Block 4

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Block 4

Expect to read the entire gamut of classroom activity, from "big picture" overviews to detailed reflections on one day's work. Posts are authored by faculty.

We took learning outside the classroom.

We dug into the math behind the art of M. C. Escher.

Shear forces and failures helped us understand the physics behind a hovercraft.

One animating question for this class is whether we should—or even can—break free of the anthropocentric view of history.

We use games and codes to make the practice of solving problems incredibly fun.

We're learning to struggle with Shakespeare and fall in love with language.

We're growing as expository writers as we explore elegant mathematical topics.

We're expanding our repertoire of maker skills while developing as resourceful learners.

Our study of gravity ranges from slinkies to black holes to sandwich throwing.

Our study of language raises questions about culture, class, and identity.

Students in Literature 1 are discovering their ideas matter.

In Language Arts, we're exploring the question, "Who are you—what makes you you?"

We're developing our ability to turn ideas into reality by pursuing increasingly ambitious projects.

Our annual math burst gives students an extended experience with open-ended mathematical discovery.

We're exploring functions and laying the foundation for future study.

Students are turning ideas into reality through projects in Big History Science.

After a week of mathematical exploration, we turned our attention to communicating our ideas.

"In the classroom" posts.

"Out" of the classroom.

"Out" of the classroom.

Orienteering is a competitive sport involving off-trail running with a detailed topographical map and a compass. The sport is well-known and quite popular in the Scandinavian countries of Norway, Sweden, and Finland. Unfortunately, in the United States, orienteering is still very much a "fringe sport," but the Bay Area Orienteering Club (BAOC) is one of the largest clubs in the country, typically organizing two or three orienteering events a month.

I attended my first orienteering event in 1993 and was immediately hooked, since it combines both physical and mental challenges. In fact, orienteering is often called "The Thinking Sport" by its participants, and mathematicians, scientists, and computer programmers tend to be attracted to the sport for this reason. Well over half the members of BAOC have careers in a technical field of some sort.

"Real" orienteering takes place in the woods, on a map that typically looks like this:

Unfortunately, it will be challenging to ever take Proof School students to this type of terrain, since not only are all of our wooded mapped areas a considerable distance from downtown San Francisco, but they are also difficult or impossible to reach by public transportation. So the two orienteering events I have organized at Proof School this year have both been urban or semi-urban in nature, but I hope that they gave our students a taste of what the sport is all about.

The first orienteering activity I organized for Proof School was on a map of downtown San Francisco, a portion of which is shown below. Can you locate Proof School on the map?

About two-thirds of the middle school hiked or ran on this map during two different Flex Fridays in Block Two, and I took another small group out on the map last Tuesday, on the day of the BAMO contest. Students were expected to navigate to some or all of the sixteen control points on the map, in any order, and to answer a multiple choice question at each control, to prove that they were there. For example, control #13 is at the fortune cookie factory in Chinatown, and the question there was:

"According to the sign above the front door, when was the Golden Gate Fortune Cookie Factory established?" (Correct answer: 1962)

The student groups who were the most serious about completing the course quickly were able to find all sixteen controls in a little under two hours. Other groups stopped more often to sightsee along the way and skipped some control points, but all the students had fun, and became a little more familiar with the city in which they attend school.

During our most recent Build Week, I organized a second event on an orienteering map of Fort Mason, shown below:

This time, students had to negotiate the course in order of the control point numbers. At each control point, I had hung a pink streamer the day before, and finding the streamer indicated to the students that they had reached the correct location. I was impressed by how well students were able to read this map, as all of the student groups found all of the control points in under one hour, and very few navigational errors were made. Perhaps Proof School has some champion orienteers in its midst!

BAOC has also produced professional-quality orienteering maps of Golden Gate Park and the Presidio. I look forward to holding informal Proof School orienteering activities on these maps in the future, or perhaps even taking students to a weekend event organized by the Bay Area Orienteering Club.

*--Steve Gregg*

In the classroom

In the classroom

M. C. Escher’s images of impossible staircases, distorted reflections, and parallelograms metamorphosing into birds and fishes are well-known. Though Escher claimed not to be a mathematician, he kept his eyes open to sources of mathematical inspiration, from the exquisite symmetries of Islamic tilework and carvings in the Alhambra to a diagram of the Poincaré hyperbolic disc model in a contemporary geometry textbook. For one afternoon this past Build Week, we dug into the math behind Escher’s works and created some “Escherian” drawings of our own.

We began by trying to identify all the polyhedra in Escher’s “Stars”. The Platonic solids, familiar to middle schoolers from Maker Studio, are all there, which led to an impromptu lesson on how to draw a good-looking icosahedron step by step. Harder to spot—and to draw—are various truncated, stellated, and intersecting pairs of polyhedra; we had a debate about whether the cuboctahedron is a cube with the corners shaved off, or an octahedron subjected to the same treatment. (It’s both.)

Next, we examined “Sphere Spirals,” which gives a surprising answer to this question: If you set a course at a constant compass bearing, say 30° north of east, what path would you trace out around the globe? A due north course will take you straight to the North Pole along a meridian; a due east course will follow a parallel of latitude; but try any noncardinal direction, and you’ll spiral in ever-tightening coils as you approach the Pole. We considered questions such as: Will you circle the Pole finitely or infinitely many times? How far will you have traveled when you get there? To show this spiral path as a straight line, what kind of distortion must a world map engage in? What’s the shortest path between two points on a spherical Earth?

Many of Escher’s drawings are based on what he called “regular divisions of the plane”—a.k.a. tessellations. These begin with simple tilings, usually triangle- or parallelogram-based, which are then modified to become sheets of interlocking crabs or seahorses or lizards. We looked at “Regular Division of the Plane I,” which demystifies the process by showing it in 12 steps: beginning with a featureless fog in the first panel, Escher shows how to carve the grid, detach small indents and reattach them elsewhere, and finally transform the tiles into creatures. By viewing many of Escher’s tessellations, we observed that the underlying grid doesn’t fully determine the pattern; there are many possible systems of symmetries, known as the 17 wallpaper groups. Using a schematic chart, students identified which group was represented by each of Escher’s symmetry drawings. Along the way, many of us recognized a type of symmetry we’d never had a name for—glide reflection.

