multijimbo

A colleague recently asked, in the course of an unrelated conversation, for my views on why society should fund pure mathematics.  Particularly during hard economics times, it’s a reasonable question to ask, why do we as a society fund the things we fund.  On the other hand, it is a question about which I have a strong vested interest and a significant bias.

She asked me as I was preparing to talk to a group of pre-university students, giving them reasons that they should consider studying mathematics at university, and the combination of the 2 questions got me thinking.

I am by inclination and training a pure mathematician.  I will admit to having dabbled in a few bits of more applied areas of mathematics, but I like geometry and I like studying geometry for its own sake.  I enjoy groups and like studying groups for their own sake.  I enjoy the abstraction and I love the art of being a mathematician. 

My first answer is the same as it would be for most academic endeavours, the subjects covered by the departments that make up universities.  It seems bizarre to me that as a society, we would ever be willing  to say, we can stop now because we’ve learned enough. It seems bizarre to me that we as a society would ever be willing to say, we have a sufficient understanding of the the inner workings of the physical realm and the realm of ideas, of the human mind and our place in the wider world, that we don’t need to learn any more.

We will never come to a point where we know everything, where we’ve learned enough.  Part of this is developing our understanding, and part of this is creating art.  There is art in exploring ideas, navigating through possibilities, making connections between hitherto unrelated ideas and concepts.

But beyond this, for mathematics of all stripes and types, there are other reasons.

Mathematicians predict the future.  We construct models that allow us to extrapolate from the past and the present, based on the information we have and based on our understanding of the mechanisms of the world, and make reasonable guesses of what the world might be like in days, or weeks, or months.  We bear witness to this every day, when we hear the weather report and learn what the future might hold for us.

Mathematicians reconstruct the past.  My favorite example here concerns phylogenetic trees.  We start with evolution and natural selection.  If we make the assumption that life began once on earth, or at least that it is only the descendants of one beginning of life, which I naively believe is the more reasonable assumption, then life is a tree.  Where the branches split from one another mark where one species gives birth to two, and so on, and so on.

Viewing things in the present day, we don’t know where are the branches in this tree.  But if we know far apart the current species are (say, in how different their DNA is one from another), then we can reconstruct this tree, and in doing so, we can reconstruct the history of all life.

Mathematicians explore the unseen.  I like the Radon transform as an example of this.  A CAT scan works by shooting beams of radiation through an object with variable density.  Some highly sophisticated mathematics allows us to reconstruct the density of the object from the absorption of radiation along each of these beams.

Mathematicians are explorers.  Our explorations link the inner world of the mind, with its notions of beauty and structure and order, with the outer world in which we live, allowing us to create models to understand the rhythms of the world.  We forge unexpected connections, and even the most abstract and obtuse piece of mathematics has the possibility of shedding light on a poorly understood (or even well understood) corner of our world. 

And that I think is sufficient reason.

Posted in mathematics

And so, we come back to spend a bit of quality time with my grand plan of fundamentally redesigning my teaching.  I think that the place to start is by looking deeply at the things I’m currently doing.  From time to time, it behooves us all to lean back, close our eyes, and examine our fundamental assumptions, those bits of bedrock on which we base everything we do.

As mathematicians have done for centuries, I lecture.  But what does this mean in practice?  It starts with all of the things that go into preparing for the act of lecturing.  I spend time in my office, or on the train, or wherever I happen to be, structuring the material.  I work through examples, some standard, some non-standard, some weird and strange to illustrate a particular point I wish to make, so that I can present them smoothly during the lecture itself.   I construct exercises to allow the students to engage with the material.

But what of the act of lecturing itself?  I am beginning to think that in some ways and at some times, lecturing is the wrong description, and that transcriber might be more appropriate.

Reading and writing are very recent acquired talents in human history, both only a few thousand years old, and both only much more recently expected of members of the general populace.  Speaking and listening, though, are much older, and so a question worth asking is, do we learn better when listening or when reading?  I don’t know the answer, but if you do or can point me in a direction, I’d be interested in getting a reference for who’s explored this.

And this is relevant for the discussion we’re having.  If it turns out that the answer to my question above is no, there is no difference in learning between listening and reading, then this begins to undercut the reason for having a standard lecture, where the lecturer talks and the students listen.  Until I learn otherwise, though, I choose to believe the answer is yes.

This is not to say that there is no place for a lecturer and students being in the same room at the same time.  Only that in order for the lecture to add some value to the student experience, there would need to be something else.  There would need to be interaction between lecturer and student.

And so there is where I am at the moment.  What are the things that I can do to add value to that time when I’m in the room with my students.  And I’ll go back to the beginning, to the choice of the material I talk through versus the material I make available; to how I present this material; to how I encourage my students to be active participants versus passive recipients.

Looking back, I’ve been experimenting with things for years now, but not in a particularly coordinated way.  And so, before I next teach, I think that quest will be to impose some coordination on my activities in and outside of the class room.

Posted in mathematics