User predrag punosevac - MathOverflow

"Mathematics talk" for five year olds

2012-10-01T01:09:23Z

I am trying to prepare a "mathematics talk" for five year olds from my daughter's elementary school. I have given many mathematics talks in my life but this one feels very tough to prepare. Could the members of the community share their experience with these kind of lectures. I was thinking to talk about some theorems of Euclidean geometry which will include some old fashion compass, straight edge construction with some kind "magical outcome" and then try to give kids some logical reasons for the "magic". Any ideas?

Edit: I would like to thank one more time to the members of the mathoverflow community for their generous input and support as well to report the outcome of my talk.

I just got out from my daughter's elementary school where I ended up teaching four class periods today instead of the one I originally prepared for. I taught two sections of 5 year olds (26 kids a section) as well as two large group of fifth graders (close to 100 kids in total). I was "over prepared" to talk to 5 year olds which came handy with fifth graders.

Inspired by the answers from this forum I chose to talk about Platonic solids and have kids mostly engaged in practical activities as oppose of "teaching" them. My assistant chair at Augusta State University Georgia has generously shared her large collection of POLYDRON blocks. I had three bags full of equilateral triangles, squares, and pentagons. I have also pre-build one set of all five Platonic solids (Tetrahedron, Cube, Octahedron, Dodecahedron, and Icosahedron). I have also printed out cut and fold maps for all solids from this website and gave them to kids together with building blocks.

We identified first the properties of polygons (number of sides, vertices, and angles) of each of building blocks were to use as well as the fact they were regular (sides and angles of equal length). I was rather surprised that five year old children have no problem identifying pentagon as it is the shape of rather important building in Washington DC.

Then we introduced the rules of our "game":

Only the same "shapes" were to be used for building solids.
Two faces could meet only in one edge.
Each vertex of the solid had to meet the same number of faces.

Five year old kids had no problem assembling Tetrahedron, Cube, Octahedron however not a single group (they were allowed to work alone of in groups of 2-3) was able to assemble Dodecahedron, and Icosahedron. This was not the case with fifth graders (older kids) where several groups (4-5 out of 100 kids) successfully assembled Dodecahedron, and Icosahedron.

Even 5 year olds were able to identify number of faces, edges, and vertices by counting from the cut and fold charts. They had harder time identifying Schläfli symbols for each Platonic surface due to the fact that they had to count them on my pre-build models but they have never the less accomplished the task. We were able to come up with Euler characteristic (magic number as I referred) but the real focus was on subtracting numbers which we did using our fingers. Obviously the kids got lost after the cube due to the size of numbers involved. I was not able to convey any information about further combinatorial properties of Platonic solids related to Schläfli symbols to five year olds.

On another hand fifth graders had no problems identifying

$$pF=2E=qV$$

but had hard time solving equations as $pF=2E$ for $F$ and $2E=qV$ for $V$ and substituting into

$$V-E+F=2$$

not a single fifth grader could follow my computation for the estimate

$$\frac{1}{p}+\frac{1}{q}>\frac{1}{2}$$

where were effectively ended our little lecture.

In both sections kids asked me to preform some more "magic tricks". I glued a long strip of paper for them creating a cylinder and Mobius bend. Many kids thought of cylinder as a circle and Mobius bend as a figure eight (few fifth graders mentioned infinity symbols) even that they could not give any logical explanation why they think that way. We cut cylinder and Mobius bend and children start cheering my name when Mobius bend "broke" into just another bigger Mobius bend.

Five year olds wanted to hug me after the lecture and sit at my table in cafeteria. The fifth graders were either indifferent or came to me after the talk to shake my hand and ask if I can teach another class. It is also worth noting that while playing with blocks many fifth graders made prisms, pyramids while some try to pass non-platonic solids for Platonic solids bending rules of our game.

Teachers were trusty for this kind of experience. They are in a bad need for professional development after years of budget cuts and fear for their jobs. The school is going to buy blocks. I hope to make visits semi-regular and help them as much as I can (obviously out of selfish interest to improve my daughter's education). I have already planed to introduce some other games like tangram, pentominoes, and Hanoi tower. I will also install GeoGebra on their computers.

