Note on Objectives
Course objectives are open-ended and well over a semester's worth of material is given. Enrichment School will tailor the pace of teaching to each cohort's needs.
Mathematics
Mathematics is known as the "unreasonably effective" subject because it's methodical thinking about abstract problems. Having a disciplined, mathematical mind unlocks an extreme range of scientific understanding. In addition to its practical applications, math is a pursuit with its own intrinsic value. Its stunning beauty elevates it to an art form that many devote their lives to.
Pre-Algebra MA 101
Postrequisites: MA 201 MA 202
Pre-algebra is the start of your mathematical journey at Enrichment School. You will learn the basics of mathematics that will support the rest of your academic journey.
Concepts and skills
- Review of natural number arithmetic
- Properties of operations (commutativity, distributivity)
- Notation of expressions, including operation precedence conventions and tree and Polish notation
- The integers
- The rational numbers, including operations and representations (decimal, ratio, percentage, Egyptian, mixed)
- Factorization, primes, GCD, LCM, Goldbach's conjecture, and Euclid's algorithm
- Exponents, roots, logarithms, real numbers, and laws thereof, number bases, and scientific notation
- Tuples (ordered pairs) and Cartesian coordinates
- Variables, equations, and the substitution principle
- Properties of equations and inequalities (reflexivity, symmetry, transitivity)
Textbook: Algebra, the Easy Way, by Douglas Downing
Elementary Algebra MA 201
Prerequisites: MA 101
Postrequisites: MA 301 MA 310 MA 350 MA 351
Algebra is a continuation of your mathematical journey. Here, you will start to think more with abstractions and use the most important and versatile tool in all of mathematics: functions.
Concepts and skills
- Review of variables, multiple variables
- Functions, function composition and operations, and graphing
- Polynomial equations, including solving quadratic and cubic equations
- Conic sections
- Rational functions
- Polar coordinates
- Finite and infinite series
- Proofs by induction
- Basic combinatorics
- Matrices and systems of equations
- Complex numbers
Textbook: Algebra, the Easy Way, by Douglas Downing
Geometry MA 202
Prerequisites: MA 101
Postrequisites: MA 301
Enrichment School's geometry course is heavily informed by the book "A Mathematician's Lament" and it emphasizes demonstration over calculation. The Socratic methods that Enrichment School employs complement the proof-centric nature of this subject very well.
Concepts and skills
- Geometric primitives (lines, segments, rays, shapes, angles)
- Area, volume, and measuring thereof
- Introduction to proofs
- Angles, measuring angles, and angle theorems
- Congruence and similarity
- Right triangles, the Pythagorean theorem, and trigonometry
- The unit circle
- Identities of trigonometric functions and their proofs
- Solid geometry: polyhedra, Euler characteristics, nets, great circles, volume and surface area
Textbook: Introduction to Geometry, by Richard Rusczyk
Calculus MA 301
Prerequisites: MA 201 MA 202
Postrequisites: MA 400 SCI 301 SCI 302
Differential equations are "the language of nature" and calculus is the bedrock they're built on. All useful science, if it doesn't explicitly require calculus, requires wielding a strong, intuitive understanding of its underlying concepts. Nothing could be farther from the truth than the belief that calculus is a rarefied pursuit of abstract nonsense in ivory towers.
Concepts and skills
- Review of functions
- Definition of a limit
- Definition of a derivative
- Rolle's and mean value theorem and proofs thereof
- Properties of derivatives
- Derivatives, graphs, and optimization
- Antiderivatives
- The fundamental theorem of calculus
- Natural logarithms, Euler's formula, and polar representations of complex numbers
- Integration methods
- Infinite series
- 3D vectors and dot and cross products
- Spherical and cylindrical coordinates
- Partial derivatives
- Multiple integrals
- Parametric equations and tangents, curvature, and torsion
- Line and surface integrals
- Divergence, curl, and associated theorems
Textbook: Calculus, the Easy Way, by Douglas Downing
Independent Mathematics Study MA 400
Prerequisites: MA 301
It is Enrichment School's most ambitious goal to have a single student that does well in this course. The intent of the course is to teach real, higher math. Content is tailored to each student, with a "light touch" approach.
Potential subjects:
- Linear algebra
- Fourier analysis
- Elementary number theory
- Algebra
- Real analysis
- Point-set topology
Potential Textbooks: A Book of Abstract Algebra, by Charles Pinter; Elements of Point Set Topology, by John Baum; Real Analysis (18.100A) in MIT OCW, by Casey Rodriguez
Statistics and Probability Theory MA 310
Prerequisites: MA 201
Postrequisites: SCI 302
Probability and statistics are simply quantitative reasoning under uncertainty. Uncertainty is simply a fact of life for any non-omniscient being, so applications of this course should be evident. Moreover, statistical methods are the foundation for machine learning, applications of which have seen some interest in recent years.
