Mathematics Courses


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Mathematics Courses


 

Math topics are taught block by block, and include five major fields of mathematics.

 

Block 1: Problem Solving + Combinatorics 
We begin the year with an introduction to classic techniques in mathematics such as induction, proof by contradiction, and invariants. More advanced classes cover graph theory, Pólya counting, and partitions.

Problem Solving 1
This course is designed for students with minimal mathematical problem solving background. Students will learn foundational methods such as formulating a simpler question, finding a unifying pattern, or working backwards. They will hypothesize, confirm or refute conjectures and develop strategies for making progress on problems when the best approach is not clear. Particular emphasis will be placed on working through a broad selection of carefully chosen problems.

Problem Solving II
This class will be an exploration of problem solving techniques, with an emphasis on proof writing. We will practice working through the complete arc of solving a math problem, articulating our reasons for believing that our solution is correct, and transforming our intuition into a rigorous proof. The particular problems that we work with will come from a variety of mathematical subject areas, including game theory, logic, and topology.

Enumerative Combinatorics
This course digs deeper into counting techniques, with a twin emphasis on the power of algebra and the methods of combinatorial proof. Students will explore the binomial and multinomial coefficients, path-counting, recursion and induction, special sequences, generating functions, probability distributions, combinatorial identities, proof by bijection, and existence proofs. Students will discover and prove formulas for sequences describing a wide range of combinatorial objects.

Graph Algorithms
This course will begin with a brief review of basic graph theory, then focus on algorithmic aspects of the subject. Students will study spanning trees, dynamic programming, shortest paths, matchings, Eulerian circuits, and the Chinese postman problem. Further topics would include max flow/min cut, linear programming and linear optimization. Implementing algorithms via programs and applying them to various types of networks will play an important role in the course.

Topics in Algebraic Combinatorics
In this course, students will add powerful tools from algebra to their repertoire of counting techniques. We will explore the many kinds of generating functions, including ordinary power series and exponential generating functions, which we will use to derive closed forms, identities, and asymptotics for a range of classic sequences. Along the way, we will learn about combinatorial objects such as integer partitions, set partitions, and trees. The second half of the course will cover group actions and Pólya theory, a suite of methods for counting in problems with complex symmetries. In addition, each student will complete a research project examining an application of generating functions.


Block 2: Algebra + Number Systems 
Our students investigate a wide range of topics, from polynomials, inequalities, and complex numbers to matrices and symmetry groups. There's a lot more to algebra than just factoring quadratics.

Algebra 1
This course provides a solid grounding in one of the most essential and ubiquitous tools used in mathematics: algebra. Students begin by understanding the concept of variable, then learn about manipulating algebraic expressions, solving equations, using ratio/percent/proportion, graphing lines, factoring to solve quadratic equations, and analyzing inequalities. The purpose of the course is to organically motivate the need for and utility of algebra, while providing adequate training in order to use algebra proficiently.

Algebra 1 Lab
In this course, we will explore applications that rely only on skills learned in Algebra 1. We will solve geometry problems using equations of lines, fit curves to data, use piecewise-linear functions to compare insurance and cell phone plans, and use linear programming to find optimal production schedules. There will also be purely mathematical investigations, such as using visual means to divide polynomials and finding connections between Pascal’s triangle and n-dimensional cubes and simplexes. In addition we will reinforce algebra skills in areas of quadratic equations, graphing, and simplifying expressions with fractions.

Algebra 2
This course continues the study of algebra, a subject on which nearly every other branch of mathematics depends. After a brief review of introductory concepts, students will study quadratic equations and their graphs, quadratic inequalities, functions, the coordinate plane, polynomials, exponents and logarithms, complex numbers, and sequences. Together with the Algebra 1, this class provides a complete introduction to the subject which should serve students well in their subsequent coursework.

Number Systems
This course is primarily an introduction to specific number systems used throughout mathematics, and secondarily an introduction to axiom systems and the general methods of algebra. Course coverage will include sets, operations, closure, cardinality, field axioms, rational numbers, real numbers and their constructions, algebraic extensions of the rationals, complex numbers, ring axioms, extensions of the integers, Euclidean algorithm and factorization theorems, integers modulo n, the Two-Squares Theorem, finite fields, and polynomials over finite fields.

