# The Friendship of Calculus: A Girl’s Journey Into the Unwavering Depths of The Third Dimension

One of my multivariable calculus students did her final project based around a book we read and discussed in class. It is called The Calculus of Friendship by Steven Strogatz. In it, the author writes each chapter about his own life and relationship with his former calculus teacher through the lens of some mathematical puzzle or concept.

My student wanted to do something similar, exploring her her multiple identities with her mathematical experience through the lens of multivariable calculus concepts. With her permission, I am putting up her three chapters here. It was a powerful experience listening to it as she read it aloud during her public presentation. I entreat you to read it. And although it may seem strange, there are many parts of it that are worth standing up and reading aloud. If you do that, you can inhabit my student’s voice for a while and really hear what she’s trying to say.

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The Friendship of Calculus: A Girl’s Journey Into the Unwavering Depths of The Third Dimension

by Brittany Boyce

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Chapter One: The Fourth Dimension

The fourth dimension as described in the dictionary is “a postulated spatial dimension additional to those determining length, area, and volume.” The key word in that definition is postulated. The fourth dimension is not something we can see, hear or touch, it comes from our imagination. In the times of early human life, the Mystics saw the fourth dimension as a place where spirits resided, since they did not inhabit our 3-dimensional world and were therefore not limited to our earthly confines. Albert Einstein, in his theory of special relativity, called the fourth-dimension time, but also concluded that time and space were inseparable. But what truly is the fourth dimension? In life, we try to make meaning of the world, what it will bring, what it will mean, how it will help us grow or not, and how it will change. Although we have a certain plan on what we want our world to look like, it is not something tangible that we can hold on to or grasp. The 4th dimension is something we can only imagine. We use the 3rd dimension, what we know and live through to help us envision the 4th. We assign colors and densities to certain points in space, and that helps us paint a picture that we can live with, but we are never truly satisfied.

In 1884 Edward A. Abbott, published a book about the problem of seeing dimensions that are not our own. In “Flatland: A Romance of Many Dimensions,” Abbott describes the life of a square living in a 2-dimensional world, which means he lives with triangles, rectangles, circles, and other two dimensional creatures, but all he sees are other lines because everything is flat. When the square finally has the chance to visit the third dimension with the help of a trusty sphere, a new world opens up to the square. Yes he is a shape like his 3rd dimensional counterparts, but he never took the chance to step out of his world and never sought to understand other worlds because he was never encouraged. At first, the square did not have the ability to comprehend the 3rd dimension, because for his whole life he only knew two dimensions. When the sphere takes the square out of the 2nd dimension, the square is finally able to see that there is a lot more to the world than just flat shapes like himself. The square was able to learn that other shapes have depth, color, height, etc. and because he was so amazed he turned to the sphere and asked what was beyond this dimension. The sphere, like the square, was appalled, unable to comprehend a world that wasn’t his own.

In this way, the sphere is like each and everyone of us. We are unable to comprehend other worlds, simply because we haven’t lived in other worlds. Our levels of privilege and different experiences explicitly prohibit us from knowing what each other’s lives are like. But does that mean we shouldn’t try? Does that mean we should just sit down and not try to understand anything simply because it is different from our experience? The answer to that question my friends, is a simple no.

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It was spring of junior year 2016 and I was sitting with my dean at the time, Mr. Brownstone, in his office going over my course registration. Now, this was it. This was the end, the last course registration I would ever have to do, the icing on the cake that would make me and my resume look appealing to the college of my dreams.

We were going over each class to make sure they looked okay, english looked good, history looked good, languages looked good, and art looked good. The only problem was if I was going to decide to jump into the deep end that was Multivariable Calculus. At first when I heard that Multivariable Calculus was an option I avoided that conversation like the plague with all my previous math teachers.

“You’re taking Multi right?” Mr. Brownstone said.

“Can I take Math Apps instead? They’re both different types of advanced math right?” I replied with a slight chuckle. He looked and me and laughed and replied with a hard “No.” There was no way he wasn’t letting me take Multivariable Calculus, and there was no way he wouldn’t make me step up to the challenge. As a kid who was already succeeding, I did not see the point in taking something extremely hard, but I went along with it anyway.

See that’s the thing about Mr. Brownstone and many other faculty members at Packer. They look out for you by pushing you to your limits and although in the moment you hate them, it’s always worth it in the end. Multivariable Calculus had already had its reputation of being a class, that would really “challenge you,” to put it nicely. Mr. Shah also already had a reputation of being one of the hardest teachers in Packer, so just thinking about this class was making my stress levels rise.

As a junior going into what would be the second half of the hardest year of my life, I didn’t think I was ready for this level of mathematics. I had always prided myself on being good at math and I enjoyed the subject as a whole but all the new variables, operators, and symbols in calculus had opened the door to a whole new side of math that scared me to be honest. Not that an integral sign is physically scary in anyway, but I was scared of the fact that I might not be able to do it. I was scared of needing help because growing up I was taught to be independent. Help was a foreign concept to me because I’ve always been told that based on my skin color no one was willing to help me and so I always had to fight for myself.They had always taught me to be independent because independence was power, and power was success.

Multivariable Calculus had always been a puzzling topic to me. What is it? I still couldn’t tell you. I was already confused by the addition of the alphabet, Greek and English, into the mathematical world, so when I heard that there could be multiple variables added into equations that I would soon be required to solve, I was even more worried. I remember thinking to myself that Mr. Shah would be too hard of a teacher for me and that the material would be too confusing. There was a part of me that thought that I would lose my status of “intelligent” and that I would let down all the people who told me I could be successful regardless of my background. In taking this class, I felt a certain pressure to do well as a poor, young, black, gay woman because not many others like me had this opportunity to study at such a high level in high school. Going to a place like The Packer Collegiate Institute, where I was one of few, always reminded me of my duty to the marginalized communities.

This type of math, meaning calculus, had always felt like a very distant topic to me. I never could picture myself being a “mathematician” because even though I was passionate about math and I had always been good at it, when I looked in the mirror, I never saw a mathematician.

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So it was September of 2016 and my first day of Multivariable Calculus with Mr. Shah had finally arrived. I had no idea what to expect and I was scared out of my mind. It was my second day of classes as a senior in highschool. The pressure was on. I had a chance to prove that I could be as great as everyone thought I could be. So here was my shot, my ticket to the big time academia.

Overall, looking at my new math teacher, Mr. Shah, he didn’t look so intimidating. However, his reputation still preceded him. See that’s the thing about Packer teachers, there are some that you can’t mess with. Some that are so passionate about what they study that they try to imbue you with that same passion in the form of school work. They expect so much of you, and give you so much work to better you, that you can’t help hate and respect them for it.

