# ESP Biography

## DANIEL ZAHAROPOL, MIT graduate and mathematics educator

Major: 18

College/Employer: The Art of Problem Solving Foundation

## Brief Biographical Sketch:

Dan graduated from MIT in June 2004 in mathematics and then got masters' degrees in mathematics and teaching mathematics from the University of Illinois. While he was at MIT he was a director of Splash, and he later co-founded the nonprofit Learning Unlimited to help Splash programs spread to universities around the country. If you've been to programs at Yale, Clark, or many other campuses, they started through LU. Dan also started BEAM, which stands for Bridge to Enter Advanced Mathematics, a program to help low-income students from New York City study advanced mathematics and get access to mathematical resources throughout high school.

Dan has been teaching for ESP since Splash 2000, has directed Splash twice, and was Chair of ESP for one and a half years. He has been teaching advanced mathematics to high school students for years at Canada/USA Mathcamp, taught for a semester at the Boston Math Circle, and taught online at the Art of Problem Solving. He also taught the introductory computer science course at MIT for two semesters. In other words, he loves teaching.

In addition to his mathematical experience, Dan has done a fair bit of theater. He has taken classes in acting and playwrighting and has had three student groups productions of his plays, one of which he directed. He is an avid reader and watcher of plays, and enjoys going up to Chicago frequently to see the latest in innovative theater.

## Past Classes

(Look at the class archive for more.)

Puzzles That Make You Think in Splash 2016 (Nov. 19 - 20, 2016)
A lot of people like Sudoku, and they're fun puzzles, but there's a problem with them: they're all the same. Once you learn some basic strategies, you're mostly doing the same thing over and over again. In this class, each puzzle will be new and different. You'll have to keep coming up with new strategies, developing your thinking and learning to tackle new situations. You'll learn to stretch your mind and be more creative when faced with a new problem. Join us for a fun time, solving at your own pace and going over all kinds of different challenges!

The Honest Definition of a Continuous Function in Splash 2016 (Nov. 19 - 20, 2016)
Maybe you've heard that a continuous function is one you can draw without lifting your pencil off the paper. Do you really think that's the kind of definition a mathematician uses? It's all right for intuition, but it doesn't let you do any actual math with it. As a math major, the first real definition of a continuous function that you'd see is called the epsilon-delta definition, and it's much more sophisticated. This class is meant as an introduction to what deeper mathematics is really like. We'll see how to really define continuous functions, what exactly they can and can't do, and how to prove a lot about what's going on underneath the hood. We'll even see what continuous functions would look like if we go to strange spaces that are not like the Euclidean plane at all.

How We Got to the Civil War in Splash 2016 (Nov. 19 - 20, 2016)
You might think that Republicans and Democrats are at each other's throats now. You have no idea! In the 1840s and 1850s, the country was a crazy place. People talked openly about slavery being a good thing and the superiority of the white race. But a growing movement started to see it as a moral wrong, and proxy wars were waged for influence in the country. The US attacked Mexico because we thought we were entitled to the land. A guy from Florida invaded Cuba. Three times. This was when the Republican Party was born; when the Democrats were strong in the South and the Republicans in the north. Find out how it happened, and the incredible factors that led to a terrible war.

Introduction to Group Theory in Splash 2015 (Nov. 21 - 22, 2015)
You can add things that are not numbers. It sounds, absurd, doesn't it? But if addition is a process for combining two objects into a new object that follows certain rules (like the associative law), then why not? In fact, here's an example. If you have two rotations of the plane through the origin, you can add them by doing one rotation, then the next. Add a 112-degree rotation to a 205-degree rotation, and you get a 317-degree rotation. But rotations aren't like numbers: add a 300-degree rotation plus a 61-degree rotation... and you get a 1-degree rotation! A set of things you can add is called a group, and groups let us study the symmetries of many kinds of spaces, which is crucial for doing physics, chemistry, and most of mathematics. The simple notion of "you can add things that aren't numbers" brings forth a huge number of concepts. We'll learn about groups and then talk about isomorphisms, quotient groups, and more. If you want to get a sense of what studying advanced math is really like, this is an excellent place to get a taste of how mathematicians think.