I might edit this post in next few days and add few details.

Most Kind Regards,

Predrag

Faculty Handbook: Mentoring Undergraduates in Research and Scholarship

2013-03-04T01:50:37Z

A few days ago I was asked by the director of the Center for Undergraduate Research and Scholarship at Georgia Regents University (formerly known as MCG and Augusta State) to contribute an article for the new Faculty Handbook: Mentoring Undergraduates in Research and Scholarship which is to be written along the lines of the University of Alaska Anchorage Handook. In my short time at Georgia Regents University I was indeed one of the first mathematicians who engaged in "Undergraduate Research" in part due to the nature of my subject Dynamical Systems and easiness with which one can do some very law quality numerical experiments. However, I think that I am far a way from the point that I can give any meaningful advise to anybody else. I hastily authored this short document more or less written for myself while applying to UN Lincoln IMMERSE program which fully exposes my lack of competence to contribute above article. To make matters worse despite 30 contributing articles to the University of Alaska Anchorage Faculty Handbook not a single one was written by a mathematician. What bit of advice or particular detail would you include if you had to write it? I don't know if I can acknowledge you properly in the handbook, but I'll refer to this question somehow.

Answer by Predrag Punosevac for Non-computational software useful to mathematicians

2012-01-12T03:10:51Z

This is my short list of math related software not used for computing. I made an effort to list software in descending order with respect to the frequency of use. I left out $\TeX$ and my version control system of choice CVS since OP was not interested in those.

A good editor. My choice goes to nvi.
A spell checker like ispell.
Regular expressions for pattern matching/search.
Make utility.
I do not know about you guys but for us who must teach and manage grade books nothing comes close to being as useful as awk.
A good file synchronizer like Unison.
Shell script(sh).
sed.
An e-mail client like mailx.
PostScript language as a page description language for printing but also a tool for teaching geometry classes. Obviously you must have PostScript interpreter on your computer like GhostScript (comes with a viewer) and a good PS manipulation tool like psutils.
A tool for manipulation of PDF files. I like Pdftk.
Python as a glue language and for instance its library Matplotlib for making illustrations.
A good image processing tool like Graphics Magic.
Vectorial drawing program as Xfig.

Answer by Predrag Punosevac for Open source mathematical software.

2012-01-12T02:49:53Z

I can't believe that Python and its vast scientific application libraries are not mentioned in this thread. Let me be little bit bias and mention projects like PyDSTool. What makes Maple or Mathematica language simpler than Python? Similarly time proven but proprietary computer algebra system MuPad now a part of MATLAB is not mentioned. FreeMat, the cleanest (other two being GNU Octabe and Scilab) reimplementation of MATLAB API, is also not mentioned. PostScript, page description language (arguably proprietary), is not mentioned. It is wonderful language for teaching geometry and programming pictures.

P.S. I was surprised that people even mentioned things like Maple and Mathematica. I personally have not meet a person who had a look at the source code of these two systems. However, I have friends who have worked for MathWorks and have seen the source code of MATLAB.

Journals which do accept non LaTeX macros submissions

2012-01-12T03:45:48Z

This question is inspired by my arguably perceived perception about a confusion at least among younger crowd about $\LaTeX$. $\LaTeX$ is just a set of macros for computer typesetting system $\TeX$ developed by Leslie Lamport. Even though it is now almost a standard set of macros in mathematics publishing, it is wrong to assume that there were no other no less useful $\TeX$ macros for writing mathematics. The first one that comes to my mind is AMS-$\TeX$ macro package developed by Michael Spivak and described in his wonderful book "Joy of $\TeX$".

I was wondering if people can name mathematics Journals which accept submissions written by $\TeX$ macros other than $\LaTeX$ or in pure $\TeX$ for that matter.

Answer by Predrag Punosevac for How to write popular mathematics well?

