Sunday, June 28, 2015

How to Fix Common Core: Tau Day Edition

Here is my 200th post on this blog. Today is Tau Day, the day that we celebrate one of the most important mathematical constants, the number tau.

But what exactly is this constant "tau," anyway? It is not defined in the U of Chicago text -- if it were, it would have appeared in Section 8-8, as follows:

tau = C/r, where C is the circumference and r the radius of a circle.

Now the text tells us that C = 2pi * r, so that C/r = 2pi. Therefore, we conclude that tau = 2pi. The decimal value of this constant is approximately 6.283185307..., or 6.28 to two decimal places. Thus the day 6/28, or June 28th, is Tau Day.

It was about ten or fifteen years ago when I first read about Pi Day, and one old site (which I believe no longer exists) suggested a few alternate Pi Days, including Pi Approximation Day, July 22nd. One day that this page mentioned was Two Pi Day, or June 28th. But this page did not suggest giving the number 2pi the name tau or any another special name.

Now about four years ago -- on another Pi Day, March 14th -- I decided to search Google for some Pi Day webpages. But one of the search results was strange -- this webpage referred to a "half tau day":

"The true circle constant is the ratio of a circle’s circumference to its radius, not to its diameter. This number, called τ (tau), is equal to 2π, so π is 12τ—and March 14 is thus Half Tau Day. (Of course, since τ=6.28, June 28, or 6/28, is Tau Day itself.) Although it is of great historical importance, the mathematical significance of π is simply that it is one-half τ."

This linked to the webpage of a physicist, Michael Hartl, who lives right here in California. He has come up with "The Tau Manifesto," where he explains the rationale for considering 2pi as a new constant, to be named after the Greek letter "tau":

In “π Is Wrong!”, Bob Palais argues persuasively in favor of the second of these two definitions [the ratio of the circumference to the radius -- dw] for the circle constant, and in my view he deserves principal credit for identifying this issue and bringing it to a broad audience. He calls the true circle constant “one turn”, and he also introduces a new symbol to represent it [the symbol pi with 3 "legs" instead of two]. As we’ll see, the description is prescient, but unfortunately the symbol is rather strange, and (as discussed in Section 4) it seems unlikely to gain wide adoption.

The Tau Manifesto is dedicated to the proposition that the proper response to “π is wrong” is “No, really.” And the true circle constant deserves a proper name. As you may have guessed by now, The Tau Manifesto proposes that this name should be the Greek letter τ (tau).

Hartl then devotes the rest of his Manifesto to explaining what would happen if mathematicians were to write their functions in terms of tau rather than pi.

In my last post from about a week ago, I mentioned how Legendre sometimes took the measure of a central angle of a circle to be the length of its associated arc. In the case of a circle of radius 1, this gives us radian measure. Now Hartl discusses radian measure in Section 2.1 of his Manifesto:

There is an intimate relationship between circles and angles.Since the concentric circles have different radii, the lines in the figure cut off different lengths of arc (or arc lengths), but the angle θ (theta) is the same in each case.In other words, the size of the angle does not depend on the radius of the circle used to define the arc. The principal task of angle measurement is to create a system that captures this radius-invariance.

Hartl then shows what happens when we write radian measures in terms of tau rather than pi:

We also see the genius of Bob Palais’ identification of the circle constant as “one turn”: τ is the radian angle measure for one turn of a circle. Moreover, note that with τ there is nothing to memorize: a twelfth of a turn is τ/12, an eighth of a turn is τ/8, and so on. Using τ gives us the best of both worlds by combining conceptual clarity with all the concrete benefits of radians; the abstract meaning of, say, τ/12 is obvious, but it is also just a number.

In the following section, Hartl moves on to the trig functions, sine and cosine:

Although radian angle measure provides some of the most compelling arguments for the true circle constant, it’s worth comparing the virtues of π and τ in some other contexts as well. We begin by considering the important elementary functions sinθ and cosθ. Known as the “circle functions” because they give the coordinates of a point on theunit circle (i.e., a circle with radius 1), sine and cosine are the fundamental functions of trigonometry.

