math and low intensity debate e.g. tau versus pi

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kirby urner

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Mar 26, 2016, 12:46:55 PM3/26/16
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This Khan Academy classic is worth projecting in a classroom.  I know I would be eager to share it, if back in high school teaching, like in the good old days:

https://www.khanacademy.org/math/trigonometry/unit-circle-trig-func/long-live-tau/v/tau-versus-pi

I'm not suggesting I'd turn my whole curriculum over to Sal, just that I'm up for sampling choice pieces by authors I admire, Vi Hart another one (good at math video as an art form).  This retrospective by Khan, combined with a polemic at the end, seems a great high point to cite.

The math classroom of the future, in one of its many forms, casts the teacher in the role of VJ (video DJ) in a lot of ways.  Most clips might be of lesser length than Khan's average, although some teachers might invest in a full series of something, with their own time at the podium, as emcee and expert, to apply local spin, relate it to actual students more directly (personalize, make it real, take questions).

It's up to the teachers and lesson planners to glue the samplings into coherent pathways that come together to produce whole concepts.  The teacher may improvise a good portion of the presentation, yet there's a known terrain, the school's agreed-upon curriculum, what it's known for.

The fact remains that "citing video" is a lot like a VJ's job and editing video is the "anime" equivalent of doing "manga" or "stills" (picture Jupyter Notebooks with still graphics courtesy of matplotlib).  In the graphical arts, manga equals comic books, anime equals cartoons.  In mathematics, one needs both.

Sal covers a lot of ground in the above video, looking back, reviewing trig functions, radians, the whole idea of pi. 

Then comes his impassioned defense of tau (as 2 pi).  He uses tau's appearance on the stage as an excuse to quickly review the whole board once again, having set us up the first time with a critique of Euler's Formula, arguably the most beautiful in math, but having a flaw, a blemish.

I don't see any either/or here really.  Use tau in place of pi when it's prettier.  Why not?

However at this juncture it's worth mentioning another low intensity tug-o-war in which tau is involved.  Versus Phi.  There's been some debate about whether tau stands for .618... i.e. the reciprocal of 1.618... 

Some authors use tau for the latter, however the convention I'm most familiar with assigns Greek letter phi to 1.618... and tau to its reciprocal (0.618...).  In English or other romanized ("ascii-fied") texts, we may use phi in place of the Greek letter.  Then comes the low intensity debate whether to pronounce it "fee" or "fie".  A contingent says it both ways.  Maybe one should say "fee" on odd days of the week?

Quoting from Wikipedia:

Since the 20th century, the golden ratio has been represented by the Greek letter φ (phi, after Phidias, a sculptor who is said to have employed it) or less commonly by τ (tau, the first letter of the ancient Greek root τομή—meaning cut).
What I suggest coming from a liberal arts background is we remind students of mathematics that debate is a feature (not a bug) of their discipline and circle these simple examples, in preparation for investigating bigger debates in lessons to come.

When Sal comes back around again, reviewing the board, the trig functions, the unit circle, he shows that Euler's formula may be made even more beautiful.   By that time some of us are ready to vote for Sal's argument, especially with the caveat that pi is not hereby banned.  These constants do not compete so much as reinforce one another.

As for phi versus tau, I think we've pretty much settled on tau being the reciprocal of the golden mean, which golden mean is itself > 1, i.e. is 1.618... or (1 + rt2(5))/2 -- note that I sometimes replace sqrt() with rt2() as I don't want to push the mental model to favor "squares" too prejudicially, given the triangle-friendly balance of this curriculum (Martian Math inspired)).

Especially if tau is going of to do yeoman's service as 2 pi, replacing 2 pi r with tau r in many a Jupyter Notebook, all the more reason, then, to not force it to do double duty as the golden ratio as well.  That'd only add to the post Babel confusion, not that I expect any posting by me to serve as an edict.  The post Babel confusion (not everyone on the same page) is more what I'm drawing attention to, as another lens for viewing math.  Tune in the debates for a change.  Listen to arguments.

Kirby

Additional reading / links:
http://mathforum.org/kb/thread.jspa?threadID=2246748&tstart=0

Peter Farrell

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Mar 27, 2016, 3:31:26 PM3/27/16
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Hi, Kirby,

For all the math geek I have in me, this is more of the "sage on the stage" kind of math class that the misguided idealist in me is imagining we're moving away from in the 21st Century. As much as we'd like to present this stuff fully edited to learners, maybe we should just be giving them the tools and challenging them to answer a Project Euler-type math problem.

As a math teacher and certainly as a CS teacher I increasingly saw myself like a guitar teacher, showing my students the chord of the day and challenging them to combine it with the chords they've already learned to make something new. The danger for us know-it-alls is the students might not incorporate it "correctly" or might come up with a song we don't like. Oh, well.

Lou Reed reminisced about his one guitar lesson, where his teacher wanted to teach him melodies like "Mary Had a Little Lamb" and Lou asked to learn rock and roll. His teacher scoffed, "You just need three chords for that." Lou then asked him to teach him the three chords and he never needed to go back.

I have an idea. Let's teach the "three chords of Mathematics" to students and trust them to go off and learn to teach themselves about Pi and Tau and Phi. I'm copying a Physicist/Professor/Author in India named Dr. C.K. Raju. I found him years ago when I was researching teaching calculus without limits and googled "calculus without limits." Come to find he put on a 5-day program of that name to teach calculus using solvers and other software. He tried it out a handful of times and considered it a success. You can read about it here:


How would the 5-day math course look? With all the technology at our disposal I'm sure we could supply the chords, I mean tools to work with all the useful and fun ideas in algebra, geometry, trigonometry and calculus in 5 days. Then the students go off and learn as they please. Like Lou, they might keep it up and change music forever, or they might not keep it up and they'll forget everything, like lots of students do today anyway.

I'm absolutely serious about this. I look forward to the replies and I promise I'll give you proper credit, like I have Dr. Raju.

Peter Farrell
San Mateo, CA

Joseph Austin

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Mar 27, 2016, 6:05:32 PM3/27/16
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In my opinion, it would be even more practical to measure angles in "revolutions",
e.g. a right-angle is 1/4, sin (1/4) = 1. This is essentially degree measure "normalized".
I noted previously somewhere in this forum that the distinction between "radians" and "degrees" is whether one measures angles as multiples of the radius or multiples of the circumference, the two quantities being an irrational pair.
I had also observed that one can calculate the area of a circle without explicitly using pi (or tau), as half the circumference times half the diameter: A=(c/2)*(d/2).

> On Mar 26, 2016, at 12:46 PM, kirby urner <kirby...@gmail.com> wrote:

kirby urner

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Mar 27, 2016, 6:23:50 PM3/27/16
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Hey Peter, what an amazing link and referral, thank you, and happy Easter. [1]

I had never heard of Dr. C. K. Raju, and yet on a first pass I found so much in his thinking that's congenial.

In some countries, a conventional mathematics classroom is unlikely to focus on even "low intensity" debates such as pi-versus-tau -- let alone any high voltage ones such as Berkeley-versus-Newton -- precisely because, in those countries, mathematics has become the poster child for a discipline with indomitable arguments.

What's the point of showcasing "debates" when the whole purpose of the training is to instill a strong faith in airtight dogmas, theorems with no possibility of refutation once on-the-face-of-it obvious givens are supposed? 

According to Raju, the goal of mathematics in Western culture is to convey the inertia of a steamroller with the figure of Euclid in the driver's seat.  I enjoyed Raju's passages suggesting even "Euclid" himself was concocted or based on a miss-translation from the Arabic, invented for the role. [2]

Raju deftly reinforces a saying I favor: "all math is ethno-math" [3], a slogan meant to counter the perception that we might dabble in "ethno-mathematics" now and then, just to prove ourselves broad-minded, but then retreat into some other math that we know is "beyond culture". 

Really?  Such retreats exist?  Most the math I come across is very Earthling-flavored, for starters, and only gets more ethnic as we zoom in.  Keith Devlin points out non-humans do mathematical things all the time and sure, maybe the universe is energized mathematics, but when it comes to human doings, I don't think trying to tease apart the "pure mathematics" from "everything else" is very wise.  To what end did we need such "purity" in anything, where were the motives?  I'm suspicious.

No, rather than fight the fact of cultural immersion, I suggest that we're each called upon to "define our own tribe" and take responsibility, not just for ourselves, but also the various groups we wish to be identified with, whatever they may be. 