2011-12-21T22:05:43Z

As a graduate students I was forced (not really, I actually enjoyed) to read several long papers/essays how to write mathematics by people like Halmos who really knew how to it well, but my favorite advice came out of Chinese fortune cookie: "Good writing is clear thinking made visible".

Answer by Predrag Punosevac for Short Course Suggestions For High School Students

2011-12-21T21:58:19Z

You have not told us where the students are coming from but it is pretty safe to say that these days no high school students (with exception of few countries on the world) are exposed to any meaningful Geometry course.

How about teaching them some real old fashion synthetic (Euclidean/Lobachevsky) geometry course based let say on Kiselev's classic

http://www.amazon.com/Kiselevs-Geometry-Book-I-Planimetry/dp/0977985202

with possible excursion into Projective geometry.

There is no more natural place to introduce the concept of groups (actions on the sets) than in Geometry (composition of isometric transformations). There is no more natural place to introduce them to the concept of measure. It is very easy to involve hard combinatorial problems and many other things. Finally, set theory is all over the geometry and axiomatic method rules.

Many of the most challenging "Olympic problems" are geometric in its nature.

Best,

Predrag

Answer by Predrag Punosevac for Suggestions for wiki farm with good latex support

2010-07-10T16:24:05Z

I was investigating the same question recently myself. This is what I came up with.

The best support for TeX input to my knowledge has Noösphere which is the engine behind PlanetMath

http://planetmath.org/

I however do not believe that you can install Noösphere on your own-servers because it is proprietary application. Please somebody correct me if I am wrong.

On the another hand I really like PmWiki

http://www.pmwiki.org/

which also has support for TeX input.

http://www.pmwiki.org/wiki/Cookbook/TrueLatex

There is another feature of PmWiki that I really like. You can store your date in plain text files. Most but not all Wikis do require some kind of database.

Answer by Predrag Punosevac for Suggestions for teaching advanced high school students

2011-06-26T18:56:09Z

Assuming that you are dealing with U.S. high school students I could say something. At Augusta State University I am in charge with AMC 12/10 and Putnam (which is indeed Collegian competition but I alway have one or two very talented high school kids). I put two websites, one for AMC 12/10 ( http://predrag.freeshell.org/AMC/amc_about.html ) and one for Putnam ( http://predrag.freeshell.org/Putnam/putnam_news.html )

Carefully open all links on the bottom page with resources. My favorite however (probably I am nostalgic a bit) is link to Kvant ( http://kvant.mirror1.mccme.ru/ ). Serge Tabachnikov of Penn State have published two books of translations of most memorable articles.

I also like to use Gelfand's high school text books

For example Algebra http://www.amazon.com/Algebra-Israel-M-Gelfand/dp/0817636773

I like very much articles from three volume translation of Mathematics: Its Content, Methods and Meaning by A. D. Aleksandrov, A. N. Kolmogorov and M. A. Lavrent'ev.

http://www.amazon.com/s/ref=nb_sb_noss?url=search-alias%3Dstripbooks&field-keywords=mathematics+its+meaning+contectst&x=0&y=0

However, the articles are very, very challenging.

There is also a very famous Encyclopedia of Mathematics for high school students in Russia but I am not sure if it is translated into English.

Look for Vinogradov's book on number theory. It very, very deep but accessible for high school kids.

Answer by Predrag Punosevac for Learning LaTeX properly

2010-07-11T08:55:52Z

I am resurrecting this eight months old thread to post hopefully useful link to a small write up about TeX resources I did for GSU guys

http://predrag.freeshell.org/SciCG/resources/tex.html

Answer by Predrag Punosevac for Teaching a pedagogy course

2011-06-24T21:31:02Z

At the University of Arizona where I received my Ph.D. just like at your institution all incoming graduate students also had to take a semester long course on pedagogy but it was always taught by a postdoc. The mandatory reading was "How to Teach Mathematics" by Steven G. Krantz but in retrospective, I wish, the mandatory reading had been his other book "A Mathematician's Survival Guide". We did lots of exercises like video recording our lectures or grading the same exam (by the whole class) and then comparing the grades.