Of course, since sine and cosine both go through one full cycle during one turn of the circle, we have [the period] T=τ; i.e., the circle functions have periods equal to the circle constant.As a result, the “special” values of θ are utterly natural: a quarter-period is τ/4, a half-period is τ/2, etc. In fact, at one point I found myself wondering about the numerical value of θ for the zero of the sine function. Since the zero occurs after half a period, and since τ6.28, a quick mental calculation led to the following result:

That’s right: I was astonished to discover that I had already forgotten that τ/2 is sometimes called “π”. Perhaps this even happened to you just now. Welcome to my world.
Later on, in Section 5.1, Hartl discusses what effect his new constant has in higher dimensions. Here is not merely referring to the sphere and its surface area and volume. Indeed, here he considers dimensions higher than 3D. The idea that there can exist a fourth dimension -- an idea anathema to Euclid and the ancient Greeks -- is a separate topic of its own. Today, we just take it for granted that 4D and the higher dimensions exist, and so we want to find the surface areas and volumes of the "hyperspheres" (the generalization of the sphere to n-dimensions).

We start our investigations with the generalization of a circle to arbitrary dimensions. This object, called a hypersphere or an n-sphere, can be defined as follows. (For convenience, we assume that these spheres are centered on the origin.)0-sphere is the empty set, and we define its “interior” to be a point. A 1-sphere is the set of all points satisfying

which consists of the two points ±r. Its interior, which satisfies

is the line segment from r to r.
Then a 2-sphere is a circle, and a 3-sphere is an ordinary sphere, and a 4-sphere would be a sphere in a 4D space (often called a "glome"). Hartl then uses calculus to derive formulas for the surface areas and volume of all of these hyperspheres, and he obtains formulas in terms of pi. I won't go into the details here -- you can read it from Hartl's Manifesto or from Wolfram MathWorld, which is where Hartl himself links:

The Wolfram formula involves things like the factorials of half-integers (which the TI-83 and higher calculators can handle) and double factorials (which aren't built in to the TI-83).

Hartl then shows that these formulas are simpler when written in terms of tau -- and they become even simpler when written in terms of "lambda" -- the measure of a right angle, which can also be written as either pi/2 or tau/4. (Some authors have proposed the name "eta" instead of "lambda" for the measure of a right angle. I prefer "lambda" because I wish to reserve "eta" for the constant e^(1/e), which appears in tetration. the next operation after exponentiation.)

Hartl concludes his Tau Manifesto as follows:

The Tau Manifesto first launched on Tau Day: June 28 (6/28), 2010. Tau Day is a time to celebrate and rejoice in all things mathematical.19 If you would like to receive updates about τ, including notifications about possible future Tau Day events, please join the Tau Manifesto mailing list below. And if you think that the circular baked goods on Pi Day are tasty, just wait—Tau Day has twice as much pi(e)!

Notice that Hartl first wrote his manifesto on Tau Day 2010. It was the following Pi Day 2011 when his Half Tau Day page appeared in my search results, and so I have been following his Tau Day page for four years now.

So we see that according to Hartl, tau is superior to pi. In his Tau Manifesto, he suggests that mathematicians should use tau instead of pi in calculations. Each year, he writes a "State of the Tau" address in which he discusses the progress he has made in converting mathematicians to tau. Here is the beginning of this year's "State of the Tau" address:

Happy Tau Day 2015! Interest in the true circle constant (τ = C/r = 6.283185…) and The Tau Manifesto continued unabated this year, highlighted by a surge of attention on the “Pi [Half Tau] Day of the Century” (3/14/15). (Tau will have its revenge on 6/28/31—party at my place!) As one of the leaders of the “opposition”, I was invited to the Pi Day festivities at the Exploratorium in San Francisco—the organization that originally created Pi Day—but I was on vacation in Barcelona at the time and was unable to attend. (I know, rough life!) That the invitation was proffered in the first place is an excellent sign, though, as it serves as proof that even the Paladins of Pi recognize tau as a legitimate rival.

Just like calendar reformers and tetraters, tauists are an esoteric bunch. Since around Pi Day I posted several videos related to pi, today I post several videos related to tau:

1. Numberphile:

Numberphile is a British mathematician who posts videos about many numbers. He has several videos about pi, as well as these two videos about tau. In the first, he explains the number tau itself, and in the other, two mathematicians debate whether pi or tau is the superior circle constant. D News has also posted a pi vs. tau debate:

2. Emmex Plusbee:

In this video, a group of high school students walk out of class to protest Pi Day. They begin to chant, "What Do We Want? Tau! When Do We Want It? Now!"

