All Posts

2025

Learning is About Bridge-Building, Not Jumping
2025 Jun, ~550 words • The way you increase your ability to make mental leaps is not actually by jumping farther, but rather, by building bridges that reduce the distance you need to jump.

Recall First, Reason Second
2025 Jun, ~150 words • To build automaticity: instead of deriving/reasoning a result before applying it, force yourself to first recall the result from memory, and then justify the it afterwards. Recall first, reason second.

The Field of Education is Due For a Copernican Revolution
2025 May, ~5750 words • You'd think that teacher training programs would focus on the mechanics of learning, but instead they typically focus on ritualistic compliance. If we trained doctors like we do teachers, then we'd still be bloodletting. Teacher credentialing severely lacks rigor, and this lack of rigor leads to a massive loss in human potential. Students suffer for it, and it drives serious educators out of the profession. It attracts and supports the type of people who think it's more important to practice sharing circles than to learn about the importance and implementation of spaced review. When you make it your mission to maximize student learning -- including leveraging the learning-enhancing practice techniques that have been known, reproduced, and yet ignored by the education system for decades -- you realize that there is a massive amount of human potential being left on the table. Students can be learning way, way, way more than they currently are.

Protect The Habit
2025 May, ~150 words • When the time comes to get back into the swing of things, it's a lot easier to speed up a slow wagon that you're on, than to get back on a wagon that you've completely fallen off of.

A Huge Component of Learning System Design
2025 Apr, ~200 words • ... is that it has to be robust to all sorts of behavior arising from the various human emotional experiences associated with learning & intense training.

How Taxing Work Becomes Fun
2025 Apr, ~400 words • "Wait, am I... cracked? No way. But I just did this thing that I've seen cracked people do and I wasn't able to that before. Holy shit I'm actually getting cracked."

Math Academy’s Eurisko Sequence, 5 Years Later: Student Outcomes Emerging From the Most Advanced High School Math/CS Sequence in the USA
2025 Mar, ~2500 words • During its operation from 2020-23, 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). It's still early and the first cohort hasn't even graduated from college yet, but there have already been some amazing student outcomes in terms of college admissions, accelerated graduate degrees, research publications, and science fairs.

The Importance of Learning Your Prerequisites
2025 Feb, ~500 words • Mastery learning -- one of the most reliable, largest-effect-size techniques for elevating student learning outcomes -- centers on learning prerequisites. In fact, the famous Two-Sigma Problem is centered around the effectiveness of mastery learning.

On Debugging
2025 Feb, ~200 words • You want to peel back layers of weirdness.

Learning is Memory
2025 Feb, ~550 words • This might feel obvious, but many learners don't fully grasp the implications, and as a result, end up not actually learning much.

Experts Perceive Differently
2025 Feb, ~250 words • It's not just that the expert thinks differently from the novice. It's also that the expert literally perceives information differently to begin with. And the driving force behind this is long-term memory.

Long-Term Retention Requires Short-Term Forgetting
2025 Feb, ~200 words • Just like successfully lifting a heavy weight forces your body to adapt to strengthen muscles, successfully recalling a fuzzy memory (lengthy wait) forces your brain to adapt to strengthen memory.

Scraping Bits Podcast #137 (Round 4): Learning Math is Hard, Proof Writing, Which Order to Learn Math
2025 Feb, ~2.75h • [0:00] How to get stuff to stick in your head. The importance of retrieval practice: comfortable fluency in consuming information is not the same as learning. Making connections to existing knowledge and/or emotions, exploring edge-cases in your own understanding. How to get stuff to actually enter your head in the first place: the importance of prerequisite knowledge.
[~19:00] Math Academy's upcoming Machine Learning and programming courses. Closing the loop on the pipeline from learning math to producing seriously cool ML/CS projects. How to get learners to persist through that pipeline at scale by breaking it up into incrementally simple steps.
[~40:00] Why it's worth learning proof-writing if you want to do any kind of mathy things in the future (including any sort of applied math). When to make the jump into proof-writing. What learners typically find challenging about proof-writing.
[~53:00] The advantages and challenges of modeling the world with differential equations. The importance of physics-y intuition about how the world works, what features actually matter enough to be incorporated into your model, and how much approximation you can get away with.
[~1:14:00] The experience of diving down the deep trench of mathematics (and also coming back to concrete everyday life).
[~1:22:00] The advantages and challenges of modeling the world with probability and game theory. The importance of understanding human nature and deviations from probabilistic / game-theoretic rationality.
[~1:33:00] The importance of getting through the grindy stage of things, especially at the beginning when you have no data points to look back at to see the transformation underway. You often need to stick with it for several months, not just several days or even several weeks, before you really see the transformation get underway.
[~1:54:00] Even after reaching a baseline level of initial mastery, it takes repeated exposures over time for knowledge to become fully ingrained. The importance of spaced review and continually layering / building new knowledge on top of old knowledge. Gaining procedural fluency opens up brainspace to think more deeply about components of the procedure.
[~2:25:00] People who hate on vs support others who are on an upskilling journey. Supporters tend to be more skilled themselves.
[~2:37:00] Progress update on the upcoming ML course. The mountain of positive sentiment online surrounding Math Academy. Our learners being incredibly supportive to each other. How calculus, linear algebra, and probability work together as prerequisites for machine learning.

Pick the Right Tool For the Job
2025 Feb, ~850 words • Avoid the vicious cycle of "I only use A because I don't like B because I can't remember how to use B because I only use A."

Q&A: But Don’t You Need 10,000 Hours To Learn Math?
2025 Jan, ~400 words • No. Math Academy's foundations series that goes from fractions to first-year university is benchmarked about 15,000 XP, about 250 hours of focused work. Of course, there's plenty of university math to dig your teeth into after that, but that's the order of magnitude of work we're talking.

What Learning Actually Is – at a Concrete, Physical Level in the Brain
2025 Jan, ~2650 words • Learning is a positive change in long-term memory. By creating strategic connections between neurons, the brain can more easily, quickly, accurately, and reliably activate more intricate patterns of neurons. Wiring induces a "domino effect" by which entire patterns of neurons are automatically activated as a result of initially activating a much smaller number of neurons in the pattern.

We’re Working On Streaks!
2025 Jan, ~200 words • Streaks are amazingly effective in just getting people to show up. It's a measure of habit/consistency, not progress -- but when effective training techniques and honest progress metrics are in place, streaks can truly push the needle on talent development.

CS Primer Show #23: MathAcademy and the efficient pursuit of mastery
2025 Jan, ~1h • Math Academy was originally built to support a school program. How come it also works so well for adults? What makes someone a student a good fit for Math Academy -- what's required to succeed? The idea of calibrating to student interest/motivation profiles in the future, just like we currently calibrate to student knowledge profiles.

Advice on Upskilling
2025 Jan, ~28600 words • Advice on consistency, skills, discipline, the grind, the journey, the team, the mission, motivation, and learning.

Retrieval Practice is F*cking Obvious
2025 Jan, ~1350 words • In the science of learning, there is absolutely no debate: practice techniques that center around retrieving information directly from one's brain produce superior learning outcomes compared to techniques that involve re-ingesting information from an external source.

