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Blog (Tier 3)

Math Trauma Isn’t Really About Math

Math trauma tends to be less about math and more about being asked to do advanced maneuvers before you’ve mastered the basics – and then being told to try harder when you inevitably fall. Read more...

Announcing our SAT Math Prep Course

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. Read more...

Learning Rate Differences Exist

A recent study measured a 2x learning rate difference between the 25th and 75th percentile – likely an underestimate due to methodological choices. The authors reported it as 1.5x and called it an “astonishing regularity.” Read more...

How Learning Debt Starts

Learning debt usually starts with adults letting compensatory hacks slide – not calling out weak fundamentals before they compound. When this happens at scale across many students and schools, it degrades the entire educational system. Read more...

Founder Involvement Is What Keeps Everything On The Rails

As Paul Graham has explained, conventional startup wisdom says to hire good people and let them work, but experienced founders know this is a recipe for hiring professional fakers and letting them derail the company. Founders need to stay in the weeds. Here’s an example of what that looked like at Math Academy. Read more...

The Cycle of AI

Progress is made in AI, people lose their shit thinking Jarvis-level AGI is just around the corner, the singularity gets canceled, then nothing is AI, until more progress is made. Rinse & repeat. Read more...

Protect The Habit

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. Read more...

Experts Perceive Differently

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. Read more...

Actively Doing is the Key to Alpha

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. Read more...

Make it So Easy a Kid Can Learn It

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. Read more...

Get On the Right Team

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. Read more...

Love What You Do

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. Read more...

Just Do The F*cking Work

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. Read more...

Pictures are Valuable in Math Learning, but They’re Often Overvalued

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. Read more...

Resolving Confusion about Deliberate Practice

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? Read more...

Book Review: Developing Talent in Young People by Benjamin Bloom

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. Read more...

Different Students Need Different Amounts of Practice

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. Read more...

The Issue with Watered-Down Math Courses

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. Read more...

Spaced Repetition vs Spiraling

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. Read more...

Leveraging Cognitive Learning Strategies Requires Technology

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. Read more...

Cognitive Science of Learning: Interleaving (Mixed Practice)

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. Read more...

Cognitive Science of Learning: Spaced Repetition (Distributed Practice)

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. Read more...

Layering: Building Structural Integrity in Knowledge

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. Read more...

Cognitive Science of Learning: Minimizing Associative Interference

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. Read more...

Cognitive Science of Learning: Developing Automaticity

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. Read more...

A Brief History of Mastery Learning

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. Read more...

Recommended Language, Tools, Path, and Curriculum for Teaching Kids to Code

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). Read more...

Myths and Realities about Educational Acceleration

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. Read more...

Effective Learning Does Not Emulate the Professional Workplace

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. Read more...

The Story of the Science of Learning

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. Read more...

Business Lessons from Science Fair

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. Read more...
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Math Education

Learning Rate Differences Exist

A recent study measured a 2x learning rate difference between the 25th and 75th percentile – likely an underestimate due to methodological choices. The authors reported it as 1.5x and called it an “astonishing regularity.” Read more...

How Learning Debt Starts

Learning debt usually starts with adults letting compensatory hacks slide – not calling out weak fundamentals before they compound. When this happens at scale across many students and schools, it degrades the entire educational system. Read more...

Why Can’t College Students Do Middle School Math?

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”. Read more...

The (Free) Textbook I Wrote While Teaching The Most Advanced High School Math/CS Sequence In The USA

It scaffolded high school students up to doing masters/PhD-level coursework: reproducing academic research papers in artificial intelligence, building everything from scratch in Python. A former student worked through it right before conducting research that won 1st place ($250,000) in the Regeneron Science Talent Search, getting personally recruited by the head of NASA (with a fighter jet ride as a signing bonus), and publishing his results, solo-author, in The Astronomical Journal. Read more...

Common Failure Modes in Early Math Education

1) Staying in block-counting land so long that the the manipulatives become a crutch. 2) Covering a bunch of special-case arithmetic strategies instead of getting the kid really solid on the standard algorithms. 3) Avoiding memorization. 4) Serving up too many easy facts during times tables practice. Read more...

What Happens When Middle School is Put to Good Use

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. Read more...

Recall First, Reason Second

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. Read more...

The Field of Education is Due For a Copernican Revolution

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. Read more...

How Taxing Work Becomes Fun

“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.” Read more...

Fortify Your F*cking Fundamentals

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. Read more...

Prereq Yo’ Self Before You Wreck Yo’ Self

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. Read more...

Make it So Easy a Kid Can Learn It

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. Read more...

Math is a Well-Defined Body of Knowledge

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. Read more...

Complete Individualization: an Often-Forgotten yet Critical Component of True Deliberate Practice

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. Read more...

ML Courses can Vary Massively in their Coverage

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.” Read more...

Pictures are Valuable in Math Learning, but They’re Often Overvalued

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. Read more...

Why Talent Development is Necessary in Math

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. Read more...

Different Students Need Different Amounts of Practice

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. Read more...

Why is the EdTech Industry So Damn Soft?

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. Read more...

The Issue with Watered-Down Math Courses

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. Read more...

Who Needs Worked Examples? You, Eventually.

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. Read more...

Higher Math Textbooks and Classes are Typically Not Aligned with the Cognitive Science of Learning

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. Read more...

What Mathematics Can Teach Us About Human Nature

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. Read more...

Spaced Repetition vs Spiraling

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. Read more...

Leveraging Cognitive Learning Strategies Requires Technology

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. Read more...

Cognitive Science of Learning: Interleaving (Mixed Practice)

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. Read more...

Cognitive Science of Learning: Spaced Repetition (Distributed Practice)

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. Read more...

Layering: Building Structural Integrity in Knowledge

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. Read more...

Cognitive Science of Learning: Minimizing Associative Interference

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. Read more...

Cognitive Science of Learning: Developing Automaticity

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. Read more...

A Brief History of Mastery Learning

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. Read more...

Deliberate Practice: The Most Effective Form of Active Learning

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. Read more...

Your Mathematical Potential Has a Limit, but it’s Likely Higher Than You Think

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. Read more...

Struggle Does Not Imply Inability

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. Read more...

Myths and Realities about Educational Acceleration

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. Read more...

Effective Learning Does Not Emulate the Professional Workplace

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. Read more...

People Differ in Learning Speed, Not Learning Style

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. Read more...

Accountability and Incentives are Necessary but Absent in Education

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. Read more...

The Story of the Science of Learning

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. Read more...

Cognitive Science of Learning: How the Brain Works

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. Read more...

Optimized, Individualized Spaced Repetition in Hierarchical Knowledge Structures

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). Read more...
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Blog

Q&A: But Don’t You Need 10,000 Hours To Learn Math?

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. Read more...

We’re Working On Streaks!

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. Read more...

On Writing Good Code

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. Read more...

ML Courses can Vary Massively in their Coverage

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.” Read more...

The Tip of Math Academy’s Technical Iceberg

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. Read more...

What Mathematics Can Teach Us About Human Nature

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. Read more...

Recreational Mathematics: Why Focus on Projects Over Puzzles

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. Read more...

Intuiting Adversarial Examples in Neural Networks via a Simple Computational Experiment

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. Read more...

