Mailman 3 January 2001 - Edu-sig

Re: Edu-sig digest, Vol 1 #216 - 1 msg
by Dorothea Salo Jan. 19, 2001

Jan. 19, 2001

> Whatever happened to kicking back and appreciating the > masters, getting a sense of what high quality mathematics > is like without actually trying to be a master yourself? > That's how we do it most music and art and literature. > For some reason, math is supposed to be different. Why? Damned if I know. Makes no sense to me either. There's a ton of math literacy that should be taught this way and isn't. Statistics is my particular bugaboo; it truly irritates me how many people have no idea what a standard deviation is, or what goes into calculation of a margin of error. Now, I would have a terrible time *calculating* a margin of error, at least without a book at hand to refresh my memory, but I damn well know what it is, why it matters, and that it relates to factors like sample size and sample randomness. In these days of InstaPolls, this kind of basic "duh"-level understanding is *important*. > "Math appreciation" means being able to follow what others > have done _without_ making the claim that you could or > should be able to do the same. We need to take away the > notion that the only math worth reading, playing with, > exploring, is math you have the capacity or ability to > reinvent yourself, from scratch, ab initio. Brilliantly stated, sir. Bravo. Dorothea -- Dorothea Salo Impressions Book and Journal Services, Inc. phone: (608) 244-6218 fax: (608) 244-7050 http://www.impressions.com

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Exploring Bernoulli Numbers
by Kirby Urner Jan. 19, 2001

