Books
The Math Academy Way
- I. PRELIMINARIES
- Chapter 1. The Two-Sigma Solution • Bloom’s Two-Sigma Problem • The Core Essence of Bloom’s Two-Sigma Problem • Talent Development vs Traditional Schooling • Talent Development is Prohibitively Expensive • Stages of Talent Development • Bloom’s 3-Stage Talent Development Process • Bloom’s Taxonomy is Often Misinterpreted • Key Papers
- Chapter 2. The Science of Learning • Cognitive Learning Strategies • The Persistence of Tradition • A Common Theme Preventing Adoption • Theme and Examples • Desirable Difficulty vs Illusion of Comprehension • The Educational System Prefers Illusion • Technology Changes Everything • Revival via Technology • Necessity of Technology • Accelerating Student Learning by 4x • Key Papers
- Chapter 3. Core Science: How the Brain Works • Sensory, Working, and Long-Term Memory • Design Constraints • Case Study: Information Flow During a Computation • Neuroscience of Working Memory • Key Papers
- Chapter 4. Core Technology: the Knowledge Graph • Understanding the Knowledge Graph • Linking Topics and Prerequisites • Zooming Out • Course Graph • Using the Knowledge Graph • Scaffolded Mastery Learning • Additional Linkages • Key Prerequisites Enable Targeted Remediation • Encompassings Enable Turbo-Boosted Learning Speed • Diagnostic Exams
- Chapter 5. Accountability and Incentives • Accountability and Incentives are Necessary but Absent in Education • What Happens in the Absence of Accountability and Incentives • Tragedy of the Commons • Grades Can’t Be Trusted • Evidence for Grade Inflation • Why Grade Inflation is a Problem • Resistance to Objective Measurement • Radical Constructivism Rejects the Idea of Measuring Learning Objectively • Radical Constructivism is a Present Force in Education • Math Academy is Held Accountable for Student Learning • Key Papers
- II. ADDRESSING CRITICAL MISCONCEPTIONS
- Chapter 6. The Persistence of Neuromyths • Neuromyths are Common Misconceptions about the Brain • Why Neuromyths Persist • Key Papers
- Chapter 7. Myths & Realities about Individual Differences • People Differ in Learning Speed, Not Learning Style • Learning Style Preferences are Irrelevant • Working Memory Capacity (WMC) Differences are Relevant • WMC Impacts Perceived Effort • WMC Impacts Abstraction Ability • WMC Impacts Learning Speed • Lack of Evidence for WMC Training • Different Students Need Different Amounts of Practice • Your Mathematical Potential Has a Limit, but it’s Likely Higher Than You Think • Levels of Math • The Abstraction Ceiling • Learning Energy vs Level of Math • All Can Learn Some, Many Can Learn More, but Few Can Learn All • Nature or Nurture? Both Matter • Speed of Skill Acquisition Matters Because Time is Limited • Why the “All Can Learn All” Myth Persists • Struggle Does Not Imply Inability • Struggle Can Be Caused by Missing Foundations • Struggle Can Be Caused by Ineffective Practice • Struggle Can Be Caused by Insufficient Practice • Struggle Can Be Caused by Lack of Motivation • Analogy to Lifespans • Student Bite Size vs Curriculum Portion Size • The Misconception: If Instruction is Done Perfectly, Won’t All Students Learn at the Same Rate? • The Resolution: Under Favorable Learning Conditions, Student Bite Size Equals Curriculum Portion Size • Consistency with Observations • Key Papers
- Chapter 8. Myths & Realities about Effective Practice • Effective Practice Does Not Emulate the Professional Workplace • Direct Instruction is Needed • Definition and Importance • Unguided Instruction has a History of Pseudoscience • Unguided Instruction is Logically and Scientifically Inconsistent • Unguided Instruction Leads to Major Issues in Practice • Many Hands Make Light Work… and Light Learning • Effective Practice Requires Effort • There is No Such Thing as Effortless Learning • To Oppose Effortful Practice is to Oppose Talent Development • Testing and Repetition are Necessary • Computation is Necessary • Competition Can be Helpful and is Unavoidable in the Big Picture • Why the Myth Persists in Education (But Not in Talent Development) • Why Talent Development is Important in Math • Key Papers
