Wednesday, April 24, 2013

Organizing your crayons

My little buddy here is intent on an important task. He is trying to organize his crayons.
Go ahead, make my gray!

You may think this is an easy task, organizing crayons, but you would be wrong. At least, that is, if you are an adult. I distributed crayons to a group of first and second graders. I set them loose in a gym and asked them to organize themselves by color. It took maybe ten or fifteen minutes, but they did it. I gave the same task to a group of managers in a large corporation that shall go nameless. They held preliminary meetings with action items for subcommittees to perform market analysis, develop a business case, and research environmental impacts. Project leaders were appointed to develop realistic estimates of timelines, and to do risk analysis. I expect them to get back to me  by mid-June with a detailed action plan. They got started back in September of 2007.

Arranging crayons in a line

One possible way to arrange crayons is by the colors of the rainbow. I pulled 13 crayons out of my favorite box of crayons and made a nice lineup which I call the "craynbow" [1]. You can see red, red-orange, orange-red, orange, yellow-orange, orange-yellow, yellow, and so on.

The craynbow

I know this is completely atypical of me, but I sort of told a lie. The last two crayons on the top, the red-violet and violet-red, are not actually in the rainbow. The most startling difference between the rainbow and the craynbow is that the rainbow is two crayons short of a full deck. This is probably obvious, but the additional two crayons form a bridge that allows us to connect the bottom of the craynbow to the top. Thus, the 13 crayons can be laid out in a circle. 

But there are just a few other crayons that didn't make the cut. Most notably, the craynbow is missing white, gray, black, pink, and brown. Also the craynbow does not have a spot for lavender, burnt sienna, periwinkle, or maroon [2]. There are 96 crayons in my box, and 83 of them don't have a spot reserved for them in the craynbow. You don't have to be the sharpest crayon in the shed to see that you can't arrange a box of crayons in a single line in a way that chromatically feels good [3].

Color is three-dimensional

There is a simple reason why we can't put all the colors in a line. Color is three-dimensional. The craynbow looks at just one of those three dimensions of color, that is, the hue. Us color theory guys have two additional attributes that we assign to colors: chroma and lightness.

Showing hue and chroma

Let's start out with those 12 crayons arranged around in a circle. (I don't know what happened to the 13th crayon, by the way. Maybe it fell behind the couch.) What happens if we mix each of those crayons with a little gray, and then with a lot of gray? The results are shown in the really nifty circle below. Feel free to drop the artist an email telling him just how much you like the circle.

This cool drawing took me four hours to draw

Now for the cool part. Any color in this circle can be identified by naming the hue (the original crayon) and the chroma (how far it is from gray). The hue in this circle is the crayon that you started with. The chroma tells us how much gray was mixed in, that is, how rich the color is.

By the way, if you happen to be into analytic geometry, you might realize that hue and chroma are polar coordinates for the color plane. If, on the other hand, you don't really care all that much about anal retentive geometry, then you probably don't care about whether your polars coordinate.

Showing lightness and chroma

This next image demonstrates how lightness (along the up and down axis) combines with chroma (along the right to left axis). All the colors in this image have the same hue, yellow-red (5YR) [4].

A page from the "Munsell HVC Color Charts"
(It took me four hours to put these paint chips in order.)

Along the left edge, we see a scale of colors that go from black up to white, all with just a tiny bit of orange flavoring to them. As we move to the right, there is more and more orange added--that is higher chroma.

Notice that we just don't get all that many shades of orange-black. As you add orange (or yellow) to black, it just has to get lighter. Thus, there is only one shade of orange on the bottom row. On the other hand, you can add orange to white without significantly decreasing its lightness. So, there are four shades of orange in the top row. The most saturated orange is basically still a pretty bright color. Blue and purple work pretty much the opposite way. Adding blue or purple to a pale gray will necessarily make it darker. It is possible to have a very saturated blue or purple that is also very dark.

These three attributes can be put together to create a color tree. The pages (like the one above called 5YR) are bound together at the left edge, and fanned out as shown in the very detailed and elaborate illustration below.

Here is another four hours of my life spent in Photoshop

Such a color tree was devised by Albert Munsell in the early 1900s. It is a testament to his genius and meticulousness that you can still buy a facsimile of this color tree almost a full century later. Not only that, but the guy actually has a blog, despite the fact that he died in 1918. He must be a ghost writer, I guess.

A very attractive model shows the Munsell color tree

Someday I will write a blog post about how this guy Munsell laid the foundation for the ever-popular color space CIELAB, and came to be known as the Father of Color Science. He was also the father of A. E. O. Munsell, who carried on his work. I don't intend to write a blog post about how Albert became the father of A. E. O.  

