The Curious Engineer

All blogs / Math

Justin Skycak

justinmath.com

MathEssays / personal

’23’24’25’26
Posts
828 from 5 Oct 2023 to 24 Oct 2026
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weekly (84 in the last year)
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personal blog
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2026 55

  1. A Core Math Course Should Not Tour the Professor’s Interests
  2. A Lot of Advanced Math Is Taught Backwards
  3. Division of Labor in a Group Project Is Division of Learning
  4. Effective Learning Requires Deliberate Practice
  5. Familiarity vs. Understanding
  6. Fundamentals Put a Ceiling on What You Can Create
  7. High School Teachers Don’t Grade Homework Anymore
  8. How to Maximize Your Progress in a Talent Domain
  9. Memory Is the Substrate of Learning
  10. People Confuse Difficulty with Rigor
  11. Your Greatest Advantage Over LLMs Is Your Long-Term Memory
  12. The Best Career Advice for Young People
  13. Math Academy Success Stories
  14. Full-Ass Everything
  15. Shoutout to My Former Student Elijah Tarr
  16. ANOTHER Math Academy High Schooler Publishes a Research Paper!
  17. Don’t Quit Your Compounding Too Early
  18. Prerequisite Knowledge is Intellectual Capital
  19. The Cure for Procrastination
  20. Learning Is a Balancing Act
  21. Self-Discovery Is an Effortful Process
  22. Recall Before Rereading
  23. Optimize Actions, Not Plans
  24. Don’t Coast
  25. How To Compress a Grade Level’s Worth of Learning Much, Much Shorter Than a Year
  26. School Is Slow Because It’s Miscalibrated
  27. Scaling a Startup Is Like Playing Call of Duty Zombies
  28. The Bar For Joining the Technical Workforce is Skyrocketing
  29. Expertise is the Base Layer and Automation is the Multiplier
  30. The Math Academy Prequel: Jason Roberts’s Origin Story
  31. The Process of Building Intuition Can Itself Be Counter-Intuitive
  32. The Most Costly Part of Failure
  33. The Longer You Wait Before Building The Life You Want, The More Likely You Are To Settle For Something Else
  34. Why I Care So Much About Self-Service Learning
  35. How to Go From 0 to 100 mph
  36. How To Get The Most Value Out Of a STEM Degree
  37. Everybody Wants to Think Creatively But Nobody Wants to Build Their Knowledge Base
  38. Confidence Opens Doors But Competence Gets You Through Them
  39. How A Laptop Changed My Life
  40. There Is So Much More Math After Calculus (That You Should Learn)
  41. Standard Milestones Are Based on an Unserious Approach to Work
  42. Pre-Learning Math Can Kick Off a Virtuous Cycle
  43. Mathematical Acceleration is Developmentally Appropriate For Students Who Have Learned The Prerequisites
  44. Math Acceleration Leads to Earlier and More Prolific Careers
  45. Educational Acceleration Helps You Win The Race Against Time
  46. Competition Math Is Not The Highest ROI For Most Mathematically Gifted Students
  47. How To Do Cognitive Weightlifting
  48. You Have Potential
  49. How To Balance Investing Vs. Spending
  50. Skill Debt is Like Any Other Kind of Debt
  51. The Biggest Danger in Both Learning and Startups: Falling Off The Rails
  52. The #1 Killer of Creativity (and the #1 Unlock)
  53. Nothing Comes Together Unless You Make It So
  54. Why Can’t College Students Do Middle School Math?
  55. Put Your Environment on Easy Mode

2025 74