- Chapter 9. Myths & Realities about Mathematical Acceleration • Acceleration is Often Misunderstood • Developmental Appropriateness • Advanced Study is Appropriate Once Prerequisites Have Been Mastered • Why the Myth of Developmental Inappropriateness Persists • Depth of Learning • Accelerated Students Learn More Material, Just as Deeply • How We Ensure Comprehensive, Deep Learning • Continuity of Courses • Accelerated Students Don’t Run out of Math Courses • Advanced Students Can Place Out of College Courses Beyond Placement Tests • Relevance to Students’ Futures • Learning Math Early Reduces Risk and Opens Doors to Opportunities • Higher-Grade Math is Typically More Productive than Grade-Level Competition Problems • Key Papers
- III. COGNITIVE LEARNING STRATEGIES
- Chapter 10. Active Learning • Definition and Importance • Case Studies • Case Study 1: Why Active Learning is Obvious • Case Study 2: Most Students Don’t Even Pay Attention During Lectures • Case Study 3: How Active Learning Saved MIT’s Physics Classes • Case Study 4: If You’re Active Half the Time, That’s Still Not Enough • Neuroscience of Active Learning • Persistence of Misconceptions • Key Papers
- [In Progress] Chapter 11. Direct Instruction • Active Learning Should Not Imply Unguided Learning or Group Work
- Chapter 12. Deliberate Practice • Definition and Importance • Deliberate vs Non-Deliberate Practice • Deliberate Practice is Effective, Non-Deliberate Practice is Not • Effort is Required • Deliberate Practice Feels Like Exercising with a Personal Trainer • Cycle of Strain and Adaptation • Discomfort is Required • Long-Term Compounding • Expertise is the Product of Incremental Improvements Over Time • Motivational Supplements are Not Substitutes for Deliberate Practice • Misinterpretations of Deliberate Practice • Key Papers • Additional Resources
- Chapter 13. Mastery Learning • Mastery Learning is Underused • Implementing Mastery Learning • Knowledge Frontier as Zone of Proximal Development • Key Papers
- Chapter 14. Minimizing Cognitive Load • The Learning Staircase • Micro-Scaffolding • The Expertise Reversal Effect • Key Papers
- Chapter 15. Developing Automaticity • Importance of Automaticity • Automaticity Frees Up Working Memory • Working Memory is Limited, but Long-Term Memory is Not • Expertise Requires Automaticity • Case Study: Computing Exponents With vs Without Automaticity on Multiplication and Addition Facts • Automaticity, Creativity, and Higher-Level Thinking • Automaticity is Necessary for Creativity • Automaticity is Necessary for Higher-Level Thinking • Automaticity is a Gatekeeper to Mathematical Literacy and Academic Success • Neuroscience of Automaticity • Key Papers
- Chapter 16. Layering • Facilitation and Structural Integrity • Facilitation • Structural Integrity • How We Layer • Key Papers
- Chapter 17. Non-Interference • Associative Interference • Non-Interference • Key Papers
- Chapter 18. Spaced Repetition (Distributed Practice) • Retaining Knowledge Indefinitely • The Spacing Effect • Spaced Repetition • Building Intuition About Spaced Repetition • Consensus Among Researchers • Spaced Repetition is Underused • Spaced Repetition Improves Generalization • Repetition Compression • Calibrating to Individual Students and Topics • Spaced Repetition vs Spiraling • Key Papers • Additional Resources
- Chapter 19. Interleaving (Mixed Practice) • Interleaving vs Blocking • Benefits of Interleaving • Efficiency • Discrimination and Category Induction Learning • Experimental Support • Desirable Difficulty: Why Interleaving is Underused • Micro- and Macro-Interleaving • Macro-Interleaving • Micro-Interleaving • Key Papers • Additional Resources
- Chapter 20. The Testing Effect (Retrieval Practice) • Retrieval is the Most Effective Method of Review • Spaced Retrieval Practice • The Testing Effect is Underused • Reducing Anxiety and Promoting a Growth Mindset • Appropriate vs Inappropriate Usage of Timed Tests • Desirable vs Undesirable Difficulties • Appropriate Timed Testing Can Reduce Math Anxiety • Implementing Appropriate Timed Testing • Key Papers • Additional Resources
- Chapter 21. Targeted Remediation • High-Granularity, High-Integrity Remediation • Corrective Remediation • Preventative Remediation • Foundational Remediation • Content Remediation
- Chapter 22. Gamification • Importance of Gamification • Increasing Learning, Engagement, and Enjoyment • Increasing Learning Efficiency • Incentivizing Quantity and Quality of Work • XP-Time Equivalence • Incentivizing Quantity of Work • Incentivizing Quality of Work • Closing Loopholes • Progress vs XP • Key Papers