Besides already mentioned Arnold's article (some of ideas from that article are also eloquently presented in his Ordinary Differential Book 1st one not the Geometric Methods) I really, really like the following writing

http://www.math.psu.edu/katok_a/reflections.html

by Anatole Katok which is indeed geared towards more advancing courses which you will probably not have opportunity to teach as a TA.

I do not agree with everything Professor Katok is saying but it is a great reading. For example:"After that the picture gets filled in, sometimes straightforwardly using the most elegant or most useful available arguments". I quite on the contrary believe that the picture should be filled by the arguments which the most instrumental in exposing key ideas. I also do not believe in well polished statements of the results as they are usually result of many iterations of the original idea/discovery. I like to state result in its original form when it was discovered (for example Stokes theorem can be stated on the square in $R^2$ in which case its proof become a trivial exercise in applying Fundamental Theorem of Calculus). My advisor Qiudong Wang is a master in delivering such courses (Dynamical Systems, ODE) which are completely revealing.

As of teaching courses taken by unmotivated learners (intro college courses) I have much less to say and in my experience you have to treat crowd on case by case basis. I am not sure that any reading can help you with that (you have to get experience).

Complex harmonic oscilator

2011-05-26T03:40:22Z

I was wondering if anybody could give me some references to already existing literature for the following open ended problem.

Namely, I am interested in studying the equation of "complex harmonic oscillator"

$$\ddot{z}(t)+q(t)z(t)=0$$

where $z:\mathbb{R}\to\mathbb{C}$.

The case when $t$ is complex is also interesting and might shed some light on to the real case. Assume that the function $q(t)$ is real (for real time) and strictly decreasing. Actually, without lost of generality let $q(t)=1/(1+t)$. I have strong numerical evidence for the following claim. There exist the set of initial conditions such that $|z|$ is a constant or strictly monotone decreasing (increasing) functions on the solutions satisfying those initial conditions.

I am considering the following auxiliary function $\rho(t)=\ln(z\cdot \bar{z})$ the real case but I still can not make a use of the fact that q(t) is positive and decreasing on $[0,\infty)$. For the curious, I am essentially play the same kind a game like in the proof of Sturm-Picone comparison theorem or Poincare-Benedixon theorem.

I started looking into the case of the complex time due to the following simple observation. Suppose that $z_1(t)$ and $z_2(t)$ are two linearly independent holomorphic solutions of the equation. The Schwarzian derivative in notation $S$ of the ratio $z_1/z_2$ satisfies

$$S(\frac{z_1}{z_2})=1/(1+t)$$

Thank you.

Answer by Predrag Punosevac for Tools for collaborative paper-writing

2010-09-19T21:50:55Z

One more vote for old trusted CVS here. Actually, I use OpenCVS since my OS of choice is OpenBSD.

I would like to bring to attention of SVN users the following package

http://www.ctan.org/tex-archive/macros/latex/contrib/svn-multi/

which allow you to create (TeX) log messages on the LaTeX document itself. There is another package for using SVN with LaTeX called svn. Disadvantage of svn is that is not playing well with the papers which are broken into multiple files. Here are couple very good articles from the PracTeX journal about SVN, LaTeX and even using it with TextMate

http://www.tug.org/pracjourn/2007-3/kalderon-svnmulti/

http://www.tug.org/pracjourn/2007-3/skiadas-svn/

http://www.tug.org/pracjourn/2007-3/scharrer/

Another interesting reading from the same issue of PracTeX Journal

http://www.tug.org/pracjourn/2007-3/henningsen/

which addresses original question.

I am not a big fun of distributed revision control systems like Git. If I would have to pick one distributed system it would be Mercurial.

I would like to point another issue. Traditional revision control systems are intended to be used with text files (source codes). Most classical revision control systems like CVS, SVN have troubles dealing with binary files. TeX files are of course text files. However, illustrations in papers unless created with the programming languages like PSTricks, MetaPost, PGF/TikZ, or Asymtote are binary files. So they are not plying well with the revision control systems. That is one more reason for young mathematicians to start programming their illustrations instead of creating them with vector graphics editor like Xfig. My favorite is PSTricks because it is essentially the easiest way to use pure PostScript inside TeX documents. As we all know PostScript is THE page description language. Therefore, I also have strong preference for Powerdot (PSTricks) over Beamer(PGF/TikZ).

