Mailman 3 March 2001 - Edu-sig

Re: switching gears (and how)
by Lloyd Hugh Allen 07 Mar '01

07 Mar '01

Arthinator(a)aol.com wrote: > Kirby writes: > >measuring cups) -- although that could be considered > >Pythagorean, as I hear the school was really into beans > >somehow. Not to make a post completely unrelated to python, but: The Pythagoreans were vegetarian, except that they did /not/ eat beans, being as how when you pop a bean open it resembles a foetus. Now if only I had a footnote for you. ------------------------ Re: Group Thery: One of my colleagues at Ohio State, Jason-whose-last-name-evades-me-right-now, had a neat explanation of group theory in terms of rotating tires--each permutation is an element of the group. To swing it back to Python, could dictionaries and tuples be used to represent groups? Had an image for a second there, but now all of the sudden it's gone. Guess I'll leave the implementation to someone else or another day.

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In praise of VPython
by Arthur_Siegel＠rsmi.com 06 Mar '01

06 Mar '01

Having instigated some unpleasantness on the list, I feel some special obligation, perhaps, to try to be as loud about what I see as positive developments as I have been about what I see as negative. To those who might see it as OT, my apologies. VPython is exciting. The more I understand its design, intentions, and implementation the more appreciative I become. Jason Cunliffe had tried to give me a clue that I was missing something. I had been guilty of selling it short as another Python scripting add-on to a graphic C/C++ app/renderer. In fact it is what its original name implied it was going to be - visual Python. Designed out of of the box by folks with a deep understanding and appreciation of Python and Numeric. Having gotten used to working with Python add-on scripting and having to write contorted Python to get the results I was looking to achieve , it took me a while to catch on to the fact that all VPython needs and wants is good, concise Python - no backbends necessary. For example, the complete "from scratch" construction of a beautfully rendered sphere from a VPython demo (there is a sphere primitive, the following actually being the pedagogical long way around in VPython) from visual import * L = [ ] for t in arange(0,2*pi,0.2): for s in arange(0,pi,0.1): L.append((cos(t)*sin(s), sin(t)*sin(s), cos(s))) convex(pos=L, color=(0.5,0,0.5)) Concise mathemathical notation for an analytical sphere, and at the same time the sphere itself - again,, beautifully rendered. >From my take, much of what I had hoped to see achieved in concrete form in connection with edu-sig comes together with the existence VPython. Hope those out in the field - teachers designing Python based curriculum - begin to take advantage of what it has to offer. ART

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re: Switching gears
by Arthinator＠aol.com 05 Mar '01

05 Mar '01

A previous contribution to the switching gears thread, I find, was returned undelivered: > edu-self(a)python.org > unknown local-part "edu-self" in domain "python.org" Either my ego showing, or spending too much time this week with Python/VPython. Kirby writes: >I'm not sure to what extent our thinking diverges outside >some set of concepts related to geometry. At this point, >I don't have an overview of how you'd approach inter- >connecting math topics outside of Euclid's Elements and >projective geometry (neither of which I have any problem >including, which isn't to say I've achieved the level of >mastery you have vis-a-vis either). I can promise you I have no level of mastery of over Euclid or anything connected with mathematics. One does not master subjects one comes to at the age of 40 - especially when one is also making a living and raising a family. I am an enthusiastic dabbler. >In some ways, I'm trying to recover classical topics that >used to be covered as a matter of course. Today with the >1st graders it was all about greek roots (tetra, octa... >hedron) and chatter about the Platonic Five -- what sets >them apart from the others etc. The only non-classical >part was the bean-fill (using lima beans and polyhedra as >measuring cups) -- although that could be considered >Pythagorean, as I hear the school was really into beans >somehow. I have lost passion for the argument, but do think it would be ruder not ot respond at all. Kirby, you know as well as I that Book I Prop I before Book I Porp II, and well before Book XI Prop I (where we first see solids) is at the very essence of Euclid. Not making Euclid sacred, but the deductive process does have some "sacredness" IMO. David Joyce's take at: http://aleph0.clarku.edu/~djoyce/java/elements/elements.html closely resembles my own. My own education as an adult started when I was able to suppress my pride, and study the old hat - the older the better. Won't be finding my way back to the current frontiers of human knowledge in this particular life-time. So it goes. But am looking to accelerate the process with whatever tools I can get my hands on. ART

