It is astounding to me that mathematics — of all school subjects — elicits such potent emotional reaction when “reform” is in the air. We’ve seen the community response to the Common Core State Standards in the U.S., the potency of the Back to Basics movement in Alberta, Canada, and the myriad of internet examples of the absolute absurdity of “new math.”

At face value, the strong reactions we see can be interpreted as paradoxical. Parents might openly admit they themselves did not understand mathematics, that they actively hate mathematics even, but insist that we don’t dare do anything different for their child in math class! Parents’ befuddlement over their child’s third-grade homework might be seen as a wrong of the new curriculum, not as evidence of the failing of their own mathematics education, that they weren’t provided the flexibility and agility of thought to see simple arithmetic in multiple lights.

It seems that previous generations were seduced to equate familiarity with understanding. For instance, our standard arithmetic algorithms are somewhat bizarre — they are the end result of a human process of codifying arithmetical thinking, designed with the extra goal of using as little of precious 17th-century ink as possible. But if one does them often enough, their routine begins to feel comfortable and familiar.

These algorithms are the comfort math of the generations of students for centuries past. And I would go further, these algorithms are the *definition* of mathematics for so many folk of the past. To not perform these procedures is to not do math. Reform is a threat to these algorithms and hence a threat to math, never mind that we’re in the 21st-century when ink, paper, and slate chalk are no longer precious. And also never mind that we don’t even need these algorithms — our smartphones will perform the paper-and-pencil computations with greater speed and greater accuracy than we humans.

(I should make the point that despite these observations, the Common Core State Standards in the U.S. still requires student fluency with these algorithms: they continue to be taught. The Standards have simply delayed the introduction of these processes until the proper number sense and familiarity with arithmetic is in place so that these algorithms make sense and can be used in both routine and in flexible and clever ways. It is actually asking *more* of students with regard to these procedures than for past generations!)

This article is part of our Professor’s Perspective series—a place for experts to share their views and opinions on current events.

I fully agree with the sentiment of the parental outrage given this interpretation.

So … Can we educators work to understand this sentiment, put the right words to it, and fully engage in conversation about it? Can we be fully transparent about our approaches and intents and listen to, honor, learn from, and respond to parental and societal reaction with clarity and grace?

Of course we can and of course, we must.

Consider the following example of “new math” absurdity that made the internet rounds a few years ago. We can learn from it. Even if this is a false write-up of some supposed classroom work, someone went to the effort to write and post this piece to make some points. We should hear what is being said: it illustrates some failings we educators can easily fall into, and sometimes do.

(For the sake of ease, I’ll use the pronoun “he” for the author of this note.)

1. In going to the effort of writing out the “old fashioned” way in the top half of the page, the author is probably telling us that he interprets the goal of this mathematics exercise as simply getting the answer 20. I suspect the teachers’ goal was not this, but to have students practice anchoring arithmetic to 5s and 10s just in the way we are expected to compute change. (For example, paying for a 57-cent item with a dollar bill yields 3 + 40 cents worth of change. That’s three pennies, a quarter, a dime, and a nickel.)

**Our lesson: We need to communicate the true goal of given exercises to parents.**

2. By writing 32–12 = 20 at the top of the page, the author is likely also making that point that the arithmetic here is straightforward: one can just see the answer is 20! The convoluted approach at the bottom of the page is thus an absurd manner for answering the problem. And the author is right!

**Our lesson: If we are going to ask students to practice mathematics ideas, we need provide interesting or meaningful examples with which to practice them.**

An example like 73–57 or 2018–1995 might be better. (By the way, if I asked you how many years are between 1995 and the current year, 2018, what approach would you naturally take?)

Even with the true goal of the exercise in mind, the approach of “anchoring to 5s” seems cumbersome: In building from 12 to 32, might it be more natural to think 8 then 10 then 2?

**Our lesson: We need to be sure not to insist on one approach when analyzing a problem. We need to encourage students to generate efficient practices.**

If the point of using such a straightforward example was simply to develop fluency in this anchoring technique, then we need to make clear to student and parent alike that this work is simply practice work and not couch it in a framework that suggests it is the required arithmetical approach.

3. The student part of the piece is written with pedantic care. Perhaps the author here is making the point that he thinks students are always required to undertake the arduous task of showing his or her work no matter how tedious that enterprise might be. Parents are right to react to this when it is indeed laborious and mind-numbing.

**Our lesson: Work to have students show their work only when there is work worth showing. Let’s honor our students’ intellectual capabilities and time!**

Sure, there is pedagogical care to be undertaken here, that students need to practice explaining mathematical thinking with (meaningful!) beginning examples, but it need not be every example, each and every time.

**Our lesson: Make showing/explaining your work interesting.**

So often we couch a mathematics task as simply to get the right arithmetical answer. If, for example, one can just see that the answer is 20, then there is no point going to any more effort than just stating that the answer is 20! So let’s not ask just computational questions, but also ask direct thinking questions with the computational answers already provided: *Describe three ways to see why *35 x 12* is *420*. *(My brain thinks 35 x 10 + 70 and it thinks 5 x 7 x 4 x 3 = 20 x 21 and it thinks of dividing a 35-by-12 rectangle into four convenient pieces. What does your mind do?)

