Summarizing some threads (KIrby again)...
Those of you frequenting this list for some years will recognize most of these themes. From time to time I like to archive a summary. Principal themes: I. Math Objects (an approach to learning math) II. Objects First (an approach to learning programming) These two go hand-in-hand. Math Objects are traditional concepts such as polynomials, polyhedra, vectors, integers, treated as Types of Thing, i.e. we're making math concepts concrete by distilling the "things" or "types" people have invented over the centuries. One place to begin, familiar to computer science, is to differentiate alpha from numeric types. Objects First means taking the object-oriented philosophy seriously, meaning we're mining everyday (ordinary) human language semantics, wherein we already think in terms of named things (nouns) having behaviors (verbs) and attributes (adjectives). My curriculum anchors Objects (things) in the biological world of biota, animals, creatures, flora and fauna. Then we move to the more abstract types of object of interest in mathematics, polyhedra especially because these are also visible and tangible, forming a bridge to that biological world. Python is especially cool as an OO language because when building a biological creature as a template, one has these special names that look somewhat like __ribs__. The methods stack up providing a backbone or rack of ribs i.e. there's a visual analogy to a creature, a snake in particular, right in the language itself. The Objects First approach doesn't buy into the "ontogeny recapitulated phylogeny" ideology, by which I mean: just because programming languages evolved a certain way doesn't mean newcomers have to traverse the discipline in that same order. Regions new to telephony don't need to install land lines before they go with cell phones -- go straight to cell (straight to OO). Another theme: III. streamlining the teaching of spatial geometry I've separated this last theme out of the mix because it's what sets me apart more than the above and makes me a marginal figure, apparently off my rocker in some way. I passionately believe that we should be taking greater advantage of the streamlining done by the geodesic dome guy, Bucky Fuller, regarding how to compact a lot of geometric information into a compressed data structure he named the concentric hierarchy of polyhedra (meaning you include them inside each other, sort of like Russian dolls -- not a new idea, but the devil is in the details). I won't go into some verbose presentation of III in this post. However I do think when you move from calculators to full fledged computers, then it's time to get off the plane and start taking advantage of those much bigger and more colorful screens. So even if you're highly skeptical of the Bucky Fuller bit, you might stay with me on this notion the polyhedra and spatial geometry will naturally come into vogue as we move beyond calculators and start taking more advantage of computers. I've invested many years developing these ideas and presenting them in cogent form. The materials are open source and on the Internet. Again, it's III that makes me moves me into the "esoteric" category, where I start questioning only using a Euclidean set of axioms, start taking up a "geometry of lumps" and making all sorts of high level connections to Karl Menger (dimension theorist) and Ludwig Wittgenstein (philosopher). I also tend to get polemical, as a lot of positive futurism attaches here, and to the extent the world seems unnecessarily hellish, I get exercised about wasting already stockpiled assets that might make a big positive difference. I inherited this long-running campaign from an earlier generation and have a lot of loyalty to some of my mentors in this area, including but not limited to Bucky Fuller himself. Kirby
Yes, you're on the right track. All of schooling is based on medieval metaphors (such as the lecture, originally for purposes of dictation in the days before printing) and the history of thought, instead of the logic of the subject matter or of the reasons for learning anything. We are still teaching pre-calculator, pre-computer math as though there was some special virtue in hand calculation with paper and pencil, instead of recognizing that the days of the counting house are over and are not coming back. (There is a virtue in understanding how arithmetic works. It is better acquired by having students teach computers how to do it, that is, by learning how to program accurately and effectively.) Also, geometry is not the only realm where we need to take a fresh look. All of math has been restructured in the last century in terms not just of objects, but of systems. Axiom systems, structures, symmetries, relationships between seemingly different branches of math, or math and an application area such as physics or crytography or...we don't even know what, yet. Category theory attempted to generalize everything by looking at systems of objects and mappings between them. It has been superseded by the even more general topose theory, which starts from your lumpengeometrie and turns into a Theory of Nearly Everything in math. I will not attempt to explain this all today, because I am supposed to be writing a book. More later. On Fri, Oct 23, 2009 at 16:11, kirby urner <kirby.urner@gmail.com> wrote:
Those of you frequenting this list for some years will recognize most of these themes. From time to time I like to archive a summary.
Principal themes:
I. Math Objects (an approach to learning math) II. Objects First (an approach to learning programming)
These two go hand-in-hand.
Math Objects are traditional concepts such as polynomials, polyhedra, vectors, integers, treated as Types of Thing, i.e. we're making math concepts concrete by distilling the "things" or "types" people have invented over the centuries. One place to begin, familiar to computer science, is to differentiate alpha from numeric types.
Objects First means taking the object-oriented philosophy seriously, meaning we're mining everyday (ordinary) human language semantics, wherein we already think in terms of named things (nouns) having behaviors (verbs) and attributes (adjectives).