2024

Fortify Your F*cking Fundamentals
2024 Dec, ~350 words • Skating around the rink will get you to a decent level of comfort in your basic skating skills, but being able to land jumps and spins will force a whole new level of robustness and fault-tolerance in those underlying skills. The same applies to knowledge in general.

Chalk and Talk Podcast #42: Math Academy: Optimizing student learning
2024 Dec, ~1.25h • The best podcast about Math Academy to date. If you want to understand what we're doing but don't have time to skim our 400+ page book, this episode sums it up in just an hour.
[~5:00] What is Bloom's two-sigma problem, how did Bloom attempt to solve it, why does it remain unsolved, and what is Math Academy's approach to solving it?
[~10:00] What is mastery learning? Why is full individualization important? What is our knowledge graph and how do we use it to implement mastery learning? How do we use data to improve our curriculum?
[~21:00] Why is it so important to be proficient on prerequisite skills? How does this relate to cognitive load? You see this same phenomenon everywhere outside of math education. Jason has a "learning staircase" analogy that elegantly encapsulates the core idea.
[~26:30] Why are worked examples so important? How do we leverage them?
[~29:30] Our perspective on memorization. Yes, students need to memorize times tables (among other things). No, they should not be expected to do this before they know what multiplication means (and how to calculate it using repeated addition).
[~33:30] Our perspective on the concrete-pictorial-abstract approach -- what it's useful for, and how it often gets misapplied.
[~41:00] What is spaced repetition? How does that work in a hierarchical body of knowledge like math? What are "encompassings" and why are they so important? How do we choose tasks that maximize learning efficiency? How do we calibrate the spaced repetition system to student performance and intrinsic difficulty in topics?
[~48:00] What is the testing effect (retrieval practice effect) and how do we leverage it? How do we gradually wean students off of reference material? How do quizzes play into this?
[~52:00] What does a student need to do to be successful on Math Academy? What does an adult need to do to facilitate their kid's success, and what are our plans to build more of this directly into the system?
[~55:30] We have a streamlined learning path specifically designed for adults, to get them up from foundational middle-school material up to university-level math in the most efficient way possible. What the learning experience often feels like for adults: it can be an emotional experience when you successfully learn math that you used to be intimidated by, and realize that the reason you struggled in the past wasn't because you're dumb but rather because you were missing prerequisites.
[~1:02:00] How did Math Academy get 8th graders getting 5's on the AP Calculus BC exam? What's our origin story? Can any student be successful on Math Academy? The students in our original Pasadena program -- what was their background, what did they learn in our program, and what are they doing now?
[~1:10:00] What's next for Math Academy? We want to become the ultimate math learning platform and empower the next generation of students with the ability to learn as much as they can.

Prereq Yo’ Self Before You Wreck Yo’ Self
2024 Dec, ~250 words • If you hammer prerequisite concepts/skills into your long-term memory, get it really solid and easy to retrieve, then you can lessen the load on your working memory, keep it below capacity, avoid getting "broken," and keep up with the game.

Actively Doing is the Key to Alpha
2024 Dec, ~100 words • Lots of people consume. Fewer people actively do. Even fewer people attempt challenging things. And even fewer people than that build up the foundational skills needed to succeed in doing those challenging things.

Top 3 Skills
2024 Nov, ~50 words • Math, coding, communication.

On Writing Good Code
2024 Nov, ~150 words • It's kind of amusing how some (novice) devs will boast/revel at how many lines of code they wrote while simultaneously cramming each line full with as much complexity as they can hold in working memory.

How I Would Go About Learning an Arbitrary Subject Where No Full-Fledged Adaptive Learning System is Available
2024 Nov, ~3600 words • I'm using an LLM to learn biology. My overall conclusion is that IF you could learn successfully, long-term, by self-studying textbooks on your own, and the only thing keeping you from learning a new subject is a slight lack of time, THEN you can probably use LLM prompting to speed up that process a bit, which can help you pull the trigger on learning some stuff you previously didn't have time for. BUT the vast, vast majority of people are going to need a full-fledged learning system. And even for that miniscule portion of people for whom the "IF" applies... whatever the efficiency gain of LLM prompting over standard textbooks, there's an even bigger efficiency gain of full-fledged learning system over LLM prompting.

The Future of Multistep Tasks on Math Academy
2024 Nov, ~35m • The primary key to motivation, goal-setting, understanding how to apply all the mad skills you’ve learned... it seems like it's all coming down to multisteps.

Demonstration of Setting Encompassing Weights
2024 Nov, ~25m • Encompassing weights control how much spaced repetition credit is propagated backwards from a more advanced topic to a simpler prerequisite topic when a student does a spaced repetition on the more advanced topic. Setting them is tedious, and it sucks, but it's completely necessary. That's sometimes what you’ve got to do when you want to build a solution that actually solves a problem. You have to put in the hard work.

Golden Nuggets Podcast #40 (Round 4): How Justin learns, new ML course, the magic of Twitter
2024 Nov, ~1h • Developing coding projects for the upcoming ML course. How would I go about learning a new subject where there's not an adaptive learning system available? The power of instructional guidance and a good curriculum Why I want to learn biology, why I haven't done so yet, how I wish that "Math Academy for biology" existed, and how I'm going to try to get myself over the hump by instructing an LLM how to tutor me at least more efficiently than a standard textbook. Strategies I use to improve my output, especially writing output. Viewing Twitter as a mode of production instead of a mode of consumption.

Make the Habit Easily Repeatable
2024 Nov, ~200 words • Start out with a volume of work that's small enough that you don't dread doing it again the next day.

Two of the Biggest Myths in Education
2024 Nov, ~150 words • Myth 1: Understanding amounts to something other than memory. Myth 2: Sudents can perform high-level skills without mastering low-level component skills.

Some Tips for Junior Devs
2024 Nov, ~500 words • 1) Learn SQL and how to use a debugger. 2) Never come up emptyhanded, even if you don't fix the bug.

Q&A #1: WM taxation, ML ETA, catching errors, coding tutorials, math vs calisthenics, foundations
2024 Nov, ~50m • When to take breaks. How to catch computational errors when working out math problems. There's a lack of resources for people who want to learn machine learning -- coding tutorials and math textbooks typically suck in their own ways. Generalizing the principles of effective learning & skill acquisition to contexts outside of math learning. What to do when you want to complete a project but your base level of knowledge is low.

Scraping Bits Podcast #116 (Round 3): Essential Math for Machine Learning, Math Intuition/Creativity, Proof Vs Computation
2024 Nov, ~3h • Why go through lots of concrete computational examples first before jumping into abstract proofs. The importance of having a zoo of concrete examples. The evolution of Math Academy's content. How to identify the right "chunks" of information and the right prerequisites for the knowledge graph. How to continue learning math as efficiently as possible after you finish all the courses on Math Academy. Frustrations with the lack of existing ML learning resources. How to know whether you're ready for ML projects or you need to learn more math. The blessing and curse of intellectual body dysmorphia. Harnessing reality distortion as a helpful tool. Journaling and documenting one's life.

Make it So Easy a Kid Can Learn It
2024 Nov, ~450 words • If you can scaffold the content so well that it creates a smooth, efficient learning experience for knucklehead kids, it's going to feel even smoother for more conscientious adults.