4. The fact that the author wrote and posted this piece to get a reaction demonstrates that change is fundamentally unsettling to us humans.

**Our lesson: Mathematics education is changing: it is working to couple understanding with all the same mathematics that was studied in the past. We need to acknowledge that there is, in fact, a change.**

Fear of change can manifest itself in all sorts of curious ways. One parent argued “You don’t need to understand how a car works in order to drive one” and another demanded “Just teach my kid the <expletive> math,” wanting a procedure demonstrated, not explained.

The 21st-century is not looking for humans who serve as calculating machines, but instead, it seeks problem-solvers and innovative thinkers. Google, Adobe, Apple and NASA need people who not only know the standard procedures of mathematics, but can also think with agility, flexibility, and innovation. They want people who have the confidence to create new ideas and explore them, tweak them, and find new paths of innovation never conceived of before. Mathematics can, and should, train for such innovative thinking.

And this leads to my final point.

5. If we truly acknowledge there is a change in mathematics education — as this internet piece purports to demonstrate — then we need to stand by what we value: understanding, flexibility of thought, innovation, problem-solving, reflection on solutions and approaches, and the search for efficiency and elegance.

**Our lesson: We need to assess what we value and precisely what we value.**

If all our assessments are about computation with speed, for which memorization without understanding is a good strategy for surviving, then we are hypocrites. Let’s be honest and true to our goals, even with testing.

Sadly, this is where the fear of parents is absolutely valid: in the high-stakes testing of college admissions, we educators have no right to “mess with” students’ chances of receiving high standardized test scores.

**Our lesson: We must acknowledge that testing agencies in most parts of the world have not yet caught up with what we educators value. We must find the means for students to experience tremendous success on all fronts — with speed testing and with deep understanding and mathematical innovation.**

We can assess and reward the latter in our classrooms at least. So let’s do so! And we can couple memorization and fluency of procedure with understanding. So let’s do that too! And we can do all this with communicated clarity and transparency, with flexibility, and with inclusive conversation.

With potent emotional reactions, it is clear that parents are fully invested in supporting K-12 mathematics learning. They are “all in.”

Think how terrific that is! What a tremendous opportunity we have to harness and clarify the emotion and foster powerful united support for our students’ education!

For more from Professor Tanton, check out Mathematical Visualization and Geometry on The Great Courses Plus!

]]>In this full lecture, we’ll study common mathematical properties of three-dimensional shapes such as cones, spheres, cylinders and pyramids.

Taught by Professor James Tanton, Ph.D.

Keep reading:

The Power of a Mathematical Picture

Pi: The Most Important Number in the Universe?

Mathematics and Plato’s Guardians

In this full lecture, Professor David Kung, Ph.D. delves into the voting paradoxes that arise in elections at national, state, and even club levels. You’ll also study Kenneth Arrow’s Nobel prize-winning impossibility theorem, and assess the U.S. Electoral College system, which is especially prone to counter intuitive results.

Taught by Professor David Kung, Ph.D.

There is an inspiring article that highlights thinking outside of the box when it comes to solving problems. The article compares a gifted class of middle school students with a remedial sort of vocational class, the article was measuring the creativity of the students. The experiment asked the students in each class one question: *How do you weigh a giraffe?*

The students in the gifted class were not so much gifted as successful, and they were used to succeeding and used to pleasing their teacher. They panicked because they didn’t know how to answer this question. This was way before the Internet, and the students couldn’t go online and look it up.

Meanwhile, in the vocational class, almost immediately some kid just blurted out and said, “Hey, I know what to do. Just take a chainsaw, and chainsaw that giraffe into chunks. Then weigh the chunks.”

Chainsawing the giraffe is an attitude that a good problem solver should have because you want to be fun, and you also want to be a little bit bad. Breaking rules is a good thing when we’re not talking about actual cruelty to animals here. We’re just talking about thinking outside the box or breaking mathematical rules.

Learn more about looking at solving problems in a whole new way with The Art and Craft of Mathematical Problem Solving

To break the mathematical rules, we need a healthy dose of the 3 C’s.

Concentration, creativity, and confidence are psychological attributes that are important for just about everything, but they’re vital for solving problems. How do we enhance them? All three of them are linked, but confidence is the least important of the three because it’s truly derived from the other two.

As your concentration ability increases, and if your creativity gets stronger, then you’ll naturally become more confident.

To master concentration, you must set aside a quiet time and place for your work. You need to relax. You need to develop good work habits, and you need to find problems to concentrate on that are interesting to you, approachable by you, and addictive. Pretty much, that will do the trick. In order to build up your concentration, you want to build up from level 1, which is a minute or so of concentration, at least to level 3, getting up to an hour.

Collect a stock of back-burner problems. Start cultivating problems that you cannot solve. Make sure they’re interesting and then you’ll think about them. If you can find a problem that’s exciting to you, that annoys you, that sort of gnaws at you, then you’ll think about it. Interesting problems will force you to become a better concentrator.

*To master concentration, you must set aside a quiet time and place for your work.*

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Concentration leads to confidence, which frees you to explore, which facilitates investigation and creativity.

