Category Archives: Teaching

Learning Programming — Finger-Painting and Killing Zombies

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Zmob, my first (and only) original game.

By the end, the breadth and depth of our collective knowledge was far beyond what anyone could expect from any high school course in any subject. 

Education Versus Exploration

I’m a lab TA for an introductory Python programming course this semester, and it’s been…depressing. I remember my early days of programming, when the possibilities seemed endless and adding new features to my programs was exciting and gratifying, and I brimmed with pride at every detail, and I boasted to my friends of the amazing things I did, and I felt powerful. The world was literally at my fingertips. I could give substance to any idea I cared to entertain and any facet of life I wanted to explore. I had developed an insatiable thirst for programming that has lasted to this very day.

My younger self, if programming were more noodley.

The ironic thing is that today I look back on the programs I wrote and cringe with unending embarrassment. My old code was the artistic equivalent of a finger-painting made by a kindergartener. Sure, it might look kind of like a horse, but only because the kid has no idea what he’s doing. The programs I wrote were bug-ridden, hard to read, poorly organized, and a veritable spaghetti-slop of logic.

But I can’t deny how much fun I had, and how much I learned. I will describe all of that in more detail below, but first I’d like to contrast my experience with the students I’m teaching this semester. Their labs are no more than mindlessly shuffling around data, and implementing dry, boring functions like Horner’s method for evaluating polynomials. And (amazingly) their projects are even less interesting. I think the biggest difference is that my students don’t have to actually solve any problems by writing programs. And so their experience is boiled down to following directions and pretending to be a computer in order to follow their own program’s logic.

I’m certainly not saying that following directions and simulating a computer in your head aren’t important things to be a good programmer. What I’m saying is that my students get no gratification from their work. Their results are just as dry as the problem, and the majority of the joy I see among them is when they finish a problem and don’t have to think about it anymore (even if their solution is completely wrong).

The course has other problems with it. For instance, the professor teaches the students C paradigms instead of Python paradigms (I don’t think he ever learned the right way to do things in Python), and he confuses them with talk of stack frames and registers and all sorts of irrelevant architectural details. Remember, these students have never programmed before, and some started the course just barely competent with a computer. I didn’t know what a stack frame was until I had three years of programming under my belt (two of those years were the early, experimental years).

All of this has gotten me thinking pretty regularly about how I might teach my own course, if I might ever have one. This post will roughly be an outline of how my own computer science education began. I’ll distill the most important aspects of it: the things that made me want to keep programming and the things that taught me deep ideas in natural contexts.

My First Time was with Java

My high school (Campolindo High, in Moraga, CA) was blessed with a computer science course. With my early exposure to computers (at 3 years old, by my parents’ accounts), my love of video games, and my basic grasp of HTML, it seemed inevitable that I belonged in the class. In retrospect, it was perhaps the most beneficial course I ever took, followed closely by Honors/AP English, German, and public policy. Not only did it provide me with the aforementioned thirst for programming, but it planted a mathematical seed in my mind that would flourish years later like a giant bean stalk, which I’m still in the process of climbing today.

Right off the bat the course was different. The teacher’s name was Mr. Maters, and by the end of the first week we ceased to have lectures. Mr. Maters showed us barely enough to get a simple program running with input and output, and then we were left to our own devices.

There were roughly two options for getting credit. The first was to follow an outline of exercises and small projects from a book on GUI programming in Java. Most students followed these exercises for the first two months or so, and I did at least until I had made a stupid little pizza shop program that let you order pizzas.

The second option was wide open. The students could do whatever they wanted, and Mr. Maters jokingly told us, “At the end of each quarter I’ll judge your worth, and if I deem you deserve an A, you’ll get an A, but if I deem otherwise… you’ll get an F!”

Of course, Mr. Maters was the nicest guy you ever met. He would calmly sit at his computer in the front of the lab, maintaining a help queue of students with questions. He would quietly and calmly listen to a student’s question, and then shed some insight into their problem. Mr. Maters would get a better idea of a student’s progress by how frequent and how poignant their questions were, and more often than not students who were waiting in the queue would solve their own problems before getting to the front.

Most of the students in the class chose the “wide open” route, and that meant designing games. I’m not talking about good games, mind you, I’m talking about games made by high schoolers. I made the ugliest Whack-a-Mole you ever saw, a lifeless AI for a Battleship game, and a video poker game that featured Mr. Maters’s face on the back of every card (and replacing the faces of the kings, queens, and jacks). For the last one, I even collaborated with other students who made card games by jointly designing the Maters-themed deck. Collaboration would become a bigger theme the second year I took the course (yes, I took the same course twice), but before we get there, there were some other indispensable features I want to mention.

First, the lab room was set up so that Mr. Maters could remotely control any computer in the room from his desk. The program he used was dubbed the reverent name, “Vision,” and the slackers feared its power. Vision allowed Mr. Maters to look at our code while we were asking him questions at the front, and also helped him monitor students’ progress. Second, we were allowed a shared drive on the school’s network so that we could instantly pass files back and forth between lab computers. This had a few direct learning benefits, like sharing code examples, sprites, and sound files we used in our programs. But more importantly it gave a sense of culture to the class. We occasionally had contests where every student in the class submitted a picture of Maters’s face photoshopped into some ridiculous and funny scene (really, MS-Painted into a scene). Recall, this was the early days of internet memes, and naturally we youngsters were at the forefront of it.

Third, we were given “exploration” days every so often, on which we were freed from any obligation to work. We could play games, browse around online, or just sit and talk. More often than not we ended up playing LAN Unreal Tournament, but by the end of my second year I chose to spend those days working on my programs too; my games turned out to be more fun to work on than others were to play.

All of these instilled a sense of relaxation in the course. We weren’t being taught to the midterms or the AP exam (although I did take the AP eventually, and I scored quite well considering I didn’t study at all for it). We weren’t even being told to work on some days. The course was more of a community, and even as we teased Mr. Maters we revered him as a mentor.

