# How to solve it! Who is continuing Polya's legacy?

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### Andrius Kulikauskas

Feb 10, 2011, 7:56:18 PM2/10/11
Greetings from Chicago, where I moved to last year from Lithuania.

I'm writing a book and learning materials, in the Public Domain, for
adult self-learners who want to learn basic math (arithmetic to
precalculus) in terms of deep ideas and classic problems. Here are my
notes so far: http://www.gospelmath.com/Math/DeepIdeas

Currently, I've been reading George Polya's classic books "How to Solve
It" and "Mathematical Discovery: On Understanding, Learning and Teaching
Problem Solving". I'm curious who today is building on his work or
pursuing similar directions?

For example, I've created a generic "playsheet" for students I tutor:
I've created spaces for them to state a problem (such as adding
fractions 1/2 + 1/3), calculate the problem, illustrate the problem
(let's say with a pie chart), make up a non-contrived word problem (You
own 1/2 of a company and I own 1/3, how much do we own together?), do
alternate answers, jot down any ideas or questions that come up, and
indicate how hard or easy the problem was. My thought is to help the
students fill out the sheet so that they can make up and do similar
problems at home. Hopefully, as they get to used to this, they can be
more original and make up their own problems or variations.

My real passion is to build a directory of "ways of figuring things out"
in all manner of fields, including math, science, engineering, medicine,
law, finance, theater, art, music, governance, agriculture, sports,
ethics, teaching, learning and many more. Here's my proposal:
http://www.selflearners.net/Business/Proposal and here are 200+ ways
that I've figured things out in my philosophy and my life:
http://www.selflearners.net/ways/ Ultimately, I want to express them as
games. Thus I'm active in the Gamestorming community for business
innovation: http://www.gogamestorm.com/?p=536

How might I best seek support, private or public, for such Public Domain
projects?

Thank you!

Andrius

Andrius Kulikauskas
http://www.selflearners.net
m...@ms.lt
+1 (773) 306-3807

### Steve Garlick

Feb 10, 2011, 8:23:34 PM2/10/11
Try Alan Schonfeld (sp), will look up the exact reference when I'm back in my library

Regards,

Steve Garlick

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### Andrius Kulikauskas

Feb 11, 2011, 9:11:37 AM2/11/11
Thank you, that helps. I wrote to Alan Schoenfield, but he hasn't
responded yet:
http://groups.yahoo.com/group/livingbytruth/message/1118
Anybody else?

And do we know of any support for publishing Public Domain learning
materials or textbooks?

I have a Ph.D. in math from UCSD.

Thank you!

Andrius

+1 (773) 306-3807

### Maria Droujkova

Feb 11, 2011, 9:29:14 AM2/11/11
On Thu, Feb 10, 2011 at 7:56 PM, Andrius Kulikauskas wrote:

My real passion is to build a directory of "ways of figuring things out" in all manner of fields, including math, science, engineering, medicine, law, finance, theater, art, music, governance, agriculture, sports, ethics, teaching, learning and many more.  Here's my proposal: http://www.selflearners.net/Business/Proposal  and here are 200+ ways that I've figured things out in my philosophy and my life: http://www.selflearners.net/ways/  Ultimately, I want to express them as games.  Thus I'm active in the Gamestorming community for business innovation: http://www.gogamestorm.com/?p=536

How might I best seek support, private or public, for such Public Domain projects?

Andrius,

What sort of support do you seek? Suppose you have all the support you need - what will the support and your tasks look like?

Cheers,
Maria Droujkova

### Sue VanHattum

Feb 11, 2011, 10:05:34 AM2/11/11
Paul Zeitz wrote The Art and Craft of Problem Solving. I love what he has to say about problem solving. Most things I've read break it down the same way Polya did, so there's nothing new, really. Zeitz breaks it down differently.

Regards,
Sue

> Date: Fri, 11 Feb 2011 08:11:37 -0600
> From: m...@ms.lt
> Subject: [Math 2.0] Re: How to solve it! Who is continuing Polya's legacy?