Every Packer upper schooler knows who I am talking about. Firstly in the sciences, there is Dr. Lurain, an exceptional chemistry teacher who often appears and often is very serious, but will light up and burst out in laughter in appreciation of a good chemistry joke. Next, in the languages there is Mr. Flannery, an inspiring Latin teacher who pushes his students to the breaking point every week with his famous tests. You will always catch one of his students learning lines, memorizing vocab, or reading some famous classical story. Mr. Flannery is no joke, but he has a devout dedication to each and every one of his students. The list goes on and on, but Mr. Shah was one of those teachers. Students told me how they were required to write essays on their tests or be so thorough in their answers to get full credit. But, he didn’t have the demeanor of a mean and strict teacher, he was very passionate about math and he didn’t look like he planned to intentionally make my life a living hell.

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In the first few days of Multivariable Calculus with Mr. Shah, I remember thinking “okay, come on hit me! I can handle it.” I was expecting some complex problem that I couldn’t handle or some other problem that required, some “higher math” that required prior knowledge I didn’t get a chance to grow up with. Instead, Mr. Shah nurtured us, all of us.  He taught us not to be scared of Multivariable Calculus. He taught us that we were prepared for 3-dimensional calculus, and the third dimension was just a step up from the second dimension. He made us aware that we already fought hard enough through Calc I & II with Mr. Rumsey, which was a battle of its own, to be sitting here together taking the same class. He never said it was going to be easy, but he made us feel like we were prepared from the bottom up. But, this comfort and reassurance is not something everyone in the world has the privilege to have.

At times, going up a dimension can seem scary. Most often, in our world things can be complicated enough, which causes us to forget that there are things that are higher than ourselves and more important than ourselves. If you’re like me, you use the fact that two-dimensional calculus was already hard enough, so why study 3-dimensions? Why go beyond what you already know? What’s the point?

The point wasn’t to solve the problem right every time or to be able to understand the most complex things first. It was to be willing to take that step into the unknown in the first place. I had an amazing opportunity to try to understand a world that didn’t necessarily welcome me with open arms. I wasn’t lucky because I had the intellectual ability to take Multivariable Calculus. I was lucky because I was one of few students who had an instructor that made me feel like I could understand the higher maths. Not many kids my age have the ability to study the higher maths, or to even believe that they could study the higher maths, especially students of color, women, and LGBTQ+ students. Today’s education system lacks mentors that have the ability to push kids in the right direction and to make them believe in themselves regardless of their social status. What is unique to my experience is that as a woman of color, low socioeconomic status, and who is proud to say that she is a part of the LGBTQ+ community, I had people around to support me. There was never one time I felt that my peers or teacher didn’t think I was worthy enough to be there taking that class because of my gender, race, sexuality, or socio economic status.

However, although my reality was brighter and more positive than other students who share my identity and do not have the same support system I do, I cannot just be grateful and move on with my life. I must think about those who have to fight harder, speak louder, and do better than I do to hold their place in the classroom and the community of the higher maths. I must bring attention to their fight even though I only know my own.

Chapter Two: Line Integrals

A line integral is essentially integration of a function along a curve. But, that means nothing to most of you. On each curve there are an infinite number of points that trace the path of the curve, determining what it will look like, how it will behave, and how it can be analyzed. Not each point is worth more than another in value or in status, but each plays an integral role in defining the curve. Let’s just say, all points are created equal. But what does that curve really mean? What can it do for us and what can we do for it? Sure it can be pretty to look at or cool to trace, but it all means nothing if we can’t make something out of it or give meaning to it.

That’s where our friend the line integral comes in. To many, it looks like a weird “s.” To my readers, three of these majestic creatures in a row means that I am switching directions or switching to a different moment in time. But to a mathematician, the line integral gives meaning to the curve. It takes the path traced by the infinite amount of points and cuts it into infinitesimally small pieces and adds it all together into the culmination of a single amount, quantity, and meaning. The line integral represents the culmination of everything we’ve been through and the addition of all those infinite moments into one big picture called life. But, while you may have all the pieces and the trajectory, solving the line integral and finding the meaning behind the trajectory, will not always be easy.

Often times, in school we as children are set on a given path or a chosen trajectory, let’s call it f(x). We are given a curve C, and we are told to follow it. We get the grades, play the sports, and be the children our parents want us to be. But what does it all mean when we have hit all the points, traced the path, and completed it? What is it supposed to mean? How are we supposed to evaluate our lives when we haven’t even begun to make any choices for ourselves? And how are we supposed to deal the the fact that we may never make meaning of our chosen path even though we might have all the tools?

The creators of calculus dared to confront this problem through math, because of course, it was the only option. To them, doing the work, solving the integral and making meaning of such a path, was more important than perhaps what the integral meant numerically. Frankly, to be the most cliche, it’s about the journey, not the destination. Not all integrals are meant to be solved in the most complex way or with calculus; sometimes it only takes simplest geometric proof or the simplest meaning of life that can propel you in the right direction, or help you move forward in the problem.

Do you ever wonder how long it takes to change your life? What measure of time is enough to be life altering? Is it four years like high school? One year? A 2-semester calculus class? A semester long, history course? Can your life change in a month? A week? A single day? We’re always in a hurry to grow up, to go places, and get ahead. But when you’re young, one hour or even 50 minutes can change everything.

Through integration, a curve becomes a series of tiny straight lines, working together towards one common quantity. Through integration, life becomes a series of tiny moments working together towards the culmination of you and what your life means. However, sometimes it may be hard to make meaning of a certain time in your life. Sometimes that moment may be unsolvable and that can be frustrating. But, the important thing to remember is that each infinitesimally small piece or small moment works to affect the meaning of your life. Each small experience adds something to your journey.

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I had made it through the first semester of Multivariable Calculus feeling like I could actually pursue mathematics in college. But, I wasn’t completely sure what helped me get here. There was some small moment along my path, where it just clicked. There was something about the elegance of Multivariable Calculus that caused me to light up during every class. Surely there were days that I was tired and defeated, and felt that I could not take anymore of Mr. Shah’s high expectations; But, something about the math itself always brought me back to that stillness I felt. The stillness that was almost calming at the sight of an elegant proof or after spending time doing hard rough algebra, fighting and wrestling with exponents, variables, and symbols to finally get an answer. I didn’t know it then, but that stillness was my ability to feel passionate about math. I had a willingness to understand the concepts behind the algebra I was doing, and had come to appreciate the conceptual approach rather than the hard hitting, laborious algebra I was used to my whole life.

For the more complex conceptual solutions, sometimes I felt cheated, when the very complex parts of the problems were reduced by simple geometric approaches. I saw the immense power of calculus, and I didn’t want it to be reduced or lessened by geometry. There was something about putting my head down and jack hammering through the hard work that always pleased me, but I soon learned that it wasn’t cheating, nor did it lessen the power of calculus in any way.

