Punya Mishra’s Web

The math-po (and sci-po) stream keeps flowing. Math Mama Writes, who started the whole math-poetry movement has some more on her blog, and here is Erin Nash with some really beautiful biological poetry. And of course, here’s her husband Sean Nash having his students writing poetry too. Of course let’s not forget my daughter Shreya (who sort of started this whole thing) and her sci-po’s at her blog Uniquely Mine.

Below are some thoughts about math-poetry – but you can ignore all that and scroll right down to the poem: The infinity of primes!

Math art by durentu

Through all this I have been plugging away at my math poetry. I know the original challenge was to write something to motivate students to learn math (and I did write one along those lines). But more interesting to me has been this theme I have picked up, which is of writing proofs as poetry. I know many people have described mathematics in poetic terms but I am trying something slightly different here. I am trying to explain theorems (as in these couple of instances, see here, here, here and here) and speficially in the poem included below, I am actually trying to construct a mathematical proof in rhyming verse.

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Image credit: eoliene_pe_campii

Mathematical Beauty: A limerick
Punya Mishra, Jan 27, 2010

Doesn’t it just gladden your heart to see

These games we can play with infinity?

How can one stay aloof

From the elegance of a proof

And remain immune to mathematics’ subtle beauty?

I recently received an email from a teacher in Poland, seeking advice for a curriculum outline for their Design Technology Section. They said, and I quote:

Unfortunately, I have minimal experience with the subject as a teacher or as a student in my younger years, consequently, I have little background as to what a DT class should look like.

As you might guess I’m struggling trying to put together some sort of DT curriculum for our Middle School.

Our small school does not not have any kind of fabrication equipment so our DT class is currently heavy on IT design aspects….(web design, research on a topic and devise a solution, book creation,etc… )

The specific request was for”some useful and practical information that I can implement fairly easily.”

Is this what design is? by StephenMitchell on Flickr

As I crafted the response I realized that (given my lazy self) that it would make a good blog post. So here is what I wrote back:

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The other day, for one reason or another, I did a Google search for the word “recursion.” According to Wikipedia, recursion

… in mathematics and computer science, is a method of defining functions in which the function being defined is applied within its own definition; specifically it is defining an infinite statement using finite components.

This is a screen shot of what Google gave me as a result of my search: (Click on the image for a larger version).

Look carefully at what Google suggests. It says.

Did you mean: recursion

The exact same word that I had searched for… For a moment or two I thought I had found a glitch in Google’s suggestion mechanism… but it suddenly hit me, that this was exactly what recursion meant! If I clicked on that link I would be taking the first step into an endless cycle, an infinite loop, which would end only when I “got it.”

This has been an inside joke amongst programmers for a while. Wikipeda provides one example from a hypothetical dictionary that goes as follows:

Recursion

If you still don’t get it, see: “Recursion”.

So in some sense Google provides a “working definition” of the word – that explains it better than just reading a definition in a dictionary. How cool is that. I think it is little games like this one that make Google so much fun to use.

(I must add that this is not a new discovery. The Wikipedia page about recursion does mention this Google trick – but it was new to me!)

A tangent, a line and a circle
A math poem

Image credit: chrstphre (on Flickr)

A point outside a circle,
shoots out two lines
one heading for the center
the other more feline
smoothly kisses the curve
That delicate swerve
of the ball and then, abruptly
turns to the center,
making an angle
and meets the first
which went on straight
To thus form a triangle.

The angle formed by the tangent
And the other line, the radius
the mathematical mind sees
is exactly 90 degrees!

And funnily enough, it doesn’t matter
where the initial point is, the starter
Where ever it may be
(as long as it is outside the circumference)
The angle will always be 90!
Did that make sense?

