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In the classroom


Block 3

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In the classroom


Block 3

 

Block 3: A look inside our teaching and learning.

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.


Advanced Euclidean Geometry

We engage course material in a wide variety of classroom formats, giving texture to the learning experience.


Language Arts 1

We are reading literature slowly, deliberately, and deeply, while developing our social-emotional learning.


Maker Studio

Polyhedra, fractals, and tessellations are all a part of Maker Studio.


Literature 1

In Literature 1, students are writing their own prompts, syllabi, and goals. It’s how they are realizing their education is theirs.


Latin 1

Students are taking the lead on learning. 


Intermediate Python

Students are designing their own 2D array projects.


Language Arts 2

Students are mapping arguments, identifying assumptions, and learning to counter-argue.


Fundamentals of Geometry

Students explore the neighborhood while solving geometric problems. 


Introductory Python

Learning standard algorithms allows students to investigate the quirks of the English language.


Physics

We're learning science through experimentation. 


Aspects of Geometry

We expanded our definition of geometry to encompass the geometry of a sphere and of negatively curved surfaces.


Elements of Geometry

Students study Euclid and learn to question assumptions.


Block 2

"In the classroom" posts.

Block 1

"In the classroom" posts.

 

Advanced Euclidean Geometry


In the classroom

Advanced Euclidean Geometry


In the classroom

 

In just two weeks, students in our Advanced Euclidean Geometry course have covered almost everything mentioned in the course description. 

These topics include the Simson line, Euler line, Ceva's Theorem, a beautiful proof of Heron's formula for the area of a triangle, and more. We are off to a fast start, and class morale is high. It is delightful to have enough of a solid geometric background to be able to present and understand results that are at once mysterious, compelling, and elegant.


incenters.jpg

We have engaged with course material in a wide variety of classroom formats. Yesterday students tackled a set of questions individually before being randomly paired up as part of a review activity; the day before that they worked in larger groups to document material presented earlier in the week.

We are using Overleaf, an online collaborative LaTeX environment, as our platform for mathematical writing. We also use GeoGebra as our dynamic geometry software, which enables students to conduct explorations and create accurate diagrams on their computers. This variety of styles provides texture to the learning experience and helps to keep our classroom lively.

--Sam Vandervelde

 

Language Arts 1


In the classroom

Language Arts 1


In the classroom

 

In Language Arts 1, we are reading literature slowly, deliberately, and deeply. We're also reading to develop our social-emotional learning. 


This block, we are reading Lois Lowry's classic, Number the Stars. The book is set in German-occupied Denmark, and tells the story of one family's escape to safety, once their young daughter is given safe harbor from the Nazis. Our reading material marks a change for us in Language Arts 1, and is introducing students to seminar-style discussions of literature and to close reading skills.

While we have read short stories and non-fiction, this is our first novel. It's terrifically written, threading the needle of speaking to the horrors of the Holocaust without indulging gratuitously in violence or needlessly peddling fear to capture our emotions. It's honest and historically informed, and captures the larger story of a state-wide rescue of approximately 92% of Danish Jews in 1943. This is literature that gives us a ground-floor perspective on living in history, giving us an opportunity for empathetic reading while learning to read analytically.   

However well-researched and well-written, Number the Stars is still literature. There is plenty of understandable dissent from scholars, survivors, and advocates that the Holocaust shouldn't be introduced through a work of fiction. The Holocaust, after all, isn't fiction, however delicately told.

That is why our first days this block focused on history. We surveyed what students already knew about World War II before digging into its history, made alive by the US Holocaust Memorial Museum (USHMM). The USHMM has an abundance of teaching materials for every age, from very young children to adult scholars, with maps, videos, and artifacts. We also welcomed a professor from USF's Jewish Study and Social Justice Program, and who has worked with secondary school students as part of the "Beyond Bridges: Israel-Palestine" program, to help us better understand the history surrounding World War II.    

