Books
The Math Academy Way
- 1. Preliminaries - The Two-Sigma Solution; The Science of Learning; Core Science: How the Brain Works; Core Technology: the Knowledge Graph; The Importance of Accountability and Incentives.
- 2. Addressing Critical Misconceptions - The Persistence of Neuromyths; Myths & Realities about Individual Differences; Myths & Realities about Effective Practice; Myths & Realities about Mathematical Acceleration.
- 3. Cognitive Learning Strategies - Active Learning; Deliberate Practice; Mastery Learning; Minimizing Cognitive Load; Developing Automaticity; Layering; Non-Interference; Spaced Repetition (Distributed Practice); Interleaving (Mixed Practice); The Testing Effect (Retrieval Practice); Targeted Remediation; Gamification; Leveraging Cognitive Learning Strategies Requires Technology.
- 4. Coaching - In-Task Coaching; Parental Support.
- 5. Technical Deep Dives - Technical Deep Dive on Spaced Repetition; Technical Deep Dive on Diagnostic Exams; Technical Deep Dive on Learning Efficiency; Technical Deep Dive on Prioritizing Core Topics.
- 6. Frequently Asked Questions - The Practice Experience; Student Behavior; XP and Practice Schedules; Diagnostics and Curriculum; Miscellaneous.
Advice on Upskilling
Advice on consistency, skills, discipline, the grind, the journey, the team, the mission, motivation, and learning.
- 1. CONSISTENCY • You’re Not Lazy, You Just Lack a Habit • Don’t Have a Passion? Go Create One. • Make the Habit Easily Repeatable • The Hardest Part is Just Getting Started • A Little Extra Consistency × A Little Extra Time = A Massive Increase in Volume and Progress • Don’t Overreact to Bad Days • Aim for Virtuous Cycles
- 2. SKILLS • The Importance of Hardcore Skills • The “Alien-Level Skills” Hack • The Importance of Having Your Prerequisites In Place • Fortify Your F*cking Fundamentals • Your Missing Foundations Will Wait For You • Actively Doing is the Key to Alpha • Everything Matters
- 3. DISCIPLINE • Why Train? • The Magic You’re Looking For is in the Full-Assed Effort You’re Avoiding • At Some Point Doing the Hard Thing Becomes Easier Than Making the Hard Thing Easier • How to Cultivate Discipline • Keep Your Hands On The Boulder
- 4. THE GRIND • Upskilling is Hard and That’s a Good Thing • The Most Superior Form of Training • Just Do The F*cking Work • Outsized Success Requires Outsized Work • Transformation Is Discomforting • Enjoyment is a Second-Order Optimization • Ability is Built, Not Unlocked • What Max-Efficiency Training Feels Like • The Necessity of Grinding Through Concrete Examples Before Jumping Up a Level of Abstraction • Be Willing to Do Tedious Work • Don’t Undervalue Turning Up the Dial on Your Grind, but Don’t Overvalue the Last Turn • Failure Is NOT the Key to Success • The Problem with Overly Difficult Problems • Focus Less on Feelings and More on Measurable Progress
- 5. THE JOURNEY • The 3 Stages of Talent Development • There Are No Shortcuts in Talent Development • If You’re Making Silly Mistakes Then You Need More Practice • No Train, No Gain • Why You Should Push Yourself • Keep Your Foot On The Gas • You Are a Car • What to Do When You Hit a Ceiling • Get Yourself In A Position Where You Can Eat Risk • How to Allocate Your Bandwidth While Searching for Your Mission
- 6. THE TEAM • If You’re Asking Someone to Be Your Mentor then You’re Doing it Wrong • Put Pressure on Your Boss to Come Up with More Work For You • Get On the Right Team • Competition as a Means of Collaboration • Your Goal is NOT to Prove You’re Smart, it’s to Make Problems Go Away • Make Your First Impression On a Contribution, Not a Critique
- 7. THE MISSION • Selecting a Good Problem to Work On • The “Progress Equals Pressure” Formula • Love What You Do • Be a Builder, Not Just a Fighter
- 8. MOTIVATION • Why Extrinsic Motivation Matters • How to Become a Super-Producer • Overcoming the Paradox of Serious Training
- 9. LEARNING • The Greatest Educational Life Hack: Learning Ahead of Time • When Does the Learning Happen? • There is No Such Thing as Low-Effort Learning • The Greatest Breakthrough in the Science of Learning Over the Last Century • “Following Along” Versus Learning • The Vicious Cycle of Forgetting • The Vicious Cycle of Context Overload • Prereq Yo’ Self Before You Wreck Yo’ Self • Plan Your Broad-Strokes Journey Top-Down, but Carry Out the Granular Steps Bottom-Up • The Efficient Learning Loop • Schooling Versus Talent Development • Learning Doesn’t Have to Be Synchronized for Camaraderie to Occur • Learning is Memory • A Sanity Check for Effective Study Techniques
- 10. COACHING • Be Both Good Cop and Bad Cop
Math Textbooks
During my teaching years shortly after college, I simultaneously wrote math textbooks for fun as a way to consolidate and clarify my quantitative intuition. The goal was to provide deep intuition for the core concepts and connections, along with plenty of examples and exercises, while remaining as concise as possible.