Responsibility means standing up for one's chosen people or peoples (perhaps a profession), admitting weaknesses, contributing strengths.  Of course in debate one may slyly divert attention from one's vulnerabilities in order to advance a position.

Given my becoming steeped in Wittgenstein's later philosophy close to the outset of my academic career, having devoured his Remarks on the Foundations of Mathematics as an undergrad, I'm predisposed to see logical foundations, those of the Principia Mathematica for example, as "painted foundations under a painted castle" (paraphrasing RFM).  The logic is not "holding up" so much as "providing an added layer of ornamentation".

Dedekind's formalisms about the "real numbers", blended with Cantor's notions about "sets", add proper decorum, providing "clothes" for what were originally more nakedly practical computations, driven more by pragmatic concerns (engineering).

Dr. Raju proposes making mathematics simpler to teach by prying it loose from those control freaks seeking merely to suppress argument. 

Lets return mathematics to practice and argue its utility on that basis, not by conveniently promising some more persuasive metaphysics down the road, always at a next higher level, requiring of more dues.

Dispensing with much of the claptrap, as just so much religious (i.e. cultural) baggage is appealing and in some courses should be carried out. 

However, considering the curriculum as a whole, I'm more inclined to bring back the dogmas, but this time more consciously opened up to continuous philosophical investigation. 

I believe even young children are capable of engaging in philosophical discourse as a guided activity, with explicit reference to various rules (rule-consciousness is already what childhood is much about).  My sister has a degree in this subject (Philosophy for Children), offered by Montclair University. [4]

We're not compelled to buy into every language game we come across, as students, but even without buying in, we're welcome to investigate what makes it tick.

Kirby

[1]  speaking of Easter, here's an algorithm for computing what day of the year it falls, pretty much cut and paste from its source, but using // as the primary division operator to mean "integer division" e.g. 4 // 3 equals 1 with no "remainder" and 3 // 4 equals 0 as 3 has no 4s in it, not even one.



Using:
http://aa.usno.navy.mil/faq/docs/easter.php

@author: kurner
"""
import datetime

def from_year(y):
    """Please note the following: This is an integer calculation.
    All variables are integers and all remainders from division
    are dropped. For example, 7 divided by 3 is equal to 2 in
    integer arithmetic.
    """
    c = y // 100
    n = y - 19 * ( y // 19 )
    k = ( c - 17 ) // 25
    i = c - c // 4 - ( c - k ) // 3 + 19 * n + 15
    i = i - 30 * ( i // 30 )
    i = i - ( i // 28 ) * ( 1 - ( i // 28 ) * ( 29 // ( i + 1 ) )
        * ( ( 21 - n ) // 11 ) )
    j = y + y // 4 + i + 2 - c + c // 4
    j = j - 7 * ( j // 7 )
    l = i - j
    m = 3 + ( l + 40 ) // 44
    d = l + 28 - 31 * ( m // 4 )
    return datetime.datetime(y, m, d)

for yr in [2000 + x for x in range(21)]:
    easter = from_year(yr)
    print(easter.strftime("%a %m-%d-%Y"))

OUTPUT:

Sun 04-23-2000
Sun 04-15-2001
Sun 03-31-2002
Sun 04-20-2003
Sun 04-11-2004
Sun 03-27-2005
Sun 04-16-2006
Sun 04-08-2007
Sun 03-23-2008
Sun 04-12-2009
Sun 04-04-2010
Sun 04-24-2011
Sun 04-08-2012
Sun 03-31-2013
Sun 04-20-2014
Sun 04-05-2015
Sun 03-27-2016
Sun 04-16-2017
Sun 04-01-2018
Sun 04-21-2019
Sun 04-12-2020


[2]  The name “Euclid” is not mentioned in Greek texts of the Elements, 1 which acknowledge other authors, such as Theon, father of Hypatia. The origin of “Euclid” in Latin texts from the 12th c. could well derive from a translation howler at Toledo—“uclides” meant “key to geometry” in Arabic. The key “evidence” for this “Euclid” is an obviously forged passage in a late rendering of Proclus’ Commentary, which otherwise speaks anonymously of “the author of the Elements”, and propagates a contrary Neoplatonist philosophy declared heretical and cursed by the church. (pp 1-2, Zeroism and Calculus without Limits, C. K. Raju)

kirby urner

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Mar 27, 2016, 7:00:39 PM3/27/16
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On Mar 27, 2016 15:05, "Joseph Austin" <drtec...@gmail.com> wrote:
>
> In my opinion, it would be even more practical to measure angles in "revolutions",
> e.g. a right-angle is 1/4, sin (1/4) = 1.  This is essentially degree measure "normalized".

That's very like with tau as 1/4 tau is indeed a right angle in radians, instead of pi / 2.

Whatever fraction of the unit circle you've gone around, that's how many times tau as well.

Euler's e to the i tau is 1. How pretty right?

I feel I have sufficient backing to use tau for 2 pi any time. I might cite Sal in a footnote.

Kirby

Joseph Austin

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Mar 27, 2016, 7:01:03 PM3/27/16
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Peter,
Those "three chords" of music follow from a mathematical theory of harmony!
So what are the "three chords" of mathematics?
I have noted that you can go a long way to formalizing arithmetic with the Peano postulates and Ackerman's function:

When I was taking Advanced Calculus in college, one of my frustrations was that calculus seemed to not be rigorously defined from axioms as plane geometry had been in high school.  It always seemed to me that the theorems involved a certain amount of "arm-waving" and "tricks" that had to be memorized.  As I see it, the real problem is that the so-called "real" numbers aren't "real", they are hypothetical abstractions having no known physical correspondent.
(Although differential equations have been quite successful in modeling the "laws" of physics.)

But I digress.  I would suggest that those "chords" you seek are axioms.
If one is willing to accept Computer Science as "mathematics," we can point to significant progress reducing computation or "algorithmics" to "axioms" or basic building blocks. Indeed, programmers equipped with a few fundamental concepts can create virtually any information-processing application, the "new music" of mathematics. 
For example, consider Google's  "blockly"  and programming systems built with those concepts. For examples, see: [https://code.org].
Joe
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Ted Kosan

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Mar 27, 2016, 7:18:25 PM3/27/16
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Joe wrote

> I would suggest that those "chords" you seek
> are axioms. If one is willing to accept Computer Science as
> "mathematics," we can point to significant progress reducing
> computation or "algorithmics" to "axioms" or basic building
> blocks.

I strongly support this view. For example, an understanding
of the logical building blocks upon which elementary algebra
is based would enable students to learn elementary algebra
in a small fraction of the time than is usually needed to
teach this subject.

The time that using this more efficient approach would free
up could be used to teach students computer programming.

Ted

kirby urner

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Mar 27, 2016, 7:58:35 PM3/27/16
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I too advocate something similar, just that I don't see a need to reduce mathematics to any "one" set of axioms as if we were supporting a single monolith on top of them.

The analogy with board games, like one buys at the store, is more than skin deep. 

Every game comes with a set of rules and definitions, e.g. the rules of chess, and these needn't be "intuitive" (what's intuitive about the knight's move?) just "axiomatic" (accepted by all players as given). 

Then, in accordance with these rules, one comes up with "legal plays" which may be towards a goal, for example a solution to some problem.  The rules define a "rule space" or "language game" to adopt Wittgenstein's term.

Board games come piled high in the store, with no limit on the number of games one might play.  Axioms are like that:  an infinitude of permutations, some leading to more "playable" (e.g. "interesting") games than others.  Mathematics as an enterprise is far from "over".  New games get invented every day.

In my current, or rather just-completed classes (last Wednesday was a last in my course), I favored introducing some elementary Group Theory based on a Permutation type, where a permutation might be a specific mapping of a set of objects to themselves in a different order.  A permutation is a bijection relating a set of elements to themselves.

Such a permutation, using lowercase letters a-z + space, for example, might be used to encrypt a phrase, treating said permutation as a substitution code:

able was i ere i saw elba  <-- original message
xtlbzhxezozbcbzozexhzbltx  <-- enciphered with substitution code
able was i ere i saw elba <-- enciphered again, but with the inverse mapping, which reverses encryption

Permutations may be multiplied and inverted, and therefore divided and powered.  They define a group, with subgroups.  They may be expressed using cyclic notation.  This territory is well-known and main stream.

I find all this abstract algebra stuff way easier than a lot of the delta calculus (differential / integral calculus) curriculum, and it exercises programming skills just as certainly. 

So why not more Group Theory between Algebra and Calculus?  Couldn't we have a math track (call it lambda calculus?) that provided an alternative to the standard college prep route?