The Brain in One Sentence

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. Read more...
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Talent Development

Learning Is a Balancing Act

Figure skating illustrates how effective learning is a balancing act: strong foundations are required before practicing advanced skills, but advanced skills also make your foundations more robust – provided you’ve mastered the basics well enough to get a grapple on the harder moves. Read more...

Prerequisite Knowledge is Intellectual Capital

It’s easy to feel untalented when you’re really just missing prerequisite skills. A Real Analysis student who thought she might fail her class had simply never gotten much practice with proof-writing – a few sessions filling that gap and she came out with an A. Read more...

Math Trauma Isn’t Really About Math

Math trauma tends to be less about math and more about being asked to do advanced maneuvers before you’ve mastered the basics – and then being told to try harder when you inevitably fall. Read more...

Summary of My Own Math Journey

Grinding it like a video game, running a secret self-study op inside traditional classes, taking talent development seriously while making every rookie mistake, and how it unlocked a life I almost certainly never would’ve found otherwise. Read more...

What Happens When Middle School is Put to Good Use

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. Read more...

I Believe You Have Potential

What people tend to need the most yet have the least in their lives is a supportive hard-ass. Not to be confused with an unsupportive hard-ass or a supportive pushover. That’s the gap I aim to fill as best I can with my writing. Read more...

What’s the Point of Being Way Ahead? It Buys You Time.

Time is the #1 killer of dreams and aspirations. When someone gives up on their dream, or gives up on figuring out what that dream is, it’s typically a result of them losing the race against time. That is the point of compressing time, of removing skill bottlenecks early. Read more...

The Metagame Podcast #39: Math Academy and The Science of Learning

The most comprehensive 2h overview of my thoughts on serious upskilling, to date. Not just how to train efficiently, but also how to find your mission. Not just the microstructure, but also the metagame. We covered tons of bases ranging from the micro level (science of learning & training efficiently) to the macro level (broader journey of finding, developing, and exploiting your personal talents).
[~0:30] 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?
[~9:00] Efficient learning feels like exercise. The point is to overcome a challenge that strains you. It is by definition unpleasant.
[~13:30] Knowledge graphs are vital when constructing efficient learning experiences. They allow you to systematically organize a learner’s performance data to identify their edge of mastery (the boundary between what they know and don’t know), what previously learned topics below the edge are in need of review, and what new topics on the edge will maximize the amount of review that’s knocked out implicitly.
[~18:00] None of this efficiency stuff matters if you don’t show up consistently. Progress equals volume times efficiency. If either of those factors are low then you don’t make much progress.
[~21:30] Getting excited about the idea of getting good provides an initial activation energy, but seeing yourself improve is what fuels you to keep playing the long game, and efficiency is vital for that.
[~26:30] Your training doesn’t have to be super efficient at the beginning. You can gradually nudge yourself into higher efficiency training even if you don’t have a whole lot of intrinsic motivation to begin with. However, there’s often a skill barrier you need to break through to really get to the fun part, and it’s advisable to do that in a timely manner so you don’t stall out. But at the same time, don’t rush it and fall off the rails.
[~34:30] A common failure mode: being unwilling to identify, accept, and start at the level you’re at.
[~41:30] Center your identity on a mission that speaks to you, that you can contribute to, and do whatever else is needed to further it, regardless of whether you perceive these other things to be “you” or not. You’ll be surprised what capabilities you develop, that you hadn’t previously perceived to be a part of your identity.
[~48:30] How to find your mission: sample wide to figure out what activities speak to you, then filter down and pick one (or a couple) that you’re willing to seriously invest your time and effort climbing up the skill tree and going on “quests”. You may not understand this early on, but skill trees branch out, and quests beget follow-up quests, and the act of climbing to these branch-points will imbue you with perspective that you can leverage to keep filtering down. If you iterate this process enough, it gradually converges into a single area that you can describe coherently and uniquely. That’s your mission.
[~55:30] Every stage in the journey to your mission is hard work, and the earlier you get to putting in that work, the better off you’re going to be. It’s never too late, but the longer you wait, the rougher it gets. At the same time, don’t make a rash decision, don’t tear the house down and build up a new house that you don’t even like. But don’t underestimate how fast you can progress when your internal motivation is aligned with your external incentives.
[~1:12:00] Focus on what matters. That’s obvious, but it’s so easy to mess up lose focus and not realize it until after you’ve wasted a bunch of time.
[~1:15:30] How to get back on the horse after you’ve fallen off. How to avoid feeling bad when something outside of your control temporarily knocks you off your horse. A good social environment can push you to get back on your horse.
[~1:26:30] If you’re a beginner, don’t feel like you have to be advanced to join a community of learners. You can do this right away. And don’t shy away from posting your progress – it’s not about where you are, it’s about where you’re going and how fast. It’s only people who are insecure who will make fun of you. Most people, especially advanced people, will be supportive.
[~1:31:30] There are numerous cognitive learning strategies that 1) can be used to massively improve learning, 2) have been reproduced so many times they might as well be laws of physics, and 3) connect all the way down to the mechanics of what’s going on in the brain. The biggest levers: active learning (as opposed to passive consumption), direct/explicit instruction (as opposed to discovery learning), the spacing effect, mixed practice (a.k.a. interleaving), retrieval practice (a.k.a. the testing effect). Read more...

Math Academy’s Eurisko Sequence, 5 Years Later: Student Outcomes Emerging From the Most Advanced High School Math/CS Track in the USA

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. Read more...

The Importance of Learning Your Prerequisites

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. Read more...

Experts Perceive Differently

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. Read more...

Q&A: But Don’t You Need 10,000 Hours To Learn Math?

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. Read more...

Advice on Upskilling

Advice on consistency, skills, discipline, the grind, the journey, the team, the mission, motivation, learning, and expertise. Read more...

Fortify Your F*cking Fundamentals

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. Read more...

Prereq Yo’ Self Before You Wreck Yo’ Self

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. Read more...

Actively Doing is the Key to Alpha

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. Read more...

How I Would Go About Learning an Arbitrary Subject Where No Full-Fledged Adaptive Learning System is Available

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. Read more...

Get On the Right Team

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. Read more...

Math is a Well-Defined Body of Knowledge

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. Read more...

How to Cultivate Discipline

Tear down the unproductive habit and build up a counter-habit whose gravity eventually becomes strong enough to completely overtake the original habit. Read more...

Love What You Do

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. Read more...

Complete Individualization: an Often-Forgotten yet Critical Component of True Deliberate Practice

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. Read more...

Just Do The F*cking Work

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. Read more...

Five Steps to Becoming a Fully-Fledged Quantitative Software Engineer

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. Read more...

Resolving Confusion about Deliberate Practice

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? Read more...

Book Review: Developing Talent in Young People by Benjamin Bloom

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. Read more...

Who Needs Worked Examples? You, Eventually.

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. Read more...

How Bloom’s Taxonomy Gets Misinterpreted

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. Read more...
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Blog (Tier 2)

The Cure for Procrastination

The cure for procrastination is often just a tiny dosage of action. Tell yourself you’ll stop after a few minutes. Usually you won’t want to, because you had built the task up in your head to be worse than it is. Read more...

Learning Is a Balancing Act

Figure skating illustrates how effective learning is a balancing act: strong foundations are required before practicing advanced skills, but advanced skills also make your foundations more robust – provided you’ve mastered the basics well enough to get a grapple on the harder moves. Read more...