These early days of the block helped students understand that our book, as a work of fiction that gets at historical significance, is really a representation, and therefore we really do need to read it metaphorically. Knowing historical context helps us understand the book's arc, and it helps us read deeply and closely. To do that, we arrange the desks each day into a big seminar table, where we are better able to see one another while we respond to both the text and to each other's ideas. The seminar format, often based on the Harkness Table, is a development on the Socratic Method

To help students understand what close reading is, and why it matters, I asked students to take us to a scene they thought was particularly important to the book. One student offered up a scene in which two girls find a familiar shop closed, with a new padlock on the door and a sign posted in German. The passage, with its mention of locks and signs, was brimming with the best kind of metaphor: one that is both physical and symbolic. 

We read the passage aloud, and then spiraled back through the passage sentence by sentence, circling what seemed to be keywords. We defined them and discussed their literal significance, but we also wanted to think about their symbolic resonance. From there, we leveraged our Block 1 activity of making word clouds with synonyms to think more abstractly. We asked:

  • What else is "closed," symbolically to these girls in this book?
  • Why might the new padlock be a metaphor for how these girls experience the German occupation?
  • What do you think it means that these girls understood the sign was in German, but didn't know the words?

As we came up with ideas for each keyword, and related those keywords to the larger story of the book, we took time to write down our thoughts—in the book. This was surprising to many students, even more so when I invited them to write in ink. It's important that students are able to take ownership of their learning, and writing in the book, even in ink, is one way for them to know this book is theirs, that their ideas are worth noting down, and that they can track their thinking alongside their reading. These thoughts will form the basis of their writing assignment this block: a close reading of the book, which has its roots in the French explication de texte.

These methods of "reading in the margins" are really methods of engagement, ways that physically manifest what so many technologies are after these days: interactivity. But writing our ideas in the margins, filling the voids of a book with our questions and ideas, is also a nice reminder that a book isn't literature until we note it, process it, and think about it. It's a nice reminder that a book without a reader forever remains closed, that the story it tells would have no ears and no eyes to fall on. Without a reader, a book is simply an inert object. It needs us to open it, and it needs us to do more than allow its words to wash over us. It's especially true of a book like Number the Stars.

A work of literature like Lois Lowry's can leave a mark on us as readers, that much is true. But we, too, can leave our mark in it.

--Zachary Sifuentes

 

Maker Studio


In the classroom

Maker Studio


In the classroom

 

Polyhedra, fractals, and tessellations are all a part of Maker Studio.

The focus of this block in Maker Studio is mathematical making: polyhedra, fractals, tessellations, geometric designs, and more. Not surprisingly, such activities are particularly exciting with Proof School students. When I ask what it means for an octahedron to be stellated, multiple hands shoot up. When we work with compasses and straight edges, impromptu math investigations happen. 

We spent our first two weeks building the platonic solids in three ways. Physically manipulating and constructing these fundamental polyhedra helps us build intuition about their form, their patterns, and the remarkably simple constraints that define them.

We started with balloons. How many continuous balloons does it take to build a tetrahedron? A dodecahedron? Can you prove it? The solution is a cute application of graph theory.



Next, we built platonic solids out of file folders, drafting their nets ourselves. Imagine a cube, slit at some edges so that it unfolds into a continuous two-dimensional shape. How would we draw a shape that folds into an octahedron? An icosahedron? One ambitious sixth grader even managed to construct a cuboctahedron. 

Finally, we spent two classes working on modular origami: folding many identical pieces that come together to form an ornate polyhedron. On our second day of folding, we folded while watching Between the Folds--a fascinating documentary about the art and science of origami. As we dabble in a range of activities, it’s inspiring to be reminded of how deeply one can dive into a single art form.

This block’s activities mark a departure from past blocks. For the first time, precision really matters. Some projects require far more diligence and patience than we’ve needed before. We need to think carefully as we work, as we have a specific end goal in mind. Our middle schoolers have risen to the challenge beautifully. 

