<div dir="ltr">On Sun, Aug 17, 2008 at 1:32 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;"> "Bill Kerr" <<a href="mailto:billkerr@gmail.com">billkerr@gmail.com</a>> writes: >   • intuition [...] >   • different ways of looking at maths (constructive and intuitive compared <div class="Ih2E3d">>     with rule driven and formal) </div>[...] >   • other mathematicians who hold similar views - Poincare, Brouwer, Godel) I'd be curious on how Cynthia relates mathematical theories (like intuitionism) to pedagogical theories.  What is the "similar views" that Poincaré, Brouwer and Gödel are holding?  Is that views about pedagogy or views about mathematics (namely intuitionism)? Can you tell me more about this? (or send me pointers?) Thanks! -- Bastien</blockquote><div> hi Bastien, Cynthia is reporting on Papert's ideas about deficiencies in School maths. eg. that focusing on number is just one way of doing maths and that there may be other ways that tap into more natural ways of children learning, eg. body syntonic turtle geometry. 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. 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. Some pointers, apart from Cynthia's book: Mindstorms by Papert, Chapter 6: Powerful Ideas in Mind-size Bites illustrates that mathematical intuition is quite central to Papert's thinking Godel, Escher, Bach by Hofstadter has several are references to intuition, programming intuition and Godel and introspection Minsky's The Emotion Machine pursues these ideas in the AI field and includes a section on Poincare (7-7 Poincare's Unconscious Processes), Minsky's book is available on line at his MIT site Brouwer - I don't know, I was just quoting from Cynthia's book here 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></div> I read Edwards reply too, which seems to be at odds with this thinking cheers </div>