The Greatest Educational Life Hack: Learning Math Ahead of Time
2024 Apr | Learning math early guards you against numerous academic risks and opens all kinds of doors to career opportunities.

The Story of Math Academy’s Eurisko Sequence: the Most Advanced High School Math/CS Sequence in the USA
2023 May | During its operation from 2020 to 2023, Eurisko was 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).

But WHERE do the Taylor Series and Lagrange Error Bound even come from?!
2019 Dec | An intuitive derivation.

Recommended Language, Tools, Path, and Curriculum for Teaching Kids to Code
2024 Jan | I'd start off with some introductory course that covers the very basics of coding in some language that is used by many professional programmers but where the syntax reads almost like plain English and lower-level details like memory management are abstracted away. Then, I'd jump right into building board games and strategic game-playing agents (so a human can play against the computer), starting with simple games (e.g. tic-tac-toe) and working upwards from there (maybe connect 4 next, then checkers, and so on).

Tips for Learning Math Effectively
2024 Jan | Solving problems, building on top of what you've learned, reviewing what you've learned, and quality, quantity, and spacing of practice.

Can You Automate a Math Teacher?
2023 Oct | For many (but not all) students, the answer is yes. And for many of those students, automation can unlock life-changing educational outcomes.

For Most Students, Competition Math is a Waste of Time
2023 Sep | If you look at the kinds of math that most quantitative professionals use on a daily basis, competition math tricks don't show up anywhere. But what does show up everywhere is university-level math subjects.

Business Lessons from Science Fair
2023 Jun | The most important things I learned from competing in science fairs had nothing to do with physics or even academics. My main takeaways were actually related to business -- in particular, sales and marketing.

My Experience with Teacher Credentialing and Professional Development
2023 Apr | It's centered around political ideology rather than the science of learning.

Why I Don’t Worship at the Altar of Neural Nets
2023 Jan | In order to justify using a more complex model, the increase in performance has to be worth the cost of integrating and maintaining the complexity.

Tips for Developing Valuable Models
2022 Oct | Stuff you don't find in math textbooks.

Learning vs Feeling
2024 Apr | The strongest people lift weights heavy enough to make them feel weak.

What Mathematics Can Teach Us About Human Nature
2024 Apr | It highlights the aversion that people have to doing hard things. People will do unbelievable mental gymnastics to convince themselves that doing an easy, enjoyable thing that is unrelated to their supposed goal somehow moves the needle more than doing a hard, unpleasant thing that is directly related to said goal.

Estimating a Visitation Interval: an Exercise in Bivariate Bayesian Statistics
2024 Mar | Loosely inspired by the German tank problem: several witnesses reported seeing a UFO during the given time intervals, and you want to quantify your certainty regarding when the UFO arrived and when it left.

Recreational Mathematics: Why Focus on Projects Over Puzzles
2024 Feb | There's only so much you can hone your math skills by working on a problem that someone else has intentionally set up to be well-posed and elegantly solvable if you think about it the right way.

Intuiting Adversarial Examples in Neural Networks via a Simple Computational Experiment
2024 Feb | The network becomes book-smart in a particular area but not street-smart in general. The training procedure is like a series of exams on material within a tiny subject area (your data subspace). The network refines its knowledge in the subject area to maximize its performance on those exams, but it doesn't refine its knowledge outside that subject area. And that leaves it gullible to adversarial examples using inputs outside the subject area.

Should Students be Asked to Regurgitate Known Proofs?
2024 Feb | Imitating without analyzing produces a robot / ape who can't think critically; analyzing without imitating produces a critic who can't act on their own advice.

Subtle Things to Watch Out For When Demonstrating Lp-Norm Regularization on a High-Degree Polynomial Regression Model
2024 Feb | Initial parameter range, data sampling range, severity of regularization.

The Only Way to Teach a More Sophisticated Technique
2024 Feb | ... is to present a problem where known simpler techniques fail.

How I Got Started with Calisthenics
2024 Jan | My training has been scattered and fuzzy until recently. Here's the whole story.

The Easiest Way to Remember Closed vs Open Interval Notation
2024 Jan | An oval () fits inside a rectangle [ ] with the same width and height.

A Common Source of Student Mistakes
2024 Jan | Many students who pattern-match will tend to prefer solutions requiring fewer and simpler operations, especially if those solutions yield ballpark-reasonable results.

Critique of Paper: An astonishing regularity in student learning rate
2023 Nov | It rests on a critical assumption that the amount of learning that occurs during initial instruction is zero or otherwise negligible, which is not true.

My Go-To Math Riddle: How Many Squares are in a 10 x 10 Grid?
2023 Nov | Q: Draw a 10 x 10 square grid. How many squares are there in total? Not just 1 x 1 squares, but also 2 x 2 squares, 3 x 3 squares, and so on. A: The total number of square shapes is the total sum of square numbers 1 + 4 + 9 + 16 + ... + 100.

Study Sessions Should be Short and Frequent as Opposed to Long and Sparse
2023 Nov | First, you want to form a habit. Second, you want to operate at peak productivity during your session. Third, you want to minimize the amount you forget between sessions.

Educational resources commonly address slant asymptotes. Why not general polynomial asymptotes?
2023 Oct | Answer: It's not very useful (not in practice, not in theory).

The Abstraction Ceiling: Why it’s Hard to Teach First-Principles Reasoning
2023 Oct | Everyone has some level of abstraction beyond which they are incapable of engaging in first-principles reasoning. That level is different for everyone, and it's not a hard threshold, but beyond it the time and mental effort required to perform first-principles reasoning skyrockets until first-principles reasoning becomes completely infeasible.

