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.