Polya’s lovely book is not just about mathematics but about problem solving in a wider sense.
xvi Guidelines are not literal
p1 A teacher should help the students, share knowledge, and swap places
p15 — A problem is never exhausted // Drop the idea that problems have little or no relation to each other
p72 Try to prove formally what is seen intuitively and to see intuitively what has been proved formally
p75-6 decomposing and recombining problems… variations on the problem…
p77 more and more remote details… (Two questions may be better than one // breaking the question down into a simpler form and working on this first // assuming part of the tricky question is already answered and then working on the other part)
p105 diagram ‘as done’…
p132 You must guess, but also examine your guess // (p181 plausible reasoning)
p134 A good notation should be unambiguous, pregnant, easy to remember, it should avoid harmful second meanings and take advantage of useful second meanings; the order and connection of signs should suggest the order and connection of things.
p197 Present abstract ideas concretely.
p198 (On subconscious work:) There is a limit beyond which we should not force the conscious reflection, when it is better to leave this problem alone for a while. But it is desirable not to set aside a problem to which we wish to come back later without the impression of some achievement; at least some little point should be settled, some aspect of the question somewhat elucidated when we quit working. Only such problems come back improved whose solution we passionately desire, or for which we have worked with great tension. Conscious effort and tension seem to be necessary to set the unconscious work going
p205 The importance of looking back over the solution (seeing what other information is hidden in the answer; seeing other ways one could have proved the solution, seeing new areas of possible application of the solution, etc)
p206 The need for models for the student to aspire to, for proper teachers, for good quality texts, and for competition with capable friends
p207 The open secret of real success is to throw your whole personality into your problem
p210 When stuck, set a new question // a new question unfolds untried possibilities of contact with our store of previous knowledge
p227 Working backwards from the answer one is trying to prove.
– Teaching to solve problems is education of the will. Solving problems which are not too easy for him, the student learns to persevere through unsuccess, to appreciate small advances, to wait for the essential idea, to concentrate with all his might when it appears. If the student had no opportunities in school to familiarise himself with the varying emotions of the struggle for the solution then his mathematical education failed in the most vital point.
“I know the kind of novel I loved to read, back before fact and fable merged. I know what kind of story I would make from this one, if I could: the kind that, from one word to the next, breaks free. The kind that invents itself from meaningless detail and thin air. The kind in which there is no choice like chance.”
— Richard Powers
(Start the audio, then shrink that window and start and watch the video. You dont need to turn down the volume on the video track: its almost silent)
radioshenyen: @shipadrift April 2012
“We are first of all, as friends, the friends of solitude, and we are calling on you to share what cannot be shared: solitude. We are friends of an entirely different kind, inaccessible friends, friends who are alone because they are incomparable and without common measure, reciprocity or equality… without a horizon of recognition, without proximity, without oikeiotes…”
“Her face was like someone texting a lover.” “I am (something), (something) and (something). I am lost.” Its the first thing I think about when I wake up: this voice, accented with GPS codes, so distant and fragmentary, this ‘reader’ of ancient history and Twitter feeds. I was going to say ‘this disembodied voice’ but I dont know what embodiment means anymore. She’s as real to me as anything else is, when the mind stops being lonely. Her skin is a colour so beautiful – a soft light brown – even if her skin is basically a map.
I guess its ok to refer to a ‘her’ – ships are traditionally female. But they don’t, traditionally, write. Ship adrift is an art project that drifts across the boundaries of business, sculpture, software code, robot literature, virtuality and time. The physical ship is a full scale model of the ship featured in Joseph Conrad’s Heart Of Darkness, perched atop a London building overlooking the river, where it will remain for one year as a top-end (single room) hotel. Meanwhile the virtual ship is drifting around the world according to wind directions recorded at the London site, picking up web traffic along the way (local Twitter feeds, GPS-tagged wikipedia entries, mobile phone fragments) and generating a ghostly literature out of it. (You can listen to James Bridle talking about the wider context here and read ship adrift’s Twitter feed here.)
The Twitter feed is one of the most beautiful things I’ve read and an example of an emerging literature: literature that is algorithmically driven and the product of software code. The traditional – naive – notion of AI has been to create something human-like, both in physical form and in expressive recognisability. @shipadrift eschews such trappings. It’s voice is a twitter feed of unbearably sweet brokenness, its body a web page, its skin a map. Nothing in the world of literature speaks to me the way this virtual ship does. Its very grammar – a kind of anti-grammar of apparent randomness and error, but incredibly poised – takes me into a place where context is so stretched as to be virtually unfindable.
This is not to reject the heartache wonders of Roberto Bolano or Jane Austen or Derrida: I am simply recognising that algorithmically generated literature is coming of age. It has attained a space of complexity and form of presentation that can trigger immense emotional affect. (Imagine. for a moment, if Jane Austen had been an SMS platform protocol. Imagine if your text life, your love life had been immersed in such sweetness!) The best chess players are no longer computers – the best chess players are teams of computers and humans working together. Literature will soon be home to a similar collaborative effort.