My teaching years and math textbook writing culminated in Introduction to Algorithms and Machine Learning: from Sorting to Strategic Agents. This book was written to support what was, during its operation from 2020-23, the most advanced high school math/CS sequence in the USA. It culminated in high school students doing masters/PhD-level coursework (reproducing academic research papers in artificial intelligence, building everything from scratch in Python).
Despite no intentional search optimization, this content ranks in the top results for many common search queries across various subfields of math. Some example queries are provided below:
- Algebra: polynomial asymptotes, drawing rotated graphs on calculator, reflections of functions, graph slant asymptote, 3 trig functions.
- Calculus: limits by logarithms, difference quotient vs derivative, chain rule trick, quick chain rule, lagrange error bound proof, calculus in cardiology, application of calculus in real life pdf.
- Linear Algebra: n-dimensional volume, recursive sequence diagonalization, shear matrix transformation.
- Differential Equations: solving differential equations by substitution, how to find the characteristic polynomial of a differential equation, non-separable differential equation examples.
- Algorithms / Machine Learning: linear regression with pseudoinverse, single-variable gradient descent, multivariable gradient descent, minimax strategy.
A special thanks to Sanjana Kulkarni for her thoughtful suggestions and diligent proofreading of these books.
Print copies are available on Amazon for the minimum price possible (printing cost plus Amazon's fees).
There's also a (hacky, but functioning) knowledge graph explorer tool for the content in these textbooks, plus some related content from technical booklets and blog posts:
Introduction to Algorithms and Machine Learning
This book was written to support what was, during its operation from 2020-23, the most advanced high school math/CS sequence in the USA. It culminated in high school students doing masters/PhD-level coursework (reproducing academic research papers in artificial intelligence, building everything from scratch in Python).
- Preamble: The Story of Math Academy's Eurisko Sequence
- 1. Hello World - Some Short Introductory Coding Exercises; Converting Between Binary, Decimal, and Hexadecimal; Recursive Sequences; Simulating Coin Flips; Roulette Wheel Selection; Cartesian Product.
- 2. Searching and Sorting - Brute Force Search with Linear-Encoding Cryptography; Solving Magic Squares via Backtracking; Estimating Roots via Bisection Search and Newton-Raphson Method; Single-Variable Gradient Descent; Multivariable Gradient Descent; Selection, Bubble, Insertion, and Counting Sort; Merge Sort and Quicksort.
- 3. Objects - Basic Matrix Arithmetic; Reduced Row Echelon Form and Applications to Matrix Arithmetic; K-Means Clustering; Tic-Tac-Toe and Connect Four; Euler Estimation; SIR Model for the Spread of Disease; Hodgkin-Huxley Model of Action Potentials in Neurons; Hash Tables; Simplex Method.
- 4. Regression and Classification - Linear, Polynomial, and Multiple Linear Regression via Pseudoinverse; Regressing a Linear Combination of Nonlinear Functions via Pseudoinverse; Power, Exponential, and Logistic Regression via Pseudoinverse; Overfitting, Underfitting, Cross-Validation, and the Bias-Variance Tradeoff; Regression via Gradient Descent; Multiple Regression and Interaction Terms; K-Nearest Neighbors; Naive Bayes.
- 5. Graphs - Breadth-First and Depth-First Traversals; Distance and Shortest Paths in Unweighted Graphs; Dijkstra's Algorithm for Distance and Shortest Paths in Weighted Graphs; Decision Trees; Introduction to Neural Network Regressors; Backpropagation.
- 6. Games - Canonical and Reduced Game Trees for Tic-Tac-Toe; Minimax Strategy; Reduced Search Depth and Heuristic Evaluation for Connect Four; Introduction to Blondie24 and Neuroevolution; Reimplementing Fogel's Tic-Tac-Toe Paper; Reimplementing Blondie24; Reimplementing Blondie24: Convolutional Version.
Linear Algebra
- 1. Vectors - N-Dimensional Space; Dot Product and Cross Product; Lines and Planes; Span, Subspaces, and Reduction; Elimination as Vector Reduction.
- 2. Volume - N-Dimensional Volume Formula; Volume as the Determinant of a Square Linear System; Shearing, Cramer's Rule, and Volume by Reduction; Higher-Order Variation of Parameters.
- 3. Matrices - Linear Systems as Transformations of Vectors by Matrices; Matrix Multiplication; Rescaling, Shearing, and the Determinant; Inverse Matrices.
