Philosophy of teaching

A very committed lecturer and friend of mine recently sat in on one of my lectures. She is taking a course essentially designed to make her an even better lecturer than she already is, and she needed to listen to someone else give a lecture, then ask some follow-up questions. I found the experience terrifying to begin with - it is so hard to be natural in front of one’s peers, where by natural I mean the completely unnatural persona one puts on in order to both entertain and inform a class. I soon forgot she was there and got into the usual swing of things.

Her follow-up questions were really illuminating for me. They centred around what one might call my philosophy of teaching. I might paraphrase them as What did I see as my job in the classroom? What is my approach to teaching? Why did I become a lecturer?

Another question can by email and I would like to share my response and invite yours. She asked

I remember you saying that the course I sat in on today is taught by mathematicians but designed by engineers. So do you feel that this influences what you teach and how you teach it? Have you ever felt competing pressures from the context vs. your personal views of teaching and learning? Have you found a way of resolving them?

My response was

You do ask interesting questions! I think my main aim is to show that mathematics is a way of thinking and working rather than a set of rules or results. Math is something you do and there is no substitute for rolling your sleeves up and getting dirty with it. (”Mathematics is not a spectator sport” someone once said.) I also think it is beautiful, challenging, pleasing, and fun, a product of hard work and creative people.

That said, in this particular course I am supposed to equip engineers with a set of tools to help them be better engineers. Modern engineering is suffused with mathematics, and they need to be fluent both in the mathematical way of thinking and in a particular set of tools. So, due to time restrictions and their own needs and interests, I skip over some of the challenging stuff, and some of the most beautiful parts (like proofs), and concentrate on using the ideas. But “in the margins”, so to speak, I always try to link the ideas to things they know, things they might have heard of, and cutting-edge stuff which builds on what we are learning. This way they at least see a hint of what drives weirdos like me to be mathematicians.

I think these questions touch on some fascinating aspects of the teaching and learning experiences. If you are a teacher, how would you respond? We are all learners; how do you think a teacher’s philosophy of teaching influences your ability to learn?

Mathematics: rigorous, tedious, fantastic, fairy-like

What strikes me about the following quote from Scientific American in 1908 is the opinionated, chatty, subjective, silly, and nearly-trivial tone which so often is blamed on the recent influence of blogging. There ain’t nothing new under the sun.

Mathematics is the most exact and the most thoroughly grounded of the sciences. And yet, in the very field explored by this rigorous and tedious method, have arisen fantastic and fairy-like structures of the imagination, which transcend all our experience. They have arrived at the conception of the fourth and higher dimensions. It would be impossible to confine a person having the secret of this dimension by the six surfaces of a prison cell. His slightest movement in the direction of a fourth dimension would put him at once out of three-dimensional space. It would be well for him to take care just what he did when in four-dimensional space, as upon coming back into space of three dimensions he might be much changed. (Quoted from SciAm here.)

Hang on: tedious? My Encarta World English Dictionary defines tedious as “boring, because of being long, monotonous, or repetetive”, and tedium as stemming from a Latin root meaning “wearisome”. Well, excuse me. Did the author not read my blog? I mean, I know connections were slower in 1908, but still . . .

Creativity: a scientific viewpoint

Last night I was lucky enough to be able to attend a wonderful two hour lecture and chat session with Margaret Boden OBE, Research Professor of Cognitive Science at the University of Sussex in my old home town of Brighton. She is here visiting the University of Canterbury as a Distinguished Erskine Fellow, and gave an evening lecture last night in the city as part of the University’s general programme of reaching out to the community. Her topic: creativity, and what science has to say about its origins, how to avoid discouraging it, and how to nurture it.

Boden is one of those brilliant speakers who brings to the stage an engaging personality, a powerful intellect, and an effortless access to a weight of experience. It is very hard to summarise the breadth and depth of her opinions, and the ingenuity of her illustrative examples, so I will confine myself to giving you the bare bones, and encourage you to track down her books (listed on her webpage linked to above).

She began by assuring us that creativity is not magic or divine, neither is it a special faculty possessed by an elite, but rather an aspect of general intelligence. Most importantly, an understanding of creativity is not beyond the reach of the scientific process. What, then, is creativity? Her answer: coming up with ideas which are new, surprising, and valuable. An “idea” here can be as broad as you please, and certainly includes physical objects.

Those key words - new, surprising, and valuable - are, she said, weasel words, because they have so many meanings. Much of the confusion people have when discussing creativity is due to subtle switches between the meanings. She unpacked them as follows.

1. New was said to have two meanings.
   1. New to the individual, what she termed psychological creativity.
   2. New in the history of human thought.

   Since an idea new in history is certainly new to the individual, we only need concern ourselves with the first meaning.