To build both your confidence and creativity, you need to be disciplined about using those interesting problems. You need to set up a problem-solving routine, some workplace, a lucky pen, and then you should keep to your routine to get your mind in a relaxed state.

Then, occasionally, deliberately break your routine. If you like to work in the morning, work late at night. If you like quiet, go to a noisy café. If you like to work in a restaurant, go sit in a library, etc.

You should also, as a strategy, specifically think about peripheral vision. The peripheral vision strategy is to realize that many problems cannot be solved with direct focus. It’s just like your eyes. Your fovea has very good focus, but it has less sensitivity than the perimeter of your eyes, the periphery of your vision.

*Peripheral vision strategy—many problems cannot be solved with direct focus. *

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Many problems need to percolate in your unconscious in this way. You need to cultivate a good supply of back burners, and just get in the habit of not solving problems. The more you do this, the more you’ll get into a state of investigative, purposeful contemplation, and the more powerful your mind will get.

Let’s look at a tool made famous by Carl Gauss. He was a prodigy, and as a teenager, he solved a problem that had been unsolved since Hellenistic times. He found a way to construct a regular heptadecagon, 17-gon, using compass and straightedge. The rest of his career was not much different. What Gauss could do in an afternoon was equivalent to what an ordinary mathematician could do in a lifetime.

When he was 10, he was faced with the problem: How do you find the sum of the numbers 1 + 2 + 3 up to 100? How do you compute this in 1787 when there are no calculators? Well, what little Gauss did was to pair the terms, the beginning term and the end term, (1 + 100); and then the second term and the next to last term, (2 + 99); and then (3 + 98); (4 + 97); and so on down to (50 + 51). Each of those pairs adds up to 101, and there are 50 such pairs. Thus, the sum is 5050, and that’s pretty clever. This is called Gaussian pairing and is an example of a powerful and useful tool.

Wishful thinking is one of your first strategies for this because pretending to solve a problem, even an easier one, keeps you happy. It allows you to keep thinking about solving problems. Even delusion helps – deluding yourself into thinking that you’ve solved a problem actually allows you to solve it later because you can relax and be happy. Making yourself happy and confident, even if it’s through such a transparent thing as delusion, is fine.

A corollary of wishful thinking is a very sensible idea, which I just call the “make it easier” strategy. The idea is completely common sense. If your problem is too hard, just make it easier by removing the hard part. Either make the size smaller or remove an element that makes it hard. For example, if it involves square roots, remove them temporarily.

What you should keep in mind is strategy and tactic is what makes someone a good problem solver, not the tools. Now, if you’ve never seen the Gaussian pairing tool, which Gauss used to sum the numbers from 1 to a 100, you are undoubtedly impressed. Gaussian pairing are quite clever, but they are tools and tools are just tricks. These are things that can be acquired. What you should keep in mind is that strategy and tactic are what makes someone a good problem solver, not the tools.

And you should use these new ideas. Any time you see a new, interesting idea, learn it, use it, and make it yours. Ideas are collective human property. They are not private property. Don’t forget that what you’re doing is chainsawing the giraffe. It’s okay to mess around and break some rules.

Learn more: The Problem Solver’s Mind-Set

Let’s look at a quickie: a problem that requires “think outside the box” thinking.

If you consider the problem of nine dots in a grid and ask how do you join them all by drawing no more than four absolutely straight lines? If you think outside the box , as demonstrated in the diagram, it’s pretty obvious what to do.

As long as you go outside the box, you’re able to get all 9 dots. It’s a fun and challenging problem if you’ve never seen it before.

*Thinking outside the box helps you become a good problem solver. *

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People may be endowed unequally with confidence, creativity, and power of concentration, but all of these are trainable skills. It’s possible to practice them and improve them, but in order to do so, you will need to see lots of creativity in action and you need lots of open-ended opportunity to experiment.

Taught by Professor Paul Zeitz, Ph.D.

In this full lecture, discover methods that teach you to visualize numbers in a whole new light.

Taught by Professor James Tanton, Ph.D.

Let’s say you buy a lottery ticket; what are the chances that you’re going to be rich for the rest of your life?

You walk across a golf course in a stormy day; what are the chances you’ll be hit by lightning?

What are the chances that your investments will allow you to live happily for the rest of your days?

You have a fever; you have a cough. What are the chances that it’s a serious disease rather than something trivial?

All these are real-life examples of situations where we’re confronted with possibilities whose outcomes we do not know. In fact, I would argue that many or most parts of our lives—and the world and trying to understand the world—involve situations where we don’t know what’s going to happen. They involve the uncertain and the unknown.

It would be nice to say, “Well, our challenge in life is to get rid of uncertainty and be in complete control of everything.” That is not going to happen. One of life’s real challenges is to deal with the uncertain and the unknown in some sort of an effective way; and that is the realm of probability.

Probability accomplishes the really amazing feat of giving a meaningful numerical description of things that we admit we do not know, of the uncertain and the unknown. It gives us information that we actually can act on.

For example, when we hear there’s an 80% chance of rain, what do we do? We take an umbrella. Of course, if it doesn’t rain, we say, “Well, there was a 20% chance it wouldn’t rain. That’s okay.” If it rains, we say, “Oh, yes, the prediction was right. There was an 80% chance of rain.”