  1. Success is the Product of Small Efforts Compounded
  2. The (Free) Textbook I Wrote While Teaching The Most Advanced High School Math/CS Sequence In The USA
  3. A Former Student Got Recruited By NASA With a Fighter Jet Ride as the Signing Bonus
  4. Vastly Underrated Predictor of Success
  5. How To Figure Out What To Do
  6. The Most Mathematically Gifted Student I Ever Worked With Still Needed To Be Pushed to Learn Calculus
  7. Incentives Don’t Have To Be Big To Be Effective
  8. Incentives Can Change Behavior Overnight
  9. The Only Reliable Path To Wealth
  10. Don’t Get Lazy
  11. How I Went from 19 to 25000 Followers on X in 18 Months
  12. You Can Explain Anything If You Break It Into Small Enough Steps
  13. The Most Important Thing to Understand About Building Educational Knowledge Graphs
  14. Common Failure Modes in Early Math Education
  15. Don’t Turn Your Blessing Into A Curse
  16. You Can Be Happy Doing Tons Of Work
  17. How to Pull Alpha Out of Your Brain You Didn’t Even Know Was There
  18. Either You Make Decisions, Or Decisions Get Made For You
  19. Litmus Test For How Well-Structured Your Code Is
  20. I Love Seeing Hardcore People Moving Into Serious Edtech
  21. The Universe Will Bend
  22. Summary of My Own Math Journey
  23. The Lie That Learning Should Feel Pleasurable
  24. For a While, YouTube Thought I Was a Gym Owner
  25. Loss of Interest Can Sometimes Be Traced Back to Lack of Foundations
  26. The Most Common Failure Mode of Stereotypically Smart People
  27. Why I Quit My Data Science Job to Tutor Full Time
  28. Find Your North Star
  29. One of the WORST Mistakes You Can Make While Studying
  30. How and Why To Become the Smart Kid
  31. Yes, You Need to be Automatic on Math Facts (and Rapid-Fire Training Is Coming)
  32. A Grade Level of Learning Can Be Compressed Much Shorter Than A Year
  33. The Power of Pre-Learning: The Greatest Educational Life Hack
  34. The State of “Blah”: Why You’re In It and How You Escape It
  35. The Origin Story of Eurisko, the Most Advanced Math/CS Track in the USA
  36. Bottom-Up Versus Top-Down Arguments in Machine Learning Education
  37. Jason Roberts’s Philosophy of Breadth-First Development
  38. What Happens When Middle School is Put to Good Use
  39. Why Pre-Learn Your College Math Classes?
  40. If You’re Coming Into a Math Major at an Elite University, Spin Up on Math Beyond AP Calc BC
  41. Too Many People Quit Before They Start
  42. Learn to Love the Games That You Were Built to Win
  43. Q&A: Thoughts on Input
  44. The Joy of Math is Like The Joy of Exercise
  45. I Believe You Have Potential
  46. The Best Tool for Writing Powerful Tweets
  47. What’s the Point of Being Way Ahead? It Buys You Time.
  48. You Have To Work Really Damn Hard to Figure Out What Fulfills You
  49. The Vicious Cycle of Half-Assery
  50. The Least Efficient Learning Strategy
  51. Consuming is Only Helpful Insofar as it Enables You to Produce
  52. Performance Training Minus Intrinsic Enjoyment Far Outranks The Opposite (But Having Both is Best)
  53. What Many People Fail to Understand about Educational Acceleration
  54. I’ve Wanted Alien-Level Edtech Ever Since I Was A Kid
  55. The Missing Middle in Test Prep
  56. You Can Often Just Make Yourself Like Things
  57. My 2 Rules of Thumb (So Far) When Using AI Coding Tools
  58. Learning is About Bridge-Building, Not Jumping
  59. Recall First, Reason Second
  60. You Want Exciting Opportunities? Learn Math and Coding.
  61. How to Generate a Specific, Actionable Upskilling Path in ANY Domain
  62. The Field of Education is Due For a Copernican Revolution
  63. It’s All About Domain Knowledge
  64. How Taxing Work Becomes Fun
  65. Math Academy’s Eurisko Sequence, 5 Years Later: Student Outcomes Emerging From the Most Advanced High School Math/CS Track in the USA
  66. The Importance of Learning Your Prerequisites
  67. Learning is Memory
  68. Comfortable Fluency in Consuming Information is Not a Proxy For Actual Learning
  69. Long-Term Retention Requires Short-Term Forgetting
  70. Pick the Right Tool For the Job
  71. The Experience of Maxing Out One’s Cognitive Horsepower
  72. What Learning Actually Is – at a Concrete, Physical Level in the Brain
  73. Advice on Upskilling
  74. Retrieval Practice is F*cking Obvious

2024 29

  1. Learning Higher-Grade Math Ahead of Time is the Greatest Educational/Career Life Hack
  2. Fortify Your F*cking Fundamentals
  3. You Can Effectively Turn Long-Term Memory Into An Extension of Working Memory
  4. Failure Modes in the Talent Development Process
  5. How To Get a Full Time Software Job During College (5-Step Roadmap)
  6. Prereq Yo’ Self Before You Wreck Yo’ Self
  7. The Magic You’re Looking For is in the Full-Assed Effort You’re Avoiding
  8. How I Would Go About Learning an Arbitrary Subject Where No Full-Fledged Adaptive Learning System is Available
  9. Competition Math is NOT an All-Encompassing Holy Grail of Math Learning or General Problem Solving
  10. You Can Use Pre-Learning to Kick Off a Virtuous Cycle
  11. The Pursuit of Real Life Superhero Training
  12. At some point, doing the hard thing becomes easier than making the hard thing easier.
  13. How Math Academy Creates its Knowledge Graph
  14. If You’re Asking Someone to Be Your Mentor then You’re Doing it Wrong
  15. Spaced repetition is so similar to weight training that it might as well be called “wait” training.
  16. You Are NOT Lazy, You Just Lack a Habit
  17. Why I Recommend Students NOT Take Notes
  18. How to get from high school math to cutting-edge ML/AI: a detailed 4-stage roadmap with links to the best learning resources that I’m aware of.
  19. I’m Writing a Book on the Science of Learning (update: 400-page working draft is freely available)
  20. The Math Death Spiral: How Knowledge Gaps Lead to Student Failure
  21. Why is the EdTech Industry So Damn Soft?
  22. The Most Superior Form of Training and the Most Hard-Hitting 2 Sentences in All of Talent Development Research
  23. The Greatest Breakthrough in the Science of Education Over the Last Century
  24. Conversational Dialogue is a Fascinating Distraction for AI in Education
  25. Want to Major in Math at an Elite University? Getting A’s in High School Math is Not Good Enough
  26. Which Cognitive Psychology Findings are Solid, That Can Be Used to Help Students Learn Better?
  27. If You Want to Learn Algebra, You Need to Have Automaticity on Basic Arithmetic
  28. Lots of People in Education Disagree with the Premise of Maximizing Learning
  29. The Greatest Educational Life Hack: Learning Math Ahead of Time

2023 1

  1. Hierarchical Spaced Repetition (HSRS): Spaced Repetition Compression and Fractional Implicit Repetition (FIRe)

Undated 669