--Kathy Lin

 

 

Literature 1


In the classroom

Literature 1


In the classroom

 

In Literature 1, students are writing their own prompts, syllabi, and goals. It’s how they are realizing their education is theirs. 
 

In conferences and in back-to-school night, in information sessions last year and this, and in check-ins with parents and the school’s humanities advisory board, I’ve often engaged in conversations about opportunity. 

In our high school literature class, that opportunity comes in many forms: students have the opportunity to take a college-level writing class as freshmen in high school; they have the opportunity to engage in long-form discussions of literature and history; they have the opportunity to think critically, to read scholarship on issues like ethics, and to write papers that perform highly sophisticated intellectual tasks. 

But they also have the opportunity to learn past the meager confines of the content of a class. This block, for instance, students are writing their own syllabi, authoring their own writing prompts for their papers, and articulating their own goals. It’s a way for students to realize their education is theirs.  

It’s a risky move, having them write their own teaching and learning materials, but it’s also calculated for success. That’s why in block two I gave them the assignments, but not the timeline for getting it done. They worked in groups to calibrate the various writing assignments and estimate how long they’d need to get each one done; they debated whether they wanted homework over the week-long Thanksgiving Break; they rationalized their choices. It introduced students to the more sophisticated independence of blocks three and four.

This block, at the start of week two, I gave them the major writing assignment for block four along with an empty timeline with hashmarks on it. There were two due dates listed: one for the draft of the major assignment, and one for the revision. What they came up with was logical, pedagogical, and rather incredible:


One example of a syllabus that a team of students devised.

One example of a syllabus that a team of students devised.


From there, we talked about what they might need to write in block three that would be a meaningful stepping stone to their block four paper. That’s when I tasked them with filling in the timeline, writing their own writing prompt, and identifying their own goals.  Here’s what one student wrote:

  • Write a 1-3 page response paper by February 12th explaining one of the three theories we read over the past few weeks. The response paper should have two main sections. The first section should explain the theory in detail, making sure you understand the theory thoroughly. The second section will explain a possible test case, and show how the test case challenges the theory you have described or casts light on a new perspective. This response paper should be a stepping stone to our larger 6-8 page paper in Block 4. The goal is to be able to understand and articulate the theory, to propose a possible test case, and to understand and be able to explain how the test case complicates the “initial reading” of the theory. 


Collectively, the class demonstrated an innate ability to think metacognitively about the intellectual task of the block four paper. They showed they can—and want to—set very high expectations of themselves. They showed that, beyond being exceptional learners, they are eager for the opportunity to learn. 

--Zachary Sifuentes

 

Latin 1


Latin 1


 

In Latin class, students are taking the lead on their learning.

Across the curriculum, our students are given the opportunity to drive their own learning. In Latin class this block, for instance, they have invented new ways of reviewing old material, moderated class discussions and activities, and written questions, both interpretive and grammatical, about the passages of Latin literature and history that we've read.

The students began the block by working in pairs to help each other review the Latin language they learned the last two blocks. Each pair came up with an interactive activity to review a particular grammar concept and set of vocabulary, and then led the whole class in participating. Some groups created out-loud verb-conjugation games that required the class to work together and listen to one another, while others used a trivia-style question-and-answer format. One group even gave a vocabulary quiz that included fake words to test the students' mastery of Latin nouns.

More recently, as we've been practicing reading comprehension of longer passages of Latin, the students worked in small groups to lead a class discussion of one of these excerpts. And rather than presenting the passage (reading aloud, translating, and analyzing it themselves), they had to lead the class in doing these things by serving as moderators and teachers. As part of their preparation, they came up with three grammatical and three interpretive questions to ask their classmates. This meant not only understanding what the text said, but identifying which aspects of the grammar or syntax were the most complicated or interesting, and even more importantly, which aspects of the language might help us explore the text through close reading and interpretation.