As I did, most students who took a first year in the course stuck around for a second year. And that was when the amazing projects started to come.

Zmob

The second year in the computer science class was all games all the time. Moreover, we started by looking at real-time games, the kinds of side-scrolling platformers we loved to play ourselves (yeah, Super Mario Brothers and Donkey Kong Country). I tried my hand at one, but before long I was lost in figuring out how to make the collisions work. Making the levels and animating the character and making the screen scroll were all challenging but not beyond my reach.

One of my early side-scrollers based on the Starfox series.

But once I got fed up with getting him to jump on blocks, I found a better project: Zmob (short for Zombie Mob). It was inspired by collaboration. I helped a friend nail down how to draw two circles in a special way: one was large and fixed, and the other was smaller, always touching the first, and the line between their two centers went through the position of the mouse. In other words, the smaller circle represented the “orientation” of the pair of circles, and it was always facing toward the mouse. It was a basic bit of trigonometry, but once I figured out how to do it I decided a top-down zombie shooting game would be fun to work on. So I started to it. Here’s the opening screen of an early version (typos and errors are completely preserved):

The intro screen to Zmob 1.0

In the game you control the black circle, and the grey circle is supposed to represent the gun. Zombies (blue circles) are spawned regularly at random positions and they travel at varying speeds directly toward your character. You can run around and if you hold down Shift you can run faster than them for a time. And of course, you shoot them until they die, and the game ends when you die. The number of zombies spawned increases as you go on, and your ammunition is limited (although you can pick up more ammo after you get a certain number of kills) so you will eventually die. To goal is to get a high score.

The game plays more like reverse-shepherding than a shooter, and while it might be hard, I don’t think anyone but me would play it for more than ten minutes at a time.

The important part was that I had a lot of ideas, and I needed to figure out how to make those ideas a reality. I wanted the zombies to not be able to overlap each other. I wanted a gun that poisoned zombies and when a poisoned zombie touched a healthy zombie, the healthy one became poisoned. I wanted all sorts of things to happen, and the solutions naturally became language features of Java that I ended up using.

The poison gun. White zombies are poisoned, while blue zombies are healthy.

For instance, at first I just represented the zombies as circles. There was no information that made any two zombies different, so I could store them as a list of x,y coordinates. Once I wanted to give them a health bar, and give them variable speeds, and poison them, I was forced to design a zombie class, so that I could give each zombie an internal state (poisoned or not, fast or slow, etc.). I followed this up by making a player class, an item class, and a bullet class.

And the bullets turned out to be my favorite part. I wanted every bullet on the screen to be updated just by me calling an “update()” function. It turns out this was the equivalent of making a bullet into an interface which each specialized bullet class inherited from. Already I saw the need and elegance behind object oriented programming, something that was totally lost on me when I made those stupid “Shape” interfaces they have you do in basic tutorials. I was solving a problem I actually needed to solve, and an understanding of inheritance was forever branded into my mind.

And the bullet logic was a joy in itself. The first three guns I made were boring: a pistol, a machine gun, and a shotgun. Each sprayed little black circles in the expected way. I wanted to break out and make a cool gun. I called my first idea the wave beam.

The wave beam: sinusoidal bullets.

The idea behind the wave beam is  that the bullets travel along a sinusoidal curve in the direction they were shot. This left me with a huge difficulty, though: how does one rotate a sine wave by an arbitrary angle? I had x and y coordinates for the bullets, but all of the convoluted formulas I randomly tried using sines and cosines and tangents ended up failing miserably. The best I could get was a sort of awkwardly-stretched sideways sine.

After about a week of trying with no success, I finally went to my statistics teacher at the time (whom I still keep in touch with) and I asked him if he knew of any sort of witchcraft mathemagic that would do what I wanted.

After a moment’s thought, he pulled out a textbook and showed me a page on rotation matrices. To my seventeen-year-old eyes, the formula was as mysterious as an ancient rune:

My particular code ended up looking like:

x += frame*Math.cos(angle) + Math.sin(frame)*Math.sin(angle)
y += frame*Math.sin(angle) + Math.sin(frame)*Math.cos(angle)

When I ran the code, it worked so perfectly I shouted out loud. After my week of struggle and botched attempts to figure this out, this solution was elegant and beautiful and miraculous. After that, I turned to calculus to make jumping look more natural in my Fox side-scroller. I experimented with other matrix operations like shearing and stretching. By the end of that year, I had a better understanding of a “change of basis” (though I didn’t know the words for it) than most of the students I took linear algebra with in college! It was just a different coordinate system for space; there were rotated coordinates, fat and stretchy coordinates, along with skinny and backward coordinates. I tried all sorts of things in search of fun gameplay.

And it wasn’t just mathematics that I learned ahead of my time. By the end of the year I had “finished” the game. I designed a chain gun that set off chain reactions when it hit zombies, I had given it a face lift with new graphics for the player and zombies. I even designed a smart tile-layout system to measure the size of the screen and display the background appropriately. I had gotten tired of measuring the sizes by hand, so I wrote a program to measure it for me. That sounds trivial, but it’s really the heart of problem solving in computer science.

Zmob, with images

The whole class “beta tested” it, i.e., spent a few days of class just playing it to have fun and find bugs. And they found lots of bugs. Overt ones (divide by zero errors making bullets go crazy) and subtler ones (if you time everything right, the zombies can’t get close enough to hurt you, and just keep bumping into each other).

One pretty important issue that came up was speed. Once I added images, I decided to use a Java library to rotate the images on every frame so they were pointing in the right direction. Now some people say Java is slow, but this part was really slow, especially when it got up to a hundred or more zombies. My solution, it just so happened, was a paradigm in computer science called caching. You pre-compute all of the rotations you’ll ever need ahead of time, and then store them somewhere. In fact, what I really did was called lazy-loading, a slightly more sophisticated technique that involved only storing the computed rotations once they were needed.

