Between the Folds

Happy New Year everyone! Almost time to get back to class, but it should be pretty fun. In the first week back I plan on showing my students the film Between the Folds, a very interesting look at origami, some of its eccentric artists, and the way it relates to math and mathematical thinking. I recommend this film to anyone interested in the arts, creativity, and/or math, from a couple very simple rules (one page, no cutting or pasting) you can produce an infinite array of figures, and in fact it is these limitations that feed the creative force behind the artform.

Classic case of Girls v. Boys

Verdict: case thrown out. Ive never really given too much trust to these types of studies of one gender performing better or worse than the other. Unless the figures consistently come out overwhelmingly lopsided either way, we dont have anything to worry about with this issue.

Sometimes it seems that people assume that these types of results have to be exactly 50/50, but that not true. In reality, one side is almost always going to come out a little bit better than the other, just like coin flips - sure each side is equally as likely on average to come up, but flip a coin 100 times and I'll be surprised if you get a perfect 50/50.

I count myself lucky to get to witness an age in which all sorts of glass ceilings are being shattered, and understand how these types of studies could have interesting facets and social implications, but it seems like they are conducted just to have a headline either way and its always implied that "something must be done because the losing gender is falling behind." I think it would be more constructive to focus on better teaching to both genders in general rather than worrying too much about one or the other.

Today I learned...

That the reason pizza becomes strengthened and doesnt drop all the cheese and toppings onto your lap when you fold a slice down the middle is due to the mathematician Gauss's Theorema Egregium. Something you always know intuitively but dont know why.

From Wikipedia:
"An application of the Theorema Egregium is seen in a common pizza-eating strategy: A slice of pizza can be seen as a surface with constant Gaussian curvature 0. Gently bending a slice must then roughly maintain this curvature (assuming the bend is roughly a local isometry). If one bends a slice horizontally along a radius, non-zero principal curvatures are created along the bend, dictating that the other principal curvature at these points must be zero. This creates rigidity in the direction perpendicular to the fold, an attribute desirable when eating pizza, as it holds its shape long enough to be consumed without a mess. This same principle is used for strengthening in corrugated materials, most familiarly corrugated fiberboard and corrugated galvanised iron."

Car Talk Algebra Puzzler

I came across a math puzzle from Click & Clack on NPR's always entertaining Car Talk, see if you can figure it out, it goes like this:

"RAY: Last month, Tommy and I decided that we were going to take a trip north to see the foliage. Tommy drove the first 40 miles. I drove the rest of the way. We looked at the foliage for three or four minutes, then decided to head home.

We took the same route home.

On the way back, Tommy drove the first leg of the trip and I drove the last 50 miles.

I got home and my wife said, "Who did the driving?"

I explained that Tommy drove the first 40 miles, then I drove the rest of the way. On the way back, Tommy drove the first leg of the trip, and I drove the last 50 miles.

She said, "But who did most of the driving?"

I told her, "You can figure it out. In fact, you can even figure out how much more of the driving was done by that person."

And that's the question. Who drove the most -- and how many more miles did that person drive?"

Creating Art Using Only Lines and a Circle

Have you ever used a Spirograph? If you have ever been drawn toward the geometric patterns and colors created by this nostalgic toy, then you should try using some of the techniques described here at MathCraft and here at Jill Britton's website. Both of these wonderful sites teach you how to create intriguing works of art using very simple supplies: just a compass and straight edge, or a needle and thread.


Straight lines can be deceptive, as you can see see in the pictures to the side, believe it or not these pictures are made using nothing but straight lines! Notice the circle in the middle of the top picture is made by the outlines of the inside of 6 different colored pentagrams. The heart-shaped cardioid below has the same kind of outline, getting its curves from the perfectly straight lines composing its border. The same principle can be used by weaving a needle and thread through a piece of paper or fabric, as demonstrated on the Jill Britton site.

Take a look and try it out on your own, at the very least youll have something to doodle in class that actually uses math.

