Monday, February 5, 2007

Why, of all the numbers in the range 1 through 20, do we like 17?

Dave Munger over at Cognitive Daily has posted the results of his poll asking readers to pick a number between 1 and 20. The number 17 was picked almost 18 percent of the time, compared to the expected 5 percent from his sample of 347 responses. Comments on the results included the following attempts at explaining why humans seem to prefer 17:

Austin wrote:
As someone who chose the number 17 perhaps I can offer my personal insight - I was looking to select the most unlikely number (sorry!). In this sense i wonder if this is actually a measure of perceived least common number rather than random selection - there is a body of research about the ability to actually generate random numbers being very difficult for human subjects (methinks Sallice, T).
Michael said:

I think the answer is bisection. Our minds first go to (20+0)/2=10 but then we think, oh that's too obvious. So we next choose (20+10)/2=15. Nope, still too obvious. Then we go (20+15)/2=17.5, round it to 17 and the loop ends, because we are getting too close to 20 to be comfortable. Nonsense, probably, but that is what comes to mind...

Craig Pennington wondered,

...if the fairly uniform distribution of n*7 modulo 10 (7, 4, 1, 8, 5, 2, 9, 6, 3, 0) has anything to do with the popularity of 7 and 17. I'd bet (x*10)+7 is preferred for most "random" ranges (more so for primes -- and just under 1 in 4 primes less than 100 is 7 modulo 10.)

i wonder what would be result of an equivalent experiment carried out on some of our primate cousins? Experiments do suggest that rhesus monkeys, at least, do posses the capacity for "spontaneous number representation."

Sunday, February 4, 2007

Stunning facts

Did you know that 111,111,111 x 111,111,111 = 12,345,678,987,654,321? It is number 87 on the Interesting Real Facts site, where i also discovered that:
  • On average, 12 newborns will be given to the wrong parents daily.
  • Leonardo da Vinci could write with one hand and draw with the other at the same time.
  • The average person's left hand does 56% of the typing.
  • You burn more calories sleeping than you do watching TV.
There are 104 other "Strange but true facts"on the page. Enjoy!

Thursday, February 1, 2007

I support the Math Insurgency

i have just found out via a link at Let's Play Math about the math insurgency that seems to be brewing in Washington state. It is led by a group called Where's the Math whose mission is
To ensure that all Washington State students have an equal opportunity to compete successfully in the international economy by aligning Washington State math standards, assessments and curricula to those of top performing nations in the world.
They are concerned that mathematics achievement in America is far below what it should be and seriously lagging most other industrialized countries in the world. One of their members, Hugh Taylor of Washington state laments that U.S. math education in general is "uniquely unsuccessful."

While sympathizing with these concerns, it is worth noting the global dimensions of the problem: For students in many 'developing countries,' poor or mediocre math education--and thus achievement--is the norm. That is why i hope the math insurgency succeeds in the US, and spreads to the developing world.

Tuesday, January 30, 2007

My Schrodinger's Cat moment

This may sound extreme, even crazy...perhaps. But just keep an open mind for a minute. You see, this morning, i experienced something of a Schrodinger's cat moment. Mercifully, it did not have to do with the life and death situation depicted here. Nevertheless, i experienced two emotional states simultaneously--a mixture of being happy and not-happy at the same time. The not-happy state came from my being forced to abandon an old but much loved calculus book, "Calculus with Analytic Geometry" by A. B. Simon. The happy state was induced by my discovery of a new, excellent alternative, Calculus by G. Strang.

The Simon had been a surprise birthday gift from a friend, back in the 80s. It served me for years as a faithful reference, coming to my rescue more than once when i needed a quick brush-up on this or that topic in a hurry. But, attached as i am to this hard-covered and heavy old tome, it has been increasingly difficult to carry it along on my frequent international travels. So i began searching for a downloadable, well-written and comprehensive textbook online. i figured that such a book, combined with the Derive software on my machine, should greatly facilitate my ongoing adventure in self-learning, without the hassle of lugging a big book around.

