Ed, do you have any more information on your Kindergarten Calculus outline?  I'd love to help you flesh out the idea as a Sugar activity. <div class="gmail_quote">On Tue, May 6, 2008 at 7:07 PM, Alan Kay <<a href="mailto:alan.nemo@yahoo.com">alan.nemo@yahoo.com</a>> wrote: <blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;"><div><div style="font-family: times new roman,new york,times,serif; font-size: 12pt;"><div style="font-family: times new roman,new york,times,serif; font-size: 12pt;"> I was comparing Bruner to Piaget ... <div style="font-family: times new roman,new york,times,serif; font-size: 12pt;"><div class="Ih2E3d">----- Original Message ---- From: Edward Cherlin <<a href="mailto:echerlin@gmail.com" target="_blank">echerlin@gmail.com</a>> </div><div><div></div><div class="Wj3C7c">To: Alan Kay <<a href="mailto:alan.nemo@yahoo.com" target="_blank">alan.nemo@yahoo.com</a>> Cc: Education <<a href="mailto:its.an.education.project@tema.lo-res.org" target="_blank">its.an.education.project@tema.lo-res.org</a>> Sent: Tuesday, May 6, 2008 10:43:36 AM Subject: Re: [Its.an.education.project] Fw: [Etoys] Some thought about Operational thinking and Squeak On Tue, May 6, 2008 at 8:45 AM, Alan Kay <<a href="mailto:alan.nemo@yahoo.com" target="_blank">alan.nemo@yahoo.com</a>> wrote: > > Good "real education" comments here by Ed Cherlin. Thank you. > By the way, Jerome Bruner's more comprehensive (and nicely more diffident) When I write research papers, I will be diffident about drawing conclusions. When I teach Buddhism, I always remember not to teach people unless they ask. (The old rule is that they must ask three times.) I don't do advocacy about life-and-death matters like education that way. > approaches to thinking about these matters have been very helpful over the > years. He inspired my mother, and she passed it on to me. "Any subject can be taught effectively in some intellectually honest form to any child at any stage of development." > Also, 6 year olds can do a very important and useful version of calculus > using both shapes, growth, numbers and simple arithmetic (cf. my > descriptions of how 1st grade teach Julia Nishijima invented and pulled off > this approach). There is almost nothing about her on the Net other than your description of her work. How can I contact her, and where can I find out more about her approach to calculus? Do you know why nobody else seems to be doing this? > The key idea here (that Montessori understood much more deeply than Piaget) > is that, for children (and most other learners) situated context really > makes huge differences in what seems to be comprehensible and can be > operated with.  One part of SC is user interface design ... but there is > much more to this ... Context makes a huge difference in what can even be noticed. The classic experiment is to put people in a room with a table and chair and a No Smoking sign on the wall, and instruct them to do something requiring string. Hardly anybody finds the string holding up the sign. If the same string holds up an empty frame, almost everybody finds it. One of the most important historical examples is the unintentional Sapir-Whorf experiment that resulted from the controversy between Newton and Leibniz over priority in the discovery of caculus. British mathematicians refused to use the Leibniz d notation (dx/dt), and continued with Newton's cumbersome dot notation (ẋ). As a result British contributions to analysis were nearly nil for more than a century. Charles Babbage founded the Analytical Society in 1816 to "replace the dot-age of Cambridge with the pure d-ism of the Continent." He and his friends succeeded in getting the notation used in Cambridge math exams changed, and British contributions to analysis soared. The Sapir-Whorf Hypothesis, that language sets limits on what people can think, remains controversial in linguistics, but is accepted as a given in mathematics. "By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and, in effect, increases the mental power of the race."--Alfred North Whitehead. He then goes on to give examples, particularly of the digit '0', which began as a notational convenience for representing numbers and later turned into the number 0, resulting in the development of whole new branches of algebra. The limits that language and notation place on thought are not absolute, because we can create new language, and new math notations, with which to construct a new understanding by expressing what we could not say or think before, and because a new notation can suggest new ideas to us. It generally turns out that the new ideas can be expressed in the old language once we know what we are trying to say. Some of Gauss's most impenetrable theorems, from the point of view of his contemporaries, came about because he used complex numbers to discover them, but presented his proofs in real numbers alone. I have constructed my own mental model of education from a variety of materials, including a number of such critical experiments and historical examples. Unfortunately, most discourse on education assumes what is to be investigated and what is to be left out. The current context of No Child Left Behind seems to make it impossible to discuss anything real in education in public, and makes it easy to sideline as "out of the mainstream". This appears to be one of the intentions of the program. I notice that the Social Conservatives have been implacable enemies of Bruner's approach for decades. > Cheers, > > Alan > > ----- Forwarded Message ---- > From: Edward Cherlin <<a href="mailto:echerlin@gmail.com" target="_blank">echerlin@gmail.com</a>> > > On Sun, May 4, 2008 at 12:55 AM, Hilaire Fernandes <<a