Jan. 19, 2001

More of the usual math-through-programming jazz. A continuation of a thread on math-teach at the Math Forum... Kirby ===================================================== >We see that the number of subtriangles in each row = 2n - 1 >(series of odds, n = 1,2,3...) and that the n+1 row will >duplicate the previous, plus add 2 more triangles i.e. > > row(n+1) = (2n - 1) + 2 = 2n + 1 > ^^^^^^^^ ^ > triangles 2 > in last row more > >See: http://www.inetarena.com/~pdx4d/ocn/graphics/triarea.gif Of course this is a good visual proof that n^2 (n=1,2,3...) is the sum of the first n consecutive odd numbers, i.e. n n^2 = SIGMA (2n-1) i=1 Kids lucky enough to have a CLI can write: >>> def odd(n): return 2*n - 1 >>> import operator >>> def g(f,n): """ n g(f,n) = SIGMA f(i) i=1 """ return reduce(operator.add,map(f,range(1,n+1))) >>> g(odd,5) 25 and so on. This brings me to another beef I have with the way K-16 math is usually taught: we make the silly rule that if the students can't actually do the math that might be relevant in a given thread, then we probably shouldn't even talk about it. What other subjects do we teach according to this rule. Don't listen to Bach until you can play like Bach, don't read Shakespeare until you can write like Shakespeare -- ridiculous. And yet text books are careful to just coast along at the level of "exercises", where the students are supposed to do, do, do. Whatever happened to kicking back and appreciating the masters, getting a sense of what high quality mathematics is like without actually trying to be a master yourself? That's how we do it most music and art and literature. For some reason, math is supposed to be different. Why? For example, here we are talking about sums of 2nd power expressions expressible as a 3rd power equation (tetrahedral and cuboctahedral numbers), or 2nd powers = sums of 1st powers (e.g. sum of consecutive odds, or consecutive integers). There's the theme of degree, and of consecutive integers (plus the linear, areal, volume relationship for powers 1,2,3). What other mathematics is relevant in this connection? A student tracking this thread might ask, or, if told, might appreciate, that people asked themselves long ago whether the sum of consecutive integers to the 5th power might be expressed as a polynomial to the 6th power, or whether 1^7 + 2^7 + 3^7 +... + (n-1)^7 + n^7 might be equivalent to some polynomial in the form A[1] n^8 + A[2] n^7 + A[3] n^6 + ... + A[n] n, where A[1]... A[n] are rational coefficients. In general, given: n SIGMA f(i) where f(i) = i^x i=1 we want to find a corresponding: n+1 SIGMA A(i) n^i A(1)..A(n) = rational numbers i=1 that gives the same result. Just to be able to understand the problem, to appreciate what's being sought, is mental exercise enough at some levels. You don't have to expect kids to then sit down and pull the Bernoulli numbers out of a hat (the Bernoullis figure in to the computation of these coefficients, which are also functions of x -- see below). What's being exercised here is a kind of math literacy. We have SIGMAs flying around, and an exponent that's also a variable. It's like trying to decipher some code, to make sense of the terrain. Even before proofs, or derivations, you just need to practice with the lingo. It probably helps in many cases to write the top SIGMA as a simple program at the CLI: >>> def sumpow(n,x): sum = 0 for i in range(1,n+1): sum = sum + i**x return sum >>> sumpow(5,5) 4425 >>> 1**5 + 2**5 + 3**5 + 4**5 + 5**5 4425 This is a departure from my usual approach of defining f(i) and SIGMA and composing them as g(f,n). That's OK. We should do it both ways. One reason I like hyperlinking to the Bernoulli numbers at this point because here's where we hook in Ada, the Countess Lovelace. We have few enough women in the narrative, but Ada makes up for a lot of what's missing. She's a perfect character to include for young students: Five weeks after Ada was born Lady Byron asked for a separation from Lord Byron, and was awarded sole custody of Ada who she brought up to be a mathematician and scientist. Lady Byron was terrified that Ada might end up being a poet like her father. Despite Lady Byron's programming Ada did not sublimate her poetical inclinations. She hoped to be "an analyst and a metaphysician". In her 30's she wrote her mother, if you can't give me poetry, can't you give me "poetical science?" Her understanding of mathematics was laced with imagination, and described in metaphors And here's the link to Bernoulli numbers: When inspired Ada could be very focused and a mathematical taskmaster. Ada suggested to Babbage writing a plan for how the engine might calculate Bernoulli numbers. This plan, is now regarded as the first "computer program." A software language developed by the U.S. Department of Defense was named "Ada" in her honor in 1979. [ http://www.well.com/user/adatoole/bio.htm ] Of course critics will say the above narrative means Ada is a heroine in computer world, and her name has no relevance to math classes -- nevermind about the