Teaching kids real math with computers: Conrad Wolfram - TED talk

[David Weksler]

Hi Folks -

Rather impressive presentation on math reform and use of computers in
math education:

http://blog.ted.com/2010/11/15/teaching-kids-real-math-with-computers-conrad-wolfram-on-ted-com/

-- David

[Dani Novak]

I find it not just impressive but also true especially near the end.  To me ti is all unfolding human consciousness and evolution but most of my colleagues do not feel that way and some students do neither.  I just read this comment from a student.  Oh Lord... at least she is honest....

I will be 100% honest with you.  Maybe it is because I have been sitting at my computer for almost 2 hours working on something that should only take 2 minutes, I am not a fan of this website at all.  Like I have said many times before, I hate computers and I feel as though the assignments that go along with doing the website do not benefit from ‘playing’ on the website.  I think that all the work can be done by hand or in your head (maybe even with the help of a calculator).  By using the computer, it takes up more time than it should, while giving me a headache.  I apologize if this offends you.  I would rather be honest.

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Dr.  Dani Novak
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[Edward Bujak]

David,

Great video.  Thanks for the link.

Excellent points are made throughout the Ted Talk by Conrad Wolfram, but I like the one at about the 12 minute mark when he said "to understand math, program it" which is certainly true for any topic when you program it, but more so for analytical subjects.  The other big take-away is that computers liberate the human from the drudgery of math intensive laborious calculations so that humans can do the more cognitive higher-level thinking that we need to do and that only humans can do.

--Ed Bujak

> Date: Mon, 15 Nov 2010 20:21:19 -0800
> Subject: [Math 2.0] Teaching kids real math with computers: Conrad Wolfram - TED talk
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> To: mathf...@googlegroups.com

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[Maria Droujkova]

We talked with John Sharp yesterday about his upcoming Math 2.0 event (January 15th, "Bridges" math art conference and his projects, including Sliceforms and anamorphic art). Among other things, math models came up. John said that up a hundred years ago, making physical models of math entities was considered important for understanding, and then pure symbolic work became more valued.

Programming, to me, is a type of modeling. It shares many of its benefits with other methods of modeling and re-presenting, from paper crafts to making a movie or storytelling. However, because programming is now seen by the humanity as a huge value and a lucrative occupation, it has more chances to find its way into curricula than, say, origami.

Another "hands-on modeling" event I want to schedule soon is with Bonita Lawrence, who builds analog machines for solving differential equations:
http://www.marshall.edu/mu-advance/news-MU-Differential-Analyzer-Grand-Opening.asp
http://www.youtube.com/watch?v=NmX151Jd3_o
Take a look at THAT, people who argue that manipulatives are for young kids or newbies only!

Cheers,
Maria Droujkova

Make math your own, to make your own math.

[Bradford Hansen-Smith]

Maria, John Sharp is right about origami;  the square is only a part of the circle and does not generate enough information to make it a worthwhile teaching tool beyond the specialized area it now occupies. I will venture to say from my experience that folding circles is and will eventually will be a valuable teaching tool for mathematics, particular for lower and middle grade level students. I do not think anyone can argue this since I do not know anyone that has spent enough time comprehensively exploring the circle to know any different Drawing pictures of circles and folding squares gives no indication of the educational possibilities of folding circles.

"However, because programming is now seen by the humanity as a huge value and a lucrative occupation, it has more chances to find its way into curricula than, say, origami."

You have put your finger on a big problem. The real incentive that drives math seems to be the money that can be made. How would one  expect any different in this money-focused culturally corrupt  planet we live on. Money is not how the universe works, nor is mathematics. We have so far shot ourselves in the foot and do not notice our leg turning green because we are looking at the abstractions of our own constructions. Don't get me wrong, I have a high regard for math as a language to better understanding this world we live in. Conrad Wolfram has made a strong case for reconsidering the curriculum from the bottom up, and there is an important place for mechanized computation but the concerns for developing human potential is not tied up with mathematics or money. Brad Bradford Hansen-Smith Wholemovement 4606 N. Elston #3 Chicago Il 60630 www.wholemovement.com wholemovement.blogspot.com/ --- On Tue, 11/16/10, Maria Droujkova <drou...@gmail.com> wrote:

[kirby urner]

I got a lot of emails about this talk too, as my associates associate it with my
long-standing position, which is that "math through programming" is a good
way to go.