To create the illusion of infinity in a bounded space, Escher based some of his tessellations on hyperbolic geometry. Members of our group who took Sachi’s Aspects of Geometry course in Block 3 were able to explain to us how this works!

Finally, we explored the distortions Escher used in works such as “Balcony,” “Hand with Reflecting Sphere,” and “Print Gallery.” The last of these, depicting a museum with a framed hanging of a city that bursts through its frame to encompass the museum itself, is so deviously complex that Escher was forced to leave a blank spot at its center; it would be almost 50 years before a group of mathematicians would analyze Escher’s technique and figure out what belongs in that space.

A common technique for copying a drawing is to superimpose a grid on the original, then copy it out square by square onto a target grid. In a twist, I gave students grids with various distortions applied to them: one- and two-point perspective, a “peephole” bulge, spherical reflection, and so on. The students then chose their own drawings to copy to Escherian effect. Although we ran short of time in the end, the few finished sketches (including two Sigma Cats) were successful beyond my expectations, and I regret that I didn’t get to see the final result of one student’s attempt to scrutinize a certain presidential candidate in a fish-eye lens.

*--Austin Shapiro*

For our February build week, we encouraged students to propose project ideas. On the request of some students and inspired by discussion with others during the commute to school, we decided to build a hoverboard.

Building a hoverboard at school is a daunting task! There are lots of models out there already. We knew we weren’t going to compete with Hendo or adopt a design where the magnetic force would provide the required lift—the students discovered that would require a special track and embarrassing volumes of liquid nitrogen, or we would have to pave the Maker Studio floor with copper.

Instead we relied on the humble leaf blower, some plywood, and a plastic sheet; our secret weapons were duct tape and a heavy-duty stapler. I cut plywood into a four-foot-wide octagon with a hole cut out for a handle. Proofniks descended on it with sandpaper of ever finer grits, smoothing it to a glass-like finish. True to the spirit of reuse and recycle, our sanding blocks were fashioned from Kathy’s children’s hand-me-down wooden blocks!

Once the plywood base was smooth, the students cut out a 6’ square sheet of plastic, then use our stapler and Gorilla tape to make it airtight. Then came the delicate process of cutting quarter-sized holes so the air could escape through the bottom of the hoverboard when the leaf blower nozzle was inserted on the rider side of the board. Upon completion, the first hoverboard adventurers took to the seat—the hardwood floors of the Maker Studio!

Our pioneering navigators wanted to push the envelope and cranked up the leaf blower, and to their dismay, the central part of the plastic cover got ripped from its moorings. This led to a wonderful discussion about design, repair, and shear forces, and students were able to bring back the maimed ship to shore and ride it again. The second group learned from the first and made interesting design tweaks, and their ventures brought up new and interesting questions about the actual mechanism behind hovering. If the Bernoulli brothers had seen us, they would have been proud.

*--Kaushik Basu*

All year in Big History, we’ve been gradually zooming in on the time and place we call home. Starting out from the Big Bang, we’ve followed the action to galaxies and stars, to our own solar system, to Earth and its biosphere; now, we’re beginning the story of humanity’s ancestors. Is this a chauvinistic way of framing the history of the cosmos? Maybe not: after all, the universe as a whole is expanding and cooling, with energy becoming ever more dispersed. There truly is something special about those little pockets of space where complexity, from the range of exotic elements forged in supernovas to the variety of life itself, manages to thrive.

Then again, there could be billions of worlds like Earth. We started the year by testing and invalidating the geocentric view of cosmology. One of the ongoing, animating questions for this class is whether we should—or even can—break free of the anthropocentric view of history.

It was in this spirit that we started our unit on biology by questioning the very definition of life. Students began by classifying a set of objects—lichen, a refrigerator, a tornado, a seedless orange tree, a computer virus, a real virus—as living or nonliving. There was near consensus of the answers, but every attempt to codify the essential traits of life in a way that agreed with our intuition seemed to break down under the weight of edge cases and counterexamples. For example, what does a refrigerator do if not metabolize energy to maintain homeostasis? Or if reproduction and adaptation are the keys to life, what are we to make of the seedless orange tree? One member of the class observed that there were different ways a criterion for life could operate: it could be a necessary condition or a sufficient condition, and the real definition might require a combination of several criteria. Even so, we weren’t able to come up with any perfect demarcation.

Pressed as to why it matters to have a definition, students responded: How else would we be able to recognize another form of life if we found it? Or if we created it? And how else can we know when life begins and ends? I am not sure if the class was more disturbed or relieved to learn that biologists have hardly fared any better than themselves in answering these questions!

Undaunted, we pressed forward under a mutual agreement that life exists and that we want to know how it works. This part of the course was guided by Dobzhansky’s famous dictum that “nothing in biology makes sense except in the light of evolution”. We began by tracing the roots of Darwin’s big idea back to earlier scientific paradigm shifts, particularly the “uniformitarianism” of Lyell that had overturned geology in the preceding decades. Scientific breakthroughs are often portrayed as outbreaks of unique genius, so it was eye-opening for students to learn that some of the ideas behind evolution had been in the air for a while before Darwin managed to fit them into a coherent whole. We also saw some of the questions Darwin couldn’t answer in his own time, such as the mechanism of heredity. (Our own discussion of heredity—much aided by post-Darwin developments—brought to bear many examples students could relate to, from camellias to Manx cats to human blood type.)