  1. list
  2. Math is a Well-Defined Body of Knowledge
  3. The Necessity of Grinding Through Concrete Examples Before Jumping Up a Level of Abstraction
  4. How to Cultivate Discipline
  5. Transformation Is Discomforting
  6. Complete Individualization: an Often-Forgotten yet Critical Component of True Deliberate Practice
  7. The Importance of Hardcore Skills
  8. Career Hack: Put Pressure on Your Boss to Come Up with More Work For You
  9. The Future of Education
  10. The Trick to Future-Proof Your Coding Career Against AI
  11. Five Steps to Becoming a Fully-Fledged Quantitative Software Engineer
  12. Why Talent Development is Necessary in Math
  13. The Future of Proof-Based Courses on Math Academy
  14. Don’t Undervalue Turning Up the Dial on Your Grind, but Don’t Overvalue the Last Turn
  15. One of the Most Amusing Errors in Teaching
  16. The “Alien-Level Skills” Hack
  17. A White Pill on Cognitive Differences
  18. One of the Weirdest, Most Treacherous Math Problems You Will Ever Encounter
  19. Fast, Correct Answers Do Matter in Mathematics
  20. What’s the Best Way to Teach Math: Explicit Instruction or Less Guided Learning?
  21. When should you do math in your head vs writing it out on paper?
  22. The Pedagogically Optimal Way to Learn Math
  23. Who Needs Worked Examples? You, Eventually.
  24. How to Crush a Standardized Math Test: SAT/ACT, AP/IB, GRE/GMAT, JEE, etc.
  25. How Bloom’s Taxonomy Gets Misinterpreted
  26. Higher Math Textbooks and Classes are Typically Not Aligned with the Cognitive Science of Learning
  27. Why Not Just Learn from a Textbook, MIT OpenCourseWare, Khan Academy, etc.?
  28. The Problem with “Think Really Hard, Struggle for a While, Eventually Solve it or Look Up The Answer” Problems
  29. What People Think Maximum-Efficiency Learning Should Feel Like, vs What it Actually Feels Like
  30. Student Bite Size vs Curriculum Portion Size
  31. Review Should Feel Challenging
  32. The Vicious Cycle of Forgetting
  33. Deliberate Practice: The Most Effective Form of Active Learning
  34. Your Mathematical Potential Has a Limit, but it’s Likely Higher Than You Think
  35. Struggle Does Not Imply Inability
  36. People Differ in Learning Speed, Not Learning Style
  37. Accountability and Incentives are Necessary but Absent in Education
  38. Cognitive Science of Learning: How the Brain Works
  39. Talent Development vs Traditional Schooling
  40. Critique of Paper: “An astonishing regularity in student learning rate”
  41. For Most Students, Competition Math is a Waste of Time
  42. The Story of Math Academy’s Eurisko Sequence: the Most Advanced High School Math/CS Track in the USA
  43. My Experience with Teacher Credentialing and Professional Development
  44. Tips for Developing Valuable Models
  45. But WHERE do the Taylor Series and Lagrange Error Bound even come from?!
  46. The Main Value Proposition of Math Academy
  47. Announcing our 8th Grade Math Course
  48. What Happens When Students Don’t Know Their Math Facts
  49. Don’t Look Back at the Reference Too Often
  50. The Reliable Path to Success in All Forms
  51. What “It’s Too Late” Usually Really Means
  52. Students Don’t Need a Million Different Explanations
  53. Being Talked at for an Hour Is Not Effective Learning
  54. AI Tools Are Multipliers, Not Equalizers
  55. “Aha” Moments Indicate Missing Prerequisites
  56. The Importance of Risk
  57. Seek Context
  58. Math Trauma Isn’t Really About Math
  59. Announcing our SAT Math Prep Course
  60. Most of My Coding is Now Agentic
  61. Obsession Is The Highest Form of Habit
  62. Most people don’t hate math. They hate the cognitive friction of missing prerequisites.
  63. Zoom Out
  64. When I took DiffEq in undergrad, it was taught as part of an Abstract Algebra class
  65. Virality vs. Value
  66. Take Action
  67. The funny thing about Differential Equations is that pretty much everyone who teaches it has a different conception of what it covers.
  68. The Most Transferable Skill
  69. The Biggest Flex
  70. How To Do a Ton of Intense Work
  71. Channel Your Anger Into Productivity
  72. Learning Rate Differences Exist
  73. How Learning Debt Starts
  74. Founder Involvement Is What Keeps Everything On The Rails
  75. Just Get Started
  76. The LLM Training Corpus is a Small Subset of All Knowledge
  77. The Fall of Olympus: Education Edition
  78. The Best Teachers are Supportive Hard-Asses
  79. Students Need To Memorize Math Facts
  80. My Rule of Thumb for Lifestyle Upgrades
  81. Every Kid Should Receive Individualized Instruction
  82. If You Want A School To Make An Exception, Then You Have to Give Them No Way Out
  83. A Little Gamification Can Go A Long Way
  84. Just Solve Today’s Problems Today
  85. How To Get Eyes On Your Writing When You Have No Audience
  86. Why Train Up Your Writing?
  87. Yes, You Need To Work Through Concrete Examples
  88. I Didn’t Realize How Fragile My Coding Chops Were Until I Started Working on Real Production Systems
  89. The Cycle of AI
  90. If You’re In Science, Learn Math and Coding
  91. The Manifold Hypothesis in SAT Prep
  92. The Effect of Debating Without Foundational Knowledge In Place
  93. The Crux of the Confusion When People Think 0.999… is Less Than 1
  94. Nothing Really Matters Unless It Matters In The Long-Term
  95. It Comes Down To Force of Will
  96. The Core of a Super-Efficient Learning Process
  97. Performance Improvement is the First Priority
  98. How Would You Teach If Your Life Depended On It?
  99. It’s Too Easy To Gas Yourself Up Thinking About Your Percentile
  100. Overhelping: A Pathology of the Over-Involved Parent/Tutor
  101. Intuition Through Repetition
  102. Spaced Repetition 2.0: Accounting For and Discouraging Reference Reliance
  103. It’s Incredible How Poor Math Instruction Can Be
  104. One of the Most Demoralizing Things That Can Happen To a Student
  105. The Most Efficient Way To Successful Creative Production
  106. The Need to Practice Retrieving From Memory
  107. If You Bypass the Lift, You Bypass Learning
  108. The Hardcore Skills That Changed My Life
  109. Self-Reports of Learning Are Ridiculously Unreliable
  110. Most People Don’t Realize They Can Climb Skill Trees
  111. There Is No Shortcut
  112. Don’t Put Off Training Because You Think Your Goals Are Too Far Away
  113. A ML Model is a Decent First-Order Approximation of a Human Learner
  114. How To Overcome The Suck
  115. Why Try When Others Could Do Better?
  116. It Always Becomes a Battle of Willpower By The End
  117. Don’t Bloat the Feedback Loop
  118. The Whole Point of Instruction
  119. Pure Math Skills Have Proven Useful in My Applied Work
  120. Teaching Isn’t About How Much Fun The Teacher Has
  121. Tie Your Comfort to Real, Tangible Value
  122. Why You Need to Develop Automaticity in Procedures Even If You Only Care About Proofs/Derivations
  123. You’re going to be sharing your pie with AI, and your feelings about this will depend on your ability to expand the pie.