Whereas in the 1900s we may have had no practical goal in mind for such number and group theory, Euler's Theorem, totients, Euclid's "extended method" and so on, today we do:  understanding public key cryptography, embedded in every web browser. 

I know at least one high school textbook that takes this approach (building towards RSA, explaining how and why it works), but said text is used in one of the most exclusive / expensive private schools, which I guess means no one else is allowed to use it?  Sure looks that way.

The publishing world has its own ways of arm-twisting that don't have much to do with what's in the best interests of the student.  That's what it looks like from my angle.

Faculties that want more pay and more respect need to insist on their commensurate right to source the school's curriculum (a way of branding) and not have it handed to them by the various know-nothing state committees.  That'd be a different job description, no question about it.

Kirby

Peter Farrell

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Mar 27, 2016, 8:38:58 PM3/27/16
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The basics don't have to be quite so basic. Last night I hosted a Raspberry Jam at a nearby Coder School where I'm a code coach and there are always a number of beginners who sit down and learn the basics of programming in Python by flipping through my book and coding along. (I even sold a couple of copies!) I've heard some noise on another forum to the effect that the repeat loop "for i in range(n)" confuses kids, and should just be "repeat(n)". I would have agreed before I taught Python to dozens of kids, very few of whom ever had a problem remembering "for i in range..." Back to the knight's move: once you use it a few times it "makes perfect sense." 

Teach people the basics and if they use it and get used to using it, they'll be able to tackle the more abstract stuff. Whether they'll want to is another question. But do we start at algebra or the other end? My cousin is a successful engineer who was a teacher's nightmare because he saw no use in any of his classes, even math, until differential equations (often the last thing you study in 16 years of math). Finally it all made sense. "Why didn't they just teach me this to begin with?" Dave asks now. Of course he was an obsessive tinkerer who still lives cheerfully among piles of electronic parts. He saw differential equations on his oscilloscope every day without knowing it.

I think learning the Three Chords would benefit those who just need to know three chords, and would also be an excellent foundation for people like me who went on to learn twice as many.

I've warned you before, I'm serious about rolling up my sleeves and creating this. Maybe Maria will agree to work with me again. This could be so good for the math-phobic out there that this morning I thought of a catchphrase to use when advertising: "Must Hate Math." The other group, who answer the "Must Love Math" ad, will get exactly the same material. Haha

Anybody interested in contributing, or in hearing the progress as it goes along?  Shit's about to get applicable.

Peter

Ted Kosan

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Mar 27, 2016, 10:36:34 PM3/27/16
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Peter wrote:

> I've warned you before, I'm serious about rolling up my sleeves and creating
> this. Maybe Maria will agree to work with me again. This could be so good
> for the math-phobic out there that this morning I thought of a catchphrase
> to use when advertising: "Must Hate Math." The other group, who answer the
> "Must Love Math" ad, will get exactly the same material. Haha

I have come to understand that the main reason most students who hate
math hate it is because what they were taught was not math. For
example, students are taught the laws of algebra, but they are not
explicitly taught the rules of algebra. What are they taught that is
being called math? I don’t know. What would the game be called that
operated exactly like chess, except the rules for how the pieces are
moved were never written down and never taught explicitly? Maybe it
would be called "c_es_"?

I think most people would grow to hate any game that they were forced
to play for years without ever having been taught the rules of this
game.



> Anybody interested in contributing, or in hearing the progress as it goes
> along? Shit's about to get applicable.

According to Donald Knuth "Science is knowledge which we understand so
well that we can teach it to a computer; and if we don't fully
understand something, it is an art to deal with it. Since the notion
of an algorithm or a computer program provides us with an extremely
useful test for the depth of our knowledge about any given subject,
the process of going from an art to a science means that we learn how
to automate something."

Before you begin work on creating a class which teaches fundamental
"chords" that are in math, I recommend that you (and any person in
this group who is thinking on contributing to this project) try to
write a computer program that solves elementary algebra equations
using a step-by-step process which is similar to the way humans solve
equations. One reason for doing this is to significantly reduce the
probability of the class having the unfortunate effect of increasing
people's hatred of mathematics.

Here are some example problems to get started with if you or any of
the other programmers in this group (either individually or as a team)
decide to pursue this recommendation:

-57 = -(-p+1) + 2(6 + 8p)
-4n + 11 = 2(1 - 8n) + 3n
-6v - 29 = -4v - 5(v + 1)
-a -5(8a-1) = 39 - 7a
-4(1+a) = 2a-8(5 + 3a)


Ted

Peter Farrell

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Mar 27, 2016, 11:30:45 PM3/27/16
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Hi, Ted,

Completely agree with you about why people come to hate math, or what their school district could afford to call math.

Maybe I wasn't completely clear in my aims. I'm not trying to teach (as in "dictate to the student") every little thing in math. That is what we imagine we're doing now, and we're clearly failing. I'd like to provide people with the tools to explore math in a potentially more powerful way. In my book Hacking Math Class with Python I already tackled solving the elementary equation

ax + b = cx + d

by showing how to create a Python function. This to me is a chord that people learning to play songs will definitely come across. People learning algebra better be able to deal with that equation. I don't know whether your more complicated equations come up all that often except in algebra books. I'm sure they can be tackled using solvers like Maxima or WolframAlpha if they do come up. 

I just put your first equation into Sympy. I had to get the -57 on the other side, change p to x and put in multiplication signs so the computer knows "8x" means "8*x." The correct answer, -4, is shown.

solve(-(-x+1) + 2*(6 + 8*x)+57,x)
[4]

Should that really be the first thing you bring up and warn me NOT to begin creating this project until I solve? You seem to be concerned about my project "having the unfortunate effect of increasing people's hatred of mathematics."

I'd be happy to send you (or anybody interested) a pdf of my book to show what I've already addressed, and perhaps you'll better understand where I'm coming from.

Thanks for the input, 

Peter

Ted Kosan

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Mar 28, 2016, 12:40:11 AM3/28/16
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Peter wrote:

> In my book Hacking Math Class with Python I already tackled solving the
> elementary equation
>
> ax + b = cx + d
>
> by showing how to create a Python function. This to me is a chord that
> people learning to play songs will definitely come across. People learning
> algebra better be able to deal with that equation. I don't know whether your
> more complicated equations come up all that often except in algebra books.
> I'm sure they can be tackled using solvers like Maxima or WolframAlpha if
> they do come up.
>
> I just put your first equation into Sympy. I had to get the -57 on the other
> side, change p to x and put in multiplication signs so the computer knows
> "8x" means "8*x." The correct answer, -4, is shown.
>
> solve(-(-x+1) + 2*(6 + 8*x)+57,x)
> [−4]

Conventional CASs like Sympy and Maxima do not solve equations using a
step-by-step process that is similar to the way humans typically solve
equations. Students need to be shown each step that is taken when
solving an equation (and why it is taken) in order for them to fully
understand the mathematics that is being performed. Here is one
solution for the first equation I listed which contains all of the
solution's steps (although it does not include why each step was
taken):

http://p1.ssucet.org/tkosan/misc/mathfuture/steps/



> Should that really be the first thing you bring up and warn me NOT to begin
> creating this project until I solve?

Yes. One of my points is (and I am saying it with a smile on my face
and a twinkle in my eye :-) if a person can't write a program that can
solve elementary algebra equations similar to the way humans do (and
have it show all of the steps it took to solve it), that person does
not understand how elementary algebra works well enough to be teaching
elementary algebra to humans.

The reason I say this with a smile on my face and a twinkle in my eye
is because I could teach almost any experienced programmer how to
write such a program in a surprisingly short amount of time.



> I'd be happy to send you (or anybody interested) a pdf of my book to show
> what I've already addressed, and perhaps you'll better understand where I'm
> coming from.

I just purchased your book from Amazon, and I am looking forward to
reading it :-)

Ted

kirby urner

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Mar 28, 2016, 1:07:43 AM3/28/16
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On Sun, Mar 27, 2016 at 9:40 PM, Ted Kosan <ted....@gmail.com> wrote:

 
> Should that really be the first thing you bring up and warn me NOT to begin
> creating this project until I solve?

Yes. One of my points is (and I am saying it with a smile on my face
and a twinkle in my eye :-) if a person can't write a program that can
solve elementary algebra equations similar to the way humans do (and
have it show all of the steps it took to solve it), that person does
not understand how elementary algebra works well enough to be teaching
elementary algebra to humans.