My curriculum anchors Objects (things) in the biological world of biota, animals, creatures, flora and fauna. Then we move to the more abstract types of object of interest in mathematics, polyhedra especially because these are also visible and tangible, forming a bridge to that biological world.
Python is especially cool as an OO language because when building a biological creature as a template, one has these special names that look somewhat like __ribs__. The methods stack up providing a backbone or rack of ribs i.e. there's a visual analogy to a creature, a snake in particular, right in the language itself.
The Objects First approach doesn't buy into the "ontogeny recapitulated phylogeny" ideology, by which I mean: just because programming languages evolved a certain way doesn't mean newcomers have to traverse the discipline in that same order. Regions new to telephony don't need to install land lines before they go with cell phones -- go straight to cell (straight to OO).
Another theme:
III. streamlining the teaching of spatial geometry
I've separated this last theme out of the mix because it's what sets me apart more than the above and makes me a marginal figure, apparently off my rocker in some way.
I passionately believe that we should be taking greater advantage of the streamlining done by the geodesic dome guy, Bucky Fuller, regarding how to compact a lot of geometric information into a compressed data structure he named the concentric hierarchy of polyhedra (meaning you include them inside each other, sort of like Russian dolls -- not a new idea, but the devil is in the details).
I won't go into some verbose presentation of III in this post. However I do think when you move from calculators to full fledged computers, then it's time to get off the plane and start taking advantage of those much bigger and more colorful screens. So even if you're highly skeptical of the Bucky Fuller bit, you might stay with me on this notion the polyhedra and spatial geometry will naturally come into vogue as we move beyond calculators and start taking more advantage of computers.
I've invested many years developing these ideas and presenting them in cogent form. The materials are open source and on the Internet.
Again, it's III that makes me moves me into the "esoteric" category, where I start questioning only using a Euclidean set of axioms, start taking up a "geometry of lumps" and making all sorts of high level connections to Karl Menger (dimension theorist) and Ludwig Wittgenstein (philosopher).
I also tend to get polemical, as a lot of positive futurism attaches here, and to the extent the world seems unnecessarily hellish, I get exercised about wasting already stockpiled assets that might make a big positive difference. I inherited this long-running campaign from an earlier generation and have a lot of loyalty to some of my mentors in this area, including but not limited to Bucky Fuller himself.
Kirby _______________________________________________ Edu-sig mailing list Edu-sig@python.org http://mail.python.org/mailman/listinfo/edu-sig
-- Edward Mokurai (默雷/धर्ममेघशब्दगर्ज/دھرممیگھشبدگر ج) Cherlin Silent Thunder is my name, and Children are my nation. The Cosmos is my dwelling place, the Truth my destination. http://www.earthtreasury.org/
On Fri, Oct 23, 2009 at 5:51 PM, Edward Cherlin <echerlin@gmail.com> wrote:
Yes, you're on the right track. All of schooling is based on medieval metaphors (such as the lecture, originally for purposes of dictation in the days before printing) and the history of thought, instead of the logic of the subject matter or of the reasons for learning anything. We are still teaching pre-calculator, pre-computer math as though there was some special virtue in hand calculation with paper and pencil, instead of recognizing that the days of the counting house are over and are not coming back. (There is a virtue in understanding how arithmetic works. It is better acquired by having students teach computers how to do it, that is, by learning how to program accurately and effectively.)
That's an interesting point about how form trumps substance. Students would show up with note books and attempt to transcribe verbatim as the lecturer spoke slowly and pedantically, at a bandwidth suitable for keeping up. Turning to the board and writing some of the notes is an exercise in transcription: the notebooks would all contain those same notes from the teacher. Fast forward to 'Sesame Street' and you get video shorts, basically YouTube clips, on the numbers 1-12 and letters A-Z -- a large and growing database with each TV show containing a mix of old and new material. Repetition is not a bad thing. Changes in topic, a kind of disjointed approach, requires viewers to connect the dots in their own minds. In philosophy, we have the aphoristic writers such as Nietzsche (in Will to Power especially) and Wittgenstein (the investigations). This is early hypertext in the sense that the reader needs to make links, find meaning in the relationships. The number of edges (relationships) between N vertices is N(N-1)/2 i.e. I shake hands with every other person here except me, but then me shaking hands with you, and you with me, is the same edge, so divide by 2. The so-called MTV generation is used to short aphoristic video clips with implied hyperlinks. When they encounter the chalk 'n talk lecture style of say Renaissance Italy (University of Bologna) it seems like a drastic drop in bandwidth sometimes. Ergo, I think to regalvanize computer science, other technical subjects, we need to put less emphasis on lecture and more on videography, computer animation. It's a positive feedback cycle in that the skills required to make these video presentations are likewise the skills we seek to transmit e.g. how to apply a rotation matrix to a set of vectors from the origin, to make a given polyhedron rotate on screen. This could be your high school math class, with vectors, polyhedra expressed as objects in Python.