2. Surprising was said to have three meanings.
   1. Statistical surprise - essentially just an unlikely event, like an outsider horse winning a race.
   2. The sort of surprise one feels when an idea, once seen, seems almost like it was waiting to be discovered all along. The idea is “not an unfamiliar sort of thing”.
   3. Impossibilistic surprise - an idea which seems completely impossible, and yet it exists!
3. Valuable has so many meanings that we couldn’t really exhaustively list them. The list would include things like utility, beauty, simplicity, rich complexity, and so on. The criteria of value were said to be (a) diverse, (b) changeable, and (c) (mostly) socio-cultural. This notion of value can thus be seen to be subjective, which rather removes it from the realm of science. No complete scientific theory of creativity seems possible. But we can still ask how creative ideas emerge, that is, how an idea new, suprising, and judged valuable has arisen.

I liked the way Boden unpacked these ideas, especially the notion of surprise. But what I found most illuminating came next, when she expanded the idea of creativity into three types. They are as follows.

1. Combinatorial creativity, consisting of an unfamiliar combination of familiar ideas. A canonical example being poetic imagery. This, apparently, is the most common definition used when people discuss creativity, and is also the most studied in the scientific literature. This type of creativity depends on the richness of associations in the human mind, on appeal to shared meanings and experiences.
2. Exploratory creativity, and
3. Transformational creativity, involving the exploration and (sometimes) transformation of a structured conceptual space. One example Boden used to illustrate exploratory creativity was a collection of perfume bottles of all sorts of shapes. She pointed out that a perfume bottle is a hollow container capable of holding a liquid, with a stopper to prevent the liquid evaporating. But the bottle artisans had explored the whole range of variations within that conceptual space - tall bottles, short bottles, round ones, square ones, small stoppers, tall stoppers, and so on. Her example of transformational creativity was Kekulé’s discovery that benzene is a ring of carbon and hydrogen atoms. Before Kerkulé, all molecules were strings, but the topological change from strings to rings transformed the conceptual space of organic chemistry.

   

Combinatorial creativity goes along with statistical surprise; exploratory creativity with the not-unfamiliar kind of surprise; and transformational creativity induces an impossibilistic surprise. The three types are not mutually exclusive, nor is there a hierarchy here.

Well, I must admit I was blown away by this categorisation. Suddenly, the aspects of the examples I carry around of inspiring creativity in the mathematical sciences neatly fitted into this list. It raises interesting questions: which kind of creativity do you notice in yourself more? Which do you find most impressive or inspiring in others? What times of day, or moods, or environments, or activities encourage which kinds of creativity in you? (Bob has asked a similar question over on Heroes Not Zombies.)

Speaking of which, Boden finished the main part of her talk with some tips for how to avoid discouraging creativity, and how to encourage it. These tips, she said, work equally well for children and adults. First, the best way to discourage creativity of all three kinds is “to slap people down when they try to do something new”. Even worse than saying “that’s wrong” is to say “that’s silly” or (even worse!) “you’re silly”. This destroys people’s ability to freely think in a creative, daring way. She pointed out that she’s not saying that there are no wrong answers, but a wrong answer comes from somewhere, from a process of thought, and if you can modify that process in an encouraging and positive way then the right answer will come, the person will learn, and will nurture their confidence and creativity.

How then to encourage creativity? Each type needs a different kind of encouragement, as follows.

1. To encourage combinatorial creativity, ensure that the person has access to a wide range of ideas, where again an “idea” can be anything from the works of Plato to a game of football. Perhaps, for a child, ask them to combine ideas in silly ways. Maybe give them two crazy ideas, and ask them to write a sentence, or a story, or paint a picture which links those ideas.
2. Exploratory creativity can only be encouraged in one way: practice! One must learn the conceptual styles, the rules and conventions of the structured conceptual space in which one works. Boden cited an interesting study in which pieces of music from various stages of the careers of famous composers were selected and played to a panel of expert judges. The composers included some who were child prodigies, composing perhaps, like Mozart, at the age of 4. The panel was not told the composer, only the year of composition, and was asked to rate the piece on how adventurous it was - in Boden’s terminology, on whether it exhibited exploratory creativity or transformational. In every single case, even for the child prodigies, it was found that their early work was exploratory, and transformational work only came after a period of around 12 years of deep and devoted immersion in musical culture and history. Practice, practice.
3. So on to transformational creativity, which obviously builds on exploratory creativity, so we at least need that first. Beyond that, encouragement to change the rules, to drop some of them, will free the individual to be transformational. They may never be, of course, but rules should be seen as breakable.