Understanding Infinity in Splash 2015 (Nov. 21 - 22, 2015)
Can there be different sizes of infinity? "Of course not!" says a friend. "Infinity is infinity. It means something goes on without end. There can't be different sizes of that." "Sure!" says another. "Say that you take the integers. They're infinite. Now take the positive integers. The integers include all the negatives, so there must be more of them!" You might think that it makes no sense to talk about different sizes of infinity. But mathematics has found a precise way to understand infinity and to measure its size. Come find out who above is right --- if anyone --- and to discover the power of a good mathematical definition.

The Reeb Foliation of the 3-Sphere in Splash 2015 (Nov. 21 - 22, 2015)
First, you get a circle. Go up a dimension and you get a "normal" sphere. Go up another dimension and you get the 3-sphere. This is a really interesting object: it has to sit inside four dimensions, because it doesn't fit in three-dimensional space, and it has a number of really interesting properties. We're going to study those properties, first by figuring out exactly what this 3-sphere thing is, and then by analyzing it by taking a "foliation." If that doesn't make sense, don't worry about it --- we'll go over it in class. But if you want to start to visualize things in four dimensions, this is a great class to do so. The main material will take about an hour and fifteen minutes. The rest of the time will be reserved for your questions about higher-dimensional geometry and topology.

How to Understand Statistics and Reasoning in Splash! 2013 (Nov. 23 - 24, 2013)
Do you want to understand the world? Statistics can give us incredible power to understand how the world works, how people work, and how we think. But statistics can also be manipulated to promote false diet fads, help one political cause over another, or just to convince you of something that's not true. We'll study ideas of correlation and what it means for a result to be "statistically significant". You'll learn how to interpret the statistics you see and get a sense of which are worth paying attention to---and which are just gimmicks. You'll also learn where you can go in more depth.

How We Got to the Civil War in Splash! 2013 (Nov. 23 - 24, 2013)
You might think that Republicans and Democrats are at each other's throats now. You have no idea! In the 1840s and 1850s, the country was a crazy place. People talked openly about slavery being a good thing and the superiority of the white race. But a growing movement started to see it as a moral wrong, and proxy wars were waged for influence in the country. The US attacked Mexico because we thought we were entitled to the land. A guy from Florida invaded Cuba. Three times. This was when the Republican Party was born; when the Democrats were strong in the South and the Republicans in the north. Find out how it happened, and the incredible factors that led to a terrible war.

Epsilons and Deltas in Splash! 2013 (Nov. 23 - 24, 2013)
So, what's a continuous function? Maybe you've heard that it's a function you can draw without lifting your pencil off the paper. Well, that definition is nonsense. It's nice for intuition, but it doesn't let you think about what continuity means, or how to prove anything about it. The first real definition of a continuous function that you'd see as a math major is the epsilon-delta definition. This is finally a chance to work with continuous functions, to understand exactly what they can and can't do, and to prove a lot about what's going on underneath the hood.

Metric Spaces in Splash! 2013 (Nov. 23 - 24, 2013)
One of the most beautiful things that mathematics can do is take an idea and generalize it. Take addition and generalize it: you get group theory. Take the idea of maps and generalize it: you get graph theory. Take the idea of distance and generalize it: you get metric spaces. Metric spaces are just collections of points where there's a consistent notion of distance. Our own world is a metric space. But take that idea and generalize it and you get an incredible theory that introduces many other kinds of spaces. By studying these spaces, we learn more about what distance really means, and we can prove general results that apply to many different kinds of spaces. We'll also get a first introduction to topology, an even more general study of different kinds of spaces.

Puzzles on a Checkerboard in Splash! 2012 (Nov. 17 - 18, 2012)
Suppose I give you dominoes that cover two squares on a checkerboard. Then it's not hard to cover the whole checkerboard with dominoes. Now remove two opposite corners. I bet that you can't do it anymore! Come try your hand at different tiling challenges and see if you can discover the secret to which ones work and which ones don't!

