end of the scale.  When we bother to, we can learn each other’s languages and we can come to understand the subtleties buried within these languages.  After all, it’s been some time since I’ve run into the English-English translation problem that frequently plagued me when I first moved to England.

Beyond this, it gets rather speculative.  I have not undertaken an exhaustive examination of all possible pieces of evidence, and in fact, if there is any properly scientific (non-anecdotal) evidence for or against any of the below, please let me know.  I’m curious.

SLIGHTLY HARDER: Something which has been explored in fiction but for which we have no strong, hard evidence is the extent to which we communicated clearly with our hominoid cousins, such as Neanderthals.  We do know that our ancestors overlapped in Europe and elsewhere in the world, and we must have interacted with Neanderthals, if for no other reason than current estimates are that modern Europeans possess roughly 4% Neanderthal DNA.   I’ll take the optimistic position that interbreeding implies meaningful conversation.

HARDER STILL: It is not my intention to insult any dog owners who might be reading, but I don’t yet believe that talking to dogs or other animals such as dolphins is as straightforward as talking to one another or to our hominoid cousins.   Significant communication clearly takes place, between dogs and people, between African grey parrots and humans, but no one has ever to my knowledge had an unambiguous conversation with a dog or a dolphin or an elephant about the weather.  We are fairly certain that elephants communicate over long distances but what are they saying?  Some people find whale song soothing, but what are they actually talking about?  As was raised in a story I read a long time ago but have forgotten almost entirely, do dolphins and whales have an oral history going back generations and might we find in their stories the distance echo of when an alien spacecraft crashed into the oceans?

MUCH HARDER STILL: In my readings, I tend not to take notes, and perhaps I should become a better note taker.  But I have the memory of reading that trees communicate.  When trees at one side of a grove are attacked by pests, trees at the other end start producing the appropriate defensive chemicals before the pests have made it across.  But could we ever have a conversation with an oak tree?  Even if we were able to decode the meanings of their communications and even if we were to decide, as we have with dolphins and elephants, that those communications carry meaning of the sort we might be able to participate in, there is the issue of time scales.  To what extent does a vast difference in life spans affect communication between species?  And what would we talk about?

There are many other things we could stick on this list.  Rocks, perhaps, if we want to take the view that rocks might be intelligent communicative beings but just working on a time scale where we see them as essentially static and they don’t notice us at all.

But after all the terrestrial possibilities have been exhausted, we finally come to extra-terrestrials.  As a wannabe writer of science fiction with but a single story to my credit (to date) (so far) I want communication with extra-terrestrials to be possible.  As a mathematician, I want to see whether mathematics really is the language through which all intelligent beings from across the universe can converse.

But I think that having a reasonable conversation with an extra-terrestrial is going to be harder than having a conversation with an oak tree, and much harder than with a dolphin.  Perhaps I’m wrong.  I suppose the best we can say at this point is, let’s hope we get the chance to find out.

Posted in bizarre and random things

academics, change and a bad light bulb joke

•3 May 2015 • 3 Comments

Before I get rolling on this one, let me say that I love working at a university.  I love the variety of things that I get to do over the course of a day or a week, a month or the whole of an academic year, from teaching and working with students, through spending some time with research projects and questions that I’ve been carrying around for various lengths of time, and on through (gasp, shock, horror) committee work and university administration and the conversations that can shape how the university functions.

However, despite being at the forefront of the discovery and creation of new knowledge and the generation of connections between different areas of human endeavour, universities can be somewhat conservative in their outlook.   I find myself pondering the why of this, sometimes encouraged by the discussions going on around me at the time, and I’ve realized that an oft-times unfair caricature can be found in an old joke.

How many academics does it take to change a light bulb?

My preferred answer, which I like to give as the respondent to whom I’m telling the joke starts running through numbers under their breath (probably more than 1, but as many as 10?), is CHANGE?!?!?! with a mock offended tone to my voice. Though the joke, and the sentiment behind it, is not unique to universities, it does play to the stereotype of the academic sitting in their cluttered office, squinting perhaps at the sheets of pictures of long-graduated students taped to their walls, muttering under their breath that today’s students couldn’t hold a candle to the students back in their day.

