Archive for the ‘Mathematics’ Category

The Top 5 Things Done Wrong in Math Class

Sorry to jump on the top-n list bandwagon, as Vi Hart deliciously parodies, but that’s just how this one shakes out. Some of the reasons why these things are done wrong are pretty advanced, but if you’re a high school student who stumbled upon this blog, please stay and read. Know that it’s okay that you won’t get everything.

All of these gripes stem from the same source: they obfuscate what ought to be clear and profound ideas. They’re why math is hard. Like a smudge on a telescope lens, these practices impair the tool used to explore the world beyond us.

EDIT: This list focuses on notation and naming. There are other “things” done wrong in math class that any good teacher will agonize over with far more subtlety and care than this or any listicle.

5. Function Composition Notation

Specifically f \circ g, which is the same as g(f(x)). No wait, f(g(x)). Probably. This notation comes with a built-in “gotcha”, which requires mechanical memorization apart from the concept of function composition itself. The only challenge is to translate between conventions. In this case, nested parentheses are ready-made to represent composition without requiring any new mechanistic knowledge. They exploit the overloading of parentheses for both order of operations and function arguments; just work outwards as you’ve always done. We should not invent new symbols to describe something adequately described by the old ones.

Nested parentheses lend themselves to function iteration, f(f(x)). These functions are described using exponents, which play nice with the parens to make the critical distinction between f^2(x) = f(f(x)) and f(x)^2 = (f(x))^2 = f(x)f(x). This distinction becomes critical when we say arcsine aka \sin^{-1} and cosecant aka \frac {1}{\sin} are both the inverses of sine. Of course, things get confusing again when we drop the parens and get \sin^2x = (\sin x)^2 because \sin x^2 = \sin (x^2). This notation also supports  first-class functions: once we define a doubling function d(x) = 2x, what is meant by d(f)? I’d much rather explore this idea, which is “integral” to calculus (and functional programming), than quibble over a symbol.

4. The Word “Quadratic”

I’m putting “quadratic” where it belongs: number four. The prefix quadri- means four in every other context, dating back to Latin. (The synonym tetra- is Greek.) So why is x^2 called “quadratic”? Because of a quadrilateral, literally a four-sided figure. But the point isn’t the number of sides, it’s the number of dimensions. And dimensionality is tightly coupled with the notion of the right angle. And since x equals itself, then we’re dealing with not just an arbitrary quadrilateral but a right-angled one with equal sides, otherwise known as a square. So just as x^3 is cubic growth, x^2 is should be called squared growth. No need for any fancy new adjectives like “biatic”, just start using “square”. (Adverb: squarely.) It’s really easy to stop saying four when you mean two.

3.14 Pi

Unfortunately, there is a case when we have to invent a new term and get people to use it. We need to replace pi, because pi isn’t the circle constant. It’s the semicircle constant.

The thrust of the argument is that circles are defined by their radius, not their diameter, so the circle constant should be defined off the radius as well. Enter tau, \tau = \frac{C}{r}. Measuring radians in tau simplifies the unit circle tremendously. A fraction time tau is just the fraction of the total distance traveled around the circle. This wasn’t obvious with pi because the factor of 2 canceled half the time, producing \frac{5}{4}\pi instead of \frac{5}{8}\tau.

If you’ve never heard of tau before, I highly recommend you read Michael Hartl’s Tau Manifesto. But my personal favorite argument comes from integrating in spherical space. Just looking at the integral bounds for a sphere radius R:

\int_{\theta=0}^{2\pi} \int_{\phi=0}^{\pi} \int_{\rho=0}^{R}

It’s immediately clear that getting rid of the factor of two for the \theta (theta) bound will introduce a factor of one-half for the \phi (phi) bound:

\int_{\theta=0}^{\tau} \int_{\phi=0}^{\frac{\tau}{2}} \int_{\rho=0}^{R}

However, theta goes all the way around the circle (think of a complete loop on the equator). Phi only goes halfway (think north pole to south pole). The half emphasizes that phi, not theta, is the weird one. It’s not about reducing the number of operations, it’s about hiding the meaningless and showing the meaningful.

2. Complex Numbers

This is a big one. My high school teacher introduced imaginary numbers as, well, imaginary. “Let’s just pretend negative one has a square root and see what happens.” This method is backwards. If you’re working with polar vectors, you’re working with complex numbers, whether you know it or not.