Get On the Right Team
2024 Nov, ~100 words • You can be the most committed and capable workhorse on the planet, but if you're on the wrong team, the only thing you'll change is your team's allocation of work.

Math is a Well-Defined Body of Knowledge
2024 Oct, ~150 words • At the end of the day, whether or not they know math comes down to whether or not they can apply techniques within that well-defined body of knowledge to solve problems within that well-defined body of knowledge.

Golden Nuggets Podcast #39 (Round 3): MA’s upcoming machine learning course
2024 Oct, ~2h • Rationale, vision, and progress on Math Academy's upcoming Machine Learning I course (and after that, Machine Learning II, and possibly a Machine Learning III). Design principles behind good math explanations (it all comes down to concrete numerical examples). Unproductive learning behaviors (and all the different categories: kids vs adults, good-faith vs bad-faith). How to get the most out of your learning tasks. Why I recommend NOT to take notes on Math Academy. What to try first before making a flashcard (which should be a last resort), and how we're planning to incorporate flashcard-style practice on math facts (not just times tables but also trig identities, derivative rules, etc). Using X/Twitter like a Twitch stream.

Failure Is NOT the Key to Success
2024 Oct, ~100 words • Failure only moves you towards success to the extent that you learn from it. Learning from failure means not making the same mistake over and over again.

How to Cultivate Discipline
2024 Oct, ~450 words • Tear down the unproductive habit and build up a counter-habit whose gravity eventually becomes strong enough to completely overtake the original habit.

Simple, Representative Concrete Examples
2024 Oct, ~150 words • When an algorithm or process feels magical, that's typically an indication you don't really understand what's happening under the hood.

Love What You Do
2024 Oct, ~250 words • If you don't love it, you'll never be able to keep up with the same volume of effective practice as someone who does have that love. You'll never outwork them.

Complete Individualization: an Often-Forgotten yet Critical Component of True Deliberate Practice
2024 Oct, ~450 words • There are many studies demonstrating a benefit of some component of deliberate practice, but these studies often get mislabeled or misinterpreted as demonstrating the full benefit of true deliberate practice. The field of education is particularly susceptible to this issue because it is impossible for a teacher with a classroom of students to provide a true deliberate practice experience without assistive technology that perfectly emulates the one-on-one pedagogical decisions that an expert tutor would make for each individual student.

ML Courses can Vary Massively in their Coverage
2024 Oct, ~400 words • I was coming in with the mindset of "we need to cover the superset of all the content covered in the major textbooks," which we're able to do quite well for traditional math. For ML, the rule will have to be amended to "we need to cover the superset of all the content covered in standard university course syllabi."

Just Do The F*cking Work
2024 Oct, ~250 words • At the end of the day you can either waste time debating your coach on the training regimen, or you can use that time to just put your head down and do some f*cking work.

Pictures are Valuable in Math Learning, but They’re Often Overvalued
2024 Sep, ~150 words • Pictures can help build mathematical intuition, but sometimes learners think they should fully visualize every single problem they solve, which actually handicaps their thinking. Math involves generalizing patterns in logically consistent ways, and the generalizations eventually go beyond what you can fully picture in your head.

The Importance of Hardcore Skills
2024 Sep, ~450 words • Hardcore skill development is necessary to do big things, it's one of the greatest social mobility hacks, and it gives you the ability/confidence to take risks knowing that you'll be okay.

Golden Nuggets Podcast #37 (Round 2): Balancing learning with creative output
2024 Sep, ~1.75h • Balancing learning math with doing projects that will get you hired. The role of mentorship. Designing social environments for learning. Why it's important to let conversations flow out of scope. Misconceptions about "slow and deep" learning. How to create career luck. The sequence of steps that led me to get involved in Math Academy (lots of people ask me about this so here's the precise timestamp: 1:13:45 - 1:24:45). Strategies to maximize your output. The "magical transition" in the spaced repetition process.

Career Hack: Put Pressure on Your Boss to Come Up with More Work For You
2024 Sep, ~250 words • One of the best career hacks -- especially for a junior dev -- is to knock out your work so quickly and so well that you put pressure on your boss to come up with more work for you. Your boss starts giving you work that they themself need to do soon, which is really the exact kind of work that's going to move your career forward.

Scraping Bits Podcast #107: Proof Writing, Discovering Math, Expert Systems, Learning Math Like a Language
2024 Sep, ~1.75h • Why aspiring math majors need to come into university with proof-writing skills. My own journey into learning math. Math as a gigantic tree of knowledge with a trunk that is tall relative to other subjects, but short relative to the length of its branches. The experience of reaching the edge of a subfield (the end of a branch): as the branch gets thinner, the learning resources get sh*tter, and making further progress feels like trudging through tar (so you have to find an area where you just love the tar). How to fall in love with a subject. How to get started with a hard subject that you don't love: starting with small, easy things and continually compound the volume of work until you're making serious progress. How to maintain focus and avoid distractions. The characteristics of a math prodigy that I've tutored/mentored for 6 years and the extent to which these characteristics can be replicated. How Math Academy's AI expert system works at a high level, the story behind how/why we created it, and the stages in its evolution into what it is now. How Math Academy's AI is different from today's conventional AI approach: expert systems, not machine learning. How to "train" an expert system by observing and rectifying its shortcomings. How to think about spaced repetition in hierarchical bodies of knowledge where partial repetition credit trickles down through the hierarchy and different topics move through the spaced repetition process at different speeds based on student performance and topic difficulty. Areas for improvement in how Math Academy can help learners get back on the workout wagon after falling off. Why you need to be fully automatic on your times tables, but you don't need to know how to do three-digit by three-digit multiplication in your head. Analogy between building fluency in math and languages. #1 piece of advice for aspiring math majors.

The Future of Education
2024 Sep, ~1350 words • To quote a Math Academy student: "The fastest and most rigorous progress will be made by individuals in front of their computers."

Five Steps to Becoming a Fully-Fledged Quantitative Software Engineer
2024 Sep, ~800 words • Once you get past steps 1-3, it's hard to find scaffolding. You can't just enroll in a course or pick up a textbook. The scaffolding comes from finding a mentor on a mission that you identify with and are well-suited to contribute to. And it can take a lot of searching to find that person and problem area that's the right fit.

Why Talent Development is Necessary in Math
2024 Sep, ~350 words • When students do the mathematical equivalent of playing kickball during class, and then are expected to do the mathematical equivalent of a backflip at the end of the year, it’s easy to see how struggle and general negative feelings can arise.

Golden Nuggets Podcast #35: Optimizing learning efficiency at Math Academy
2024 Sep, ~2.5h • Why are people quitting their jobs to study math? How to study math like an Olympic athlete. Spaced repetition is like "wait"-lifting. Desirable difficulties. Why achieving automaticity in low-level skills is a necessary for creativity. Why it's still necessary to learn math in a world with AI. Abstraction ceilings as a result of cognitive differences between individuals and practical constraints in life. How much faster and more efficiently we can learn math (as evidenced by Math Academy's original school program in Pasadena). Math Academy's vision and roadmap.