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Re: [Edu-sig] Hitting two targets: OO + group theory (pedagogy)
by Kirby Urner 04 Mar '01

04 Mar '01

At 06:07 PM 03/02/2001 -0800, you wrote: > >============== OREGON CURRICULUM NETWORK ===================== > >RE: HITTING TWO TARGETS: OO + MATH TOPIC Note: this had some terminological problems i.e. no such thing as a 'relative prime', only integers 'relatively prime' to others. Plus there's no need to stipulate the modulus be a prime -- in fact one of my examples on math-learn used a modulus of 20. Thanks to David Ullrich on sci.math for these tips, plus urging me to promote Group to a superclass so that we might subclass _specific_ kinds of group, like these groups of coprimes, as subclasses thereof. I have since incorporated these suggestions into my draft code, with an eye to doing another web essay later. Kirby

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Hitting two targets: OO + group theory (pedagogy)
by Kirby Urner 02 Mar '01

02 Mar '01

============== OREGON CURRICULUM NETWORK ===================== RE: HITTING TWO TARGETS: OO + MATH TOPIC OO + POLYHEDRA In earlier writings, I've discussed how object-oriented (OO) syntax might be melded with math notations with regard to polyhedra.[see links] A Polyhedron superclass, subclassed as Tetrahedron, Octahedron etc., would demonstrate the concepts of class hierarchy and instantiated selves, or objects. Many intro-to-OO books use already flatlander shapes (superclass: shape; subclasses: triangle, circle, square...). However the Polyhedron case is mathematically more interesting and takes better advantage of the computer's graphical display capabilities. Rotation, translation and scaling may be implemented using matrices, or we might swap in quaternions to accomplish rotation, or we might develop use some other Clifford Algebra ala Hestenes et al. OO + GROUP THEORY More recently, I've been looking to group theory as another paradigm case for hitting two targets with one stone: OO concepts and number-theoretic math concepts. Each set of concepts helps to reinforce the other, thereby bootstrapping students into a synergetic mindset which sets the stage for elaboration both in programming and in abstract mathematics. Whereas this might at first sound like a college level approach, my contention is that newcomers to programming will leap-frog those who cut their teeth in the hay day of procedural programming -- which doesn't mean the procedural approach has gone out of style, only that it's no longer relevant to consider the object-oriented as "advanced", with mastery of earlier forms a prerequisite. Functional programming methods will also make their debut earlier in a student's career. And group theory has always been accessible at a variety of levels. It makes sense to start phasing in some of these concepts early, so that later plantings of these ideas find fertile ground. PYTHON AS INTRO LANGUAGE The computer language of choice in these explorations is Python. We want a language which is interactive at the command line, like Logo, such that programmed methods may be accessed and combined on the fly, with the screen itself serving as a principal site for i/o. Whereas we'll want the option to read in text files for processing, there should be no demand on students to compile standalone executables which must either support their own GUI for interactivity, or rely on command line arguments pointing to files containing the raw data of interest. ISETL, DrScheme, APL and others share these positive features with Python and there's no stipulation here that only Python is the appropriate vehicle for these kinds of activities (given the OO focus, some of these others do not lend themselves to hitting our double-target) -- only that this author has found the Python environment highly congenial for this kind of undertaking. Let's define our sandbox to be groups of relative primes less than a particular prime modulus n. These are groups under multiplication, where a*b is defined as (a*b) mod n. In Python, the mod operator is %, such that a%b returns the remainder a (mod b). SYNTAX BASICS The primitive built-in function pow() takes a mininum of two arguments a,b such that pow(a,b) = a^b (or a**b in Python, as exponentiation is signified with a double-asterisk). The optional 3rd argument provides a modulus, such that pow(a,b,n) = (a**b) mod n. This 3-argument form of pow() is internally optimized such that