Monthly Archives: April 2013

Education compared to a Penny-farthing Bicycle

‘A growing body of literature has linked the ability to delay gratification to a host of other positive outcomes, including academic success, physical health, psychological health, and social competence.’ – Wikipedia article

I don’t know about the rest, but I find the link with academic success interesting: I can see why it might be the case, but I wonder if it is an encouraging sign.

There was a time –  for roughly two decades, between 1870 and 1890 – when a man’s inside leg measurement was strongly linked with positive outcomes in cycling – the longer it was, the more likely he was to be successful. The reason was simple: on the bicycle of the day – now variously called the Ordinary, the Penny-farthing or the Hi(gh)-wheeler – the rider sat astride a huge directly-driven wheel of anything from from 55 to 60 inches (140-152 cm) in diameter, so that a single rotation of the cranks would propel him some five yards and more (4.7m) along the road; the hour record for these formidable machines (paced) is a remarkable 23.72 miles. A man called Tom Stevens rode one round the world between April 1884 and December 1886.


With the advent of the ‘safety’ bicycle and its geared chain drive, the fact of being long-legged ceased to be an advantage – on a six-speed Sunbeam (available in 1907) a rider of average leg could drive the equivalent of a 129” (328cm) wheel (the legacy of the penny farthing is that (in Britain, at least) bicycle gears are still measured in inches, as the equivalent of a directly-driven wheel of that diameter). If you were to turn such a gear at a modest sixty rotations per minute for an hour – no great feat on a level road – you would travel a shade over 23 miles. Thus a different way of doing things can bring feats once reserved to the few within the compass of the many.*

This might tell us something about education. Despite great advances in the way we understand teaching and learning, the high school system in this country is still – like the diamond-frame ‘safety’ bicycle that dethroned the Ordinary – essentially a Victorian design. While the diamond frame bicycle might be likened to the shark – having early evolved a form perfectly adapted to its purpose, there has been no need to alter it – I do not think the same can be said of our high school system.

That it is a system is perhaps the first point to note: it all hangs together, from the design of the buildings, the division and delivery of the curriculum, the staff structure, the central importance of texts – which is why it has been so difficult to change. It is, in essence, conceived as an economic method of knowledge transfer: large groups of students are taught by single teachers in rooms designed expressly for that purpose. The curriculum that is delivered is divided into separate subjects, each with its expert, and the content is ordered to allow a graded progress over a period of years. Language, mainly written language, is the principal vehicle of instruction. It is, in a word, rational. It is a system that works best (and it can work very well) when the students are grouped according to ability, literate, biddable and with a capacity for deferred gratification.

The deferred gratification is needed because learning in these conditions offers little in the way of enjoyment and requires a fair degree of self-denial: it is something of a slog, and although it is rational (indeed, perhaps because it is rational) the point of it is not always obvious, and to keep at it you must believe what your teachers and parents tell you when you complain, that ‘it will all be worth it in the end’ and ‘some day you’ll be grateful’.

Yet learning can be enjoyable and exciting in itself, when it rouses the curiosity and feeds the passions – but reason and logical progression are seldom key motivators: they facilitate learning for the experienced learner, i.e. the person who has already (through years of deferred gratification) learned how to play the game. The illuminating analogy here is to consider how we learn language, and indeed how language learning has changed.

The grammar I learned in primary school was largely derived from Latin grammar,  (which is the grammar for which Grammar schools are named) despite the very considerable differences in character between Latin, a highly inflected language where word-order is relatively unimportant, and English, a largely uninflected language where word order matters a lot. We did parsing and analysis – dividing sentences into the various parts of speech (noun, verb, adjective, etc) clauses (main, co-ordinate and subordinate) and identifying their grammatical relations (subject and predicate, direct and indirect objects). 

All this gave me a command of written English and stood me in good stead when I went on to high school and actually learned a little Latin and rather more of its modern descendants, French and Italian – so I’m not complaining: it was an approach that served me well, though it has still left me a lot better at reading French and Italian (and writing them to some extent) than I am at speaking either and – particularly – understanding them when they are spoken. And even now, if I consider learning a language, my first impulse is to buy a grammar book: I feel safe with that, I know my way around. I shy away from speaking to people though: I’d rather acquire some degree of expertise first.