  124. Domain Expertise, Math/Coding, Communication.
  125. Filling In Your Foundations is the Difference Between Omitted Steps Being Minor Potholes vs Uncrossable Chasms
  126. If You Struggle to Train Consistently, Do It Immediately After Waking Up
  127. Don’t Get Hung Up on Youth Competitions
  128. Q&A: Is there a maximum daily math study duration after which one risks causing more harm than good?
  129. Turn The Magical Into The Mechanical
  130. Never Come Up Empty-Handed
  131. Protect The Habit
  132. Critique of Article: “Why Tacit Knowledge is More Important Than Deliberate Practice”
  133. You Don’t Need a Million Different Explanations
  134. Models Are Just Data Interpolators
  135. A Huge Component of Learning System Design
  136. Variety by Way of Repetition
  137. The Pull Towards Mediocrity in Education
  138. The Quickest Way to an Existential Crisis in Your Math Education
  139. Plan Your Broad-Strokes Journey Top-Down, But Carry Out the Granular Steps Bottom-Up
  140. A little extra consistency × a little extra time = a massive increase in volume and progress.
  141. Keep Your Foot On The Gas
  142. Why I’m So Against Discovery Learning
  143. How to Get Learners to Perceive Their Progress
  144. Experts Perceive Differently
  145. Why Learning Efficiency is Such a Big Lever in Maximizing Student Potential
  146. The #1 Killer of Math Motivation is Not Disinterest But Overwhelm
  147. An LLM is like a Well-Read, Skilled Junior Who is Missing a LOT of Context
  148. Deliberate Practice Has a Strict Definition
  149. The Biggest Lever You Can Pull to Make Learning Fun is to Not Overwhelm the Learner
  150. Self-Reported Study Time is Incredibly Unreliable
  151. The Best Mental Model for Serious Structured Learning
  152. The Worst Possible Way to Fail at Learning Math
  153. Why Math Educators Should Care About Talent Development
  154. The Goal is NOT to Prove You’re Smart, it’s to Make Problems Go Away
  155. Yes, you need to spin up on foundational knowledge. No, you are not an exception.
  156. A Sanity Check for Effective Study Techniques
  157. No Train, No Gain
  158. Active Problem-Solving is Where The Learning Happens
  159. The #1 Trick for Super-Productivity
  160. Schooling vs Talent Development
  161. 3 Common Areas of Confusion in Talent Development
  162. Don’t have a passion? Go create one.
  163. There Are No Shortcuts in Talent Development
  164. The Vicious Cycle of Context Overload
  165. Actively Doing is the Key to Alpha
  166. It’s Amusing How Fake Some University Syllabi Are
  167. How to Get High on Math
  168. How To Get Stuff To Stick In Your Brain
  169. It’s Memory All The Way Down
  170. How to Mitigate Intellectual Body Dysmorphia
  171. Efficient Lab Activities are About Demonstration, Not Discovery
  172. It’s Not About the Type of Motivation, It’s About the Total Amount of Motivation
  173. The Future of Math Facts Practice on Math Academy
  174. Make the Habit Easily Repeatable
  175. Some Tips for Junior Devs
  176. It’s Rare to Find Computation Walkthroughs in ML Learning Resources
  177. Math Learning Shouldn’t Be Left to Luck
  178. Make it So Easy a Kid Can Learn It
  179. Get Yourself In A Position Where You Can Eat Risk
  180. Get On the Right Team
  181. The cost of college is so out of whack that it would actually be cheaper to hire a personal tutor for every hour you’d spend in a college lecture.
  182. It’s Good to Self-Explain, But You Still Need To Solve Problems
  183. Selected Blog Posts About the State of Math Education
  184. Do. Not. Transcribe. From. The. Reference.
  185. Max-Efficiency Learning in a Nutshell
  186. If You’re Making Silly Mistakes Then You Need More Practice
  187. How To Mitigate Nonsense from Lazy/Adversarial Students
  188. Why I Haven’t Been Using AI Coding Agents, and Why I Code Quant/Algo-Heavy Infra in Node.js
  189. How to Allocate Your Bandwidth While Searching for Your Mission
  190. Failure Is NOT the Key to Success
  191. Obsession is the Root of Both Greatness and Demise
  192. Don’t Overreact to Bad Days
  193. Failure Modes in People Who Develop Math Skills but Don’t Capitalize On Them via Coding
  194. What Math To Learn Next After Calculus
  195. Good Cop, Bad Cop: One of the Most Challenging Parts of Teaching/Coaching
  196. Efficient Learning is About Balancing a Tradeoff
  197. Why It’s So Important to Actively Work Out Problems, Not Just Consume Content, During the Learning Process
  198. Why Being Out Of Your Depth Is So Bad
  199. When Should Students Memorize Math Facts?
  200. “Learning” Info Without Practicing Reproducing it is Not Really Learning
  201. If you don’t practice retrieving information from memory, it dissipates almost entirely.
  202. The Importance of Automaticity is Blatantly Obvious Yet Weirdly Controversial In Education
  203. Mistakes That Knowledgeable People Make When Teaching
  204. I think incrementally working through a knowledge graph (KG) is the right mental model for CS education too, not just math.