the modulus is used to reduce the product as powering proceeds -- far more efficient than applying the syntax (a**b)%n, which can take indefinitely longer to complete for large numbers. Where object-oriented syntax shines in this application is when we go to define multiplication. We're able to override the meaning of * (multiplication operator) in such a way that operator.mul will likewise behave consistently, i.e. will respect the definition we provide for multiplication. This is done be defining __mul__ in the class definition. Similar overriding is done with respect to ** and pow(), by defining __pow__, and for relational operators <, >, and ==, by defining __lt__, __gt__, and __eq__ respectively. I won't go into a full-blown dissection of source code here, but rather will conclude with an interactive command line session which gives the flavor of the final result. In class, or over the web (if a distance learning opportunity), students would be boning up on Python and some of the basics of group theory via supplementary materials. Some of the preliminary prototypical writing is available in draft form on math-learn, an egroup focussed on K-12 math education, with emphasis on the middle school years. [see links below] Further forays into group theory + Python may be found in the 'For further reading' section of my intro page on cryptography. COMMAND LINE INTERFACE (CLI) Here's from the Python command line: >>> from groups import * >>> residues17 = factory(17) # make objects using coprimes of 17 >>> residues17 # display as list of integers [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16] >>> table(residues17) # generate a multiplication table 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ---------------------------------------------------- 1| 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 2| 2 4 6 8 10 12 14 16 1 3 5 7 9 11 13 15 3| 3 6 9 12 15 1 4 7 10 13 16 2 5 8 11 14 4| 4 8 12 16 3 7 11 15 2 6 10 14 1 5 9 13 5| 5 10 15 3 8 13 1 6 11 16 4 9 14 2 7 12 6| 6 12 1 7 13 2 8 14 3 9 15 4 10 16 5 11 7| 7 14 4 11 1 8 15 5 12 2 9 16 6 13 3 10 8| 8 16 7 15 6 14 5 13 4 12 3 11 2 10 1 9 9| 9 1 10 2 11 3 12 4 13 5 14 6 15 7 16 8 10| 10 3 13 6 16 9 2 12 5 15 8 1 11 4 14 7 11| 11 5 16 10 4 15 9 3 14 8 2 13 7 1 12 6 12| 12 7 2 14 9 4 16 11 6 1 13 8 3 15 10 5 13| 13 9 5 1 14 10 6 2 15 11 7 3 16 12 8 4 14| 14 11 8 5 2 16 13 10 7 4 1 15 12 9 6 3 15| 15 13 11 9 7 5 3 1 16 14 12 10 8 6 4 2 16| 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 >>> a = residues17[7] # select a = specific object >>> a 8 >>> a.inv() # compute a's inverse 15 >>> a*a.inv() # a * a^-1 = 1 1 >>> powers(residues17) # compute g to successive powers 1 = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] 2 = [1, 2, 4, 8, 16, 15, 13, 9, 1, 2, 4, 8, 16, 15, 13, 9] 3 = [1, 3, 9, 10, 13, 5, 15, 11, 16, 14, 8, 7, 4, 12, 2, 6] 4 = [1, 4, 16, 13, 1, 4, 16, 13, 1, 4, 16, 13, 1, 4, 16, 13] 5 = [1, 5, 8, 6, 13, 14, 2, 10, 16, 12, 9, 11, 4, 3, 15, 7] 6 = [1, 6, 2, 12, 4, 7, 8, 14, 16, 11, 15, 5, 13, 10, 9, 3] 7 = [1, 7, 15, 3, 4, 11, 9, 12, 16, 10, 2, 14, 13, 6, 8, 5] 8 = [1, 8, 13, 2, 16, 9, 4, 15, 1, 8, 13, 2, 16, 9, 4, 15] 9 = [1, 9, 13, 15, 16, 8, 4, 2, 1, 9, 13, 15, 16, 8, 4, 2] 10 = [1, 10, 15, 14, 4, 6, 9, 5, 16, 7, 2, 3, 13, 11, 8, 12] 11 = [1, 11, 2, 5, 4, 10, 8, 3, 16, 6, 15, 12, 13, 7, 9, 14] 12 = [1, 12, 8, 11, 13, 3, 2, 7, 16, 5, 9, 6, 4, 14, 15, 10] 13 = [1, 13, 16, 4, 1, 13, 16, 4, 1, 13, 16, 4, 1, 13, 16, 4] 14 = [1, 14, 9, 7, 13, 12, 15, 6, 16, 3, 8, 10, 4, 5, 2, 11] 15 = [1, 15, 4, 9, 16, 2, 13, 8, 1, 15, 4, 9, 16, 2, 13, 8] 16 = [1, 16, 1, 16, 1, 16, 1, 16, 1, 16, 1, 16, 1, 16, 1, 16] >>> b = residues17[12] # pick a g the period < totient >>> b 13 >>> sg = subgroup(b) # generate the subgroup using b >>> sg [1, 13, 16, 4] >>> table(sg) # generate the multiplication table 1 4 13 16 ---------------- 1| 1 4 13 16 4| 4 16 1 13 13| 13 1 16 4 16| 16 13 4 1 The source code I'm using is here: http://www.inetarena.com/~pdx4d/ocn/python/groups_draft1.html Links: OO + Polyhedron: http://www.inetarena.com/~pdx4d/ocn/oop.html also: http://www.inetarena.com/~pdx4d/ocn/trends2000.html See bottom of http://www.inetarena.com/~pdx4d/ocn/overcome.html for links to the math-learn postings (these were rough drafts -- a better version for the web is in the works) Also see: http://www.inetarena.com/~pdx4d/ocn/clubhouse.html ============== OREGON CURRICULUM NETWORK ===================== By Kirby Urner, 4D Solutions, Portland, Oregon, USA