Bernoullis or that "computer science" wasn't even invented as a discipline yet (in those days, you weren't considered "interdisciplinary" just because you thought about computing machinery in connection with computing). Again, I think in K-12 at least, there's no need to carve learning into math without much programming on the one hand, and programming without much math on the other -- they synergize so well, that they deserve to be fused more deeply. Convergence is a trend, or should be a trend. It's a symptom of overspecialization that so many schools have fancy computer labs, but in math class kids have to make do with calculators. Back to Python, kids can import, and play with the Bernoullis. As rational numbers, they may be expressed as (numerator, denominator) pairs. Here's real.py in action (slightly tweeked by me to fin Python 2.0 syntax), a powerful, jewel of a module by Jurjen N.E. Bos: >>> from real import * >>> map(bern,range(0,20)) [(1L, 1L), (-1, 2), (1L, 6L), (0, 1), (-1L, 30L), (0, 1), (1L, 42L), (0, 1), (-1L, 30L), (0, 1), (5L, 66L), (0, 1), (-691L, 2730L), (0, 1), (7L, 6L), (0, 1), (-3617L, 510L), (0, 1), (43867L, 798L), (0, 1)] ... which is to say that B(i) is 0 for all odds except B(1), and B(2) = 1/6, B(4) = -1/30 B(6) = 1/42, B(8) = -1/30 and so on. The "L"s you see in the above tuples stand for "long": these are what we call BigNumbers, e.g.: >>> bern(200) (-498384049428333414764928632140399662108495887 4572066749680558226172636696215236875688658023 0221099913260141269761327939105865452714534051 5840099290478026350382802884371712359337984274 122861159800280019110197888555893671151L, 1366530L) We don't have our coefficients yet, but there's light at the end of the tunnel at last. For my coefficients, I turn to Chapter 20, The Bernoulli Era, in 'The History of Mathematics' by Carl Boyer (2nd edition, John Wiley and Sons, 1991), pg. 419. Again, this isn't something I can necessarily derive on my own, not being a mathematician of Jacques Bernoulli's caliber (I might be able to follow a proof -- none given in this context). But does this mean such math is uninteresting to me, or that I shouldn't be exposed to it? No way. Nor should my students be so insulated. "Math appreciation" means being able to follow what others have done _without_ making the claim that you could or should be able to do the same. We need to take away the notion that the only math worth reading, playing with, exploring, is math you have the capacity or ability to reinvent yourself, from scratch, ab initio. That doesn't mean we should teach mindless rule-following without comprehension at every turn. But it does mean there should be phases of study in which we're testing and verifying, applying and trying out -- because these activities also involve a level of comprehension and exercise. Math is a communal activity. No single human could reinvent it all, not even Gauss. That's life in the real world. For example I, or my student, can translate the mathematical notation provided by Boyer into executable, and therefore testable code. I've done this in my bernoulli.py, which contains coeffs(), a method for returning the sought-for coefficients -- for which the Bernoulli numbers were needed: from real import bern def coeffs(x): A = [(1,x+1),(1,2)] if x<2: return A num,den,k = x,2,2 while x>1: b = bern(k) A.append((num*b[0],den*b[1])) num = num*(x-1)*(x-2) den = den*(k+1)*(k+2) k = k+2 x = x-2 return A If I'm summing numbers to the 1st power, what are my coefficients? >>> bernoulli.coeffs(1) [(1, 2), (1, 2)] i.e. n SIGMA n = (1/2)n^2 + (1/2)n i=1 -- same as n(n+1)/2 derived in earlier posts. How about if I'm summing numbers to the 2nd power, what are my coefficients? >>> bernoulli.coeffs(2) [(1, 3), (1, 2), (2L, 12L)] i.e. n SIGMA n^2 = (1/2)n^3 + (1/2)n^2 + (1/6)^n i=1 So, to answer one of our earlier questions, what's 1^7 + 2^7 + 3^7 +... + (n-1)^7 + n^7 expressed as an 8th degree polynomial in terms of n? = (1/8)n^8 + (1/2)n^7 + (7/12)n^6 - (210/720)n^4 + (2520/30240)n^2 I have a function, poly() that will actually apply the coefficients to powers of n and compute this sum. Let's compute a few values: >>> bernoulli.poly(10,7) # 1^7 + ... + 10^7 18080424.9999999999999999999999+-2 >>> bernoulli.poly(11,7) # 1^7 + ... + 11^7 37567595.9999999999999999999999+-2 >>> bernoulli.poly(12,7) # 1^7 + ... + 12^7 73399404.0000000000000000000000+-2 The slightly off results come from working with decimals of limited precision. Now let's compute these sums more directly, by defining f(x) = x^7 and passing f as a parameter to our usual: n g(f,n) = SIGMA f(i) i=1 >>> bernoulli.g(f,10) 18080425 >>> bernoulli.g(f,11) 37567596 >>> bernoulli.g(f,12) 73399404 Same answers, approximately. If we're persuade the algorithm is correct, we could convert to integers as a last step in poly(), thereby assuring whole number answers. Kirby [1] module by Jurjen N.E. Bos: ftp://ftp.python.org/pub/python/contrib-09-Dec-1999/ DataStructures/real-accurate.pyar [2] bernoulli.py by K. Urner http://www.inetarena.com/~pdx4d/ocn/python/bernoulli.py