Self-Discovery Is an Effortful Process

Self-discovery doesn’t feel pleasant every step of the way – that’s the point. You discover what you’re good at and love by working hard at various challenges until the signal emerges from the noise. There is no shortcut. Read more...

Optimize Actions, Not Plans

Optimization is best spent on actions, not plans. Take action and then optimize the next rep. No plan survives contact with reality, so there’s no point layering optimizations on scenarios that may not play out. Read more...

How To Get The Most Value Out Of a STEM Degree

Pre-learn your major before college. The compound effect: placing out of intro courses leads to a reputation as the advanced student, which opens research and internship doors immediately and lets you widen the gap throughout college. Read more...

How To Balance Investing Vs. Spending

Two rules of thumb: don’t upgrade your lifestyle until you can sustain it at 3-4% of your net worth annually. And once an upgrade becomes a rounding error, force yourself to do it – but only if it solves a real pain point. Read more...

Common Failure Modes in Early Math Education

1) Staying in block-counting land so long that the the manipulatives become a crutch. 2) Covering a bunch of special-case arithmetic strategies instead of getting the kid really solid on the standard algorithms. 3) Avoiding memorization. 4) Serving up too many easy facts during times tables practice. Read more...

I Love Seeing Hardcore People Moving Into Serious Edtech

We put man on the moon with computers weaker than a digital watch. Why don’t we have efficient learning at scale? We overcame Earth’s gravity half a century ago, but we can’t overcome the gravity of educational mediocrity? Bullshit. That’s why I get so excited seeing hardcore people moving into serious edtech. People who don’t take bullshit for an answer. Read more...

I Believe You Have Potential

What people tend to need the most yet have the least in their lives is a supportive hard-ass. Not to be confused with an unsupportive hard-ass or a supportive pushover. That’s the gap I aim to fill as best I can with my writing. Read more...

The Missing Middle in Test Prep

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. Read more...

Recall First, Reason Second

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. Read more...

How Taxing Work Becomes Fun

“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.” Read more...

The Importance of Learning Your Prerequisites

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. Read more...

What Learning Actually Is – at a Concrete, Physical Level in the Brain

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. Read more...

Retrieval Practice is F*cking Obvious

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. Read more...

Fortify Your F*cking Fundamentals

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. Read more...

Prereq Yo’ Self Before You Wreck Yo’ Self

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. Read more...

How I Would Go About Learning an Arbitrary Subject Where No Full-Fledged Adaptive Learning System is Available

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. Read more...

Math is a Well-Defined Body of Knowledge

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. Read more...

How to Cultivate Discipline

Tear down the unproductive habit and build up a counter-habit whose gravity eventually becomes strong enough to completely overtake the original habit. Read more...

Complete Individualization: an Often-Forgotten yet Critical Component of True Deliberate Practice

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. Read more...

The Importance of Hardcore Skills

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. Read more...

Career Hack: Put Pressure on Your Boss to Come Up with More Work For You

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. Read more...

The Future of Education

To quote a Math Academy student: “The fastest and most rigorous progress will be made by individuals in front of their computers.” Read more...

Five Steps to Becoming a Fully-Fledged Quantitative Software Engineer

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. Read more...

Why Talent Development is Necessary in Math

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. Read more...

A White Pill on Cognitive Differences

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. Read more...

Who Needs Worked Examples? You, Eventually.

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. Read more...

How Bloom’s Taxonomy Gets Misinterpreted

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. Read more...

Higher Math Textbooks and Classes are Typically Not Aligned with the Cognitive Science of Learning

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. Read more...

Student Bite Size vs Curriculum Portion Size

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.) Read more...

Deliberate Practice: The Most Effective Form of Active Learning

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. Read more...

Your Mathematical Potential Has a Limit, but it’s Likely Higher Than You Think

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. Read more...

Struggle Does Not Imply Inability

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. Read more...

People Differ in Learning Speed, Not Learning Style

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. Read more...

Accountability and Incentives are Necessary but Absent in Education

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. Read more...

Cognitive Science of Learning: How the Brain Works

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. Read more...

Critique of Paper: “An astonishing regularity in student learning rate”

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. Read more...
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Math Academy

Deep Learning With Yacine Podcast #1: The Science of Learning Math (and Anything Else)

[2:10] My background: growing up in a non-technical family and finding math on my own.
[5:45] Self-studying 3,000 hours of college math in high school: starting with calculus the summer after 10th grade and continuing through undergraduate-level math for the rest of high school.
[16:10] Whether the same ground could have been covered more efficiently – and how being responsible for other people’s learning eventually crystallized the underlying principles.
[29:55] How having math foundations in place paid off in research: getting into Fermilab and CERN research projects at university labs.
[43:10] What the Math Academy learning system looks like: adaptive diagnostic, custom knowledge graph, minimum effective doses of instruction followed immediately by problem-solving, mastery before advancing.
[47:34] How we built the knowledge graph: years of manual work by domain experts, refined with analytics for nearly a decade.
[1:10:46] How the FIRE (Fractional Implicit REpetition) algorithm works: solving a harder problem implicitly reviews the sub-skills it encompasses, compressing the review pile significantly.
[1:35:50] Math and sport. Cognitive science principles – mastery before advancing, spaced practice, interleaving – are often easier to see in sport than in math.
[1:42:00] Does doing math well require different skills than teaching it well?
[1:56:25] Automaticity as a prerequisite for deeper understanding.
[2:05:35] The anatomy of “aha” moments.
[2:14:11] Learning math as an adult: the amount of work doesn’t change, only your free time does. Math Academy’s Mathematical Foundations sequence covers the prerequisite stack for university math in roughly 15,000 minutes.
[2:24:10] Balancing fundamentals and exploration: exploration pays off most at the frontier of a subject.
[2:33:55] Is it ever too late?
[2:46:00] Bottom-up versus top-down learning.
[2:56:30] Students with ADHD often feel the effects of inefficient pedagogy more strongly. Interleaving minimum effective doses of guided instruction and active problem-solving is better for everyone.
[3:06:20] AI tools as a multiplier on existing ability: the more you know, the more useful they are; the less you know, the harder it is to detect when they’ve gone wrong.
[3:14:37] What I’m most focused on right now: taking Math Academy from workshop to factory – producing courses at scale without sacrificing quality. Read more...

Failure Modes in Teaching - Math Academy Podcast #8, Part 2

What we covered:
– In elementary school, there’s often an intense focus on conceptual understanding, but not enough time spent building real fluency with core skills. And this has left many kids without automaticity on basic things like multiplication facts. Math is extremely hierarchical, and when students don’t have the basic facts at their fingertips, they quickly run into bottlenecks as the material gets more complex.
– Sure, drills can be made more fun, but the bottom line is that they have to get done. In high school and college, most of the class time is spent copying notes from the board – and these notes are often copied by the instructor from a textbook or from other source material. This game of telephone through transcribing is just a performative activity. It’s theater. It’s passive and it does next to nothing for learning and retention.
– In upper level college math courses especially, students may only receive short weekly problem sets, which really aren’t enough to build mastery, even if the problems are really hard, because students just spend most of their time flailing around.
– The bottom line is that students need reps: lots of them, building up scaffolding to the highest, hardest levels that they’re expected to reach. High school assignments tend to be better in that regard, but students frequently don’t receive timely feedback, and often their work isn’t even graded for accuracy. That feedback loop is so critical: without it, students won’t know what they’re doing wrong or how to improve.
– So rather than just pattern matching to how math has traditionally been taught, what actually makes training effective? There’s a few core principles:
1) Maximize the amount of time spent actively learning, interleaving minimum effective doses of explicit guided instruction active practice.
2) Make sure students are consistently working at the edge of their abilities: not bored, but not overwhelmed.
3) Provide frequent, timely feedback so students can adjust and improve.
These principles should be applied to math education and training environments everywhere.