For such a “simple” idea, evolution is notoriously prone to misunderstandings. To get a visceral sense of how evolution works, we played a game in which students played the role of animal species trying to survive in an environment buffeted by change. Card draws and rolls of the dice determined who died, who reproduced, and what mutations their offspring carried. While the game left plenty to be desired as a simulation of actual evolution, it nevertheless served to illustrate many surprising phenomena. Students saw traits favored by conditions early in the game become liabilities as conditions changed; saw the role of extinction in creating opportunities for adaptive radiation; saw the fragility of communities when biodiversity dropped, or when the rate of environmental disruption overwhelmed the pace of adaptation. Above all, the game served to make clear that adaptation is a matter of luck—the right chance mutation at the right time and place.

I wrote the evolution game in a weekend, and one of the most fun experiences of the Big History course to date has been watching the students find ways to improve on it. One student proposed a scientific naming system to track the species that were being born and killed off, using linear descent as the basis for classification. Another (responding to the uniformitarian idea) suggested a more realistic distribution of event magnitudes, with small environmental changes accumulating constantly and major catastrophes being rare. A third student noted that the game was, in one way at least, unfixable: to properly simulate evolution would require billions of organisms undergoing billions of events, something that groups of six students cannot pull off in half an hour. Not content merely to raise this objection, he did something about it, writing his own evolution simulator as a computer program that could take advantage of the rapid pace of CPU cycles.

To close our unit on biology, we touched on two more big questions. The first of these—what is a species?—was directly inspired by a student’s comment about the inadequacy of the common definition that a species is a population of individuals that can interbreed and produce fertile offspring. Where, the student asked, does this leave bacteria that reproduce by cloning themselves? Just as in our previous discussion of what distinguishes life from non-life, we found that no answer to this question is entirely satisfactory, and we must navigate with several criteria that produce similar answers… most of the time. The second big question was, how are human beings influencing the course of evolution on Earth? This question brings us full circle, and will follow us through the rest of the year as we try to understand what makes our own epoch truly unique.

*--Austin Shapiro*

Algebra 1 began with Mira in block 2; after a break for geometry in block 3, we're returning to algebra to hone our skills and tackle more complex problems.

While designing the curricula for the second part of Algebra 1, I looked at a broad range of textbooks and online math education resources. I found that there are many topics that are included in every Algebra I class, such as solving and simplifying equations in one variable, graphing lines, and exponents. Beyond these core topics, though, what constitutes “Algebra 1” is up to interpretation. Some courses include systems of linear equations, and some go right to quadratics after lines. Some introduce function notation, and others leave that to Algebra 2 or Precalculus. Mira started our Algebra 1 class off with a strong foundation in the the basics in block 2, but the path from that foundation forks in many interesting directions this block.

Algebra 1 is really about building a toolbox to tackle difficult problems with confidence and speed. Whether we get there by studying problems about multiple unknowns or by examining quadratics is less important than whether students can translate complex problems into algebraic equations to solve, and solve those equations skillfully. With that in mind, I try to balance teaching students efficient methods for solving problems with encouraging them to generate and pursue their own ideas for approaching problems—even if their ideas are not the most direct way to get to the answer.

Even so, Algebra 1 depends on practice! Understanding a concept is half the battle; getting to the point where it is easy, automatic, and free of mistakes takes a lot of repetition. The only way to get there is to do dozens of problems. But, there’s no reason that can’t be fun! We practice by solving problems to unlock secret codes: students tear through problems which give answers that range from 1 to 26, each encoding a letter that will later reveal a message. Or, we practice by reading fairy tales with silly characters who run into a lot more algebra than the average dragon in search of a princess. On other days, we solve Japanese puzzles like sudoku or shikaku puzzles with variables instead of numbers.

Students are doing a fantastic job mastering the skills they started learning in block 2, and as a class we are having a lot of fun along the way.

*--Sachi Hashimoto*

Nothing captures the artful evolution of the English language from the time of Shakespeare to our contemporary usage like the radical transformation of “artificial.” While it has gone on to acquire negative connotations (signifying contrived or false), in Shakespeare’s time, it meant something altogether different: artfulness, indicating craftsmanship. Its pejorative associations were once taken as laudatory.

I open with this shift in the way a word is used because it gets at the heart of the difficulty of reading Shakespeare. Words in his works had different meanings, and sometimes radically so. He wrote with poetic declaration, even in the midst of the most prosaic moment. And when it comes to his plays, he wrote for the stage, fully aware of the wild stunts patrons were capable of in the Globe Theater.

Our focus on Shakespeare this block in Language Arts 1, then, is a focus on translating “A Midsummer’s Night Dream.” We fully engage the various ways we can translate. Students are translating words and phrases, figuring out the panoply of thou’s and doth’s that dot his play, and rewriting key passages into modern English. But we’ll also do larger translations, turning the play into a fold-out book replete with images and graphics that document our smaller translations of the play’s language.

On that note, one student brought me a graphic novel version of the play, with the original text of Shakespeare. I found the graphic novel to be a great example of another kind of translation, so I bought copies for all the students. I also wanted to reward students for their effort to “struggle with it!” (our class motto this block, thanks to Kathy’s Maker Studio). So I arrived one Thursday morning on campus at 6am, gift wrapped each book, and wrote a personal note to each student. I then had “Shakespeare” write a letter with that day’s project, sealed them up in envelops with “Air Mail” stickers, and asked Dr. V. to do a mail call during our morning meeting. Sometimes, a kid just needs small surprises like these to work up the energy to “struggle with it!”

Wrapping books for each student and writing them personal notes.

There are, of course, many examples that make Shakespeare hard for contemporary readers—and worth struggling with it! Words have changed, syntax was different (and deliberately made more “artificial”), and even the orthography of writing behaved by different rules. For instance, in the original folios of Shakespeare, students would find f’s in place of many s’s, u and v virtually interchangeable as different ways to write the same letter, and no distinction between i and j. (This is why those letters look so similar to one another!)

Indeed, during Shakespeare’s time, the entire English language was under vast and fast evolution. With the advent of the printing press, publication became rampant, equivalent in effect to the boom we’ve seen with our age of the internet. There weren’t yet many standardized rules for spelling or writing. On a bigger scale, the English language was at the tail end of the Great Vowel Shift, which began in the 1350s.