Also, geometry is not the only realm where we need to take a fresh look. All of math has been restructured in the last century in terms not just of objects, but of systems. Axiom systems, structures, symmetries, relationships between seemingly different branches of math, or math and an application area such as physics or crytography or...we don't even know what, yet. Category theory attempted to generalize everything by looking at systems of objects and mappings between them. It has been superseded by the even more general topose theory, which starts from your lumpengeometrie and turns into a Theory of Nearly Everything in math.
A key word here is accessibility. The math pipeline is considered broken in that relatively few students stick with it or choose technical fields, thanks to feeling turned off in high school. Turtle Art is supposed to help fix this. Ray tracing could be considered like Turtle Art in that each beam of light is the path of some turtle. Front loading with better eye candy and better audio, returning music to the math curriculum (rhythm as fractions, intervals, the various scales and their frequencies) is a way to keep the bandwidth closer to what students are getting from television. Teachers may object that this is just fluff, pandering to attention spans made short by television. This is where computer programming comes in, requiring concentration and focus. The math lab is closer to an art studio. Students build a portfolio, using open source tools. VRML (x3D), Anti-Prism by Adrian Rossiter, Springie by Tim Tyler, your Turtle Art, Gregor's turtle module, Qhull, various Java applets. Speaking of Java applets, here's an interesting one in that it's written in Java but then run through GWT (Google Widgets Toolkit) so that what's running in your browser is actually Javascript. You don't need Java runtime to use this: http://interisland.net/johngilbrough/Space/ (features different ways of achieving stereo) The teacher still has a role of course, as lesson planner, transmitter of skills, tour guide to this huge world of free resources. A teacher is but another student further along in mastery, role modeling life-long learning. The ideal math lab has a projector, is connected to the Internet. Students take turns giving presentations as well. Show and Tell. Lightning talks.
I will not attempt to explain this all today, because I am supposed to be writing a book. More later.
Looking forward.
I passionately believe that we should be taking greater advantage of the streamlining done by the geodesic dome guy, Bucky Fuller, regarding how to compact a lot of geometric information into a compressed data structure he named the concentric hierarchy of polyhedra (meaning you include them inside each other, sort of like Russian dolls -- not a new idea, but the devil is in the details).
A key innovation is taking a regular tetrahedron as one's unit of volume, thinking in terms of "tetravolumes". Instead of the messy irrational volumes we're used to teaching about, this compressed data structure supplies easily memorable whole number volumes for many basic polyhedra. This isn't about replacing or displacing the traditional approach (based on the unit-volume cube). It's simply mind-expanding and informative to realize one's freedoms. Here's a whole different approach, like some Math from Mars. You could even teach it that way for marketing purposes: Martian Math. I explain this more here, noting that all the graphics were done using a combination of Python and the free open source ray tracing program POV-ray. These pages go back a decade by now, interpreting source material from the late 1970s: http://www.grunch.net/synergetics/volumes.html http://www.grunch.net/synergetics/volumes2.html More background: http://www.4dsolutions.net/ocn/numeracy0.html (the first of a four part series, all developed in Python). Kirby
I won't go into some verbose presentation of III in this post. However I do think when you move from calculators to full fledged computers, then it's time to get off the plane and start taking advantage of those much bigger and more colorful screens. So even if you're highly skeptical of the Bucky Fuller bit, you might stay with me on this notion the polyhedra and spatial geometry will naturally come into vogue as we move beyond calculators and start taking more advantage of computers.
I've invested many years developing these ideas and presenting them in cogent form. The materials are open source and on the Internet.
Again, it's III that makes me moves me into the "esoteric" category, where I start questioning only using a Euclidean set of axioms, start taking up a "geometry of lumps" and making all sorts of high level connections to Karl Menger (dimension theorist) and Ludwig Wittgenstein (philosopher).
I also tend to get polemical, as a lot of positive futurism attaches here, and to the extent the world seems unnecessarily hellish, I get exercised about wasting already stockpiled assets that might make a big positive difference. I inherited this long-running campaign from an earlier generation and have a lot of loyalty to some of my mentors in this area, including but not limited to Bucky Fuller himself.
Kirby _______________________________________________ Edu-sig mailing list Edu-sig@python.org http://mail.python.org/mailman/listinfo/edu-sig
-- Edward Mokurai (默雷/धर्ममेघशब्दगर्ज/دھرممیگھشبدگر ج) Cherlin Silent Thunder is my name, and Children are my nation. The Cosmos is my dwelling place, the Truth my destination. http://www.earthtreasury.org/
participants (2)
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Edward Cherlin -
kirby urner