  205. Love What You Do
  206. Go Through the Question Bank Breadth-First, not Depth-First
  207. Top-down’s fine for playing around. You’ll run into walls, but don’t give up — go bottom-up to get unstuck.
  208. Just Do The F*cking Work
  209. How Do You Increase a Student’s Ability to Make Mental Leaps?
  210. Pictures are Valuable in Math Learning, but They’re Often Overvalued
  211. A Thermodynamic Engine that Converts Mental Effort into Skills/Knowledge
  212. The “Progress Equals Pressure” Formula
  213. The Day of Reckoning in Math Learning
  214. Why is there sometimes resistance to automaticity in education?
  215. Competition as a Means of Collaboration
  216. Writing is a Skill that Can Be Trained
  217. To date, the total amount that Math Academy has spent on paid X/Twitter posts is…
  218. What Implies Learning
  219. Some Pitfalls to Watch Out For when Learning From Projects
  220. Enjoyment is a Second-Order Optimization
  221. Critique of Article: “The Problems With Deliberate Practice”
  222. Resolving Confusion about Deliberate Practice
  223. My Next Big Modeling Project: Behavior Coaching
  224. True active learning means…
  225. Spaced repetition is more than memorization – it’s also generalization.
  226. How “Kicking the Can Down the Road” Happens in Education
  227. How to Decide What Features to Build
  228. How to Learn Machine Learning: Top Down or Bottom Up?
  229. Book Review: Developing Talent in Young People by Benjamin Bloom
  230. Ability is Built, Not Unlocked
  231. The Best Description of Explicit Instruction I’ve Ever Heard
  232. The 2 Most Common Ways that People Get Retrieval Practice Wrong
  233. Why Extrinsic Motivation Matters
  234. The Anatomy of Effective Practice
  235. Different Students Need Different Amounts of Practice
  236. Overcoming the Paradox of Serious Training
  237. Hard Work vs Luck is a False Dichotomy
  238. Math is Overpowered When Combined with Other Expertise but Underpowered Alone
  239. “Following Along” vs Learning
  240. The Issue with Watered-Down Math Courses
  241. If You Want to Learn Math, You Can’t Shy Away from Computation
  242. Silly Mistakes are Still Mistakes
  243. Why 4x8 and 6x8 Are, Perhaps Surprisingly, Some of the Hardest Multiplication Facts for Students to Remember
  244. There is No Such Thing as Low-Effort Learning
  245. Spaced Repetition vs Spiraling
  246. Learning vs Feeling
  247. Leveraging Cognitive Learning Strategies Requires Technology
  248. The Utility of Gamification in Learning
  249. Cognitive Science of Learning: The Testing Effect (Retrieval Practice)
  250. Cognitive Science of Learning: Interleaving (Mixed Practice)
  251. Cognitive Science of Learning: Spaced Repetition (Distributed Practice)
  252. Layering: Building Structural Integrity in Knowledge
  253. Cognitive Science of Learning: Minimizing Associative Interference
  254. Cognitive Science of Learning: Developing Automaticity
  255. Cognitive Science of Learning: Minimizing Cognitive Load
  256. A Brief History of Mastery Learning
  257. The Neuroscience of Active Learning and Automaticity
  258. Active Learning: If You’re Active Half the Time, That’s Still Not Enough
  259. What Counts as Active Learning?
  260. The Double-Edged Nature of Hierarchical Knowledge
  261. You Know it’s Edutainment When…
  262. Recommended Language, Tools, Path, and Curriculum for Teaching Kids to Code
  263. Tips for Learning Math Effectively
  264. Myths and Realities about Educational Acceleration
  265. Effective Learning Requires Intense Effort
  266. Effective Learning Does Not Emulate the Professional Workplace
  267. The Story of the Science of Learning
  268. The Tragedy of the Commons in Education
  269. How I Won a Heat Capacitor Competition Without a Heat Capacitor
  270. According to Feynman himself, his classes were a failure for 90% of his students.
  271. Business Lessons from Science Fair
  272. Why I Don’t Worship at the Altar of Neural Nets
  273. Selecting a Good Problem to Work On
  274. The Counterintuitive Nature of Effective Learning Strategies
  275. The Past Year of Scaling Has Been Transformative For Us
  276. Announcing our ACT Math Fundamentals Course
  277. An Example of Grade Inflation
  278. 8 Failure Modes to Watch Out for in Math Learning Resources
  279. Q&A: How I Feel About AI in Math
  280. Think of Motivation as Something You Earn
  281. If It’s Good For You, Just Do It
  282. A Really Inefficient Way To Learn
  283. How To Manufacture Self-Sustaining Motivation
  284. It’s true in relationships, it’s true in work.
  285. The Secret To Success in Life is Consistent Effort
  286. Never Underestimate the Value of Being Pointed in the Right Direction
  287. Mathematics Is Closer to Weightlifting Than to a Movie
  288. It’s Wild How Fast Kids Can Progress
  289. A Huge Number of Capable Kids Never Discover Who They Could Have Become
  290. Get Right Back on the Horse
  291. Decrease Your Cycle Time
  292. You Don’t Have to Be Born an Outlier to Become an Outlier
  293. Most students could learn several times faster than most schools assume.
  294. Anyone who has been enjoying the Math Academy Podcast should also check out this one
  295. You Can Only Outsource The Thinking You’ve Already Thought
  296. Wages Are Not For Spending
  297. Q&A: Have you started using AI coding tools?
  298. Pure Vibe Coding is Like a Ponzi Scheme
  299. Replace Your Addiction With Math
  300. Q&A: Do I Still Learn New Things Outside Of Work?
  301. Q&A: How I Find Time To Tweet
  302. Pain of Action Versus Pain of Regret
  303. The Bar is Being Lowered
  304. Your Story is a Hero’s Journey
  305. The Utility of Hard Truths
  306. What’s Even Harder is NOT Doing The Work
  307. Don’t Dwell On The Past
  308. The Highest Job Security
  309. Learning Probability Makes You See The World Differently
  310. Meta-Work is Not The Work
  311. Entrepreneurs Have to Be Self-Deluding
  312. Q&A: Do you use LLMs at all in the writing process?
  313. Effective Teaching Puts Business First, Fun Second
  314. Pascal’s Wager: Training Edition
  315. Integrating Coding Into Our Math Curriculum
  316. Why Skill-Building Needs To Come First in Math Education
  317. Edge Cases Are Interesting If You’re In A Domain You Love
  318. Math Academy High Schoolers Taking Math Courses At Caltech
  319. Q&A: Why focus on filling in prerequisite knowledge? Why not focus on improving generalization ability directly?
  320. Q&A: How do you, personally, balance work and upskilling while interacting on X?
  321. Q&A: Is Mental Overtraining A Thing?
  322. Q&A: What’s the limiting factor to MA producing more courses more quickly?
  323. Q&A: Reinventing vs. Reimplementing The Wheel
  324. There is Always a Bad Excuse to Rationalize Not Skilling Up
  325. The Performance Formula
  326. Q&A: What does your day-to-day work look like these days? Do you find yourself doing math explicitly?
  327. Q&A: Combining Math and LLMs
  328. A Recent Lightbulb Moment in my Math Academy Work
  329. Q&A: Is there really no evidence of a method to increase working memory?
  330. Compound Hard Work and Luck
  331. John Wooden Might Be the Most Quotable Figure in the Entire Field of Talent Development
  332. Random Graphs are Complicated
  333. If You’re Not Both Technical and a Domain Expert, Then You’re Underpowered
  334. Why Would You Use An LLM For That
  335. Learning Doesn’t Have to Be Synchronized for Cameraderie to Occur
  336. Your Missing Foundations Will Wait For You
  337. Challenges One Might Encounter While Leveraging Effective Pedagogical Techniques
  338. A Rant About Typical ML Learning Resources
  339. On Debugging
  340. Coding is All About Making Locally Correct Decisions
  341. What’s the Value-Add of University?
  342. Information Doesn’t Stay In Your Brain Unless You Practice Retrieving It From Memory
  343. Does Math Have a Genetic Component?
  344. Exploration and Skill Development
  345. An Interview Question for LLM Research Assistants
  346. Everything Matters
  347. Failure Modes Shared Between Human Learners and Machine Learning Models
  348. The Funniest Math Academy Username of All Time
  349. Combining Math and Coding
  350. Q&A: But Don’t You Need 10,000 Hours To Learn Math?
  351. We’re Working On Streaks!
  352. What Raw Intuition Looks Like, Without Articulative Clothing
  353. The Most Powerful Machine Learning Model You Will Ever Use Is Your Brain
  354. “I am twenty-something years old, is it too late for me to learn math/coding/etc.?”