href="mailto:hilaire@ofset.org" target="_blank">hilaire@ofset.org</a>> > wrote: >  > > <a href="http://blog.ofset.org/hilaire/index.php?post/2008/05/01/Operational-thinking" target="_blank">http://blog.ofset.org/hilaire/index.php?post/2008/05/01/Operational-thinking</a> > > It would be much easier to evaluate this contribution if it included > specific examples. > > I have been working on some examples in DrGeo, and I disagree with the > author on its unsuitability. Certainly you can't expect children to > discover much with DrGeo if left entirely to their own devices. The > question is what guidance the teacher gives to the child in discovery. > > I can build geometric models to illustrate a wide variety of concepts, > and then let children vary the diagram in many ways to see which > relationships remain the same through all variations. For example, > take any triangle and connect the midpoints to divide it into four > smaller triangles. What are the necessary relationships among them? If > you move any vertex of the original triangle, you change its shape and > size. What of the four smaller triangles? Which relationships change, > and which remain constant? > > Similarly for many other constructions, and for symmetries, > tesselations, and other forms that lead to fundamental concepts of > math and science. We will not teach primary schoolers the details of > Emmy Noether's theorem that every symmetry in physics is equivalent to > a conservation law, but we can and should lay the groundwork for a > deeper understanding of this essential discovery at an appropriate > age. > > I have the outline of a practical Kindergarten Calculus program, in > which we would teach concepts visually without the algebraic and > numerical apparatus that is essential for calculus calculations. It > can all be done in DrGeo, as well as with physical objects. > > The deepest understanding in math and physics, and in many other > areas, comes when we can see and use two or more representations of > the same ideas, and also see why they are equivalent, and how to turn > any of them into the others. The whole recent proof of Fermat's Last > Theorem came down to an instance of this called the Taniyama-Shimura > conjecture, now proven as the Modularity Theorem, that all elliptic > curves over the rational numbers are modular. This gives us mappings > between three realms: elliptic curves, modular functions, and > L-series, that were once seen as quite distinct. We can't even explain > what the theorem is about to young children, or even to most adults, > but we can show them other such mappings within geometry and > arithmetic. Analytic geometry, connecting numbers and images, is one of those most often taught in secondary schools. > It turns out that in physics, it is necessary to connect the two quite > different realms of mathematical models and experimental results in a > fairly specific way in order to have an effective theory. One of the > greatest and at the same time most familiar and most misunderstood > examples is how the shift from Galilean to Einsteinian relativity, > based on the single painstakingly tested experimental result that the > speed of light is the same for all observers, requires the equivalence > of mass and energy. > > If any of this fails to make sense to you, I recommend that you look > on that fact as a sign of some of the greatest failings in > conventional education. For anybody who would like an explanation of > any of this, I can answer some questions and refer to to excellent > published expositions for many more. I will not attempt to walk your > through the proofs, but I can demonstrate the relationships I > describe. > > What we mostly don't have is a path by which children can be guided to > discover much of this themselves. But we have bits and pieces of that > path in work by Alan Kay, Seymour Papert and many others. I have > thought of a few other bits that I hope will add to the enterprise > when I get a chance to work them out in more detail and try them out > on children. > > I think that the hard question is how to get teachers to discover > enough of this to be able to use is effectively. Nobel laureate > Richard Feynman said that we don't really understand a subject unless > we can create freshman lecture on it. Mathematicians suggest trying to > explain ideas to your grandmother. I propose that we find out how much > of what we think we know we can explain to children and to teachers. > >  >  Hilaire >  > >  >  -- >  >  <a href="http://blog.ofset.org/hilaire" target="_blank">http://blog.ofset.org/hilaire</a> >  >  _______________________________________________ >  >  Etoys mailing list >  >  <a href="mailto:Etoys@lists.laptop.org" target="_blank">Etoys@lists.laptop.org</a> >  >  <a href="http://lists.laptop.org/listinfo/etoys" target="_blank">http://lists.laptop.org/listinfo/etoys</a> -- Edward Cherlin End Poverty at a Profit by teaching children business <a href="http://www.EarthTreasury.org/" target="_blank">http://www..EarthTreasury.org/</a> "The best way to predict the future is to invent it."--Alan Kay </div></div></div></div></div> <hr size="1"><div class="Ih2E3d">Be a better friend, newshound, and know-it-all with Yahoo! Mobile. <a href="http://us.rd.yahoo.com/evt=51733/*http://mobile.yahoo.com/;_ylt=Ahu06i62sR8HDtDypao8Wcj9tAcJ" target="_blank"> Try it now.</a></div></div> _______________________________________________ Its.an.education.project mailing list <a href="mailto:Its.an.education.project@tema.lo-res.org">Its.an.education.project@tema.lo-res.org</a> <a href="http://lists.lo-res.org/mailman/listinfo/its.an.education.project" target="_blank">http://lists.lo-res.org/mailman/listinfo/its.an.education.project</a> </blockquote></div>