And I never learned the name of this technique until I got to a third-year college course in dynamic web programming when we discussed the Hibernate object-relational mapping for databases! Just like with linear algebra, my personalized problems resulted in me reinventing or rediscovering important concepts far earlier than I would have learned them otherwise. I was giving myself a deep understanding of the concepts and what sorts of problems they could solve.  This is distinctly different from the sort of studying that goes on in college: students memorize the name of a concept and what it means, but only the top students get a feel for why it’s important and when to use it.

An Honest Evaluation in Retrospect

I’ll admit it, I was more dedicated to my work than the average kid. A small portion of the class was only engaged in the silly stuff. Some students didn’t have a goal in mind, or they did but didn’t pursue the issue with my kind of vigor. We didn’t have access to many good examples outside of our own web browsing and the mediocre quality of the books Mr. Maters had on hand. The choice of Java was perhaps a steep learning curve for some, but I think in the end it did more good than harm.

But on the other hand, those of us that did work well got to explore, and absorb the material at our own pace. We got to struggle with problems we actually wanted to solve, leading to new insights. One of my classmates made a chat client and a networked version of Tron, while others made role-playing games, musical applications, encryption algorithms, painting programs, and much more. By the end, the breadth and depth of our collective knowledge was far beyond what anyone could expect from any high school course in any subject. I don’t say that lightly; I spent a lot of time analyzing literature and debating contemporary issues and memorizing German vocabulary and fine-tuning essays and doing biology experiments, but programming was different. It was engaging and artistic and technical and logical and visceral. Moreover, it was a skill that makes me marketable. I could drop out of graduate school today and find a comfortable job as a software engineer in any major city and probably in any industry that makes software. That class was truly what set me on the path to where I am today.

And worst of all, it absolutely breaks my heart  to hear my students say “I didn’t think programming would be like this. I’m just not cut out for it.” The best response I can muster is “Don’t judge programming by this class. It can be fun, truly.”

What They Need

It’s become woefully clear to me that to keep students interested in programming, they need a couple of things:

1. Instant gratification

My students spend way too much time confused about their code. They need have some way to make a change and see the effects immediately. They need teaching tools designed by Bret Victor (skip ahead to 10:30 in the video to see what I mean, but the whole thing is worth watching). And they need to work on visual programs. Drawing programs and games and music. Programs whose effects they can experience in a non-intellectual way, as opposed to checking whether they’re computing polynomial derivatives correctly.

2. Projects that are relevant, or at least fun.

Just like when I was learning, student need to be able to explore. Let them work on their own projects, and have enough knowledge as a teacher to instruct them when they get stuck (or better yet, brainstorm with them). If everyone having a customized project is out of the question, at least have them work on something relevant. The last two projects in the class I teach were regrettably based on file input/output and matrix sums. Why not let them work on a video game, or a search engine (it might sound complicated, but that’s the introductory course over at udacity), or some drawing/animation, a chat client, solve Sudoku puzzles, or even show them how to get data from Facebook with the Graph API. All of these things can be sufficiently abstracted so that a student at any level can handle it, and each requires the ability to use certain constructs (basic networking for a chat client, matrix work for a sudoku, file I/O in parts of a search engine, etc.). Despite the wealth of interesting things they could have students do, it seems to me that the teachers just don’t want to come up with interesting projects, so they just have their students compute matrix sums over and over and over again.

3. The ability to read others’ code.

This is an integral part of learning. Not only should students be able to write code, but they must be able to read foreign code. They have to be able to look at examples and extract the important parts to use in their own original work. They have to be able to collaborate with their classmates, work on a shared project, and brainstorm new ideas while discussing bugs. They have to be able to criticize code as they might criticize a movie or a restaurant. Students should be very opinionated about software, and they should strive to find the right way to do things, openly lampooning pieces of code that are bloated or disorganized (okay, maybe not too harshly, but they should be mentally aware).

These three things lie at the heart of computer science and software development, and all of the other crap (the stack frames and lazy-loading and linux shells) can wait until students are already hooked and eager to learn more. Moreover, it can wait until they have a choice to pursue the area that requires knowledge of the linux shell or web frameworks or networking security or graphics processing. I learned all of the basics and then some without ever touching a linux terminal or knowing what a bit was. I don’t doubt my current students could do the same.

And once students get neck deep in code (after spending a year or two writing spaghetti code programs like I did), they can start to see beauty in the elegant ways one can organize things, and the expressive power one has to write useful programs. In some sense programming is like architecture: a good program has beauty in form and function. That’s the time when they should start thinking about systems programming and networking, because that’s the time when they can weigh the new paradigms against their own. They can criticize and discuss and innovate, or at least appreciate how nice it is and apply the ideas to whatever zombie-related project they might be working on at the time.

I hold the contention that every computer science curriculum should have multiple courses that function as blank canvases, and they should have one early on in the pipeline (if not for part of the very first course). I think that the reason classes aren’t taught this way is the same reason that mathematics education is what it is: teaching things right is hard work! As sad as it sounds, professors (especially at a research institution) don’t have time to design elaborate projects for their students.

And as long as I’m in the business of teaching, I’ll work to change that. I’ll design courses to be fun, and help my future coworkers who fail to do so. Even in highly structured courses, I’ll give students an open-ended project.

So add that onto my wish list as a high school teacher: next to “Math Soup for the Teenage Soul,” a class called “Finger-paint Programming.” (or “Programming on a Canvas”? “How to Kill Zombies”? Other suggested titles are welcome in the comments :))

How to Take a Calculus Test

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A Detailed Reference

As my readers may know, part of my TA duties include teaching a discussion section, and this semester it’s Calculus. We’re in week 9 now, closing in on the second midterm exam, and frankly put: my students suck at tests. I know they know the material, and they’re certainly not any dumber than any other random group of students, but they just can’t seem to score well on in-class exams and quizzes.