Probability is a rather subtle kind of a concept because it can come out one way or the other, and still a probabilistic prediction can be viewed as correct—but decisions made on probability have all sorts of ramifications.

When we make medical decisions, for example, we are making decisions that are based on probabilities, and yet they have extremely serious consequences, including life and death consequences.

In the case of the rain, all we risk is getting wet. But in many areas of making decisions on the basis of probability, there are very serious consequences. When we make medical decisions, for example, we are making decisions that are based on probabilities, and yet they have extremely serious consequences, including life and death consequences.

Back before probability was viewed as commonplace as it is today— between 1750 and 1770 in Paris, there was a smallpox epidemic for which a vaccine was developed. Unfortunately, the inoculations were rather risky. They reckoned that there was a 1 in 200 chance of death from taking the inoculation, but on the other hand, there was a 1 in 7 chance of dying eventually from the disease. So making that kind of decision is a very dramatic question where we’re weighing probabilities.

If you took that inoculation and you died immediately from smallpox, did you make the right decision or not? Well, of course, you don’t want to be among the 1 in 200 that died from the inoculation. On the other hand, on the basis of probability, it was the right decision. There are many controversies about this kind of thing and in today’s world with lawsuits and all this would be a very serious kind of an issue to undertake.

Well, in many arenas of life, our understanding of the world comes down to understanding processes and outcomes that are probabilistic in nature, that really come about from random chance, that things are happening by randomness alone. Over the last couple of centuries, the scientific descriptions of our world increasingly have included probabilistic components in them.

Many aspects of physics all involve questions of probability. Things we imagine—molecules causing things to happen by the aggregate force of probabilistic occurrences—quantum mechanics — thermodynamics. At the very foundations of our knowledge of these studies is the theory of probability.

Biology, genetics, and evolution are all based very centrally on random behavior, as well. In fact, in all of these areas, the goal is to make definite, predictable, measurable statements about what’s going to happen that are the result of describing random behavior.

The description of random behavior is how we, as scientists and mathematicians, define the world. This is a major paradigm shift in the way science has worked for the last 150 years. As time goes on, there continues to be an increase in the role of probability and randomness at the center of scientific descriptions.

Probability gives us a specific statement about what to expect when things happen at random. But how can it be effective when, by definition, random outcomes of one trial or one experiment are completely unknown? Well, if you repeat those trials many, many times and look at them in the aggregate, that’s when you begin to see glimses of regularity. It’s the job of probability to put a meaningful numerical value on the things that we admit we don’t know.

Taught by Professor Michael Starbird, Ph.D.

When used in statistics, the word population refers to the entirety of the collection of people or things that are of interest. A sample is a subset of the total population.

In general, the goal is to infer information about the whole population from information about the sample. In other words, it’s not in our interest to know only about the people who are asked in the sample. What we’re really interested in is those aspects of the entire population.

Learn More: Induction in Polls and Science

If you choose the sample randomly, the advantage is that using probability you can make inferences about how well the opinions of the sample do, in fact, represent the opinions of the whole population.

On the other hand, if you intentionally choose certain groups to reflect what you believe to be reflective of reality, you may bring your own biases to the selection process, and those biases are then going to be reflected in the people whom you ask. Representative of the whole population means that the sample should have the same characteristics that the whole population does.

The whole concept of choosing the sample randomly is that you have a better chance that the proportion of people in the sample with a certain opinion will be, in fact, the same as the entire population.

The most familiar occasion where this comes up is before an election, when pollsters try to find out what proportion of the voters will vote for the Democratic candidate and what proportion will vote for the Republican candidate.

There are several major pitfalls in the way sampling can be done. In the 1936 U.S. presidential election, the two primary contenders for the presidency were the incumbent, Franklin Delano Roosevelt, and the Republican opponent, Alfred Landon. At the time, the magazine *The* *Literary Digest* had for several elections conducted polls to predict who would win the coming election. They had successfully predicted the outcomes in several elections, so this was a major poll.

Learn More: Political Polls—How Weighted Averaging Wins

In the 1936 election, *The Literary Digest* sent out 10 million voting surveys, and they received 2.4 million replies. Based on those surveys, *The* *Literary Digest* predicted that Landon would win in a landslide, with 370 electoral votes to Roosevelt’s 161.

Well, you may not recall reading about President Landon in your American history books. Obviously he did not win the presidency.

In fact, the only correct aspect of *The* *Literary Digest*’s prediction was that the election was a landslide, but unfortunately for them, the landslide was the other way. Roosevelt won the election with 62 percent of the popular vote and by an incredible 523 electoral votes to 8 for Landon.

Obviously, *The* *Literary Digest*’s sampling method was not representative of the whole population.

What went wrong? Well, one thing was that *The* *Literary Digest* got their samples from several different kinds of lists. One list was the subscribers to their own magazine. They also looked at car registration records, and that was an available list of a lot of names, and they sent their surveys to those people. They also used telephones.

The people to whom

TheLiterary Digesthad sent their survey were likely wealthy people and obviously their opinions were not representative of the population at large.

The people to whom *The* *Literary Digest* had sent their survey were likely wealthy people and obviously their opinions were not representative of the population at large.