It's not the only way to go, though, and I'm not a big fan of "one size fits all"
approaches.  If technology means anything good, it means helping to provide
more alternatives.

For those who do elect to take the programming route, there are a lot of 

details to consider, such as "programming what?", and "programming how?".

In my various classes over the years, I've put some emphasis on working 

with pre-written programs, which students are able to read and change.  

This in contrast to "blank canvas" programming where you start with a 

blank slate.  In human language learning, we know recognition is easier 

than recall.  This applies in mathematics as well.  Learn to read from those

already fluent, just as children learn from fluently native-speaker adults 

(of whatever language).

A lot of math-with-programming people tend to focus on the very young and 

go for a lot that's colorful and glitzy.  I'm not against doing this, however my 

own focus is a more mature audience with an active imagination.  A lot of the 

"picturing" of what's happening in the code needs to go on in the student's 

head, in the mind's eye.  The output may nevertheless be colorful and textured

(in the case of polyhedra especially -- ray tracers, or real time rendering 

may be used).

A kind of practice to aim for is one that requires a lot of concentration that 

is willingly sustained, along with a clear problem for which results are achieved 

and felt to be rewarding.  I feel that's the same gold standard we've always 

aimed for.

In the case of this TED talk, there's a lot of showing off of Mathematica in
particular, so you can see it as a sales talk as well as a more general stance.

The climax of the talk comes when we see Mathematica in all its glory 

doing so many things in so many little windows -- a kind of epiphany.

My classes haven't been using Mathematica, as we prefer to use free 

and open source when possible.  I do favor languages and environments with 

an interactive / conversational mode (which Mathematica famously has, 

as does APL, Logo and J).  I'm also not averse to using Mathematica 

when its available -- or Mathcad as the case may be.  Exposure to more 

than one language is better than just looking at one.

I've taken a more detailed and nuanced position versus how a specific 

approach to programming (the object oriented approach) might be used to 

instill a deeper appreciation for:

(a) abstract algebra concepts and
(b) spatial geometry

These go together of course, as rotational symmetry groups and polyhedra

go together i.e. there's already a well developed literature linking 

(a) and (b) that we're able to explore.  Some of my writings on cryptography

take this well trodden path, building up to RSA, likewise a topic in this 

recent textbook Mathematics for the Digital Age and Programming in Python

by Maria and Gary Litvin (Skylit Publishing).

My use of the object oriented paradigm sometimes nets me criticisms from
the functional programmer camp.  You'll find some hostility towards my
ideas (and those of some of my peers, including Gary) on math-thinking-l for 

example.  Having this difference on the computer side might help explain 

the frustrations many would-be funders experience when trying to get to 

the bottom of this whole "math and programming" set of memes.  They're
confused by the raging debates, which tend to get quite technical.

I think the best solution is to incorporate these debates right into the
learning experience, i.e. cue students as to what the running controversies
might be and make it clear that mathematics has always fractured into
schools of thought with only partially overlapping agendas.

Also along the lines of (a), I introduce operator overloading, meaning students
get to define the guts of addition and multiplication.  That's a framework for
defining subtraction and division in terms of inverses and neutral elements
(abstract algebra ideas).  In the Python language, which I tend to favor, types
of "math object" (the blueprints for these), tend to look somewhat biological,
like creatures, as I will briefly illustrate.

class Snake:
 __rib__
 __rib__
 __rib__
 __rib__

.... is sort of the template (snakes have lots of __ribs__).  The "reflexes" of an
object trace back to this "back bone" and/or "rib cage".  One of the __ribs__
might be called __add__ meaning it defines "addition behavior".  So if
you're defining a Rational Number class Q, you'll write code for __add__
that tells a Rational how to add with another of the same kind.  Likewise
for __mul__ etc.