>
> Thank you, that helps. I wrote to Alan Schoenfield, but he hasn't
> responded yet:
> http://groups.yahoo.com/group/livingbytruth/message/1118
> Anybody else?
>
> And do we know of any support for publishing Public Domain learning
> materials or textbooks?
>
> I have a Ph.D. in math from UCSD.
>
> Thank you!
>
> Andrius
>
> Andrius Kulikauskas
> http://www.selflearners.net
> m...@ms.lt
> +1 (773) 306-3807
>
>
> 2011.02.10 19:23, Steve Garlick raīŋŊīŋŊ:

### Mike South

Feb 11, 2011, 10:51:48 AM2/11/11
On Fri, Feb 11, 2011 at 8:11 AM, Andrius Kulikauskas wrote:
Thank you, that helps. I wrote to Alan Schoenfield, but he hasn't responded yet:
http://groups.yahoo.com/group/livingbytruth/message/1118
Anybody else?

And do we know of any support for publishing Public Domain learning materials or textbooks?

Have you looked at lulu.com?

mike

### Mike South

Feb 11, 2011, 10:56:19 AM2/11/11
Something that may be of use is first person narrative of an actual problem solution by various people.

Usually by the time you write up a problem solution you've figured out optimizations in your logic and completely forgotten about many of the blind alleys you went down (I think the brain aggressively forgets dead ends...or maybe just my brain), so you just about have to keep a log of your thinking as you go.

My guess is that if you got a lot of people to contribute those you could read through them and find some kind of insight toward general principles/techniques that would be worth distilling and reporting on.  Even the narratives themselves would probably be of value.  So, maybe a book "How I Solved It" as a sort of homage/sequel to "How to Solve It".

Just an idea.

mike

### Linda Fahlberg-Stojanovska

Feb 11, 2011, 11:07:50 AM2/11/11

Mike South wrote: Something that may be of use is first person narrative of an actual problem solution by various people.

Usually by the time you write up a problem solution you've figured out optimizations in your logic and completely forgotten about many of the blind alleys you went down.

I totally agree. Screencasts of people writing using tablets like scratch paper would be great. Linda

### Sue VanHattum

Feb 11, 2011, 11:18:11 AM2/11/11
Ooh, I like that. Does anyone have examples of that? If not, maybe I can try to record the problem-solving session I plan to go to tomorrow.

Warmly,
Sue

From: lfah...@gmail.com
Subject: RE: [Math 2.0] How to solve it! Who is continuing Polya's legacy?
Date: Fri, 11 Feb 2011 17:07:50 +0100

Mike South wrote: Something that may be of use is first person narrative of an actual problem solution by various people.

Usually by the time you write up a problem solution you've figured out optimizations in your logic and completely forgotten about many of the blind alleys you went down.

I totally agree. Screencasts of people writing using tablets like scratch paper would be great. Linda

### Tomatsfu

Feb 11, 2011, 11:46:21 AM2/11/11
Morning,
Would this site work for your publishing needs:
http://en.flossmanuals.net/
~tom

Thank you!

Andrius

--

### Linda Fahlberg-Stojanovska

Feb 11, 2011, 12:06:10 PM2/11/11

Sue – do you want a screencast of me thinking about and solving a problem on my tablet?  If so, I can do this – do you have anything particular in mind?

If not, no worries.

Linda

### shaun

Feb 11, 2011, 4:15:06 PM2/11/11
to mathfuture, alans
Ck12 and Khan Academy are both great places to start. I also create free online classes...
--
Shaun Errichiello
Salk School of Science
212-614-8786
http://shaunteaches.blogspot.com/

### Dani Novak

Feb 11, 2011, 5:16:53 PM2/11/11
I think this is a wonderful project.  Did you consider to connect it with actual tools like GeoGebra NLVM and others?  From my experience many people learn math by doing it in an active way.  This way of looking at Math is the way Math is emerging from the depth of our collective consciousness and it is a language of reason, logic and beauty and your approach is in harmony with that.

Another related way is to offer free courses through http://p2pu.org/  or http://www.wiziq.com/  because when you teach you develop it further and the text book you are writing could be an anchor.

Cheers,

--Dani

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http://faculty.ithaca.edu/novak/
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### Edward Cherlin

Feb 11, 2011, 6:59:42 PM2/11/11
On Fri, Feb 11, 2011 at 10:51, Mike South <mso...@gmail.com> wrote:
>
>
> On Fri, Feb 11, 2011 at 8:11 AM, Andrius Kulikauskas <m...@ms.lt> wrote:
>>
>> Thank you, that helps. I wrote to Alan Schoenfield, but he hasn't
>> responded yet:
>> http://groups.yahoo.com/group/livingbytruth/message/1118
>> Anybody else?
>>
>> And do we know of any support for publishing Public Domain learning
>> materials or textbooks?