This group also posted another video -- Blowing Up a Pi(e) on Tau Day -- but unfortunately, I haven't been able to locate it on YouTube. I just remember this part, paraphrased:

Girl: Why are we blowing up a pie again?
Boy: Because you don't define a circle by the diameter!
Girl: Huh?
Other Boy: Because it's fun!

After the students blow up the pie, the resulting fire display lead them to call out, "Happy Early Fourth of July!" Incidentally, the first day on which fireworks may legally be sold in many cities here in California is Tau Day.

By the way, most schools, even Labor Day Start Schools, are out by Tau Day. In other posts, I've mentioned that Pi Approximation Day, July 22nd, is a great alternative to Pi Day for summer school Geometry classes. But at some Early Start Schools, even summer school has ended by July 22nd. I point out that the best alternative day for a Pi Day in an Early Start Summer School class would in fact be Tau Day.

3. Michael Blake

As I mentioned back on Pi Day, Michael Blake is one of several musicians who created pi music simply by taking the digits of pi and applying them to the C major scale -- so 3 denotes the third note of the scale or E, 1 is the first note or C, 4 is the fourth note or F, and so on. In this video, Blake does the same with tau.

4. Vi Hart

Vi Hart -- a self-described "mathemusician" -- also has a song about the circle constant tau, based on the notes of the major scale. There is also a video -- first posted on that very same Pi Day when I first discovered the number tau -- where Hart explains why tau is superior to pi.

Hart isn't as prolific video post as she was in the past, but every year on Pi Day, Hart posts her annual "Anti-Pi Rant." Some commenters have joked that Pi Day should be renamed "Vi Day" because it's the only day that Vi is guaranteed to post a new video.

Today Vi Hart posted a special 360-degree video for Tau Day:

Numberphile -- inspired by Vi Hart -- also has a video called "The Tau of Phi," which features a song that combines the digits of tau with those of the golden ratio phi.

5. Xtra Normal and Why Tau is Right and Pi Is Wrong:

These are both brief videos about the number tau. Both of these have to do with a special equation known as Euler's identity: e ^ (i*pi) + 1 = 0. (Recall that Euler was the same 18th-century Swiss mathematician who solved the Bridges of Konigsberg problem.) This formula is often considered elegant as it includes the five important constants 0, 1, e, i, and pi, along with the (first) three operations addition, multiplication, and exponentiation (sorry, no tetration). Tauists note that if we replace pi with tau, we obtain e ^ (i*tau) = 1 + 0, so it still contains five important constants and the (first) three operations.

6. Sal Khan:

Sal Khan is perhaps the best known mathematician on YouTube. Even though most of Khan's video is about the number pi, at the end he shows how some of these formulas can be simplified by replacing pi with tau.

6.28. Michael Hartl himself posts a video where he recites pi to 40 digits, beating Futurama creator Matt Groening, who gave only two digits. I don't link to this video as it is not on YouTube, but you can see the video on the Tau Manifesto itself. And besides, it's about reciting pi, not tau.

In his "State of the Tau" address, Hartl notes that he was able to take advantage of this year's Pi Day of the Century to raise awareness of tau -- but of course, he notes that the Tau Day of the Century will occur on 6/28/31, or June 28th, 2031, since tau begins 6.2831....

I agree that Hartl makes a strong case in favor of tau. By the way, I once tried to tell a Pre-calculus student I was tutoring about tau. I'd notice that he kept referring to 180 degrees as pi/2 and 90 degrees as pi/4, and told them that if we were to use tau, the radian measures tau/2 and tau/4 would be correct after all. I'm not sure whether this helped him or not -- after all, all my discussion about tau didn't change the fact that his teacher and tests will use pi, not tau.

Based on this, I'd argue that if I were to ask a student to draw (freehand) an angle measuring pi/4 radians, the student would first convert this to 45 degrees, then draw the angle. But if students were to learn about tau instead of pi, if I were to ask someone to draw an angle of tau/4 radians, one could tell that this is 1/4 of a circle and thus a right angle without converting to degrees at all. (Then again, I'm not sure whether tau/12 is that much easier than pi/6 -- both are likely to result in a student needing to convert to degrees before drawing.) Still, I bet that he would have fared better if he had learned tau all along and not have to use pi ever, since tau fit better with his intuition.