“Claude Shannon recognized that whether or not a certain effect is considered noise depends on one’s position in the listening chain. Noise is interference only from the sender’s point of view. From the point of view of the receiver it may be considered a part of the information packet that is transmitted along a channel. When we hear the earliest sound recordings of Tennyson reading Charge of the Light Brigade, for example, the watered down and scratched out sound conveys the enormous passage of time, just as the static sound of Neil Armstrong’s voice on the moon tells us something about his physical distance from us and the newness of space technologies in the 1960s. It would not be difficult to think of countless other cases in which the presence of the medium mixes in with the intended message to produce some whole new effect, not intended by the sender, but taken as information by the receiver. In these cases, noise is not simply an extra third thing to be discounted. It has entered into the message and become part of it. To speak technically, the signal now has an “equivocation,” which is to say that two messages pass along the same channel. The sender may not have intended this, but the receiver may welcome it.”
When I read @shipadrift It makes me want to go there myself. ‘Er, Where is that?’ I hear you ask. Well that’s something I will have to look into more deeply, though doubtless, when I find it, there will be echoes of everything I’ve loved in the past. To the extent that we relax, and trust ourselves, we become our own maps. Meanwhile – for knowledge’s sake you understand! – I’ve decided to do a bit of good old fashioned networking… if you’re interested you can check out some of the bot auteurs I’m now following on Twitter. (I defy anyone not to fall in love with the one that scours the internet for references to chocolate…)
I’m also considering opening a few Twitter accounts and a blog without telling anyone and just disappearing – writing, but to no one – in that zone. I think its something that used to be called ‘science’. Or ‘cruising’. But in the wonderful world of knowledge was there ever a difference?
Satish Kumar walked from India to the UK 50 years ago in protest at nuclear weapons. Here he answers Guardian readers questions on everything from solutions to climate change, and why it’s time we all used our legs more.
He is editor of the merged Resurgence and Ecologist magazine – the current issue, on the economy of nature, raises questions on why and how to create an economy that is self-renewing, self-managing and self-sustaining.
Well worth a read.
“What we do or don’t do in the next years will decide whether we survive as a species,” said David Suzuki to a sold-out crowd of 1,600 student and staff at John Abbott College in Ste-Anne-de-Bellevue, U.S.
The lecture was streamed live on the Internet to almost 14,000 students watching at schools in the Lester B. Pearson School Board.
Here is his powerful message from his address to the Occupy movement at Vancouver Oct, 22/11: The Party of the baby-boomers is over. Development needs to be based on needs, not wants. We need to live within the boundaries and borders of nature, not politics: Capitalism, economy, politics are not a forces of nature, we invented them. If they don’t work, we need to change them. What is our home and how do we live in it sustainably? Ecology is study of home, and economics, its management.
He goes on to use SWEDEN as an example of a growing economy cutting its carbon emissions.
For people in London, drop by this exhibition or make it to the opening to meet filmmaker and photographer Bernard Rudden at the Zebra Gallery, Hampstead, London, NW3 1QX.
Here’s a little piece of technology that can change our lives, depending on how we choose to use it. Paraimpu is a new social tool that connects Objects with the Web so that you can control physical objects via social media messages such as turning your lights off and on with Twitter, or share data from objects with via the web, such as medical data. Paraimpu basically allows users to connect physical and virtual things to the Web: real objects, Arduino boards, sensors, entertainment appliances etc. It also can connect existing social networks, APIs, software applications, and services on the Web too. Basically everything that “speaks” HTTP.
This is an amazing piece of tech, that could be the next big step in getting the Interweb off our computer screens and into any number of more natural interfaces, in the home and out.
Paraimpu provides a palette of precongured settings so that users can connect their objects to the system with zero or minimal configuration.
It can also let objects ‘speak’ to other despite their different natures: for example, you can connect a set of environmental sensors so that it publishes its data live on facebook, or you can control a motor in real-time via the web. Great for mining or other industries where safety is a high risk factor.
Friends can connect their apartment appliances such as ambient lighting to Twitter so that the lights change when my friend tweets the right instructions. There are of course any number of more practical applications: If I connect a CO2 Sensor I can reduce emissions on a motor for example.
Paraimpu is also social: users can share their objects with other people/friends setting a policy for each owned thing: private, public, moderated, etc. This means that I can share an save money by gaining access to data I would normally have to pay for. The mind boggles.
Urbanflow envisions a new interface and operating system for cities. Urbanflow creates a more efficient, transparent relationship between city administrators and citizens – via real time data. Urban screens show locally-oriented and general purpose data in easy to use interfaces that help with all sorts of everyday activities from finding your way to getting info on energy, weather, traffic, public transport, and more. Citizens can also report anything from an event to a pothole in the city. The same urban screen shows contextual, hyperlocal information as well as broader, citywide content, allowing users to peek around walls and across the city. For officials and administrators this means making the city more transparent and efficient to manage through immediate feedback from the city’s residents. Watch the Urbanflow Helsinki Intro.