How about Creative Commons? For example,

http://www.clrn.org/fdti/

has several free math and science textbooks.

> Have you looked at lulu.com?

Not just lulu.com, but booki for writing and translating books. FLOSS

http://booki.flossmanuals.net/

and then uses lulu.com for on-demand printing. Booki is Free Software
available for anyone to use. There is a public instance on the Web
where you can write any sort of book you like.

http://www.booki.cc/

I'm helping to organize Spanish book translations for FLOSS Manuals.

--
Edward Mokurai (默雷/धर्ममेघशब्दगर्ज/دھرممیگھشبدگر ج) Cherlin
Silent Thunder is my name, and Children are my nation.
The Cosmos is my dwelling place, the Truth my destination.
http://www.earthtreasury.org/

### Sue VanHattum

Feb 12, 2011, 12:28:41 AM2/12/11
I do best with text, Linda, but other people might like that.

Do you know of a problem you've never solved that you'd like to think about? (I might think about this problem sometime soon:
http://divisbyzero.files.wordpress.com/2011/02/nbyn.png?w=260&h=260  - How many triangles of any size are in this figure, or in an mxm square like this? His post seems to have disappeared for now.)

Warmly,
Sue

From: lfah...@gmail.com
Subject: RE: [Math 2.0] How to solve it! Who is continuing Polya's legacy?
Date: Fri, 11 Feb 2011 18:06:10 +0100

### Linda Fahlberg-Stojanovska

Feb 12, 2011, 5:25:34 AM2/12/11

Hi Sue – wasn’t sure whether this is anything like what you had in mind. http://mathcasts.org/mc/lfs/trianglecountS/trianglecountS.html

A.      I tried really hard to just “jump right in” and not think at all about it as I think the point was to see “how we start to organize our thoughts”.  (I am sure others organize their thoughts much better, but I like to think that there are a lot of people out there like me that meander their way through problems.)

B.      I left in my erroneous thinking about T3-s since I thought this important.

C.      If I was actually producing this formally, I would correct my “typos”, e.g. (towards the end) there are 2 places I say “triangle” and mean “square” and there are 2 places I write “T3” and mean “T4”.

D.      Finally, clearly I didn’t finish the problem because (1) ran out of time, (2) wasn’t sure this was what we were looking for and (3) don’t really like these problems. (Guess which one is the real factor – Ok Maria, if this counts as a rich problem, I admit I don’t like it personally :)).

Happy Saturday to all! Linda

P.S. Done with cheap graphics tablet on a regular laptop;  recorded/produced with camtasia studio.

### Dani Novak

Feb 12, 2011, 7:04:32 AM2/12/11
Sue,

Your post inspired me to create this Lesson and I think this approach has a lot of potential to help kids learn the concept of a function in a fun way (no pun intended).  I co-designed this language with David Rosenthal who coded it in Java.

--Dani

### Alexander Bogomolny

Feb 12, 2011, 8:42:49 AM2/12/11
Dani,

my little boy has a 7-9 am swim practice today. I am sitting there not entirely awake yet, with no worthy thoughts to occupy myself. So, for nothing better to do, I have reproduced (believe me I have) the diagrams from your lesson plan. I wish to state for the record that it took me just a little under 2 minutes with copy&paste in MS Paint. Naturally, I also got the case of N = 1. You should include this case in your table.

Alex

### Sue VanHattum

Feb 12, 2011, 9:52:08 AM2/12/11
Linda and Dani, I don't want to look at your cool stuff until after I've played with the puzzle. I'll do that today, while I'm on BART. (I'm going over to SF to the Escape From the Textbook conference.

Warmly,
Sue

Date: Sat, 12 Feb 2011 07:04:32 -0500
Subject: Re: [Math 2.0] How to solve it! Who is continuing Polya's legacy?
From: dan...@gmail.com
CC: dav.ro...@gmail.com

### Linda Fahlberg-Stojanovska

Feb 12, 2011, 11:23:35 AM2/12/11

Hiya Dani, et.al.