Along the lines of (b) spatial geometry, I'm one of the very few teachers 

out there who takes the trouble to explore something called "tetrahedral 

mensuration" meaning we use a regular tetrahedron for a unit of volume 

and model of 3rd powering.  

This is not done to replace or substitute for the more orthodox cube-based
approach, but does provide valuable contrast and, in the process, a deeper
understanding of the role of axioms and definitions (having Euclidean and
non-Euclidean geometries going in parallel helps sharpen the meanings on
both sides).

Back to (a), I've developed the abstract algebra piece in more detail on
this list recently:

http://groups.google.com/group/mathfuture/msg/b0f524a87bb0b4e9?hl=en

I've got a lot more at my web site:

http://www.4dsolutions.net/ocn/cp4e.html

The tetrahedral stuff (b) links to some positive futurism, hopeful visions (ala
Bioneers etc.), transcendentalist heritage (ala Emerson etc.)...  world game
(Afghanistan etc.).  

Although I'm one of the few, I'm not the only one covering this and our bevy 

of "rad math" teachers gets a lot of mileage out of having an inside track.

For example, we tend to use these curriculum segments as leverage to show 

off our better grasp on recent history.  This is useful in an overseas context, 

especially when seeking to boost credibility among diplomat families.  

I've branded some of these more futuristic aspects of the digital math
curriculum "Martian Math" which helps keep it from seeming too parochial
(also, I used to go by "the martian" in high school, here's a link to that
story:  http://mybizmo.blogspot.com/2008/11/smiley-guy.html  ).

The above may sound somewhat elitist I realize, however the same cards
may be played within a district to enhance the reputation of any public or
private school.  However, the incentive may not be as strong in this context,
given most domestic colleges aren't competing for these same overseas 

candidates.

Note: geodesic domes and spheres may occur anywhere, including in non-human
engineering, is in the morphology of the virus (another a theme of martian
math):  http://www.4dsolutions.net/satacad/martianmath/mm30.html

Kirby

[kirby urner]

It's not either/or with origami and computer programming.  On the contrary, the computer algorithms for defining folding patterns have developed by leaps and bounds recently.  I went to a talk on that at OSCON (open source conference) and was much impressed.  Lots to learn about.  Amazing origami results.

Kirby

[Bradford Hansen-Smith]

Kirby, I agree with you, computer programing has advanced origami tremendously in the beauty and complexity of what can be done folding the square.  Consider the square is one of five parts of the circle, and yet so much of math can be traced back to the circle, which is after all, treated as only an image. My question and challenge is to see what can be generated if as much time and effort went into programming the same level of folding with the circle as is done with the square. That would mean people would have to start seriously folding circles to know the difference and I don't think that is going to happen very soon. There is too much comfort in the level of confusion about mathematics.

Brad Bradford Hansen-Smith Wholemovement 4606 N. Elston #3 Chicago Il 60630 www.wholemovement.com wholemovement.blogspot.com/

--- On Tue, 11/16/10, kirby urner <kirby...@gmail.com> wrote:

[kirby urner]

Your focus on the circle as a starting point is at least well demarcated and easy to comprehend.  Sharp distinctions make useful niches in the ecosystem of ideas, you have that going for you.  Likewise this tetrahedron stuff I'm doing, and branding as Martian Math, Gnu Math, Pythonic Math, depending on context, has an easier time gaining traction than just another brand of blah.

Folding circles is a part of the literature I study, per this Figure for example:

http://www.rwgrayprojects.com/synergetics/s09/figs/f3720.html

Kirby

[Bradford Hansen-Smith]

Kirby, I am well grounded in Fuller, and the bit of folding circles he did. Ten years into Fuller and seeing how simply and elegantly he folded the spherical VE using four circles, it occurred to me that everything else could be folded from folding and joining circles. I have yet to believe any different. I think "another brand of blah." is rather harsh. What larger context is there than the unity of a sphere, and when compressed reforms to circle unity? We believe the image better suits our reasoning as a unit of nothing rather than unity. We do not even acknowledge that maybe the circle is both, or that it might be something different than the image we call circle.