  355. New Year’s Resolution Motivation Compilation (2025)
  356. It’s Nobody’s Responsibility to Continue Developing Your Talent
  357. A student doesn’t have to favor math over video games to be a successful math learner.
  358. Lots of students study way more than necessary while simultaneously learning way less than necessary.
  359. You Have to Concentrate Your Efforts Directly On The Goal
  360. Try to Recall From Memory
  361. How Grade Inflation Flies Under the Radar
  362. Playing Around Does NOT Speed Up the Learning Process
  363. Comfortable Fluency in Consumption is Not a Proxy for Learning
  364. Introduction to the Expertise Reversal Effect
  365. Thoughts About Prerequisite Knowledge
  366. You Are a Car
  367. The Best Neural Nets Textbook That I’ve Seen So Far
  368. Progress is Enjoyable In Itself
  369. It’s not about how much you put in, it’s about how much you get out.
  370. The Image I Want to Put in People’s Minds When They Think About Edtech
  371. Schooling and Talent Development are Completely Different Things
  372. Top 3 Skills
  373. Start simple. Whatever gets the ball rolling.
  374. On Writing Good Code
  375. The One Skill You Can Acquire By Passively Consuming Information
  376. What Math To Learn for Skill Stacking
  377. Automaticity is the Road to Intuition
  378. Learning Advanced Math/Coding Opens Doors You Don’t Even Know Exist
  379. The Most Common Failure Mode in Stereotypically Smart People
  380. Two of the Biggest Myths in Education
  381. Why I Prefer the Name “Talent Development” over “Education”
  382. Wish I had more RAM
  383. Seek Virtuous Cycles
  384. The Importance of High Standards for Students
  385. Hard things only remain hard to the extent that you avoid them and do easy things instead.
  386. The Key to Confidence
  387. Why Train?
  388. Why Type Checking is Not so Helpful in Quant/Algo-Heavy Coding
  389. Timescales For Tracking
  390. Learning Math is Like Climbing a Ladder
  391. Analytics Projects Can Be Tricky to Debug
  392. The Key to Learning Fast
  393. Don’t Worry About the Long Term Too Much
  394. Be a Builder, Not Just a Fighter
  395. Why Do So Few People Know About the Greatest Educational/Career Life Hack?
  396. If You Can’t Find a Job That Really Excites You…
  397. Passing a Class at School Does Not Imply Having Learned All The Material
  398. It’s kind of disappointing how many shitty takes on learning come from within the field of education itself.
  399. Hand Computation, Conceptual Debugging, and Coding Projects
  400. Simple, Representative Concrete Examples
  401. ML Courses can Vary Massively in their Coverage
  402. Math Education is a War Zone
  403. Music Concerts: An Unexpected Setting to See Startup/Product Ideas in Action
  404. Frequent, low-stakes quizzes are such a powerful learning tool.
  405. Q&A: Who Does What at Math Academy
  406. It’s easy to think you know the prerequisites when in fact you don’t.
  407. How to Maximize Performance on a Standardized Math Test
  408. What Math Students Need Beyond the “Why”
  409. You will never achieve your goals unless you transform yourself into a person who is capable of achieving them.
  410. The Most Effective Way to Motivate Students to Learn Math
  411. Intuition Behind Polynomial Numerators in Partial Fractions
  412. One Of the Most Comforting Things That a Struggling Math Learner Can Hear
  413. On “Hitting a Wall”
  414. Q&A: When Did You Start Thinking About Optimizing Learning?
  415. Rationale Behind My Choice of Calisthenics Exercises
  416. Sources of Motivation in Successful Math Learners
  417. How do you apply math to CS when so many software engineers say that there is not that much math in coding?
  418. What’s the Highest Sustainable Daily XP on Math Academy?
  419. How to Know When You are Practicing at the Edge of Your Ability
  420. Trick to Check Equality of Expression Containing Subscripts Using a Basic LaTeX Expression Evaluator
  421. Learning Loss, Grade Inflation, and Radical Constructivism
  422. Want to know about how the science of learning is missing from teacher education?
  423. Record for Most Work Done on Math Academy on a Single Date (as of July 2024)
  424. Levels of Mathematics
  425. What is learning, at a physical level in the brain?
  426. Individual Variation in Working Memory Capacity (WMC): a First Step Down the Research Rabbit Hole
  427. The Value of Foundational Math Knowledge in Machine Learning
  428. The Tip of Math Academy’s Technical Iceberg
  429. The Worst Segment of the K-12 Mediocrity
  430. Paper Idea: A Theory of Optimal Learning Efficiency in Hierarchical Knowledge Structures
  431. When Does the Learning Happen?
  432. The Goal of Active Learning is NOT to Increase Cognitive Load