The year 1936 was in the middle of the Great Depression, and many people were having financial problems and were cutting back on their budgets. Probably one of the first things to go in tight times would be one’s subscription to *The* *Literary Digest*. In addition, not many people owned cars or telephones. These were luxury items for a lot of people in 1936. Because of this, the people to whom *The* *Literary Digest* had sent their survey were likely wealthy people and obviously their opinions were not representative of the population at large.

*The* *Literary Digest* poll’s second pitfall was that it was a voluntary response survey.

This is a transcript from the video series

Meaning from Data: Statistics Made Clear. It’s available for audio and video download here.

Because of this story, *The* *Literary Digest*, which otherwise would simply be lost in the dustbin of history, will now live on forever in statistics textbooks as a great example of bias in sampling.

A success that came from this *Literary Digest* fiasco is the story of George Gallup.

At the time, Gallup was a young statistician just starting out, and he did his own poll for the 1936 election. He took a survey of 50,000 people and made two predictions of his own for the election.

- He correctly predicted that Roosevelt would win the election.
- He also predicted that
*The**Literary Digest*poll would be wrong and estimated how wrong they would be before their poll came out.

He was one of the people who introduced the concept of randomness in political polling as a key feature of sampling techniques. That is absolutely one of the fundamental criteria to look for when you’re evaluating whether a sample survey is, in fact, a good one.

Learn More: Samples—The Few, The Chosen

Randomness is a basic ingredient of essentially all of the standard statistical techniques, and the reason it’s an ingredient is because the analysis of randomness and probability that allow us to apply mathematics to the understanding of the results that we get.

The most basic way to get an accurate sample is to take a sample that’s called a simple random sample, which is, as the name implies, simply to take the entire population you’re interested in, and say how many people you want to survey and randomly select them from that group, and then get the answer from each member of that selected sample.

Of course, there are lots of problems in getting the answer from that selected sample. But the simple random sample is the gold standard for finding a representative sample.

The definition of π centers around circles. In fact, it’s the ratio of the circumference of a circle to its diameter—a number just a little bit bigger than 3. We’ll explore humankind’s odyssey to compute, approximate, and understand this enigmatic number, π. These attempts throughout the ages truly transcend cultures.

The constant π helps us understand our universe with greater clarity. In fact, the definition of π inspired a new notion of measurement of angles, a new unit of measurement. This important angle measure is known as “radian measure” and gave rise to many important insights in our physical world. As for π itself, Johann Lambert showed in 1761 that π is an irrational number, and later, in 1882, Ferdinand von Lindemann proved that π is not a solution to any polynomial equation with integers. However, many questions about π remain unanswered.

Learn More: Geometry—Polygons and Circles

Any discussion of the origins of pi must begin with an interesting experiment involving circles that we can all try. Take any circle at all and take the length of the circumference—which is the length around—and measure it in terms of the diameter, which is the length across. You will end up with three diameters and just a little bit more, and if you look really closely, it’s actually a little bit more than 1/10 of the way extra. So, this little experiment shows us that that ratio of the circumference to the diameter is going to be a number that’s around, or a little bit bigger than, 3.1. No matter what the size of the circle is, the circumference is slightly greater than three times its diameter.

We’ve given this fixed, constant value a name, and we call it π. So, let’s say this more precisely. The number π is defined to equal the ratio of the circumference of any circle to its diameter across. This ratio is constant. No matter what size of circle we try this with, that number will be always the same. It begins 3.141592653589, and it keeps going.

We’ll first take a look at the early history of π and the ancient struggle to pin down its exact value—first, a word about the symbol π. We use the Greek letter π for this number, because the Greek word for “periphery” begins with the Greek letter π. Now, the periphery of a circle was the precursor to the perimeter of a circle, which today we call circumference. The symbol π first appears in William Jones’s 1709 text *A New Introduction to Mathematics*. The symbol was later made popular by the great 18^{th}-century Swiss mathematician Leonhard Euler around 1737.

Learn More: Number Theory—Prime Numbers and Divisors

Moving from its name to its value, we have evidence that the Babylonians approximated π in base 60 around 1800 B.C.E. In fact, they believed that π = 25/8, or 3.125—an amazing approximation for so early in human history. The ancient Egyptian scribe Ahmes, who is associated with the famous Rhind Papyrus, offered the approximation 256/81, which works out to be 3.16049. Again, we see very impressive approximation to this constant. There’s even an implicit value of π given in the Bible. In 1 Kings 7:23, a round basin is said to have 30-cubit circumference and 10-cubit diameter. Thus, in the Bible, implicitly it states that π equals 3 (30/10).

The Indian mathematician and astronomer Aryabhata approximated π, in c. 500 CE, with the fraction 62,832/20,000, which is 3.1416—a truly amazing estimate.

Not surprisingly, as humankind’s understanding of number evolved, so did its ability to better understand and thus estimate π itself. In the year 263, the Chinese mathematician Liu Hui believed that π = 3.141014. Approximately 200 years later, the Indian mathematician and astronomer Aryabhata approximated π with the fraction 62,832/20,000, which is 3.1416—a truly amazing estimate. Around 1400, the Persian astronomer Kashani computed π correctly to 16 digits.