Visualizing Global Population Trends

In the video to the left, NPR explains visually how much the world's population has exploded in recent history. Its interesting to think about, it almost seems like we are in an age of flux, between an old time and a new future. How will our global culture as a species change just within the next hundred years or so? Will individuals naturally decide to have less children as a conscious or subconscious reaction to longer lifespans, better medical services, less food, water, & other resources to divide between an ever growing population? Is there a self-regulating limit that nature silently exerts on all of us, like an inflating balloon aware of the pop?

The Garden of Cosmic Speculation

I found this awesome mathematically-designed garden at reckon.posterous.com, check it out!

Technology & the Classroom

In this article by Ian Jukes of The 21st Century Fluency Project, he discusses the role of technology in the classroom. The phrase that stuck out at me was: "Because the most powerful technology in the classroom was, is and will remain...a classroom teacher. But not just any classroom teacher - it has to be a classroom teacher with a love of learning, an appreciation of the aesthetic, the esoteric, the ethical, and the moral - a teacher who understands Bloom and Gardner."

I have had discussions with people and read articles saying that we need to switch to a educational system based around ideas such as the Kahn Academy, in which learning is a very individualized, self-motivated, "high-tech" path, all but getting rid of physical classrooms, if not teachers.

I agree to a certain extent, and I love sites like the Kahn Academy, but the problem is that the only students that will really benefit from this method are those who are self-motivated, genuinely curious learners in the first place, the rest would not suddenly start to care about, say, Algebra just because its now on a computer screen.

So I agree with this article to a certain extent in the same manner; yes, I think that the younger generations do have a certain intuitive link with computers, the internet, and technology, and yes I agree that not only should schools be trying to utilize these tools to their advantage, but that it is inevitable that schools will naturally have to change technologically in this direction just like every other industry on the planet. We should absolutely be educating our students to be technologically savvy to be able to lead the way in the ever globally connected world. On the other hand, I sometimes think that articles like this one can sound a bit exaggerated in that I dont know that technology will suddenly make a higher % of students more interested in learning, because this problem of lack of self-motivation and caring about school has existed long before 20 or so years ago. This % of students, I would have to guess, wont be a drastically different % in a classroom loaded to the brim with technology - though I do believe technology implementation definitely does have a positive effect, if any, I think its more than anything the quality of teacher that matters more at inspiring students.

I do not necessarily think that children in America, on average, seem to be declining in motivation, inspiration, creativity, intelligence, or curiosity relative to the general American populace of any age, nor that they are so ADD or electronically "rewired" that they need to be surrounded by the stuff in order to learn. I do, however, see this trend becoming a growing problem in our society in general, not just in children who are growing up with it. This problem is more complex than I care to go into.

What has seemed fascinating to me is putting our education system into historical relevance, and I havent heard too many people doing that. Correct me if Im wrong, but the last century or two have been a significant historical aberration in regard to to % of educated populace. Before that, it was only a very small % that were fortunate enough to have an education as we know it available to them, and now its not only available, but mandatory, in our country! This is a great thing, but we forget how new this experiment is. I imagine (though just guessing) that those who did receive an education more than a couple centuries ago, would be comparable to our modern classroom's most self-motivated, interested learners.

So now we are faced with the challenge of educating the entire populace - and in an exponentially ever-changing technological world. Keeping up with all the tech is absolutely positive and unavoidable, but it doesnt erase more complex underlying social problems, and it doesnt trump what I believe to be the 2 greatest factors: an inspirational, creative, understanding, intelligent teacher and as small class sizes as possible.

So for SASIC, I think we are doing a great job with infusing technology, and we have some really awesome teachers, and have an atmosphere of creativity and room for the students to become inspired, and stretch their legs out, so to speak. So I think we should keep up the great work, and try to keep in mind that inspiring the students to want to learn throughout their lives, be constantly curious, and express their creativity constructively are the best gifts we can give to them and to our society, species.