Echoing what Donald Knuth was saying: even if we don't have it distilled to algorithms we may still teach it (whatever "it" is) as an Art, if not a Science. 

Passing on the art of chess playing, like doing algebra, required no "program" before computers, beyond following whatever rules *could* be taught.  We teach the liberal arts (math one of them?), always leaving room for intuition.

I suppose that's why it's 'The Art of Computer Programming' that Knuth wrote, and not the Science thereof. Our programs do not write themselves.  Our algorithms are not capable of inventing themselves.[1] 

Sir Roger Penrose thinks our innate ability to create new mathematics demonstrates our freedom from deterministic / algorithmic processes. 

We *really* think, and therefore perform non-computable tasks, whereas "artificial intelligence" is precisely what we're able to automate.

Kirby

[1]  a point I'm making at the end of this Youtube, about the role of the "first person":
https://youtu.be/6xQxhD29Rdc 
(some pro tips for Python teachers)


Ted Kosan

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Mar 28, 2016, 3:49:31 AM3/28/16
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Kirby wrote:

> Echoing what Donald Knuth was saying: even if we don't have it distilled to
> algorithms we may still teach it (whatever "it" is) as an Art, if not a
> Science.
>
> Passing on the art of chess playing, like doing algebra, required no
> "program" before computers, beyond following whatever rules *could* be
> taught. We teach the liberal arts (math one of them?), always leaving room
> for intuition.
>
> I suppose that's why it's 'The Art of Computer Programming' that Knuth
> wrote, and not the Science thereof. <snip>

Donald Knuth has stated “I'm not strong on logic, so TAOCP treads
lightly on this sort of thing.”
(http://www.informit.com/articles/article.aspx?p=2213858). I think the
main reason Knuth teaches programming as an art instead of a science
is because one needs to understand mathematical logic at a deep level
in order to teach programming as a science.

A person who does have a deep understanding of mathematical logic is
David Gries, and this understanding enabled him to write a book titled
"The Science of Programming". Here is a passage from this book:

"Programming began as an art, and even today [1981] most people learn
only by watching others perform (e.g. a lecturer, a friend) and
through habit, with little direction as to the principles involved. In
the past 10 years, however, research has uncovered some useful theory
and principles, and we are reaching the point where we can begin to
teach the principles so that they can be consciously applied." p.vii.

The fact that humans have written programs that can do step-by-step
elementary algebra equation solving proves that a science of
elementary algebra equation solving exists. Since a science of
elementary algebra equation solving exists, it is a disservice to
students to ignore this science and teach them the obsolete art of
elementary equation solving instead. To me, this is like doctors using
obsolete medical practices like bloodletting instead of modern
approaches to medicine when trying to cure a patient.

Ted

kirby urner

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Mar 28, 2016, 10:52:40 AM3/28/16
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On Mon, Mar 28, 2016 at 12:49 AM, Ted Kosan <ted....@gmail.com> wrote:
 
The fact that humans have written programs that can do step-by-step
elementary algebra equation solving proves that a science of
elementary algebra equation solving exists. Since a science of
elementary algebra equation solving exists, it is a disservice to
students to ignore this science and teach them the obsolete art of
elementary equation solving instead. To me, this is like doctors using
obsolete medical practices like bloodletting instead of modern
approaches to medicine when trying to cure a patient.

Ted

I agree that Computer Algebra Systems (CAS) are a cool technology and many aspects of mathematics are subject to algorithmic treatment. 

Euclid's Method, so-called, for finding the greatest common divisor of two natural numbers (which may be one if they're "strangers" i.e. relatively prime) is circled in one of Knuth's volumes as the very paradigm of what we mean by "algorithm" (named for our Bagdad House of Wisdom guy, Al Khwarizmi [1]).

I'd like to have a little Python or ClojureScript program that eats the equations you give and spits out solutions.  Sounds like you have one. 

Mostly we do not show students how to write such scripts, you're correct.  We show them a job we also know how to automate.  I don't blame them for squirming, feeling treated like robots, made to learn things machines will do faster and more accurately. 

But then a lot of things humans can do, machines can do also, yet that's not a reason to withhold the skill, nor a reason to hide how machines do it, either.  Lets both teach the skill and show how it's automated.  It's not either / or.

As for your more general claim that one needs a really deep appreciation of logic to automate their processes, I only somewhat agree. 

Keep in mind that hand-me-down numerical recipes (and cooking recipes too) are sometimes empirically fine tuned with no deep understanding.  They just work for some reason, and that's enough to code something on a computer, and it works there too, but exactly why, or how, no one knows for sure.  In sum, an automated process does not always require deep understanding to implement.

Drawing from C.K. Raju's ammo chest, one might also argue that no mathematician, living or dead, has a deep enough appreciation of logic to truly systematize mathematics ala the ill-fated Bourbaki project, given Western logic is two valued, whereas much of Eastern logic is four valued.[2] 

Raju insists that no one has yet united these logic systems or generalized from one to the others sufficiently to shift the weight of anything purely two-valued off its parochial foundation.  Among those castles "too fragile to move" would in Raju's assessment include most if not all of what we currently call "math" in the West. 

Our grasp of logic is still superficial in 2016 AD.  We have a long way to go.  We're in a Mystified Age.

Nevertheless we have many islands of Science whereon we may automate everything with clear understanding, such as algebraic equations in a CAS.  Most of what students encounter is ocean, with little ships dotting the landscape, the presumptive pundits (the teachers), captains of their own little testaments to pure reason. 

Some of these contraptions are actually sea worthy and convey valuable lessons about what it takes to survive at sea (students are lucky to take the journey), yet all have a half-life and/or TTL (time to live). 

Teachers themselves, the captains, not just students, need to jump ship on occasion, leaving what they maybe used to call a Science for some spanking new vessel. 

What once was a fancy Science in its day becomes yet another sinking belief system -- which belief systems also rise, like undersea volcanoes, to make our new Science islands.

Lets keep focusing on including more CAS.  Wolfram Alpha will solve an equation and then, for an extra fee, show it in steps.  I've footnoted a web page with examples. [3]  But Wolfram's is not the only CAS out there.  How to code these is knowledge that should spread.

Kirby

[1] a joke I tell:  "You know how they say Al Gore invented the Internet?" (smirks and nods), "Well it's actually more than that; why do you think they call it an "Al-Gore-ithm")".

[2] Raju doesn't harp on this but most computer languages, including SQL, are three value, having True, False and None (or Nil) plus come with a boolean mapping of just about anything but zero and empty objects to True.

[3]  https://www.wolframalpha.com/examples/StepByStepSolutions.html 

Ted Kosan

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Mar 28, 2016, 4:17:39 PM3/28/16
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Kirby wrote:

> Lets keep focusing on including more CAS. Wolfram Alpha will solve an
> equation and then, for an extra fee, show it in steps. I've footnoted a web
> page with examples. [3]

Wolfram Alpha can show the steps it took to solve an equation.
However, I have not seen any evidence that it is capable of explaining
why it took the steps it did. Without this capability, Wolfram Alpha
is not very useful for teaching the principles of elementary equation
solving.



> I'd like to have a little Python or ClojureScript program that eats the
> equations you give and spits out solutions. Sounds like you have one.
>
> <...>
>
> How to code these is knowledge that should spread.

I am indeed writing a step-by-step equation solver, and I agree that
the knowledge of how to write one should be spread. Are you interested
in being one of the first people to learn and then spread this
knowledge? From what I have seen, you have the right mix of computer
programming and mathematics background which is needed to teach this
knowledge effectively.

Ted

Joseph Austin

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Mar 28, 2016, 5:28:49 PM3/28/16
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Peter, Ted, Kirby,
I applaud your ideas.

1. Algebra is the formalization of word problems. Why confuse the issue with meaningless letters?
Use words (and axioms) from the start.
(When I studied algebra, we started with the axioms, but alas, also with abstract letters.)

2. I've believed we teach scientific math backwards. 
(One semester of calculus is as pointless as one semester of foreign language.) 
We should start with differential equations.
1. exponential growth. 2. constant acceleration: F = d(mv)/dt; 3. harmonic oscillator.  Etc.
Use difference equations and numerical methods to approach the limit.

Back in the 1960's we offered a course that took several physics problems, 
developed the math, then created nunerical simulations in FORTRAN.
It covered ordinary and partial differential equations, matrices and systems of linear equations, 
probability and monte carlo simulation.
[Did I mention this was a summer enrichment program for high school students?]
I have a copy of a text similar to one we used:
Herbert Peckham, "Computers, Basic, and Physics," Addison Wesley 1971

3. Group theory is fundamental. The permutation group is the fundamental (or universal?) group.
BTW, group theory figures in elemeentary particle physics.
As does differential equations.