Let’s break away from this historical hunt for the digits of π for a moment, and consider π as an important number in our universe. Given π’s connection with measuring circumferences of circles, scholars were inspired to use it as a measure of angle distance. Now, let’s consider a circle having radius 1. Radius is just the measure from the center out to the side. It’s half the diameter.

The traditional units for measures of angles are, of course, degrees. With degrees, one complete rotation around the circle has a measure of 360 degrees, which, by the way, happens to approximately equal the number of days in one complete year and which might explain why we think of once around as 360.

This is a transcript from the video series

Zero to Infinity: A History of Numbers. It’s available for audio and video download here.

Once around will be 2π. One full rotation around, which is an angle of 360 degrees, would be swept out with circumference length of 2π in this particular circle. In fact, what would be halfway around? Well, that would be 180 degrees, and we would sweep out half of the circumference, which, in this case, would be π. Ninety degrees would sweep out a quarter of the circle, and for this particular circle, that would have length π/2, or one-half π.

We’re beginning to see that every angle corresponds to a distance measured partway or all the way around this particular circle of radius 1. In other words, for any angle, we can measure the length of the arc of this circle swept out by that angle.

This arc length provides a new way of representing the measure of an angle, and we call this measure of angles “radian measure.” So, for example, 360 degrees = 2π radians, those are the units; 180 degrees equals π radians, and 90 degrees would equal π/2 radians. Remember, all these measures are always based on a special circle that has radius 1.

Learn More: Geometry and the Transformation Tactic

**Radian Measures and the Power of Pi**

It turns out that this radian measure is much more useful in measuring angles for mathematics and physics than the more familiar degree measure. This fact is not too surprising, since radian measure is naturally connected through the circumference length with the angle, rather than the more arbitrary degree measure that has no mathematical underpinnings, but just represents an approximation through a complete year.

The term radian first appeared in print in the 1870s, but by that time, great mathematicians, including the great mathematician Leonhard Euler, had been using angles measured in radians for over a hundred years. Well, beyond angle measures, π is central in our understanding of our universe.

In fact, the number π appears in countless important formulas and theories, including the Heisenberg uncertainty principle and Einstein’s field equation from general relativity. So it’s a very, very important formula, a very important number indeed.

Taught by Professor Edward B. Burger, Southwestern University

Which would you say is bigger: the complete works of Shakespeare or an ordinary DVD? The complete works of Shakespeare fit in a big book, of roughly 10 million bytes. But any DVD, or any digital camera, for that matter, will hold upwards of four gigabytes, which is 4 billion bytes. A DVD is 400 times bigger. All the printed words in the Library of Congress would be 10 trillion bytes, 10 terabytes. That’s one very large wall full of DVDs, but it’s also about the size of a single high-end personal hard drive. That is, you might carry all the books in the Library of Congress on a single device the size of just one book.

Big Data: How Data Analytics Is Transforming the World – Stream it now on The Great Courses Plus

And data is not merely being stored: We access a lot of data over and over. Google alone returns to the web each day, to process another 20 petabytes. What’s that? It’s 20,000 terabytes, 20 million gigabytes, 20 quadrillion bytes. How big do you want to go? Google’s daily processing gets us to one exabyte every 50 days. And 250 days of Google processing may be equivalent to all the words ever spoken by humankind to date, which have been estimated at five exabytes. And nearly one thousand times bigger is the entire content of the World Wide Web, estimated at upwards of one zettabyte, which is 1 trillion gigabytes. That’s 100 million times larger than the Library of Congress. Of course, there is a great deal more that is not on the web.

But let’s turn to the velocity of data. Let’s start a clock, to see what this feels like. Not only is there a lot of data, it’s coming at very high rates. High-speed Internet connections offer speeds 1,000 times faster than dial-up modems connected by ordinary phone lines. Here are some things that are happening every minute of the day. YouTube users upload 72 hours of new video content. In the United States alone, there are 100,000 credit card transactions. Google receives over 2 million search queries. And 200 million email messages are sent. It can be hard to wrap one’s mind around such numbers. How much data is being generated? Let’s turn to Facebook. In only 15 minutes, the amount of photos uploaded to Facebook is greater than the number of photographs stored in the New York public photo archives. That’s every 15 minutes! Now think about the data over a day, a week, or a month.

The cost of a gigabyte in the 1980s was about a million dollars. So, a smartphone with 16 gigabytes of memory would be a $16 million device

Finally, there is variety. One reason for this can stem from the need to look at historical data. But data today may be more complete than data of yesterday. The cost of a gigabyte in the 1980s was about a million dollars. So, a smartphone with 16 gigabytes of memory would be a $16 million device. Today, someone might comment that 16 gigabytes really isn’t that much memory. This is why yesterday’s data may not have been stored or have been stored in a suitable format compared to what can be stored today. Now, consider satellite imagery. The images come in large variety of aspect ratios. While I know that a satellite image will contain pixels, I don’t necessarily know what is in the picture, or not in the picture. I don’t necessarily know where to look. I may not even know what to look for.