Ideals and Varieties
This course will introduce big ideas in algebraic geometry, studying how geometric objects can be described using algebraic equations. Students will study fields, affine space, algebraic varieties, ideals, and division algorithms for polynomials. The course will culminate in a study of the Hilbert Basis theorem and Groebner bases. Emphasis will be placed on computational fluency and geometric intuition. Students will also engage in an independent project on a specific area in algebraic geometry such as projective space, elliptic curves, or conics.


Block 3: Geometry + Topology 
Proof School provides students the opportunity to discover the real depth and elegance of geometry, whether Euclidean, projective, or spherical geometry. During this block we also offer electives such as topology and differential calculus.

Fundamentals of Geometry
This course introduces students to standard concepts in geometry. Topics will include parallel lines, angles, triangles, circles, parallelograms and other common shapes. Students will study metric properties of geometric objects such as length, perimeter, area, angle measure and volume. Finally, the course will cover the fundamental notions of similarity, proportion and right triangles.

Elements of Geometry
This course will present the art of proof in the context of introductory Euclidean geometry. Students will develop their skill at mathematical writing as they study the topics of angles, parallel and perpendicular lines, triangle congruence, similar triangles, parallelograms, other common quadrilaterals, circles, geometric inequalities and constructions. Emphasis will be placed on sound arguments and clearly written proofs.

Advanced Euclidean Geometry
Beginning with a careful development of the relationship between circles, arcs and angle measure, students will explore classic results in Euclidean geometry including cyclic quadrilaterals, the Simson line, concurrence, the Euler line, and the nine-point circle. Students will work collaboratively to document all class material, which will serve as a resource and reference during the course. Emphasis will be placed on discovery and rigorous mathematical writing. We will implement GeoGebra software for classroom exploration and diagram creation.

Aspects of Geometry
This course will explore many facets of geometry that lie beyond high school Euclidean geometry. The aim is to expose students to the many different aspects of geometry, in order to give them a sense of the broad scope of the field. Topics may include constructions, projective geometry, homogeneous coordinates, hyperbolic geometry, tilings, and transformations groups.


Block 4: Analysis + Statistics 
This familiar sequence of courses begins with elementary functions (trigonometric, exponential, and logarithmic), continues through calculus of one or more variables, and extends into the realm of differential equations or real analysis. We also bring our Algebra 1 class to conclusion.

Algebra 1b
This course provides a continuation of and conclusion to Algebra 1a. Students will cover topics including the Cartesian plane, linear inequalities, systems of two equations in two variables, and ratios/percent/proportion. The course will culminate with a first look at quadratic equations. The purpose of the course is to organically motivate the need for and utility of algebra, while providing adequate training in order to use algebra proficiently.

Exponentials, Logs, and Trig
This course begins with a foundation in function notation, coordinate transformations, and families of functions. Students will next obtain a solid understanding of the laws of and relationship between exponential functions and logarithmic functions. Along the way they will apply these enormously useful functions to understand a variety of natural phenomena, such as population growth and compound interest. The bulk of the course will be devoted to a comprehensive study of trigonometry, including right triangle trigonometry, the Law of Sines and Law of Cosines, the unit circle, graphs of trigonometric functions, and applications to periodic motion.

Mathematical Laboratory
There are a number of marvelous mathematical results that require a knowledge of precalculus topics before they become accessible to students. The purpose of this course is to delve into a wide variety of such topics, which rely on and in turn reinforce tools from trigonometry, exponentials, logarithms, and complex numbers. Our labs will explore the plethora of contexts in which the mathematical constant e arises in probability, the manner in which quaternions provide a means of composing rotations in three-space, and Marden's Theorem, to name a few.

Real and p-adic Numbers
This course introduces students to the concept of metric spaces and related elementary techniques of analysis, using the examples of real and p-adic numbers as completions of the rationals under different norms. The course begins with a rigorous construction of the real numbers as equivalence classes of Cauchy sequences of the rationals. We will then be able to prove such basic calculus results as the Intermediate Value Theorem and the Extreme Value Theorem, whose proofs are almost always omitted from calculus courses. Finally, we will construct and explore a different family of number systems called the p-adic numbers. Throughout the course, students will build skills in working with the epsilon-delta definitions of limits and continuity, which are fundamental to any further study of analysis. 