Graph Theory: Modeling the Internet, Facebook, and our Social Lives in Splash! 2012 (Nov. 17 - 18, 2012)
There's a simple mathematical object called a graph. It's not a graph of a function or anything like that, but just a bunch of dots connected with lines. Graphs can be used to model web pages and the internet, friendships and dating, the subway and street maps, and how to schedule a bunch of different events. We'll get a first look at graphs and understand how coloring graphs can tell us something interesting about them!

How to Start a Splash in Splash! 2012 (Nov. 17 - 18, 2012)
Splash is run by students — undergraduates and graduates at MIT. Beyond MIT, there are Splash programs running at over a dozen universities nationwide and when you go to college, wherever you go to college, you have the opportunity to start another Splash yourself. Come learn about all of the intricacies that go into running a massive program like Splash and find out how you can do it too.

Group Theory in Spark! 2012 (Mar. 10, 2012)
You can add things that are not numbers. It sounds, absurd, doesn't it? But remarkably, you can generalize the core principles of addition to work on things that are not like numbers at all. This produces the concept of a group, a set of objects with a binary operation that is associative and has a couple of other nice properties. Groups allow us to study symmetry, crucial for physics, chemistry, and most of mathematics. The simple notion of "you can add things that aren't numbers" brings forth a huge number of concepts in mathematics. We'll talk about isomorphisms, homomorphisms, kernels, quotient groups, and more. If you want to get a sense of what studying advanced math is really like, this is an excellent place to get a taste of how mathematicians think.

The Most Challenging Puzzles in Spark! 2012 (Mar. 10, 2012)
Join us as we solve some of the deepest and most challenging puzzles around. These puzzles, seen in competitions such as the MIT Mystery Hunt, require careful analysis and deep logic, and they’re often given without any instructions! We’ll go through a sample of these puzzles and try to work through them together. Stretch your brain and grow your skills for finding deep patterns, examining open-ended questions, and pulling out solutions without nearly enough information.

Convergence, Continuity, and Space-Filling Curves in Splash! 2011 (Nov. 19 - 20, 2011)
Intuitively, a continuous curve is one that has no holes: you can draw it without lifting your pen from the paper. Perhaps you've heard this kind of definition before. But it's not very satisfying! What does it mean that you can draw it without lifting your pen from the paper? What does it mean that it has no holes? Amazingly, mathematics has a way of precisely defining what it means for a curve to be continuous. We can define other amazing things too, like a mathematically-correct (with proof!) way of adding up infinitely many numbers. We'll explore these definitions and then apply them to get a remarkable result: a continuous curve that covers the whole plane.

What is Intelligence? in Splash! 2011 (Nov. 19 - 20, 2011)
“If the Aborigine drafted an I.Q. test, all of Western civilization would presumably flunk it,” wrote anthropologist Stanley Garn. What is intelligence, really? Can we measure it? If so, what does it tell us about the human mind? Is it something that’s born into us by our genes, or does it depend on how we’re raised? What makes someone smart? Psychologists and neuroscientists have been doing research into the remarkable processes that go on in our brain, trying to understand what gives us the ability to think. Together, we'll explore some experiments and try to better understand what "intelligence" means.

Metric Spaces and Topology in Spark! 2011 (Mar. 12, 2011)
Deep mathematics comes from taking simple concepts and generalizing them. Maybe you think you understand the idea of distance. Generalize it, and you get a "metric space," a new object that satisfies all of the basic properties of distance but does far more interesting things than happen in our world. With metric spaces, we can understand the idea of continuous functions more deeply; we can start to understand the mathematical field of topology; we can prove the existence and uniqueness of solutions to differential equations; and much more. This class will offer a fast-paced introduction to the theory of metric spaces and a look at what you can do with them.

How to Run a Splash in Splash! 2010 (Nov. 20 - 21, 2010)
Ever wondered what goes on behind the scenes to make a Splash happen? Come see a completely accurate portrayal of exactly what we do to make Splash happen every year, presented by the directors of Splash 2009.