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Switching gears..
by Arthur_Siegel＠rsmi.com 01 Mar '01

01 Mar '01

Kirby wrote - >I think I'll give up pursuing the "math through programming" >thread with the math teachers. I'll stay in contact with >the few with an interest who got in touch, but most of the >time it's like talking to a brick wall. If any consolation, you're doing well at least when preaching to the converted. The concept in "Overcome" of looking at code as an alternative form of math notation - with the added feature that it not only says what it means, but does what it says it means - certainly sells me. It is interesting that our agendas do diverge - I think you consider yourself an innovator looking to revamp math education from the ground-up with new fundamental concepts, and I consider myself in many ways a reactionary, looking to restore something that I believe once was, and got lost . And yet we agree on so much. Not that that fact gets either of our agendas anywhere. ART

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Re: Switching Gears (long)
by Dethe Elza 01 Mar '01

01 Mar '01

Re: Teachers I grew up travelling a lot, changing schools, always being the outsider. I got to compare a lot of school systems (including private, public, priviledged suburban, inner city, Catholic, and alternative) and a lot of teachers. I had a fair number of really excellent teachers over the years. One math teacher who taught me a lot more than math, for instance. I also had teachers who varied from incompetent to apathetic to downright hostile (one gym teacher who tried to get in physical confrontations with the kids, and I'm talking 4th-5th graders). A good teacher can make a big difference. Unfortunately, the system is flawed on so many levels. Even with great teachers, school was very traumatic. By changing schools so often I was always the outsider, which may have been part of it. Part of was that I didn't clique well. Part of it was the essential meaningless of what we were forced to do. Here's an example of how bad it got: One day I woke up with a spontaeous nosebleed and my mom let me stay home from school. For the next couple of weeks I spent part of the morning punching myself in the nose to try (unsuccessfully) to recreate it. But I survived the system (the alternative schools helped) and even went on to finish a bachelor's degree and a master's. Now I have two kids, a girl (Mina) age 4, and a boy (Azlen) age 6 months. My wife (Daniela) and I have discussed what to do if they don't like school--we won't force them to go. So far it's not a problem. Mina is in daycare and Montessori, and loves it. She is very social and needs to be around other kids, and to have lots of structured learning time (like most kids if allowed, she soaks up learning like a sponge). She starts kindergarten in the fall, and we've found a very nice K-3 primary school. There are only 100 kids, and the principal, who showed us around, seemed to know them all by name. So Mina will be going to school. But I still consider her homeschooled. Daniela spent three years home with Mina and it shows--she can read more than 30 words, write her name, and do simple math in her head (as long as it involves Fig Newtons). She's going to school for social skills -- like surviving bad teachers. We'll try to cushion the blow of the real world, but we have to expose her to it eventually. If she absolutely hates school, she can stay home and get a better education and we'll find other paths for socializing, but I don't think that will happen. She's got a very different personality than me, very extroverted, so to her school will just be a bigger audience for her, and she'll probably continue to do additional "academic" learning at home. After all, she has two over-educated parents. What I'm trying to say, is that the schools can't and won't be the solution. School, especially in the US, is designed to keep kids off the streets as long as possible, train them to sit stil and endure