0:00 - Introduction
3:13 - Professors often wing pedagogy
5:37 - Too much class time is spent transcribing notes
7:41 - College problem sets are too short
12:53 - A lot of homework isn’t even graded for accuracy
18:22 - Copying notes in class is performative productivity
22:29 - Alex taught math courses at University College London
25:35 - Teaching is often an annoying obligation for research professors
30:03 - The bar for teaching is on the floor
32:34 - Even football practices often waste players’ time
34:20 - Most training is inefficient because people pattern match to the status quo
34:57 - First principles for effective training
37:13 - Too many models can paralyze and become a crutch for kids
39:24 - Kids can get stuck using training wheels in math forever
42:05 - Non-standard methods are often distracting and inefficient
46:18 - Designing 6th-8th grade courses to align with school curricula
52:30 - Conceptual understanding without ability is useless
55:17 - Skills practice can and should be gamified Read more...

Role of Teachers in the Math Academy Classroom - Math Academy Podcast #8, Part 1

What we covered:
– A lot of schools have recently begun using Math Academy in their classrooms. And one of the biggest benefits of using Math Academy is that it automates all the mechanical parts of teaching, like writing questions, keeping track of what students know and what they don’t know, monitoring student progress, assigning extra practice when needed, grading, all that grindy stuff.
– None of these tasks is enjoyable. They suck. Just ask any teacher. I mean, we grinded through all that back when we were teaching ourselves, and it takes so much effort just to get even a halfway decent approximation of doing it right. And there’s just a limit to how well that you can do it if you’re doing it manually. It’s the whole reason why we built the system.
– And what that system does, what Math Academy we does is it frees up teacher bandwidth to focus on the human elements of teaching: building relationships, connecting what students are working on to their own unique interests. Those kind of things that enhance the learning experience, but that really can’t replace skills practice.
– I mean, in-class projects can be great, but only if students have the prerequisite knowledge to be successful with them. If they don’t, then projects are frustrating, and the students who understand the material will end up doing all the work and carrying everybody else, who will learn next to nothing. It’s inefficient and frustrating all around unless students have their skills in place.
– Ultimately, if students don’t master the math in each class, they’ll be unprepared for the next one. And in a subject as hierarchical as math, these gaps compound quickly. True empowerment isn’t simply telling students they have potential. It’s making sure they actually have the real skills to move forward and realize that potential.

0:00 - Introduction
2:56 - What is the teacher’s role alongside Math Academy?
5:37 - Math Academy frees up teachers to do the human parts of teaching
7:03 - Projects are great if students have the prerequisite skills
7:42 - Drills without context are boring
8:43 - Games without skills are inefficient
11:14 - Build fun activities on top of a solid foundation of skills
12:15 - Teachers can tailor the class to the students’ preferences
13:28 - Implementing mastery learning is too much work for a single teacher
15:27 - Doing projects without prerequisites is frustrating
16:57 - True empowerment is giving kids the skills they need to succeed
19:30 - Missing skills compound in hierarchical skill trees
24:06 - Lack of automaticity in lower level skills slows down higher level tasks
27:14 - The MA team builds and improves courses through experience
29:21 - The MA team targets tasks with low pass rates for additional scaffolding
31:03 - Alex built knowledge graph intuition through years of experience
37:40 - Social media enforces hyper-accountability
39:19 - Differential equations courses are often a hodgepodge of disjointed techniques
43:20 - Math Academy university courses are a superset of elite university content
45:18 - Differential equations is a highly branching subject
49:21 - The breadth of Differential Equations makes it often poorly taught Read more...

Announcing our SAT Math Prep Course

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. Read more...

OpenEd Podcast: The Real Reason Your Kid Hates Math (It’s Not Them)

Most kids hate math because it’s taught inefficiently, through one-size-fits-all lectures, where they’re constantly asked to learn new things despite not having mastered the prerequisites. Students can learn far faster and with less stress when instruction is individualized, mastery-based, interleaved, and reinforced through spaced repetition. Math becomes frustrating when students are pushed ahead with gaps in foundational skills or cognitively overloaded; it becomes motivating when they experience small, frequent wins at the edge of their “knowledge frontier.” Math Academy operationalizes this through an adaptive diagnostic that detects missing prerequisites and constructs a custom course to cover them, short alternating bursts of instruction and practice to ensure that students master material before moving on, and cumulative spaced review & quizzes to prevent forgetting, creating an individualized “glove fit” to each individual student. The broader vision is a shift away from lockstep classrooms toward individualized, coach-like learning. Read more...

Earning the Right to Scale - Math Academy Podcast #7, Part 2

What we covered:
– As Math Academy has grown over the past year, we’re getting a better sense of general do’s and don’ts when scaling a startup. We’ve learned hard lessons about overloading the database, the task processor, and our team, requiring numerous infrastructure and process updates.
– Schools have been using the system and we’ve built plenty of additional features to, among other things, accommodate unique billing schemes and make it easy for teachers to manage classes on the system.
– We’ve intentionally grown organically and were self-funded, which forced us to do things manually at the beginning. Years ago, we taught math classes in person and Jason onboarded our first online users on hundreds of hour-long individuals and calls. These were crucial experiences to learn who our customers are, what they want from the product, and common failure modes.
– In our experience, doing things manually at the beginning ensures that you 1) build a product that customers actually get value from, and 2) you don’t clutter your product with unnecessary bells and whistles that don’t add value. In other words, you have to do the manual work to earn the right to scale.

0:00 - Introduction
2:18 - Building infrastructure to handle increasing load
3:41 - Bringing on AWS expertise to robustify the backend
4:22 - An overloaded database enters a new realm of physics
5:50 - Prioritizing execution over perfection in start-ups
6:33 - Paying the bill for accumulated infrastructure debt
7:53 - Improving job prioritization of the task processor
9:52 - Benefits of scaling organically
11:42 - Wisdom is the result of failures
12:18 - There is no substitute for experience
13:17 - Focusing on solving problems, not advertising
14:48 - Upgrading with surgical precision
15:35 - The pain-point compass
17:04 - Managing finite time and resources
18:27 - Development of the gravity feature
20:42 - Gravity is a suggestion, not a hard override
22:25 - Limiting gravity to avoid cognitive overload
28:29 - Balancing customization and customer confusion
31:28 - The feature sandbox
33:58 - Increasing volume of customer support emails
35:22 - Additional infrastructure requirements for schools
36:18 - Learning about the customer through direct interaction
38:14 - Step 1: Manually added schools using spreadsheets
40:22 - Step 2: Developed tools to handle specialized school requests
41:23 - Step 3: Goal is 100% self-service sign-ups for schools
42:32 - Solve the problem manually first, then automate it
43:44 - Why focus on schools?
46:15 - Math Academy goes to college
49:37 - You can’t anticipate every edge case
52:14 - Letting user behavior build the product roadmap
58:54 - Becoming successful means working harder
1:00:24 - The customer support hurdle
1:03:27 - How Justin’s expanding roles drove growth (both personal & company)
1:09:03 - Teaching as market research for Math Academy
1:10:52 - The value of having been inside the trade Read more...