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Natural Language Programming
by Arthur_Siegel＠rsmi.com Jan. 14, 2001

Jan. 14, 2001

Kirby wrote - >I don't have any lines or planes of so-called "infinite" >extent. Having those would be challenging, and I think >you've done that in PyGeo somehow. What has been particularly challenging is having the visual being sufficiently suggestive of what is going on analytically. Your can't really *draw* an infinite plane in 3d space without mucking up the screen. You have to suggest this oriented square is really a visual symbol for an infinite plane in space. So even working purely visually one is still in the realm of creating a symbolic language of a kind. Sort of interesting. ART

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KineticsKit - A Physics-Based General 3D Mass-Spring System
by Markus Gritsch Jan. 14, 2001

Jan. 14, 2001

Hi! I'd like to present you KineticsKit - A Physics-Based General 3D Mass-Spring System. It's purely written in Python and has several example programs included. Just give it a try - you will like it. Requirements: ------------- Python 2.0 or higher VPython available from http://virtualphoton.pc.cc.cmu.edu/projects/visual/ Objective: ---------- KineticsKit provides some classes to simulate the behaviour of a 3D mass-spring-system. It can be used to investigate some basic kinetics, but I wrote it mostly for the fun of programming it and playing with it. Usage: ------ First a System has to be generated. The System has several attributes like gravity, viscosity and timestep which must be given in the constructor. Some of them are mandatory (timestep), the rest of them are optional. Next Masses are generated and inserted into the System. Masses also have lots of attributes like m, pos, if it should be fixed at its initial location, its initial velocity and the color. Most of them are implemented as keyword arguments and can thus be omitted if they are not used. After this, the Masses can be connected with Springs. There is no restriction in making the connections so one Mass can be connected with several others. A Spring needs to know which two Masses it should connect, and its spring-constant. The rest of the constructor arguments are again implemented as keyword arguments, but I recommend to take a look at them. They are useful to tense the Spring, specify a damping value or give it a nicer color. Finally the System has to be started and kept running with an infinite loop. Interaction with the system is possible with the mouse. A Mass is picked by clicking at it. It can be moved around and the rest of the system responds to the movement. To release the Mass the mouse must be clicked again. """A Physics-Based General 3D Mass-Spring System. by Markus Gritsch (gritsch(a)iue.tuwien.ac.at) Objective: ---------- KineticsKit provides some classes to simulate the behaviour of a 3D mass-spring-system. It can be used to investigate some basic kinetics, but I wrote it mostly for the fun of programming it and playing with it. Usage: ------ First a System has to be generated. The System has several attributes like gravity, viscosity and timestep which must be given in the constructor. Some of them are mandatory (timestep), the rest of them are optional. Next Masses are generated and inserted into the System. Masses also have lots of attributes like m, pos, if it should be fixed at its initial location, its initial velocity and the color. Most of them are implemented as keyword arguments and can thus be omitted if they are not used. After this, the Masses can be connected with Springs. There is no restriction in making the connections so one Mass can be connected with several others. A Spring needs to know which two Masses it should connect, and its spring-constant. The rest of the constructor arguments are again implemented as keyword arguments, but I recommend to take a look at them. They are useful to tense the Spring, specify a damping value or give it a nicer color. Finally the System has to be started and kept running with an infinite loop. Interaction with the system is possible with the mouse. A Mass is picked by clicking at it. It can be moved around and the rest of the system responds to the movement. To release the Mass the mouse must be clicked again. Zooming / rotating of the scene is also done with the mouse by holding the left / right mouse button down. For further details please have a look at the example programs and the constructors of the classes System, Mass and Spring. Have fun! -- |\/\/\/| /------------------------------------------------------------------\ | | | Markus GRITSCH | phone: +43 / 1 / 58801-36015 | | | | Institute for Microelectronics | cellular: +43 / 676 / 4973431 | | (o)(o) | Technical University of Vienna | fax: +43 / 1 / 58801-36099 | C _) | Gusshausstrasse 27-29 / E360 | email: gritsch(a)iue.tuwien.ac.at | | ,___| | A-1040 Vienna / AUSTRIA | SMS: 436764973431(a)max.mail.at | | / \------------------------------------------------------------------/ /____\ / \ "Computers let you make more mistakes faster than any other invention in human history, with the possible exception of handguns and tequila." Mitch Radcliffe

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webpage for KineticsKit
by Markus Gritsch Jan. 13, 2001

Jan. 13, 2001

Hi! I've put together a litte webpage for KineticsKit. KineticsKit can now be downloaded from this address: http://stud4.tuwien.ac.at/~e9326522/KineticsKit/ Code enhancements: Added a second visual representation of a spring (double-helix), made a small modification to the spring constructor, added another example (picture of it can be seen at the top of the webpage). enjoy, Markus

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inline python for education
by Bruno Vernier Jan. 11, 2001

Jan. 11, 2001

<delurking> Hello everyone, I've followed this mailing list for several month. I am a math/computer science intructor (high school level). I have just written the first version of a zope product called Eduml which is an extension of an existing product called ZWiki. (everything is GPL) This package allows anyone to quickly include python, including PIL (python imaging library) , SQL queries, gnuplot commands, and LaTeX instructions in what appears to be a plain old web page (and oh yes, we can include HTML too but not directly inside the eduml elements) for example, <h1>a simple example</h1> <python> A = randint(0,9) B = randint(0,9) </python> <math> simplify $$ \A \over \B $$ </math> <sql> select * from lessons </sql> <draw> line ((0,0,100,100)) ellipse ((0,0,50,50)) </draw> <plot> sin(x) </plot> I've posted a showcase/sandbox area on http://ess.vancouver.bc.ca/zope/bruno/eduml Thanks, Bruno p.s. the new elements are part of an XML for education project I've been involved with for 2 years called EDUml (Educational Markup Language) but that is another story. </delurking> --

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GIS, CAD, and mapping daily news provided by Spatial News
by Jason Cunliffe Jan. 11, 2001

Jan. 11, 2001

Hands High Software's New Teachers' Aide Delivers a Staff-Full of Help From a Hand-Held Computer http://spatialnews.geocomm.com/dailynews/2001/jan/11/news4.html

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Natural Language Programming
by Arthur_Siegel＠rsmi.com Jan. 11, 2001

Jan. 11, 2001

>>With regard to vectors, it's already standard. Agreed We've being doing too much of that lately. Should we talk Euclid again, to give us some spice.. For vector stuff it absolutely makes sense to use the operator override functions. End up with a visual calculator . Neat I think.