Why is there sometimes resistance to automaticity in education?
2024 Sep, ~200 words • The need for automaticity on low-level skills is obvious to anyone with experience learning a sport or instrument. So why is there sometimes resistance in education? It makes sense if you think about what people usually find persuasive.

Competition as a Means of Collaboration
2024 Sep, ~250 words • The whole idea is that you want the other person to raise the bar on competition and pass you up, so that you're motivated to come right back and do the same to them.

Writing is a Skill that Can Be Trained
2024 Sep, ~450 words • Every time you put out a post, get feedback, make improvements, and carry those improvements forward into future posts, that's essentially a "rep" of deliberate practice.

Scraping Bits Podcast #102: Learning Mathematics Like an Athlete
2024 Sep, ~1h • My background. Why learn advanced math early. Thinking mathematically. A "mathematical" / "first principles" approach to getting in shape with minimalist strength training. Benefits of building up knowledge from scratch & how to motivate yourself to do that. Goal-setting & gamification in math & fitness. Maintaining motivation by looking back at long-term progress (what used to be hard is now easy). Traits of successful math learners. How does greatness arise & what are some multipliers on one's chance of achieving it. How to build habits, solidify them into your identity, and have fun with it.

Road to Reading Podcast #23: Discussing Cognitive Science
2024 Sep, ~1.5h • [0:00] What is the science of learning?
[~7:00] Students learn better when they're actively solving problems and explicitly being told how to solve them.
[~13:00] Students retain information longer when they space out their review with expanding intervals.
[~19:00] Spaced repetition is so similar to weightlifting that you might as well call it "wait"-lifting. The wait creates the weight.
[~22:00] Desirable difficulties: making the task harder in a way that overcoming the difficulty produces more learning -- but not all difficulties are desirable, and no difficulty is desirable if the student is unable to overcome it in a timely manner. Other desirable difficulties include interleaving (mixed practice) and the testing effect (retrieval practice).
[~32:00] The testing effect (retrieval practice effect): students retain information longer when they're made to practice retrieving it from memory. Again, it's just like weightlifting. The way to build long-term memory is to use long-term memory. You're picking up a weight off of the ground of long-term memory and lifting it up into working memory.
[~36:00] The power of automaticity, the ability to execute low-level actions without them exhausting your mental bandwidth. It's important to develop automaticity because we all have limited working memory capacity. Automaticity helps us overcome that limit.
[~44:00] Automaticity is a critical component of creativity. It frees up space for creative thinking.
[~48:00] The expertise reversal effect: the difficulty of the task needs to be calibrated to the ability of the learner. If expert-level tasks are given to non-experts (or vice versa), little learning will occur.
[~55:00] Why it's important to transition from massed/blocked practice (repeating the same exercise consecutively) to interleaving (mixing/varying up the exercises).
[~1:02:00] Effective learning strategies can feel counterintuitive / unnatural because the point is to increase effort, not to reduce effort. It's completely different from typical work or chores that you might do in batch. It's completely different from reading a fluent story from start to finish. It's about interrupting the flow of thought and coming back to it later.
[~1:09:00] Deliberate practice: a high-level description of the most effective form of practice identified by the academic field of talent development.
[~1:15:00] To what extent does the accumulated volume of deliberate practice predict whether someone is going to become an expert? Deliberate practice is the primary factor, but genetics is an important secondary factor.
[~1:17:00] NON-examples of deliberate practice. Common pitfalls when people try and fail to do deliberate practice, and how to avoid them.
[~1:23:00] How to learn more about the science of learning.
[~1:29:00] The #1 takeaway: use interleaved spaced retrieval practice. You can use this in the classroom.

You Are NOT Lazy, You Just Lack a Habit
2024 Aug, ~700 words • The habit is a psychological force field that protects you from all sorts of negative feelings that try to dissuade you from training.

Why I Recommend Students NOT Take Notes
2024 Aug, ~1100 words • If you try to keep information close by taking great notes that you can reference all the time... that just PREVENTS you from truly retaining it.

A White Pill on Cognitive Differences
2024 Aug, ~1350 words • It's a hard truth that some people have more advantageous cognitive differences than others -- e.g., higher working memory capacity, higher generalization ability, slower forgetting rate. However, there are two sources of hope: 1) automaticity can effectively turn your long-term memory into an extension of your working memory, and 2) many sources of friction in the learning process can be not only remedied but also exploited to increase learning speed beyond the status quo.

Resolving Confusion about Deliberate Practice
2024 Aug, ~950 words • Doesn't "beyond the edge of one's capabilities" mean that you can't do it? How can you practice it if you can't do it? Also, "performance-improving adjustments on every single repetition" is hard to understand in some realms of performance. For instance, does each step a runner takes involve feedback and improvement?

My Next Big Modeling Project: Behavior Coaching
2024 Aug, ~650 words • Even if students are working on exactly the right things, they need to be working exactly the right way to capture the most learning from their time spent working.

Book Review: Developing Talent in Young People by Benjamin Bloom
2024 Aug, ~700 words • Bloom studied the training backgrounds of 120 world-class talented individuals across 6 talent domains: piano, sculpting, swimming, tennis, math, & neurology, and what he discovered was that talent development occurs through a similar general process, no matter what talent domain. In other words, there is a "formula" for developing talent -- though executing it is a lot harder than simply understanding it.

Ability is Built, Not Unlocked
2024 Aug, ~150 words • Curiosity/interest motivates people to engage in deliberate practice, which is what builds ability.

Different Students Need Different Amounts of Practice
2024 Jul, ~450 words • The amount of practice should be determined on the basis of each student's individual performance on each individual topic. Some students may end up having to do more work, but this ultimately empowers them to learn and continue learning into the future.

When should you do math in your head vs writing it out on paper?
2024 Jul, ~900 words • There is an asymmetric tradeoff between 1) blowing your working memory capacity and leaving yourself unable to make progress, versus 2) wasting a couple extra seconds writing down a bit more work than you need to. When in doubt, write it out.

Why is the EdTech Industry So Damn Soft?
2024 Jul, ~2500 words • The hard truth is that if you want to build a serious educational product, you can't be afraid to charge money for it. You can't back yourself into a corner where you depend on a massive userbase. Why? Because most people are not serious about learning, and if you depend on a massive base of unserious learners, then you have to employ ineffective learning strategies that do not repel unserious students. Which makes your product suck.

The Issue with Watered-Down Math Courses
2024 Jul, ~400 words • When students are not given the opportunity to learn math seriously, and are instead presented with watered-down courses and told that they’re doing a great job, they’re being set up for failure later in life when it matters most.

Who Needs Worked Examples? You, Eventually.
2024 Jul, ~1550 words • Math gets hard for different students at different levels. If you don't have worked examples to help carry you through once math becomes hard for you, then every problem basically blows up into a "research project" for you. Sometimes people advocate for unguided struggle as a way to improve general problem-solving ability, but this idea lacks empirical support. Worked examples won't prevent you from developing deep understanding (actually, it's the opposite: worked examples can help you quickly layer on more skills, which forces a structural integrity in the lower levels of your knowledge). Even if you decide against using worked examples for now, continually re-evaluate to make sure you're getting enough productive training volume.