In truth, the reasons that universities struggle with change are the same reasons that cause everyone else to struggle with change.   Change is harder than stability.  For instance, physical infrastructure, once put into place, can be difficult and expensive to change, and in a world that is increasingly virtual, we need to spend more of our time and effort being clever about how we use these.  And we’re doing this, in part because we have to, like everyone else, and in part because we want to.

There is one part of dealing with change that I personally find fascinating.  As a mathematician, some of what I teach are things that are not particularly new.  Mathematics is a beautifully hierarchical subject (about which I’m sure I’ll say more in future posts) and we need to have an appreciation for the older stuff so that the newer stuff makes sense.  But we can still and so bring in examples from yesterday and today.  We can also make use of the technology that changes from one instance of a module to the next a year later.  I like using Twitter in large lectures to give the students a different avenue for asking questions and correcting perceived mistakes (always deliberate on my part, don’t you know).  But enough for now and more to come.

Posted in mathematics

the notion of edition in the digital age

•2 March 2015 • Leave a Comment

In exploring the dark and dusty corners of the drafts folder, I found a piece that I’d started writing back in 2015. I’m not sure how I lost it or why I stopped working through the idea, but it’s something I’ve been thinking about recently.

In part, this revisitation comes from the materials I’ve been preparing for my class this semester and thinking through how they change from one year to the next, as I continue to work through the examples and navigate the ever-changing structure of the notes.

Another reason comes from the thinking I’ve been doing about a possible third edition of the Hyperbolic Geometry text book and what that might look like. There is a permanence to a book that I very much like; words on paper and once the words are printed, effort is required to shift them into other words. This is especially relevant in a field that’s moving quickly.

I’ve also been thinking about the couple of stories I’ve had published, as I’ve been persuaded to look for reprint possibilities for them, and this raises the question of the extent to which I might want to tinker.

This is a problem for readers as well as authors, and perhaps even a larger issue for readers. I’m not sure I’d want to buy a book knowing that the book might well shift and change, as the authors continues to shape and refine their work. It’s tricky, because I also want a book as well crafted as possible.

This brings to mind the old joke, that a paper or a story or a thesis are never finished, they are merely submitted. There is a variation on this theme, that a paper or a story is never finished, merely published. (And I do hate writing ‘merely’ and ‘published’ as consecutive words in any sentence.)

And so an answer to the question implicit in the title might then be, an edition is a snapshot, a moment in the life of a work, setting down a marker of sorts.

I do have great sympathy for the librarians and archivists, trying as best they can to track and keep a record of what’s being published. Trying as best they can to keep a record of these moments.

As a reader, I can see the benefits of constant updating, constant refining, but as noted above, there is also the frustration of the reader with having a constantly shifting piece of reading. I can also see the attraction to the author, wanting their work to be as good as it can possibly be.

But since we started making marks in clay tablets, we have been keepers of records. We have the notion of canon and we have a fondness of the version of record. I’m not sure what happens to all of these basic assumptions we make of the permanence of published works.

Perhaps it will depend on the purpose. I can see an educational text book shifting to reflect the class as taught; this might also reflect a change in teaching towards more of a challenge based structure on top of a basic syllabus, and so the underlying work might well need to shift to reflect this style of delivery.

l can see fiction having a more rigid structure, but it would be a curious thing to read, a story that shifted with the reader, almost like the old adventure books (‘there are two doors’). I’m not sure a human author would be able to keep up with the authorial demands, but might a rigid published story or novel find itself someday replaced by an extended conversation with an artificial intelligence? I’ll be interested to see.

Posted in bizarre and random things, writing ephemera

a second meditation on being a teacher

•14 February 2015 • Leave a Comment

I just attended the first TeachMeet #tmsotonuni earlier this week at the University of Southampton. Though I wasn’t able to attend the whole of the event, the talks I did see fill me with enormous hope and joy.  I love the energy of teachers.