4. Speaking of fundamental:
I've noticed a lot of "three operator" systems:
Logic: and, or, not
Boolean algebra: union, intersection, complement
Structured programming/Regular Expressions: sequence, alternative, repetition
Algebra: add, multiply, power

But conventional "algberas" typically restrict themselves to two operators.
Is there any fundamental theory of such "three-operator" systems?
Or, perhaps they can all be reduced to a single operator, as "and or not" can be reduced to "nand" or "nor".

As for textbooks, I once taught at a school that wouldn't allow changing textbooks in less than five years,
to save money.  It may have worked for Shakespeare, but wasn't very practical for computer science.

Peter, Not sure what I could contribute to your project, but I'm game.
And I'd love to have a copy of your book draft.

Joe Austin
DrTechDaddy at gmail dot com

Joseph Austin

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Mar 28, 2016, 5:45:02 PM3/28/16
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Ted, I'm catching up with this thread post by post.

When I was teaching programming back in the 70's,
I gave my data structures class an assignment to write a program to solve a linear algebraic equation for x.
They were to build the precedence tree, then prune and transpose the branches to isolate x.
I had just introduced the concept of structured programming, and most students had no difficulty completing the assignment in a week.

So this was basically an example of "playing the game by rules"
and "making the problem so simple a computer can do it."

BTW, transforming an equation from linear text to operator tree form is a practice familiar in computing
but I'm not sure it's used in math class.
(Speaking of which, when I was in elementary school, we "diagrammed sentences" by essentially transforming them into a "grammar tree" form.
I don't recall that this was being taught when my kids were in school.)

Joe

> On Mar 27, 2016, at 10:36 PM, Ted Kosan <ted....@gmail.com> wrote:

Joseph Austin

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Mar 28, 2016, 6:02:03 PM3/28/16
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On Mar 28, 2016, at 3:49 AM, Ted Kosan wrote:

Since a science of
elementary algebra equation solving exists, it is a disservice to
students to ignore this science and teach them the obsolete art of
elementary equation solving instead.

I'd go a step further. Since elementary algebra equation solving can be automated,
it is a disservice to spend time teaching students to solve algebra equations "by hand".
We spend  years drilling children on decimal-notation arithmetic algorithms that can be performed by a $5 machine.
I'd say, teach them the principles in binary notation (for basic understanding), 
then teach them how to set up problems to use a calculator.
As for algebra, give them an app for that, and teach them art of setting up word problems to use it.

Joe

kirby urner

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Mar 29, 2016, 1:29:06 AM3/29/16
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On Mon, Mar 28, 2016 at 1:17 PM, Ted Kosan <ted....@gmail.com> wrote:
 
> How to code these is knowledge that should spread.

I am indeed writing a step-by-step equation solver, and I agree that
the knowledge of how to write one should be spread. Are you interested
in being one of the first people to learn and then spread this
knowledge? From what I have seen, you have the right mix of computer
programming and mathematics background which is needed to teach this
knowledge effectively.

Ted

I appreciate your confidence in my abilities.

As Joseph remarks, turning a linear (in the sense of character string such as "-57 = -(-p+1) + 2(6 + 8p)") into operator tree form. I was a philosophy student studying logic more than a computer science student learning to write a compiler, back when surrounded by professors in university, so might end up reinventing some wheels. Does your solution come from a literature I should bone up on?

Let me scan Youtube for video squibs on CAS:

https://youtu.be/DWU2tKlacZE 

I learned a lot from this gent, charming young guy. 

Symengine replaces SymPy with faster C++ code harnessed using Cython. However I only have the latter on my Mac, perhaps not available through the conda infrastructure -- I see it only for Windows and Linux64.

Here's a screen shot from copying one of his examples:
https://flic.kr/p/ERETXX

Then this guy does a quick tour using some in-determinant CAS:
https://youtu.be/yZR5QGlX_9M

... And why not Wolfram himself, lets listen:
https://youtu.be/n2XDQ08dyXk
(he battles his computer quite a bit, manages to win)

He says the "step by step" view in Wolfram Alpha is "completely fake" meaning it's not showing the steps it actually uses, but the steps a human might use. That still doesn't mean it provides line by line justifications.

Minutes 47-54 have video and audio issues.  He gets into his new kind of science cellular automata theme / meme.  I wrote some Python based on his book when it first came out, a handsome volume.

OK, I know a little more than I did when I started this homework assignment.  Lots to know huh?  And I've only just scratched the surface.

I went down to the code school tonight for the Monday Python meetup.  I was a volunteer mentor for a few people.  I showed one guy my thekirbster.pythonanywhere.com which lets you move the knights around -- that's all that move.  Then back here to study CAS concepts.  Is this a boot camp or what. :-D

Kirby

Ted Kosan

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Mar 29, 2016, 4:51:48 AM3/29/16
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Joe wrote:

> BTW, transforming an equation from linear text to operator
> tree form is a practice familiar in computing but I'm not
> sure it's used in math class.

My understanding is this kind of transformation is not
taught in math classes. However, I think it should be.



> [...] when I was in elementary school, we "diagrammed sentences"
> by essentially transforming them into a "grammar tree" form.
> I don't recall that this was being taught when my kids were
> in school.

Mathematical expression trees, natural language grammar
trees, and computer programming abstract syntax trees have
so many features in common that I think they should be
taught together.



> I'd go a step further. Since elementary algebra equation
> solving can be automated, it is a disservice to spend time
> teaching students to solve algebra equations "by hand".

Wolfram Research's "Computer Based Math"
(http://computerbasedmath.org/) approach to teaching
mathematics appears to use this philosophy. I went
through a period when I thought this way too. However, I
changed my mind and decided that people should not use
software that automates a given area of mathematics until
they understand the principles of how this software works.



> We spend years drilling children on decimal-notation
> arithmetic algorithms that can be performed by a $5 machine.
> I'd say, teach them the principles in binary notation (for
> basic understanding), then teach them how to set up problems
> to use a calculator.

I agree with this.


Ted

Joseph Austin

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Mar 29, 2016, 1:01:36 PM3/29/16
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On Mar 27, 2016, at 7:58 PM, kirby urner <kirby...@gmail.com> wrote:

The analogy with board games, like one buys at the store, is more than skin deep.  

Every game comes with a set of rules and definitions, e.g. the rules of chess, and these needn't be "intuitive" (what's intuitive about the knight's move?) just "axiomatic" (accepted by all players as given).  

Then, in accordance with these rules, one comes up with "legal plays" which may be towards a goal, for example a solution to some problem.  The rules define a "rule space" or "language game" to adopt Wittgenstein's term.

Board games come piled high in the store, with no limit on the number of games one might play.  Axioms are like that:  an infinitude of permutations, some leading to more "playable" (e.g. "interesting") games than others.  Mathematics as an enterprise is far from "over".  New games get invented every day.

Kirby, and Ted et al,

re Axiom Games

I'm envisioning applying axioms and turtles to "old math" and "new math".

Axiom systems may be thought of as rewriting rules for arrangements of symbols.
Rewriting systems figure prominently in computer science, particularly in the theory of programming languages.
The fundamental theories of computing are also expressed as rewriting (axiom) systems:
Lambda calculus and the Turing machine.
A significant difference between traditional mathematical axioms and computing algorithms is that axioms are declarative 
but algorithms are imperative or generative.  

Recall that a Turing machine consists of 
a "turtle" that remembers s single-character "state" 
and roams back and forth on 
a string or unbounded "tape" of characters,
following a table of rules or instructions that specifies, 
based on the its current state 
and the character on the currently occupied cell of the tape, 
what character to write on the tope 
and which direction to move to occupy the next space.

It has been hypothesized and never disproven that anything that can be computed, can be computed with such a device and an unbounded tape.

So, since the Turing machine (an axiom system) can compute "anything", it can surely do arithmetic:

Old Math:

I imagine that, if the Turing turtle were to roam over a cartesian grid of squares,
that all the algorithms of conventional decimal-system arithmetic taught in elementary school could be modeled with such a device.
That is, do it the way the digits are laid out and written down in rows and columns on the blackboard or paper, 

So, what if those elementary school children were taught the Turing Game,
and challenged to create a Turing Turtle to do the arithmetic problems?
(For practicality, I would start with binary arithmetic.)