How Bloom’s Taxonomy Gets Misinterpreted
2024 Jul, ~1000 words • Many educators think that the makeup of every year in a student's education should be balanced the same way across Bloom's taxonomy, whereas Bloom's 3-stage talent development process suggests that the time allocation should change drastically as a student progresses through their education.

Higher Math Textbooks and Classes are Typically Not Aligned with the Cognitive Science of Learning
2024 Jul, ~400 words • Research indicates the best way to improve your problem-solving ability in any domain is simply by acquiring more foundational skills in that domain. The way you increase your ability to make mental leaps is not actually by jumping farther, but rather, by building bridges that reduce the distance you need to jump. Yet, higher math textbooks & courses seem to focus on trying to train jumping distance instead of bridge-building.

The Value of Foundational Math Knowledge in Machine Learning
2024 Jul, ~400 words • If you start to flail (or, more subtly, doubt yourself and lose interest) after jumping into ML without a baseline level of foundational knowledge, then you need to put your ego aside and re-allocate your time into shoring up your foundations.

Conversational Dialogue is a Fascinating Distraction for AI in Education
2024 Jun, ~350 words • Hard-coding explanations feels tedious, takes a lot of work, and isn't "sexy" like an AI that generates responses from scratch – but at least it's not a pipe dream. It’s a practical solution that lets you move on to other components of the AI that are just as important.

The Tip of Math Academy’s Technical Iceberg
2024 Jun, ~500 words • Our AI expert system is one of those things that sounds intuitive enough at a high level, but if you start trying to implement it yourself, you quickly run into a mountain of complexity, numerous edge cases, lots of counterintuitive low-level phenomena that take a while to fully wrap your head around.

Student Bite Size vs Curriculum Portion Size
2024 Jun, ~750 words • Students eat meals of information at similar bite rates when each spoonful fed to them is sized appropriately relative to the size of their mouth. (Note that equal bite rates does not imply equal rates of food volume intake.)

Review Should Feel Challenging
2024 May, ~350 words • It's the act of successfully retrieving fuzzy memory, not clear memory, that extends the memory duration.

The Vicious Cycle of Forgetting
2024 May, ~250 words • To transfer information into long-term memory, you need to practice retrieving it without assistance.

If You Want to Learn Algebra, You Need to Have Automaticity on Basic Arithmetic
2024 May, ~1850 words • Solving equations feels smooth when basic arithmetic is automatic -- it's like moving puzzle pieces around, and you just need to identify how they fit together. But without automaticity on basic arithmetic, each puzzle piece is a heavy weight. You struggle to move them at all, much less figure out where they're supposed to go.

Bloom’s 3 Stages of Talent Development
2024 Apr, ~300 words • First, fun and exciting playtime. Then, intense and strenuous skill development. Finally, developing one's individual style while pushing the boundaries of the field.

What Mathematics Can Teach Us About Human Nature
2024 Apr, ~150 words • 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.

What to Do When Math Gets Too Hard
2024 Mar, ~450 words • In general, when you feel yourself running up against a ceiling in life, the solution is typically to pivot and into a direction where the ceiling is higher.

Spaced Repetition vs Spiraling
2024 Mar, ~500 words • By periodically revisiting content, a spiral curriculum periodically restores forgotten knowledge and leverages the spacing effect to slow the decay of that knowledge. Spaced repetition takes this line of thought to its fullest extent by fully optimizing the review process.

Learning vs Feeling
2024 Mar, ~50 words • The strongest people lift weights heavy enough to make them feel weak.

Recreational Mathematics: Why Focus on Projects Over Puzzles
2024 Feb, ~500 words • There's only so much fun you can have trying to follow another person's footsteps to arrive at a known solution. There's only so much confidence you can build from fighting against 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, ~750 words • 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.

Leveraging Cognitive Learning Strategies Requires Technology
2024 Feb, ~4050 words • While there is plenty of room for teachers to make better use of cognitive learning strategies in the classroom, teachers are victims of circumstance in a profession lacking effective accountability and incentive structures, and the end result is that students continue to receive mediocre educational experiences. Given a sufficient degree of accountability and incentives, there is no law of physics preventing a teacher from putting forth the work needed to deliver an optimal learning experience to a single student. However, in the absence of technology, it is impossible for a single human teacher to deliver an optimal learning experience to a classroom of many students with heterogeneous knowledge profiles, each of whom needs to work on different types of problems and receive immediate feedback on each of their attempts. This is why technology is necessary.

The Utility of Gamification in Learning
2024 Feb, ~600 words • Gamification, integrating game-like elements into learning environments, proves effective in increasing student learning, engagement, and enjoyment.

Cognitive Science of Learning: The Testing Effect (Retrieval Practice)
2024 Feb, ~4000 words • The testing effect (or the retrieval practice effect) emphasizes that recalling information from memory, rather than repeated reading, enhances learning. It can be combined with spaced repetition to produce an even more potent learning technique known as spaced retrieval practice.

Cognitive Science of Learning: Interleaving (Mixed Practice)
2024 Feb, ~3350 words • Interleaving (or mixed practice) involves spreading minimal effective doses of practice across various skills, in contrast to blocked practice, which involves extensive consecutive repetition of a single skill. Blocked practice can give a false sense of mastery and fluency because it allows students to settle into a robotic rhythm of mindlessly applying one type of solution to one type of problem. Interleaving, on the other hand, creates a "desirable difficulty" that promotes vastly superior retention and generalization, making it a more effective review strategy. But despite its proven efficacy, interleaving faces resistance in classrooms due to a preference for practice that feels easier and appears to produce immediate performance gains, even if those performance gains quickly vanish afterwards and do not carry over to test performance.

Cognitive Science of Learning: Spaced Repetition (Distributed Practice)
2024 Feb, ~5150 words • When reviews are spaced out or distributed over multiple sessions (as opposed to being crammed or massed into a single session), memory is not only restored, but also further consolidated into long-term storage, which slows its decay. This is known as the spacing effect. A profound consequence of the spacing effect is that the more reviews are completed (with appropriate spacing), the longer the memory will be retained, and the longer one can wait until the next review is needed. This observation gives rise to a systematic method for reviewing previously-learned material called spaced repetition (or distributed practice). A repetition is a successful review at the appropriate time.

Layering: Building Structural Integrity in Knowledge
2024 Feb, ~400 words • Layering is the act of continually building on top of existing knowledge -- that is, continually acquiring new knowledge that exercises prerequisite or component knowledge. This causes existing knowledge to become more ingrained, organized, and deeply understood, thereby increasing the structural integrity of a student's knowledge base and making it easier to assimilate new knowledge.

Cognitive Science of Learning: Minimizing Associative Interference
2024 Feb, ~400 words • Associative interference occurs when related knowledge interferes with recall. It is more likely to occur when highly related pieces of knowledge are learned simultaneously or in close succession. However, the effects of interference can be mitigated by teaching dissimilar concepts simultaneously and spacing out related pieces of knowledge over time.