A poem on the fact that a tangent to a circle is perpendicular to the radius drawn to the point of tangency. See image below

Image from ETC, an online service of Florida’s Educational Technology Clearinghouse

I guess ’tis the season of Math-Po’s! Sue VanHattum, whose challenge started all this, commented on my recent Math-Po (Math-Po (Mathematical Poetry): Goldbach’s Conjecture) by providing an example of her own writing, a poem titled Imaginary Numbers Do the Trick. That piece so inspired me that I spent the next hour (and a good part of a faculty meeting), writing one on the same idea. A close read of both these poems (hers and mine) will reveal that I was more than inspired… some phrases and words from Sue’s work insinuated themselves into my pre-frontal cortex and ended up in my poems. As they say, plagiarism is the sincerest form of flattery. Read, The Mathematical i, after the cartoon…

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My previous post (Poetry, Science & Math, OR why I love the web) mentioned a challenge by Sue VanHattum of “Math Mama Writes” to “write a little kids’ poem … and that tells of the beauty of math, or, that mentions math and challenge, both in a positive way.” Well, I got inspired and took a part of my lunch break today to write something up. I am not sure it technically fits Sue’s challenge but here it is none the less. [If you are interested in learning more about the history and mathematics behind Goldbach's Conjecture, one of the oldest unsolved problems in mathematics, check Goldbach's Conjecture on Wikipedia.]

Goldbach’s Conjecture
Goldbach, a mathematician, serious and stern
Many years ago noticed a pattern
He wrote, to Euler, the math genius
Here is something, he said, to excite us!

I have seen, he scribbled, with my imagination
That every even digit
(except two, which doesn’t fit)
Can be broken into a partition
Of two primes which add
To the original even digit
(Now, Euler, don’t fidget!)
But isn’t that totally rad!

Now since that day this simple thesis
Remains just that, a hypothesis
Forcing number lovers to lose their slumber
As they try to prove, primes in pairs can add up to any even number.

(Two is the exception, as we said before
Which is, come to think of it, a bit of a bore).

You can see an original of the letter that Goldbach wrote to Euler at mathisgoodforyou.com.

A 5th grade science assignment, transformed. A rant about Mother Goose. A math poetry challenge!  How did that come to be? And what does that have to do with loving the Interwebs? Read on…

I had written earlier about how my 10 year-old daughter had been writing poems on science (Scientific Poems or Sci-Po’s for short). It all started with an extra-credit assignment she needed to do for her science class, and a need, I perceived, to keep her blog (Uniquely Mine) up-to-date. She has quite a few written now. For instance here is one about a news item about scientists finding dinosaur eggs (and other dino-stuff) in India (Cluster of dinosaur eggs found in southern India), and here’s the poem:

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I had written earlier about the idea of “postdictable” which was defined as something that is “surprising initially, but then understandable with a bit of thought.” It lies at the spot between predictability and total chaos. The movie Sixth Sense is postdictable in the best sense of the world. Good teaching I believe needs to be postdictable. That is what keeps us engaged, keeps us waiting for more, the payoff as it were. And best of all, once all the pieces are in, we can’t wait to go back and review everything again, to see just how beautifully the whole thing holds together. There is a strong aesthetic component to this – a sense of wholeness, closure, elegance, and inevitability. Good poems have this quality, as do mathematical theorems. A well crafted lecture or a lesson plan has this quality as well. In my mind these ideas are closely tied to the Dewey’s idea of experience and to the idea of design. Hopefully I will have a chance to explore these connections in a later post but for now, here are a couple of commercials that I think were postdictable in a really cool kind of way.
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My daughter, whose creative exploits have been featured here before (for instance see her design for a math-music game), now has a blog, titled Uniquely Mine. It features original writing (poems, stories) by her. Do check it out. You can find regular updates on this blog via the beauty of RSS feeds on the right column (just scroll down).

The one piece by Shreya I would like to draw attention to is a story titled Obtuse can be right. She wrote this as a part of a fourth grade assignment, and it is pretty cool, with interesting geometry-related wordplay. Enjoy.

My friend Gaurav Bhatnagar (whose doggerel on ambigrams is featured on my blog as well) gave it high praise in his Facebook update, saying, “This is a masterpiece. Highly recommender (sic!). As good as Asimov’s short shorts.” I haven’t had a chance to see Asimov’s short shorts but I guess they are cool!