One simple solution to a complex integral we often faced in class was the integral of cos^2(x) from zero to kπ, k being a multiple of ½. Now for all you mathematicians out there, you know that this integral is no joke. There is no simple u-substitution or power rule you can use to solve this, it must be solved with integration by parts, which is a method that requires some of that “jack hammering” I loved so much.

The proof of ∫ cos^2(x)dx using integration by parts, goes as follows:

Using this integral, the area under the curve on the interval 0 to π/2, makes the integral equal to π/4. While this solution did take intuition and elegance, such as turning the ∫ sin^2(x)dx into ∫ 1 – cos^2(x)dx. Then, adding the ∫ cos^2(x) to both sides to make the proof simple algebra. We learned to tackle complex integrals like this using integration by parts in Calculus I & II,  but 3-dimensional calculus builds on 2-dimensional calculus, so complex integrals always popped up in daily problems. I admired the hard work that calculus required, and the instincts that one could gain from solving such problems, but let’s be real, no one is going to remember the solution to a random integral because one random integral is not that important to all of math. So the question we’re faced with is do we fill our minds with random memorizations of quantities representing areas under curves or do we find another way to remember?

One day, Mr. Shah gave me and my fellow peers a new tool to add to our mathematician’s tool belt. He gaves us geometry. He took us back to our roots and showed us that sometimes simplicity is the ultimate sophistication. So we tackled the same solution.

What is ∫ cos^2(x)dx?

We were essentially tasked with finding the area shaded above. Sometimes when you’re in the middle of solving a problem, and this integral pops up, you can’t result to algebra every time. Sometimes the matter is too urgent and the problem can’t wait for you to do all this algebra. So Mr. Shah showed us one single shape that would change the way we would approach any integral for the rest of our math careers.

Now look closely. The area under the curve is equivalent to exactly ½ the area of the blue rectangle. Now the graph tells us that the length of the rectangle is 1 and the width of the rectangle is π/2. That makes the area of the rectangle  π/2 • 1 =  π/2, making half the area of the rectangle π/4. BOOM. One complex integral simplified with the power of geometry. This proof amazed me. I was astounded by the elegance of such a simple solution. I mean a seventh grader could do this.

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Calculus was never meant to be unreachable. Renowned mathematician Edward Frenkel once said, “mathematics directs the flow of the universe, lurks behind its shapes and curves, and holds the reins of everything from tiny atoms to the biggest stars.” The beauty that math holds has become a privilege unreachable to those who are marginalized everyday for their skin color, race, and sexuality. Everyday students of color and women are told that they cannot or should not see this beauty, the beauty of math is held from them until they climb and fight to the point where they are so bruised, broken and beaten that they give up. There are increasingly low percentages of black and Latinos in high-paying, high-status jobs in finance, science and technology. Since  perceived intelligence in the higher math communities are increasingly influenced by racial prejudices it is getting harder and harder for students of color to believe that they can be something more than the stereotypes. Fundamentally, this is a question about power in society.

Being a student of color who had to claw tooth and nail and go to highly selective programs to even be in a place like Packer, I have experienced that loss of a love for education. Being a black girl who was able to show her intelligence at such a young age, I was set on the path to success. Do your school work, get a good job, be successful. But at the time, I didn’t really know what it meant to be successful. I still don’t know what it means. Most of the time, success is dependent on whether or not I beat the system. I was never told to do what makes me happy. I was told to do what makes me money. I never had the privilege of growing up studying what interested me, or what I was passionate about, and I never knew that having the chance to delve into European history or a new language was a privilege. I was too busy preparing for survival. I was busy getting a head start on the material I needed for the future, so the pressure and the rigor of a predominantly white and male setting wouldn’t defeat me.

There are kids out there who don’t get to enjoy and love knowledge because they are not taught that knowledge is beautiful, they are taught that knowledge is power, and that power is the key to success. Academics never become leisure activity because survival is more important than leisure. They are set on their own path, and asked to make meaning of that path without loving the path in the first place. At the end of their trajectory, they are left at a crossroads, choose another path that they truly love with the possibility of failure or never love a path at all.

Chapter Three: Path Independence

Path Independence shows that the value of a line integral of a conservative vector field along a piecewise smooth path is independent of the path; that is the value of the integral depends on the endpoints and not the actual path C. Now wait a second, am I hearing that vector calculus thinks that it’s about the destination not the journey? Frankly, I don’t blame the creators of this theorem. Most of our world thinks life is path independent. People think that they can see past their privileges and just go on with their lives and that every accomplishment they achieve is independent of a third party. But is our world truly conservative? No pun intended. Do we live in a world where, as one of my favorite bloggers puts it, “instead of recognizing our unfair privileges, we just build walls around us and project out way of life as normal. Any story you tell about how you got where you are that doesn’t include land theft, profiting off of forced, unpaid labor, illegal occupation, murder, assault, theft, psychological and physical warfare, exploitations, and a culture of complicity is, you know, a lie.”

If it is then what’s the point of me fighting so hard to hold onto my passion for mathematics? What’s the point if my journey, which might be ten times harder than someone else’s is recognized in the same or even a lesser fashion than someone who got to the same endpoint. Isn’t there supposed to be beauty in the struggle? Value in someone’s journey? What’s the point in finding the meaning of your path if it is weighed the same as everyone else’s path who started and finished at the same places you did? How are we supposed to try to learn and value the experience of others if we just value where we’ve ended up? Does this mean that the situation you are born in, something that you can’t control, has some type of influence on the overall meaning of your path? It shouldn’t.

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The problem of the conservative vector field that is our world always had a place in our Multivariable Calculus classroom although we sometimes didn’t know it. Every Day 4, when we had class for 90 min, we would hold a book club. Mr. Shah would assign us a different piece of literature to read regarding math, whether it was Flatland by Edward A. Abbott, or The Calculus of Friendship by Steven Strogatz, or Love and Math by Edward Frenkel. As a senior, already up to my eyeballs in work, I disliked him for giving me this reading on top of all the math problems he had already assigned me. I never knew it then, but what Mr. Shah was doing was important work. He was showing us what is was like not to be path independent. He made us value the stories of the mathematicians before us, so that we could know how hard it could be for the person sitting right next to us to be successful in the mathematical community. He made sure to make us feel the responsibility we had to the ethics of the math community. We discussed the politics of math, the religion of math, and the inequities of math every week.

He showed us that while learning the material itself was important, the story behind the material is just as important. In life and in math, there are multiple approaches to solving problems. Often times in math class, Mr. Shah highlighted when two students had different approaches to the same answer and would even have them write it on the board for the whole class to experience. Each approach would have something different. Maybe a trick, a new tool, or even a slight adjustment. When I thought about the way a problem was solved, I never really saw the value in the different approaches, all that mattered to me was that the same answer was achieved.