The Truth About the Complex Numbers in Splash! 2010 (Nov. 20 - 21, 2010)
The complex numbers are an amazing mathematical object. But they are also an amazingly hard space to deal with. You can't define a consistent square root, for example --- the square root has to take on two different values. And taking logarithms is even worse: a logarithm has infinitely many values to it! We're going to study the complex numbers and uncover how they really work. Our exploits will take us through what it means to take the derivative of a complex-valued function, on to a bit about integration, and finally talking about Riemann surfaces. Along the way, I'll mention interesting things that come up such as the Riemann hypothesis.

How Your Brain Lies To You in Splash! 2010 (Nov. 20 - 21, 2010)
Think you’re perfectly logical? Think that you see everything around you? That you remember things just how they happened? Turns out, you don’t. We’ll see just how your brain doesn’t work the way you think it does. It misleads you. It takes shortcuts, and tells you things that aren’t true. Be aware of where your brain goes wrong, and you’ll be smarter, better able to avoid being misled, and more aware of what’s around you.

A Bit Of Everything! in Spark! 2010 (Mar. 13, 2010)

The Real Numbers in Spark! 2010 (Mar. 13, 2010)
What is a real number? Probably no one's ever told you. Is it just anything that can be written with an infinite decimal expansion? How can you even define something like that? We're going to define the real numbers, from scratch. Along the way, I'll briefly explain how to construct the natural numbers, the integers, and the rational numbers. You'll see how to define arithmetic, from scratch, on these sets, as well as multiplication, and you'll see how mathematicians can precisely define what a number really is. This class is going to be *very* hard and abstract. You should only come to it if you're looking for a real mathematical challenge, something entirely outside the realm of what you see in school. If you've ever asked yourself, "how could such a thing exist?" then this may be the class for you.

Infinite Series, Decimal Expansions, e, and the Complex Numbers in Spark! 2010 (Mar. 13, 2010)
You're probably familiar with the idea that $$1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots = 2.$$ Amazingly enough, we can sum together infinitely many numbers --- and we can define what this means rigorously, and prove it. (Of course, you need to know what the real numbers are first, so you should have taken "The Real Numbers!") We're going to explore these infinite sums, using them to explain what a decimal expansion really is, how to define the transcendental number e, and finally apply what we've learned to sum together infinitely many complex numbers and understand some fundamental properties of this field. This class is going to be *very* hard. You should only come to it if you're looking for a real mathematical challenge, something entirely outside the realm of what you see in school.

The Most Challenging Puzzles in Spark! 2010 (Mar. 13, 2010)
Join us as we solve some of the deepest and most challenging puzzles around. These puzzles, seen in competitions such as the MIT Mystery Hunt, require careful analysis and deep logic, and they’re often given without any instructions! We’ll go through a sample of these puzzles and try to work through them together. Stretch your brain and grow your skills for finding deep patterns, examining open-ended questions, and pulling out solutions without nearly enough information.

Metric Spaces, Compactness, and the Fundamental Theorem of Algebra Part I in Splash! 2009 (Nov. 21 - 22, 2009)
This class is about several different concepts in mathematics, and how they interact to produce some really stunning results. It’s about the power of generalization. And it all starts with one simple question. What is distance? We’ll ask what the most important ideas about the notion of “distance” are, and then find a way to generalize them to places far different than just the distance between two points in space. This will lead us to define mathematical objects called “metric spaces,” sets of points where we can tell how far apart two points are, but nothing else. Yet even with just that – just a notion of distance – we’ll be able to come up with a huge host of results, including, finally, the idea of “compactness,” one of the most fundamental notions in mathematics. These ideas are extremely abstract, and you should come prepared for a very difficult math class --- and a long one, lasting a total of six hours. However, when we’re done, we’ll be able to prove a truly amazing result: every polynomial has a root in the complex numbers. You'll also be grounded in an advanced field of mathematics possibly unlike anything you've ever seen before.

Metric Spaces, Compactness, and the Fundamental Theorem of Algebra Part II in Splash! 2009 (Nov. 21 - 22, 2009)
This is Part II of a three-part class. See the description under Part I!

Metric Spaces, Compactness, and the Fundamental Theorem of Algebra Part III in Splash! 2009 (Nov. 21 - 22, 2009)
This is Part III of a three-part class. See the description under Part I for more information!