So today I staged an intervention, called “How to Take a Math Test.” In it I provided a list of ten items about test-taking strategies for math, but I like to think of them as an insight into how graders grade for these elementary college courses. A quick Google search for “how to take a math test” gives a pitiful list of unhelpful pointers, mostly about how to study for a test, and not what to write down on a test to show your knowledge.

What I’m not going to cover are the stupid obvious things, like: go to class, do your homework, study, and don’t fall asleep during an exam. If you have no idea what’s going on in lecture or discussion, then you have much bigger problems than test-taking skills, and you should go read your course textbook before reading this article. Here we care more about the question of what to do when you have no idea how to approach a problem, but know the general techniques for solving other problems.

Here is the complete list, without explanation.

How to Take a Calculus Test

  1. Show what you know.
  2. Don’t invent new math.
  3. Don’t contradict yourself.
  4. Do the easy questions first.
  5. If you don’t know how to do a problem, start by writing down relevant things that you know are true in general.
  6. Break difficult problems into manageable pieces.
  7. Know what a function is, and know what things are functions.
  8. If you aren’t taking a derivative, it’s probably wrong. (see the explanation below)
  9. If you’re doing obscene amounts of computation, it’s probably wrong.
  10. Don’t care about the final answer.

Point by Point

1. Show what you know.

This is really the most important part. As a grader, we start by assuming that a student knows nothing about Calculus, and it’s the student’s job to prove us wrong. For instance, if you’re being quizzed on the derivative of $ 9^x / 3^x$, and you don’t see any nice simplifications, then you should be writing down something like the following:

$ \frac{d}{dx} (9^x) = 9^x \ln(9)$
$ \frac{d}{dx} (3^x) = 3^x \ln(3)$
$ \displaystyle (u / v)’ = \frac{u’v – uv’}{v^2}$

Suppose at this point you have a brain aneurysm or mess up horribly on the subsequent algebra, whichever is more frightening. Suppose further that your professor still wants to grade your test. Even though you left the rest blank, or completed it with a boatload of errors, you’ve pointed out to the grader that you understand how to take the important derivatives, and you know how to piece them together. That accounts for the majority of credit on this problem.

2. Don’t invent new math.

In other words, use what you know, and only what you know. I had a student make an very creative “simplification”:

$ \textup{arccot}(xy) = \textup{arccot}(x) \cdot \textup{arccot}(y)$

I know this student has never seen such an identity before, because it’s simply false (try, for example, $ x = y = 1$). He seemed to be confused with properties of logarithms and properties of trigonometric functions, but in any event he couldn’t have been too confident that this was true. The same goes for any identities: if you aren’t 100% sure they work, either try plugging in easy numbers to verify they don’t work, or just ignore that idea. Not all problems need an algebraic trick before the calculus comes in. At least in a first course on calculus, algebraic “tricks” are usually never needed to solve a problem.

3. Don’t contradict yourself.

Sometimes this is obvious, and students do it anyway. For instance, on a midterm a very large number of students approached the following problem by plugging in numbers and claiming the conclusion. Here is a mock version of a student’s answer.

Use the intermediate value theorem to show that there exists a solution to the equation $ \sin(x) = 1-x$ on the interval $ [0, \pi]$.

$ \sin(0) = 1 – 0 \ \checkmark$
$ \sin(\pi) = 1 – \pi \ \checkmark$
So a solution exists.

This answer makes mathematicians cry. If this student knows trigonometry (and they are assumed to know it, as a prerequisite to this class), then they certainly see that they just claimed 0=1! From a grading standpoint, there is nothing in this solution that implies the student has any idea what’s going on. He might vaguely remember that he’s supposed to plug in the endpoints somewhere, but he didn’t even stop to check if what he wrote down made sense.

Here’s a subtler example. My students freaked out when they had to find the derivative of $ |x|$ on the midterm, so every day in lecture the professor reminded them how to do it. On the next week’s quiz, he put the same problem on there, and here’s what many students wrote:

$ \displaystyle \frac{d}{dx}(|x|) = 1$ if $ x \geq 0$,
$ \displaystyle \frac{d}{dx}(|x|) = -1$ if $ x < 0$,
and it does not exist when $ x = 0$.

Here the contradiction is that the student claimed that the derivative both exists and does not exist at $ x=0$! Note the “or equal” part of $ \geq$ above. If this were a history essay, the analogue would be claiming Richard Nixon was the 37th and 38th president of the United States! It doesn’t matter what the correct answer is, because what you wrote is always false.

4. Do the easy questions first.

This one should be self-explanatory, but it still causes issues. Pick off the rote derivatives at the beginning of the test, and focus on the word problems later.

5. If you don’t know how to do a problem, start by writing down relevant things that you know are true in general.

This one is in the same realm as showing what you know, but let me express it in a different way. Suppose you are asked to find the values of $ x$ for which the following ridiculous function is increasing and decreasing:

$ \displaystyle f(x) = \ln(7^{\sqrt{\arctan(x^{99})}})$

Of course, you’ve probably just had a heart attack, but if you’re still alive you should start by writing down what’s true about functions in general:

A differentiable function is increasing when its derivative is positive, and decreasing when its derivative is negative. So we need to find when $ f'(x) > 0$, and $ f'(x) < 0$.

The hope on the educators part is that this will help you figure out on the spot what to do next. Oh, so we need to take a derivative? How would I start taking the derivative of this massive thing. Well I know the derivative of $ \arctan(x)$ is $ 1/(1+x^2)$, but that’s an $ x^99$ so there will be a chain rule somewhere… The more little pieces of information you can get across that you know, you’ll both get a higher score and closer to a correct solution without scribbling down a bunch of nonsense and hoping the grader doesn’t look.