I found this discussion encouraging for mathematicians, and I hope it encourages you in your field too. It is often said of mathematicians that they produce their best work before the age of 30. I don’t know a single mathematician who thinks it is true of themselves, and yet one encounters the idea regularly. I imagine, though, that a small group of transformationally-creative mathematicians, who get the most attention, have immersed themselves in maths since they were young children, have stuck diligently with it, working hard, and so around 12 years have elapsed when they find themselves in their twenties, with few commitments and lots of energy. No wonder they work well. Some of us come to fall in love with maths much later, and perhaps never put in the sweat and tears required to reach transformational creativity, and so explore and combine throughout our whole careers. Who knows? (The host of the talk, Denis Dutton, said that he recently read a paper by the psychologist Colin Martindale in which it was said that of course creativity can be taught. Simply find something you want to study 18 hours a day every day for fun, then do it for 20 years and you will be creative.)

Finally, in the question session which followed, I asked (waaaay too many questions, one of which was) about the role of the subconscious in creative thought, especially in the sciences. I pointed out that, anecdotally at least, mathematicians seem to have their best ideas when not doing maths, such as taking a shower or walking the dog. There seems, I said, to be a period of intense cogitation, followed by incubation, followed by sudden realisation. (This has been called by Guy Claxton the “undermind” at work - see a post of mine here.)

Boden doesn’t think so. She argues, citing some wonderful studies, that what counts is rest, time away from the issue at hand. The problem is that in the period of intense thinking we get trapped in ways of looking at the issue, trapped in cycles of thinking. Time away resets our thought processes, so when we come back to the problem, our fresh eyes (and presumably the deeper understanding of the problem provided by the intense thinking) see the solution immediately.

How does that resonate with you? It makes a lot of sense to me, especially when I think back to those flashes of inspiration in the shower. They often did not come straight out of the blue during a not-thinking phase, but followed a little voice which said “let’s get back to the problem” - and then the solution appeared, usually by a feeling of stupidity for not seeing it earlier!

Me in six words (a meme)

The rules of the game can be read over here. The idea is to write a six word autobiography, or memoir, then “tag” five other bloggers to do the same. I was tagged by the tea-rrific Pepsoid at The Art of Tea.

The idea is deliciously ridiculous. How can one possibly summarise even a day of one’s life in six words? But I found myself searching for a common theme in my experience of the world, something ever-present in the way I interacted with people, or with things. An aspect of my life which remained changeless throughout the changing years. In the end, I came up with

Looking at others looking at me.

The people I would like to tag to do the same are:

1. Bob Leckridge at Heroes not Zombies (who has commented on this meme but not released his memoir!)
2. Robert Talbert at Casting Out Nines.
3. David Corfield at Why Do People Get Ill?
4. Clifford V Johnson at Asymptotia. (Who is almost certainly too busy.)
5. Cecilia over at PhDComics. (How unlikely . . . )

Illegality of military robots and questions of identity

Military robots - milbots, anyone? - play an increasing role in modern warfare. Robotic bomb disposal units have been around for some time, as have computer-assisted aircraft controls, but now the aim is to have robots autonomously plan and make kills. One example is the class of robots known as Unmanned Combat Aerial Vehicles, or UCAVs.

 I’m going to swallow hard and sweep aside my objections to this hideous use of technology and my visions of a hellish future first of “our machines versus your people”, then “our machines versus your machines”, before finally “all machines versus all people”, because the idea of a robot killer raises some important issues which we are going to have to confront sooner or later.

Who has accountability for the kill? Currently, no weapons are released autonomously, or at least that is the official line cited on Wikipedia’s UCAV link above - a human will always be to blame (if something goes wrong) or praiseworthy (if things go “well”). But a post on New Scientists’ tech blog discusses the question can a robot commit a war crime? At what point does an autonomous robot have culpability? If we declare a robot guilty, have we judged it to be human? What is the line between organic life and a robotic system, shall we say between carbon and silicon life? What, indeed, is the line between life and non-life? If such a line exists, do we make a mistake when we try to extend our notion of non-life to include life - should we extend our notion of life to include non-life?

More importantly, who will be the first human to marry a robot?

Butterflies remember being caterpillars

For as long as I can remember I’ve been fascinated by the process by which a caterpillar dissolves itself into a butterfly. It seemed astonishing that a life would destroy itself to birth a totally different creature with none of the characteristics or memories of its progenitor. But, how could memory possibly survive the mushification in the cocoon?

No-one knows, but now there is some evidence that somehow it does. New Scientist magazine has a nice little piece about the work here.

The meaning of scientific literacy

The promotion of scientific and mathematical literacy is always to be applauded. Looking at the world honestly, and drawing conclusions based on the evidence, are two important skills for making a human whole. Alone, they lead to a lifeless vision of terrifying sterility, but combined with compassion, creativity, imagination, and love, they have the power to transform.