  433. A Quick Trick for Finding a Matrix Transformation Formula
  434. Bloom’s 3 Stages of Talent Development
  435. What Mathematics Can Teach Us About Human Nature
  436. What to Do When Math Gets Too Hard
  437. Estimating a Visitation Interval: an Exercise in Bivariate Bayesian Statistics
  438. Recreational Mathematics: Why Focus on Projects Over Puzzles
  439. Intuiting Adversarial Examples in Neural Networks via a Simple Computational Experiment
  440. Should Students be Asked to Regurgitate Known Proofs?
  441. Most Students Don’t Even Pay Attention During Lectures
  442. What To Do Leading Up to a Standardized Exam Like AP Calculus BC
  443. Subtle Things to Watch Out For When Demonstrating Lp-Norm Regularization on a High-Degree Polynomial Regression Model
  444. Why Poking Around Wikipedia Doesn’t Move The Needle on Math Learning
  445. How Much Math Do You Need to Know for Machine Learning?
  446. The Only Way to Teach a More Sophisticated Technique
  447. How I Got Started with Calisthenics
  448. The Easiest Way to Remember Closed vs Open Interval Notation
  449. Bloom’s Two-Sigma Problem
  450. A Common Source of Student Mistakes
  451. Ambiguous Absolute Value Expressions
  452. My Go-To Math Riddle: How Many Squares are in a 10 x 10 Grid?
  453. Study Sessions Should be Short and Frequent as Opposed to Long and Sparse
  454. Educational resources commonly address slant asymptotes. Why not general polynomial asymptotes?
  455. Can You Automate a Math Teacher?
  456. The Abstraction Ceiling: Why it’s Hard to Teach First-Principles Reasoning
  457. When Can You Manipulate Differentials Like Fractions?
  458. How to Look Up the Meaning of an Unknown Math Symbol or Expression
  459. How to Remember Type I, II, and III Regions in Multivariable Calculus
  460. Minimalist Strength Training, Phase 2: Gaining Mass
  461. Minimalist Strength Training, Phase 1: Getting Ripped
  462. Quants vs Systems Coders
  463. Memory vs Time Graphs
  464. The 5 Breeds of Quants
  465. From Procedures to Objects
  466. The Ultimate High School Computer Science Sequence: 9 Months In
  467. Tips for LaTeX Math Formatting
  468. Path Dependency in Multivariable Limits
  469. Thales’ Theorem
  470. Trick to Apply the Chain Rule FAST - Peeling the Onion
  471. CheckMySteps: A Web App to Help Students Fix their Algebraic Mistakes
  472. Solving Tower of Hanoi with General Problem Solver
  473. A Game-Theoretic Analysis of Social Distancing During Epidemics
  474. Making Indirect Interactions Explicit in Networks
  475. Book Summary: Memory Evolutive Systems
  476. Introduction to Computers
  477. The Brain in One Sentence
  478. Shaping STDP Neural Networks with Periodic Stimulation: a Theoretical Analysis for the Case of Tree Networks
  479. On the Contrasting Educations and Outcomes of Ben Franklin and Montaigne
  480. A Brief Overview of Spike-Timing Dependent Plasticity (STDP) Learning During Neural Simulation
  481. A Visual, Inductive Proof of Sharkovsky’s Theorem
  482. Building an Iron Man Suit: A Physics Workbook
  483. The Physics Behind an Egg Drop: A Lively Story
  484. A Formula for the Partial Fractions Decomposition of $x^n/(x-a)^k$
  485. Sound Waves
  486. Detecting Dark Matter
  487. Evidence for the Existence of Dark Matter
  488. The Math Academy Way
  489. Some Short Introductory Coding Exercises
  490. Converting Between Binary, Decimal, and Hexadecimal
  491. Recursive Sequences
  492. Simulating Coin Flips
  493. Roulette Wheel Selection
  494. Cartesian Product
  495. Brute Force Search with Linear-Encoding Cryptography
  496. Solving Magic Squares via Backtracking
  497. Estimating Roots via Bisection Search and Newton-Raphson Method
  498. Single-Variable Gradient Descent
  499. Multivariable Gradient Descent
  500. Selection, Bubble, Insertion, and Counting Sort
  501. Merge Sort and Quicksort
  502. Basic Matrix Arithmetic
  503. Reduced Row Echelon Form and Applications to Matrix Arithmetic
  504. K-Means Clustering
  505. Tic-Tac-Toe and Connect Four
  506. Euler Estimation
  507. SIR Model for the Spread of Disease
  508. Hodgkin-Huxley Model of Action Potentials in Neurons
  509. Hash Tables
  510. Simplex Method
  511. Linear, Polynomial, and Multiple Linear Regression via Pseudoinverse
  512. Regressing a Linear Combination of Nonlinear Functions via Pseudoinverse
  513. Power, Exponential, and Logistic Regression via Pseudoinverse
  514. Overfitting, Underfitting, Cross-Validation, and the Bias-Variance Tradeoff
  515. Regression via Gradient Descent
  516. Multiple Regression and Interaction Terms
  517. K-Nearest Neighbors
  518. Naive Bayes
  519. Breadth-First and Depth-First Traversals
  520. Distance and Shortest Paths in Unweighted Graphs