So, we stand in a data deluge that is showering large **volumes** of data at high **velocities** with a lot of **variety**. With all this data comes information and with that information comes the potential for innovation. Steve Jobs, charismatic co-founder of Apple, was diagnosed with a pancreatic cancer in 2003. He became one of the first people in the world to have his entire DNA sequenced, as well as that of his tumor. It cost him a six-figure sum but now he had his entire DNA. Why? When doctors pick medication, they hope the patient’s DNA is sufficiently similar to the patient in the drug trial. Steve Jobs’s doctors knew his genetic makeup and could carefully pick treatments. When one treatment became ineffective, they could move to another. While Jobs eventual died from his illness, having all the data and all that information added years to his life.

Human beings tend to distribute information through what is called a transactive memory system, and we used to do this by asking each other

We all have immense amounts of data available to us every day. Search engines almost instantly return information on what can seem like a boundless array of topics. For millennia, humans have relied on each other to recall information. The Internet is changing that and how we perceive and recall details in the world. Human beings tend to distribute information through what is called a transactive memory system, and we used to do this by asking each other. Now, we also have lots of transactions with smartphones and other computers. They can even talk to us. In a study covered in *Scientific American*, Daniel Wegner and Adrian Ward discuss how the Internet can deliver information quicker than our own memories can. Have you tried to remember something and meanwhile a friend types it into a smartphone, gets the answer, and if it is a place already has directions? In a sense, the Internet is an external hard drive for our memories.

So, we have a lot of data, with more coming. We aren’t just interested in the data; we are looking at data analysis, and we want to learn something valuable we didn’t already know. For example, UPS must decide on a delivery route for packages to save time and gas. Consider 20 drop-off points; which route is the best? Seems simple enough, but checking all possible routes isn’t that easy. You have 20 choices for the first stop, 19 for the second, and so forth. In all, there are about 2 times 10 to the 18^{th} power. How big is that number? That’s five times the estimated age of the universe. Clearly, we aren’t checking that number of combinations on a computer each time a driver needs a route. Keep in mind, that’s only 20 stops.

UPS has about 55,000 drivers every day. Until recently, UPS drivers had a general route to follow. It allowed for decisions on the part of the driver. UPS now has a program called ORION, or On-Road Integrated Optimization and Navigation to help. It uses math to decide on routes. They can be counterintuitive but save time in the end. It doesn’t find* the* best route, but a lot of research has been done to find good solutions to this problem. Keep in mind, UPS has a harder problem than simply finding a route to save time. They also must consider other variables like promised delivery times. How much can this save? Consider these two numbers. Thirty million dollars: that’s the cost to UPS per year if each driver drives just one more mile each day than necessary. Eighty-five million: the number of miles the analytics tools of UPS are saving per year. Data analysis doesn’t always involve exploring a data set that is given. Sometimes, questions arise and data hasn’t even been gathered. Then, the key is knowing what question to ask, and what data to collect.

As an example, let’s join Oren Etzioni on a flight from Seattle to Los Angeles for his younger brother’s wedding. Wanting to save money, Oren bought his ticket months before the “I dos” were said. During the flight, Oren asked neighboring passengers about their ticket price. Most had paid less, even though many had bought their tickets later. For some of us, this might simply tell us not to worry so much about choosing close to the date of a flight. But Oren was Harvard’s first undergraduate to major in computer science. He graduated in 1986. To him, this was a problem for a computer to solve. He’d seen the world this way before. He helped build MetaCrawler, which was one of the first search engines. InfoSpace bought it. He made a comparison-shopping website, also snatched up. Another startup was bought by Reuters.

So, Oren gave 12,000 price observations grabbed by his computer programs from a travel website over 41 days. He ended up with something that could save customers money, and not just by comparing current prices. It didn’t know why airlines were pricing the way they did, but it could help predict whether fares were more likely to go up or down in the near future. When it became a venture capital-backed startup called Farecast, it began crunching 200 billion flight-price records. Then? Microsoft bought it in 2008, for $110 million, and integrated it into the Bing search engine. What made it possible to predict future fares? Data—lots of it. How big and what’s big enough depends, in part, on what you are asking and how much data you can handle. Then, you must consider how you can approach the question. UPS can’t look for the optimal answer. But they can save millions of dollars finding much better answers. Again, they can do this by asking questions only answerable with the data that is streaming in and available in today’s data explosion.

Taught by Professor Tim Chartier, Ph.D.

Phot of Steve Jobs Matthew Yohe [CC BY-SA 3.0 (http://creativecommons.org/licenses/by-sa/3.0) or GFDL (http://www.gnu.org/copyleft/fdl.html)], via Wikimedia Commons

Imagine an experiment in randomness. Take a coin and flip it 200 times, and each time record whether it’s a heads or a tails, putting down Hs for the heads and Ts for the tails. Now, suppose you ask a person to just write down a random list of 200 Hs and Ts, and you put up both lists on a blackboard, one made by actually flipping a coin, and the other made by a human. Even though they may both look like an ocean of Hs and Ts, there is a way to tell which one is truly random, and which is human generated.

Learn more: Our Random World—Probability Defined

The thing to do is look for strings of long sequences where there are all Hs in a row or all Ts in a row. In the 200 Hs and Ts generated by randomly flipping a coin, you might see at least four or five long sequences of Hs or Ts: six Hs in a row here, five Ts there—a lot of streaks of many things in a row.

How often will a human being write more than four strings of the same letter in a row when they’re trying to be random?