Anybody who knows me (and/or reads this blog) will know of my love of issues related to representation (see all postings under that category). So I am always looking out for new and interesting representations. An lovely example sent to me by Patrick Dickson is website titled Information is beautiful. This site, and much of the work showcased there is the creation of David McCandless, a free-lance information designer. He describes his interests as being in “how designed information can help us understand the world, cut through BS and reveal hidden connections, patterns and stories underneath.”

For instance check out If Twitter was a community of 100 people

An example of how just visualizing and representing data can change one’s prior conception of a idea can be found at How I learned to stop worrying and love the bomb (sort of).

The one that made the best connection to me personally was Caffeine & Calories.

Enjoy.

I just came across these lovely visual mathematical proofs.

For instance consider the following sequence:
1/2 + 1/4 + 1/8 + 1/16 + … = 1

and then see the following image on the blog!!

How cool is that!!!!

I had posted about something similar earlier (see visualizing mathematics).

Note: Before someone gets technical on me, I should admit that these are not “proofs” in the rigorous mathematical sense of the term, but they do provide visual evidence / explanation that helps us understand the underlying patterns.

I also recommend the blog (http://www.billthelizard.com/) on which these “proofs” appeared for a bunch of other interesting stuff about mathematics and programming.

I love how these interconnected pipes called the Intertubes lead to serendipitous discoveries. Here are two videos, the first I went looking for, and the second, fell into my lap, so to speak, due to YouTubes related videos section.

The video I went looking for was based on a delightful book I had picked up at a garage sale a few years ago. “The Dot and the Line: A Romance in Lower Mathematics” is a little book (20 pages or so) with an intriguing story-line and its geometrical illustrations. The main character is a straight line who is in love with a dot – but sadly she is more attracted to a wild, unruly squiggle. How the simple line develops his talents and wins the love of the dot is told through whimsical (and mathematically sound) illustrations.

I learned later that the famous animator Chuck Jones had made this into a short film. Here it is (thanks to YouTube).

This is just a wonderful example of how mathematics and art, perception and recognition, creativity and design can come together. This book (and the movie) speak to me at so many different levels. What is most amazing is the ability we humans have to see purpose and meaning in the simplest of lines and curves. So much of art and science depend on this ability to perceive / construct patterns.

Nowhere is this more beautifully (and humorously) illustrated than in this other video I discovered. Written and narrated by Mel Brooks (yes THE Mel Brooks) this animated short film, The Critic, takes a different interpretive stance (crankier and edgier) than the previous narration. That this short animation captures, powerfully how we as humans both seek, and question, the meanings of the patterns we see around us.

I just finished reading parts of Sheri Turkle’s latest book, Simulation and its discontents, and the parallels to what she is writing about and Mel Brook’s Critic are quite strong. The cranky one man in the short recognizes or “sees” meaning is some of the abstract images he sees on the screen and yet he questions their value. The scientists and designers quoted in Turkle’s book echo some of the same concerns.

What is amazing is that the Mel Brooks short was made in 1963, the Chuck Jones movie was made in 1965 and Turkle’s book was published just this year, in 2009!

Matt Koehler and I just published an article in Learning & Leading with Technology, the membership magazine of the International Society for Technology in Education (ISTE).

The complete citation is as follows:
Mishra, P., & Koehler, M. J. (2009, May). Too Cool for School? No Way! Learning & Leading with Technology, (36)7. 14-18. [PDF download].

This article includes a few examples of work done by my students as a part of a doctoral seminar. I had given them an assignment titled, How can a technology become an educational technology? and the work of three of them made it to the paper. I had written previously about Noah’s idea for using microblogging in the classroom (see here, here & most recently here). I haven’t blogged about the other two (though I have discussed them in presentations I have made) so it is good to have them represented here. Paul’s idea was to use specialized search engines (particularly visual search engines) to help students understand the idea of inter-textuality (the idea that texts often refer to each other in complex and intricate ways to create webs of meaning). Erik Byker, on the other hand, looked at how freely available DJ software can be used to teach mathematical concepts such as ratios, fractions, and percentages. Cool stuff!