I had the opportunity to say a few words at the beginning.  Reflecting on the words I would say, I realized that I’ve been teaching for almost 30 years, and that surprised me.  I don’t know why it should surprise me, time being what time is and given the age that I am, but nonetheless it did surprise me.

I was preaching to the converted when I said that teaching is hard.  I have always found it to be hard, from the days when I first started.   Part of why I find it hard is the nature of the beast, taking some moderately complicated mathematics and structuring it for the consumption of people who have never seen it before.  But that’s only part of it.

The other part, the main part, is the nature of teaching.  As teachers, we are trying to affect change in our students.   Part of the change is the knowledge located in the heads of our students.  Between the beginning and the end of a semester, or the beginning and the end of a single session, I want the mathematical knowledge in my students’ head to be different.  I want them to know more at the end than they did at the beginning.

But it’s more than just facts.  Part of it is the machinery we use to process these facts.  And this is the more important part.  There was a time, not all that long ago, when the acquisition and storage of facts was an expensive proposition.  One of the functions of universities was to act as repositories for these facts.

But facts are now cheap.  We have access to vast quantities of facts.  We are overwhelmed with facts and so just having facts is no longer as distinctive a proposition as it once was.  The tools to process facts, though, to take facts and to generate something interesting from them, that is distinctive.  And this is what we are trying to teach our students.  And this is hard.

How do I meaningfully move through this immersive ocean of facts in which I find myself?  I need to have tools to use, to help me make sense of it all.  And these are the tools I am trying to give to my students.

Posted in mathematics, Uncategorized

the numerology of unrelated constants 1

•13 October 2014 • 3 Comments

I recently gave a mathematics talk at the Science Cafe at the Southwestern Arms in Southampton, on the most beautiful formula in mathematics.  The formula is simple, involving only 7 symbols:

e^{i π} + 1 = 0

(I know that there appear to be 10 symbols in the above, but that’s an artefact of how we express a formula without the fancy tools that mathematicians use to make their formulas appear to the eye as works of art.  The symbols ^, { and } should be viewed as invisible, with ^ denoting exponentiation and { } merely delineating what terms form the exponent.)

So, why is this formula beautiful?  And what does beautiful even mean in the context of a formula?   This is one of those questions that if you ask half a dozen mathematicians, you might well get half a dozen reasonable answers.  (In fact, you might get upwards of a dozen.)  I have an answer with which I am comfortable, but this is just me.

I would like to start with the statement that e and π are complicated numbers, and I would like to explain this statement in a bit of detail.  Mathematicians like to categorise and characterise things, and in this case we characterise numbers.  There are lots of different ways of doing this, but this way is very well known.

Think of it as a sort of sifting process.  We begin with the positive numbers, 1 and 2, 3 and 1005, 8 and 10126341763912362, which we can view as the boulders that we sift out first.  Such numbers are often referred to as counting numbers, as they are the numbers of things.  For an explanation of this, I refer to the expert known as the Count.

Contained in the act of counting is one of the great abstractions of mathematics, which is the abstraction of number.   Specifically, how we move from 3 apples or 3 oranges to the concept of 3, divorced from 3 of any specific objects. Closely related to this is the concept of definition.  How do we define 3 as separate from 3 of something or somethings?  All I will say about this is that we mathematicians have ways, but they are not ways to be described here.

What can we do with positive integers?  One direction is to extend to negative integers, but that’s a conversation and a construction for another day and another time.  A different direction to go is introduce complex numbers, but that’s also a conversation for a completely other day.  A third direction is to use the positive integers break the set of all positive real numbers into different pieces.

The first step is to consider the rational number, the fractions, which are the quotients of integers.  As simple as the positive integers are, the rational numbers are almost as simple, in the sense that each rational number is completely specified by 2 integers, the numerator and denominator, to use terms to take us back to warm spring days with mathematics at the front of the room and sun shining outside the window, tempting us from our lessons.