So, when i stumbled upon a PDF version of Calculus by Strang freely available on the MIT OCW site, what else could i do but gleefully grab the whole shebang. Granted, the quality of the PDFed Strang leaves something to be desired, but that minor blemish can immediately be dismissed as a small price to pay for the sheer clarity and elegant brevity with which Strang handles the introduction to the subject. For example, he starts Chapter 1 with "The right way to begin a calculus book is with calculus." Strang follows this with a quick introduction to the central question of calculus:
Notice that the units of measurement are different for v (velocity) and f (distance). The distance f is measured in kilometers or miles (it is easier to say miles). The velocity v is measured in km/hr or miles per hour. A unit of time enters the velocity but not the distance. Every formula to compute v from f will have f divided by time.

The central question of calculus is the relation between v and f.
....We need to know how to find the velocity from a record of the distance. (That is called differentiation, and it is the central idea of differential calculus). We also want to compute the distance from a history of the velocity. (That is integration, and it is the goal of integral calculus.) Differentiation goes from f to v; integration goes from v to f. We look first at examples in which these pairs can be computed and understood.
As i delve deeper into Strang's Calculus, it seems as if i am reading my old Simon! Both authors seem to be one of a rare kind, possessing the same ability to communicate deep truths to the uninitiated in simple, unambiguous language. As this realization of sameness deepens, i realize that, far from betraying Simon, my download of Strang actually honors both authors while enriching my library! And, just like that, my Schrodinger's Cat moment suddenly passes, and i feel myself collapse, thankfully, into a single happy state!

Monday, January 29, 2007

My take on mathematical software tools for adult learners

Last Saturday, i posted my first article to Helium's user-generated article database. i found out about Helium last year when it was covered by Marshall Kirkpatrick on Techcrunch. My article on math software tools is reproduced below (slightly edited). It draws on my on-going experience as an adult self-learner engaged in a year-long calculus refresher course:
Are you a working adult seeking to improve your knowledge of essential mathematical concepts and techniques? Whether you are enrolled in a formal course or trying to teach yourself, one of the first things you need to accept is that mathematics cannot be learned simply by reading about concepts and techniques. It takes practice: the study of new concepts must go hand-in-hand with practice at applying those concepts to the solutions of problems expressed in mathematical forms.

Most adults-especially those with liberal arts or non-science educational backgrounds-are often discouraged and put off by the prospect of performing time-consuming mathematical calculations. The good news is that help is now at hand in the form of mathematical software that can handle most of the mechanical or algorithmic parts of problem-solving. One such software tool is Derive 6, which perform numeric and symbolic computations, algebra, trigonometry, calculus, and plots graphs in two and three dimensions. There are several other tools of comparable quality such as CalcCenter 3, a product of Wolfram Research.

By integrating one or more of these technologies into the process of learning mathematics, adult learners are freed to concentrate on the mathematical meaning of concepts. This in turn could dramatically shorten the time required to learn and master the applications of those concepts to real life problems, while appreciating the inherent beauty of mathematics.

Saturday, January 27, 2007

Update: Higgstory in the Making?

John Conway has posted Part 2 of his exciting report on the hunt for the Higgs boson. Although he is still very cautious about the significance of their experimental results, one cannot help but sense that a major discovery is about to be announced; what John and his colleagues are seeing is real sunshine at the end of the tunnel--and not the headlight of a train coming from the opposite direction:
In the end, some day we are going to have something new right there in our data, and we cannot shrink from it. We’ve gone a very long time with no truly new discovery in particle physics, no observation that truly changes the paradigm. We’ve gotten used to fluctuations coming and going, and are justly skeptical of any new ones that come along. But I think I got a glimpse that Saturday morning of what it will feel like when we do have something new, and real, and it’s a sensation that I hope I’ll have again some day soon.
We interested onlookers can only, well, look on--and wish them well! Meanwhile, bravo to John, his team and all others involved in the making of this particular Higgstory!

Friday, January 26, 2007

Higgstory in the making?

How can anyone read the opening lines of John Conway's post about the hunt for Higgs boson and not get glued to their monitor?

"I’ve been looking for the Higgs boson for almost 20 years.
So there I was, on a Saturday morning in December, at CERN as it so happened, when I saw the graph we’d been working towards all year. At first I thought it was some mistake - the hair literally rose up on the back of my neck, and I said: 'Holy crap! What’s that?'"