Surfaces and Low-Dimensional Topology in Splash! 2009 (Nov. 21 - 22, 2009)
There are some amazing things that happen in two dimensions. “Two dimensions” doesn’t mean the plane. It means all spaces that *look* two dimensional, like spheres and the surfaces of donuts (what mathematicians call “tori”). It also means strange, less familiar spaces, like the Klein bottle – which is two-dimensional everywhere, but you can’t fit a copy of it in our three-dimensional world; you need four-dimensions to comfortably fit it at all. We’re going to explore the many (infinitely many!) kinds of “surfaces,” two-dimensional objects like these. Even though the objects are two-dimensional, they might not fit even in our three-dimensional world, yet we'll still be able to develop a way to study them. By the end, we’ll understand what all of them look like, even the ones that don’t fit in three dimensions at all. I’ll also talk a little bit about what happens in higher dimensions, studying a kind of object called a “manifold.” This class will require a lot of thinking, so please come prepared for some very challenging abstract thought.

An Introduction to Current Events in Splash! 2009 (Nov. 21 - 22, 2009)
Interested in news and politics? We'll talk about all the latest events, focused on major activity in the last week before Splash. What happened to health care reform? What about the stimulus effort? What about Afghanistan? Iran? North Korea? In this whirlwind overview, we'll touch on a lot of the happenings, with an emphasis on US politics. We'll do things from the beginning; if you're not up on the news, that's OK. (It's what the class is for!)

A Discussion of Current Events in Splash! 2009 (Nov. 21 - 22, 2009)
We'll pick 1-2 topics from current events and have a freeform discussion. We'll talk about what's been happening, who the major players are and what their plans are, and what we think the "right" course is. Be prepared to contribute: this is for you to discuss, and there's no agenda or lesson plan for the class!

Number Tricks in Droplet Spring 2009 (May. 01, 2009)
Want to be able to tell in your head if 48302853453 is divisible by 9? What about 3? What about 11? We'll learn some tricks to tell quickly which numbers are divisible by 2, 3, 5, 9, and 11. But I'm not just going to tell you a rule: I'll also show you *why* that rule is true. We'll learn a little bit of modular arithmetic and understand something really interesting about numbers --- all while you learn some quick tricks in math.

Playwright's Workshop in HSSP Summer 2009 (Jul. 12, 2009)

Discovering Astrophysics in HSSP Spring 2009 (Mar. 14, 2009)
Pinpricks of light. That's all we can see in the sky, yet somehow astronomers can still understand phenomena like galaxies and supernovae and quasars. Even more basic than that, how can we tell how far away a star is? How do we know what stars are made of, or their temperature, or their age? How can we understand black holes, or come up with the big bang theory for the formation of the universe? In this class, you'll be directing experiments and interpreting the results. We'll give you data, but you'll come up with new ways to collect it, you'll find the patterns, and you'll make conclusions about just what's going on out there in the universe. Meanwhile, we'll present material that will help you in your investigations or explain more of the background behind what you're seeing, from the physics of black holes to general relativity to the formation of the universe.

Habits of Mind in HSSP Spring 2009 (Mar. 14, 2009)
What do mathematics, philosophy, linguistics, biology, history, and some of the hardest puzzles in the world have in common? Underlying them all are common patterns of thought, habits of mind that allow you to appreciate the deepest aspects of these disciplines and make progress in their study. Each week, we'll investigate a different area. We'll construct historical events from original sources, we'll try to reason about abstract mathematical structures, and we'll try to understand some ideas behind deep philosophical arguments. Join us as we try to stretch our reasoning skills and our knowledge of these and other disparate topics.

Constructing Numbers in Spark! Spring 2009 (Mar. 07, 2009)
What are numbers, really? I mean, what *are* they? As children, we were taught how to count, as if numbers had always been there and were obvious. When you got to fractions, well, those were supposed to be clear too. And then real numbers? $$\pi$$? It was always brushed under the rug... it's just some weird decimal that goes on forever, right? Well, you can't prove anything about numbers if you don't know what they really are. How do we know that mathematical constructions actually work? What basis tells us that even something as simple as addition makes sense --- how do you even define it? What could it possibly mean to take something like $$\pi^{\sqrt{2}}$$? Well, the numbers can be built out of something much, much simpler. You can work your way right up from almost nothing to the full complexity of the real numbers. Come and find out how a mathematician thinks about a concept you might have thought was simple.