Oh yes – I see now how to draw it with MuMart. Thank-you. I REALLY like in you Jing how you do it with numbers and then add the “parameter”  making it clear what the x is for. (It seems to me that the commands: DECLARE x param x{2,8} are defining a whole number slider).

I thought I would clarify that I don’t particularly like these problems as I have a problem “seeing” like others. Thus I consider these “trick” problems since I never see pictures the way other people see.  I never “see all the triangles”, I don’t get the right count and there is no way to check. So even though I understand all the principles of problem solving, I easily get these problems wrong. This really annoys me.

### Sue VanHattum

Feb 13, 2011, 12:04:11 AM2/13/11
Linda, Dani and Mike, may I quote you and/or point to your creations as part of my blog post about this problem? (I'll post it here too when I'm done.)

Warmly,
Sue

### Linda Fahlberg-Stojanovska

Feb 13, 2011, 2:13:38 AM2/13/11

Sure. Linda

Sent: Sunday, February 13, 2011 6:04 AM
Subject: RE: [Math 2.0] How to solve it! Who is continuing Polya's legacy?

Linda, Dani and Mike, may I quote you and/or point to your creations as part of my blog post about this problem? (I'll post it here too when I'm done.)

Warmly,
Sue

--

### Sue VanHattum

Feb 13, 2011, 1:23:44 PM2/13/11
I've posted on my blog, regarding my progress on the triangle problem. I tried to capture all of my thinking. I'm not done with it. Don't know when I'll have time to finish. I hope soon. I'd love to see lots of folks show their thinking on this.

Warmly,
Sue

### kirby urner

Feb 13, 2011, 2:10:51 PM2/13/11
I've been studying your Gospel Economics.  You might find this link of interest:

A lot of problem solving, step by step, using algebraic styles of thinking, goes
on in computer programming.

The desire to keep programming from becoming too much a part of some math
classes is balanced by a complementary desire to make programming a core
activity if other math classes, likewise for credit.

The Oregon legislature has set some standards for Discrete Math which are
compatible with a high school semester, year-long, or even four-year track
that would include more CS topics.  The problem with CS in the past is it was
elective, i.e. did not count towards fulfilling the 3 year math requirement.

In the new model, a student might go:  Algebra -> Geometry -> Digital Math
or even Algebra -> Trig -> Digital Math.  With multiple Digital Math modules
(under design) we'll be able to interpolate it in more places.

Digital Math hallmarks:  more use of GIS/GPS concepts, lat/long, ties
with geography (surveying, earth measure), more use of spatial geometry
and computerized display techniques (ray tracers, graphics engines),
more use of simulations (e.g. Sims).  Regionally:  more delving into
biochemistry for rich narratives (different from rich problems, which
we also tackle).

Kirby

### Cooper Macbeth

Feb 15, 2011, 11:04:59 AM2/15/11
Using Sue's descriptions for the number of small triangles: T1, I have:
T1                     T2                     T4                  T8                    T9                    T16                Total      1st diff       2nd diff     3rd diff
0x0       0                         0                      0                    0                      0                        0                      0          8              28             20

1x1       4                         4                      0                    0                      0                        0                      8         36             48             20
4 (1x1)
2x2      16                       16                     8                    4                      0                        0                     44         84             68
9(1x1)                                         4(2x2)
3x3      36                       36                    32                   16                    8                        0                    128       152
16(1x1)                                        9(2x2)                         4(3x3)
4x4      64                        64                   72                   36                  32                      12                     280

We see that the 3rd difference has become a constant, we could do a 5x5 and get 520, but the 3rd difference will still be 20.  Therefore we have a cubic and finishing the Polynomial Derivation spreadsheet yields the general formula:
+10/3 m^3 +4 m^2 +2/3 m, for any m x m.  Hope I didn't kill too many synapses.  :-)

### mathmama

Jun 6, 2014, 12:09:37 AM6/6/14
Three years later, after someone commented on my blog about this, I finally finished working it out. Cooper, I don't think you'll get common differences, because there's a floor or ceiling function involved. The 4x4 case should get you 268 triangles, not 280. Post is at: http://mathmamawrites.blogspot.com/2011/02/tin-ceilings-triangles-and-loving-math.html, with solution in the comments.