What this all means

The first message is that you need three dimensions to organize a box of crayons. That's cool, but the more important message is that with the colors organized and quantified like this, it is possible to put a unique identifier on each color. All you need to do is specify the hue, chroma, and lightness, and you can unambiguously label any color. This might not be as romantic as using words like vermilion or aubergine, but it's a darn sight more precise. What young lady wouldn't prefer to hear poetry that scientifically specifies the color of her rosy cheeks?
[1]  A major supercomputer company is developing a computer just to compute all the colors in the universe. The computer will be called the Cray-Ola.

[2]  A ship loaded with red paint collided with a ship loaded with blue paint. All the crew were marooned.

[3] I used to tell my color classes just that you can't arrange your crayons all in a line in a way that made sense. Some smart alec said "Sure you can. Alphabetically!"

[4] I understand that some people might be so silly as to call the color "5YR" by the pedestrian name "orange". Please try not to laugh at them, ok? (My wife told me that I should point out that I'm being sarcastic here.)

Wednesday, April 17, 2013

Scientists don't need math?

The Wall Street Journal recently published an essay called: "Great Scientist ≠ Good at Math". The essay is by a scientist by the name of E. O. Wilson. This gent is considered the world's expert on myrmecology. I am sure most of my readers already know what this is, but just in case there are a few people who stumbled on this blog who aren't aware of the field... myrmecology is the study of ants. Dr. Wilson is an important enough guy that someone has taken the time to make a Wiki entry on him [1].
I am not a myrmecologist, but I play one in old SciFi B movies

Dr. Wilson had some controversial things to say about the amount of math needed for scientists: "Many of the most successful scientists in the world today are mathematically no more than semiliterate." Wow. Them's fighting words. Especially after my last blog post where I bemoaned the innumeracy of the USA.

I took a few of my scientist buddies out for a beer and asked them what they thought about whether math is important for a scientist. Here is what they had to say:
Whoever despises the high wisdom of mathematics nourishes himself on delusion.
Da Vinci

The Universe is a grand book which cannot be read until one first learns to comprehend the language and become familiar with the characters in which it is composed. It is written in the language of mathematics.

What science can there be more noble, more excellent, more useful for men, more admirably high and demonstrative than mathematics.
Benjamin Franklin

Every new body of discovery is mathematical in form, because there is no other guidance we can have.
Charles Darwin

When you can measure what you are talking about and express it in numbers, you know something about it.
Lord Kelvin

But there is another reason for the high repute of mathematics: it is mathematics that offers the exact natural sciences a certain measure of security which, without mathematics, they could not attain.
Albert Einstein

A first fact should surprise us, or rather would surprise us if we were not used to it. How does it happen there are people who do not understand mathematics? If mathematics invokes only the rules of logic, such as are accepted by all normal minds ... how does it come about that so many persons are here refractory?

Henri Poincare

What? Math? Oh yeah. Ummm... I think it's important to scientists. You don't think they can afford to pay accountants to do their taxes, do you?
John the Math Guy 

So... sorry, Dr. Wilson. My buddies don't agree with you.

[1] Just in case you're wondering, I have not yet taken the time to write my auto-Wiki-ography. I'll get around to it some day. Certainly before anyone else will add me to Wikipedia!!!

Wednesday, April 10, 2013

Why all the problems with Math?

Nine out of every seven adults have trouble with fractions. It's a sad, but true, statistic.

Earlier this week, I was sent a link to an info graphic about America's Math Problem from my good friend Allison of, Well, I haven't actually met her yet, but I am sure she would be a good friend if we ever did meet. She sure seemed nice in the email. Here is one graphic from the webpage she sent me to:
Yet another depressing statistic

Anyway... the infographic shows a bunch of the usual depressing statistics about math education and the state of math abilities here in the U.S. For example, eighth graders in the U.S. rank 25th in the world in math skills. Only one in three are considered proficient.

One in five adults are innumerate. Why does this happen?

John the Math Teacher Guy

Years ago, I had a hobby job teaching remedial math at a local university. This zero-credit class was basically a review of high school algebra. Passing this class or testing out of it was a requirement for getting a diploma at this university.

It was not hard for me to get this job. Why? They were desperate for teachers. Every semester, 7% of the students at this university of 20-some thousand students were taking this class. (If my math is correct, that's 1,400 students.) The need for teachers was so great that basically anyone within 50 miles of the university who could conjugate a quadratic equation was asked to teach.

Over the course of eight semesters, I developed some opinions on why so many people have trouble with math, and of course, I have some opinions on how to fix it.

"It is obvious"
My eighth grade math teacher

There is an old joke, told by old math guys. A professor was going along in his lecture and said "It is obvious, then, that the cosine of this angle is..." A student toward the front timidly raised a hand. "I'm sorry, I don't understand. Why is that obvious?"

The professor started to speak, and then checked himself. He took out a piece of paper and started alternating between scribbling, pacing, and tugging at his beard [1]. After a few minutes, he ran out of the room to his office. Ten minutes later, he came racing back into the room, waving a piece of paper, saying "YES!!  It is obvious!!"