All of this is to say that, during Shakespeare’s time, the English language was just beginning to emerge out of its past to form the beginnings of the modern era. This “renaissance” period, in fact, is better known to scholars as the Early Modern Period, which helps us see the time period’s connection, but in admitted infancy, to our modernity. All of these forces conspire to make reading Shakespeare much more like reading a foreign language, even if it has a much stronger and direct lineage to our current use of English.

To "struggle with it” with energy and smiles, we’ve broken up into teams once again. There is very little in the way of “teaching" going on; instead, I have given each team a set of puzzles and sheets for them to work on and then present to me, and once I see they have solved that particular puzzle, they get the next one. You wouldn’t believe the energy in the room when smart, motivated kids are given puzzles to solve and a very long runway.

It captures, in large part, what I mean when I say education is about opportunity. With kids like these, they need structure and goals, that’s for sure. But what they also need is the chance to shine. Each group has done incredible work. Most important, each student knows that he or she made the accomplishment. Students know they own what they’ve learned about the structure of Shakespeare’s plays, the structure of his phrases and sentences, and the use of his words.

In the end, struggling with Shakespeare is worth it, because he was a master of the artful use of language, of human nature, of his cultural and political era. But how do you animate a young middle schooler to see it's worth it? In our class, struggling with Shakespeare is worth it when students don’t feel they’re being directed by me but by their own internal compass. Struggling with Shakespeare is worth it when they see his language is a code that has a key—and it's a key they can help write. Struggling with Shakespeare is worth it when they see they are doing it together, when learning is made social and interactive.

*--Zachary Sifuentes*

In the Classroom

In the Classroom

Although not always apparent to students, the canon of topics introduced in a standard precalculus course (exponentials, logarithms, trigonometry, and complex numbers) leads in a variety of delightful directions; they do not exist solely as a set of functions with which to apply the theory of calculus. One of the main purposes of this course has been to enjoy and explore these elegant topics in their own right, prior to seeing them in other contexts. At the same time, my intent is for students to review and reinforce their knowledge of these topics on their way to using them in the service of calculus.

For example, we have discovered a plethora of ways in which the mathematical constant e arises. This value, approximately equal to 2.718281828, appears when considering problems related to compound interest, when playing a guessing game against our computers, even in as innocent a setting as figuring out how many random numbers it takes, on average, to obtain a sum greater than 1. Along the way we have reviewed laws of exponents and stumbled upon one of the most legendary relationships in all of mathematics relating e, pi, and i.

The centerpiece of this course, however, has been the lab reports that students complete every other week. My goal is to introduce students to the art and craft of writing a self-contained, extended discussion of a single main result. We have talked about how to organize a two to three page article, how to use LaTeX to typeset mathematical expressions, how to use proper notation to assemble a coherent proof, and how to incorporate figures and examples into a discussion to illuminate a result for the reader. It is an ambitious undertaking for my students, but they are rising to the challenge beautifully and already (as I peruse the second set of lab reports coming in) making significant strides in their ability to communicate effectively. I anticipate adding many of these polished lab reports to the corresponding student portfolios.

*-- Sam Vandervelde*

In the Classroom

In the Classroom

For the last two blocks of the year, we’re tackling four machine-enabled units: 3D printing, Arduino, machine sewing, and digital cutting. Students are rotating through these units in small groups, spending six full classes on each. These units are possible thanks to five families, who collectively donated our 3D printer, four Silhouette cutting machines, and ten sewing machines. Thank you so much for your generosity!

One main goal, as always, is to develop as makers. We’re honing our 2D and 3D digital design skills in ways that lead to tangible products, transforming fresh sheets of fabric into creations that range from whimsical to useful, and using electronics to make our creations responsive and interactive. We’re making things that we wouldn’t be able to make with just our bare hands, and we’re expanding our repertoire of skills that enable us to turn our ideas into reality.

Equally important are our disposition goals. With six classes devoted to each unit, students are able to dive more deeply into each unit than they have been in past blocks. To use that time meaningfully, they need stamina, focus, and creativity.

As part of our school’s broader goal of helping students build independence, we’re explicitly focused on developing as resourceful makers these two blocks--makers who draw upon a range of resources and strategies to tackle challenges. We’re learning by tinkering, experimenting, seeking help from classmates, and looking for answers online and in books.

We aim for students to benefit from teachers without relying on them--to be eager and able to learn new things regardless of whether an “expert” is guiding the way. I purposely reply to student questions by asking them, “What have you tried?” This prompts students to develop the habit of first thinking through possible solutions on their own. We explicitly encourage experimentation; failed experiments are now a routine part of our learning process, and “the next one will be better” is a common refrain. We half-jokingly use a “Struggle with it” motivational poster that we made in class last block--a reminder that struggle is a valuable part of the learning process.

Of course, Sachi and I don’t just leave kids to their own devices. We have deliberately chosen units that are simultaneously enticing and accessible. We have carefully pored through tutorials to provide a curated set of materials and resources. We also set up a structure within which students are able to learn, succeed, and be creative. We check in with students throughout class, and we’re available to help when students need a nudge in the right direction. We look at their creations with a constructive eye, simultaneously celebrating their successes and brainstorming with them about next steps.

There are undoubtedly moments of frustration. Sometimes software doesn’t download correctly; 3D prints topple halfway through printing; our edge-stitching feels far inferior to what’s pictured; or the tiniest error on circuitry or coding keeps our Arduino creations from working. But for every frustrating moment, there’s eventually a small moment of triumph when that 3D print does work out, when our second edge-stitched project looks far better than our first, or as sometimes is necessary, we cut our losses and tackle a new project with fresh energy.