The Most Challenging Puzzles in Spark! Spring 2009 (Mar. 07, 2009)
Each year, around a thousand people gather at MIT for the annual "Mystery Hunt," a contest in which teams compete to solve some of the deepest and most challenging puzzles around --- often given without any instructions! We'll go through a sample of these puzzles and try to work through them together. Stretch your brain and grow your skills for finding deep patterns, examining open-ended questions, and pulling out solutions without nearly enough information.

Bookbinding in Splash! 2008 (Nov. 22 - 23, 2008)
Ever wonder how books are made? Ever think it would be cool if you could have your creative writing in hardcover? Ever wanted to use the word "signature" in a way that none of your friends understand? Come bind copies of the novel written in "How to Write a Novel in 3 hours." You will practice some basic techniques and learn about more of them in the course of this class.

Actors' Workshop in Splash! 2008 (Nov. 22 - 23, 2008)
Do you love acting, or want to get into it? Want to learn the basics and explore a few interesting parts? Or do you want to see how to make your roles more specific, how to add power and truth to the lines you utter? While we may not be able to answer every acting question, what we can do is create a supportive environment where we can all work on scenes and develop our skills with guidance from some experienced directors. We'll talk about acting, the process that goes into it and how to improve. Together, we'll work on some scenes: we'll split up, each work on a role, and then present them to each other at the end. From this class, you'll see some new perspectives on what goes into portraying a role convincingly, and you'll also get to work on your acting skills in a focused but low-stress environment. If you're new to acting, or if you've had years of experience, you are welcome here!

Playwrights' Workshop in Splash! 2008 (Nov. 22 - 23, 2008)
Drama has great power. While movies and books have their own advantages, there is something special about being able to connect emotionally with real people that are right in front of you. But writing for the theater is hard. It needs to sound authentic, but it also needs to move the plot forwards. Characters must be true to themselves and each other. Each scene must have a point, it must have action, it must be engaging. Come join us as we explore some excerpts from plays, write our own short scenes, and discuss what it is that goes into a good play. If you've written any kind of play (or just a short scene) before, you are encouraged to bring several copies. But this class is open to anyone, including and especially those who have never written before!

Write a Novel In 3 Hours in Splash! 2008 (Nov. 22 - 23, 2008)
Sound insane? Well, yeah. And yet, we shall boldly venture forth, writing without a prayer of proofreading, hacking out our chapters as fast as our fingers can type them. Be a part of this crazy experiment! In the first 90 minutes, we will come up with a plot, sketch out chapter summaries, and divide them up. In the second 90 minutes, we will write, write, write! If you take bookbinding, you'll get to bind this novel afterwards. Those who expect quality may be disappointed. Those few, brave, creative souls who want to see what happens anyway are warmly invited!

How You're Being Lied to With Statistics, and How To Tell in Splash! 2008 (Nov. 22 - 23, 2008)
On June 13, 2007, the New York Times reported that New York City students had made huge gains in math: as many as 11% more were passing the state math exam than the year previously. Does that mean that students had really gotten that much better in one year? It has been found that countries that use fluoride in their drinking water have a higher cancer rate than other nations. Should we stop using fluoride in our water? It has been reported in the media and elsewhere that 150,000 young American women die of anorexia each year. (I'll give this one away: only about 60,000 women under the age of 50 die in the US at all each year, making this statistic totally impossible.) Sometimes the math and numbers scare people. Come see the many subtleties of statistics, and get a step closer to being able to discriminate the good from the lies.

A Discussion on Mathematics Education in Splash! 2008 (Nov. 22 - 23, 2008)
How should teachers teach? What, if anything, should tests test? We'll try to understand these questions through our own experiences learning math in school, and by studying debates that are ongoing in the education community, about topics such as the "Math Wars," the No Child Left Behind Act, unequal access to good teaching, and what it means to attract the best teachers. This will be a discussion-oriented class. I'll provide some context and information about research, and together we'll better understand the many difficult issues in creating a good mathematics education for everyone.