"New Math"

Traditional math has been divided into silos.
Arithmetic  about quantity: number, counting, etc.
Geometry is about shape.
Algebra is about symbols and  and rearranging symbols according to axioms.
Computing is about sequencing operations: algorithms.
Calculus (differential equations) is about inferring global macro characteristics from local micro properties and their compatible boundary conditions.

Meanwhile, nature offers a system involving quantities, shapes, arrangements, sequential transformations: molecular biology.

I would suggest that molecular biology should serve as an organizing model for the "new mathematics".
The cell operates with a helical molecule of DNA to construct proteins from amino acids which are coded in triples of four nucleic acid "digits".
The DNA "digits" are not merely representational, but operational, in that their bonding properties actually effect the transformations that they represent.

("Three apples plus two apples" don't "make" five apples;
but one apple seed plus soil and water and air do literally make bushels of apples!)

So suppose we extend our concept of "mathematics" to include rules (axioms) for creating and transforming general 3D arrangements of symbols, 
somewhat akin to the way nature arranges Hydrogen, Oxygen, Carbon, etc. into DNA, proteins, etc. 
Such a system would combine and subsume features of counting, shape, symbol arrangement, algorithms, and generation through boundary matching that characterize current mathematical sub-disciplines.

Now I need to find a good elementary book on molecular biology!

Joe











Peter Farrell

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Mar 29, 2016, 1:22:39 PM3/29/16
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Great ideas, everybody, even the ones I don't agree with (yet).

What if you only had 5 days to get somebody up and running in math (algebra to trig) using tech? Assume they are not math-friendly but want to change that.

My goal is to teach the "three chords of math" as quickly as possible and give the participants the time to practice using them. There's no reason it has to be 3 chords, but that's the traditional blues/rock stereotype.

I've come up with: 

Solvers (writing programs to solve equations, convert units and generally return values)

Graphers (visualization in general, including turtles, 3D graphics packages like VPython and Pi3D)

Storage (variables store 1 value, lists more than one, and arrays store multidimensional values)

We bounce back and forth between showing them how to put together the chords and challenging them to do it themselves.

Again, the time limit is going to rule out lots of otherwise important material. I have never forgotten this scene from Apollo 13 where the ground crew had to quickly come up with a rescue plan using only the materials the rocket crew had available. 

Thanks!

Peter

kirby urner

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Mar 29, 2016, 3:22:46 PM3/29/16
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All good thinking Joe, got me remembering back to my days at FreeGeek trying to wrap my head around  mod_rewrite in apache web server (connecting to your axioms as "rewriting" rules).

Genetic code:  yes!  8x8 = 64 codons, all triplets of GATC, 4 to the 3rd (a tetrahedron for me, given a penchant for tetravolumes).

http://images.slideplayer.com/24/7225764/slides/slide_12.jpg
http://beacon-center.org/wp-content/uploads/2013/07/Expanded_Genetic_Code.png

I lean on Turtle class turtles and LOGO (great history!) to bring in the more adult-sounding "Tractor class" and/or Python co-routine (built around yield and send in some Python5 implementations).

Tractor-as-avatar brings in agriculture and my code schools seem tilting towards "Internet of Things meets food growing" (as in Cuba [1]) i.e. software as an adjunct to farm management as a core curriculum template, but extensible to myriad other story problem universes.

LOGO had just the one Turtle so FD 20 was unambiguous.  Nowadays we "instance a Turtle" meaning create an instance of the Turtle type, as many as we like.  the_turtle.forward(20) then.

Our code school grids are not always of XY squares as sometimes we navigate on a plane of hexagons (many board games feature these, especially war simulations, to allow flanking movements).

We zoom back to show the "high frequency hexapent" on the cloud server:

http://www.4dsolutions.net/ocn/hexapent.html
http://hexagonalawarenessproject.tumblr.com/
http://www.dividedspheres.com/

Your talk of Turing Machines got me thinking back to last nights viewing. 

After the Wolfram-in-Mumbai tape, this one with Sir Roger Penrose queued itself up and played:

https://youtu.be/4YYWUIxGdl4  (he talks about a Turing Machine quite a bit).

I've seen him talk live a few times too, great presenter, really good with those overheads.

I even ran the mic in one of his sessions, at the Oregon Math Summit '97, where I too was a presenter:

http://math.oregonstate.edu/oregon-math-summit

Your molecular-based approach syncs well with one Linus Pauling on my end, a character much blogged about in my on-line journals. 

You've likely heard of him.  One of our study groups meets in his boyhood home, just blocks from our home office.

Kirby

[1]  http://worldgame.blogspot.com/2015/12/wanderers-20151215.html


Joseph Austin

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Mar 29, 2016, 8:15:04 PM3/29/16
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Kirby, 
I enjoyed the Penrose on the Big Bang, but didn't hear much about Turing machines.

I never "got" the second law of thermodynamics.
Isn't it a tautology:  "more-probable states are more probable"?

Neither of which have much to do with teaching math to kids.
Perhaps we have the same dilemma as teaching computing.

Most people want to learn how to *use* computers to solve other real-world problems.
Computer Scientists (those who teach computing) on the other hand, are more interested in how computers actually work.

As mathematicians, we delight in discovering the implications of our axioms.
But our students (will) want to know how much the prom will cost, or how soon their college loan will be paid off.

The part of math I was never taught, but which is the part most sorely needed (IMHO),
is how to turn a real-word problem, a "word problem" if you will, into mathematical formulas.
Or a computer algorithm.

If our politicians knew how to do that, we might be able to balance our budgets!
As it is, one group wants to balance the budget by cutting taxes,
the other wants to balance it by increasing spending,
then they compromise and do both!

(An Excel textbook I once used had an example of computing profit as sales minus expenses.
Expenses were budgeted as fixed or various percentages of sales.
(The way, unfortunately, some businesses do budgets.)
But there were no formulas relating sales to "expense", i.e., investment.
So ROI would increase when you reduced advertising and bonuses!)

Joe

kirby urner

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Mar 30, 2016, 11:28:28 AM3/30/16
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On Tue, Mar 29, 2016 at 5:14 PM, Joseph Austin <drtec...@gmail.com> wrote:
Kirby, 
I enjoyed the Penrose on the Big Bang, but didn't hear much about Turing machines.


You're very right Joe, my mistake.  I went back in my browser history and found this one:

https://youtu.be/eJjydSLEVlU

He starts into the Turing Machine at about 12 minutes.  I was confusing this one with the Before the Big Bang one.  My apologies.

 
I never "got" the second law of thermodynamics.
Isn't it a tautology:  "more-probable states are more probable"?


Plus it seems like quantum mechanics is telling us that things are tilted towards the improbable in the sense that our theory of randomness reflects a lower probability than we think it should (quantum events are correlated more than they "should be" based on pre-QM statistical analysis).  So does that suggest we're not the authorities, when it comes to what's most probable?  Our physics is still too weak?  Wouldn't be the first time, looking back.

 

Neither of which have much to do with teaching math to kids.
Perhaps we have the same dilemma as teaching computing.

Most people want to learn how to *use* computers to solve other real-world problems.
Computer Scientists (those who teach computing) on the other hand, are more interested in how computers actually work.


There's a lot of "trying to meet the customer half-way".  For example the hype around Object Oriented Programming was you could finally focus on the target knowledge domain and stop trying to think so hard about computer architecture.  This was spun as a kind of liberation theology.  We are no long slaves to the registers.  We get to forget what a CPU is.

To some extent that was not just hype.  A high level programming language offers libraries for digging down into hardware but does not insist on this focus as much as the lower level ones do.


 
As mathematicians, we delight in discovering the implications of our axioms.
But our students (will) want to know how much the prom will cost, or how soon their college loan will be paid off.


C.K. Raju was suggesting most our math never came from axioms in the first place but was post hoc provided with that theoretical underpinning to make it prettier and more persuasive.  A lot of the substance came through the door at first used, but as yet unproved.  Proof follows practice. 

We still have some rather unproved-looking math in the mix, e.g. some of those convergences by Ramanujan.  They just work, he never explained step by step how to get them.  Modern math suggests this is true:  axiomatic foundations may provide an incomplete map to what's used in the higher reaches.

What I tend to put stress in is the value of a theorem or algorithm, its utility, as distinct from its possibly several proofs.  How does V + F == E + 2 help us?  Then we can talk about "why so".
 