Over the past four weeks, students have designed a line of 3D-printed Proof gear, including the keychains that we gave out at the admissions event on March 12. One student is on the cusp of finishing a 3D-printed mechanical hand. Placemats, tote bags, pencil cases, and even a stuffed duck have materialized among the machine sewing group. Several students took it upon themselves to design their own sewing patterns! In the Arduino group, students have produced an Etch-a-Sketch, a drawing arm, a walking contraption, and more. These days, students in the digital cutting group are making stickers, designing T-shirts, and preparing to make intricate pop-up cards.

It has been such a pleasure working with our middle schoolers this year: making together, helping them grow, and admiring their creativity.

*-- Kathy Lin*

In the Classroom

In the Classroom

The Physics class has been busy this block investigating gravitation, oscillations, and waves. While we can't bring black holes into the lab, we have reveled with fascination at the recent detection of the gravitational signature of colliding black holes in the LIGO experiment. The students have been doing the calculation of orbiting binary systems similar to these and have had their first introduction to the idea of a field. They have found the far-reaching conclusions from Newton’s universal law of gravitation and how it explains the astronomical observations made by Kepler. Celestial motion has become a natural extension to applying their ideas about energy and angular momentum conservation.

Among the really interesting results of gravitation is a terrestrial puzzle which the students have discovered in a problem set. If a sandwich is dropped through a hole drilled through the center of earth, it oscillates and arrives back at its starting point well before you’re hungry again, in a little more than an hour. It turns out that this is true no matter which two points you drill a hole through. There have been some rumors about trying to reduce commute time times by digging straight tunnels between Post Street and student residences. If you notice any unusual activity in your backyard, don't be alarmed: it's for the benefit of science!

As we were studying oscillations as projections of circular motion, it was wonderful to have observations that linked our study of the sandwich to the circular motion of a sandwich thrown with such speed that it begins to orbit the earth. Again, it was a voyage of discovery when students found the thrown sandwich takes the same time to go around the earth, except now the meeting was not quite as gentle. Though we have understood the mathematics, a physical connection drawn between the projections of circular motion to this problem is still an interesting work in progress.

We have studied how oscillations are ubiquitous, and why small angle approximations enable physical systems to be mathematically tractable. This has naturally led to our study of waves, and this is the first major mathematical abstraction that the students are working with, as we deal with wave functions which vary both in space and time. We have been exploring this with motions of a really long slinky, and building demos of mechanical wave machines, and we will wind up the block with an analysis of musical instruments and sound.

*-- Kaushik Basu*

In the Classroom

In the Classroom

*Caesar salutem dicit Ciceroni*—Caesar says greetings to Cicero. This oddly third person construction is characteristic of the way Ancient Romans would begin a letter—the equivalent of “Dear so-and-so.” In Latin class this block, students have been learning some of the particular historical valences and conventions of Latin words (such as *dico *here - to say or speak) and some of the ways in which language relates the members of a society to one another—not only through communication but through social hierarchies and distinctions of class and identity.

Students studied Roman letter-writing conventions by reading excerpts from real letters and noticing patterns in syntax and word choice. We talked about the differences from letter-writing etiquette today, and then students wrote their own letters in Latin using the grammar and vocabulary they’ve learned this year. One student wrote a fictional love letter, while another wrote a letter to Donald Trump!

We continued our exploration of Roman history by reading a selection of primary and secondary sources about slavery and citizenship in the Roman world. Students were quick to make comparisons between this complex hierarchy of status and the very different ways in which class seems to work in our society today. We also read a short excerpt from Howard Zinn’s *A People’s History of the United States* to compare Roman slavery to slavery in the United States. In our readings, students encountered one particular Latin word—*officium—*which they had learned from their textbook meant “duty,” but which also had a very specific historical meaning: *officium* was the term for a set of legally enforceable duties owed by a freed slave to his former master, which he had to accept as a condition of his freedom. In this way, students learned the limitations of word-to-word translation as a way of understanding a foreign language; as in this case, a single word can carry a whole constellation of meaning contingent on its use in a particular historical moment.

Language can also be taken out of its original context and given a new context and constellation of meaning: we saw an example of this when we listened to Cat Stevens’ 1972 song, “O Caritas,” which is almost entirely in Latin! Students were given the lyrics but with a number of Latin words left blank, which they had to try to figure out from listening to the song. This activity helped students to become more attuned to the sounds of the Latin language and to remember that originally this language was spoken aloud. We also interpreted the song, close reading the lyrics and the musical accompaniment, and talked about how it might have related to the historical moment in which it was written, including the Vietnam war.

*-- Sydney Cochran*

In the Classroom

In the Classroom

These days, casual observers of Proof School’s high school literature class might be underwhelmed by the sight of students tucked into various corners of the Maker Studio, working away on their computers. These casual observers are likely to see me talking with students for varying lengths of time: five minutes with one student, 20 minutes with another, 10 minutes with a third.

When the gong rings, signaling the end of class and the beginning of lunch, here’s what those same casual observers might see: students tucked away in those same corners, still at work on their computers.

So what’s going on here?

The answer to that question is deceptively simple and deceptively complicated. On the one hand, students are writing their papers. I like giving them class time to work and run into the inevitable roadblocks. I like being there for them at the moment they need it.

On the other hand, what they are doing is quite complicated: they are using test cases to examine a theory put forth by a contemporary scholar. These scholars are experts in their fields and on the topics we are investigating as a class. Those topics include the ethics of representing victims of the Holocaust, the verbal representation of visual art, the role of beauty in defining justice, and a strategy known as “antimorphosis” that artists and writers use to know the unknowable.

It’s a sophisticated kind of essay, one that I helped to develop and integrate into the Harvard College Writing Program’s regular pedagogy for all first-year students. It involves saying something back to that scholar, by highlighting an unwritten assumption that makes the theory valid or by articulating a necessary condition for the theory that the scholar does not make explicit.