Yet I learned my own language when I was too young to be aware of doing it, without the aid of books and with no knowledge of grammar; and I learned it by talking to people who talked to me; and if my observation of young children since is anything to go by, I think the experience probably afforded me a great deal of enjoyment and even outright hilarity. Thankfully, language teaching now makes more use of these ‘natural’ methods than in my day.

Might it not be better if we devised an education system that was geared to our natural propensities for learning, rather than one which – however effective it might be for some (like me) – achieves its end by stifling those natural propensities, and with them, spontaneity and enjoyment? Must we defer gratification to learn?

*always provided you had the 19 guineas (£19-19s-0d or £19.95) that a Sunbeam A6 would cost you – the equivalent of about £1100 today.

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Why is a raven like a writing desk? The power of abstraction.

ImageAbstraction is an interesting notion. The word itself is derived from the Latin preposition ‘ab’ meaning ‘from’ or ‘away from’ combined with the verb ‘trahere’ ‘to pull or draw’ (which also gives us our word ‘tractor’) – thus it means, literally, ‘to pull or drag away from’ so that it conveys the sense of separation, something that was embedded being removed from its original surroundings.

The first thing I would like to consider is abstraction in relation to numbers, which strikes me as being fundamental to the discovery of mathematics. To help us here we might imagine, in the style of Wittgenstein, a couple of ‘primitive societies’ whose use of number differs from our own.

In the first, counting is only ever done in the presence of objects: these people have no notion of mental arithmetic or ‘counting in your head’ – to say ‘1, 2, 3, 4’ when there is nothing there to count would seem to them as bizarre as if I were to say ‘monkey hat potato bicycle’ out of the blue, with none of these things being present. For them, numbers are not at all abstract – they are a property of concrete objects, unthinkable apart from them.

The next society takes the view (not unreasonably) that only similar things can be counted: sacks of wheat, say, or earthenware oil jars in a storehouse can be enumerated, but if someone is confronted by a dozen objects all different, he will point to each in turn and say ‘one, one, one…’  on the grounds that they are all different, so don’t amount to anything together. After all, no-one can say that a jar of oil is the same as a sack of wheat, can they? That would be absurd! Then one day someone, in a flash of inspiration, sees that although the objects are different, the ‘ones’ are the same, so a second teller is introduced to count the number of times the first says ‘one’ (perhaps by making marks on a slate). He is then able to say ‘there are twelve ones there’.

This will give rise to a joke, a riddle: ‘why is a monkey like a hat?’ to which the answer will be ‘because they’re both one!’ (at which all laugh heartily). (You might even imagine quick children demonstrating this to their slow parents – monkey = 1; hat = 1; 1=1, so monkey = hat! – and the parents being amazed at their children’s sagacity – ‘the things they learn in school these days!’). (Come to think of it, that might be the solution to the Mad Hatter’s riddle, ‘why is a raven like a writing desk?’ – Lewis Carroll (or rather C L Dodgson) was a mathematician)

But what interests me here is that this is the beginning of abstraction, to see that one of anything is equal to one of anything else, at least as far as counting goes – it is the dawning of the realisation that you can carry out operations with numbers without reference to what they are ‘numbers of’.

Full abstraction is the emancipation of number: it is no longer the property of objects, it is something in its own right. Nobody asks ‘what does 1 stand for here?’ or on being told to add five and five, demands ‘five of what?’ And it is when you reach that stage that you begin to discover the amazing things that you can do with numbers, the complex network of relationships that exists between them.

Another interesting thing has happened here: we have extended the meaning of ‘existence’, discovered a new order of reality. Numbers seem to exist just as surely as ravens and writing desks – if not more so – yet not in the same way. We feel sure that the propositions of arithmetic, and all the other properties of numbers, hold good always and everywhere; if all knowledge of them was lost, we feel sure they could be discovered again and that they would be the same – in fact, they seem to be unchanging and imperishable. It is little wonder that Pythagoras and his followers made a religion of them.

Which gives rise to some interesting speculation: what other properties can we abstract from their original setting and treat as things in their own right? will those exist in the same way that numbers seem to? and will they form part of a complex network of relationships that stands to them as mathematics does to number? In particular, is this something that we can do with language and words?

Could be worth investigating further, maybe. But meantime, at least you know why a raven is like a writing desk.

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