John's post--and you don't have to be a particle physicist to comprehend it--provides a fascinating story of his personal involvement in the search for this ellusive particle. Along the way, he offers a basic description of what it is: "The Higgs boson is a particle which is essentially a by-product of the Standard Model, a sort of physical manifestation of a hypothetical 'Higgs field' which permeates all space-time and with which all particles have some level of interaction."

John compares the search for Higgs boson with the scientific quest that began, a hundred years ago, to work out the periodic table of the elements, an elegant framework that orders elements into into neat rows and columns according to their chemical properties.

"It took...thirty years of experimenting and theorizing to figure it out. That led to quantum mechanics, the solution to the hydrogen atom, and then the understanding of more complex atoms and molecules. Then it all broke open: nuclear energy, silicon electronics, computers, cell phones..."

So, are we on the verge of a history-making discovery that will similarly break open a new pandora's box full of: nuclear fusion, quantum computers and artificial intelligence, anti-gravity, and...and (gasp) faster-than-light travel...? i can't wait to read Part II John's Higgstory as it unfolds.

Wednesday, January 24, 2007

The Happy Planet Index

The folks over at the Happy Planet Index (HPI) have developed a new way to rank countries: by their performance in supporting good life for their citizens "whilst respecting the environmental resource limits upon which our lives depend." Vanuatu tops the list of countries with an HPI of 68.2, closely followed by Colombia (67.2) and Costa Rica (66.1). Zimbabwe is at the bottom at 16.6. Nigeria at 31.1 does better than the USA, which shares an HPI of 28.8 with the West African state of Cote d'Ivoire. So how, you may ask, is the HPI calculated? "The statistical calculations that underlie the HPI are quite complex. However conceptually, it is straight forward and intuitive: HPI=(life satisfaction + life expectancy)/Ecological footprint. No country in the world has yet achieved the "reasonable ideal" of of 83.5. HPI is a creation of the New Economics Foundation, supported by Friends of the Earth.

Tuesday, January 23, 2007

New schools in a new century...

America's New Commission on the Skills of the American Workforce, a high-powered, bipartisan assembly of Education Secretaries and business, government and other education leaders has concluded that "we need to bring what we teach and how we teach into the 21st century." Well, its about time, according to Claudia Wallis and Sonja Steptoe in their article, "How to bring our schools out of the 20th century," published in Time Magazine. They propose the following list of "21st Century Skills" that American schools should cultivate in their students:
  1. Knowing more about the world. "Kids are global citizens now, even in small-town America, and they must learn to act that way."
  2. Thinking outside the box. "Kids also must learn to think across disciplines, since that's where most new breakthroughs are made. It's interdisciplinary combinations--design and technology, mathematics and art--"that produce YouTube and Google," says Thomas Friedman, the best-selling author of The World Is Flat."
  3. Becoming smarter about new sources of information. "In an age of overflowing information and proliferating media, kids need to rapidly process what's coming at them and distinguish between what's reliable and what isn't."
  4. Developing good people skills. "Most innovations today involve large teams of people," says former Lockheed Martin CEO Norman Augustine. "We have to emphasize communication skills, the ability to work in teams and with people from different cultures."
The full article contains interesting case studies and sound recommendations for re-aligning public schools--originally designed to educate workers for agrarian life and industrial-age factories--to make the necessary shifts. All well and good, and I would add: while making these needed changes in the aims and content of education, America should also bring educational policies into the 21st Century. At the very least, these policies should ensure equal opportunities for every citizen to access the educational resources of the 21st century.

Monday, January 22, 2007

I fall in the average, therefore I am?

Thanks to John Daly for pointing us to a new book by Sarah E. Igo titled "The Averaged American: Surveys, Citizens, and the Making of a Mass Public." In it she demonstrates "how opinion polls, man-in-the-street interviews, sex surveys, community studies, and consumer research transformed the United States public." The American public, she argues, is currently awash in aggregate data. One result is that survey techniques and findings have become the vocabulary of mass society--and essential to understanding who modern Americans think they are. Huh? So as people become data-intensive, they become statistical in their sense of self? Is that freedom? Reminds me of a sub-plot in the movie "The Shawshank Redemption" with all those prisoners, many there for life with no hope of escape. The only one who managed to escape was an above-average kind of person who could observe and think beyond what the prison authorities and his fellow prisoners tried to tell him: you cannot escape from this jail.