Here is my opinion... There is a tendency for professors and teachers to present a lesson so as to make the exposition seamless. When the teacher is at the board, the class does not get to hear the wrong turns that were taken the first time he or she saw this problem. "Hmmm... maybe I can make a perfect square if I add seven to both sides? Wait, that doesn't work. Oh fudge. Wasn't there some fudge left in the fridge?" I always gain weight when I do math anywhere near a fridge.

The student is left with the impression that the teacher just automatically knew that factoring the denominator would lead to the solution. This inevitably leads the student to the next thought... "I didn't immediately see that, so I guess I don't have a math brain." 
Normal brain (on left) compared with the math brain (on right)

As much as I would like to claim superiority over mere mortals for my stupendous math brain, the whole "math brain" thing is a myth [2]. Twice this past week, I found myself helping folks with Excel. Both of these folks were giving me variations on the "I can't do math" theme. And the thing is, both of these people actually did understand what they were asking me about. All I did, in both cases, was say "yeah, you're doing it right."

Don't get me wrong here -- I think most math teachers are good. A typical student has 13 teachers who learn them math by the time they get a high school diploma. If only 10% of math teachers manage to intimidate, then a typical student has a 130% chance of running into a teacher who derails them [3]. Once the student learns that they can't do math, it is very hard to get them back in the saddle.

Problems with stories
Glen had five story problems and Gail had six migraines. If the hypotenuse went on sale for 30% off, then how long would it take Glen and Gail to go crazy together? Story problems are the source of much anxiety. It is a little known fact that story problems led to the Crimean War. Absolutely true. And story problems also caused the bubonic plague.

Edward MacNeal argues that at least part of the problem is that the linguistic aspect of math is often separated from the math side of math. At least that's what I think he is saying. I dunno. I don't do words so good. 

The non-statistical human brain
I have blogged before about how people just don't make good statisticians. Surprisingly, I am not the first to bemoan this problem. John Allen Paulos did a whole bunch of bemoaning this in his book "Innumeracy". Here he talks about the traps that many of us fall into:

"[This] book is largely concerned with various inadequacies - a lack of numerical perspective, an exaggerated appreciation for meaningless coincidence, a credulous acceptance of pseudosciences, an inability to recognize social trade-offs, and so on."

"In an increasingly complex world full of senseless coincidence, what's required in many situations is not more facts - we're inundated already - but a better command of known facts, and for this, a course in probability is invaluable."

As an example of our over-willingness to attach meaning to coincidences, here is a quote from the brilliant scientist and mathematician, Michele Bachmann: "I find it interesting that it was back in the 1970s that the swine flu broke out under another, then under another Democrat president, Jimmy Carter. I'm not blaming this on President Obama, I just think it's an interesting coincidence."  Gerald Ford was president at that time, but there are just all kinds of flaws in her logic.

Here is another quote, from author H.G. Wells: "Statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write."

How do we solve this problem? I think this concept must be taught not once, but throughout our education. Intuition is a good thing, and can be helpful in leading us to solving problems. But, intuition is not knowledge. I am just kinda thinking out loud here, but maybe math teachers need to reward students when they guess, rather than discouraging them from guessing.

Math Anxiety
Each semester that I taught, my first assignment was a touchy-feely one. I asked the students to write about one of their experiences of being taught math. The next class day, I asked the students to share their stories (if they were comfortable sharing). The most heart-wrenching story was one woman who told of being forced to do long multiplication at the board. She was so anxious that she wet her pants.
OMG!  How do I add these two fractions?!?!!?

In my opinion, math anxiety is the single most pervasive cause of innumeracy. The anxious brain can't be expected to function well. Below is a quote from a pair of authors who agree with me. Their book is something of a self-help book on math anxiety.

"Math anxiety, which is widespread and poses a serious problem in our society, often stems from childhood experiences, including intimidation and humiliation. It is reinforced by cultural and family messages and transmitted by teachers who suffer from it themselves. More frequent in women than in men, it is manifest in self-sabotaging behavior that leads to unsatisfactory work lives."
"Where Do I Put the Decimal Point?", by Elisabeth Ruedy and Sue Nirenberg

How do we deal with it?  As a teacher, I had a lot of ways to get students beyond the anxiety. First off, humor. It is not possible for a person to be anxious while they are laughing. I know it may be hard for readers of my blog to believe this, but I am capable of being funny once in a while.

Another useful technique is just plain being honest about math anxiety. That was the whole point at the start of my class--to let everyone know that math anxiety is not rare. We all have it. I might get my membership in the Society for Pompous Math Guys revoked for saying this, but even I get a little math anxiety once in a while.