The CITE Journal had a recent special issue devoted to TPACK. You can access the special issue (edited by Judi Harris and Matt Koehler) here or individual articles below.
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I recently blogged (here and here) about the experiment conducted by students in Italy that allowed them to use publicly available NASA audio recordings from the moon landings to determine the distance between the earth and the moon. I bit more online research led to me to the original paper published on arXiv. arXiv is described as an “e-print service in the fields of physics, mathematics, non-linear science, computer science, quantitative biology and statistics” and is hosted and supported by Cornell University. Reading the paper led me to a writing this posting, since i see this as one of the best examples of the TPACK framework in use that I have come across.

I provide the abstract and a link to the original paper (in pdf format) below and follow that with my thoughts.
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A followup to my previous posting about the Italian kids calculating the distance to the moon using recordings from the Apollo Space program.

As I read the story on the technology Review website, I came to the comments made by readers. One stuck out. This is what somebody had said:

Wow, they took the speed of light and multiplied by 2.62 then divided by 2. Interesting method of doing it, but it doesn’t take a rocket scientist for sure.

By focusing on the surface aspect of the math this person misses the point of the story almost completely. Misses, it I may add by almost the distance from the Earth to the moon. Read the rest of this entry »

For one reason or another, I have three consecutive posts regarding the earth and sun and moon – i.e. the local area in the solar system. I had just completed my previous postings (on on seeing through eclipses and measuring the radius of the earth) when I came across this news story (h/t Geekpress): School kids measure distance to the Moon Read the rest of this entry »

While searching for information for my previous posting on using eclipses to see, I came across an interesting paper that provided yet another way of figuring out the shape of the Earth.

Lynch, D.K. (2005). Turbulent ship wakes: further evidence that the Earth is round. Applied Optics, 44(27). 5759 – 5762.

First some context, see this image below:

View aft (south): ship’s wake extending to southern horizon (20050802a11).
Photo credit: B. Cuthbertson

I have come across some new TPACK related videos/podcasts (either on youtube or elsewhere) that I feel may be worth sharing.
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Here is a cool video about a “a mechanical, binary adding machine that uses marbles to flip the bits” – in other words a computer made of wood, that works at a pace that we can grasp! Marvelous. (HT: Collision Detection). Check out the video:

A fascinating series of illusions to reveal just how complicated a phenomenon perception is. I was particularly impressed by the “rubber hand” illusion.

And then, from Gizmodo, comes the Yoshimoto cube. Words are not enough to describe what this mind-bending object can do, you just have to watch the video.
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Gaurav Bhatnagar is one of my oldest friends – going back to 8th grade in Modern School, Barakhamba Road. He recently published his first book on Mathematics for kids, titled, Get Smart: Maths Concepts, published by Penguin India. The book also has an associated blog – though it is rather sparsely populated with posts at this moment. This makes him the third school friend of mine to have written a mathematics related book. How cool is that! Congratulations Gaurav!

You can read about the other two authors and their book here.

Political debates are heavily analyzed – by pundits and laypeople alike. I had my own minor visual contribution to this discourse through this WordMap/Cloud of the third and final debate between McCain and Obama . Such wordmaps are fun to create and see but are not terribly insightful. Yes you can see that Obama used the word “see” more often than McCain but how far does that really take you in terms of interpreting and making sense of the campaign. And then comes this! Read the rest of this entry »

I do not understand David Brooks. Brooks is an op-ed columnist for the NYTimes. For the most part his columns are right-of-the-political wing nuttiness, garbed in some erudite clothing. I am not linking to them here but his past few op-eds suggesting that McCain would make a great president despite the shallow, erratic and negative campaign he has run have become somewhat repetitive and tiresome.

And then, once in a while, when I have just about decided not to read his columns any more, he throws out some really cool and interesting stuff. Read the rest of this entry »

Title says it all… check it out.

Here are some links that came up during our discussion today regarding web-based software for education. Read the rest of this entry »

I love visual proofs of mathematical theorems. One visual proof I use quite often in my design courses (CEP817 or CEP917) is a visual proof of the fact that the sum of consecutive odd numbers is a square number.
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Understanding 4D while living in a 3D world.
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