Block 5: Number Theory + Computer Science 
Either of these disciplines could occupy a lifetime of study; we let students go as far as they are able to in the subject of their choice while making sure that all students have a solid foundation in both. Electives in both subjects may make an occasional appearance in earlier blocks to provide additional coverage.

Intro to Number Theory
Mathematicians have long been fascinated by the unexpectedly rich theory and difficult questions that arise from the interplay between the seemingly innocent operations of addition and multiplication among integers. In this first course we will cover the classical canon of topics, including divisibility, divisibility tests for small integers, counting divisors, simple proofs involving divisors, perfect numbers, number bases, primes (including factorization and the infinitude of the primes), GCD and LCM, the Euclidean algorithm, Pythagorean triples, and irrational numbers. We will conclude with a week on modular arithmetic and congruences.
 

Multiplicative Number Theory
Gauss refers to his result on quadratic reciprocity as the Aureum Theorema, or the “Golden Theorem.” It is one of the most elegant results in all of number theory and serves as a pinnacle for this intermediate level course. To reach this apex, we will review modular arithmetic and linear congruences, then study Fermat's Little Theorem, Wilson's Theorem, RSA encryption, the Chinese Remainder Theorem, unique factorization of primes, primes as the sum of two squares, the Euler totient function, the Mobius function, Mobius inversion, primitive roots, and Legendre symbols, before finally reaching quadratic reciprocity.


Discrete Probability Distributions
This course begins with major concepts in probability theory including conditional probability, independence, and Bayes' Theorem. Students will learn to work with discrete random variables and their probability distributions, including common types such as uniform, binomial, geometric, and Poisson. We will find creative applications of expected value, discover concentration phenomena such as the Law of Large Numbers, and explore classic problems in probability such as gambler's ruin, the coupon-collector problem, and the German tank problem. The last part of the course will be dedicated to the study of information theory, with applications to guessing games and data compression.
 

Data Structures and Java
One of the essential tools for any programmer is a working knowledge of the various classical data structures, along with an understanding of which types of data are best suited for each means of organization. This course will provide a thorough introduction to the most commonly used structures, including stacks, queues, linked lists, binary trees, hash tables, heaps, and graphs, as time and the experience of the students in the class allows. In order to expose students to one of the most important and widely used coding languages currently employed, this course will be taught in Java.

 

 

Computer Science Courses


Computer Science Courses


 

 

Introduction to Python
Students use Python for their introduction to the world of programming. This course covers basics such as variables, input/output, graphics, branching, loops, strings, functions, lists, and arrays. Throughout the year an emphasis is placed on using programs to answer compelling questions, create entertaining games, and display original artwork.


Intermediate Python
This second course in Python review some of the topics presented in the introductory course but in greater depth and with more sophisticated applications, and also introduces a number of new topics. This course covers object-oriented programming, recursion, two-dimensional arrays, sets and dictionaries, searching and sorting, and first-class functions. Students learn to use Tkinter to produce graphical user interfaces, which are used to visually illustrate many of the concepts of the course.


Data Structures
This course introduces students to a wide variety of techniques for organizing, sorting, and generally working with various types of data. We cover stacks, queues, lists, sequences, trees, dictionaries, hash tables, and sorting algorithms. Along the way we consider issues of running time, efficiency, and suitability of different structures for various types of data.


Graphics Studio
In the first part of this course, students use their knowledge of Python to produce artwork and graphical games, using turtle graphics, Tkinter, VPython, and Pygame. In the second part of the course, students learn the basics of web programming, using HTML in conjunction with JavaScript to create interactive web pages. Finally, as time permits, we explore some development environments for creating cell phone apps.


Each year, we anticipate developing new computer science electives ranging from App Design to Robotics to Machine Learning.

 

Science Courses


Science Courses


 

We study and practice science in dedicated classes as well as in classes that explore particular scientific principles, methodologies, and materials.