Now consider the list generated by the human being. How often will a human being write more than four strings of the same letter in a row when they’re trying to be random? Well, we sort of resist this, because we don’t think that’s very random. They think you’ve got to sort of alternate—H-T-H-T—and so here in a human generated one you would see very few strings of Hs and Ts in a row.

As a matter of fact, when you flip a coin 200 times, the probability of having at least one string of six or longer of Hs or Ts is roughly 96 percent—very likely. The probability of having at least one string of five is 99.9 percent—it’s essentially certain. You’d be very unlikely to flip a coin that many times without getting these long strings, and if you actually simulate this on the computer, you’ll see that this plays out, that you just almost always get long strings.

One of the common misconceptions that a lot of people have about randomness is illustrated by the coin flipping experiment. Let’s say that you flip a coin many times, and just randomly it happened that 10 times in a row you got heads. Well, doesn’t it seem like the next time it’s more apt to be a tails? It does to most people. And the answer is, of course, that the coin doesn’t know what it’s just done. To the coin, every flip is a new flip, and it’s just as likely to be a heads as a tails after it’s done 10 heads in a row, as it was to get a heads than a tails if it had done none of them.

Take a coin, and more than a million times, you flip the coin 11 times. Obviously you do this with a computer.

To demonstrate this, you can simulate the following experiment. Take a coin, and more than a million times, you flip the coin 11 times. Obviously you do this with a computer. Computers are great, by the way; they don’t care—a million times, they’ll just go ahead and do it. So you just do it a million times, and what do you get? To make it easy, you actually flip the coin 11 times for 1,024,000 times, because every 1,024 times is the probability of getting 10 heads in a row. In other words, if you do the experiment of flipping the coin 1,024,000 times, and each time you flip it 11 times, you expect that the first 10 will all be heads about 1,000 times.

Learn more: Probability Is in Our Genes

So you run the computer simulation a first time, and the number of times you get 10 heads in the first simulation is 1,008: extremely close to 1,000. What happened to the 11^{th} coin? Well, 521 times it turned out to be a head also, and 487 times it turned out to be a tail. There’s no memory. Approximately half the time heads, half the time tails.

If you do it again, the first 10 might be heads 983 times, and then the 11^{th} flip heads 473 times and tails 510 times. During a third experiment, 1,031 times it came out heads 10 times in a row, and of those, 502 had the next coin be a heads, and 529 a tails. The coin has no memory. After it’s gotten 10 heads in a row, it’s just as likely to be heads the next time as it was the first time you flipped that coin.

There is another counterintuitive aspect of probability, and it’s really interesting to think about what is rare, and how we view rarity in probability. Suppose you got dealt the following hand: the two of spades, the nine of spades, the jack of clubs, the eight of spades, and the five of hearts. Well, it probably doesn’t strike you as an impressive hand, one you write home about, but it is. One out of 2,598,960—that’s the probability of getting that hand.

Now if you were dealt the ace, king, queen, jack, ten of spades—a royal flush in spades—what’s the probability of getting this royal flush in spades? Exactly the same—1 out of 2,598,960—and yet you would write home to your mother about this hand for sure. Your previous hand was just an average hand, and yet in your whole life of playing cards, you know what? You will probably never get that hand again, because its probability is almost zero—1 out of 2,598,960. So this is one of the counterintuitive concepts of probability: that rare events happen all the time, but you may not recognize them as significant.

Learn more: Probability Everywhere

Rare events absolutely happen by chance alone. The most-common rare event that you see mentioned in the newspapers every day is the lottery. The probability of winning the Powerball lottery is approximately 1 out of 146,000,000. This is the big multistate lottery in some states. One out of 146,000,000. That chance is so remote you’d think it would never happen; but it happens regularly. Why? Because a lot of people try. A lot of people buy random numbers and some of them then occasionally win. If you try something that’s rare often enough, then it will actually come to pass.

This concept—that rare things will actually happen if you repeat them enough and you look for them enough—was encapsulated in an observation that was first made by the astronomer Sir Arthur Eddington in 1929, and he was describing some features of the second law of thermodynamics. He wrote the following:

If I let my fingers wander idly over the keys of a typewriter it might happen that my screed made an intelligible sentence. If an army of monkeys were strumming on typewriters they might write all the books in the British Museum. The chance of their doing so is decidedly more favourable than the chance of the molecules returning to one half of the vessel.

However, you can find patterns in random writing, and in fact an enterprising author made a lot of money a few years ago when he wrote *The Bible Code*. What the author of *The Bible Code* did was take the Bible, written in Hebrew, and find patterns of words by skipping a certain number of letters, and in that pattern of skips they would find words written out. One example was “Atomic holocaust Japan 1945.” He said that this was an example of how the Bible showed the future.

The truth is that this is just a matter of probability. If you take all possible sequences of different lengths, you can by randomness alone find surprising things, and just to demonstrate it, people debunking this analysis found patterns in *War and Peace* and so on. This is another challenging part of probability, namely that if you look for rare things but you have a lot of places to look, you’ll tend to find them.

These are some of the challenges of looking at and asking what is random in the world.

Taught by Professor Michael Starbird, The University of Texas at Austin

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