Being able to specify a rational number by 2 integers raises the (not completely) obvious question, which asks, what numbers can we specify using finitely many integers, and how might this act of specifying work.  There are many ways, but one is to build a polynomial, which is an expression of the form

p(x) = a_n x^n + a_{n-1} x^{n-1} + … + a_1 x + a_0

where the integers in questions are n, a_n, a_{n-1}, …, a_1 and a_0.  The a_n, …, a_0 are the coefficients of the polynomial and n is its degree.  We can specify a number Z (almost completely) by asking the question, does there exist a polynomial p(x) of some degree n with some collection a_n, a_{n-1}, a_1, a_0 of coefficients, so that p(Z) = 0.  It is an old fact, not as old as Euclid and his geometry but old nonetheless, that for such a polynomial p(x), there are only finitely many numbers which have the property that they yield the value 0 when they take on the role of x in the polynomial p(x). (Yes, I am performing a bit of a fudge here, but I am trying to be a bit informal.)

These numbers Z, for which there exists some polynomial p(x) as described above, with integer coefficients, so that p(Z) = 0, are the algebraic numbers.  Rational numbers are always algebraic (a proof left to the reader, but one I’ll be happy to discuss at a later point should a request come in), and some irrational numbers are also algebraic.  The classical example of an irrational but nonetheless algebraic number, going back to a different ancient Greek mathematician known as Pythagoras, is the square root of 2.

Why go through all this sifting, from integers to rational numbers and from there to algebraic numbers, is that the algebraic numbers form the largest class of real numbers that can be specified by a finite collection of integers.  Beyond the algebraic number lies the vast wilderness of the transcendental numbers, the jungle where π and e both live.

Enough for one day.  I’ll finish this particular story soon and we’ll see where we go from there.

Posted in mathematics
Tags: , ,

dooms day devices 1: money

•20 July 2014 • 9 Comments

We each have things that we like to contemplate in the quiet moments, those little things that distract us when we should be focusing on something else.

I like to contemplate dooms day devices.  I suppose this goes back to the first time I watched Dr Strangelove or How I Learned to Stop Worrying and Love the Bomb, which is one of my favourite movies, as well as to innumerable science fiction movies and stories.

There are different types of dooms day devices.  There is the Dr Strangelove dooms day device, a collection of nuclear bombs sheathed in Cobalt Thorium G, which will detonate and cover the Earth in a life killing shroud of fall out should the Soviet Union be attacked.  So, the attack us and die dooms day device.

There is also the keep me happy and fed or die dooms day device, which is the one I wish to scree a bit on today.  I’m sure that it’s possible to construct a formal hierarchy of dooms day devices, but that’s a project for a rainy day.  Which, given that I live in England, might be tomorrow.

The hypothesis that I would like to put forward in the following is that money is a dooms day device.  This thought came to me when watching some documentaries on events that led to the financial crisis, the worst in several generations, that we are currently crawling our way out of.

Obviously, money is not an attack me and die dooms day device.  But I have come to believe that money is a keep me happy and fed or die dooms day device.  Money is ubiquitous in modern society.  We use money as a way of avoiding barter by having a neutral means of determining value.

Don’t get me wrong.  I like money.  Having money makes my life easier.  And given my chosen profession of teaching mathematics in a university, I would have great difficulty coming up with something worth bartering, once society crumbles and university teaching is no longer the most highly valued of professions.

So why do I think that money is a dooms day device?  One reason, and this is the reasoning of an amateur, an unstudied casual reader of economics, is that we have found ourselves in the position where money seems to have gained control over us.  Money seems to have the upper hand.

We live at a time when the world is changing.  For the first time in human history, it seems that we have finally and completely divorced money from any physical asset.  Gold still has value, and diamonds still have value, but we no longer require that our money be linked to any physical asset.  This is the problem that we are working our way out of, slowly, that money is based on trust and very little else.

And so, money spirals out of control.  Particularly in our current days of near 0% interest rates, where it is cheaper (for some) to borrow, and where economists and pundits periodically speculate on the next asset bubble to burst and endanger the global economy, it seems that this unlinking of money and anything physical to stand behind it might not be the best idea that we as humans have ever had.

Or perhaps I’m just learning how expensive kids can really be.

Posted in bizarre and random things
Tags: ,

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