Students often were able to combine their grammatical and interpretive questions, asking, for instance, about the effect of shifting verb tenses in a dedication by the Roman poet Catullus. Students also brought up important issues of cultural comparison when examining, for instance, an excerpt by the Roman historian Livy about Lucretia, a woman who became famous for her virtue—an even an impetus for political revolution—when she committed suicide after being raped by the Roman king. Here the students even made connections to their Literature class, discussing the different meanings attributed to death in different societies and time periods.

Next week we'll continue our exploration of Roman history and life by reading a few texts in translation, including a fascinating legal treatise from 5th century B.C.E. We'll focus particularly on the hierarchical structures of class in Ancient Rome and the conflicts that arose between the different classes. These issues will tie in well with the high schoolers' topic of poverty and homelessness during last block's build week and will allow them the opportunity to think about class conflicts in America today through historical comparison.

By leading and creating their own activities and discussions, the Latin students this block are practicing synthesizing, reorganizing, and reconceptualizing their knowledge, and are coming to appreciate the depth and complexity of this new language they're learning, relating to it as a sort of sphinx who asks questions of those who come seeking answers.

--Sydney Cochran

 

Intermediate Python


Intermediate Python


 

In Intermediate Python, students are designing their own 2D array projects.

We have explored three major topics in Intermediate Python this year: object-oriented programming, recursion, and 2D arrays. In this block, students have just completed a graphical 2D array project of their own design, in which they were able to further explore these fundamental ideas of computer science. Several students coded up Conway's "Game of Life":

GameOfLife.jpg

 

Others wrote code to display one-dimensional cellular automata, systematically studied by Stephen Wolfram in his book "A New Kind of Science":

We also explored several recursive 2D array algorithms as a class, including 2D array "flood fill," which is used in paint programs to fill in connected white areas of the screen:
 

AFTER

Before



We looked at several different ways to traverse a 2D array to solve a maze, in particular, "depth-first search," which makes recursive calls to explore as far as possible in a given direction before backtracking.  

 

Later in the block we will shift gears and work with Python dictionaries. We will use them to solve interesting questions such as "What word in the English language has the greatest number of anagrams which are also words?" Stay tuned for the answer!

--Steve Gregg

 

Language Arts 2


Language Arts 2


 

In Language Arts 2, students are mapping arguments, identifying assumptions, and learning to counter-argue.

 

In “Staging Literature” this block, students read the novel Passing, which grapples with important and interrelated social issues (race, gender, class). It does so by presenting the inherent complexity of these topics rather than providing “answers” about them. In discussion, then, students used the close reading skills they've been developing this year to consider what questions the text was raising and the extent to which the text did or did not answer these questions.

One issue we've discussed is race in the novel. Is it something biological like the color of your skin, or something socially constructed such as how you self-identify, how others perceive you, or what community you live in? We saw that each of the novel’s characters had a different conception of race, just as we ourselves had different ideas based on our own experience and understanding. The have gotten very good at referring directly to passages in the text to provide evidence for their ideas—a skill they worked on in blocks 1-2.

Beyond the questions we considered together, students practiced having discussions on their own, without teacher intervention. They even set “disposition goals” for themselves; some students indicated they wanted to speak up more while others wanted to listen more attentively to their classmates or pose questions to the group. We've used writing to reflect on how well they accomplished their goals. 

Becoming aware of their goals helped them on our next task: reading two pieces of literary criticism about the novel. This kind of non-fiction helped us better understand the challenges of Passing, but it also helped challenge their literacy skills. Literary criticism is a  different type of text, one that presents explicit answers to the novel’s questions in the form of an argument. Students mapped out each argument, tracing its movements paragraph by paragraph, and they discussed connections and similarities between the two apparently disparate critiques. In week six we’ll return to the novel—full of questions that our two critical essays answered in very different ways—and we’ll consider what allows there to be multiple arguments about a piece of literature.