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In my last quarter of my Packer career and a Multivariable Calculus student, we returned to pure learning. We watched a series of lectures, which was considered our preparation for college math and the whole host of difficulties that would come with it. As the time was winding down, and I began to think about what my final project would look like, I admired Mr. Shah for making us do something that we were interested in and that was meaningful to us. I had never gotten the chance to do so, while still preparing for “survival.” Once again, I hated all the stress and work it brought me. But, I was very grateful that I had gotten a chance to make meaning out of my experience. While the culmination of my mathematical trajectory or “path” at Packer was not completely numerical or quantitative, the experience of finding meaning through math has been life changing.

Math is beautiful, and I only hope that seeing this beauty no longer becomes a privilege in this world, but a necessity. Everyone deserves to believe that they can be passionate about something and not be deemed a failure. No kid should have to carry the weight of their struggle alone. We must not be path independent, we must be aware of the stories that are around us.

# Multivariable Calculus Projects 2015-2016

Each year, I have students in my multivariable calculus class do “fourth quarter projects.” We continue working with the material during classtime, they have regular nightly work, but I cancel all problem sets and tests. Instead, students choose a project topic they are interested in pursuing that has some relationship to the course (even if the relationship is a bit tenuous). I want the project to be one of passion. Their entire fourth quarter grade is based on these projects. This year, my kids came up with some amazing projects — some of the best I’ve seen in my eight years of teaching this course. (Some previous years projects are here, here, here, and here.)

## An Augmented Reality Sandbox

Earlier in the year, I showed my student a video of an augmented reality sandbox that I stumbled across online. She showed interested in making it. It takes in a mapping of a surface (in this case sand in a sandbox) and projects onto the surface colors representing the height of the sand over time (so red is “high” and blue is “low”). The cool part about this is that the projection changes live — so if you change the sand height, the projection updates with new colors. Level curves are also “drawn” on the sand.

Here are some videos of it in action (apologies for the music… I had to put music on it so the conversations happening during the playing with the sand were drowned out):

The student was going to design lesson plans around this to highlight concepts in multivariable calculus (directional derivative, gradients, gradient field, reading contour maps) but ran out of time. However upon my suggestion, during her presentation, she did give students contour maps of surfaces, turned off the projector, had students try to form the sand so it matched the contour map, and then turned the projector on to have students see if they were right or not.

During the presentation, one student who I taught last year (but not this year) said: “This is the coolest thing I’ve seen all year!” and then when playing with the sand: “I AM A GOD!” Entrancing!

## Harmonograph

In my first year of teaching this course, a student was entranced by lissajous curves when we encountered them. These are simple parametric equations which create beautiful graphs. I then suggested for his final project that he create a harmonograph, which he did. Seven years later, I had another student see the original video of my student’s harmonograph, and he wanted to build his own! But he wanted his to have a rotary component, in addition to two pendulums which swung laterally. So he found instructions online and built it!

Here are some of the images it produced:

And here is a video of the harmonograph in motion:

(You can watch another video here.)

During the presentation, the student talked about the damping effect, how the pendulum amplitudes and periods had an effect on the outcome, and how lissajous curves were simply shadows of lissajous knots that exist in 3-space. Because of the presentation, I had some insights into these curves that I hadn’t had before! (I still don’t know how mathematically to account for how the rotary pendulum in the student’s harmonograph affects the equations… I do know that it has the harmonograph — in essence — graph the lissajous curves on a somewhat rotating sphere (instead of a flat plane). And that’s interesting!

## Teaching Devices for Multivariable Calculus

A student was interested in creating tools for teachers to illustrate “big” multivariable calculus ideas… Contour lines, directional derivatives, double integrals, etc. So she made a set of five of super awesome teaching manipulatives.  Here are three of them.

The first is a strange shaped cutout of poster-cardboard-ish material, with four animals hanging from it. Then there is string connected to a magnet on top, and another magnet on the bottom. If you hold up the string and you aren’t at the center of mass, the mobile won’t balance. But if you move the magnet around (and the student used felt around the magnet so it moves seamlessly!), you can change the position of the string, until it balances. This is a manipulative to talk about center of mass/torque.

Another is a set of figures that form “level curves.” At first I was skeptical. The student said the manipulative elow was to help students understand countour plots. I wanted to know how… Then the moment of genius…

You can change the height of the level curves to make the “hill” steeper and steeper, and then look straight down at the manipulative. If you have a shallow “hill,” you have contour lines which will look far apart. If you have a tall “hill,” you have contour lines which look close together.

Finally, a third manipulative showcases the tangent plane (and it can move around the surface because of magnets also). I can see this also being useful for normal vectors and even surface integrals!

## Cartographic Mapping

Two students decided to work together on a project dealing with cartographic mapping. They were intrigued by the idea that the surface of the earth can’t perfectly be represented on a flat plane. (They had to learn about why — a theorem by Euler in 1777.) They chose two projections: the Gall Peters projection and the Stereographic projection.

They did a fantastic job of showing and explaining the equations for these projections — and in their paper, they went into even more depth (talking about the Jacobian!). It was marvelous. But they had two more surprises. They used the 3D printer (something I know nothing about, but I told them that they might want to consider using to to create a model to illustrate their projections to their audience) and in two different live demos, showed how these projections work. I didn’t get good pictures, but I did take a video after the fact showing the stereographic projection in action. Notice at the end, all the squares have equal area, but the quadrilaterals on the surface most definitely do not have equal area.

An added bonus, which actually turned out to be a huge part of their project, was writing an extensive paper on the history of cartography, and a critical analysis of the uses of cartography. They concluded by stating:

We have attempted, in this paper, to provide our readers with a brief historical overview of cartography and its biases.  This paper is also an attempt to impress upon the reader the subjective nature of a deeply mathematical endeavor.  While most maps are based around mathematical projections, this does not exclude them from carrying biases.  In fact, we believe there is no separation between mathematical applications and subjectivity; one cannot divorce math from perspective nor maps from their biases.  We believe it is important to incorporate reflections such as this one into any mathematical study.  It is dangerous to believe in the objectivity of scientific and numerical thought and in the separation between the user and her objective tools, because it vests us, mathematicians and scientists, with arbitrary power to claim Truth where there is only perspective.

Beautiful. And well-evidenced.