Cognitive Science of Learning: Developing Automaticity
2024 Feb, ~4450 words • Automaticity is the ability to perform low-level skills without conscious effort. Analogous to a basketball player effortlessly dribbling while strategizing, automaticity allows individuals to avoid spending limited cognitive resources on low-level tasks and instead devote those cognitive resources to higher-order reasoning. In this way, automaticity is the gateway to expertise, creativity, and general academic success. However, insufficient automaticity, particularly in basic skills, inflates the cognitive load of tasks, making it exceedingly difficult for students to learn and perform.

Cognitive Science of Learning: Minimizing Cognitive Load
2024 Feb, ~800 words • Different students have different working memory capacities. When the cognitive load of a learning task exceeds a student's working memory capacity, the student experiences cognitive overload and is not able to complete the task.

A Brief History of Mastery Learning
2024 Feb, ~1250 words • Mastery learning is a strategy in which students demonstrate proficiency on prerequisites before advancing. While even loose approximations of mastery learning have been shown to produce massive gains in student learning, mastery learning faces limited adoption due to clashing with traditional teaching methods and placing increased demands on educators. True mastery learning at a fully granular level requires fully individualized instruction and is only attainable through one-on-one tutoring.

Deliberate Practice: The Most Effective Form of Active Learning
2024 Feb, ~3750 words • Deliberate practice is the most effective form of active learning. It consists of individualized training activities specially chosen to improve specific aspects of a student's performance through repetition and successive refinement. It is mindful repetition at the edge of one’s ability, the opposite of mindless repetition within one’s repertoire. The amount of deliberate practice has been shown to be one of the most prominent underlying factors responsible for individual differences in performance across numerous fields, even among highly talented elite performers. Deliberate practice demands effort and intensity, and may be discomforting, but its long-term commitment compounds incremental improvements, leading to expertise.

The Neuroscience of Active Learning and Automaticity
2024 Feb, ~900 words • Active learning leads to more neural activation than passive learning. Automaticity involves developing strategic neural connections that reduce the amount of effort that the brain has to expend to activate patterns of neurons.

Most Students Don’t Even Pay Attention During Lectures
2024 Feb, ~600 words • A startup spent months building a sophisticated lecture tool and raising over half a million dollars in investments -- but after observing students in the lecture hall, they completely abandoned the product and called up their investors to return the money.

What Counts as Active Learning?
2024 Feb, ~2300 words • True active learning requires every individual student to be actively engaged on every piece of the material to be learned.

Recommended Language, Tools, Path, and Curriculum for Teaching Kids to Code
2024 Jan, ~950 words • 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, ~1250 words • Solving problems, building on top of what you've learned, reviewing what you've learned, and quality, quantity, and spacing of practice.

Your Mathematical Potential Has a Limit, but it’s Likely Higher Than You Think
2024 Jan, ~5750 words • Not everybody can learn every level of math, but most people can learn the basics. In practice, however, few people actually reach their full mathematical potential because they get knocked off course early on by factors such as missing foundations, ineffective practice habits, inability or unwillingness to engage in additional practice, or lack of motivation.

The Greatest Educational Life Hack: Learning Math Ahead of Time
2024 Jan, ~3050 words • Learning math early guards you against numerous academic risks, opens all kinds of doors to career opportunities, and allows you to enter those doors earlier in life (which in turn allows you to accomplish more over the course of your career).

Myths and Realities about Educational Acceleration
2024 Jan, ~3400 words • Acceleration does not lead to adverse psychological consequences in capable students; rather, whether a student is ready for advanced mathematics depends solely on whether they have mastered the prerequisites. Acceleration does not imply shallowness of learning; rather, students undergoing acceleration generally learn – in a shorter time – as much as they would otherwise in a non-accelerated environment over a proportionally longer period of time. Accelerated students do not run out of courses to take and are often able to place out of college math courses even beyond what is tested on placement exams. Lastly, for students who have the potential to capitalize on it, acceleration is the greatest educational life hack: the resulting skills and opportunities can rocket students into some of the most interesting, meaningful, and lucrative careers, and the early start can lead to greater career success.

Effective Learning Does Not Emulate the Professional Workplace
2024 Jan, ~3850 words • The most effective learning techniques require substantial cognitive effort from students and typically do not emulate what experts do in the professional workplace. Direct instruction is necessary to maximize student learning, whereas unguided instruction and group projects are typically very inefficient.

People Differ in Learning Speed, Not Learning Style
2024 Jan, ~4350 words • Different people generally have different working memory capacities and learn at different rates, but people do not actually learn better in their preferred "learning style." Instead, different people need the same form of practice but in different amounts.

Accountability and Incentives are Necessary but Absent in Education
2024 Jan, ~5350 words • Students and teachers are often not aligned with the goal of maximizing learning, which means that in the absence of accountability and incentives, classrooms are pulled towards a state of mediocrity. Accountability and incentives are typically absent in education, which leads to a "tragedy of the commons" situation where students pass courses (often with high grades) despite severely lacking knowledge of the content.

The Story of the Science of Learning
2024 Jan, ~4400 words • In terms of improving educational outcomes, science is not where the bottleneck is. The bottleneck is in practice. The science of learning has advanced significantly over the past century, yet the practice of education has barely changed.

Cognitive Science of Learning: How the Brain Works
2024 Jan, ~3000 words • Cognition involves the flow of information through sensory, working, and long-term memory banks in the brain. Sensory memory temporarily holds raw data, working memory manipulates and organizes information, and long-term memory stores it indefinitely by creating strategic electrical wiring between neurons. Learning amounts to increasing the quantity, depth, retrievability, and generalizability of concepts and skills in a student's long-term memory. Limited working memory capacity creates a bottleneck in the transfer of information into long-term memory, but cognitive learning strategies can be used to mitigate the effects of this bottleneck.

A Common Source of Student Mistakes
2024 Jan, ~350 words • Many students who pattern-match will tend to prefer solutions requiring fewer and simpler operations, especially if those solutions yield ballpark-reasonable results.

2023

Critique of Paper: “An astonishing regularity in student learning rate”
2023 Nov, ~9650 words • 1) The reported learning rates are actually as quantitatively similar as is suggested by the language used to describe them. 2) The learning rates are measured in a way that rests on a critical assumption that students learn nothing from the initial instruction preceding the practice problems -- i.e., you can have one student who learns a lot more from the initial instruction and requires far fewer practice problems, and when you calculate their learning rate, it can come out the same as for a student who learns a lot less from the initial instruction and requires far more practice problems.

Ambiguous Absolute Value Expressions
2023 Nov, ~400 words • Is there a standard "order of operations" for parallel vs nested absolute value expressions, in the absence of clarifying notation?

My Go-To Math Riddle: How Many Squares are in a 10 x 10 Grid?
2023 Nov, ~350 words • 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.

Can You Automate a Math Teacher?
2023 Oct, ~1700 words • For many (but not all) students, the answer is yes. And for many of those students, automation can unlock life-changing educational outcomes.

