[Its.an.education.project] Fw: [Etoys] Some thought about Operational thinking and Squeak

Wed May 7 08:53:10 CEST 2008

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.

On Tue, May 6, 2008 at 7:07 PM, Alan Kay <alan.nemo at yahoo.com> wrote:

> I was comparing Bruner to Piaget ...
>
> ----- Original Message ----
> From: Edward Cherlin <echerlin at gmail.com>
> To: Alan Kay <alan.nemo at yahoo.com>
> Cc: Education <its.an.education.project at tema.lo-res.org>
> 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 <alan.nemo at yahoo.com> 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 <echerlin at gmail.com>
> >
> > On Sun, May 4, 2008 at 12:55 AM, Hilaire Fernandes <hilaire at ofset.org>
> > wrote:
> >  >
> >
> http://blog.ofset.org/hilaire/index.php?post/2008/05/01/Operational-thinking
> >
> > 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
> >  >
> >  >  --
> >  >  http://blog.ofset.org/hilaire
> >  >  _______________________________________________
> >  >  Etoys mailing list
> >  >  Etoys at lists.laptop.org
> >  >  http://lists.laptop.org/listinfo/etoys
>
>
> --
> Edward Cherlin
> End Poverty at a Profit by teaching children business
> http://www..EarthTreasury.org/ <http://www.EarthTreasury.org/>
> "The best way to predict the future is to invent it."--Alan Kay
>
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