Math is Kind of Important

Bad math from Germany's "Bad Bank" leads to $79,000,000,000 mistake, I bet some jobs were lost over that one.

Distribution

Linked here is a site (www.algebrahelp.com) that gives a pretty decent explanation of distribution, which we are learning this week in my Algebra class. For any students needing a second explanation, try reading through the 4 pages and trying the review at the end - bring any questions to me in class or post them below!

Vote For My Students' Photos

My Photography students entered a competition for Raise Your Hand Texas. The assignment was to have each student in their picture answering the question, “What two things inspire you to achieve success?”

They have a chance to win an iPad2, so please vote for their pictures here at Raise Your Hand Texas's facebook, just "like" their pic to vote. My students are in the ESC 20 section of their Photos, toward the bottom of the list.

Algebra Basketball

Algebra basketball game

Check out the link above for a cool basketball game that uses algebra, we will be playing this on Monday!

Vihart's Infinity Elephants

Visualizing infinite recursions and limits, from Vihart's blog.

Michael Harrison's "Revelation"

Composer Michael Harrison has reversed the progression of music history with his 2007 experimental work "Revelation."

Throughout musical history, musicians had been working to find a way to eliminate the dissonant sounds of bad intervals, called "wolf tones" or "wolf intervals." These intervals are the sloppy leftovers that are unavoidable when building instruments using Just Intonation scales or any other scales that utilize perfect ratios, which would include practically every instrument made before the 1700s when Equal Temperment hit the scene.

Every musician back then would have loved to find an instrument that would get rid of these ugly Wolf notes, and the Equal Temperment provided them with this answer and the freedom to easily play in any key... but it came at a price. The beautiful, sparkling sounds of the pure ratios needed to be compromised in the process, so now every guitar, piano, or other "fretted" instrument today plays intervals of notes that are only approximations and are slightly dissonant.

Notes played based on the pure ratios are more rich, resonant, natural, and beautiful sounding in every way, but many of us rarely get to hear them played. Though there still do exist experimental purists out there, such as Michael Harrison. In his "Revelation", he uses a piano tuned to a Just Intonation, and not only uses these usually avoided Wolf Notes, he embraces them and makes them the centerpiece of his song.

These otherwise dissonant intervals create a kind of drone with an audible "beat" at which the pitches fall in and out of alignment. Normally musicians will tune until they cant hear the "beat" anymore, but Harrison uses it to set the tempo for "Revelation", and it provides an almost eerie droning canvas for which to paint over with the sparkling pure tones provided by the Just-ly tuned piano. It works, somehow, and is hauntingly beautiful,

Predictive Policing Using Math Algorithms

Has anyone else seen and/or read Philip K. Dick's 'Minority Report'?

According to this ABC News report, police in Santa Cruz and other cities in California have started using a computer algorithm to predict where and when - down to the time of day - a crime is probable to happen. The program, developed by 2 mathematicians and an anthropologist, is based on similar programs used to predict earthquakes.

"...since the program's launch, the algorithm had correctly predicted 40 percent of crimes and had led to five arrests. In the last six weeks, Santa Cruz also saw a reduction in property crimes including car and home burglaries. Police said burglaries were down 27 percent in July compared to the same month last year."

I do believe that things which we normally perceive as random and impossible to predict, including human actions such as crime, actually do follow certain statistical patterns of which even the participants might not be conscious. Very interesting stuff, though there is a thin line between preventing crime, which is good, and the "Pre-crime Thought Police" in Minority Report.

My 1st Attempt at Casting Pods

Podcast Powered By Podbean

Podcasting & Education

I just finished listening to a pod-conference (did I just make that up?) between the SASIC teachers, exploring ways to use podcasting and other technologies in and out of the classroom. Im disappointed I wasnt able to join in on the live discussion, but as a podcast listener myself (Im especially crazy about radiolab), Im pretty excited about the possibilities that are opened by using these innovations.