## Deriving the Hagen-Poiseuille Equation from the Navier-Stokes Equations

One student was interested in fluid dynamics. So I introduced him to the Navier Stokes equations, and set him loose. This turned out to be a challenging project for the student because most of the texts out there require a high level of understanding. Even when I looked at my fluid dynamics book from college when I was giving it to him as a reference, I realized following most of it would be almost impossible. As he worked through the terms and equations, he found a perfect entree. He learned about an equation that predicts the change in pressure from one end of a tube of small radius to another (if the fluid flow in the tube is laminar). And so using all he had learned in his investigation of the field, he could actually understand and explain algebraically and conceptually how the derivation worked. Some of his slides…

It was beautiful because he got to learn about partial differential equations, and ton of ideas in fluid dynamics (viscosity, pressure, rotational velocity, sheer, laminar flow, turbulence, etc.), but even needed to calculate a double integral in cylindrical coordinates in his derivation!

## The Wave Equation and Schrodinger’s Equation

This student works in a lab for his science research class — and the lab does something with lasers and quantum tunneling. But the student didn’t know the math behind quantum mechanics. So he spent a lot of time working to understand the wave equation, and then some time trying to understand the parts of Schrodinger’s equation.

In his paper, he derived the wave equation. And then he applied his understanding of the wave equation to a particular problem:

He then tackled Schrodinger’s Wave Equation and saw how energy is quantized! Most importantly, how the math suggests that! I remember wondering how in the world we could ever go from continuousness to discreteness, and this was the type of problem where I was like “WHOA!” I’m glad he could see that too! Part of this derivation is below.

Overall, I was blown away by the creativity and deep thinking that went into these final projects. Most significantly, I need to emphasize that I can’t take credit for them. I was incredibly hands off. My standard practice involves: having students submit three ideas, I sit down with students and help them — with my understanding of their topics and what’s doable versus not doable — narrow it down to a single topic. Students submit a prospectus and timeline. Then I let them go running. I don’t even do regular formal check-ins (there are too many of them for me to do that). So I have them see me if they need help, are stuck, need guidance or motivation, whatever. I met with most of them once or twice, but that’s about it. This is all them. I wish I could claim credit, but I can’t. I just got out of their way and let them figure things out.

# Pitching college math courses

Ooops. This turned out to be a post with no images. So here’s a TL;DR to whet your appetite: I wanted to expose my seniors to what college mathematics is, but instead of lecturing, I had them “pitch” a college course to the rest of the class.

My multivariable calculus courses was coming to an end, and I got some questions about what college courses in math are about. It reminded me of a comic strip I read years ago, which I frustratingly can’t find again. It has an undergraduate going to meet with his math professor adviser, saying something like “I want to major in triple integrals.” Which is crazy-sounding — but maybe not to a high school student who has only ever seen math as a path that culminates in calculus. What more is out there? What is higher level math about? (These questions are related to this post I wrote.)

So here’s what I told my students to do. They were asked to go onto their future college math department websites (or course catalog), scour the course offerings, and find 3-4 courses that looked interesting and throw these courses down on a google doc.

It was awesome, and made me jealous that they had the opportunities to take all these awesome classes. Some examples?

After looking through all the courses, I highlighted one per student that seemed like it involved topics that other students had also chosen — but so that all the courses were different branches/types of math. I told each student to spend 10-15 minutes researching their highlighted course — looking up what the words meant, what the big ideas were, finding interesting videos that might illustrate the ideas — so they can “pitch the course to the class” (read: explain what cool math is involved to make others want to take the course).

I’m fairly certain my kids spent more than 10-15 minutes researching the courses (I’m glad!). Each day, I reserved time for 2-3 students to “pitch” their courses. And since some of the ideas were beyond them, after the pitches, I would spend 5 or so minutes giving examples or elaborating on some of the ideas they covered.

If you want to see the research they did for their pitches, the google doc they chucked their information into is here.

Some fun things we did during the pitches?

(1) We watched a short clip of a video about how to solve the heat equation (that was for a course in partial differential equations)

(2) I showed students how to turn a communication network into a matrix, and explained the meaning of squaring or cubing the matrix (this was for a course on network theory)

(3) A student had us play games on a torus (a maze, tic tac toe) (this was for a course on topology)

(4) I had students store $x=0.3$ on their calculators. Then I had each student store a different “r” value (carefully chosen by me) and then type $r*x*(1-x)->x$ in their calculators. They then pressed enter a lot of times. (In other words, they were iterating $x_{n+1}=rx_n(1-x_n)$ with the same initial conditions but slightly different systems. Some students, depending on their r value, saw after a while their x values settle down. Some had x values that bounced between two values. Some had x values that bounced between four values. And one had x values that never seemed to settle down. In other words, I introduced them to a simple system with wacky wacky outcomes! (If you don’t know about it, try it!) (This was for a course on chaos theory)

(5) A student introduced us to Godel’s incompleteness theorem and the halting problem (through a youtube video)

It was good fun. It was an “on the spot” idea that turned out to work. I think it was because students were genuinely interested in the courses they chose! If I taught a course like AP Calculus, I could see myself doing something similar. I’m not sure how I would adapt this for other classes… I’m thinking of my 9th grade Advanced Geometry class… I could see doing something similar with them. In fact, it would be a great idea because then they could start getting a sense of some of the big ideas in non-high school mathematics. Kay, my brain is whirring. Must stop now.

If anyone knows of a great and fun introduction to the branches of college level math (or big questions of research/investigation), I’d love to know about it. Something like this is fine, but it doesn’t get me excited about the math. I want something that makes me ooh and ahh and say “These are great avenues of inquiry! I want to do all of them!” I think those things that elicit oohs and ahhs might be the paradoxes, the unintuitive results, the beautiful images, the powerful applications, the open questions… If none exists, maybe we can crowdsource a google doc which can do this…

This year, our school adopted this weird rotating schedule where we see our classes 5 times out of every 7 days. And four of those times are 50 minute classes and one of those times is a 90 minute class.

I didn’t have a clear idea of what to do in multivariable calculus for the block. I still had to cover content, but I wanted it to be “different” also. After many hours of brainstorming, I came up with a solution that has worked out pretty well this year.

The 90 minute block was divided into 50 minutes of traditional class, and 40 minutes of book club. (Or 60 minutes of class, and 30 minutes of book club.)

Now, to be clear, this is a class of seven seniors who are highly motivated and interested in mathematics. I can see ways to adapt it in a more limited way to other courses, with more students, but this post is about my class this year.

## BOOKS

We started out reading Edwin Abbott’s Flatland.

Why? Because after they read this, they understand why I can’t help them visualize the fourth (spatial) dimension! But it convinces them that they can still understand what it is (by analogy) and makes them agree: if we can believe in the first, second, and third spatial dimensions, why wouldn’t we believe in higher spatial dimensions too? It’s more ludicrous not to believe they exist than to believe they don’t exist! A perfect entree into multivariable calculus, wouldn’t you say?

After reading this, we read the article “The Paradox of Proof” by Caroline Chen on the proposed solution to the ABC conjecture.