I’m always a bit worried, however, when scientific literacy is reduced to “numeracy”. It may surprise some of you to learn that I don’t think numeracy is very important, beyond a certain level. It goes without saying that to avoid being ripped off, and to feel in control of one’s finances, the ability to work with numbers is important. But science and mathematics are far more than mere numbers, in the same way that language and culture are more than letters. Galileo wrote that the book of nature is written in the language of mathematics, and this elegant statement has lead to much confusion.

The comparison is made between what links books and language with what links science and mathematics. But we all know that a book is a medium of communication of ideas. A book transmits ideas, shapes the lives of readers, and therefore carries and moulds culture. To do so, writer and reader must agree on a set of rules and conventions for the arrangement of symbols so that meaning can be extracted from them. Likewise, mathematics has a set of rules and conventions for the arrangement of symbols, but more than that it has rules for the manipulation of said symbols in order to extract new information. The analogy would be a set of rules for manipulating letters until new ideas were formed: nonesuch exists. Moreover, the mathematical statements have meaning in themselves, as mathematical statements. It cannot be said that a sentence has any meaning other than what it conveys to the reader. The mathematical symbols may then convey a further meaning in the context of the scientific culture in which they are employed. This equation may tell us that a disease will spread at a certain speed; this other may tell us that black holes are points of zero radius and infinite density. This scientific meaning is like the meaning of a sentence in a book, in that it also only “means something” in the context of what else is known, the context of a shared culture.

The point I am attempting to reach is that science and mathematics are about far more than a fluency with the rules for manipulating numbers. They are about pattern, consistency, interrelation, and meaning.

Having said all of that, I would like to bring this exercise to your attention. A global record-breaking attempt to ask young people to solve as many mental arithmetic problems as possible in a given time.

Radical shifts in perspective

A few minutes ago I posted about a wonderful Scientific American slideshow in which the brilliant science plays second fiddle to the new and jaw-dropping perspective on humankind it produces. Then there is another article also over at SciAm, asking “Do Microbes Make Snow?” It turns out that the answer could be yes, and they may make rain to boot. Quite apart from the boon this may be to the Gaia Hypothesis, the final sentence should stop you in your tracks. It reads “. . . living microbes in the clouds might just be perpetuating themselves—and spreading—when it snows or rains.” Here we are thinking that the Earth is our playground, and that if we are vulnerable to forces beyond our control then surely other beings are, and then comes the realisation that tiny microbes may in fact be in control of some of the most powerful forces on the planet.

The long view of our night view

Scientific American have surpassed themselves again with a beautiful slideshow taking the long view of how the night sky seen from Earth has changed in the past and will change in the future. The images themselves are striking as works of art, but what really hits home is the brevity of recorded human history in the face of these immense stretches of time. There is an old Anglo-Saxon story in which the lifetime of a man is compared to the brief flight of a sparrow through the warm and noisy atmosphere of a mead-hall in the depths of winter. The sparrow enters from the swirling snowy maelstrom, and returns all too soon to the void beyond. How briefer still seems the life of humankind in the face of predictions running 100 trillion years into the future. Even to think 1 million years into the future - should we survive, what would it be like to live with a million years of recorded history? How could you be sure of originality? Would the weight of accumulated knowledge oppress creativity? Could our descendants even relate to what we did to each other, how we saw the world and the universe?

One unforgettable lesson from this slideshow and the accompanying article is the realisation that, if we are correct about the future evolution of the universe, all evidence of the big bang, the expansion of the universe, and of its sheer immensity and dynamism will disappear. What do we not know, indeed, cannot know, about the universe because the evidence is now hidden beyond all reach?

Narrative in teaching mathematics

Bob has yet another fascinating post over at Heroes Not Zombies, this time explaining how writing about their experience of the disease helped cancer patients to change their thinking of the illness.

In the face of suffering and death, mathematics seems very trivial. But I have noticed that many of the best moments in a lecture theatre, those wonderful times when the audience seems rapt and suddenly there is a collective “a-ha!” look which sweeps the rows, are when I set the dry theory in the context of a story, or at least a concrete situation which can be held in the student’s mind. Moreover, I consciously try to link new ideas or results to early work, to build up a web of connections between the things we learn, and indeed the things we may know and learn from other areas of life. I explain that this strange mathematical symbol is named after an ancient Hebrew harp, or that this result is like walking into a cave at the same moment that someone pelts you with a snowball.

These images, and the tale which weaves them together, seems to lead to a more rewarding learning experience for the students. The jury is still out on whether it helps them to pass with a better mark. But the idea of narrative in mathematics is even being explored at the research level.