This is what I mean when I say that the answer is deceptively complicated. What might look like students quietly working alone is actually the sight of students actively engaging a community of scholars and a community of ideas. It’s also deceptively complicated because their ability to engage these scholars and ideas is predicated on the run-up to Block 4, including the weeks in Block 1 and Block 2 we spent actively debating each other and discussing ideas; the weeks we spent in Block 2 and Block 3 working in shorter sprints on writing during class; and the weeks students spent in Block 3 planning how they would take advantage of the wide and long runway I’ve set up for them.

In other words, casual observers of the course in Block 4 would miss the proverbial 90% of the iceberg under the surface.

We now have finished our last week of the block. This past week, on Tuesday, students spent the entire class period reading and commenting on each other’s essays. We had 4 micro-discussions going in small groups, with some of the best insights about writing I’ve heard in 15 years of teaching writing. They were talking about structure and signposts, questions and argument, and evidence and analysis.

That’s why, at the end of class, students often don’t get up to go to lunch. They continue to work on their essays because they are invested in what they are doing and what their peers are doing. They are invested in the community of ideas. As one student put it this week, it’s because she feels for the first time that she's working with an idea that matters. Or, as another student put it, it's the first time that *his* ideas matter.

*--Zachary Sifuentes*

In the Classroom

In the Classroom

This block in language arts class, students explored the question, “Who are you—what makes you you?” We read a series of literary texts about people in the process of becoming—constructing themselves into something new or being shaped by the people or circumstances around them. Students then thought about how they themselves are constructed as people, and they created self-narrative maps, in which they had to represent some aspect of who they are through a creative combination of words and images and then present that map to their classmates.

We started by reading Ovid’s “Pygmalion,” an ancient Roman story about a man who sculpts a woman out of ivory and then falls in love with his own construction. We followed this with George Bernard Shaw's 1913 play Pygmalion, a very different take on Ovid’s story, about a common flower girl who is transformed into a “lady” when a phonetics professor teaches her how to speak like one. Lastly, we watched clips of the musical film My Fair Lady, based on the Shaw play, and talked about film as a form, how it’s both similar to and different from theater. These texts are all about the socially constructed self, specifically with respect to class and to spoken language—a topic that tied directly to our discussions last block about race and gender and whether those things are socially constructed or inherent/biological.

For the past two weeks, students have had to take what they’ve learned about “construction of the self” in literary characters, through our analytical discussions and written assignments, and apply those ideas to a more creative project of self-reflection. They worked independently to brainstorm, create their map, reflect on it, and plan an oral presentation for the class. We started by discussing how when we refer to "who we are," we usually talk about things around us (the negative space)—what we do, what things we like, what we notice, the places where we spend our time. We even did a visual activity that artists use, in which we drew a shape and then manipulated the paper in a few ways in order to perceive the negative space around our drawn shape as having its own shape, and one that’s complementary to the positive shape. Students spent a weekend journaling every day but only talking about things outside of themselves (things they saw, heard, touched, smelled); when they reflected analytically on these journal entries, they saw patterns in what they noticed, and we discussed how the world around us (the negative space) is in many ways constructed by us through our unique perspective.

With the only constraint being to use both words and images to represent some aspect of who they are, students came up with some wonderfully creative map pieces! One student’s project was made of miniature wire sculptures hanging from string, which each showed something she enjoys or finds beautiful, and these were interspersed with narration on hanging slips of paper. Another student used the school’s 3D printer to map his “Peaks of Pique” (his words!), tracking his curiosity about various topics at different moments in his life and how this was influenced by where in the world he was living at the time. Students shared their maps with classmates through oral presentations, which allowed them to practice their public speaking skills—and to get to know their classmates a little better!

*--Sydney Cochran*

In the classroom

In the classroom

During Block Four, middle school students spent time working on projects in all of their morning classes. In both Introductory and Intermediate Python, this two-week project period occurred at the very start of the block. I decided to keep things simple and gave students free rein to choose a coding project they would like to work on during that time. My only request was that students design a project that is modular in nature, so that if they were not able to complete the full project in the allotted time, the part of the project that they were able to complete would still be interesting in its own right.

I admit to having concern that students would have difficulty designing projects appropriate to their current ability level, but for the most part these fears were unfounded, as the students demonstrated their ability to choose suitable projects and to write interesting, creative code. In Introductory Python, one very nice project was a Morse Code translation program. This was not just a simple 1-1 correspondence between the 26 letters of the alphabet and their Morse Code equivalents, but also included standard Morse Code abbreviations, and the ideas of "Prosigns" and "Q Codes", which I had never heard of before. One learns something new every day at Proof School!

Students in the Intermediate Python class possess a greater amount of coding experience, so perhaps not surprisingly, they were more likely to design overly ambitious projects. I received a fairly large number of partially completed games, which nonetheless contained a great deal of nicely written code. One such example was a Pacman game. The full project proposal was to recreate the arcade game exactly, but five hours of coding time was not nearly enough time to allow that to happen. However, the student was able to code up the playing field, complete with dots and walls, and had Pacman move around the screen, eating the dots, and not being allowed to move through the walls. The final product is shown below:

Overall, I thought this two-week project period was class time well spent. The students appreciated having complete freedom to choose what they wanted to work on, and got the opportunity to design a "mini-project". This experience should serve the students well as they start to design their larger projects for Block Five, regardless of whether or not those Block Five projects are related to computer science.

*--Steve Gregg*

In the Classroom

In the Classroom

During the mornings of the build week following block four, our students took part in an unprecedented activity—their first ever Proof School Math Burst. We divided the entire student body into seven groups, typically six students per group, and presented each with a different starting point. One group played a particular combinatorial game, another considered ways to optimally color the edges of a graph while satisfying a certain constraint, while a third wrote code to investigate an unexpectedly rich question involving iterating a process on piles of markers. My group began the week by counting how many ways there are to place up-arrows or right-arrows in some of the squares of a 2x2 grid so that no arrow points at another. (In case you’re wondering, the answer is 31.)