Something Unexpected in Splash! 2008 (Nov. 22 - 23, 2008)
Join us on a romp through some crazy, interesting mathematics. The catch? I don't know what it is yet. You'll get to vote. Maybe we'll build arithmetic from the ground up, or maybe you'll discover the true meaning of "algebra" ... the graduate-student class. Maybe we'll study some topology, or build the real numbers, or study interesting phenomena in higher dimensions. It's up to you to decide. Here's my guarantee: we'll do something cool, we'll do something hard, and you won't have seen whatever it is. Come and see a crazy piece of mathematics!

Algorithms in Spark! Spring 2008 (Mar. 08, 2008)
Programming languages are the boring part of writing a computer program. Yes, you heard me right, the _boring_ part. The exciting part comes with the ideas behind the program, the sequence of steps that could be written in any language. This is called an algorithm, and understanding the many kinds of algorithms along with how fast they run is a critical part of making good code. We'll see some sample algorithms and carefully understand which are efficient and which aren't.

The Nature of Infinity in Spark! Spring 2008 (Mar. 08, 2008)
In 1874, the mathematician Georg Cantor first came up with the profound ideas that led to "transfinite numbers." His insights allowed mathematicians to look at precisely what infinity means, to work with it, to understand exactly what they can do with this improbable concept. Now we can answer questions such as "when are two infinite collections of objects the same size?" We can understand how to compare the infinite set consisting of all integers with the infinite set consisting of all rational numbers (all fractions). And we can determine just how many sizes of infinity there are. Be prepared to have all your preconceptions thrown out the window.

Playwriting in Spark! Spring 2008 (Mar. 08, 2008)
We'll explore the elements of drama that make for a good play. In this discussion-focused class, we will read from other works, do some writing exercises, and share the results. This class is open to anyone with an interest in writing a play, including beginners as well as those who already have something written which they might want to share.

Understanding Games in DROPLET (2009)
We'll play some simple games and then find the best strategies to win them. What happens if you take tic-tac-toe ...

Algorithms: Just how fast do computers really work? in SPLASHONWHEELS (2008)
Let's say that you're working on a computer and you need to do a lot of addition problems. That's great ...

How You're Being Lied to With Statistics, and How To Tell in SPLASHONWHEELS (2008)
On June 13, 2007, the New York Times reported that New York City students had made huge gains in math: ...

The Reeb Foliation of the 3-Sphere in SPLASHONWHEELS (2008)
Start with a circle, which lives in the 2-dimensional plane. Go up a dimension, and you get the sphere, which ...

Actors' Workshop in SPLASH (2007)
Do you love acting, or want to get into it? Want to learn the basics and explore a few interesting ...

How You're Being Lied to With Statistics, and How to Tell in SPLASH (2007)
On June 13, 2007, the New York Times reported that New York City students had made huge gains in math: ...

Playwriting in SPLASH (2007)
Drama has great power. It's one thing to read a book, where you're separated from the characters by words, or ...

Playwriting II in SPLASH (2007)
How do characters listen to each other? What drives a scene to be successful? We're going to study characters and ...

The Amazing Things That Happen in 2 Dimensions in SPLASH (2007)
Imagine you were a two-dimensional creature, and you lived on the surface of the sphere. Look out from the north ...

The Reeb Foliation of the 3-Sphere in SPLASH (2007)
Welcome to the fourth dimension. What lives here? A circle and a sphere (the outside shell of a ball) seem ...

Visualizing Our World: Ray Tracing and Computer Graphics in SPLASH (2007)
The best computer images---those used in movies and high-quality renderings---come from a process known as "ray tracing," literally the notion ...

Complexity Theory in SPLASHONWHEELS (2006)
What problems can computers solve? The "halting problem" is simple: can I write a computer program that checks if the ...

Infinity in SPLASHONWHEELS (2006)
In this class we'll start to analyze and understand what it means for there to be "infinitely many" of something. ...

The Reeb Foliation of the 3-Sphere in SPLASHONWHEELS (2006)
First, you get a circle. Go up a dimension and you get a "normal" sphere. Go up another dimension and ...