But even with the rest of the problem left blank, this is close to half credit (depending on the grader). In general, half the credit goes to conceptual understanding, and the other half goes to technical ability (your ability to arrive at an answer and do algebra). Usually students know these general facts, but for some reason never write them down on a test. They think that if they can’t use it to find the answer, then they can’t get any credit for the knowledge. As I’ve been saying all along, it’s quite the opposite.

That being said, if you just write an essay about how to do problems in general and don’t even make an attempt at using it to solve the problem at hand, you can’t expect that much credit.

6. Break difficult problems into manageable pieces.

Using the same example above in 5., even if you don’t have the stomach to compute the entire derivative (who does, really?) you can still show your knowledge by computing parts. For instance, you might start by writing down the following

$ d/dx(x^{99}) = 99x^{98}$
$ d/dx(\arctan(x)) = 1/(1+x^2)$
$ \displaystyle d/dx(\arctan(x^{99})) = \frac{99x^{98}}{1+(x^{99})^2}$
$ \vdots$

With just those three lines, you’ve shown you know the chain rule, the power rule, and the derivative of inverse tangent. Even if you don’t finish, you’ve got most of the points already.

7. Know what a function is, and know what things are functions.

This one is always alarming. I had a student who wrote the following thing:

$ \arctan(xy) = \textup{arc}(xy) \cdot \tan(xy)$
$ x = y = 1, \textup{ so } \arctan(1 \cdot 1) = \textup{arc}\ 1 \cdot \tan \ 1 = \textup{arc} \cdot \textup{tan}$

It’s alarming how often something like this crops up. If this doesn’t make you laugh or feel pity or shock (and you’re about to take a math test), then you should seriously consider taking a few minutes to figure out why this makes absolutely no sense.

On a finer point, this is the same as what is quite often written below:

$ d/dx(\sin) = \cos$

This is nonsensical for just the same reason. Sine and cosine are functions, not your average values you can do algebra with. The argument to a function is crucial, and omitting it as above leads down the slippery slope of saying $ \arctan = \textup{arc} \cdot \tan$.

8. If you aren’t taking a derivative, it’s probably wrong.

More precisely, if you aren’t doing what you’re supposed to be tested on, it’s probably wrong. On the quiz that inspired this intervention, many of my students took the entire quiz (and got answers!) without taking a single derivative. Tests, and quizzes especially, are mostly meant to be a straightforward verification of your comprehension of the course material. If you don’t use the material at all, you’re quite likely to fail.

9. If you’re doing obscene amounts of computation, it’s probably wrong.

Tests are designed to be doable and to work out nicely. If you find yourself writing an essay’s worth of square-root manipulations and analyzing eight cases, you probably made a mistake in your initial calculations. For my students, it was mixing up inverse trigonometric derivatives.

10. Don’t care about the final answer.

This might be the hardest for students to swallow, but it’s true. Nobody cares about the derivative of some contrived function at $ x = 4$. As a grader, I don’t even look at the final answer until I’ve verified your steps are correct, and only then to make sure you didn’t forget a negative sign. The truth is that mathematics is about the argument, not the answer.

For instance, if you find yourself with no time left and you boiled down your algebra to the alarming expression $ 1/(1-1)$, don’t don’t don’t just ignore it and call it equal to 0. You will get more partial credit by recognizing that your result is nonsensical and saying that than you will by ignoring the obvious problem. What you should say is something like:

I recognize that this doesn’t make sense, because you can’t divide by zero, but I don’t have time to fix it. I think mixed up the derivatives of secant and tangent.

Of course, if you have a little extra time, you could continue to write what the derivatives should have been. This shows the grader that you are smart enough to recognize your own mistakes. In fact, if I were grading such a problem, I would go out of my way to give more credit to a student who wrote such an explanation and got no final answer than the student who just wrote 0 or “does not exist.”

A Word About Math Education

A first course in calculus is not interesting mathematics. In fact, I was among those students that abhorred calculus because it was nothing but monotonous repetition. It’s an unfortunate truth of education in the United States that students are tortured in this boring, dry way.

The only time in my life I actually enjoyed calculus was when I programmed a video game in high school, in which the characters needed to jump. I modeled their velocity as the derivative of an appropriately constructed parabola. Now that was cool, and it got me inspired enough to seek my math teacher’s help when I wanted to rotate a curve representing a bullet’s trajectory to an arbitrary angle. That’s when I started to pay attention to how math can lead to elegant computer programs.

There are a lot of things one can do with these elementary ideas in calculus. Unfortunately, the subject is shoved down our throats before we have an appetite for it. So the advice I have is to be strong, and get through it. At the very least, you’ll be smarter for juggling all of these ideas and details in your head.

Möbius Transformations are Isometries of a Sphere

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We present a video on Möbius transformations and the geometry of the sphere. Anyone who has taken or will take complex analysis (that means you engineers!) should watch this. It shows not only the beautiful correspondence between the two, but it reveals the intuition behind a lot of complex analysis, when more often than not a student is left in the dust of rigorous formulas.

In short, this is a proof without words that the Möbius transformations are in correspondence with rigid motions of the unit sphere in $ \mathbb{R}^3$. This 1-1 mapping is precisely Riemann’s stereographic projection. While we usually might see this in one context (showing $ \mathbb{R}^2$ is the one-point compactification of the plane), it is not often connected to all of our interesting transformations in such a picturesque way.

Möbius transformations lay the foundation for much of hyperbolic geometry and other advanced topics in complex analysis. They’re quite fascinating.

Teaching Mathematics – Graph Theory

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Community Service

Mathematics is supposed to be a process of discovery. Definitions, propositions, and methods of proof don’t come from nowhere, although after the fact (when presented in a textbook) they often seem to. As opposed to a textbook, real maths is highly non-linear. It took mathematicians quite a lot of fuss to come up with the quadratic formula, and even simple geometric conjectures were for the longest time the subject of hot debate.

I feel like if I’m going to be a teacher worth anyone’s time, I have to let students in on the secret that questions guide mathematics. This urge to teach is especially strong at the high school level, where it is generally agreed that “mathematics education” is a farce.