The Scientific Method
This middle school course is a deep dive into the practice of the scientific method and how it varies between fields, such as physical, synthetic, biological, and social sciences. We’ll compare approaches to evidence and experiment in different contexts to develop an understanding of what we know and don’t know, how we know it, how well we know it, and how to assess how well we know. Specific questions to address can be determined by student interests. Block themes will likely include Measurements and Errors, Thinking and Learning, Biochemicals, Evolution, and The Anthropocene Age. Along the way, we’ll read excerpts from the primary scientific literature, and discuss the philosophy and ethics of experimentation. 


Physics
Students begin the high school science sequence with physics, in a manner tailored for kids who love math. Topics covered include motion in a gravitational field, momentum, energy, collisions, electricity and magnetism, fluid dynamics, waves and periodic phenomena, and models of the atom. Laboratory work motivates and accompanies the various main topics.
 


Chemistry
The second course in our high school science sequence builds on physical principles to understand the properties of materials. We introduce topics in historical order, studying original experiments, data and writing, to get a feel for how the science developed. We also challenge students to use their math skills, going deeper into topics like atomic structure and kinetics. In the lab portion of the class, students practice designing, conducting and improving lab procedures, and interpreting data. We also address the thorny issue of chemical safety in lab and life: how do you decide how scared to be of a certain chemical, and how do you apply this in practice? Major topics include: reactivity patterns, stoichiometry, thermodynamics, atomic and molecular structure, equilibrium, kinetics and electrochemistry. 


Biology
Our standard high school science sequence concludes with the study of how principles from physics and chemistry inform our understanding of life, particularly at the cellular level. Students consider topics such as molecular biology, genetics, the role and structure of DNA, and physiology. Laboratory work features prominently both as a means of illustrating concepts and providing an opportunity to further develop lab techniques and practices. Expected to be offered in the 2018-2019 school year.


Electives in Science
Advanced students may choose to take electives and engage directed research projects in areas determined jointly by student interest and faculty expertise.

 

History Courses


History Courses


 

We teach histories big and small, and integrate historical thinking and reading across the curriculum. 

Our history curriculum involves year-long instruction in the oral presentation of research, and emphasize skills in researching, listening, and speaking.


Ancient Athens
This middle school history course follows the multi-faceted story of the rise, flourishing, and fall of Classical Athens. What factors gave rise to this astonishingly fertile society? What were its achievements? Why did it decline? We especially consider how the legacy of Classical Athens continues to shape our world today. In each block we delve into a specific aspect of Classical Athens: the Olympics, the birth of democracy, theater, philosophy, and architecture. We read selections (in translation) from the works of Homer, Herodotus, Thucydides, Pausanius, Plutarch, Plato, Aristotle, Euripides, and Aristophanes. Through the words of these great thinkers, we gain an understanding and appreciation of different approaches to the study of “history.” Beyond content, the course emphasizes developing transferable skills, especially with regard to reading, comprehending, and analyzing a variety of texts and carrying out different kinds of writing assignments. We foster skills such as researching, speaking, and performing in front of others. We work collaboratively as a class, while at the same time challenging each student to meet his or her individual potential.

 


World History: Agency and Change
In this 9th grade course on world history, we study four troubling chapters of inequality and prejudice at different points in time across the globe: the transatlantic slave trade, the Holocaust, colonialism and independence in India, and apartheid in South Africa. Through these case studies, students gain an understanding of how systems of oppression are constructed, but they also learn how these systems can be dismantled by the strength of the human spirit. The course, then, focuses on the possibilities for positive change in the world. Along the way, students piece together the fragments of history—original documents, images, and objects—and respond to different historians’ interpretations of the past. Our goal is to help students grow as empathetic, global citizens who understand the role human agency can play in worldwide events.  
 


US History: Designing Memorials
This high school course studies US history through the lens of days, sculptures, and sites of remembrance. Students build toward presenting memorial designs of their own at the end of the year. To learn both design principles and the history of the United States, we learn to read memorials as “texts” that tell us explicit and implicit messages about their subjects. Readings include contrasting selections from books like A People’s History of the United States and A Patriot’s History of the United States; scholarship like Memorial Mania and Poetry After Auschwitz; newspaper and magazine articles on the controversies behind recently opened sites (such as the Martin Luther King, Jr. memorial on the National Mall); and primary source materials like the Declaration of Independence, the Gettysburg Address, and Letter from Birmingham Jail. Students write a design proposal modeled on an open call for a design competition, and present their findings and designs in a year-end symposium. This course is expected to be offered in the 2017-2018 academic year.