Each group was guided by a faculty advisor, but the teacher's role was much more about facilitating and encouraging progress rather than dictating the direction in which the exploration unfolded. At one point I gave all my students a particular question to answer as a group, suspecting that it would help to open the door to further headway. On another occasion I moderated a brainstorming session regarding what we might look at next, without offering any particular ideas myself. Towards the end of the week I joined one student who was interested in proving one case of a conjecture the group had formulated. In an effort to be supportive I followed this student’s lead, even though he set the computation up in a different manner than I would have, and to my surprise we proved the entire general theorem!

We had several purposes in mind by giving students eight hours of supervised but open-ended time to pursue a specific topic or problem. Certainly a primary goal was to allow students to participate in mathematical discovery: exploring new topics, asking questions, making conjectures, learning tools, and finding explanations. It’s an exciting process that I’ve been eager to share with our students ever since the very early stages of planning for Proof School.

But there are many other benefits to setting aside time for a math burst. For starters, students develop stamina—the ability to stick with a problem for an extended period of time. I’m also sure that these investigations will form the basis for future papers, science fair projects, and so forth. In the more immediate future, we will spend the week following Spring Break learning the art of presenting results to a general audience, whether via poster or oral presentation. Of all the skills we plan to impart to our students, the ability to effectively communicate complex ideas ranks up near the top.

At the end of that week, parents and classmates are invited to attend our first ever Math Mini-Symposium, to hear all about what our students discovered during this Math Burst. Regardless, it has been a remarkable week. Speaking for myself, I’m still playing around with some of the mathematics our group explored. I believe that it was an energizing and rewarding experience for all of our students.

*-- Sam Vandervelde*

In the classroom

In the classroom

The gap between algebra and calculus can be difficult to size up. Consider the common name for the bridge between the two subjects: “pre-calculus” sounds less like a subject in its own right and more like a basic training to prepare students for the rigors that lie ahead. Yet, precisely because pre-calculus is not a proper subject, it is also often a time to explore bits and pieces of topics that might otherwise go by the wayside. This vagueness of purpose is a challenge to teachers as it is to students; one sees pre-calculus classes that lurch from trigonometry to matrices to probability.

Thanks to the block schedule at Proof School, we’re able to take a more granular approach. Exponentials, Logs, and Trig might be the most straightforwardly descriptive title of any course I’ve taught: this was a class to get students comfortable with the major functions that form the background not only for calculus, but for statistics, physics, and many other endeavors to come.

We began by exploring functions in general. Viewed from the wrong angle, a function can be a bewilderingly complex object, with a formula like f(t) = 2.6*sin(2π(t–0.22))+12.2. (That’s an actual function we would use later in an astronomical application. Can you guess what it models? Ask a student!) Yet most of the functions we encounter repeat a few basic themes; the theme of this function is “wave”. We learned to decode the language of such formulas in terms of graphs and their geometric transformations, allowing students to recognize the basic forms and then parse out the influence of each element in the formula. In this part of the course, students heard echoes of graphing conics with Dr. V in Algebra II.

Next, we took up exponents and logs. Students sometimes experience logarithms as a formality, whereas, to me, they are a means of passage between two mental number lines: one based on addition and differences, the other on multiplication and ratios. Students have both mental number lines already—the second is implicit in geometric sequences, including the familiar progression of tens, hundreds, thousands (and eventually millions, billions, trillions). Exponents let us jump from the additive to the multiplicative number line; logarithms take us on the return trip. To make sure this point would not be lost on the class, I introduced the number lines before the functions themselves, challenging students to slot in the numbers 1 through 10 on a multiplicative scale (hint: the intervals between them decrease, and the distance from 1 to 2 is the same as the distance from 5 to 10). One student observed that if we just knew the positions of all the primes, we could locate all the rational numbers; others exclaimed over the remarkable coincidence that 2^10 is about the same as 10^3. Through one principle—“equal distances on the multiplicative number line represent equal ratios”—students were able to gradually discover and justify all the major properties of logarithms.

A gift to the school of eight multi-scale slide rules proved to be a perfect fit to this approach. I provided no direct instruction on the use of the slide rule, but merely indicated what could be done and let students find the algorithms. As their enthusiasm for this project showed, it is one thing to have a mental model of the two number lines, and another to be able to manipulate that model with one’s hands.

To celebrate the acquisition of these vital ideas, we closed our unit on exponents and logs with a Festival of Applications. Students got to choose among three mini-investigations on population growth, hyperinflation, and the musical scale. Some did all three! They presented to their peers what they had learned, then we embarked on a second round with a “detective work” theme: one group used carbon dating to decide if a painting had been forged; another studied the origins of a surprising pattern in the first digits of data, which has been used to expose financial fraud; and a third group gleefully played the role of coroner, using Newton’s Law of Cooling to pinpoint the time at which one Trollface B. Hedgehog shuffled off his mortal coil.

The remainder of the course was devoted to trigonometry. Continuing the theme of conversion between scales, we started by exploring the relationship between angles and slopes. For instance, if we have two wedges that each have slope 1/2 on a flat surface, what slope do we get by stacking them? The answer is not 1 as one might expect. Evidently slopes do not add as angles do. By working out a series of progressively more general examples, students discovered the double-angle identity for the tangent function—all before the word “tangent” (let alone “sine” or “cosine”) had been spoken in the classroom. Members of the class recognized some very clever uses of this identity, too; by computing that three slopes of 1/2 “add up” to a slope of 11/2, while four slopes of 1/2 “add up” to a negative slope, one student was able to deduce that these wedges must have an angle of between 22.5 and 30 degrees. She subsequently refined the lower bound to 25.7 degrees, using an invented method that could in principle determine the angle to any desired precision. It is thrilling to realize that the arctangent function is not something that only a calculator “knows” how to calculate!