For their final written assignment, the students will formulate their own arguments about the novel. More specifically, they’ll write a short essay in which they must argue against themselves. They will first summarize the argument of one of the critical essays, identify what assumptions that argument is predicated on, and articulate their instinctive reaction to the argument (agree or disagree); then they will identify what assumptions they themselves are making in their reaction; lastly, they will present an argument against their original ideas and then reflect on how their thoughts on the critic’s argument or on the novel have changed.

In addition to practicing their analytical writing skills in expressing their ideas clearly and persuasively with evidence, they will be working on identifying their own assumptions. In class we practiced identifying assumptions in various statements about the novel, and we also reflected philosophically on the consequences of not being aware of one’s own assumptions, in literature class and also in everyday life.

--Sydney Cochran

 

Fundamentals of Geometry


In the classroom

Fundamentals of Geometry


In the classroom

 

In Fundamentals of Geometry, students have found the slope of a steep San Francisco hill, the length of the labyrinth at Grace Cathedral, and the height of a lamppost from its shadow.

 

As a mathematician whose area of expertise is “probabilistic phenomena in high-dimensional convex geometry," I approached Fundamentals of Geometry with some trepidation. Would I be able to make angles and polygons and parallel lines interesting to my 6th and 7th and 8th graders? Would they struggle to understand what I’ve taken for granted for so long that I can’t remember learning it?

I needn’t have worried: put a rich enough puzzle in front of these kids, and they will make the discoveries all on their own. The hardest part for me is to stay out of the way! In this spirit, we’ve used paper strips and hinges to explore what makes a shape rigid or flexible; tiled tabletops with “pattern blocks” to find out what combinations of angles would work; and worked out trick billiards shots in theory before trying them in practice. (One of the younger members of the class turned out to be quite the latent pool shark.)

In one of the most popular activities, we walked around the Proof School neighborhood and solved problems as we went—finding the slope of a steep San Francisco hill, the length of the labyrinth at Grace Cathedral, the height of a lamppost from its shadow, and so forth. It has been delightful to rediscover all the “simple” geometry in our immediate environment right alongside my students.



Yet, for all my attempts to make geometry relevant, I’m even more pleased to see my students getting satisfaction from the conjectures they make and the theories they are building. When the class found out that side-side-angle wasn’t quite enough information to decide that two triangles are congruent—except when it is!—they competed to figure out the exact circumstances under which they could rescue this “invalid” congruence test. When we worked out the exact height that billboards would have to stand above the road to line up from a special vantage point, I held out the payoff of actually getting to build a model at the end from chopsticks and clothespins. The model turned out not to be very satisfying (it seems my choice of materials did not support precision), but no one seemed to mind, since the class felt intrinsic pleasure at getting the math right.

Along with making theorems (and even the occasional proof), we’ve also made up our own problems. The students deconstructed some textbook algebra problems (How many cows and chickens have X heads and Y legs?), figured out what makes them work as problems, and then wrote some in their own inimitable styles: “Some hydras get 2 new heads when they lose an old one; others get 3. Hercules cuts off all their heads twice more, then runs away at the sight of 213 heads…” “There are two types of dinosaurs: naustidactyls, which have 1-inch noses, and pteranodons, which have 20-inch noses…”

While the class sometimes has to buckle down to master basic skills, they’re also looking ahead with eagerness to bigger challenges ahead. Students have made observations that anticipate trigonometry, integer sequences and combinatorics. In trying to persuade others of their assertions about geometric figures, they are learning how to distinguish what looks true to the eye from what they know to be true. Even at its most fundamental level, geometry turns out to be about much more than angles and polygons and parallel lines.

--Austin Shapiro

 

Introduction to Python


Introduction to Python


 

Learning standard algorithms in Introduction to Python allows students to investigate the quirks of the English language.

We have two main goals in Introduction to Python, and sometimes it’s difficult to achieve these simultaneously. On the one hand, we want students to have fun, be creative, and enjoy the subject! Graphics-based programming is one way to reach this goal, and in my previous post for this class, I described some of the work we were doing with Python turtle graphics.