Complex addition is exactly the the same as adding vectors in the xy plane. It’s also the same as just adding two numbers and then another two numbers, and then writing i afterwards. In this case, you might as well just work in R^2. (Oh hey, another use of exponents.) You can use the unit vectors \hat{x} and \hat{y}, rather than i and j which will get mixed up with the imaginary unit, and besides, you defined that hat to mean a unit vector. Use the notation you define, or don’t define it.

Complex numbers are natively polar. Every high school student (and teacher) should read and play through Steven Witten’s jaw-dropping exploration of rotating vectors. (Again students, the point isn’t to understand it all, the point is to have your mind blown.) Once we’ve defined complex multiplication – angles add, lengths multiply – then 1 \angle 90^{\circ} falls out as the square root of 1 \angle 180^{\circ} completely naturally. You can’t help but define it. And moreover, (1 \angle -90^{\circ})^2 goes around the other way, and its alternate representation (1 \angle 270^{\circ})^2 goes around twice, but they all wind up at negative one. Complex numbers aren’t arbitrary and forced; they’re a natural consequence of simple rules.

Even complex conjugates work better with angles. Instead of an algebraic argument and a formula to memorize, we can geometrically see that we we need to add an angle that brings us back to horizontal, which is just the negative of the angle we already have. This is mathematically equivalent to changing the sign on the imaginary component of the vector, but cognitively it’s very different. You can, with clarity and precision, see what you are doing in a way numerals can never express.

1. Boxplots

Boxplots make the top of the list because they’re taught at a young age and never challenged. They are brought up as a standard way to visualize data, when the boxplot was a relatively recent invention of one statistician, John Tukey. Edward Tufte has proposed variants which dramatically reduce the ink on the page. They are much easier to draw, which is important when you want to convince children that math isn’t about meticulous marks on the page. They have no horizontal component, so in addition to being more compact, they also do not encode non-information in their width.

tufte

Boxplots infuriate me because they indoctrinate the idea that there is one way to do it, and that it is not up for discussion. More time is spent on where to draw the lines than why quartiles are important, or how to read what a boxplot says about that data. Boxplots epitomize math as a recipebook, where your ideas are invalid by default and improvisation is prohibited. Nothing could be further from the truth. Moreover, boxplots slap a one-size-fits-all visualization on the data without bothering to ask what other things we could do with them. Tukey’s plots don’t just obscure the data, they obscure data science.

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How to save the world

The end of World War I was a bad time to be an optimist. It wasn’t that millions of young men had died or that western Europe had been transfigured into a hellish bombed-out landscape, although that was certainly true. It was the inescapable philosophical consideration that civilization had done this to itself. The “progress” of the industrial revolution and German unification led inexorably to total war. Civilization itself was fundamentally flawed and unsustainable; the only alternative was to admit Rousseau was right and go back to the trees.

Of course, that’s not what happened, and twenty years later they were at it again. The technology changed dramatically, but it didn’t change the fact that people were still killing each other, only how they did it. The changes that mattered were the social institutions built afterwards. Instead of the outrageous reparations in the Treaty of Versailles, there was the conciliatory Marshall Plan. Instead of the League of Nations, there was the United Nations. It wasn’t technological improvements that saved lives and improved the quality of living after the war. It was the people, with their resiliency, their forgiveness, and their intent not to make the same mistake twice.

We now find ourselves, once again, on the brink of destruction. It is not destruction by military means, but rather, economic and environmental means. Natural resources are being depleted faster than they can be renewed, if they can be renewed at all. Industrialization has spread concrete, steel, and chemicals across previously untouched land. The established political institutions are being challenged by forces as diverse as the Arab Spring and the Occupy movement. The economy is still largely in shambles. And then there’s the small matter of climate change. And so on. We’ve heard it all before. At TED 2012, this grim view was presented by Paul Gilding (talk, follow-up blog post). He’s pretty blunt about it: the earth is full.

Around a third of the world lives on less the two dollars a day. They have dramatically different cultures, education, living conditions, access to technology than the typical American or European. You honestly think that they’re the ones that are going to fix the problems? The people who are illiterate, innumerate, and don’t know where their next meal is coming from are going to fix climate change?

Depending on your answer, I have two different responses. I’ll give both of them, but you might want to think about it first. Continue reading

A new place of activists: math

This article originally appeared in the Tufts Daily on March 14, 2012.

Remember the unit circle? Of course you don’t. It’s a bunch of numbers lost in the fog of high school geometry. But it’s not your fault. It’s pi’s fault. Pi is wrong, and I want you to help make it right.