When I was doing a math problem for the class, I would often put a magnifying glass on my own anxiety, going into public panic attack mode when I started to get confused. The first instructive element: we all get moments of self doubt and confusion. The second instructive element was for me to demonstrate techniques to overcome that anxiety.

The authors of the infographic that inspired this blog post point out that math anxiety can be cured by inspiring confidence. Here is another snip, from the end of their page:
The first quiz of every class I taught, there would be at least one student, usually in the back, who I saw crying. Literally shedding tears. I didn't have to look at the actual quiz to see that they did not do well. 

When I saw this, I would ask the student to work the quiz with me, one-on-one, during my office hours. I would ask them to share their brain with me for the moment, and tell me everything that they were thinking as they approached the problem. That's when the second guessing and negative self-talk came rushing out. All they needed was a little assurance that they were thinking things through "correctly". 

I am happy to say that all these students passed the class. The lowest grade any of them got was an A-.

Popular girls don't do math
Big hair, or big brains?

I have a favorite book on overcoming math anxiety. It's called "Overcoming Math Anxiety", by Sheila Tobias. Her main thesis of the book is that math anxiety is the single most pervasive cause of innumeracy[5],  and that math anxiety can be cured.

All that aside, the main thesis of the book is that women have more of an issue with math anxiety than men do. She gives all sorts of facts and figures, and explains that it is not an issue of capability. There is no math brain, right?  The issue, she explains, is cultural. When young women hit puberty, there is a barbaric onslaught of peer pressure that says that being a math genius is uncool. "Genius" in the case, applies to anyone who gets a passing grade in either algebra or geometry.

Not everyone agrees with the assessment that there is a difference in math performance among females. Here are two articles that cite contrary evidence, one from Huffington Post, and the other from Time magazine. Perhaps Tobias' research was flawed, or perhaps times are changing. At any rate, I am encouraged. It is still disappointing that there are so many people tormented by math, but at least there are signs that it isn't so much a gender issue anymore.

[1] All math professors have beards. John the Math Guy has a beard. Therefore John the Math Guy is a math professor.

[2] While the mythical math brain has been debunked as an urban legend, the raw sexual magnetism of applied mathematicians has been demonstrated again and again in both the lab and in field studies.

[3] I just thought I would throw in that little bit of innumeracy. It's not really a 130% chance. That would be silly. Hopefully, this is a key to a person who thinks they are not good at math to reconsider.

One path to the correct answer is to look at the chance that a student has 13 teachers who are not intimidating. This chance (assuming there is no correlation from one grade to the next) is 90% raised to the 13th power. This is about 25%. The other 75% of students can be expected to have at least one math teacher who teaches them to be a mathaphobe.

Stepping back from the math problem for just a moment... I realize that I just did what I said teachers shouldn't do. I did the "It's obvious" step. It's obvious that the way to solve the problem is to turn it around the other way. It was obvious to me only because I have seen variants on this problem umpty-googol times.

Another way of looking at the problem is to take it one step at a time. What's the chances involved in two years of school? There are four possibilities: good teacher for first grade and good teacher for second grade, and so on. Multiply the probabilities together, and now you know the likelihood of all combinations.

That doesn't solve the question yet, so take it to three years. There are 8 possible outcomes... you can compute the probabilities, and away you go. Eventually, in going through this approach, the student sees the pattern.

In my opinion... that would be a good way to teach the solution to this problem.

[4] Contrary to popular belief, Paulos did not coin the word innumeracy. He wrote a book about it and made the word famous. It was first coined by Douglas Hofstadter.

[5] Sounds like something I just got done saying. Guess who got me thinking along those lines?

Monday, April 1, 2013

Genetically modified dogs

(April1 1, 2013, St. Louis, MO)  Monsanto has announced today that it has successfully spliced genes from the rabbit (Leporidae Eastrus Bunnyearis) into the domesticated dog (Canis Lupus Familiarus). According to spokesperson Jen Ettick, the genetic modification "corrects a breeding mistake made in the Middle Ages that replaced the pointed ears of wolves with floppy ears that cover the dog's inner ear." According to Ettick, these flaps decreases the hearing of the domesticated dog. "The rabbit was the obvious choice to return proper hearing to man's best friend."

Amid this announcement, there has been widespread panic regarding countless sightings of these genetically modified dogs. Thousands of pictures appeared on Facebook on Easter Sunday, leading to speculation that two of the chimera had escaped Monsanto's labs, and have been, in the words of St. Louis resident Ben T. Oudashape, "f-ing like bunnies".

Widespread sightings of Canis Bunnyearis

Monsanto denies allegations that the widespread appearance of these mutants is anything but a hoax. "At no time was the public in any danger." 

When asked to comment on rumors that Monsanto is working on crossing sheep with a popular 1980's computer programming language for next spring, Ettick replied "I cannot confirm or deny that rumor."
Rumored Pascal Lamb

Happy April Fool's Day!