2025 in Review: Content Production - Math Academy Podcast #7, Part 1

This is a Math Academy “Wrapped” for 2025, focusing on the content side of things. In summary, here’s the good:
– We released a Discrete Mathematics course.
– We added hundreds of “missing middle” topics to our SAT Math Fundamentals course to bridge the chasm between what’s in standard school curricula versus what’s tested on the SAT.
– We soft-launched a SAT Math Prep course that students automatically promote into after finishing the fundamentals course, where they see their estimated SAT score instead of a progress percentage, and they do even more SAT-specific training such as taking frequent mock SAT practice exams and doing rapid-fire problem practice to build up speed and comfort with all the slight variations in the ways that questions could be phrased on the test.
– We added tens of thousands of free response questions throughout our middle school and high school courses.
– We developed all the content including coding projects for our first machine learning course, to be released once the coding interface is ready. (If you’re waiting on that course and absolutely must start your ML journey right this moment, note that there’s a freely available 400+ page textbook that I wrote while teaching this stuff manually in the school program – it’s called “Introduction to Algorithms and Machine Learning.”)

Of course, we’re under no illusion that we need to ramp up our rate of course production and transition from a workshop to a factory. We started pursuing that goal last year, and while there has been much pain from hitting our heads on basically every ledge possible, we’ve learned a lot and have just recently, in the past couple weeks, hit an inflection point where the factory transition is finally coming together. As Alex summarized in a recent post on X: we’re working tirelessly to upgrade our course development pipeline, building new tools and processes to help us manage a higher volume of courses so we can increase output while maintaining the quality you’ve all come to expect. In particular, we’re using our nearly-finished Differential Equations course as a guinea pig to test-drive some of our new tools and processes. This is the year that Math Academy comes out of the basement and onto the factory floor.

0:00 - Introduction
3:57 - Added 115 “Missing Middle” topics to SAT Prep
6:06 - Integrating the SAT Missing Middle topics into other courses
9:42 - Added tens of thousands of free response questions
10:34 - Free response questions are useful because they don’t prime you
13:33 - When to use free response vs. multiple choice questions
14:54 - Too many free response questions taxes learners
16:39 - Limiting the length of free response answers
18:08 - Building infrastructure for free response questions was a beast
20:42 - SAT test prep course
22:22 - Machine Learning has been the hardest course to develop so far.
23:12 - People who know machine learning, math, and how to teach them are rare
25:06 - The Eurisko book was the best resource for developing the Machine Learning course
28:51 - Balancing repetition and computational load in Machine Learning problems
29:43 - Designing minimum viable problems for Machine Learning
33:53 - Building the infrastructure for dynamic select questions was a nightmare
36:12 - Dynamic select questions are good for proofs and university-level math
38:03 - The Differential Equations course is almost finished
40:23 - Iterating on course development to make better courses
42:00 - 2026 is the year of scaling up course production
43:03 - How to scale up the team without sacrificing course quality
44:39 - Learning the hard way about hiring too quickly
46:20 - Challenges of managing a fully remote, geographically dispersed team
48:54 - Building tools to measure company output
50:06 - Optimizing content writer performance is like optimizing student learning
52:31 - Incentivizing content creation to improve output
56:36 - Courses planned for the longer term
58:01 - You need to learn concrete computations before abstract proofs
59:32 - Why we separate university-level courses into computational vs proof-based
1:01:07 - The best textbooks for beginners are NOT the most complex
1:02:37 - Teaching proofs and computations at the same time overloads most students
1:04:16 - Intuition through repetition
1:04:49 - Wisdom is the abstract compression of lived experiences
1:07:39 - Mastering details before abstracting Read more...

Learning Debt and Skill Insolvency - Math Academy Podcast #6, Part 3

What we covered:
– The dangers of accumulating learning debt: the gap between what you can do and what you need to be able to do.
– If you miss building up your foundational skills in school or sports, you can get by for a while. You develop some compensatory strategies, like favoring your forehand over your backhand, or using ChatGPT to write all your school essays.
– But learning debt is like any other kind of debt: it accrues interest and eventually comes due. Over time, the workarounds become more complex. The cognitive load increases. You start avoiding situations that expose the gap, and this is where you hit your ceiling. You can’t pursue an engineering degree if you can’t do algebra. You can’t be competitive in tennis if you can’t hit with your backhand.
– Learning debt often begins because of a lack of oversight by adults. Parents, teachers, and even coaches sometimes think they’re being nice not telling you that you need to work on your weaker side, or you need to stop using a calculator on your math problems. It feels like nagging, and it can create conflict between adults and learners. So they let it slide.
– But this failure to hold the line early on inhibits students’ future potential. And when it occurs across many students across many schools, it degrades the whole educational system – leading to the current situation in which many students are totally unprepared for the rigors of college.

0:00 - Introduction
2:04 - Course phases: instruction, final review, final exam, remediation if needed
5:25 - Generating full-length SAT exams for our prep course
6:53 - Loosening up the gravity throttle for high-performing students
14:59 - Aptitude is measured by accuracy rate
18:07 - Accuracy correlates first with aptitude, second with conscientiousness
21:35 - Assessment vs. non-assessment accuracies
23:43 - Propagating accuracy through the knowledge graph
24:27 - Hidden skill gaps force bad compensations
25:27 - Sports make skill deficits and bad compensations obvious
33:38 - The Math Academy system holds you accountable for every skill
34:18 - Completing the square: a common skill deficit with temporary workarounds
36:15 - Reliance on Desmos undermines students’ ability to graph functions
37:38 - You need to know your multiplication facts for factoring
38:13 - Foundational deficits are usually caused by lack of adult oversight
38:52 - Shoring up foundations is effortful but has huge ROI
40:40 - Filling in missing foundations makes kids so much more confident
41:12 - Missing foundations stall learning and drive cheating
42:12 - Faking competence backfires downstream
45:33 - The truth hurts but is the kindest thing in the long run
46:26 - Learning debt eventually comes due, with students paying the biggest price
47:12 - Kicking the can down the road in education
49:46 - The cost of a broken education system Read more...

On The Rails and Out Of Scope - Math Academy Podcast #6, Part 2

What we covered:
– The benefits of short problems. Math Academy problems typically take only a minute or two. This way, students can stay on the rails with lots of reps, successfully building up complexity instead of getting crushed by it from the start.
– What goes wrong in college math classes: they tend not to scaffold content very well, forcing students to build their own bridges across knowledge & skill gaps. Weekly problem sets often consist of a handful of hour-long problems that instructors hope students will “self-scaffold” up to. In reality, what happens more often is that students fall off the rails.
– Founders of growing start-ups cannot be hands-off. “Things falling off the rails” is the most realistic and most dangerous failure mode, not micromanaging. Founders of small, scaling companies need to be in “founder mode,” not the “manager mode” that CEOs of huge, well-established companies are in.
– Within teams, it’s important to let conversations flow out of scope. Every innovation, every solved problem, requires relevant background context, and you often don’t know what the full context is beforehand. It’s easy to let conversations flow out of scope when you like who you’re working with and what you’re working on.