Concepts and skills
- Introduction to probability
- Elementary statistics: mode, median, mean, standard deviation
- Random variables: discrete, continuous, and categorical
- Venn diagrams, set operations, and Bayes' theorem
- Joint and conditional probabilities
- Probability distributions: normal, binomial, exponential, poisson, geometric, hypergeometric
- Sample versus population statistics
- Central Limit Theorem
- Z-scores
- Student's t-test
- Hypothesis testing: p values and confidence intervals
- Chi-squared tests
- Correlations and covariance
Textbook: The Practice of Statistics, by Daren Starnes and Josh Tabor
UNIX/Linux Usage MA 150
Postrequisites: MA 250
Linux and its predecessor UNIX are ubiquitous. If you're an Android user, you have Linux in your phone. It is the de facto operating system for all tech companies. This text was written on a machine running Linux and served on a computer running Linux. Only personal computing has proven impervious to Linux. Therefore, to operate computers in industry you must be familiar with the UNIX paradigm that Linux is based off of. This course will broaden your computing horizons by guiding you on how to use this powerful operating system.
Concepts and skills
- What is a computer?
- Concepts of Operating Systems
- Filesystems as data organization
- Kernel and shell
- Bash usage
- Coreutils
- OS-level virtualization
Textbook: None, student is expected to use OS X, Cygwin, or Linux to tinker and reference Linux documentation at linux.die.net
Introduction to Python MA 250
Prerequisites: MA 150
Postrequisites: MA 350 MA 351
Concepts and skills
- Variables and operations
- Debugging
- Control structures
- Functions
- Classes
- The module system, package management, and libraries
- Introspection
- Closures
- Generators
- Coroutines and multithreading
Textbook: None, student is expected to use the Python interpreter to tinker and reference the official documentation at python.org
Algorithms and Data Structures MA 350
Prerequisites: MA 250, MA 201
Concepts and skills
- C
- Pointers and array notation as dereference
- Time complexity notation
- Sorting lists
- Stacks and queues
- Linked lists
- Trees
Textbook: CLRS
Computer Graphics MA 351
Prerequisites: MA 250 MA 201
Concepts and skills
- Pygame
- Project: rendering the Mandelbrot set
- Graphing linear equations for lines and polygons
- Linear transformations
- Texture mapping
- Affine transformations
- Transformations for 3D rendering
- Basics of lighting
Textbook: None, student is expected to tinker
Science
We as human beings have a deep-seated need to commune with nature and a strong desire to make sense of the world. Science is our approach to achieving both by using empirical data and systematic inquiry. It is both a body and a fount of knowledge. It helps us to make sense of nature from distant stars to the hidden, pullulating world of microbes to mundane questions like why cats purr (we still don't know the answer to that last one). Each scientific insight you learn is another step into a larger world.
Introductory Physics SCI 101
Postrequisites: SCI 301 SCI 302
Physics is the most fundamental science, but presents a dilemma. In order to build intuition for the other branches of science it must be taught early, but a true understanding of Newtonian mechanics also depends on calculus. As a compromise, this course treats Newtonian mechanics at a broad, conceptual level. For completeness with the AP Physics 1 exam, density and buoyancy are included as topics.
Concepts and skills
- Newton's laws: mass and inertia
- Vectors: displacement, velocity, and acceleration
- Basic kinematics
- Momentum
- Work and energy: kinetic and potential
- Pressure and force
- Angular momentum
- Density and Buoyancy
Labs
- Deriving freefall with a ticker tape timer
- Inertial reckoning in an elevator with a scale
- Measuring centripital force with a scale
- Period of a pendulum
- Measuring latitude by using the Coriolis effect on a Foucault pendulum
- Measuring fluid pressure with a more precise Cartesian diver
Textbook: College Physics by Raymond Serway and Chris Vuille
Chemistry SCI 102
Postrequisites: SCI 201 SCI 302
Everyone who doesn't know already is curious about what stuff really is. That's the purview of chemistry. Nestled between physics and biology, chemistry gives justification for the material world.
Concepts and skills
- What is matter?
