Thanks to Michael Pershan for writing a lovely piece about ancient Greek mathematics. He links to and comments on aspects of an article, The two cultures of mathematics in ancient Greece by Markus Asper.
The words of, for example, Euclid or Archimedes appear to be of timeless brilliance, their assumptions, methods, and proofs, even after Hilbert, of almost eternal elegance…Recently, however, a consensus has emerged that Greek mathematics was heterogenous and that the famous mathematicians are only the tip of an iceberg that must have consisted of several coexisting and partly overlapping fields of mathematical practices.
The part of the iceberg that is underwater, so to speak, is practical mathematics, the kind that builders need for construction, that commercial calculators performed on a sand tray with pebbles and that surveyors used when stretching out ropes to measure land.
Some of the books of the theoretical, abstract mathematicians have come down to us; the everyday mathematics of craftsmen, engineers, accountants and surveyors is all but lost to us.
Euclid distilled the discoveries of the two centuries before him into a careful tower of propositions, each careful built on the one below, logically building upwards by detailing one step at a time:
The fragments of builder wisdom that have turned up in rubbish tips in Egypt couldn't be more different: recipes for how much stone you'll need for a particular sized house. There's not even a general formula.
High Context, Low Context
At this point I want to bring in the anthropologist Edward T Hall's description of high context and low context cultures. For Hall, “a high-context communication or message is one in which most of the information is either in the physical context or internalised in the person, while very little is in the coded, explicit, or transmitted part of the message”. In a low-context communication on the other hand, meanings are explicitly stated through language. Hall was interested in cross-cultural mis-understanding and understanding. I want to mention maths education in the light of his distinction.
Euclid is low-context: he tries to make every little step explicit for us.
The builder's recipes on the other hand were high-context. The builder works with his apprentices on site and such recipes as were written down would be adjusted and reworked within a social, oral, gestural, practical context.
Can you see where I'm going with this?
Euclid has a privileged position in the history of mathematics and of learning mathematics, and it's style, impersonal, logical, abstract, has permeated mathematical writing. Moreover, the textbook and the test have, for additional reasons of their own, adopted the same low-context mode - everything you need to know must be contained in the writing.
This kind of low-context writing, that has to contain all the information that would normally be carried by a social situation, the possibility of clarification, questioning, gradually tuning in, is hard to read for the uninitiated. It's not our normal kind of communication.
Here I want to mention Robert Kaplinsky's webinar, Why We Should Reconsider Using Word Problems (And What We Should Be Doing Instead).
Why do we funnel children through a low-context communication, the word problem, the odd-one-out verbal reasoning question, the test question, when we're in a high-context situation? We're like the builders, the students are like our apprentices, we have a social, oral, gestural context in the classroom, we have lots of materials. We can present fragments - an image, a numberless problem, an equation. There doesn't need to be a question. Through the unfolding of the lesson in the high-context culture of the classroom we can gradually, through dialogue and practical work, build the knowledge together.