However, it is also important for introductory computer science students to learn some of the standard algorithms used to accomplish various tasks, and that was the emphasis of this class for the first part of block 3. In particular, given a list of integers, students learned how to write code to answer the following questions about the integers in the list:

  1. What is the arithmetic mean of all the integers?
  2. How many integers in the list are positive?
  3. What are the smallest and largest integers in the list?
  4. Are the integers in the list ordered from smallest to largest?
  5. Does the list contain duplicates? Or are all the integers in the list distinct?

Later in the block, to make this material more interesting and to allow for some creativity, students learned how to have their Python programs read information from an external text file. Students then wrote programs to read text files consisting of all the words in an English language dictionary, and answer various questions about those words. Some of these questions appear below. I wrote some of them, and others were created by the students. Answers (at least for the dictionary we used) appear immediately below if you'd like to try some of them!

Questions

  1. What is the longest word that is a palindrome?
  2. What is the longest word containing none of the letters 'a', 'e', 'i', 'o', or 'u'?
  3. What is the longest word whose letters are all in alphabetical order?
  4. In our dictionary, no word contains a double x. However, another letter appears doubled in exactly one word. What letter is that, and what is the word?
  5. What is the longest word consisting only of letters in the first half of the alphabet?

Answers

  1. "malayalam" (a language spoken in India)
  2. "symphysy" (the fusion of two bodies, or two parts of a body)
  3. "adelops” (a genus of beetles)
  4. Exactly one word contains a double q. It's "zaqqum", a tree referred to in the Quran.
  5. "hamamelidaceae" (a family of shrubs and trees)

--Steve Gregg

 

Physics


Physics


 

We're learning science through experimentation.

This block in Physics we have transitioned from one-dimensional motion to two, developing the tools for analyzing motion with vectors, and investigating how vertical motion can be described independently of horizontal motion. We are also taking the big step of combining ideas from energy and momentum with applications of Newton’s laws to solve complex problems.

In our labs, students have flown motorized model pigs (complete with flapping wings!) to model an iconic problem, the conical pendulum, to come to grips with centripetal force. In order for students not to identify this with a macroscopic force like contact, gravity, or friction, we are also banning some words, like centrifugal forces, which only exist in special reference frames. One of the central themes of this course is to transition to Newtonian ideas to model problems, and a big step in that direction is to attribute specific meaning to everyday words, or to unlearn their meaning, in some sense, so that they have a more precise physical context.

As we study rotational motion with strong analogs to translation, we are revisiting an old experiment with LEDs attached to carpet sections (and then flung across the room) to make a different connection this time, demonstrating the independence between rotation about the center of mass, and the translational motion of the center of mass itself.

Some of our classes have been spent entirely with students discussing small conceptual problems in groups, and then presenting their ideas. Our small lecture time is interspersed with clicker questions, which the students have to sometimes hazard a guess, thinking independently, and then have a group discussion surrounding anomalous answers. This method of peer instruction has been developed by the author of the text we have adopted, and is a central element in our class.

Being wrong about their physical intuition helps students conflate their physical reasoning with an accurate mathematical model, and it generates animated discussion! Our studies on oscillations ties these themes together at the end of the block, as we step into thermodynamics and kinetic theory.

--Kaushik Basu

 

Aspects of Geometry


Aspects of Geometry


 

We expanded our definition of geometry to encompass the geometry of a sphere and of negatively curved surfaces.

 

Our goal this block was to explore many facets of geometry not usually seen in high school or college classes, but which are important to many parts of modern mathematics. We started out by revisiting Euclidean geometry, but using the language of vectors. We saw that thinking about vectors instead of points and line segments could be a powerful new tool, which allowed us to succinctly state and prove theorems. Students tackled problem sets on vector geometry, solving and presenting all of the theorems themselves.