And so, let me propose the following attempt to fulfill the Tau Manifesto. The goal is for all mathematicians to use tau instead of pi as the circle constant. Math teachers would use tau when teaching the circle, so the high school students in the above posted video would get what they want, and the student I tutored wouldn't have been so confused about radian measure. And let's give a date by which we want the goal to be accomplished -- let's say Tau Day of the Century, or June 28th, 2031, exactly sixteen years from today.

Notice that I consider this blog post to be part of my "How to Fix Common Core." Aha -- if Michael Hartl himself, or any reader of his Tau Manifesto, really wanted to get a new generation of students to learn using tau instead of pi, he might have tried to convince David Coleman to incorporate tau in the Common Core Standards. Let's look at the relevant standard in the Core:

Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

We notice that this standard doesn't actually state any formula for the area or circumference -- neither the pi formula nor the tau formula. So technically, this standard alone doesn't rule out teaching the students C = tau * r instead of C = 2pi * r. But if we really want to enforce the new constant tau, let's explicitly require the students to learn it:

CCSS.MATH.CONTENT.6.G.B.4 (proposed)
Know that tau is defined as the ratio of the circumference of a circle to its radius; know the formulas for the area and circumference of a circle in terms of tau and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

Notice that I dropped this standard from seventh grade to sixth grade, since I already stated that the circle measurement formulas used to be part of the 6th grade standards in California. In fact, since we can't make the students learn about tau on Tau Day (unless we want to shorten summer vacation), why don't we teach tau to 6th graders 28% into the school year -- that is, at grade level 6.28 or tau.

And of course, PARCC/SBAC questions would have to be written so that they use tau instead of pi -- otherwise there's no point to including it in the standards. Here's Question 3 of the PARCC Practice Exam that I posted two months ago, except with tau instead of pi:

3. The circle with center F is divided into sectors. In circle FEB is a diameter. The length of FB is 3 units.

(In the figure, the measures of three central angles are given. Angle AFB = 120, BFC = 45, and CFD = 30 degrees.)

Select the correct expression that represents the arc length of arc AED.

(A) tau/2
(B) 11tau/8
(C) 13tau/8
(D) 7tau/8

On the surface, this proposal should work. Over the next 16 years, students would learn only about tau and not about pi -- and these would become our future mathematicians. With an entire generation of students now aware of tau, we should attain our goal and fulfill the Tau Manifesto by 2031.

But what would really happen if we included tau in the Common Core? This idea of mine is not a serious proposal at all -- instead I want to make about Common Core and why so many people are opposed to the standards.

For starters, there will be a backlash to including tau in the standards. There is already a Pi Manifesto written as a response to the Tau Manifesto:

In the Pi Manifesto, the author, who only goes by the initials MSC, begins by giving the reason that pi was defined using the diameter in the first place:

So why did mathematicians define it using the diameter? Likely because it is easier to measure the diameter of a circular object than it is to measure its radius. In practice, the only way to measure the radius of a circle is to first measure the diameter and divide by 2.

In Section 2.2, MSC gives another definition of pi:

Another definition for π is to define it to be twice the smallest positive x for which cos(x)=0 , or the smallest positive x for which sin(x)=0.
Indeed, we see that this is exactly how Metamath defines pi:

"Define pi = 3.14159..., which is the smallest positive number whose sine is zero."

Some of the reasons in defense of pi are admittedly weak -- even MSC states in Section 3.2 of his manifesto that the impossibility of squaring the circle is a silly argument for pi.

But here's the point I'm trying to make here -- if tau were included in the Common Core Standards, then Common Core opponents would try as hard as they could to show that pi is better than tau -- and that includes arguments such as the one in Section 3.2. That is, if a math teacher first learns about tau by reading Hartl's Tau Manifesto, the teacher is more likely to agree with Hartl's position that if she first learns about tau by having it forced upon her in the Common Core Standards.

Most calculators, including the TI-83 and higher graphing calculators, currently have pi built in, but of course they don't have tau built in. If tau were included in the Common Core, I could easily see TI coming up with a new calculator with tau built in. But then people will say that tau was created just so that Texas Instruments could make more money selling new calculators. (Interestingly enough, Hartl notes that the Google calculator now recognizes tau. Google simply treats "tau" as a word and merely gives search results, but if at least one operation is typed in, such as "tau/1," then Google returns the correct value 6.28318530718. And of course "tau/2" returns 3.14159265359.)