I think a radiolab-esque sort of format (merging stories and discussion into segments around a central theme, while mixing in music, interviews, and other noise-scapes) would be a great way for students to present a project, sort of like a cross between journalism and art. But like anything of substance, it definitely takes a good deal of time and effort, so I will try to see what I can do and look forward to seeing what others can produce as well.

Its not a question of if technology will transform the classroom, the better questions are how and when.

Google+ for educators

I came across this article at a site called ReadWriteWeb discussing the educational potential for  Google+, the new social networking site by Google that is currently in the testing, or beta, phase (limited to a certain amount of invites).

This site should be a bit similar to facebook and/or twitter, except you will be connecting to other people in separate groups, or circles. So, for instance, people will probably have a circle for family, one for friends, one for co-workers, one for fellow hobbyists, etc., and educators could presumably have a circle for each of their classes, one with their colleagues, departments, and so on.

I will have to wait and see Google+ for myself to make any conclusions about it, but I have a feeling that within a couple generations from now post-facebook communities like these are going to start having some big impacts on education, if not other parts of our lives too.

Jackson Pollock - Artistic Physicist?

I came a cross a good article that suggests that Jackson Pollock, one of the first and most influential artists in abstract painting, may have had an intuitive sense for the physical and mathematical sciences that shows up in his works.

Pollock chose the type of paints that he used very carefully, seeking specific desired effects, textures, and colors. The researchers studying his paintings believe that his paint drippings exhibit certain properties of fluid dynamics that he might have been only subconsciously aware of, such as what is called a "coiling" effect on viscous liquids (thick slow moving liquids like honey, or thick paints).

This video demostrates the coiling phenomenon on a viscous liquid being poured slowly onto a moving belt. The zig-zags you see it start to make arent because the pouring container is moving, it is stationary, it starts doing this because the belt starts moving slower. So if Pollock slowly dripped a thick, viscous paint, it would land in a way that creates seemingly random patterns.

This makes sense, as Pollocks works are a kind of controlled chaos, seeming to be simultaneously random and yet also carefully and intricately designed. The researchers studying his paintings also suggest that his compositions show similarities to fractal geometry, which our minds see as pleasant as it is evident in the beauty of nature, such as clouds, snowflakes, mountains, and plant life.

For more of Jackson Pollock's works, click here.

A Brief Explanation of Math & Music

The connection between math and music is profound and pretty extensive, but here are some of the basics. It started with Pythagoras - yes, the same guy with the triangle theorem - a Greek man in the year 500 B.C. who had an entire philosophy, kind of a religion, with a large number of followers. His philosophies are considered among the most influential in history, and they believed, among other things, that everything in existence is composed of numbers.

So, probably also a fan of music, he tried to figure out the numbers in music. He experimented with a monochord, which is basically like a single guitar string, and found that by changing the length of the string (like holding down a guitar string) has a direct effect on the note that sounds.

For instance, putting your finger on the exact middle of a string on a guitar (the 12th fret), you are changing the length of the string to half the length that it is normally, the ratio for is 1/2 (or 1 to 2) since it is half. At this point, these two notes are the same, except one is an octave higher. Since sound is a wave in the air - like a wave in the water, except we cant see it - and the frequency determines what note it is, the frequencies of the notes behave in the same way that that the length of the string does. If you play the open (or full length) "A" string on a guitar, its frequency is 440 Hertz and is the "A" note - but then if you put your finger on the middle of the string, its still an "A" note, except an octave higher, and its frequency will be 880 Hertz. So, 2 times 440 = 880 and the ratio of the string lengths is 1/2!

The same can be said for the other notes in the scale. For instance, the ratio of string length from the same open "A" string to the "E" note (7th fret) is 3 to 2 (or 3/2) and the frequency of the "E" note is 660 Hertz. So, thats 660 to 440, or 660 divided by 440, which is equal to 3/2!!

For some reason our ears recognize these ratios without thinking about it, because simple ratios (such as 2/1 and 3/2 and 5/4) are the notes that sound pleasant to us - and our scales are built around these - whereas complicated ratios sound painful and grating to us.