Sunday, January 21, 2007

Between our rock and a cold place

The Big Sky Astronomy Club has the above photo of our earth taken by Voyager 1 in 1990 as it sailed away from Earth, more than 4 billion miles away (apologies for that obstruction on the photo caused by my image capturing software). "To my mind," said a deeply moved Dr. Carl Sagan, the famous astronomer at a public lecture on October 13, 1994 where he showed the photo, "there is perhaps no better demonstration of the folly of human conceits than this distant image of our tiny world. To me, it underscores our responsibility to deal more kindly and compassionately with one another and to preserve and cherish that pale blue dot, the only home we've ever known." Looking at it, one cannot help but think again just how fragile our situation really is--only a thin atmosphere stands between us on this tiny rock, and that cold, dark but strangely attractive expanse of zero point energy.

A bone of prime value

Uh, my name is not Ishango if you thought that's where the URL of this blog came from. I chose it after reading the fascinating story of the Ishango Bone, an ancient bone tool that is believed to be the oldest table of prime numbers. It was discovered in 1950 by Belgian explorer Jean de Heinzelin de Braucourt near the headwaters of the Nile River at Lake Edward (now on the border between modern-day Uganda and Congo). More than 20,000 years old, the Ishango Bone has
...a sharp piece of quartz affixed to one end, perhaps for engraving or writing. It was first thought to be a tally stick, as it has a series of tally marks carved in three columns running the length of the tool, but some scientists have suggested that the groupings of notches indicate a mathematical understanding that goes beyond counting.
Update: An even older object--the Lembombo Bone--was found in Swaziland. This one is actually 10,000 years older than the Ishango bone, and seems to have been the work of an iron ore mining civilization dating back 43,000 years!.

Saturday, January 20, 2007

Like it or not, math matters. A lot.

As science and technology penetrate ever more deeply into many spheres of public and private life, only the most die-hard of 'math-haters' will fail to appreciate the increasing role of mathematics in the development of our contemporary society. Early in my career as an urban and regional planner with a liberal arts background, it was necessary every now and then for me to fall back on earlier training in statistics and basic math in order to understand some of the more technical documents I came across in the course of my work.

With time, however, my job required me to deal increasingly with people from other disciplines including climate science, ecology, finance, energy. Then I found myself rapidly approaching the limit of what I can know--and do--about issues that fall outside my field. For example, I could barely follow the more technical aspects of climate science discussions because they routinely utilize the language of advanced mathematics, beyond the rudimentary knowledge and skills I had learned in my undergrad years.

The lesson was: any professional with a liberal arts background whose work intersects with climate change issues, needs to significantly upgrade his/her knowledge of advanced mathematics, or risk being lost in the conversation. At a more general level, one might argue that society as a whole stands to lose if the search for solutions to the complex problems facing us today are left in the hands of a relatively small group of "insiders." We need to democratize our problem-solving system by raising the general level of numeracy (as distinct from literacy) worldwide.

With these thoughts in mind, I began a long-term study of relevant topics in mathematics. Given my full-time job, I could not do this by enrolling in traditional classroom courses. And none of the online courses I came across satisfied my particular needs. On the other hand, I discovered hundreds of online tools and resources by which to teach myself what I needed to know. And it wasn't all work and no play. I have been pleasantly surprised at the number of fun things you can find in the math world, like this visualization of pi that was posted on Wikipedia today, and that guy who is claiming that he's found a solution to the troublesome zero.

In starting this blog, I hope to benefit from the "collective intelligence" of the Web in the form of readers comments and suggestions for locating the best math learning resources, interesting news and the like. It is a social learning experiment of sorts. I'm sure there are several people who also feel the need to improve their knowledge and skills in math for personal or professional reasons. May they find inspiration from my on-going account, and hopefully, (re) start their own journeys.

The problems facing humanity today are large, and currently revolve around human technology, society and nature relations. Many of these problems are best expressed in the language of mathematics; a language that many people do not understand mainly because they did not consider it relevant to their needs. But if one seeks to be part of the solution to the troubles we face, one needs to be able to communicate clearly in as many languages as possible, including mathematics. To know or not to be, that is the question.