And of course, the biggest problem would be in the transition itself. Students who learned pi in middle school will be confused the first year of the new standard when they were supposed to be using tau instead. Critics would say that the new standard was designed to keep the top students down, since these students who already know about pi would be confused when they see tau on the test instead of pi. Sixth graders may ask their parents for help on their math homework -- and then the parents would be confused, since they don't know what tau means. Schools might even have to hold parent training nights so that the parents can learn about tau in order to help their children. I can almost see a parent's complaint now: "My child used to love math back when he learned about pi. But now they want to treat him like a guinea pig and force tau on him, and now my child hates math."

So we see the problem here. We wanted to include tau in the Common Core Standards, because we believe that tau makes learning math -- especially trig and radian measure -- simpler. But far from convincing the public that tau is superior to pi, our plan would end up alienating students and parents, who now believe that tau makes math much worse.

Of course, the Common Core doesn't really require students to learn about tau -- the above is simply a thought exercise. But what if we replace "tau" above with, say, "transformation geometry"? Many teachers, parents, and students feel the same way about transformation geometry in reality as they would about tau in our hypothetical scenario.

The problem is that innovation is difficult -- there may be a new way of teaching math or another subject that's much, much superior to current methods, but teachers, parents, and students will be skeptical about claims of superiority if the first place they see the new methods is in Common Core or similar national standards.

So let's try again. Suppose we want everyone to use tau instead of pi, and convince them that tau will help students learn math better than pi. We know now that simply including tau in Common Core won't convince anyone of anything. So how could we accomplish this goal?

Some people oppose Common Core because they believe that the Tenth Amendment requires there to be fifty separate sets of state standards. We could, therefore, use federalism to achieve our goal by having a single state include tau in its standards. If people in other states see that the tau state is educating its students more successfully than pi states, then the pi states will rush to adopt tau as well.

Some people oppose not just Common Core, but public schools in general, due to capitalism. They believe that whoever pays for a school has the most power, so if all schools were private, the parents, not the government, have all the power over the children's education. Of course, then the problem is that not everyone can afford private school, and then the question is raised of who pays for the education of those who can't afford it, which quickly gets into politics.

Here, we avoid politics by only considering those who can afford private school for the purposes of this thought experiment. Before accepting the tuition payment, a private school can inform the parents that they are teaching the students a new method of learning geometry, using tau instead of pi. They can show the prospective parents Hartl's Tau Manifesto and inform them that if they are successful, maybe someday pi will be obsolete and every mathematician will be using tau instead -- and their children will be the leaders of the new movement.

Innovation in education is difficult because someone's children must be among the first to study under the new teaching method, and whoever's children they are will fear that they are being treating like "guinea pigs" because they are the first to learn it. Here the private school avoids the "guinea pig" problem because the parents knew that their students were learning tau before writing the tuition check in the first place. Once the private school demonstrates success, public schools, states, and the nation can adopt tau into their standards without making the children feel like guinea pigs, since tau would no longer be an unproven teaching method, but instead it would have been proven by the success of the private school.

Also, notice that adoption of tau by some private school is unnecessary to avoid the "guinea pig" problem -- a private tutoring company can teach its students tau (along with pi). We've already seen the Khan Academy video in which tau is mentioned, so that's a start.

Avoiding the transition problem is very tough. We already know that some schools grandfathered in the upperclassmen and teach Integrated Math only to the freshmen last year. Some people believe that the grandfathering should go all the way down to first grade -- in the first year of the new program, only kindergartners should be taught using the new program. But even that doesn't avoid the difficulty of having parents who learned the old way being unable to help their children learn the new way.

So we see that trying to fix these problems with Common Core is not easy indeed. I used the debate of pi vs. tau as an example to show the problems with Common Core, but we see that the Common Core debate will be much more difficult to solve.

Interestingly enough, Dr. Hung-Hsi Wu, in his Common Core lessons, gave an argument which favors pi over tau. Dr. Wu defines pi as the area of the unit disk. This is mentioned in Section 3.1 of the Pi Manifesto -- we can't get an area of tau unless the radius is sqrt(2), and we can't get an area of lambda unless the radius of sqrt(2)/2. Only an area pi gives a simple radius of 1.

Happy Tau Day, everyone! My next post will be in about a week.