This led us to the notion of “modern mathematics” (mathematics is not just done by dead white guys) and raised interesting questions of fairness, and what it means to be part of a profession. Does being a mathematician come with responsibilities? What does clear writing have to do with mathematics? (Which helps me justify all the writing I ask for on their problem sets!) It also started to raise deep philosophical questions about mathematical Truth and whether it exists external to the human mind. (If someone claims a proof but no one verifies it, is it True? If someone claims a proof and fifty people verify it, is it True? When do we get Truth? Is it ever attainable? Are we certain that 2+2=4?)

At this point, I wanted us to read a book that continued on with the themes of the course – implicitly, if not explicitly. So we read Steve Strogatz’s The Calculus of Friendship:

What was extra cool is that Steve agreed to sign and inscribe the book to my kids! The book involves a decades long correspondence between Steve and one of his high school math teachers. There are wonderful calculus tricks and beautiful problems with explanations intertwined with a very human story about a young man who was finding his way. Struggling with choosing a major in college. Feelings of pride and inadequacy. The kids found a lot to latch onto both emotionally and mathematically. Two things: we learned and practiced “differentiating under the integral sign” (a Feynman trick) and talked about the complex relationship that exists between teachers and students.

After students finished this book, I had each student write a letter to the author. I gave very little guidelines, but I figured the book is all about letters, so it would be fitting to have my kids write letters to Steve! (And I mailed the letters to Steve, of course, who graciously wrote the class a letter back in return.)

Our penultimate reading was G.H. Hardy’s A Mathematician’s Apology:

I went back and forth about this reading, but I figured it is such a classic, why not? It turned out to be a perfect foil to Strogatz’s book — especially in terms of the authorial voice. (Hardy often sounds like a pompous jerk.)  It even brought up some of the ideas in the “Paradox of Proof” article. What is a mathematician’s purpose? What are the responsibilities of a mathematician? Why does one do mathematics? And for kids, it really raised questions about how math can be “beautiful.” How can we talk about something that is seen as Objective and Distant to be “beautiful”? What does beauty even mean? Every section in this essay raises points of discussion, whether it be clarification or points that students are ready to debate.

What is perfect about this reading is at the same time we were doing it, the movie about G.H. Hardy and S. Ramanujan was released: The Man Who Knew Infinity (based on the book of the same name).

Finally, we read half of Edward Frenkel’s Love and Math:

Why? Because I wanted my students to see what a modern mathematician does. That the landscape of modern mathematics isn’t what they have seen in high school, but so much bigger, with grand questions. And through Frenkel’s engaging telling of his life starting in the oppressive Russia and ending up in the United States, and his desire to describe the Langland’s program understandably to the reader, I figured we’d get doses of both what modern mathematics looks like, and simultaneously, how the pursuit of mathematics is a fully human endeavor, constrained by social circumstances, with ups and downs. Theorems do not come out of nowhere.Mathematicians aren’t the blurbs we read in the textbooks. They are so much more. (Sadly, we didn’t read the whole thing because the year came to a close too quickly.)

## STRUCTURE OF THE BOOK CLUBS

I broke the books into smaller chunks and assigned only them. For Flatland, it might have been 20-30 pages. For Love and Math or A Mathematician’s Apology, it might have been 30-50 pages. We have our long block every 7 school days, so that’s how much time they had to read the text.

At the start, with Flatland, students were simply asked to do the reading. Two students were assigned to be “leaders” who were to come in with a set of discussions ready, maybe an activity based on something they read. And they led, while I intervened as necessary.

For every book club, students who weren’t leading were asked to bring food and drink for the class, and we had a nice and relaxing time. On that note,  never did I mention anything about grades. Or that they were being graded during book club. (And they weren’t.) It was done purely for fun.

Later in the year, I had students each come to class with 3-4 discussion questions prepared, and one person was asked to lead after everyone read their questions aloud.

The discussions were usually moderated by students, but I — depending on how the moderation was going — would jump in. There were numerous times I had to hold back sharing my thoughts even though I desperately wanted to concur or disagree with a statement a student had made. And to be fair, there were numerous times when I should have held back before throwing my two cents in. But my main intervention was getting kids to go back to the texts. If they made a claim that was textually based, I would have them find where and we’d all turn there.

Sometimes the conversations veered away from the texts. Often. But it was because students were wondering about something, or had a larger philosophical point to make (“Is math created or discovered?”) which was prompted by something they read. And most of the times, to keep the relaxed atmosphere and let student interest to guide the conversation, I allowed it. But every so often I would jump in because we had strayed so far that I felt we weren’t doing the text we had read justice (and we needed to honor that) or we were just getting to vague/general/abstract to say anything useful.

## EXAMPLES OF DISCUSSION QUESTIONS

I mentioned students generated discussion questions on their own. Here are some, randomly chosen, to share:

•  Strogatz talks about how math is a very social activity. We see this exemplified in the letters between Steve and Mr. Joffray, but where else do we see this exemplified in math? (papers, etc.) How do you think Strogatz might have felt about Shinichi Mochizuki’s unwillingness to explain his paper and proof to the math community?
• What do you think about Strogatz and Joff using computer programs to give answers to their problems? Are computers props, and their answers unsatisfying? Or are they just another method, like Feynman’s differentiating under the integral?
• Do you like A Square? In what ways is he a product of his society? Does he earn any redeeming qualities by the end of the book?
• Can you draw any connections between things in Flatland and religion? Do you think Abbott is religious? Why/why not?
• When we first read about Mochizuki’s ABC Conjecture, we debated whether or not math is a “social” subject. Perhaps many mathematicians do much of the “grind” work on their own, however, throughout everything we’ve read this year, there has been one common link when it comes to the social aspects of math: mentorship. It appears to me that all of the great mathematicians we know about have been mentored by, or were mentors others. In what ways have Frenkel’s mentors – he’s had a few – had an influence on the path of his mathematical career? Do you think he would/could be where he is today without all of those people along the way? Can you think of any mentors that have had a profound influence on your life? (The last one can just be a thought, not a share.)
• Frenkel talks about the way in which math, particularly interpretations of space and higher dimensions, began to influence other sectors of society, specifically the cubist movement in modern art. This movement was certainly not the first time math and science influenced art and culture – think about the advent of perspective in the Renaissance and the use of technology on modern art now – however math and art are often thought as opposites and highly incompatible. Why do you think that people rarely associate the two subjects? Would you agree that the two are incompatible? Can you think of other examples of math/science influence art/culture/society?

## REFLECTIONS

In many ways, I felt like this was a perfect way to use 30 minutes of the long block. After doing it for the year, there are a few things that stood out to me, that I want to record before summer hits and I forget:

(a) I think students really enjoyed. It isn’t only a vague impression, but when I gave a written survey to the class to take the temperature of things, quite a few kids noted how much they are enjoying the book clubs.