- Substances, purity, mixtures, solutions, properties
- Significant figures and dimensional analysis
- Temperature and heat capacity
- States of matter, heats of phase changes, and phase diagrams
- Atoms, molecules, elements, and compounds
- Avogadro's number and counting molecules
- Ideal gas equation
- The periodic table and properties of atoms: number, group, mass
- Molecular bonds: ionic, covalent, and metallic
- Chemical reactions and stoichiometry
- Quantifying solution strength and colligative properties
- Chemical reactions as free energy minimization
- Chemical equilibrium and the law of mass action
- Acids and bases
- Redox reactions
- Electrochemistry
- Introduction to organic chemistry
Textbooks: Chemistry: The Central Science by Theodore Brown, H. LeMay, Bruce Bursten, Catherine Murphy, Patrick Woodward, and Matthew Stoltzfus
Electricity and Magnetism SCI 301
Prerequisites: SCI 101, MA 301
Of the four fundamental forces of nature, electricity is the one that we as a species have most skillfully harnessed and directed. Electrical circuits can have components with lengths of nanometers that process signals with periods of nanoseconds, or can be as large as the continent-spanning electrical grid which directs hundreds of gigawatts of power. This course will teach you the basics of that force, with a focus on using electrical circuits to process information and direct energy.
Concepts and skills
- Charge and electric fields
- Coulomb's law
- Voltage and electric fields
- Current
- Circuit Analysis: Kirchhoff's laws, series and parallel, circuit as graph
- Thevenin and Norton equivalents and superposition
- Capacitors and inductors
- Phasors
Labs
- Ohm's Law: series and parallel
- Potentiometer as voltage divider
- Operational amplifier applications
- RC circuits and relaxation time
- Voltage controlled oscillation and LC bandpass filters
Textbook: Linear Circuit Analysis: Time Domain, Phasor, and Laplace Transform Approaches, by Raymond DeCarlo and Pen-Min Lin
Thermal Physics SCI 302
Prerequisites: SCI 101, SCI 102, MA 301, MA 310
"Thermodynamics has something to say about everything but does not tell us everything about anything." - Martin Goldstein and Inge F. Goldstein, The Refrigerator and the Universe
Concepts and skills
- What is temperature?
- Heat and work
- Isothermal and adiabatic compression
- Einstein model of a solid
- Entropy and the laws of thermodynamics
- "What is temperature?" redux
- Heat engines, refrigerators, and Carnot efficiency
Textbook: Introduction to Thermal Physics, by Daniel Schroeder
Biology SCI 201
Prerequisites: SCI 102
Biologists study the most awe-inspiring scientific phenomenon of all: life. Because the biological world is built on the adversarial process of Darwinian evolution, it features intricate complexity on every level. For over three and a half billion years organisms have transformed lifeless molecules to self-replicating, self-regulating survival machines. But beyond all this, biology also tells our story. How our body works, why we get sick, why we recover, how our brains work, and our place on the tree of life are all profound and personal puzzles that biology explores. This course is your chance to learn the basics of what makes us, and the entire biosphere, alive.
Concepts and skills
- What is life?
- Ecosystem, organism, organ system, organ, tissue, cell
- Room temperature life at the scale of Boltzmann's constant (reading)
- Proteins: the cell's molecular machines, residues from primary to quaternary structure
- The central dogma: from DNA to protein, including exceptions and gene regulation
- Metabolic pathways: respiration, photosynthesis, and more
- Cell biology, including organelles
- Cellular reproduction, incluing mitosis and meiosis
- Immunology: innate and adaptive
- Immunology: vaccination as a species-wide immune system
- Evolution by natural selection
- Deep time: eons, eras, periods
- The tree of life: domain models in flux, zoology, and botany
- Basic population dynamics: logistic growth and carrying capacity
Labs
- DNA extraction
- Genetically modifying bacteria
- Osmosis and molarity
- Microscope observations
Textbook: Biology for AP® Courses, by Julianne Zedalis, John Eggebrecht, et al with OpenStax
English
Achieving a mastery of language is the same kind of accomplishment as learning to speak. Your language determines what kinds of ideas you can consider, who will take you seriously, and how well you will be able to set the ideas in your head free from your skull into the wider world.
English Literature and Language ENG 101
Reading and writing are skills and, like all other skills, are built by practice. That's why the focus of Enrichment School's English language course is reading and writing. Students will read literary works varying in difficulty from The Time Machine to Silas Marner and be asked to write and discuss their interpretations of each work. In addition, some direct instruction in grammar and vocabulary will be given.
Concepts and skills
- Reading short stories, novellas, and novels
- Authors include Ernest Hemmingway, H.G. Wells, George Eliot, and others
- Rhetorical devices
- Composition
- Vocabulary
- Parts of speech: functions of words, sentences
- Types of sentences and sentence diagrams
- Basic IPA
- Reading poetry: Plath, Kipling, Auden, Cummings, and others
- Poetic devices
Social Studies
Economics SOC 101
Known as "the dismal science," which it's anything but, economics is the classification of rigorous attempts to understand human behavior in the social studies. By examining incentives to model human behavior, economists have systematically learned principles that help us to understand the most violently discontinuous increase in productivity, happiness, and wealth in history.
Concepts and skills
Textbook: Basic Economics, by Thomas Sowell; Modern Principles: Microeconomics, by Alex Tabarrock and Tyler Cowen