0:00 - Introduction
1:32 - Why Math Academy problems are short by design
9:48 - Long problems dilute reps on the skill that actually matters
11:00 - Isolate the new skill first, then recombine into full problems
14:10 - Typical undergrad math classes: too few problems, too complex from the start
18:07 - The proof skills gap: often assumed and not taught
29:32 - Alignment decay: teams naturally drift out of sync unless continually aligned
35:04 - Small misalignments compound fast
38:28 - Founder mode: stay in the weeds to stay in sync
49:07 - Early, frequent parent communication avoids end-of-term blowups
50:48 - High-trust collaboration requires relentless communication
57:42 - Out-of-scope conversation enables context sharing
59:14 - Over-scoping kills context sharing
1:00:51 - Enjoyment & trust fuel context sharing
1:06:13 - Missing context produces confidently wrong outcomes
1:10:01 - LLMs fail when context is missing
1:11:38 - Humans fail when context is missing
1:14:19 - Online discourse fails when context is missing Read more...

Why Can’t College Students Do Middle School Math? - Math Academy Podcast #6, Part 1

What we covered:
– A recent report from the University of California San Diego revealed that 1 in 12 incoming freshmen were not proficient in middle school math – basically, anything above arithmetic with fractions. Their existing remedial math course was too advanced for these students, so they had to design even lower remedial remedial math courses. Even crazier, over a quarter of these students had a perfect 4.0 GPA in their high school math courses.
– It’s not just UCSD. This is everywhere. A similar thing happened at Harvard, too, having to add remedial support to their entry-level calculus courses. It’s like that movie Olympus Has Fallen, except this time it’s Harvard. It’s a catastrophe.
– How did things get this bad? Teachers and administrators face relentless pressure to inflate grades, and during the pandemic many universities went test-optional, removing the only signal that reliably correlated with actual math readiness. That decision simultaneously elevated high school grades to the sole gatekeeping metric, intensifying incentives to inflate them.
– This has all coincided with the advent of LLMs, which make it increasingly easy for students to cheat. The result was predictable: grades became untethered from real competence, and multiple cohorts of students entered college without ever having to demonstrate foundational math skills.
– Teachers have to play both good cop and bad cop, and there is no avoiding the latter. If you refuse to play bad cop at all, you eventually end up playing it constantly. The best teachers are strict from the start and ease up later, once students understand that hard, honest work is non-negotiable.

Timestamps:
0:00 - Introduction
2:11 - Freshmen math collapse: 1 in 12 UCSD freshmen don’t know middle school math
6:45 - Remedial remedial math: UCSD created remediation for remedial math
8:40 - Inflated grades: 25% of remedial-remedial students had perfect GPA in HS math
10:06 - Test-optional admissions removed the last objective metric
12:13 - Pandemic inflation: GPAs skyrocketed
14:37 - Removing tests pressures teachers to inflate grades
16:52 - Grade-grubbing: endless negotiating, complaining, accusations
19:01 - Then vs. now: parents, tests, accountability
27:38 - Crisis opportunism: “Never let an emergency go to waste”
29:33 - No tests = no knowledge requirements
33:28 - Elite collapse: Harvard has the same problem
36:31 - No enforcement means no standards
37:40 - LLM cheating is trivially easy
38:25 - Catching a cheater and turning him around
48:46 - Cheating is like taking mob money. Now you’re in, you’re never out.
50:41 - Assessments must be done in person
55:06 - LLM cheating is often obvious yet hard to prove
57:17 - How to prevent cheating on long papers
58:28 - Start hardcore, then lighten up gradually
1:01:37 - Good teachers play bad cop when needed Read more...

The (Free) Textbook I Wrote While Teaching The Most Advanced High School Math/CS Sequence In The USA

It scaffolded high school students up to doing masters/PhD-level coursework: reproducing academic research papers in artificial intelligence, building everything from scratch in Python. A former student worked through it right before conducting research that won 1st place ($250,000) in the Regeneron Science Talent Search, getting personally recruited by the head of NASA (with a fighter jet ride as a signing bonus), and publishing his results, solo-author, in The Astronomical Journal. Read more...

Getting Kids To Do Hard Things - Math Academy Podcast #5, Part 2

What we covered:
– Most kids are not intrinsically motivated to do the hard things: practice their soccer drills, do their math homework, eat their broccoli. Getting them to do the hard things often requires gamification and/or incentives.
– A little gamification goes a long way. Jason gamified drills for his kids’ soccer team to get the most out of each practice (e.g., “zombie attack”), and it was unreasonably effective. The XP leaderboards on Math Academy are also unreasonably effective.
– A good incentive can change kids’ behavior overnight. The incentive doesn’t need to be big; it just needs to be something the kid really cares about. Find the thing the kid would rather be doing, and use it to motivate them to do what they’re supposed to be doing. They won’t need the incentive forever; as the kid gets used to the feeling of a new behavior, it gradually turns into a habit that they can maintain on their own.
– Even when you’re doing what you love, there will be grindy phases. But kids typically don’t understand this. They might get interested in a talent domain and want to become good enough to build a life around it, while simultaneously resisting doing the hard work to make that happen (i.e., stage 2 in Bloom’s talent development process). It’s often up to parents, who can see the long game, to push their kids through the difficult parts in paths that they find rewarding.
– For instance, the most mathematically gifted student I ever worked with, who was drawn into math by his own intrinsic interest, still needed to be pushed to learn calculus. Now he’s having the time of his life working on physics-y, calculus-heavy research-level math problems in high school. Even after finding something he loves and is good at, he still needed to be pushed to do the hard work to unlock more of it.

Timestamps:
00:00:00 - Most kids are not intrinsically motivated to do hard things – homework, drills, practice. They usually need incentives to get through.
00:08:16 - A little gamification goes a long way. Jason gamified drills for his kids’ soccer team to get the most out of each practice (e.g., “zombie attack”).
00:14:05 - A good incentive can change behavior overnight. It doesn’t need to be big, just something the kid really cares about, and they won’t need it forever. It’s about building a habit until they can maintain it on their own.
00:54:16 - The most mathematically gifted student Justin ever worked with needed to be pushed to learn calculus, and now he’s having the time of his life working on calculus-heavy research-level math problems.
01:11:54 - Even when you’re doing what you love, there will be grindy phases. It’s important for parents to help kids push through those grindy phases so that they can unlock more of what they love. Read more...