The part of math I was never taught, but which is the part most sorely needed (IMHO),
is how to turn a real-word problem, a "word problem" if you will, into mathematical formulas.
Or a computer algorithm.


Yes, and it helps to see a lot of such word problems.  At some point you might write a computer program to spit them out:

story_stuff = {}  # <-- key:value pairs
story_stuff["actor"]=choice(["Farmer John", "Sally", "Betty", "Elmo"]) # first key:value
# steps elided
print("{} had {} {} and wanted to divide them into {} equal groups.  Will there be a remainder?  What will it be if there is one?".format(*story_stuff)

Example output:
Farmer John had 12 elderberries and wanted to divide them into 4 equal groups.  Will there be a remainder? What will it be if there is one?"  Answer:  no remainder.


 
If our politicians knew how to do that, we might be able to balance our budgets!

I doubt it.  Debt is the whip that drives the action.  Nations must owe, big time.  That way we own them.
 
As it is, one group wants to balance the budget by cutting taxes,
the other wants to balance it by increasing spending,
then they compromise and do both!


Increasing borrowing is what actually happened.  And not "from ourselves" (that never meant anything).

 
(An Excel textbook I once used had an example of computing profit as sales minus expenses.
Expenses were budgeted as fixed or various percentages of sales.
(The way, unfortunately, some businesses do budgets.)

I saw an Economics textbook that wrote off "expected profit" as not worthy of counting as investor return i.e. of course there's that 3% we'd expect but what about the real deal?  Meanings change over time.
 
But there were no formulas relating sales to "expense", i.e., investment.
So ROI would increase when you reduced advertising and bonuses!)

Joe


I've open sourced a business model wherein patrons winning computer games get to commit real money to a cause or group on the management's list (one knows going in what the choices will be); with patrons building their profiles based on philanthropic giving. 

You buy a cupcake, win a game, and as a result get to donate ten cents to the World Wildlife Fund, that ten cents coming from the cupcake company's charitable giving fund.  You're their champion within the scope of play.  It's like being the state when winning the lottery money (more than players do) and getting to earmark the funds towards a worthy cause, like a state legislator.

That's a starting point.  I call it Coffee Shops Network.

Kirby

kirby urner

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Mar 30, 2016, 1:51:57 PM3/30/16
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I really like Peter's idea of "three chords" as a basis for ongoing invention, of what we might call "rock music" or whatever.

Three chords that resonate for me would be:

(A)  e to the i tau equals one
(B)  data structures (same as Peter)
(C)  dx/dt or equivalently manga versus anime (time and motion, versus still or static)

What I get from (A) is high school math up through complex numbers, including polyhedrons as graphs and rotation matrices / vectors.  We've got trig, convergence, divergence, oscillation and chaotic sequences.

(B) gives me persistence, formats for saving and sharing data, as well as algorithms (programs as data).  I have permission to speak of file trees and databases (SQL and noSQL).

Whereas:

(C) introduces the calculus, formalisms around change, both determinate and not, and in the form of physical units such as "action" = mvd per time unit.  Weights and measures.  Money.  The "real world".

Another way to write those units: (mvd)(f) as in hf = E where h is Planck's Constant (unit of action) and f is frequency (1/t, as in Hertz), E is energy or work (I'm just talking units here with mvd = m(d/t)(d), again the unit of action.  mvdf = mvd/t = E so mvd = Et (f = 1/t).  That's where we get the joule-seconds.

"Action has the dimensions of [energy]·[time] or [momentum]·[length], and its SI unit is joule-second." says Wikipedia. 

https://en.wikipedia.org/wiki/Action_%28physics%29#Mathematical_definition

Not to be confused with, and yet creatively connected with:
https://en.wikipedia.org/wiki/Unit_of_action  (as in theater, scripts being algorithms along which work is expended, results obtained -- as in a programme).

Even in theater lets think of a particle of action as a frame of film, a delta-t (timedelta), in we have some action going i.e. some mvd per frame-going-by t, some E = mvv. 

I'm suggesting a frame of film is like an "energy bucket" and the rate at which these buckets go by is E/t or formally Power (such as wattage). 

When we see something happening "too fast" (as when a film is sped up) we might object that a car, for example, is "not that powerful" (science fiction permits selective breaking of rules, "what if" projections).

Kirby

Joseph Austin

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Mar 31, 2016, 12:49:15 PM3/31/16
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I'm not clear to me what you hope the end result of the "five days" to be.
Is this "remedial math" to get them ready for calculus, or "college" algebra, or just pass the math requirement?
Or is it more along the lines of "even liberal arts majors can learn to love math"?

A couple of times, I had five days with a group of kids, to teach "computing".

The first time, I taught them Logo and Star Logo, among other things.
In Star Logo, you can do a lot of "differential equation" type stuff: rockets, diffusion, and probability, predator-prey problems.
My goal was to give them a "taste" of computing in hopes of luring some into Computer Science.

Another time, I taught Lego Robots.
Actually, I had an extra weekend before the kids came to teach the parents,
so I would have enough adults to do one-on-one.
(I told them, if your kids can do this, so can you.)
Lego Robots is essentially LOGO made real.
My goal here was to give some disadvantaged kids a taste of "tech" 
and perhaps some hope that they could "make it" in a tech world.

Another time, in a classroom situation, I did a unit on recursive functions with Scheme.
IMHO, that's about as deep into "real math" as you can get--deeper than most standard math curricula!
Build math, solvers, etc. from Peano up.

In all of this, I was trying to be relevant to kids whose "math" skills and/or motivation were weak,
in hopes of luring them in with non-traditional applications of math concepts.

FWIW, these sorts of projects play well to grant-givers.

Joe Austin


Peter Farrell

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Mar 31, 2016, 2:15:35 PM3/31/16
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Hi, Joe,


On Thursday, March 31, 2016 at 9:49:15 AM UTC-7, Joseph Austin wrote:
I'm not clear to me what you hope the end result of the "five days" to be.
Is this "remedial math" to get them ready for calculus, or "college" algebra, or just pass the math requirement?
Or is it more along the lines of "even liberal arts majors can learn to love math"?

 I'm thinking more along the lines of the "liberal arts" one, but it should be a great intro for anyone. Given the right Python tools and ensuring they do their classwork/homework, more math-friendly folks can take it as far as they want. Then we sell them the second course.

The "Must Hate Math" contingent can learn to own math by creating programs to do solving and graphing. They may never love it (or they might!) but if they stick it out (and practice) they'll get confident enough to use it.

Peter

Peter Farrell

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Mar 31, 2016, 2:40:30 PM3/31/16
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On Thursday, March 31, 2016 at 9:49:15 AM UTC-7, Joseph Austin wrote:

The first time, I taught them Logo and Star Logo, among other things.
In Star Logo, you can do a lot of "differential equation" type stuff: rockets, diffusion, and probability, predator-prey problems.
My goal was to give them a "taste" of computing in hopes of luring some into Computer Science.

StarLogo was my first experience with Logo programming! I sat and copied the code out of Papert's Mindstorms page by page. StarLogo froze too much, so I switched to NetLogo, but yes, I've used Logo to do all kinds of fun models. I really like the idea of a variable as a slider: you can change the value of a variable manually! 

Right now I'm also working on the documents for a summer camp in Python, of course using turtles (in Trinket windows).

kirby urner

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Mar 31, 2016, 4:07:18 PM3/31/16
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On Wed, Mar 30, 2016 at 10:51 AM, kirby urner <kirby...@gmail.com> wrote:

I really like Peter's idea of "three chords" as a basis for ongoing invention, of what we might call "rock music" or whatever.

Three chords that resonate for me would be:

(A)  e to the i tau equals one
(B)  data structures (same as Peter)
(C)  dx/dt or equivalently manga versus anime (time and motion, versus still or static)


Here's a simple Jupyter Notebook themed around (A). 

I'm mostly just showing how Greek letters may be used directly (these symbols are a part of Unicode) and how plotting comes into it (using matplotlib).

https://github.com/4dsolutions/Python5/blob/master/Euler's%20Formula%20Using%20Tau.ipynb

(renders directly at my public Github account)

The average profile of one of my students is an adult switching to a new career and drawing on math remembered from high school to get up to speed in a coding language. 

They might be in a 12 week boot camp, some 60 hours of immersive training.

If they don't remember the math from high school, or never had it, they learn it newly. 

We might just call it "STEM stuff" as it's a wide mix of topics, including genetics. 