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Natural Language Programming
by Arthur_Siegel＠rsmi.com Jan. 11, 2001

Jan. 11, 2001

>And let's not forget, while we're at it, that a >synonym for a word might by another symbol, like *, >thanks to the "operator override" feature. Had also been playing with the feature with PyGeo, again trying to keep to "standard" symbolism, to the extent it has been developed. So for example: p1 + p2 might be the line connecting the two points. Line1^Line2 might be the point of intersection of the two lines. But to put my money where my mouth is, it was only interesting to me to the extent I could tie it to some recognized geometric notation. Its out there, but not too developed (or probably more accurately what had been developed has fallen into disuse). A PyGeo specific notation was and is not of much interest to me, so I pretty much dropped it..

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RE: [Edu-sig] Natural Language Programming
by Morris, Steve Jan. 11, 2001

Jan. 11, 2001

Hi Art, > One last effort - One last reply. :-) > Anyone who has worked at all with 3d graphics knows that > a pretty standard vocabulary has developed, by a process > of natural selection, over numbers of years. > > 99.5% (please don't quote me on the statistic and find other > exceptions) of the 3d graphics world has settled on a vocabulary. > Presumably lots of very smart people where involved in the > trial and error process. And what we have is what I suspect is > exactly what has been determined to be the most natural > language (in standard English) necessary to do a particular > job, while avoiding ambiguities and other evils. Could certain > vocabulary words turned out diffierently. Probably. But the > point is its done, and without *COMPELLING* evidence there > is no reason to change it. > > I would serious suggest that anyone with a deep interest > in the subject of natual language technology study the development. > of 3d vocabulary. Because in the end what we are left with is > substanital (though not perfectly - we do go to third and fourth > defintions of standard English words) natural language. > And vocabulary is about nothing if it is not about communication. Here is where I disagree or at least disagree with your conclusions. Language is about communicating in the broad sense. Sometimes the communication is on many levels and often one of those levels is a level of exclusion. Vocabulary can be used to say "I am an insider and you are an outsider." Take legal language for example. This has also evolved by a process of natural selection, over a number of years. It exists the way it does (opaque) for several reasons 1) It creates a level of precision. 2) It evolved this way from archaic forms of English. 3) There is resistance to change for fear that the precision will be lost. If it ain't broke don't fix it. 4) It creates a private club with a language understood only by the practicioners thus inreasing the value of their skills. It is part of what sets apart those special people who have "passed the bar." 5) Practicioners are not bothered by the vocabulary once they have entered the priesthood and have little incentive to change it. 4 is a not inconsiderable issue. There is a lot of legal pomp and circumstance that is intended merely to emphasize the importance and solemnity of the occasion and to create respect for the process. There no legal reason that the judge needs to sit on his high bench. The bench represents his authority. "All rise." Why the heck should we rise when the judge enters. These proceedures and the language of the law in general exist at least partially to reduce communication. The entire legal process is an excercise in controlled communication (ie limiting communication.) Many things are not allowed to be said in the courtroom, Perry Mason to the contrary. In other words the language of the law is about limiting communication. And how about music. Music notation is a kludge. Only music in the key of C makes sense. Sharps and flats are repairs to the defective system. In a properly designed music notation system intervals (the relationship between notes) would be implicit in the notation. Music is interval and timing. An octave should leap out at you just from the notation. A fifth should shout its existance. Music notation was designed for playing, not composition or analysis. The structure of music (what sounds interesting and why) is highly mathematical. I defy anyone to see the mathematics in music notation. The closely held secret of music theory is that it is simple. It is only the language that is more opaque than it could be. There's many a potential talented musician who couldn't learn to read music. Is the defect in them or the notation? I suggest that it is the notation, hundreds of years of acceptance or not. An M4E program would have to address this. Just because there is an accepted vocabulary doesn't mean that it is easy to learn or that there aren't better alternatitives. What we have here is what economists call lock in. (VCS vs. Beta is the classic example.) Current practice is often far from best known practice. This is true of any evolutionary system. Why this happens is a large, well studied subject that I don't want to go into to deeply here but I'll make a simple pass. Physicists