And so, as the only community service I do regularly (and too seldom, at that), I go to local high schools and middle schools and give “lectures” on mathematics. Though I have ideas for a lot of lectures I could give, and wish I had more than just an hour to work with a class, I usually stick to a particularly intuitive lecture on graph theory. I will reproduce one such lecture here, picking out the best of the student’s innovation that I can remember. Regular text paraphrases what I speak and what is written on the board, quoted text is student response, and square brackets [ ] contain commentary.

As a note to the reader, this will serve as a very detailed introduction to Graph Theory, as opposed to the terse primers I’ve been providing thus far.

Two Puzzles

Today we are going to do three things:

  1. Think about some puzzles,
  2. Do some mathematics, and
  3. Use math to change the world.

So here are the two puzzles:

First, [after asking a student to provide her name, I invent a city name based on it] imagine you’re the mayor of Erintown. In Erintown there are seven very old and beautiful bridges, and as mayor you’d like to promote their prominence in tourism. To do this, you wish to provide a route through the city which crosses every bridge exactly once, never visiting the same bridge twice. The seven bridges are arranged as follows [a much more detailed picture than what one draws on a white board]:

Bridges of Erintown

[The informed reader will recognize this immediately as the Seven Bridges of Königsberg problem, which historically founded graph theory, and was solved by Leonard Euler in the 18th century. But honestly, what (potentially immature) high school student is interested in a problem with a name like that? As we will see throughout the post, personalization (and the engagement inherent in it) is essential to the success of the lecture.]

Unfortunately, after a few tries you are unable to find a route which works. Hence the first puzzle is: does such a route exist? If not, how can we prove it?

[At this point, we clarify some rules of the puzzle. High school students are adept at producing loopholes, and rightfully they enjoy doing so. So typically we talk about swimming, aircraft, traversing bridges halfway, teleportation, etc., banishing each possibility as it comes up. This is an important step, because in part the whole point of the mathematical formulation of this problem is to eliminate these possibilities from consideration. We very much need to rephrase this problem entirely in our minds to extract the aspects we care about and discard the rest. Even when in real life swimming is an option, our mathematical formulation must ignore swimming, and hence we must design it appropriately (and hopefully elegantly).]

For the second problem, say you’re at a party of one hundred people. At this party, someone decides to start tallying up who at the party is friends with whomelse (he’s one of those guys, a drama king). He shows his list to you, and you notice that there are two people at the party who have the same number of friends at the party. The thought occurs to you that this will always be the case, no matter how many people attend and who is friends with whom.

So the second puzzle is: at a party of $ n$ people, must it be true that there exist two people with the same number of friends at the party?

[Again, we have the requisite loopholes, like whether there are stalkers at the party, and whether you are friends with yourself. The former drives us to distinguish that we want “symmetric” friendships, i.e. if you are friends with someone then they are friends with you. The former translates to undirected edges later, while the latter hints at simple graphs. Both are usually made clear by appealing to the rules of Facebook friendship. Finally, we might make the clarification that there are at least two people at the party, in order to prevent a discussion of vacuously true statements.]

Now take five minutes and try to solve these problems, by yourself, with a friend, or with a group, however you feel most comfortable tackling a problem. [They never get very far, but at least once I’ve encountered a student who knew of the Seven Bridges problem ahead of time, spoiling much of the fun and thoroughly confusing the rest of the class.]

New Mathematics

[After five to ten minutes pass and the group is quiet again] So, who thinks they’ve solved the first problem? [hands raise, most proclaiming impossibility; those who try to explain their reasoning mostly resign to awkward case-checking or saying they just couldn’t find one that worked] And what about the second? [nobody raises a hand, most enjoy thinking about the bridges problem because it is very visual. In classrooms blessed with a SmartBoard, I can have a number of them come up to the front and attempt to draw a route with their finger (and hitting undo when they invariably fail, so that I don’t have to redraw the diagram every time).]

So, it appears that we haven’t come up with a good solution for either problem. Now a mathematician might say at this point, “screw this, I’m going to make up my own math to solve it!” And that’s what we’re going to try to do.

The first step is to compare the problems: what is similar and what is different? [Discussion ensues, but often times the students don’t understand what I’m looking for, and usually the problem is that they’re trying to come up with the “correct” answer instead of making observations; it is a curse of the schooling paradigm. Additional leading questions include:] what are the subjects of our study? How are they related? does it matter where you walk on a landmass between visiting bridges? Does it matter where the people in the party are standing? [And the most important question] Is there a better way to draw these problems?

[Soon enough students make the right observations, that our drawings of the two problems are almost identical!] It doesn’t actually matter how big or where the landmasses are, since all we care about is the order in which we cross bridges. Hence we can compress the landmasses down to dots! Additionally, we can just draw people as dots, and arrange them in any way we wish. Then, the bridges and friendships become lines connecting the dots. This yields a much nicer picture for the bridges problem, and a similar one for the party problem.

Our new form of the seven bridges problem.

By writing the problem this way, we have distilled out the relevant facts: all we really care about is the structure of how these things are connected. Unfortunately we have one problem: we don’t have names for these things! We certainly don’t want to call them bridges and landmasses, or people and friendships, because we want the picture to apply to both problems at the same time.

So to start, what would we call the picture as a whole? Appeal to your imagination about what it looks like. [Though this part is sometimes difficult, especially at the middle-school level, eventually someone calls out something truly clever] “How about a constellation?”

I like that! So here we are, this is our invention:

Constellation Theory

What are we going to call the individual dots? “Stars!” And what about the lines connecting them? “How about…connections?” Okay. So here is our first definition:

Definition: A constellation has three parts:

  1. A set of stars $ S$ [we just accept the intuitive definition of a set without issue],
  2. A set of connections $ C$,
  3. A function $ f: C \to S \times S$ which accepts a connection and tells us which two stars it is connected to.

