Essays


(Note: there is also a blog with plenty more shorter pieces.)


Tier 1: Pinned

Math Academy’s Eurisko Sequence, 5 Years Later: Student Outcomes Emerging From the Most Advanced High School Math/CS Track in the USA
2025 Mar, ~2700 words • During its operation from 2020-23, Eurisko was the most advanced high school math/CS track 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.

Why is the EdTech Industry So Damn Soft?
2024 Jul, ~2450 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 Greatest Educational Life Hack: Learning Math Ahead of Time
2024 Jan, ~3450 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).

Optimized, Individualized Spaced Repetition in Hierarchical Knowledge Structures
2023 Oct, ~5200 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).


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Why Can’t College Students Do Middle School Math?
2026 Jan, ~1800 words • 1 in 12 incoming UCSD freshmen don’t know middle school math, and the remedial math course was too advanced, so UCSD had to create a remedial remedial math course covering elementary and middle school math, and a quarter of the students placing into it had a perfect 4.0 GPA in their high school math courses, which included calculus or precalculus for nearly half of those remedial remedial students. And it's not just a UCSD problem -- the disease has spread so far that even Harvard had to had to add remedial support to their entry-level calculus courses to deal with a "lack of foundational algebra skills among students".

What Happens When Middle School is Put to Good Use
2025 Oct, ~3150 words • Typical honors students can learn all of high school math plus calculus *in middle school* if they are taught efficiently. They don't have to be geniuses, don't even have to spend more time on school. Just need to use time efficiently. Few people understand this, as well as the kinds of opportunities that get unlocked when a student learns advanced math ahead of time. The road doesn't end at calculus, that's just an early milestone, table stakes for the core university math that empowers students to do awesome projects.

The Field of Education is Due For a Copernican Revolution
2025 May, ~6150 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.

Advice on Upskilling
2025 Jan, ~42150 words • Advice on consistency, skills, discipline, the grind, the journey, the team, the mission, motivation, learning, and expertise.

Why I Recommend Students NOT Take Notes
2024 Aug, ~1350 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.

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.



Tier 2: More posts I'd recommend

I’ve Wanted Alien-Level Edtech Ever Since I Was A Kid
2025 Aug, ~1100 words • That's why I'm so excited by the prospect of eliminating educational friction, solving the thermodynamic efficiency of education, and building machines that make people insanely skilled as efficiently as possible.

The Missing Middle in Test Prep
2025 Aug, ~1650 words • There's a large gap between the standard math curriculum that students learn at school, and the additional skills that show up on standardized exams like the SAT, ACT, etc. We're working to fill it.

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.

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.

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 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."

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.

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.

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.

Your Mathematical Potential Has a Limit, but it’s Likely Higher Than You Think
2024 Jan, ~6750 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.

Struggle Does Not Imply Inability
2024 Jan, ~1200 words • If you do poorly in a math class, it doesn’t necessarily mean that you are incapable of learning that level of math. There are a number of reasons that could be the root cause of your struggle.

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.

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.

Critique of Paper: “An astonishing regularity in student learning rate”
2023 Nov, ~10600 words • 1) The reported learning rates are not 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.

For Most Students, Competition Math is a Waste of Time
2023 Sep, ~1300 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.



Tier 3: If you've scrolled this far and are looking for more recommendations...

Announcing our SAT Math Prep Course
2026 Mar, ~1700 words • There's a gigantic "missing middle" between the standard math curriculum and what actually appears on the SAT -- skills most students won't pick up even if they ace every math class. We identified it, mapped it to a knowledge graph, and built a course to teach it explicitly.

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.

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.

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.

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.

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.

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.

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.

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.