(b) For the post-Flatland book club meetings, I need to come up with multiple “structures” to vary what the meetings look like. Right now they are: everyone reads their discussion questions, the leader looks for where to start the discussion, the discussion happens. But I wonder if there aren’t other ways to go about things.

One example  I was thinking was students write (beforehand) their discussion questions beforehand on posterpaper and bring it to class. We hang them up, and students silently walk around the room writing responses and thoughts on the whiteboard. Then we start having a discussion.

Or we break into smaller groups and have specific discussions (that I or students have preplanned) and then present the main points of the discussion to the entire class.

Clearly, I need to get some ideas from English teachers. :)

(c) I love close readings of texts. I think it shows focus, and calls on tough critical thinking skills. At the same time, I need to remember that this is not what the book clubs are fundamentally about. They are — at the heart, for me — inspiration for kids. So although for Flatland I need to keep the critical thinking skills and close readings happening, I need to remember (like I did this year) to keep things informal.

(d) Fairly frequently, I will know something that is relevant to the conversation. For example, I might talk about of the math ideas that were going over their heads, or about fin de siecle Vienna, or branches of math that might show how the line between “theoretical” and “applied” math is blurry at best. I have to remember to be judicious about what I talk about, when, and why. We only have limited time in book club, so a five minute tangent is significant. And one thing I could try out is jot down notes each time I want to talk about something, and then at the end of the book club (or the beginning of the next class), I could say them all at once.

(e) I usually reserve 30 minutes for book club. But truthfully, for most, 40 minutes turned out to be necessary. So I have to keep that in mind next year when planning class.

(f) Should we come up with collaborative book club norms? Should I have formal training on how to be a book club leader? Should we give feedback to the leaders after each book club? Can we get the space to feel “safe” where feedback could actually work?

And… that’s all!

# Multivariable Calculus Projects 2014-2015

At the end of each year in Multivariable Calculus, I have students develop and execute their own “final project.” It’s fairly open-ended and students end up finding something they are personally interested/invested in and they go for it.

This year I had six students and these are their projects.

“Exploring the Normal Distribution Through the Box-Muller Transform and Visualizing It Using Computer Science” (GT)

This student had never taken a statistics course but was interested in that. We also talked about how to find the area under the normal distribution using multivariable calculus (and showed it was 1). Armed with those two things, this student who likes computer science found a way to pick independently two numbers (one each from two uniform distributions), and have them undergo a few transformations involving square roots and sine/cosines, and then those two numbers would generate two new numbers. Doing this a bunch of times will create a whole pile of new numbers, and it turns out that those square roots and sine/cosines somehow create a bunch of numbers that exactly follow a normal distribution. So weird. So cool.

“XRayField: Detecting Minecraft Cheating using Physics and Calculus” (W.M.)

This student loves Minecraft and hosts a Minecraft server where tons of kids at our school play. Earlier in the year, there was a big scandal because there were people cheating when playing on this server — using modifications to give themselves additional advantages. (This was even chronicled by the school newspaper.) One of the modifications allows players to see where the diamonds are hidden, so they can dig right to them. So this student who runs the server wanted to find a way to detect cheaters. So he created a force field around each diamond (using the inverse square law in 3D), and then essentially calculated the work done by the force field on the motion of a player. A player moving directly with the force field (like on the left in the image above) will get a higher “work score” than someone on the right (which is moving sometimes with the forcefield, sometimes not). In other words, he’s calculating a line integral in a field. His data was impressive. He had some students cheat to see what would happen, and others not. And in this process, he even caught a cheater who had been cheating undetected. Honestly, this might be one of my favorite projects of all time because of how unique it was, and how perfectly it fit in with the course.

“Space Filling Curves” (L.S.)

This student with a more artistic bent was interested by “Space Filling Curves” (we saw some of them when I started talking about parametric curves in three dimensions, and we fiddled around with Lissajous curves to end up with some space filling curves). This student created three art pieces. The first was a 2D Hilbert curve which is space filling. The second was a 3D Hilbert curve which is space filling (pictured above). The third was writing a computer program to actually generate (live) a space filling curve which involves a parametrically defined curve, where each of the x(t) and y(t) equations involved an infinite sum (where each term in this infinite sum was reliant on this other weird piecewise and periodic function). I wish I had a video showing this program execute in real time, and how it graphed for us — live — a curve which was drawing itself and how that curve being drawn truly filled space. It blew my mind.

“The Math Is Right: The Math Behind Game Shows” (J.S.)

This student, since a young age, loved watching the Game Show Network with his mother. So for his final project, he wanted to analyze game shows — specifically Deal or No Deal, and the big wheel in the Price is Right. I had never thought deeply about the mathematics of both, but he addressed the question: “When should you take the deal? Is there an optimal time to do so?” (with Deal or No Deal) and “If you’re the second player spinning the big wheel (out of three players), how do you decide whether to spin a second time or not?” (for the Price is Right). As I saw him work through this project — especially the Price is Right problem — I saw so much rich mathematics unfold, involving generating functions, combining distributions, and simulating. It’s a deceptively simple question, with a beautifully rich analysis that hides behind it. And that can be extended in so many ways.

“The Art of Balance” (M.S.)

This photograph may make it look like the books are touching the wine holder. That is not the case. This wine holder is standing up — quite robustly as we tested — through it’s own volition. And — importantly — because the student who built it understood the principle behind the center of mass. This student’s project started out with him analyzing the “book stacking problem” (which involves how much “overhang” you can create while stacking books at the edge of the table. For example, with one book, you can put it halfway over the table and it will not fall. It turns out that you can actually get infinite overhang… you just need a lot of books. This analysis centered around the center of mass of these books, and actually had this student construct a giant tower of books. The second part of this project involved the creation of this wine holder, which was initially conceived of mathematically using center of mass, then that got complicated so the student started playing around with torque which got more complicated, so the student eventually used intuition and guess and check (based on his general understanding of center of mass). Finally he got it to work. The one thing this student wanted to do for his project was “build/create something” and he did!

“Visualizing Calculus” (T.J.)

This student wanted to make visualizations of some of the things we’ve learned about this year. So he took it upon himself to learn some of the code needed to make Wolfram Demonstrations, and then went forth to do it. He first was fascinated with the idea of fractional derivatives, so he made a visualization of that. Then he wanted to illustrate the idea of the gradient and how the gradient of a 2D surface in 3D space sort of defined a plane tangent to the surface if you zoomed in enough. Finally, he created an applet where the user enters a 2D vector field, and then it calculates the divergence and curl at every point of the vector field. His description for what the divergence was was interesting, and new to me. About the point chosen on the applet, he drew a circle (and the vector field was illustrated in the background). He said “imagine you have a light sprinkling of sand on this whole x-y plane… and then wind started pushing it around — where the wind is represented by the vector field, so the direction and strength of the wind is determined by the vector field. If more sand is coming into the circle and leaving it, then the divergence is negative, if more sand is leaving the circle than coming into it, then the divergence is positive, and if equal amounts of sand are coming in and leaving the circle, then the divergence is zero.”