Optimized, Individualized Spaced Repetition in Hierarchical Knowledge Structures
2023 Oct, ~5000 words • Spaced repetition is complicated in hierarchical bodies of knowledge, like mathematics, because repetitions on advanced topics should "trickle down" to update the repetition schedules of simpler topics that are implicitly practiced (while being discounted appropriately since these repetitions are often too early to count for full credit towards the next repetition). However, I developed a model of Fractional Implicit Repetition (FIRe) that not only accounts for implicit "trickle-down" repetitions but also minimizes the number of reviews by choosing reviews whose implicit repetitions "knock out" other due reviews (like dominos), and calibrates the speed of the spaced repetition process to each individual student on each individual topic (student ability and topic difficulty are competing factors).

For Most Students, Competition Math is a Waste of Time
2023 Sep, ~1200 words • 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, ~350 words • 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, ~450 words • Speaking as someone who had to suffer through a teacher credentialing program... it's actually an anti-signal when someone references their teaching credential as a qualification to speak about how learning happens. It's centered around political ideology rather than the science of learning.

Selecting a Good Problem to Work On
2023 Jan, ~800 words • Good problem = intersection between your own interests/talents, the realm of what's feasible, and the desires of the external world.

2022

Minimax Strategy
2022 Mar, ~800 words • Repeatedly choosing the action with the best worst-case scenario.

Backpropagation
2022 Feb, ~4400 words • A convenient technique for computing gradients in neural networks.

Decision Trees
2022 Feb, ~3800 words • We can algorithmically build classifiers that use a sequence of nested "if-then" decision rules.

Naive Bayes
2022 Jan, ~1050 words • A simple classification algorithm grounded in Bayesian probability.

Regression via Gradient Descent
2022 Jan, ~1250 words • Gradient descent can help us avoid pitfalls that occur when fitting nonlinear models using the pseudoinverse.

2021

Simplex Method
2021 Mar, ~2950 words • A technique for maximizing linear expressions subject to linear constraints.

Hash Tables
2021 Mar, ~550 words • Under the hood, dictionaries are hash tables.

Euler Estimation
2021 Mar, ~400 words • Arrays can be used to implement more than just matrices. We can also implement other mathematical procedures like Euler estimation.

Tic-Tac-Toe and Connect Four
2021 Mar, ~600 words • One of the best ways to get practice with object-oriented programming is implementing games.

K-Means Clustering
2021 Mar, ~1300 words • Guess some initial clusters in the data, and then repeatedly update the guesses to make the clusters more cohesive.

Basic Matrix Arithmetic
2021 Mar, ~550 words • We can use arrays to implement matrices and their associated mathematical operations.

Merge Sort and Quicksort
2021 Feb, ~500 words • Merge sort and quicksort are generally faster than selection, bubble, and insertion sort. And unlike counting sort, they are not susceptible to blowup in the amount of memory required.

Multivariable Gradient Descent
2021 Feb, ~900 words • Just like single-variable gradient descent, except that we replace the derivative with the gradient vector.

Single-Variable Gradient Descent
2021 Feb, ~1900 words • We take an initial guess as to what the minimum is, and then repeatedly use the gradient to nudge that guess further and further "downhill" into an actual minimum.

Cartesian Product
2021 Jan, ~400 words • Implementing the Cartesian product provides good practice working with arrays.

Simulating Coin Flips
2021 Jan, ~450 words • Estimating probabilities by simulating a large number of random experiments.

Recursive Sequences
2021 Jan, ~400 words • Sequences where each term is a function of the previous terms.

2020

2019

Thales’ Theorem
2019 Dec, ~400 words • Every inscribed triangle whose hypotenuse is a diameter is a right triangle.

Eigenvalues, Eigenvectors, and Diagonalization
2019 Sep, ~5200 words • The eigenvectors of a matrix are those vectors that the matrix simply rescales, and the factor by which an eigenvector is rescaled is called its eigenvalue. These concepts can be used to quickly calculate large powers of matrices.

Inverse Matrices
2019 Aug, ~2350 words • The inverse of a matrix is a second matrix which undoes the transformation of the first matrix.

N-Dimensional Volume Formula
2019 Jul, ~2200 words • N-dimensional volume generalizes the idea of the space occupied by an object. We can think about N-dimensional volume as being enclosed by N-dimensional vectors.

Elimination as Vector Reduction
2019 Jun, ~750 words • If we interpret linear systems as sets of vectors, then elimination corresponds to vector reduction.

Span, Subspaces, and Reduction
2019 Jun, ~2550 words • The span of a set of vectors consists of all vectors that can be made by adding multiples of vectors in the set. We can often reduce a set of vectors to a simpler set with the same span.

Lines and Planes
2019 Jun, ~1650 words • A line starts at an initial point and proceeds straight in a constant direction. A plane is a flat sheet that makes a right angle with some particular vector.

N-Dimensional Space
2019 Jun, ~1100 words • N-dimensional space consists of points that have N components.

Cutting Through the Hype of AI
2019 May, ~400 words • Media outlets often make the mistake of anthropomorphizing or attributing human-like characteristics to computer programs.

What is AI?
2019 May, ~450 words • Turing test, games, hype, narrow vs general AI.

Introductory Python: Functions
2019 May, ~650 words • Rather than duplicating such code each time we want to use it, it is more efficient to store the code in a function.

Solving Differential Equations with Taylor Series
2019 Apr, ~650 words • Many differential equations don't have solutions that can be expressed in terms of finite combinations of familiar functions. However, we can often solve for the Taylor series of the solution.

Manipulating Taylor Series
2019 Apr, ~1250 words • To find the Taylor series of complicated functions, it's often easiest to manipulate the Taylor series of simpler functions.

Taylor Series
2019 Apr, ~1050 words • Many non-polynomial functions can be represented by infinite polynomials.

Tests for Convergence
2019 Apr, ~1300 words • Various tricks for determining whether a series converges or diverges.

Geometric Series
2019 Apr, ~1000 words • A geometric series is a sum where each term is some constant times the previous term.

Variation of Parameters
2019 Mar, ~1000 words • When we know the solutions of a linear differential equation with constant coefficients and right hand side equal to zero, we can use variation of parameters to find a solution when the right hand side is not equal to zero.

Integrating Factors
2019 Mar, ~600 words • Integrating factors can be used to solve first-order differential equations with non-constant coefficients.

Undetermined Coefficients
2019 Mar, ~1350 words • Undetermined coefficients can help us find a solution to a linear differential equation with constant coefficients when the right hand side is not equal to zero.

Slope Fields and Euler Approximation
2019 Mar, ~800 words • When faced with a differential equation that we don't know how to solve, we can sometimes still approximate the solution.

Separation of Variables
2019 Mar, ~650 words • The simplest differential equations can be solved by separation of variables, in which we move the derivative to one side of the equation and take the antiderivative.

Improper Integrals
2019 Feb, ~800 words • Improper integrals have bounds or function values that extend to positive or negative infinity.

Integration by Parts
2019 Feb, ~1000 words • We can apply integration by parts whenever an integral would be made simpler by differentiating some expression within the integral, at the cost of anti-differentiating another expression within the integral.

Integration by Substitution
2019 Feb, ~800 words • Substitution involves condensing an expression of into a single new variable, and then expressing the integral in terms of that new variable.

Finding Area Using Integrals
2019 Feb, ~1650 words • To evaluate a definite integral, we find the antiderivative, evaluate it at the indicated bounds, and then take the difference.