Next, we explored projective geometry.  We began with a reading by Jordan Ellenberg on how to win the lottery with geometry.  In it, Ellenberg describes using the Fano plane, a geometry with only seven points and seven lines, to win the lottery.  The Fano plane is a small example of a projective plane, which is a plane where two lines always meet at a point.  In other words, there are no parallel lines.  Students came up with their own example of a projective plane with thirteen points, and we spent some time making and proving conjectures about what these tiny geometries look like.



We saw that projective geometry is also the geometry of artists who want to paint pictures with realistic perspective.  Students tested out their understanding of projective geometry by making perspective drawings of tiled floors.  They also figured out how to measure the height of a person in a photograph.  We tested out this theory by taking a picture of one student, and seeing if we could accurately guess his height.  Our estimates centered around the right number, but we were off by a couple inches-- pretty good!

In our final unit, we used paper triangles, taped seven around every point, to make models of the hyperbolic plane. We drew straight lines and noticed that given a line and a point, we could find infinitely many other lines which go through the point, and don’t intersect the line, negating Euclid’s Parallel Postulate.  We discovered that the bigger our triangles were, the smaller their angle sum.  Students built other models of the hyperbolic plane out of squares, taped five to a point, and annuli.  By working with many physical models of the hyperbolic plane, we were able to develop intuition about what might be true in this seemingly paradoxical kind of geometry.

Over the course of the block, we experimented with new kinds of geometry, made art, proved beautiful theorems, and stretched our ideas of what geometry can encompass.

--Sachi Hashimoto

 

Elements of Geometry


Elements of Geometry


 

In Elements of Geometry, students study Euclid and learn to question assumptions.
 

For more than 2200 years, there was only one geometry textbook in the Western world: Euclid's Elements. It was second only to the Bible in the number of translations and editions published. From the Middle Ages until the early 20th century, the Elements was required reading for every educated person and was considered a model of logical reasoning and argument by mathematicians and non-mathematicians alike. For example, here is Abraham Lincoln's account of his student years:


In the course of my law reading, I constantly came upon the word ‘demonstrate.’ I thought, at first, that I understood its meaning, but soon became satisfied that I did not. ... I consulted all the dictionaries and books of reference I could find, but with no better results... At last I said,—Lincoln, you never can make a lawyer if you do not understand what ‘demonstrate’ means; and I left my situation in Springfield, went home to my father’s house, and stayed there till I could give any proposition in the six books of Euclid at sight. I then found out what ‘demonstrate’ means, and went back to my law studies.
— Abraham Lincoln

In our class, we did not get quite as far as Lincoln: we only made it three quarters of the way through Book 1 of the Elements, after which we switched to a more modern treatment. (At some point, the fact that Euclid did not have the modern concept of "number" becomes too much of a distraction from the geometric content of the book.) Still, it was thrilling to guide the students in an exploration of this ancient work that seems remarkably relevant and radical, even today. 

In American high schools, geometry is traditionally the course in which students learn to write proofs. This is Euclid's legacy: his great contribution in the Elements is the axiomatic method, which underlies all of mathematics as we know it today.  Starting from just five geometric "postulates" (such as "there exists a unique line segment through any two points") and some general "common notions" (such as "two quantities equal to the same quantity are equal to each other"), Euclid builds up the entire edifice of geometry, brick by logical brick, completely from scratch. It is an awe-inspiring demonstration of the power of reason to create something from virtually nothing. No wonder the book has been a bestseller for two millennia.

Our first few weeks of class were spent getting used to the axiomatic method, which meant learning to question our intuitions at every step. No matter how obvious something seemed, if it was not rigorously provable from the postulates, then we could not take it as true. While to a non-mathematician it may seem silly to keep insisting on proving the obvious, this is exactly the kind of mental discipline that ultimately gives mathematics its power. Abraham Lincoln understood its appeal, and after a couple of weeks, so did our Proofniks. Occasionally they outdid Euclid himself in noticing places where he takes things for granted that he should not. (Alas, even Euclid is not infallible; I had to defend him to the kids by reminding them of the many generations of giants on whose shoulders we are standing.)