# Multivariable Calculus Final Projects 2013-2014

Instead of doing traditional problem sets and tests during the fourth quarter, I have kids work on an individual project on something that relates to multivariable calculus that interests them. (During the year, I have them keep track of interesting tidbits or facts or something I go off on a tangent about [pun] that they find could be a possible final project. I also have this list of ideas I’ve culled to help them come up with a topic.)

I have them come up with a prospectus and I individually talk with kids about their proposed project and timeline for completion. Then when they get started and start envisioning a final product, they are asked to write a description of the final product out clearly, and come up with a rubric for grading that product. They are also asked to make a 20-25 minute presentation to their classmates, their parents (if they choose to invite them), math teachers, and administrators. This year, they wanted to give their presentations during senior thesis week, which means that lots of their friends could come to their talks.

And they have been! In the past week, students have given their talks and I have been way impressed by them. Honestly they’ve been more independent than in years’s past, so I was unsure of whether they were putting together a solid final project or not. They did.

M.C.
Title: Mathematical Change We Can Believe In
Description: This presentation shows how one region can be manipulated to form something more interesting, a process called Transformation of Axes. The 2D and 3D analogues, use of rectangular and rounded shapes, and proofs of the properties of transformations abound in this exciting journey through the wonders of the world of multiple (MANY) variables.

B.W.
Title: Pursuit Curves: The Ultimate Game of Tag
Description: Pursuit curves are the paths formed when one point chases another point. In this program, we will be looking at the mathematical explanations of pursuit curves, and then using a computer program I have built to model a few.

J.B.
Title: What’s Our Vector, Victor
Description: This will be an investigation into the history, origins, and evolution of vectors, their analysis, and notation.

I.E.
Title: Economists working with Models: Understanding the Utility Function
Description: Firstly, we will gain a foundational understanding of economics as a discipline. Secondly we will discuss the utility function and the questions which it raises.

C.D.
Title: From Chemistry to Calculus: a study of gas laws
Description: For my project I have constructed a “textbook” that analyzes the idael and real gas law through the lens of multivariable calculus. In my “textbook” I compare and constrast these two laws by means of graphical and derivative analysis.

E.F.
Title: Knot Theory
Description: Knots are everywhere around us, from how we tie our shoes to how the proteins in our body wind themselves up. My presentation will give an overview of their place not only in the “real world,” but also the world of classroom math and calculus.

# Green’s Theorem and Polygons

Two nights ago, I assigned my multivariable calculus class a problem from our textbook (Anton, Section 15.4, Problem 38). Even though I’ve stopped using Anton for my non-AP Calculus class, I have found that Anton does a good job treating the multivariable calculus material. I think the problems are quite nice.

Anyway, the problem was in the section on Green’s theorem, and stated:

(a) Let $C$ be the line segment from a point $(a,b)$ to a point $(c,d).$ Show that:

$\int_C -y\text{ }dx+x\text{ }dy=ad-bc$

(b) Use the result in part (a) to show that the area $A$ of a triangle with successive vertices $(x_1,y_1),\text{ }(x_2,y_2),$ and $(x_3,y_3)$ going counterclockwise is:

$A=\frac{1}{2}[(x_1y_2-x_2y_1)+(x_2y_3-x_3y_2)+(x_3y_1-x_1y_3)]$

(c) Find a formula for the area of a polygon with successive vertices $(x_1,y_1),\text{ }(x_2,y_2),...,(x_n,y_n)$ going counter-clockwise.

Today we started talking about our solutions. We all were fine with part (a). But part (b) was the exciting part, because of the variation in approaches. We had five different ways we were able to get the area of the triangle.

• There was the expected way, which one student got using part (a). This was the way the book intended the students to solve the problem — and I checked using the solution manual to confirm this. What was awesome was that even though we as a class understood the algebra behind this answer, a student still asked for a conceptualgeometric understanding of what the heck that line integral really meant. I knew the answer, but I left it as an exercise for the class to think about. So we’re not done with this problem.

• There was a way where a student made a drawing of an arbitrary triangle and then used three line integrals of the form $\int_C y\text{ }dx$ to solve it. In essence, this student was taking the area of a large trapezoid (calculated by using a line integral) and subtracting out the area of two smaller trapezoids (again calculated by using line integrals). Another student astutely pointed out that even though we had an arbitrary triangle, the way we set up the integral was based on the way we drew the triangle — and to be general, we’d have to draw all possibilities. You don’t need to understand precisely what this means — because I know I”m not being clear. The point is, we had a short discussion about what would need to be done to actually have a rigorous proof.

• There was a way where a student translated the triangle so that the three vertices weren’t $(x_1,y_1),\text{ }(x_2,y_2),\text{ }(x_3,y_3)$ anymore… but instead $(0,0),\text{ }(x_2-x_1,y_2-y_1),\text{ }(x_3-x_1,y_3-y_1)$. Then he used something we proved earlier, that the area of a triangle defined by the origin and two points would involve a simple determinant (divided by 2). And when he did this, he got the right answer.

• Another two students drew the triangle, put it in a rectangle, and then calculated the area of the triangle by breaking up the rectangle into pieces and subtracting out all parts of the rectangle that weren’t in the triangle. A simple geometric method.

• My solution involved noticing that $\frac{1}{2}(ad-bc)$ is the area of a triangle with vertices $(0,0),\text{ }(a,b),\text{ }(c,d)$. And so I constructed a solution where a triangle is the sum of the areas of two larger triangles, but then with subtracting out another triangle.

The point of this isn’t to share with you the solutions themselves, or how to solve the problem. The point is to say: I really liked this problem because it generated so many different approaches. We ended up spending pretty much the whole period discussing it and it’s varied forms (when I had only planned 10 or 15 minutes for it). I liked how these kids made a connection between a previous problem we had solved (#28) and used that to undergird their conceptual understanding. I loved how these approaches gave rise to some awesome questions — including “what the heck is the physical interpretation of that line integral in part (a)?” In fact, at the end of class, we were drawing on paper, tearing areas apart, trying to make sense of that line integral. All because a student suggested that’s what we do. (Again, I have made sense of it… but I wanted the kids to go through the sense making process themselves… their weekend work is to understanding the meaning of this line integral.)

I don’t know the real point of posting this — except that I wanted to archive this unexpectedly rich problem. Because it’s not that it is algebraically intensive (though some approaches did get algebraically intensive). Rather, it’s because it is conceptually deep.