  521. Dijkstra's Algorithm for Distance and Shortest Paths in Weighted Graphs
  522. Decision Trees
  523. Introduction to Neural Network Regressors
  524. Backpropagation
  525. Canonical and Reduced Game Trees for Tic-Tac-Toe
  526. Minimax Strategy
  527. Reduced Search Depth and Heuristic Evaluation for Connect Four
  528. Introduction to Blondie24 and Neuroevolution
  529. Reimplementing Fogel's Tic-Tac-Toe Paper
  530. Reimplementing Blondie24
  531. Reimplementing Blondie24: Convolutional Version
  532. N-Dimensional Space
  533. Dot Product and Cross Product
  534. Lines and Planes
  535. Span, Subspaces, and Reduction
  536. Elimination as Vector Reduction
  537. N-Dimensional Volume Formula
  538. Volume as the Determinant of a Square Linear System
  539. Shearing, Cramer's Rule, and Volume by Reduction
  540. Higher-Order Variation of Parameters
  541. Linear Systems as Transformations of Vectors by Matrices
  542. Matrix Multiplication
  543. Rescaling, Shearing, and the Determinant
  544. Inverse Matrices
  545. Eigenvalues, Eigenvectors, and Diagonalization
  546. Recursive Sequence Formulas via Diagonalization
  547. Generalized Eigenvectors and Jordan Form
  548. Matrix Exponential and Systems of Linear Differential Equations
  549. Evaluating Limits
  550. Limits by Logarithms, Squeeze Theorem, and Euler's Consant
  551. Derivatives and the Difference Quotient
  552. Power Rule
  553. Chain Rule
  554. Properties of Derivatives
  555. Derivatives of Non-Polynomial Functions
  556. Finding Local Extrema
  557. Differentials and Approximation
  558. L'Hôpital's Rule
  559. Antiderivatives
  560. Finding Area
  561. Substitution
  562. Integration by Parts
  563. Improper Integrals
  564. Separation of Variables
  565. Slope Fields and Euler Approximation
  566. Substitution
  567. Characteristic Polynomial
  568. Undetermined Coefficients
  569. Integrating Factors
  570. Variation of Parameters
  571. Geometric Series
  572. Tests for Convergence
  573. Taylor Series
  574. Manipulating Taylor Series
  575. Solving Differential Equations with Taylor Series
  576. Solving Linear Equations
  577. Slope-Intercept Form
  578. Point-Slope Form
  579. Standard Form
  580. Linear Systems
  581. Standard Form
  582. Factoring
  583. Quadratic Formula
  584. Completing the Square
  585. Vertex Form
  586. Quadratic Systems
  587. Linear Inequalities in the Number Line
  588. Linear Inequalities in the Plane
  589. Quadratic Inequalities
  590. Systems of Inequalities
  591. Standard Form and End Behavior
  592. Zeros of polynomials
  593. Rational Roots and Synthetic Division
  594. Sketching Graphs
  595. Polynomial Long Division
  596. Horizontal Asymptotes
  597. Vertical Asymptotes
  598. Graphing with Horizontal and Vertical Asymptotes
  599. Graphing with Slant and Polynomial Asymptotes
  600. Radical Functions
  601. Exponential and Logarithmic Functions
  602. Absolute Value
  603. Trigonometric Functions
  604. Piecewise Functions
  605. Shifts of functions
  606. Rescalings
  607. Reflections
  608. Inverse Functions
  609. Compositions
  610. course page
  611. Horizontal and Vertical Lines
  612. Slanted Lines
  613. Absolute Value
  614. Parabolas
  615. Sine Waves
  616. Graphing calculator drawing roots
  617. Shading with Sine
  618. Euclidean Ellipses
  619. Non-Euclidean Ellipses
  620. Rotation
  621. Lissajous Curves
  622. Composition Waves and Implicit Trig Patterns
  623. What is AI
  624. The First Wave: Reasoning as Search
  625. The Second Wave: Expert Systems
  626. The Third Wave: Computation Power and Neural Networks
  627. Cutting Through the Hype
  628. Strings, Ints, Floats, and Booleans
  629. Lists, Dictionaries, and Arrays
  630. If, While, and For
  631. Functions
  632. Naive Bayes
  633. MAP and MLE
  634. Linear Regression
  635. Support Vector Machines
  636. Neural Networks
  637. Decision Trees
  638. Ensemble Models
  639. Preamble
  640. Algorithm
  641. Software
  642. Use-Cases at Ayasdi
  643. Use-Cases at Aunalytics
  644. Homotopy; Approximation; Homology; Persistence
  645. Software
  646. Cardiac Output
  647. Understanding Plaque Buildup
  648. Modeling Tumor Growth
  649. Rocket Propulsion
  650. Rendering 3D Computer Graphics
  651. Physics Engines in Video Games
  652. Optimization via Gradient Descent
  653. Maximizing Profit
  654. Continuously Compounded Interest
  655. The Man who "Broke" Math
  656. The Newton-Leibniz Controversy
  657. A Failure of Intuition
  658. Derivatives in String Art
  659. Calculating the Horsepower of an Offensive Lineman
  660. Functions
  661. Intuiting limits
  662. Derivatives
  663. Integrals
  664. Sequences
  665. Calisthenics
  666. Music
  667. tutoring
  668. toy projects
  669. self study