I just invested in this tome:

https://flic.kr/p/EUmhCN  (Python Programming for Biology:  Bioinformatics and Beyond)

Kirby


Joseph Austin

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Mar 31, 2016, 5:38:34 PM3/31/16
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Suggestion:
1. visit the future employers of your students
2. ask the employees (not the boss) what "math"--i.e. "quantitative reasoning" or more generally "problem solving"--they do in their jobs
3. teach that
Or more specifically, teach them how to do that "problem solving" with available technology.

(When I was in TX we had a "community advisory board" to advise the local tech college what to teach.
The idea was to train the local population for local jobs.)

Joe Austin

kirby urner

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Mar 31, 2016, 9:23:58 PM3/31/16
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On Thu, Mar 31, 2016 at 2:38 PM, Joseph Austin <drtec...@gmail.com> wrote:
Suggestion:
1. visit the future employers of your students


Exactly right. 

I was sharing with my peer Patrick just hours ago that I'd like to interview the bioinformatics heads at Oregon Sciences Health University (OHSU) about their future employee needs. 

You're telepathic!

OHSU has been advertising for analyst jobs that have gone begging since December.  There's a disconnect twixt what we teach, and what industry is begging for.

 
2. ask the employees (not the boss) what "math"--i.e. "quantitative reasoning" or more generally "problem solving"--they do in their jobs


We know stuff around genetics is important, in that line of work.  Bioinformatics.

So... have we all seen the movie Gataca yet?  www.imdb.com/title/tt0119177/

Starring Uma Thurman. 

( Her dad's a lama dontcha know.  https://en.wikipedia.org/wiki/Robert_Thurman )
 
3. teach that
Or more specifically, teach them how to do that "problem solving" with available technology.


Reading PDB files may be important.  I'm just starting to tune those in.

https://en.wikipedia.org/wiki/Protein_Data_Bank_%28file_format%29

Regardless, saving files that contain enough information to construct a polyhedron is uber-basic.

Using algorithms for finding a convex hull, given points in space -- that's been an area I've worked in.

http://4dsolutions.net/ocn/wgraphics.html

 
(When I was in TX we had a "community advisory board" to advise the local tech college what to teach.
The idea was to train the local population for local jobs.)

Joe Austin

Exactly right.

A good beginning but we need to expand.


Kirby


Joseph Austin

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Apr 1, 2016, 7:32:46 PM4/1/16
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Peter,
Some thoughts on 5-day math course.

On Mar 27, 2016, at 3:31 PM, Peter Farrell <peterfa...@gmail.com> wrote:

kirby urner

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Apr 1, 2016, 8:22:37 PM4/1/16
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On Fri, Apr 1, 2016 at 4:32 PM, Joseph Austin <drtec...@gmail.com> wrote:
Peter,
Some thoughts on 5-day math course.



I see you're quoting my posting in your write-up:

“An adult switching to a new career and drawing on math remembered from high school to get up to speed. If they don’t remember the math from high school, or never had it, they learn it.”

My impression is Peter is working more with younger people.  I'm the one who switched from pedagogy to andragogy awhile back -- not that the techniques or content is always that different.

With adults, we tend to bare down hard on SQL, which one may treat as entirely divorced from mathematics but which is all about the union and intersection of sets, so I don't see a reason to.

Pretty much all reading and writing of records at scale involves databases of some kind, just as communications involve web page creation. 

Adults without reading or writing skills (i.e. SQL) or basic communications skills (i.e. web) have a harder time getting employment.  We have to get to these topics immediately, per advisors. 

Fortunately, the story problems are mathematical in nature, so we can leverage high school math, reinforcing something they may already know in order to tackle something that may seem more alien.

Games, as in board games, are also important, as these double as "simulations" and involve random chance.  Learning to code is all about learning to model in terms of following rules i.e. what delta over here will feed into that delta (change) over there? 

Computer science has its fingerprints all over this area, but then so does mathematics.

Kirby


Joseph Austin

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Apr 2, 2016, 11:18:35 AM4/2/16
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On Apr 1, 2016, at 8:22 PM, kirby urner <kirby...@gmail.com> wrote:

I see you're quoting my posting in your write-up:

“An adult switching to a new career and drawing on math remembered from high school to get up to speed. If they don’t remember the math from high school, or never had it, they learn it.”

My impression is Peter is working more with younger people.  I'm the one who switched from pedagogy to andragogy awhile back -- not that the techniques or content is always that different.

I'm supposing the sort of person who would sign up for a 5-day math course not interested in a standard 16-week curriculum.
Regardless of age, I expect they would respond to being treated as an adult with a purposeful goal, not as a child being required to jump thru academic hoops.

Joe

Joseph Austin

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Apr 2, 2016, 11:34:04 AM4/2/16
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On Apr 1, 2016, at 8:22 PM, kirby urner <kirby...@gmail.com> wrote:

With adults, we tend to bare down hard on SQL, which one may treat as entirely divorced from mathematics but which is all about the union and intersection of sets, so I don't see a reason to.

Pretty much all reading and writing of records at scale involves databases of some kind, just as communications involve web page creation.  

Adults without reading or writing skills (i.e. SQL) or basic communications skills (i.e. web) have a harder time getting employment.  We have to get to these topics immediately, per advisors.  

Relational Algebra is another tack.  It's not well-integrated into most programming languages of my era,
though perhaps it could be integrated into a language like APL, or an expanded LISP/SCHEME.

What CS emphasizes that "math" tends to leave out is the conditional: SELECT, or "such that".
Probably it doesn't fit well with Analysis because it renders functions discontinuous.

So bottom line, I think there's a gulf between "discrete sets" and "numbers".
My previous offering was for "numerical" math, 
but I agree a "discrete set" math is also needed.
And perhaps the two meet in "statistics".

Classification. Quantification. Probability.

A three-word history of human thought!

Joe



kirby urner

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Apr 2, 2016, 12:59:08 PM4/2/16
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On Apr 2, 2016 8:34 AM, "Joseph Austin" <drtec...@gmail.com> wrote:
>
>
>> On Apr 1, 2016, at 8:22 PM, kirby urner

>> Adults without reading or writing skills (i.e. SQL) or basic communications skills (i.e. web) have a harder time getting employment.  We have to get to these topics immediately, per advisors.  


>
>
> Relational Algebra is another tack.  It's not well-integrated into most programming languages of my era,
> though perhaps it could be integrated into a language like APL, or an expanded LISP/SCHEME.
>

I don't think we're waiting for more adequate computer languages. More adequate languages will come along but that's not the bottleneck. The current crop of languages are sufficiently capable, as well as teachable.

> What CS emphasizes that "math" tends to leave out is the conditional: SELECT, or "such that".
> Probably it doesn't fit well with Analysis because it renders functions discontinuous.
>

Discrete math,  Boolean Algebra, stuff like that.  It's part of ordinary language, to filter on criteria so SQL doesn't need to come across as highly abstract. It's about as concrete an application of a discrete math language as it gets.

> So bottom line, I think there's a gulf between "discrete sets" and "numbers".

Not a problem except in theoretical foundations maybe.

But we're not that focussed on, or obstructed by, foundations (taking a cue from C.K. Raju).

The main goals are
(A) explaining how things work and
(B) gaining the ability to manage records on an industrial scale (a scale schools have traditionally been concerned with).

We need to get work done. These students need jobs.

> My previous offering was for "numerical" math, 
> but I agree a "discrete set" math is also needed.
> And perhaps the two meet in "statistics".
>

I don't think this is an insurmountable issue. Discrete Math. Digital Math. Computational Math. Or just call it all Lambda Calculus, as distinct from continuous Delta Calculus (resurrecting an under-used term in the process).

We don't have to adopt the NCTM belief system that veering into discrete math is a dive into a "not math" subject area (computer science). It's all STEM. Turf battles need not occlude our thinking.

> Classification. Quantification. Probability.
>
> A three-word history of human thought!
>
> Joe
>

In a framework I wrote, the four main math categories are (idiosyncratically, not meant for uniform imposition on vast territories):

Martian Math (futuristic);

Neolithic Math (timeline back, to early beginnings);

Supermarket Math (business, commerce, accounting, transportation, communications);

Casino Math (risk management, stats and probability, speculation and venturing).

Neolithic to Martian is a full timeline so in principle extends to any STEM toxic we wish.

The Supermarket-Casino axis is the practical math of any age. I don't mention warfare or military math per se. The same challenges of risky venturing and supply chains, gamefication, apply.

Some code school could snarf this up as one more mapping among many. My own students tend to hear about this approach.  Some of my students are likewise teachers.

Kirby

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