American Enterprise Institute’s “The Report Card” Podcast: Math Academy

You know the kind of trainer an actor gets to prep them for the superhero role in a Marvel film? We’re that, for math. 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 about an hour.
Here’s a summary of what we covered:
[1:34] The big problem in math education is a lack of individualized instruction. In a classroom with one teacher teaching the same thing to all the students, it’s way too easy for the top half of the class and way too hard for the bottom class. What we do is pinpoint the exact problem that each student should be working on right at this moment to make maximum progress in their math learning.
[4:46] So much difficulty in math learning can be traced back to missing prerequisite knowledge. That’s why it’s important to start each student off with a diagnostic that combs through many years of prerequisite knowledge that they need to know to succeed in their chosen course. If we find any knowledge gaps, we fill them in before asking the student to learn any more advanced material that depends on it.
[6:50] We get a very high-resolution picture of the student’s knowledge profile by overlaying every question/answer event onto a structure called a “knowledge graph”. The knowledge graph encodes all the dependency relationships between mathematical topics. We leverage it to squeeze a ton of information out of every single question that we ask the student – not just figuring out what they know and don’t know, but also figuring out exactly what learning tasks they should be working on to maximize their learning efficiency every step of the way.
[8:44] Elsewhere, lots of students struggle with calculus due to gaps in prerequisite knowledge. Good teachers know this, and try to fill those gaps, but there’s a limit to how well the teacher can do this because all the students have knowledge gaps in different places and the teacher can only teach one thing at a time to all the students. But we can target these gaps precisely, backfill them, and move on based on what each individual student knows – fully individualized instruction for all students in parallel, delivering exactly what they need to work on, focusing on those weaknesses and not wasting time on things they already know cold.
[12:05] If you have more talent/aptitude, then you’re going to get more bang for your buck out of practice. You’re going to require fewer reps before getting solid enough to move on, and you’re going to generalize more naturally. However, of the students who get all the way up to calculus before struggle really sets in, the biggest roadblock is typically not talent/aptitude but rather gaps/weaknesses in prerequisite knowledge, an issue that can be resolved with fully individualized instruction.
[14:50] Math Academy origin story: Jason & Sandy coached their son’s 4th grade math field day team. That turned into a pull-out class 3 days per week the following year. The superintendent came by and was shocked to see 5th graders doing trigonometry, advanced algebra, even a little bit of calculus. So he asked Jason & Sandy to create a pilot school program in the Pasadena Unified School District.
In the program, which came to be called Math Academy, students learned all of high school math (prealgebra through precalculus) during 6th/7th grade and took the AP Calculus BC exam in 8th. Students were invited to the program by scoring in the top 7-8% of a middle school math placement that all students in the district took at the end of 5th grade. Keep in mind that about two-thirds of students in the district were on free or reduced lunch, and also, nearly half of Pasadena K-12 students are educated in private schools, compared to the California average of ~10%. In other words, generally speaking, these were not smartest kids in California, and their parents were not Caltech professors.
Jason developed software to automate the process of assigning/grading homework, and together during the pandemic we upgraded it to figure out what each individual student should work on and teach it to them directly without any human intervention. We worked like maniacs to get it ready before school went fully remote the next year, and once we put the school program on it, educational outcomes (including AP Calculus BC scores) skyrocketed. Because of the software, our students experienced a massive learning GAIN, not a loss, during the pandemic. Naturally, it only made sense to keep the school program using the software even after class returned in-person.
[21:55] We have spent thousands and thousands of hours over the years building and fine-tuning our knowledge graph. It’s not off-the-shelf, it’s not automatically generated. It’s the hard work from domain experts, primarily our director of content Alex Smith for the forwards graph (what are all the prerequisites you need to learn in order to unlock a topic) and myself for the backwards graph (when you practice a topic, what component sub-skills are implicitly getting practiced and to what extent).
[25:04] We analyze our knowledge graph by overlaying a big heatmap of where students are doing well or struggling at various parts in the graph. It’s almost like traffic intersections in a city – which ones are where most accidents happen? Let’s go make those safer. We’ve been building and refining the knowledge graph for nearly a decade now with all these analytics.
[27:22] We have a wide variety of user segments. We can help anyone who seriously wants to learn math. Basically, anyone in any sort of educational situation, kids, adults, public school, private school, charter school, homeschool, grade school, high school, college, students who are accelerating, students who are just trying to keep up, “math team” people, people who don’t yet think of themselves as “math people”, adults changing careers to a math-ier field or pursuing a math-ier subfield within their current career, the list goes on and on.
[29:31] The best predictor of how long someone will use the system and how much math they’ll learn is what kind of habit structure they have in place. Students who are consistent, as opposed to sporadic, go much further. It’s that simple.
[34:13] The only math learner persona we can’t help is the crammer – the student who has an exam in a week, is nowhere near prepared, and wants a “quick fix”. We are like a gym, and there’s always people who walk in the gym and think they’re going to work out really hard for a week and look like Thor by the weekend. There is no way to make that happen in a week, no matter how hard you work out. If you show up consistently, like 3-6 times per week for a 30-90 minute session, and then you keep that up for months, then you’re going to come out looking like the mathematical equivalent of a Greek god. But if you are looking for some kind of easy, “how can I change my life in one week,” then I’m sorry, I don’t know what to tell you.
[37:16] We alternate between minimum effective doses of text-based guided instruction followed by active problem-solving. It’s the mathematical equivalent of a tennis instructor showing a quick demonstration of how to hit a ball, just for a minute, and then students practice hitting the ball with that technique until they’re solid enough to move onto the next technique.
[40:16] Real-time reactions and hot takes: Jason on collaborating with school districts, my thoughts on the edtech industry, Jason founding a company with his wife, my experience interacting/growing on X, Jason’s impression of Waymo, my impression of math textbooks, Jason’s thoughts on the “move fast and break things” ethos, Justin’s thoughts on people’s screen time concerns.
[52:10] People say, “just give me the intuition.” But intuition comes through repetition. That’s how you get the automaticity, the natural feel, and that’s what intuition is.
At the same time, it’s important to be efficient. Don’t work 100 problems of the same type in one day. Maybe do 10 to start, then 5 the next day, another 5 a week later, and so on, while you’re filling the empty space with practice on a ton of other skills. You have to get your reps, but you also have to distribute them out over time. That’s how you learn efficiently and build long-term retention.
When people want their math learning to be less skill-heavy and more concept-oriented, what they’re often really saying is that they want a fast overview of a subject without going into the details, without really getting your reps on everything. A video that explains all of calculus in an hour, or how neural networks work in 20 minutes.
But what we’re focused on is building up a true level of mastery. Not surface-level, not shallow. The optimization problem we’re solving is NOT “how fast can we imbue you with a shallow level of understanding, enough that you can tell your friend something cool or that you think you have opinions about it.” What we’re focused on is how quickly we can get you to operating mathematically almost like a professional musician plays their instrument, or a professional athlete plays their sport.
[55:51] As a rule of thumb, if it wouldn’t work in sports, it’s not going to work in math.
[57:31] Students on our system typically learn about 3-4x as fast as a normal class. That’s why, in our school program, the students could go from pre-algebra through AP Calculus BC in 3 years, from 6th-8th grade. When that first happened, and the Washington Post wrote articles about it, lots of people couldn’t believe it. Which is why we had them take the AP Calculus BC exam so we actually have results.
[58:49] We hear all the time about students who are behind in their school class, and then use our system to catch up, and then start crushing their class, and then go well beyond their school class – as well as the resulting change in the student’s level of confidence. In just one year or less, just months, a student can go from thinking “I’m not a math person, I’ll never be good at it” to “I’m crushing my school class, it’s so easy.” That change in the student’s experience does wonders for their confidence.
[1:01:28] Is there an upper limit to how much math you can do per day and have it carry over into real learning results? Think about it like going to the gym. If you work out for 45 minutes, 5-6 days per week, you’ll get in incredible shape. You can do more if you want, but there is a point where you hit diminishing returns. Whether it’s Math Academy or the gym, it really comes down to how long you can sustain a productive full-intensity effort. It’s hard to keep that up for multiple hours, though you might be able to get better mileage by splitting up a multi-hour session into a shorter morning session and evening session. But every person is kind of different in their breaking point, how much they can stay focused and work intensely on the system. In general, one hour per weekday is what we’ve found to be the upper end of a sustainable approach for most students.
[1:03:38] We make students do review problems indefinitely into the future, but with expanding intervals – spaced repetition. It’s the optimal way to keep your knowledge base fresh enough to keep building on it without constantly having to go back and re-learn things. But we make this review process as efficient as possible by tracking all the subskills that are implicitly reviewed when you do a review problem, and we’re always trying to select tasks that kill many birds with one stone by exercising many subskills in need of review.
[1:08:41] Lots of people mistakenly think that students need a million different explanations of the same thing, and that one of those explanations is going to stick, and it’s different for each student. But really, all you need is one really good explanation that’s been battle-tested across a large number of students, and the students need to come into that explanation with all their prerequisite knowledge in place.
If you do that then you can get students learning the skills really well – students pass our lessons over 95% of the time on the first attempt, and over 99% of the time within two attempts, without any additional remediation (because enough knowledge has consolidated into their brain from the first attempt that it makes it cognitively easier for them to get over the hump the second time around).
That’s often surprising to people who think that every student needs a different explanation, but typically what they’re seeing is a symptom of the student lacking prerequisite knowledge, and you’re trying to come up with some explanation that allows them to grasp “enough” of the topic (not the whole thing) while at the same time not requiring too much in the way of prerequisite knowledge they’re missing.
[1:11:08] What makes math hard is the same thing that makes climbing a mountain hard: the steepness of the gradient. What we do is break every steep section of math into smaller steps. If you break things into small enough steps, anyone can learn. And that’s what we do with our analytics: where are the congestion points? Where are students struggling? It’s always where we’re trying to do too much at one time, so we break it up into more steps.
[1:12:25] It’s so important to have a reliable source of truth about what a student really knows, and grades are no longer a good source of truth. You remove test scores from the admissions process, the last objective metric and the last Jenga block, and you get bad situations like at UCSD where 8% of students were not proficient in middle school math. So many issues in education stem from a student having a piece of paper that says they’ve learned something when they actually haven’t. Read more...