Acting Workshop in SPLASH (2006)
We'll talk about acting, the process that goes into it and how to improve. Together, we'll work on some scenes; ...

Astronomy and Astrophysics in SPLASH (2006)
From those pinpricks that you're told are "stars," astronomers can analyze amazing details. Using wavelengths of light, changing position in ...

Playwriting in SPLASH (2006)
Drama has great power. It's one thing to read a book, where you're separated from the characters by words, or ...

Rational, Irrational, Algebraic, Transcendental, and Complex: About Numbers in SPLASH (2006)
One way to define a "real number" is as any infinite decimal. So you get stuff like pi = 3.14159265..., ...

The Nature of Infinity in SPLASH (2006)
You already know about a lot of infinite sets. There are the integers, for example. Or the rational numbers (all ...

The Reeb Foliation of the 3-Sphere in SPLASH (2006)
Welcome to the fourth dimension. What lives here? Here's an interesting specimen. Start with a circle. Go up a dimension ...

The Splash Math Program in SPLASH (2006)
Join us for the Splash Math Program, a unique new opportunity to get a focused and in-depth mathematical experience at ...

Infinity in SPLASH (2005)
Infinity is not some wishy-washy thing that isn't well-defined. There is a real, rigorous way to talk about infinity mathematically, ...

Infinity in SPLASH (2005)
Infinity is not some wishy-washy thing that isn't well-defined. There is a real, rigorous way to talk about infinity mathematically, ...

Playwrighting in SPLASH (2005)
Crafting a good play is hard, but incredibly rewarding. My hope is to impart to you some of the basics ...

Something Unexpected in SPLASH (2005)
I don't know what I'm going to teach here. No one, including me, will know until after class begins. We'll ...

The Reeb Foliation of the 3-Sphere in SPLASH (2005)
If two-dimensional space can fit a circle, and three-dimensional space can fit a sphere, surely four-dimensional space can fit something ...

The Splash Theatre Program in SPLASH (2005)
Join us in one of the wildest Splash experiments ever. We're going to start from scratch and write, act in, ...

Basic Applications of Quantum Mechanics in SPLASH (2004)
Continues the sequence begun in History and Experimental Basis of Quantum Mechanics. Begins with a brief outline of single-variable calculus. ...

History and Experimental Basis of Quantum Mechanics in SPLASH (2004)
A historical overview of experimental results leading to the formulation of quantum mechanics. Covers wave/particle duality of light and electrons, ...

Playwright's Workshop in SPLASH (2004)
One of the most effective ways to tell stories is through the theater. Actors on stage can connect emotionally with ...

Something Unexpected in SPLASH (2004)
This class was a big hit in 2002, so I'm bringing it back. Here's the deal: I like teaching hard ...

What is Distance? in SPLASH (2004)
The notion of a metric space is one of the most beautiful generalizations in mathematics. It lets you consider spaces ...

Abstractions of Mathematics in SPLASH (2003)
Prerequisites: This class willbe extremely fast; you should be comfortable with very general ideas and be willing to be thinking ...

Theory of Computation in SPLASH (2003)
Prerequisites: No programming background is necessary, but having a logical mind will help! At some fundamental level, much of computer ...

Generating Functions and Ladder Graphs in SPLASH (2002)
Math is meant to be playful. Take some recurrence (a sequence where terms depend on previous terms), do something silly ...

Something Unexpected in SPLASH (2002)
I have this failing. I always like to teach one really hard math course at Splash. I can't teach nearly ...

Algorithms in Computer Programming in HSSP (2001)
The difficult part in computer programming isn't in learning a programming language, but in learning to use it effectively. How ...

What's Cool About Math in HSSP (2001)
Ever wonder if there's something more to math? This course is going to touch on some of the really cool ...

Visualizing Our World: Ray Tracing & Computer Graphics in SPLASH (2000)
The best computer images - those used in movies and high-quality renderings - come from a process known as "ray ...

What's Cool about Math in SPLASH (2000)
High school classes almost never teach the true "cool stuff" in math where you can see brilliant proofs of neat ...