[Before the third, I ask the class whether the first two alone are enough. If I get nods, I draw a random collection of dots and lines, with the lines not at all connected to the dots, and they see we need some statement of incidence.]

Don’t be afraid of the third part (even if you don’t know what a function is), it’s just a formality that uses other (well-established) maths to make our definition consistent. Math can sometimes be a notational nightmare, but all this means is that we can take any connection and easily say which two stars it connects. Since we will always draw constellations as a picture, we can just use the picture as our “function.”

Now can someone remind me again what we were looking for in the bridges problem? Right, a route through the city that hits every bridge exactly once. First we need to translate a “route” into our language of constellations. Does anyone have a good name? [After a few generic suggestions like “trail,” “path,” and “route,” we settle on the imaginative “waltz”.] This gives our second definition:

Definition: A waltz through a constellation $ (S,C,f)$ is a list of alternating stars and connections, which we label $ (s_1, c_1, s_2, c_2, \dots, s_{n-1}, c_{n-1}, s_n)$, where the $ i$th connection $ c_i$ is connected to its neighboring stars in the list $ s_i, s_{i+1}$. In terms of our function, $ f(c_i) = (s_i,s_{i+1})$ for each $ i=1 \dots n-1$.

This is just a way to write out on paper what the waltz is. [I label the seven bridges picture and provide an example.] You who suggested the name “waltz,” what is your name? “Phil.” What is your last name, Phil? “Osman.” Great! Now we have another definition:

Definition: An Osmannian waltz through a constellation is a waltz which uses each connection in $ C$ exactly once.

[A few giggles resound when they realize I’m incorporating the student’s name into the definition.] Now can somebody rephrase the original problem in terms of constellation theory? “We want to find out if there is an Osmannian waltz in that particular constellation.” Excellent!

Now let’s turn our attention to the party problem. Can someone remind me what we were trying to find out about parties? “Whether there are two people who have the same number of friends.” Right, whether that has to be the case for any party. Now in terms of constellations, what is that? “The number of connections at each star.” Great. What’s your name? “Olivia.” Olivia, what’s your last name? “Bisel.” Okay. Here’s another definition:

Definition: The Bisel-degree of a star is the number of connections in $ C$ connected to it.

Now there are a couple of other details we have to consider. Specifically, in a general constellation we never ruled out multiple connections connecting the same two stars. And we never said a connection can’t go from a star to itself. But we must rule these out to make a sensible party problem. So we will call a constellation which rules out doubled connections and self-connections simple. [For the sake of time, we just provide a name, and it’s not that imaginative of a property anyway.]

So can someone translate the party problem into the language of simple constellations? “It’s whether every simple constellation has to have two stars with equal Bisel-degree.” Wonderful!

Now that we have a working language, let’s take another ten minutes to try to solve the problems. But this time, you aren’t allowed to use “bridges,” “landmasses,” “people,” or “friendships” anymore, you have to use the terms we invented. [The students still don’t get far, especially on the bridges problem, but every now and then a student solves the party problem. As they work, I give subtle hints, like, “what happens if you add extra connections or remove some? Does it work then? What aspect of the intrinsic structure have you altered by doing this? Try lots of examples!”]

[After bringing the class back together] So who thinks they’ve solved the first problem? [a few hands raise] “I think it has something to do with whether the Bisel-degree is even or odd.” Interesting. Did you get much further than that? “No…” Okay. What about the party problem? “I think I have it. So if everybody had a different number of friends, then one person would have to have no friends and someone would have to be everyone’s friend, but that can’t happen.”

Did everyone hear that? [I reiterate on the board in detail, explaining the idea behind proof by contradiction, and drawing a picture of the resulting constellation.] This is a very elegant proof. And if anyone can come up with a solid proof of the bridge problem, I have no doubt your teacher would give ample extra credit.

Changing the World

Now, for the mathematician this is enough. This new mathematical object, a constellation, is full of wonderful patterns that we could spend our entire lives thinking about (and many have done just this). However, it’s probably true that most of you aren’t going to become mathematicians. So let’s try to think of things in the real world that we can model as constellations. Any ideas?

[The students suggest a variety of different (and usually complex) ideas, including trade between nation-states, distributions of power among people, and the structure of galaxies. For each example, I usually have to verbally augment our representation of a graph (for the sake of time), bringing constellations with numerically labeled edges or directed connections. Since I always tell the students to “think bigger,” they inevitably say “galaxies,” and I have to explain why that doesn’t work, because the whole point of constellations (the mathematical ones) is that the relative sizes and positions of the stars don’t matter, whereas at the galactic level that completely determines the connection (which is invariably gravitational pull). We apologize for the terminological confusion.

[Eventually, we might get to the cases of modelling all roads and intersections, after which I claim that is exactly how Google Maps (and all other mapping/directions software) works. Sometimes they take the “friendship” hint and recognize Facebook as a constellation, and we often begin to talk about friend suggestions and degrees of separation. Finally, (and this is the main example I wish to work toward), we model the entire internet as a constellation, with directed connections corresponding to links between web sites. Then I talk about how Google based their company on the soundness of this particular model, making 25 billion dollars and changing the world. We do not discuss software representations of constellations, nor algorithms to extract data from them (this would be a whole course worth of information, at least).

[Depending on the amount of remaining time, I either provide the proof of the party problem, if the students didn’t solve it on their own, or continue with anecdotes about Google’s PageRank and its pitfalls. I don’t usually give the proof of the seven bridges problem, but if pressed a short sketch of the proof is easy. More often than not, the bell ends my lecture before I’m ready anyway.]

[I should emphasize that the proof of the party problem is the most exciting moment of the entire lecture. There is often a few students that have an audible “whoa!” and I’ve even received a standing ovation. This tells me that basic elegant proofs are easily within reach of a freshman high school student, and even the obviously “popular” girls have admitted to me it’s cool.]