Antiderivatives
2019 Feb, ~1350 words • The antiderivative of a function is a second function whose derivative is the first function.

L’Hôpital’s Rule
2019 Jan, ~1500 words • When a limit takes the indeterminate form of zero divided by zero or infinity divided by infinity, we can differentiate the numerator and denominator separately without changing the actual value of the limit.

Differentials and Approximation
2019 Jan, ~750 words • We can interpret the derivative as an approximation for how a function's output changes, when the function input is changed by a small amount.

Finding Extrema
2019 Jan, ~1950 words • Derivatives can be used to find a function's local extreme values, its peaks and valleys.

Properties of Derivatives
2019 Jan, ~800 words • Given a sum, we can differentiate each term individually. But why are we able to do this? Does multiplication work the same way? What about division?

Chain Rule
2019 Jan, ~400 words • When taking derivatives of compositions of functions, we can ignore the inside of a function as long as we multiply by the derivative of the inside afterwards.

Power Rule for Derivatives
2019 Jan, ~600 words • There are some patterns that allow us to compute derivatives without having to compute the limit of the difference quotient.

Derivatives and the Difference Quotient
2019 Jan, ~600 words • The derivative of a function is the function's slope at a particular point, and can be computed as the limit of the difference quotient.

Evaluating Limits
2019 Jan, ~1750 words • The limit of a function, as the input approaches some value, is the output we would expect if we saw only the surrounding portion of the graph.

2018

Compositions of Functions
2018 Jul, ~500 words • Compositions of functions consist of multiple functions linked together, where the output of one function becomes the input of another function.

Inverse Functions
2018 Jul, ~600 words • Inverting a function entails reversing the outputs and inputs of the function.

Reflections of Functions
2018 Jul, ~300 words • When a function is reflected, it flips across one of the axes to become its mirror image.

Rescalings of Functions
2018 Jul, ~650 words • When a function is rescaled, it is stretched or compressed along one of the axes, like a slinky.

Shifts of Functions
2018 Jul, ~350 words • When a function is shifted, all of its points move vertically and/or horizontally by the same amount.

Piecewise Functions
2018 Jun, ~500 words • A piecewise function is pieced together from multiple different functions.

Trigonometric Functions
2018 Jun, ~1800 words • Trigonometric functions represent the relationship between sides and angles in right triangles.

Absolute Value
2018 Jun, ~1100 words • Absolute value represents the magnitude of a number, i.e. its distance from zero.

Radical Functions
2018 Jun, ~950 words • Radical functions involve roots: square roots, cube roots, or any kind of fractional exponent in general.

Horizontal Asymptotes of Rational Functions
2018 May, ~600 words • Rational functions can have a form of end behavior in which they become flat, approaching (but never quite reaching) a horizontal line known as a horizontal asymptote.

Polynomial Long Division
2018 May, ~550 words • Polynomial long division works the same way as the long division algorithm that's familiar from simple arithmetic.

Zeros of Polynomials
2018 Apr, ~900 words • The zeros of a polynomial are the inputs that cause it to evaluate to zero.

Systems of Inequalities
2018 Mar, ~300 words • To solve a system of inequalities, we need to solve each individual inequality and find where all their solutions overlap.

Linear Inequalities in the Number Line
2018 Mar, ~800 words • An inequality is similar to an equation, but instead of saying two quantities are equal, it says that one quantity is greater than or less than another.

Quadratic Systems
2018 Feb, ~350 words • Systems of quadratic equations can be solved via substitution.

Vertex Form
2018 Feb, ~300 words • To easily graph a quadratic equation, we can convert it to vertex form.

Completing the Square
2018 Feb, ~700 words • Completing the square helps us gain a better intuition for quadratic equations and understand where the quadratic formula comes from.

Quadratic Formula
2018 Feb, ~650 words • To solve hard-to-factor quadratic equations, it's easiest to use the quadratic formula.

Linear Systems
2018 Feb, ~1150 words • A linear system consists of multiple linear equations, and the solution of a linear system consists of the pairs that satisfy all of the equations.

Point-Slope Form
2018 Feb, ~750 words • An easy way to write the equation of a line if we know the slope and a point on a line.

Solving Linear Equations
2018 Feb, ~500 words • Loosely speaking, a linear equation is an equality statement containing only addition, subtraction, multiplication, and division.

Intuiting Ensemble Methods
2018 Jan, ~400 words • The type of ensemble model that wins most data science competitions is the stacked model, which consists of an ensemble of entirely different species of models together with some combiner algorithm.

Intuiting Decision Trees
2018 Jan, ~500 words • Decision trees are able to model nonlinear data while remaining interpretable.

Intuiting Neural Networks
2018 Jan, ~1400 words • NNs are similar to SVMs in that they project the data to a higher-dimensional space and fit a hyperplane to the data in the projected space. However, whereas SVMs use a predetermined kernel to project the data, NNs automatically construct their own projection.

Intuiting Support Vector Machines
2018 Jan, ~800 words • A Support Vector Machine (SVM) computes the "best" separation between classes as the maximum-margin hyperplane.

Intuiting Linear Regression
2018 Jan, ~400 words • In linear regression, we model the target as a random variable whose expected value depends on a linear combination of the predictors (including a bias term).

Intuiting Naive Bayes
2018 Jan, ~250 words • Naive Bayes classification naively assumes that the presence of a particular feature in a class is unrelated to the presence of any other feature.

2017

Applications of Calculus: Maximizing Profit
2017 Nov, ~300 words • If we know the revenue and costs associated with producing any number of units, then we can use calculus to figure out the number of units to produce for maximum profit.

Intuiting Sequences
2017 Oct, ~350 words • A sequence is a list of numbers that has some pattern.

Intuiting Integrals
2017 Oct, ~250 words • Integrals give the area under a portion of a function.

Intuiting Derivatives
2017 Oct, ~450 words • The derivative tells the steepness of a function at a given point, kind of like a carpenter's level.

Intuiting Limits
2017 Oct, ~650 words • The limit of a function is the height where it looks like the scribble is going to hit a particular vertical line.

Intuiting Functions
2017 Oct, ~150 words • A function is a scribble that crosses each vertical line only once.

Intuiting Persistent Homology
2017 Sep, ~1900 words • Persistent homology provides a way to quantify the topological features that persist over our a data set's full range of scale.

Mapper Use-Cases at Ayasdi
2017 Sep, ~900 words • Ayasdi developed commercial Mapper software and sells a subscription service to clients who wish to create topological network visualizations of their data.

2016

2015

The Brain in One Sentence
2015 Nov, ~2500 words • 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.

On the Contrasting Educations and Outcomes of Ben Franklin and Montaigne
2015 May, ~950 words • 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.

2014

2013

2012

Sound Waves
2012 Dec, ~1100 words • A brief overview of sound waves and how they interact with things.

Detecting Dark Matter
2012 Dec, ~1900 words • A brief overview of the experimental search for dark matter (XENON, CDMS, PICASSO, COUPP).

Evidence for the Existence of Dark Matter
2012 Dec, ~1750 words • Mass discrepancies in galaxies and clusters, cosmic background radiation, the structure of the universe, and big bang nucleosynthesis's impact on baryon density.