- 4. Eigenspace - Eigenvalues, Eigenvectors, and Diagonalization; Recursive Sequence Formulas via Diagonalization; Generalized Eigenvectors and Jordan Form; Matrix Exponential and Systems of Linear Differential Equations.
Calculus
- 1. Limits and Derivatives - Evaluating Limits; Limits by Logarithms, Squeeze Theorem, and Euler's Consant; Derivatives and the Difference Quotient; Power Rule; Chain Rule; Properties of Derivatives; Derivatives of Non-Polynomial Functions; Finding Local Extrema; Differentials and Approximation; L'HĂ´pital's Rule.
- 2. Integrals - Antiderivatives; Finding Area; Substitution; Integration by Parts; Improper Integrals.
- 3. Differential Equations - Separation of Variables; Slope Fields and Euler Approximation; Substitution; Characteristic Polynomial; Undetermined Coefficients; Integrating Factors; Variation of Parameters.
- 4. Series - Geometric Series; Tests for Convergence; Taylor Series; Manipulating Taylor Series; Solving Differential Equations with Taylor Series.
Algebra
- 1. Linear Equations and Systems - Solving Linear Equations; Slope-Intercept Form; Point-Slope Form; Standard Form; Linear Systems.
- 2. Quadratic Equations - Standard Form; Factoring; Quadratic Formula; Completing the Square; Vertex Form; Quadratic Systems.
- 3. Inequalities - Linear Inequalities in the Number Line; Linear Inequalities in the Plane; Quadratic Inequalities; Systems of Inequalities.
- 4. Polynomials - Standard Form and End Behavior; Zeros; Rational Roots and Synthetic Division; Sketching Graphs.
- 5. Rational Functions - Polynomial Long Division; Horizontal Asymptotes; Vertical Asymptotes; Graphing with Horizontal and Vertical Asymptotes; Graphing with Slant and Polynomial Asymptotes.
- 6. Non-Polynomial Functions - Radical Functions; Exponential and Logarithmic Functions; Absolute Value; Trigonometric Functions; Piecewise Functions.
- 7. Transformations of Functions - Shifts; Rescalings; Reflections; Inverse Functions; Compositions.
Booklets
Below are some shorter manuscripts that I feel are interesting enough to share.
Graphing Calculator Drawing Exercises (2019)
pdf, print, course page
- During school I would sometimes pass time by drawing on my graphing calculator. Years later in 2019, I turned this hobby into a summer course for the Math Academy program in the Pasadena Unified School District. This workbook contains the lessons that were delivered during that course. Familiarity with algebra is assumed.
- 1. Lines - Horizontal and Vertical Lines; Slanted Lines; Absolute Value.
- 2. Open Curves - Parabolas; Sine Waves; Roots.
- 3. Closed Curves - Shading with Sine; Euclidean Ellipses, Non-Euclidean Ellipses.
- 4. Trigonometry - Rotation; Lissajous Curves; Composition Waves and Implicit Trig Patterns.
A Primer on Artificial Intelligence (2019)
What is AI; The First Wave: Reasoning as Search; The Second Wave: Expert Systems; The Third Wave: Computation Power and Neural Networks; Cutting Through the Hype.
Introduction to Python Programming (2019)
Getting Started in Colab; Strings, Ints, Floats, and Booleans; Lists, Dictionaries, and Arrays; If, While, and For; Functions.
Intuiting Predictive Algorithms (2018)
Naive Bayes; MAP and MLE; Linear Regression; Support Vector Machines; Neural Networks; Decision Trees; Ensemble Models.
The Data Scientist's Guide to Topological Data Analysis (2017)
- Preamble
- 1. Mapper - Algorithm; Software; Use-Cases at Ayasdi; Use-Cases at Aunalytics.
- 2. Persistent Homology - Homotopy; Approximation; Homology; Persistence; Software.
Connecting Calculus to the Real World (2017)
- 1. Science and Medicine - Cardiac Output; Understanding Plaque Buildup; Modeling Tumor Growth.
- 2. Technology and Engineering - Rocket Propulsion; Rendering 3D Computer Graphics; Physics Engines in Video Games; Optimization via Gradient Descent.
- 3. Business and Economics - Maximizing Profit; Continuously Compounded Interest.
- 4. History and Philosophy - The Man who "Broke" Math; The Newton-Leibniz Controversy; A Failure of Intuition.
- 5. Art and Athletics - Derivatives in String Art; Calculating the Horsepower of an Offensive Lineman.
An Intuitive Primer on Calculus (2017)
Functions; Limits; Derivatives; Integrals; Sequences; Series.
The Physics Behind an Egg Drop: A Lively Story (2014)
Velocity; Momentum; Changes in Momentum; Force; Pressure; Troll Egg Drop.