Reflections

This lecture has generally been successful among students for three obvious reasons.

First, it is exploratory. People intrinsically like puzzles. In sharp contrast to the typical high-school style of memorization and repetition, the students drive the method! Of course, they do so in a discordant, chaotic way, but this can just as easily be said of the same students English essays or history papers. They simply have less practice with this particular kind of argument, and so they are expectantly less coherent.

Unfortunately the shortage of time forces me to guide them much more directly than I should. The amount of content we cover in an hour lecture really deserves a week of discussion, formulation and reformulation, and debate, with as little intervention as possible. I would absolutely love a chance to work with them for an extended period of time to see how it plays out. And of course, it would be much less linear, and we’d explore the questions of the students interest. However, I do fear that they might prefer a more rigid structure, being used to the humdrum of their education heretofore.

Second, the lecture is personal. I don’t have them make up names for nothing. Mathematics is a generative subject. The students not only need to see that, but also experience the process of taking an intuitive idea and nailing it down (often with more logical rigor than anything they’ve experienced in math to date). Inventing a name for the resulting definition secures the idea in their memory, and accentuates the notion that this thing is unique to their special classroom community. Since they don’t yet have the practice or motivation to make their own proofs (the ultimate mathematical self-expression), this is the next best thing. But most of all, naming the concepts is fun!

So I frame the problems in their imaginations, not history. I give them no false pretense for why we are doing this. The puzzle is a means to its own end, and we only later discover that our work is applicable. For a large portion of mathematical history, one might argue, this is how progress worked. Certainly Leonhard Euler anticipated neither Facebook’s social graph nor Google Maps in the mid 18th century. This is the easiest way to impress upon them that many different things have similar patterns in their structure, so even studying a trivial thing can be very enlightening. The puzzles really are worth doing for their own sake.

Compare this to being given the definitions and propositions in the established mathematical language. To an untrained, uninterested student, this is not only confusing, but boring beyond belief! They don’t have the prerequisite intuition for why the definition is needed, and so they are left mindlessly following along at best, and dozing off at worst.

Third, the lecture is a conversation. While ultimately I have to dish out judgement on their suggestions (this name doesn’t make sense, that idea doesn’t pan out for this reason, etc.), I make an honest effort to explain why and reiterate our goals, showing the discrepancy, and then requesting another suggestion. Unfortunately that keeps it as a bona-fide lecture, but if I had a week, the students would ideally critique each other’s work.

At the same time, I do (to some bounded level) entertain their admittedly immature suggestions. When they keep insisting swimming or long jumping across the river, I usually quip with something like, “Pretend the tourist is your grandmother. Would she swim that far?” If not to just deny their question, this reminds them that the original puzzle was meant for all tourists, including the “weakest link” as far as swimming goes. Even when they riposte with “Yes. My grandmother is a body-builder,” I dismiss it with a smirk and a wave of the hand, perhaps sarcastically saying, “Okay.” I feel that such a level of humorous improvisation is necessary, both to keep the students on their toes and to mirror their creativity with my own, thus fueling it and directing it toward the mathematics.

Of course, I might not always spend so much time with such verbal fencing. But for a first exposure to real mathematics, and to establish my role in part as an equal but more so as an obstacle, I deem it necessary. I need to be the out-witter, so that when they exhaust their loopholes, they have no choice but to beat me by solving the problem. Given a lengthier period of time (taking into consideration the students’ maturity level), I would gradually transition to a more pointed focus on the problems, and make it clear when silliness is appropriate. With luck and planning, interest in the problems would compensate for a perhaps dull state of order.

If I Had a Class

Sometimes I entertain the thought that I might end up teaching high school, and that with the providence of the school’s administration I could have my own elective course called “Real Math,” or something perhaps more enticing to the skeptical student (“Math Soup for the Teenage Soul”? “Math as Art”? “Mathematical Composition”?).

This course would start with a week of lectures similar to the one detailed above, and then alternate each week between some objective curricula (likely the basics of set theory and methods of proof) and an exploratory topic. The latter might likely start off as more explorations into graph theory (I’d debate whether to replace their invented names with the established language) and then continue into other basic topics. During the exploratory week, students would present and critique arguments in front of the class. The problems come from a list of problems given at the beginning of the week or the students’ own minds. And though I’d prefer most of the problems being the students’ own, it’s likely that initially most of the problems would come from me. Partial solutions, interesting observations, and even the process of an incorrect solution would all be presentation-worthy.

And finally, perhaps the only original idea I would have for this course, the students would each keep a journal. It would double as a notebook for their own investigation of the material brought up during exploratory weeks and a portfolio to turn in for grading. Its grading would be largely subjective, but the students would have to display some level of effort in terms of the depth to which they explore a particular problem and the number of problems attempted. As the year would progress, I would get to know the students’ ability levels and work tendencies much more clearly, and would thus have a more refined and personalized grading method.

Of course, not long after building up these ideas in my imagination, I came across the essay A Mathematician’s Lament, in which Paul Lockhart mercilessly (and rightfully) berates the current state of mathematics education in America. He more or less advocates the kind of teaching style I propose, and then argues that today’s mathematics teachers cannot play such a role for a lack of their own love for mathematics, and would not want to, because teaching this way is extremely hard! It requires much more work than the average teacher is paid to do, and there is no time for it when you’re berated by scripted curricula and standards and have six classes a day.

After repeating the lecture above for five classes of students in the course of a single day, I certainly agree with Lockhart on the difficulty of this teaching method. Though I feel I have a natural knack for presentation and engagement, to handle the standard number of students per day expected of American teachers is quite tiring (and I am a lively and strapping young lad!). It’s clear that if mathematics education is going to be fixed, one big part will be in taking teachers out of the classroom for preparation, training, and reflection. That being said, I consider it my duty to take every opportunity to do a lecture, while it provides me both with intellectual joy and the satisfaction of beneficence.