Electives and Directed Research
Advanced students may choose to take electives and engage directed research projects in areas determined jointly by student interest and faculty expertise.

 

Literature Courses


Literature Courses


 

Language Arts 1: Building Things with Books
The long-term project of this class asks students to design, build, and outfit the Proof School Library. We visit local libraries to map their floor plans, patterns of use, and systems of organization, and we conduct research into the history of libraries. Students work extensively with the basic building block of any library: the book. We learn classical bookmaking skills, redesign books, and create interactive versions of our favorite literature, all of which activate primary analytical functions like interpretation and explanation; along the way, we develop foundational writing skills that focus on thesis, structure, evidence, and complex and compound sentences. We develop a maker mentality in the class by turning books into sculptures and using books to make spaces. This inquiry-based class discovers its reading list each year as students learn interviewing skills, and talk with peers, faculty, and family about their favorite books. The class features regular, small-group conferences and workshops in reading and writing. Writing projects vary in length and purpose, and include book reviews, letters to the editor, instruction manuals, design proposals, and new chapters to books we love.


Language Arts 2 & 3
After Literature 1, the curriculum cycles between two courses: “Staging Literature” and “Codes, Ciphers, and Keys.” Both courses reflect our goals for developing more sophisticated skills in analytical reading and writing, in creative making, and in oral communication.

> Staging Literature
Students in this class write, direct, and stage a play. We visit with local theater companies, consult with actors and directors, and make short videos explaining the nuances of character, plot, and structure. Projects include modernizing Shakespeare, adapting literature for the stage, and learning how to tell stories nonverbally through the use of gesture, music, and lighting. These acts of translation require students to read like never before, and activate critical thinking and writing skills. Readings span time and genres, and include A Midsummer’s Night Dream, The Diary of a Young Girl, A Wrinkle in Time, and selections from poets likes Robert Frost and Rita Dove. The class features regular, small-group workshops in critical communications skills, from oral presentation to analytical writing. Writing projects include reviews, director memos that defend dramatic choices, playbills, and character analyses.

> Codes, Ciphers, and Keys
This course asks students to decrypt literature in various ways, from breaking down foundational elements like plot and character to piecing together clues embedded in narration and language. We design codes for reading and writing, research the history of encryption, and create maps of books. Along the way, we learn that poems, abstract paintings, and even social customs are themselves “codes” that can be written and solved. These projects position students to think proactively about the mechanisms embedded in literature, art, and social interactions. Readings include selections from Sherlock Holmes, The Giver, and The Book Thief; writing projects include interpretations of artistic codes, a research report on cryptography, and expository essays on maps and graphs. Extended small-group workshops develop key skills in reading, writing, and collaboration.


Literature 1: Literature and Its Limits
This seminar-based high school course reads literature that responds to historical events, especially in times of war. We ask whether literature can adequately represent the unfathomable, with readings culled from the likes of Pablo Neruda, Ernest Hemingway, Homer, and Shakespeare. In addition, we look at art and music as primary sources, with supplemental materials coming from independent research into the questions raised by our class discussions. Field trips to local memorials and to the Presidio help us see that our understanding of the past is more complicated than we might think. Students come away understanding not only how depictions of war have changed, but also how the writing of history has changed. Expect regular short writing prompts, one-on-one and small-group conferences, class-wide writing workshops, and directed research. Major writing assignments include making arguments based on close readings, and testing competing theories of art, history, and “Just War” with new evidence.


Literature 2: Close Reading in the Digital Humanities
Proof School’s introduction to digital humanities surveys the tools, methods, and information technologies that fields like literature and history have started to use in research. Digital humanities is a broad practice that includes database inquiry, language frequency analysis, and data visualization. Our goal in this class is to leverage information technology to see literature in new light; projects include visualizing linguistic trends in poems from the likes of Emily Dickinson and T.S. Eliot; geo-mapping complex narratives like The Adventures of Huckleberry Finn and 100 Years of Solitude; and building an interactive poetry kiosk. Writing projects build toward a research-based argument, and include response papers, exploratory analyses, and annotated bibliographies. The class features peer reviews and regular one-on-one conferences to develop individual projects and individual skills.   