- Chapter 23. Leveraging Cognitive Learning Strategies Requires Technology • High-Level Context • The Problem: Cognitive Learning Strategies Remain Underused • The Blame: Teachers are Victims of Circumstance • The Solution: Technology Changes Everything • Resistance to Additional Effort • Active Learning • Non-Interference, Interleaving, and Spaced Repetition • Shuffling Instructional Material • Opening a Can of Worms on Forgetting • Testing Effect • Gamification • Tutoring the King’s Kid: How Would You Teach if Your Life Depended On It? • Heterogeneity of Student Knowledge Profiles • Tutoring the King’s Kid vs Teaching Many Kings’ Kids • Differences in Background Knowledge • Student Knowledge Profiles Naturally Tend Towards Heterogeneity • Every Student in the Class Effectively Needs a Private Tutor
- IV. COACHING
- [In Progress] Chapter 24. Parental Support • Why Parental Support is (Usually) Necessary • The Bare Minimum: Incentives and Accountability • The Platonic Ideal
- [In Progress] Chapter 25. In-Task Coaching • The Vicious Cycle of Forgetting
- V. TECHNICAL DEEP DIVES
- Chapter 26. Technical Deep Dive on Spaced Repetition • Fractional Implicit Repetition (FIRe) • Concrete Example • Visualizing Repetition Flow • Partial Encompassings • Setting Encompassing Weights Manually • Direct and Key Prerequisites are Sufficient • Non-Ancestor Encompassings and Mastery Floors • Student-Topic Learning Speeds • Ratio of Student Ability and Topic Difficulty • Measuring Student Ability at the Level of Individual Topics • Measuring Topic Difficulty • High-Level Structure
- Chapter 27. Technical Deep Dive on Diagnostic Exams • Minimizing the Number of Questions • Knowledge Confidence and Conditional Completion • Theory • Example • Implementation • Conservative vs. Aggressive Edge of Mastery • Supplemental Diagnostics • Selecting Good Diagnostic Questions
- Chapter 28. Technical Deep Dive on Learning Efficiency • What is Learning Efficiency? • Theoretical Maximum Learning Efficiency • Theoretical Minimum Learning Efficiency • Factors that Impact Learning Efficiency • Performance • Pace • Learning Efficiency vs Pace • Pace vs Time to Completion
- Chapter 29. Technical Deep Dive on Prioritizing Core Topics • Core and Supplemental Topics • The Mathematical Foundations Sequence
- Frequently Asked Questions • Glossary • References
Advice on Upskilling
Advice on consistency, skills, discipline, the grind, the journey, the team, the mission, motivation, and learning.
Note: If you want to improve your upskilling regimen, this book would be a good place to start -- it's short, easy reading, addresses the big ideas, and gets you in the right mindset -- but I would highly recommend to go on and read The Math Academy Way afterwards to get a complete, comprehensive understanding of everything down to the nuts and bolts.
- 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 • If You Struggle to Train Consistently, Do It Immediately After Waking Up • Training Sessions Should be Short and Frequent as Opposed to Long and Sparse • 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 • Protect The Habit
- 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 • You Want Exciting Opportunities? Learn Math and Coding • If You're Not Both Technical and a Domain Expert, Then You're Underpowered • Domain Expertise, Math, Coding, Communication
- 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 • When More Volume Equals More Progress • 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 • Don't Get Hung Up on Youth Competitions • 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 • Compound Hard Work and Luck • 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 • Never Come Up Empty-Handed • You Need a Berserker At The Helm
- 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 • Build Where Building Creates More Opportunities To Build
- 8. MOTIVATION • Why Extrinsic Motivation Matters • How to Become a Super-Producer • How The Highest Performers Sustain a Massive Workload • Overcoming the Paradox of Serious Training • How Taxing Work Becomes Fun
- 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 • Review Should Feel Challenging • Learn Like You Lift • Schooling Versus Talent Development • Learning Doesn’t Have to Be Synchronized for Camaraderie to Occur • The Driving Force Behind Expertise is Long-Term Memory • Learning is Memory • Turn The Magical Into The Mechanical • 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.