So, many instruments after Pythagoras were built around his mathematical ratios. But there was a problem - it was hard to switch keys on these instruments (Like from the key of C to the key of E) because the relative ratios just didnt add up right. Along came Sebastian Bach who devised a way of dividing the frequencies into what he called the "well-tempered scale." Bach used this system to write songs in which he could easily switch between many different keys in the same song. But the sacrifice that he made for this ability was that the notes were no longer the pure mathematical ratios, but were instead close approximations.

This "well-tempered scale" was eventually developed into the "even-tempered scale," in which all the 12 half steps in the octave are the same distance apart. This "even-tempered scale" is the system that our modern "fretted" instruments use, like the piano and guitar, and this is what allows our music writers so much freedom in composition, though the notes you hear are only approximations of the pure notes that your ears want to naturally hear.

Profound theorems:

Theorem 1. A sheet of writing paper is a lazy dog.

Proof: A sheet of paper is an ink-lined plane. An inclined plane is a slope up. A slow pup is a lazy dog. Therefore, a sheet of writing paper is a lazy dog.

Theorem 2. A peanut butter sandwich is better than eternal happiness.

Proof: A peanut butter sandwich is better than nothing. But nothing is better than eternal happiness. Therefore, a peanut butter sandwich is better than eternal happiness.

The Wisdom of the Crowd

In this excerpt from The Wisdom of Crowds by James Surowiecki, a brief story of British scientist Francis Galton is told about his witnessing of a contest that challenged people to guess the weight of an ox. Much like the guessing games that you might see in a mall that ask you to guess the number of gumballs in a giant container, the person with the closest guess to the actual weight of the ox won a prize.

Now Mr. Galton was a pretty skeptical person, believing that the majority of people were uneducated and therefore were a detriment to society and/or a democracy. Most of the people entering the contest were not experts on ox weights, so he was curious to take a look at the results of the contest, stating “The average competitor was probably as well fitted for making a just estimate of the dressed weight of the ox, as an average voter is of judging the merits of most political issues on which he votes."

But what he found was surprising... he took the average of all the 800 or so guesses and, it came to 1,197 lbs., which is an almost exact match to the correct answer of 1,198 lbs.!! Much like a colony of ants can achieve much more than any single ant can alone, the entire group of humans averaged as a whole had better insight than any single person.

This is routinely proven time and again in any contest of this kind - following the rule that the greater the number of people participating, the more accurate the average guess will be (so only a handful of people is probably not big enough). This makes me wonder about the collective power that we have as a species, and about how much goes to waste or unnoticed.

Being a democracy consisting of mainly 2 parties of political ideologies, both seem to push toward either extreme, repelling like opposite poled magnets, and these extremes are usually how they are represented in the media. But if we could train ourselves as a society to see the power of the average, I think we might find that the answer is almost always a compromise found somewhere in the middle.

I dont know how much faith I put into polling, but if only we could express the big, dividing issues of the day into a mathematical vote, and go with the average, maybe we would be better off, or at least cut through some of the bureaucracy tying down our representatives. Or maybe that's what American Idol already is, a sophisticated test to see if the masses can come to the right conclusion in the end... nah...

Benford's Law

Wikipedia - Benford's Law

This bizarre mathematical theory suggests that the numbers you encounter in the everyday world might have some hidden patterns in everything from your electricity bill, to your street address, to your license plate number.

From Wikipedia: "According to this law, the first digit is 1 about 30% of the time, and larger digits occur as the leading digit with lower and lower frequency, to the point where 9 as a first digit occurs less than 5% of the time. This distribution of first digits is the same as the widths of gridlines on the logarithmic scale."

This particular law is used to spot all kinds of financial fraud, as those who are making up numbers tend to do so following a uniform distribution. There are probabilities at work all around us, if you look for them you will be surprised at the degree in which they determine our lives.