When Can You Manipulate Differentials Like Fractions?
2023 Oct | In general, you can manipulate total derivatives like fractions, but you can't do the same with partial derivatives.

How I Won a Heat Capacitor Competition Without a Heat Capacitor
2023 Oct | Won first place in a state-level competition by finding and exploiting a loophole in the points scoring logic.

How to Look Up the Meaning of an Unknown Math Symbol or Expression
2023 Sep | Drawing --> Latex commands --> ChatGPT summary --> Google more info

According to Feynman himself, his classes were a failure for 90% of his students.
2023 Sep | While some may view Feynman-style pedagogy as supporting inclusive learning for all students across varying levels of ability, Feynman himself acknowledged that his methods only worked for the top 10% of his students.

How to Remember Type I, II, and III Regions in Multivariable Calculus
2023 May | Type I pairs with the variable that runs vertically in the usual representation of the coordinate system. The remaining types are paired with the rest of the variables in ascending order.

Minimalist Strength Training, Phase 2: Gaining Mass
2023 Apr | Minor changes to increase workout intensity and caloric surplus.

Selecting a Good Problem to Work On
2023 Jan | Good problem = intersection between your own interests/talents, the realm of what's feasible, and the desires of the external world.

Minimalist Strength Training, Phase 1: Getting Ripped
2022 Oct | Daily 20-30 minute bedroom workout with gymnastic rings hanging from pull-up bar -- just as much challenge as weights, but inexpensive and easily portable.

Quants vs Systems Coders
2022 Oct | Two subtypes of coders that I watched students grow into.

The 5 Breeds of Quants
2022 Aug | ... are summarized in the following table.

From Procedures to Objects
2022 Aug | An aha moment with object-oriented programming.

The Ultimate High School Computer Science Sequence: 9 Months In
2021 Feb | In 9 months, these students went from initially not knowing how to write helper functions to building a machine learning library from scratch.

Tips for LaTeX Math Formatting
2020 Sep | How to avoid some of the most common pitfalls leading to ugly LaTeX.

Path Dependency in Multivariable Limits
2019 Dec | The behavior of a multivariable function can be highly specific to the path taken.

Thales’ Theorem
2019 Dec | Every inscribed triangle whose hypotenuse is a diameter is a right triangle.

Trick to Apply the Chain Rule FAST - Peeling the Onion
2019 Dec | A simple mnemonic trick for quickly differentiating complicated functions.

CheckMySteps: A Web App to Help Students Fix their Algebraic Mistakes
2019 May | A prototype web app to automatically assist students in self-correcting small errors and minor misconceptions.

Solving Tower of Hanoi with General Problem Solver
2019 May | A walkthrough of solving Tower of Hanoi using the approach of one of the earliest AI systems.

A Game-Theoretic Analysis of Social Distancing During Epidemics
2016 May | In a simplified problem framing, we investigate the (game-theoretical) usefulness of limiting the number of social connections per person.

Making Indirect Interactions Explicit in Networks
2016 Mar | Category theory provides a language for explicitly describing indirect relationships in graphs.

Book Summary: Memory Evolutive Systems
2016 Mar | Framing complex systems in the language of category theory.

Introduction to Computers
2015 Nov | The main ideas behind computers can be understood by anyone.

The Brain in One Sentence
2015 Nov | The brain is a neuronal network integrating specialized subsystems that use local competition and thresholding to sparsify input, spike-timing dependent plasticity to learn inference, and layering to implement hierarchical predictive learning.

Shaping STDP Neural Networks with Periodic Stimulation: a Theoretical Analysis for the Case of Tree Networks
2015 Aug | We solve a special case of how to periodically stimulate a biological neural network to obtain a desired connectivity (in theory).

On the Contrasting Educations and Outcomes of Ben Franklin and Montaigne
2015 May | Montaigne's education, strictly dictated by his parents and university studies, resulted in an isolative work with scholarly impact but limited public reach. Conversely, Benjamin Franklin's goal-oriented self-teaching led to influential creations and roles benefiting his community and nation.

A Brief Overview of Spike-Timing Dependent Plasticity (STDP) Learning During Neural Simulation
2015 Feb | Implementation notes for STDP learning in a network of Hodgkin-Huxley simulated neurons.

A Visual, Inductive Proof of Sharkovsky’s Theorem
2015 Jan | Many existing proofs are not accessible to young mathematicians or those without experience in the realm of dynamic systems.

Building an Iron Man Suit: A Physics Workbook
2014 Jul | A workbook I created to explain the math and physics behind an Iron Man suit to a student who was interested in the comics / movies.

The Physics Behind an Egg Drop: A Lively Story
2014 Jul | A workbook I created to explain the math and physics behind an egg drop experiment to a student who was interested in Lord of the Rings and Star Wars.

A Formula for the Partial Fractions Decomposition of $x^n/(x-a)^k$
2013 Aug | And a proof via double induction.

Sound Waves
2012 Dec | A brief overview of sound waves and how they interact with things.

Detecting Dark Matter
2012 Dec | A brief overview of the experimental search for dark matter (XENON, CDMS, PICASSO, COUPP).

Evidence for the Existence of Dark Matter
2012 Dec | Mass discrepancies in galaxies and clusters, cosmic background radiation, the structure of the universe, and big bang nucleosynthesis's impact on baryon density.