Our study of Euclid culminated in Week 3 of the block with an in-depth look at his famous Fifth Postulate (also known as the Parallel Postulate). The fate of Euclid's Fifth Postulate is probably the most vivid demonstration of the power of the axiomatic method in the history of mathematics, and I cannot resist the opportunity to tell it here. Please bear with me: I promise that it does all get back to the students in the end!

Euclid's first four postulates are very straightforward statements that seem like obvious candidates for foundational geometric truths. The Fifth is different. At first reading, it is almost inscrutable. What it actually asserts is that under certain conditions, two lines are guaranteed to cross -- a statement that (once you figure out what it's saying) seems obviously true, but too complicated to be a postulate. It feels more like a theorem to be proved than like a foundational truth.  Euclid himself clearly felt uncomfortable with having to take it as given: he goes out of his way not to rely on the Fifth Postulate in his proofs for as long as he can. 

For the next two thousand years, generations of mathematicians devoted hundreds of pages to "proving" that the Fifth Postulate follows logically from the other four and therefore does not need to be stated as a postulate. None of these arguments were valid. It was not until the 19th century that mathematicans finally realized that the Fifth Postulate can never be proved, because, despite being "obvious", it is not necessarily true. The search for its proof had led several people to discover alternative (non-Euclidean) forms of geometry that are perfectly self-consistent, yet in which the Fifth Postulate is false. The first of these geometries to be described was hyperbolic geometry, a bizarre world in which everything is completely at odds with our intuition: straight lines can get arbitrarily close to each other without intersecting, the sum of the angles of a triangle can be as small as you want, rectangles do not exist, and the Pythagorean Theorem is false. Yet it was the discovery of hyperbolic geometry that ultimately paved the way for the mathematical framework underlying Einstein's general theory of relativity.  

Without the practice of suspending one's intuitive thinking as demanded by Euclid's axiomatic method, it would have been impossible for anyone to stumble upon this "strange new universe" (as one of the discoverers of hyperbolic geometry described it). We, in the twenty-first century, have gotten used to the fact that science can outstrip our intuition in describing reality; we throw around words like "quantum effect" and "black hole" without a second thought. Yet few of us get to experience first-hand the invigorating discomfort of having our fundamental intuitions and assumptions shattered by reason.

This was the experience I wanted the students to have at the end of our Euclid unit, when we spent a week teasing out the many "obvious" things that, in a world without the Fifth Postulate, are simply false. What I hope the students got out of this exercise is the sense that in mathematics, one's unexamined intuitions can be limiting and misleading. On the other hand, when one takes the time to think things through carefully, one's intuition can adapt and expand, to encompass a larger, more flexible, more interesting version of the universe.

(If all of this seems like a metaphor for social or moral progress, it is perhaps not a coincidence. This may be a good time to mention that Euclid's influence on Abraham Lincoln was not limited to legal arguments. Lincoln explicitly appeals to Euclid as a model in several anti-slavery debates, where he enjoins the audience to think through the logical implications of arguments and to question assumptions.)

But back to geometry. In the second half of the block, we returned to the familiar world of Euclidean geometry, where the Fifth Postulate is safely in place and one can more or less believe the pictures that one draws. Here, we discovered another payoff to the axiomatic method: increasingly, we found that it had the power to prove not only things that seemed intuitively obvious, but also things that seemed at first glance quite mysterious:

  • Why is it that, when you connect the midpoints of the adjacent sides of any quadrilateral, the result is always a parallelogram? 

  • Why is it that, when you draw lines connecting the three vertices of a triangle to the midpoints of the opposite sides, these three lines always intersect at a single point? 

  • Why is it that, if you choose a pair of opposite points on a circle and draw lines connecting them to a third point on the same circle, the two lines will always meet at a right angle? 

By the end of the block, we could answer all these questions, and others, with nothing more at our disposal than what we got from Euclid: five postulates, a few common notions, rigorous logical thinking, and an eagerness to question the obvious.

--Mira Bernstein