Literature 3: Distant Reading in the Digital Humanities
This course moves students toward distant reading practices in the digital humanities. Distant reading involves processing information about very large literary databases rather than analyzing a specific novel; data mining such large numbers of books tends to yield evidence from which students can draw conclusions about shifts in culture and genre. We use Project Gutenberg as our database. Our readings for discussion center on key scholarship in the digital humanities, such as Moretti’s Distant Reading, Jockers' Macroanalysis, and Cohen and Gibbs’ “A Conversation with Data,” which analyzes 1,681,161 titles in Victorian Literature. Writing projects involve explaining methodology and interpreting results, reviewing existing literature on a topic of the student's choosing, and making a research-based argument. The class features regular one-on-one conferences and peer workshops to develop each student's project and multi-modal communication skills. Students give a formal presentation of their research at the end of the year. Expected to be offered in the 2017-18 school year.


Literature 4: Directed Research in Literature
This class functions as a directed research collective that investigates an annual theme established by the faculty. Students are expected to interpret the theme according to individual interests; conduct independent surveys of the relevant literature; narrow the scope of their research; write a research-based argument; and give a compelling talk on their findings. The year-long course focuses on key skills for success in college: independence, peer review, advanced research and writing methods, and the ability to engage in cross-disciplinary discussions. Expected to be offered during the 2018-2019 school year.

 

Art Courses


Art Courses


 

The arts help students make connections across the curriculum. 

 

Maker Studio
This course introduces students to the maker culture at Proof School, and investigates the basic building blocks that lead to more complex projects. Students explore open-mindedly with primary materials like blocks, magnets, strings, and straws; expect plenty of builder kits like Legos, Zoob, and Make-Do cardboard prototyping. We tap into the sciences by building projects involving paper circuits, marble runs, and homemade contraptions, but we also engage the fine arts by making musical instruments and origami. Students share their work throughout the year via pin-ups and galleries. At the end of each year, communication becomes a main focus as students design and stage exhibits for the year-end symposium. To get a sense of the possibilities, we visit various studios across the city, including the Exploratorium, TechShop, and the San Francisco Center for the Book.


Making Art with Language, Letters & Type
This class is for advanced high school students, and shifts our maker culture from building machines that function like art to using machines that make art. The course challenges students to see letters not only as representing things but as things in themselves. We do this by designing type, working with letterpress printing, and creating palimpsests. In the process, we challenge students to see language as a material that does not always follow our will or mean what we intend. Throughout the year, we run Proof School Press, designing and printing broadsides for Emory University’s renowned poetry reading series that features Pulitzer Prize winners, Nobel Laureates, and other internationally celebrated poets. Expected to be offered during the 2017-2018 school year.


Electives and Directed Research
Advanced students may choose to take electives and engage directed research projects in areas determined jointly by student interest and faculty expertise. 

 

Language Courses


Language Courses


 

Latin 1
Students learn the underlying framework of modern Indo-European languages, including English. This means that Latin helps our students understand more complex grammatical structures, increase their vocabulary, and deepen their reading comprehension. There are many wonderful works by which to understand Western Civilization, as well as math and science texts, written in Latin well into the 19th century.


Latin 2
Ancient Rome and its language, Latin, are long dead; and yet, we encounter its legacy many times in our everyday lives. Ancient Rome and Latin are vibrant and alive in our language, culture, political system, science, technology, and architecture. In Latin 2 we gain an understanding of all the myriad ways we are the benefactors of Ancient Rome. We continue to build our knowledge and understanding of Latin vocabulary, syntax, and forms. To increase ability and confidence, we regularly practice Latin grammar. We translate both long and short excerpts from great Roman authors, such as Vergil, Horace, Cicero, Livy, and Ovid, to gain an appreciation of Latin in the context of Ancient Rome. Further, these readings are springboards to discuss Roman culture, society, and politics. We also have a special focus on the history of rhetoric and oratory in Ancient Greece and Rome, studying such thinkers as Plato, Aristotle, Isocrates, Quintilian, and Cicero. We consider specifically how the ideas and models of these thinkers influence rhetoric and oratory today.