I don’t mean that pi is factually wrong; the ratio of a circle’s circumference to its diameter hasn’t changed. I mean that it’s the wrong choice of the circle constant because it leads to weird and unnatural situations. Let me explain.

Mathematicians don’t like to measure circles in degrees. They prefer radians, which are just a way of making every circle look like the unit circle, regardless of size. Because the unit circle has a radius of one, its diameter is two and its circumference is two−pi. Therefore, every circle has a circumference of two−pi radians. Pi radians is only half a circle. That’s all the math you need. I promise.

So, in classic textbook tradition, let’s apply math to a real−world situation where you would never actually need it. Say you’re cutting up your favorite circular fruit−filled pastry and your friend wants a mathematically precise amount. Where do you cut? The problem is that one pie isn’t one−pi — it’s two−pi. If you want an eighth of a pie, it’s a quarter pi, measured along the crust. It’s also really confusing, measured from anywhere.

Continue reading

Deranged

No, I’m not checking into a psych ward. A derangement is the mathematical term for a question, such as this, that appeared in my twitter feed:

How many ways can I rearrange the letters ABCDEFGH so that no letter is in its original position? What a deceptively simple question!

We eventually found the answer (14833), but the more interesting point is how we found the answer. R. Wright had an analytical solution, while Andy Rundquist used brute force. I, however, saw this as a member of a set of problems, namely, how can I rearrange the first n letters of the alphabet so each one gets moved? I then proceeded to try small values of n.

For n=1, you’re stuck with A, so f(1)=0. For n=2, AB, becomes BA, so f(2)=1. For n=3, ABC becomes CAB and BCA, so f(3)=2. But {0,1,2…} isn’t distinct enough. I needed another term.

I started writing an organized list: ABCD, ABDC, ADBC, ADCB… but I got bored so I googled and found someone had already typed up all 24 permutations. So I copied them into a text document and eliminated the bad ones, the same process I was going to do on paper. I was left with nine.

Then I surfed over to oeis.org, the Online Encyclopedia or Integer Sequences, and entered 1,2,9. The first item to come up was sequence 000166. There was, as usual for the OEIS, a lot of math jargon, and I didn’t know the term derangement yet. I could tell the sequence had something to do with factorials and combinatorics and had a recursive definition, but I wasn’t sure, so I checked the next sequence. I knew the correct sequence would have to increase O(n!), never dropping to 0 nor exceeding n! as n got big. The second sequence listed was not bounded above by n!, which meant it said there were more permutations meeting the no repeats criteria than there were permutations total, so that couldn’t be it. The next sequences had the numbers in no particular order, so they were out too. I tweeted that I “suspected” the first sequence. If I had thought to google “subfactorial” or “derangements”, I could have found less technical information telling me this was indeed the case.

We live in the 21st century. There has been a lot discovered already, and in the last 10 years a lot of it has been put online for all to see. We do not need to reinvent the wheel, as long as we know when to ignore the used wheel salesman. (“Ugg not sure about square. That work?”) Notice how my thought process includes sanity checks, broadening or narrowing as appropriate, and using existing resources? I call this the exploration of the known. (Known to humanity, but not to you.)

The truth is out there. You just have to know where and how to look.

Tau and Pi

Interlude

You’re at a fancy restaurant and have just finished a sumptuous feast. You don’t think you could eat another bite when the waitress brings out the desert cart, and lo and behold there’s a steaming hot fresh apple pie. The waitress looks at you and asks, “how much pie would you like?”

Let’s say you want an eighth of a pie. You – who we’ll call the pie-eater – of course meant an eighth of the area of the pie. But the waitress, who we’ll call the pie-cutter, can’t cut the pie like a rectangular cake, in straight parallel and perpendicular lines. She has to cut it like a circle, using what we’ll call the pie-cutter’s algorithm:

  1. Pick a starting point on the edge of the circle.
  2. Go some fraction of the total distance around the circle, around the circle.
  3. Cut from that new point to the center.
  4. Cut from the center to the starting point.

The question of course is how far around the edge of the pie (circumference of the circle) does the pie-cutter go? Continue reading

Welcome home, Atlantis.

For the last time, that is.

If it’s any consolation that the shuttle program is ending, we got a neat picture out of it (courtesy NASA):

There goes the shuttle and – is that thin layer the atmosphere? Continue reading