Brad Bradford Hansen-Smith Wholemovement 4606 N. Elston #3 Chicago Il 60630 www.wholemovement.com wholemovement.blogspot.com/

--- On Tue, 11/16/10, kirby urner <kirby...@gmail.com> wrote:

From: kirby urner <kirby...@gmail.com> Subject: Re: [Math 2.0] Teaching kids real math with computers: Conrad Wolfram - TED talk To: mathf...@googlegroups.com

[kirby urner]

On Tue, Nov 16, 2010 at 12:52 PM, Bradford Hansen-Smith <wholem...@sbcglobal.net> wrote:

Kirby, I am well grounded in Fuller, and the bit of folding circles he did. Ten years into Fuller and seeing how simply and elegantly he folded the spherical VE using four circles, it occurred to me that everything else could be folded from folding and joining circles. I have yet to believe any different.

That's cool Brad, I was wondering how much exploring you'd done, in his so-called "geometry of thinking" -- a new way of writing philosophy, using a starkly geometry vocabulary.

 

I think "another brand of blah." is rather harsh. What larger context is there than the unity of a sphere, and when compressed reforms to circle unity? We believe the image better suits our reasoning as a unit of nothing rather than unity. We do not even acknowledge that maybe the circle is both, or that it might be something different than the image we call circle.

"...another brand of blah" was not with reference to your approach, which I was characterizing as sharp and well-defined (starting with a circle instead of a square).  

I was referring to so many other approaches to curriculum writing, which have a really hard time gaining traction or visibility because they're really not that distinguishable from the background hodge podge.  

I was not being disrespectful of your circle-based approach, quite the contrary.

Kirby

[Bradford Hansen-Smith]

Kirby wrote......"I was referring to so many other approaches to curriculum writing, which have a really hard time gaining traction or visibility because they're really not that distinguishable from the background hodge podge." Thank you Kirby for clarification and support. My efforts are not to lead the mathematical world to a better mouse trap; I wonder about the computer snap-down that will pin us to the circuit board, so to speak. Wolfram did not talk about that. A digital universe seems real because we are developing capability to take it apart and reassemble in many interesting and unusual distortions. What is a second life when we are not capable of giving order and value to our first life? The “background hodge bodge" unfortunately has become the most visible; much like slag rising to the top in melting a crucible of metal. The impurities will not just go away unless purposefully skimmed off and disposed of to reveal the purity of liquid beneath. I see we are in a process of trying to pour the metal into a mathematical mold of our own making and have not yet thought about skimming off the hodge podge of impurities.  Skimming is a choice we must make otherwise the casting will be of poor quality. The word geometry refers to earth measure, measuring things of the earth. We have been on the moon; of recent we are measuring universes we did not know existed five years ago. This old word concept creates confusion when realizing it no longer reflects our understanding and experience today. It carries with it centuries of impurities, confused thinking, concepts no longer relevant to our progressing and changing world. So with consideration to the past I respectfully updated this word to the world I live in. Geo is earth and is spherical, the only form demonstrating unity of the whole; and to measure is simple keeping tract of movement from one place to another. Wholemovement is then a word that refers to a comprehensive understanding of the word geometry; the unity of  self-referencing, self generating nature of the Whole. This word concept embraces fractals and all kinds of other geometries and at the same time offers a way to begin to skim off conceptual impurities to begin to glimpse the beauty of the metal we are working with that lies beneath the surface of all the hodge bodge floating on top. Joe, I invite your interest in Wholemovement.  PlanetMath may indeed have the potential allowing people to sift thought what has value, and to be more than another dictionary of possibilities. My take on all this is in the books I have written documenting my exploration into folding circles. I do not need to test drive to know what twenty years of in and out of many schools with teachers and students has revealed. That it works is for others to find out for themselves as it makes sense to what they are trying to achieve. You can get more information from my website. But that is simply information; it is not the experience of understanding that comes from folding the circle, observing and reflecting on what is generated.  Wholemovement is about finding out what is there that will benefit, sustain, and uplift human potential; it is not about what can I do with it.