Having whetted their appetites, the class dug into the full set of trig functions (well, not the full set: there are innumerable arcana like the versine and the sagitta that I managed to avoid mentioning, but my restraint didn’t stop one student from trying to work out the properties of the hyperbolic sine and cosine on his own). Though we began with right triangles, our discussion of “slope addition” made it inevitable that questions about angles bigger than 90 degrees would soon arise, leading us to the unit circle and periodic functions. Not wanting to force too many trigonometric rules on students during their first exposure to the subject, but knowing they would uncover some in the course of solving problems, I gave them a blank sheet on which to record these “trig relationships”. Many of the class filled this sheet with gleanings from their own work, from other students’ presentations, and from my own less discreet comments; a couple even requested a second page.

Scientific calculators, provided as a practical tool, turned out to be a tool of discovery as well. I had a nice conversation with one student who had noticed that sin(cos(tan(sin(cos(tan(sin(

sin(sin(cos(cos(cos(tan(tan(tan(x))))))))))))))) always seemed to be the same mysterious number, about 0.017. This sequence of key-presses was likely inspired by a moment of boredom, but it turned out to be as good a first brush with radians as any. (Reader, can you identify the reason for the mysterious 0.017?) Graphing technology was even more useful for gathering inspiration. When I asked the class if they could find a formula to replicate a graph of a “big wave made of little waves”, I wondered if I was asking too much; I never imagined that several students would independently figure out an exact match, down to the numerical parameters I had chosen, and one kid would throw in a summation for good measure to create a genuine fractal sinewave. Of course, the path to a solution did not run straight for most, and we ended up having an impromptu show and tell to discuss some of the weird graphs the class had inadvertently created.

Although most of the class have a few more stepping stones to cross before they are ready for calculus, we did manage to touch on what calculus is all about, through problems investigating the limit of sin(θ)/θ at the origin and the average value of sin(θ) over a quarter-period. Given a hint or two, a few students were able to intuit the area under an arch of the sinewave. We may not have the tools to prove such things yet, but a child’s reach should exceed their grasp!

With so many new ideas in this class, and very little lecture between bouts of exploration and problem-solving, it would have been easy for somebody to fall behind. Indeed, not everybody progressed at the same pace on a given day. Thus, after each segment of the class was “over”, we continued to reinforce the topics from that unit through sporadic Ten-Minute Workouts. Students were able to self-select either a regular or advanced problem on the topic of the day. (These being Proof School kids, many attempted to do both.) I added this component to the course in an ad-hoc way and didn’t have time for it every day, but after seeing the results and hearing positive reactions from students at both ends of the quiz score scale, I’m encouraged to continue using it in the future... and confident that the class’s newfound knowledge of functions will last.

*--Austin Shapiro*

In the Classroom

In the Classroom

This block, Big History Science students worked on independent projects of their own choice, with the option of tying their work into what we have learned in class so far. The variety of ideas that came from the students was astounding, ranging from making a planetary model with gears, to a gravity well, to a card game based on the elements in the periodic table, to a rail gun, to instant ice!

The students went through a process of refining the scope of their projects and weighing in practical considerations before diving into their projects. They researched methods and materials and came up with novel modifications. Some of the initial proposals, like producing graphene, sent us along a journey to assess whether such a project could be scaled down to the classroom. When we realized we couldn't produce graphene in the classroom, that group of students was flexible enough to try something new and set out on a very interesting project, but this time, with materials in hand.

Another group worked on building a Gauss Cannon and discovered the many design parameters on which a successful launch depended. They experimented with wedging magnets, tried configurations with stacking magnets when the chrome coating on the disc-shaped magnets ruptured on impact, and came up with wonderful solutions.

Yet another group fabricated a gravity well from wrapping stretchable fabric around an embroidery hoop. This is an exhibit that is typically seen at science museums, but the extension to the idea was to put the whole assembly on a rotating carousel, so it more closely resembled what happens to planets as they orbit the sun. "They don’t just fall in!," as one student exclaimed.

The project activities brought students together in unique ways. They got to work closely with each other, consulting, researching, and learning how to proceed in spite of disagreements. We look forward to doing more such projects in the future!

*--Kaushik Basu*

In the Classroom

In the Classroom

This past Friday marked the first ever Math Burst Symposium at Proof School. Over the course of two hours we were transported to a realm of combinatorial game strategies, patterns among Fibonacci numbers, a Ramsey theory question about graphs, and much more. A lively poster session punctuated pairs of talks and the time flew by. As one parent remarked, this didn’t feel like a secondary school science fair so much as a conference for first-year graduate students!

The Math Burst Symposium was the culmination of two solid weeks of mathematical exploration, discovery, and preparation by Proof Schoolers. Following a very successful Math Burst, students used each of the afternoons during the past week to prepare slides or posters that would communicate their findings from the week before. Our math faculty worked closely with students to hear practice talks, offer feedback on poster design, and share tips on how to get complicated ideas across to a general audience.

As one of the faculty members who worked with the students preparing talks, I can confidently report that they made enormous strides in their ability to craft a fifteen minute presentation. On average each team of students gave two practice talks; with each successive delivery the quality of the talk improved dramatically. I’ve worked with college students for years in the art of developing effective oral presentations. It is not an exaggeration to say that many of our students are operating at a comparable level: in the sophistication of the topics they are presenting, the clarity of their explanations, and the quality of their slides.

I’m not hesitant to acknowledge that our talks and posters were not uniformly polished. It always takes a great deal of time and effort (not to mention a willingness to heed advice) for a presentation to truly sparkle. We guided the kids, but did not tell them exactly what to say or write. We also did not just showcase the all-star presentations; every student was featured at the symposium. The take-home message is this: when a poster or talk was great, the team of students responsible for it knew that they had accomplished something excellent. They owned it, from the original math they had thought to explore all the way to the creative means they found to share it. We, their teachers, are even more proud of them than the guests who were present, because we know for a fact that they deserve virtually all the credit.

Math Burst Symposium 2016 was an exhilarating day. We are looking forward to continuing and building on this tradition for years to come. Thanks once again to all the parents who found a way to attend on a Friday morning—you made this event truly memorable.

*-- Sam Vandervelde*