think of these as local minima. A marble rolling down a hill doesn't always reach the lowest point in spite of the driving force of gravity. The path is critical. Local dips in the terrain grab it and prevent it from going lower. The same is true of the evolution of vocabulary. > I have a simple premise. If something is easy for me to > comprehend, it > is easy for the next guy. I am not sure who I would be if I thought > otherwise. > Not someone I would like very much. Strange premise. There are things that are easy for you that are hard for me. This is a simple fact and no insult. I am baffled as to why you would dislike yourself for acknowledging this simple fact. > By calling the next guy stupid, you're calling me stupid > because in some > other area of endeavor I am going to be the novice. And I am not even > going to have the opportunity to tell you - "translate", > "move" - got it, > next. Since you like being personal in these discussions let me offer myself as one of several examples. Perhaps this personification will make my point clearer to you. I do not consider myself stupid even though my ability to acquire vocabulary is limited. I have strengths that I quite immodestly think are more important and make me a quite capapable person. We all have different blends of strengths and weaknesses. The world would be boring if this were not true. People come in all shapes and sizes and capapabilities. For some, new vocabularies are a snap. My father in law was an amazing amateur linguist. He was fluent in 5 languages and did technical translation in 20 more. If he new the subject he could translate in languages he had never seen before as long as it shared common roots with one or more of the languages he did know. (He claimed that most technical words are the same in all languages anyway.) If he wanted to read something he could pick up a working capability for a language in a few weeks. He didn't work hard at this. It came naturally to him. I think the real key to his success was an almost eidetic memory for vocabulary. He had it in large measure. I have it almost not at all. You are probably somewhere in the middle. CP4E needs to address people at my end of the scale. It is an insult to no one to say that my vocabulary learning skills are less than yours. It is a simple fact. My proceedural memory and skills just might be better than yours. They certainly have helped me compensate. I test high in langauge skills because my comprehension of grammer and syntax is high. In spite of that I have never been able to learn other verbal languages because I can't get over the vocabulary hurdle. I have tried again and again. On the other hand I know over 20 programming languages, maybe as many as 30 (it's hard to count because I keep forgetting about the ones I rarely use.) There are generally fewer than 10 key words (basic vocabularly) in programming languages. These are langueages with high syntatic and grammatical content but are almost vocabulary free. So I'm a great programmer but a lousy linguist. I am suggesting very simply and personally that "translate" instead of "move" (and similar words) could have been a barrier to me. By refusing to acknowledge this difference between people you would eliminate people with my strengths and weaknesses from the people you would recruit into a CP4E type effort. Just because you don't have a problem with vocabulary doesn't mean no one else does. It is no insult to me to say so. Special vocabularies are not a barrier to everyone but they are a barrier to some. I am one. Your insistance that my abilities are the same as yours is exclusionary. You deal with my weaknesses by denying that they exist and thus don't help me work around them. That's like saying we should design programs only for the rich because everyone has the same ability to acquire money. I obviously don't mean me personally. I have successfully compensated for my problems with vocabulary, partially by my choice of careers. I over compensate perhaps in working on acquiring and using English vocablary. However CP4E, if the E means anything, should include people to whom vocabulary is a barrier. I assure you that there are more of us than you might think. CP4E (in the broad sense, not just Guido's grant projects) should be about identifying barriers to entry and addressing them. Programming is clearly something that anyone can learn. They don't largely because of various barriers to entry. The correct way to handle this particular barrier is probably with transitional vocabularies (perhaps simple aliases to the less 'naturally' named commands) just the way kindergarden uses transitional spelling stratagies to teach spelling. If you are going to teach 3d programming, eventually the student needs to know the standard vocabulary or they can't benefit from the current body of knowledge. That is no reason to say they shouldn't be allowed into the temple of 3D learning with out jumping this vocabulary hurdle first. I think the heat you see on this subject is from people like me, to whom specialized vocabulary IS a barrier, taking personally your suggestion that our problem doesn't exist or deserve to be addressed. CP4E doesn't mean CP4 everyone whose learning abilities exactly match Arthur Siegel's. Regards, Steve Morris

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