<div dir="ltr">On Sun, Aug 17, 2008 at 11:47 AM, Bastien <<a href="mailto:bastienguerry@googlemail.com">bastienguerry@googlemail.com</a>> wrote: <div class="gmail_quote"><blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;"> Hi Bill, <div class="Ih2E3d"> "Bill Kerr" <<a href="mailto:billkerr@gmail.com">billkerr@gmail.com</a>> writes: </div><div class="Ih2E3d">> She relates this to a philosophical divide in approaching maths dating > back to Whitehead and Russell's Principia Mathematica, in which all > maths is reduced to logic. Hence the contrast between and intuitive > and constructive approach on the one hand compared with a rule driven > and formal approach on the other. </div>Very clear, thanks. <div class="Ih2E3d"> > So, Cynthia, is reporting on Papert's views that the content of school > maths needs to be changed as well as the process of how it is taught > (computers offering new opportunities here). > > I think that's a very strong and positive feature of her book, that > she situates the discussion as part of a historical and philosophical > debate on the nature of maths. </div>I should definitely read her book! <div class="Ih2E3d"> > So, the answer to your question is that it's about both the nature of > mathematics and pedagogy but arising from Papert's view of the nature of > mathematics (contrasted to the purely logical, rule driven approach) and that > the learning of mathematics could be structured better to fit the natural ways > by which children learn, as discovered by Piaget. </div>Another bit of personal cautiousness here. Education is both about what teachers should teach and what children can learn.  The first side is partly conventional and partly adapted to the second one.  The scientific inquiry on how the mind works might provide new ways of understanding how student naturally learn, but it does not provide directly with /what/ they should learn, even less with what and how the teachers should teach. I'm mentioning this because sometimes, when we think of the natural ways of learning, we more or less implicitly infer that the content of what we teach and the methods of teaching should consequently evolve.  Which is partly obvious and partly misleading: it's obvious in the sense that, if you discover that your teaching methods are not efficient for you goals, then you should amend them; but it's also misleading because it masks the role played by the conventions in deciding what and how you want to teach.  As if the arbitrary parts of conventions should dissolve themselves by the sheer discovery of the "natural ways" of learning... (And I was not arguing with what you said, I just took the opportunity of saying what came to my mind!) Ok, rant's over :) -- Bastien</blockquote><div> As well as the practical and traditional limitations imposed by existing school systems I think alan kay's description of the "non universals" provides a good counterbalance to the natural learning approach </div></div><ul><li>reading and writing</li><li><a class="wiki_link" href="http://learningevolves.wikispaces.com/nonUniversals#maths">deductive abstract mathematics</a></li><li><a class="wiki_link" href="http://learningevolves.wikispaces.com/nonUniversals#science">model based science</a></li> <li>equal rights</li><li>democracy</li><li>perspective drawing</li><li><a class="wiki_link" href="http://learningevolves.wikispaces.com/nonUniversals#thHarmony">theory of harmony</a></li><li><a class="wiki_link" href="http://learningevolves.wikispaces.com/nonUniversals#simDiff">similarities over differences</a></li> <li>slow deep thinking</li><li>agriculture</li><li>legal systems</li></ul> "What's interesting is to look for things that are not universal, that seems to have some importance as well. Most people have lived and died on this Earth for 100,000 years without reading and writing, without having deductive maths and model based science .... (goes through non universals list) These are a little harder to learn than the ones on the left because we are not directly wired to learn them. These things are actually inventions which are difficult to invent. And the rise of Schools going all the way back to the Sumerian and Egyptian times came about to start helping children learn some of these things that aren't easy to learn. It can be argued that if you are trying to be utopian about education what we should be doing is helping the children of the world learn these hard to learn things. Equal rights is a really good one to help children learn. No culture in the world is particularly good at it" - transcribed from alan kay's europython 2006 keynote more about this at <a href="http://learningevolves.wikispaces.com/nonUniversals">http://learningevolves.wikispaces.com/nonUniversals</a> (work in progress) cheers </div>