boredom for hours at a time to prepare them for the work force, and to follow meaningless instructions without question or creativity. And, on the scale of education institutions in the US, that's how they *have* to be. The criticisms that are levelled against the schools/teachers/curriculum are the same now as they were 100 years ago. Literally. In that time there have been any number of suggestions for improving the situation, any number of attempts to change. Some really great ideas. But the system has proven to be very resillient to significant change. The best hope for schools is small, localized change -- that can work and is working in many places. But trying to change the way math is taught, in toto, is tilting at a very big windmill. There have always been some great teachers, who rise above the system. Even great (usually very small) schools. And there will continue to be good reasons for kids to go to school. The three Rs just might not be among them. --Dethe

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RE: [Edu-sig] Switching gears...
by Morris, Steve 28 Feb '01

28 Feb '01

Funny you should say so. I have a similar reaction when trying to help my 5th grader with math. Unfortunately I don't have the option of giving up on the teachers. I have to work around them. I'm beginning to hate the "Chicago curriculum" even though the only thing I know about it is the math problem sheets that get sent home. The biggest problem is the stone wall that appears whenever I try to get more information about the curriculum so I can try to anticipate. The teachers treat me like an adversary that they need to handle with kid gloves but for sure don't give any real information to. They are polite but unhelpful. I'm afraid to push because of potential adverse reaction towards my student child. Sorry for the off topic rant but I am at wits end. I watch my 5th graders motivation trickle away daily. I think I understand how to make math clear and interesting but all my energy is wasted trying to compensate for the weakness of his classroom work. I was hoping to give him an edge but instead am trying not to lose him altogether. Oh well, maybe next years teacher will be better. > -----Original Message----- > From: Kirby Urner [mailto:pdx4d@teleport.com] > Sent: Wednesday, February 28, 2001 2:40 PM > To: edu-sig(a)python.org > Subject: [Edu-sig] Switching gears... > > > > I think I'll give up pursuing the "math through programming" > thread with the math teachers. I'll stay in contact with > the few with an interest who got in touch, but most of the > time it's like talking to a brick wall. > > Fortunately, the computer science track covers a lot of the > same material ('Concrete Mathematics', 'Art of Computer > Programming'), so in some respects we can make up for > deficiencies in the math curriculum by bolstering the > computer programming curriculum. > > I see a large homeschooler market developing here, as many > kids have computers at home but no competent teachers in > school (Yorktown kids etc. are lucky). I hope those > computer science/programming teachers who share this stuff > with kids will continue putting a lot of it on the web in > an accessible form, as for every kid in the classroom, > there will be 100 at home with fewer options to learn in > a group setting with a live teacher. > > I feel sorry for kids who just get the regular K-12 > math curriculum these days. It's not relevant to their > needs. But it's what the teachers are programmed to > teach, and there's no sense arguing with essentially > mindless automatons. I can think of better and more > effective ways to rescue kids from a dead end curriculum > and will be heading down those avenues instead. > > Kirby > > More in a similar vein: > http://groups.yahoo.com/group/synergeo/message/1876 > > > > _______________________________________________ > Edu-sig mailing list > Edu-sig(a)python.org > http://mail.python.org/mailman/listinfo/edu-sig >

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