Monthly Archive for August, 2011

By Rishidev Chaudhuri, 3 Quarks Daily

It is an oddly well-kept secret that mathematical learning is a very active process, and almost always involves a struggle with ideas. To a large extent, this is due to the nature of mathematical intuition: grasping a mathematical idea involves seeing it from multiple angles, understanding why it’s true in a broader context and understanding its connections with neighboring ideas. And so, when you sit down to read through a proof or the description of an idea, you rarely do just that. Instead, digestion more often involves settling down with a pen and a piece of paper and interrogating the concept in front of you: “What is this statement saying? Can I translate it into something else? Can I find a simpler case that will help me gain insight into this general context? What about this makes it true? What would be the consequences if this statement were false? What contradictions would I encounter if I tried to disprove it? How does this concept reflect those that have gone before? How do the various assumptions used to prove this statement factor in? Are all of them necessary? Are there other ways to frame this fact that seem fundamentally different?” And so on. And this interrogation often involves taking your pencil and paper on long digressions, slow rambling explorations of ideas that help clarify the one you’re trying to understand.

Similarly, proving a mathematical statement or solving a problem is an unfolding of false sallies and blind alleys, of ideas that seem to work but fail in very particular ways, of realizing that you don’t understand a problem or a concept as well as you thought. And again, these are not wasted. In almost every case, if someone were to just give you a proof or a solution and you didn’t either try to come up with it first or actively interrogate it once you had it (which is almost the same thing), you’d learn that the statement was true, but learn very little about why it was true or what it meant for that statement to be true. And much of the learning in a math class happens not in the lectures but afterwards, in the time spent on problem sets (and, if you had a choice between attending the lectures and doing the problem sets, you should always pick the latter).

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Melissa Benn, The Guardian

Watching the anarchic, frightening scenes erupt on the streets of London, Birmingham, Manchester, and many more urban areas over the last week, a single word kept reverberating around my brain. Savage. It was a few moments before I remembered the source of that single, stark adjective. It appeared in a newspaper interview last January in which former children’s commissioner Sir Al Aynsley Green spoke of his fears for this generation, and for a rise in crime in the wake of savage – yes, that’s the word he used – coalition cuts to youth and education services.

Eight months on, Green’s words come as if from a different era – more benign, more socially perceptive – but are all the more prescient for that. He spoke of his fears that we are “witnessing the destruction of many of the building bricks of support for children and young people to achieve their full potential in life. It is desperately worrying. I see little in their place to inspire confidence that this generation will be looked after by government. It could spell the end of hope and expectation for many of them.”

Green touched on both the decimation of youth provision, such as the Connexions career service, and the the Future Jobs and Working Neighbourhood funds and education cuts, such as the scrapping of the Educational Maintenance Allowance. To that, we could add the botched handling of cuts to the school-building programme, BSF, and the steady erosion of national education spending in real terms.

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Photo: Wellington College, courtesy of Angelo Hornak /Alamy/Alamy