Unfortunately, there is a disturbing trend at least when it comes to grant proposals of using proprietary binary formats like doc. Those formats do not play well with revision control systems which makes collaboration on grant writing very painful. Hopefully, as the "new" docx format which is default format of MS Office 2007 becomes more wide spread this problem will slowly disappear (At least for the people that are familiar that docx is just bunch of XML files zipped together, only two of the files are actually important, one being XML version of document itself and another being Style Sheet file. People familiar with DocBook (apparently MS people were not one of them since they reinvented the wheel invented 1992) will know what I am talking about).

Cheers,

Predrag

Answer by Predrag Punosevac for Different forms of compactness and their relation

2010-08-30T22:50:37Z

You should get Munkres' First Course in Topology and read it if you have not done already. This is not the question for Mathoverflow. It is actually a fair general topology homework assignment for a solid undergraduate student.

Answer by Predrag Punosevac for Books you would like to see translated into English.

2010-07-19T07:37:59Z

Two volume introduction to Complex Analysis by B.V.Shabat. Actually, I have already translated about 150 pages of the first volume which is about as much as one can cover in Complex Variable undergraduate course offered by a typical U.S. university. I did give the translation as a hand out to my students last year when I taught Complex variables class. I did translation out of frustration with the book of Churchill and Brown.

Answer by Predrag Punosevac for Are nets and filters useful in geometry and topology?

2010-07-15T16:55:23Z

If a topological space is not first countable sequences can not be used for instance to verify the continuity of the function. In that case you can use nets instead. Topological spaces which are not first countable are commonly encountered in Functional Analysis.

Answer by Predrag Punosevac for Typo/grammar checker for LaTeX

2010-07-11T16:08:11Z

I would cold-heartedly agree with the first post. The best way to check your grammar is to have somebody else proofread your paper for you. For the sake of completeness I will add that there were two old Unix tools for checking writing: style and diction.

http://dsl.org/cookbook/cookbook_15.html#SEC220

I personally have never used them.

@Yoo Removing LaTeX is fairly easy with sed for instance but there is a tool called detex which will do exactly that for you. However it is not 100% successful and I would still suggest that you read text document.

Answer by Predrag Punosevac for Analytic ODE with complex time

2010-07-11T08:42:51Z

Beside the book of late professor Hille I would suggest that you check

Lectures on Analytic Differential Equations by Yulij Iljuashenko and Sergi Yakovenko

http://www.amazon.com/Lectures-Analytic-Differential-Equations-Mathematics/dp/0821836676

Professor Iljuashenko

http://www.math.cornell.edu/People/Faculty/ilyashenko.html

who is now at Cornell has delivered many times courses on differential equations in complex domain back in Moscow and is one of the greatest experts in the field.

As with any question on dynamical systems I would also suggest that you have handy nine volumes of Encyclopaedia of Mathematical Sciences dedicated to Dynamical Systems. The series is edited by Arnold, Anosov, Sinai, Novikov, and few other outstanding Soviet mathematicians.

http://www.amazon.com/s?ie=UTF8&rh=i%3Astripbooks%2Cp_27%3AD.V. Anosov&field-author=D.V. Anosov&page=1

For starters I think you should check the first volume (if I recall correctly).

Answer by Predrag Punosevac for Approximation with continuous functions

2010-07-10T06:14:17Z

The question is ill-posed. A function continuous in one topology might not be continuous in another topology. The same goes for convergence. When you precisely formulate your question, I am sure you will be able to come with the answer.

Comment by Predrag Punosevac

2013-03-06T17:08:27Z

@Timoty Chow This is exactly the kind of the answer I was hopping to get by posting the question on mathoverflow. Thank you so much for taking the time to read that garbage I wrote and point me in the right direction.

Comment by Predrag Punosevac

2013-03-04T03:57:19Z

@Gerhard Paseman I edited the question per your advise. I am a bit surprised that it looks like it is going to get closed. I could not be the only guy on this portal who is forced to think about "undergraduate research".

Comment by Predrag Punosevac

2012-10-03T20:46:08Z

@Philip van Reeuwijk Serbian Cyrillic alphabet has 30 letters. I have been living in U.S. for the past 17 years so 26 letters will definitely work here in Georgia:)

Comment by Predrag Punosevac

2012-10-02T01:58:00Z

@unknown (google) Outstanding!!! Thank you so much!

Comment by Predrag Punosevac

2012-10-02T01:54:51Z

@Tom Leinster Bagdala is a neighborhood of my home town Krusevac in Serbia. I moved out of that neighborhood 23 years today but it is still where my childhood dreams live. As of me, I have been leaving in U.S. for the past 17 years (I am even listed as U.S. mathematician on math genealogy). A quick search will show that I am a faculty at Augusta State University in GA with a house in South Carolina.

Comment by Predrag Punosevac

2012-10-01T03:21:22Z

@Amritanshu Prasad Yes, I love transformational congruency and my girls favorite toy is tangram!

Comment by Predrag Punosevac

2012-10-01T03:17:29Z

@Douglas Zare You sound like my wife but I can assure you that both my 5 year old and my 4 year old know how to use compass even though I would not live them alone in a room with the one. Among my "greatest achievements" with them was to teach them about ordering of natural numbers using balancing scale. That happed after their Ukrainian grandmother (my mother in-law) was disgusted how little mathematics they know comparing to kids of the same age in old Soviet Union.

Comment by Predrag Punosevac

2012-03-19T05:10:16Z

I think that you guys are little bit naive about this and similar posts. Did you bother check user profile of the guy who asked the question? He is using mathoverflow web site to increase Google page rank of his stupid on-line tutoring services. Such posts should not be just closed but pruned and user who asked the question should be immediately banned from the forum.

Comment by Predrag Punosevac

2011-12-22T20:52:24Z

Also Rick Mockel (another former Ph.D. student of Charles Conley) and his students have found many interesting applications of Algebraic Topology in Celestial Mechanic. One of Conley's students (Kris MeCoord) wrote a bunch of celestial mechanics papers with Ken Mayer and my advisor Qiudong Wang at the University of Cincinnati using nothing but Algebraic Topology techniques.

Comment by Predrag Punosevac

2011-12-22T20:48:43Z

I wanted to mention Konstantin's group at Rutgers (previously) at Georgia Tech which does essentially computational homology theory with applications to dynamical systems but only after I mention the the God Father, the late Charles Conley former professor at the University of Wisconsin and the Ph.D. thesis advisor of Konstantin Mischaikow. His survey paper on now famous Conley's Index theory should be the first read for anybody interested in involving heavy algebraic topology in dynamical systems.

Comment by Predrag Punosevac

2011-09-01T18:03:54Z

As many here I was schooled on the Rudin's book. However, for a person like me who doesn't work directly in functional analysis but by the nature of his business (dynamical systems) needs often quickly to refresh his memory I warmly recommend Zimmer's book. Probably an appropriate title for the book would be a "Functional Analysis for a working mathematician". It should be on everybody shelf.

Comment by Predrag Punosevac

2011-08-02T16:40:15Z

You have written up some really nice stuff here math.uconn.edu/~kconrad Thank you so much for making it freely available!

Comment by Predrag Punosevac

2011-07-08T18:10:50Z

I should be more careful before dismissing answers. For the past two-three days I have carefully read chapter five of the Hille's book at least dozen of times as well as skimming through several other chapters. The book contains exactly the kind of information I was craving for!

Comment by Predrag Punosevac

2011-06-26T22:12:19Z

I am not resurrecting old thread. I just changed the recommendation for PmWiki add-on for TeX support based on my own experience in the past couple of months. I am using PmWiki on several sites and it is an outstanding peace of software!

Comment by Predrag Punosevac

2011-06-24T22:31:21Z

Actually, I posted this answer 10 months ago. I just updated the link.