Building Without Bloat - Math Academy Podcast #5, Part 1

What we covered:
– Any successful endeavor requires a great team: capable people, who like and trust each other, and have complementary skillsets and ways of thinking. Some modes of thinking cannot be performed at the same time within a single brain.
– Accountability requires control. You can’t hold someone responsible for outcomes unless you also give them control over the system that produces those outcomes (though you can set reasonable operational boundaries).
– Solve today’s problems today. Smart people can invent endless hypotheticals and build giant solutions to fake problems. Not only does this waste time, but it also burdens the system with complexity that becomes a future straitjacket. Everything you build must be carried forward, so focus on what’s present in front of you, not on imagined futures five steps away.
– In a scaling system, the sheer volume of interactions will expose a long tail of bizarre scenarios, almost like rare diseases you’d never anticipate. Users will often try to repurpose software beyond its design, like hauling a trailer with a motorcycle.

Timestamps:
00:00 - Introduction
03:48 - The importance of finding your complements
24:07 - The origin story of Math Academy’s content team
43:36 - No meta-work; just solve the problems in front of you
54:26 - Jason time vs real time (real time is longer)
59:00 - The long tail of rare edge cases and unexpected user behavior Read more...

Knowledge Graph Engineering: Mental Models & War Stories - Math Academy Podcast #4, Part 2

What we covered:
– Building a knowledge graph is like city planning & road construction. Too many prerequisites leading into a single topic creates a cognitive traffic jam.
– Elegantly rewiring a live knowledge graph: the evolution of our tooling and automatic validations. How to avoid staging servers & migrations and NOT have it blow up in your face.
– UI work takes time and adds complexity, so we spend it on the customer. Internal tools are almost entirely command-line; clickable buttons are for customers.
– Justin’s transition from research coding to real-time systems. He started with mathy, notebook-driven quant code and had to learn production engineering the hard way. Once he did, it was a massive level-up.
– Alex’s plan for dealing with “content papercuts” - small issues that pile up. Inspired by Amazon’s “papercuts team.”
– Our upcoming differential equations course, the last course in the core undergrad engineering math sequence.

Timestamps:
00:00:00 - Building a production-grade knowledge graph is like city planning and road construction
00:07:26 - Elegantly rewiring a live knowledge graph: the evolution of our tooling and automatic validations
00:24:47 - Justin’s transition from research coding to real-time systems
00:44:51 - Alex’s plan for dealing with “content papercuts” - small issues that pile up
00:58:02 - Our upcoming differential equations course Read more...

The Unreasonable Effectiveness of the Knowledge Graph - Math Academy Podcast #4, Part 1

What we covered:
– Why “problem solving” is often just a vague label people use when they haven’t explicitly enumerated the underlying skills, and how those skills can in fact be exhaustively mapped in a knowledge graph.
– How to approach research problems: Alex’s PhD journey, top-down familiarity vs bottom-up mastery.
– If you have natural talent, use it, but not as a crutch, otherwise you’ll stunt your long-term development. Don’t turn your blessing into a curse.
– The story behind building our SAT prep curriculum: realizing that the standard school curriculum leaves a massive “missing middle” unaddressed; identifying 115+ missing topics to bridge the gap between textbook math and the hardest SAT questions.
– Watching the manifold hypothesis play out in test prep: the SAT may appear to allow an astronomical space of possible problem types, but in reality the actual problems live on a compact, highly structured manifold that can be fully enumerated and scaffolded in a knowledge graph

Timestamps:
00:00:00 - Intro: “problem solving” is what you call it when you don’t really know what it is (i.e. you haven’t explicitly enumerated the skills)
00:04:11 - How to approach research problems: Alex’s PhD journey, top-down familiarity vs bottom-up mastery
00:20:28 - If you have natural talent, don’t use it as a crutch. Don’t turn your blessing into a curse.
00:29:06 - SAT prep, iteration 1: Realizing that the standard school curriculum leaves a massive “missing middle” unaddressed
00:33:45 - SAT prep, iteration 2: Covering the “missing middle” problems
00:53:38 - SAT prep, iteration 3: Building the “missing middle” knowledge graph
01:08:11 - Watching the manifold hypothesis play out in SAT prep
01:16:42 - The unreasonable effectiveness of the knowledge graph Read more...

Summary of My Own Math Journey

Grinding it like a video game, running a secret self-study op inside traditional classes, taking talent development seriously while making every rookie mistake, and how it unlocked a life I almost certainly never would’ve found otherwise. Read more...

Waging War on Mediocrity: Tales From the Trenches - Math Academy Podcast #3

What we covered:
– How bureaucracies instinctively reject new ideas like an immune system attacking a foreign organ, and what it takes to keep your project from being “spit out.” Concrete example: how Jason & Sandy muscled past institutional resistance to get 8th graders passing AP Calc BC.
– Every system inevitably decays into mediocrity unless someone fights to keep the standards high. The way you keep people, systems, and projects moving is by “horsing” them forward. Concrete example: how Justin kept 8th graders passing AP Calc BC, and what it looks like when a school succumbs to the gravity of mediocrity.
– Justin’s math self-study journey in high school: grinding math like a video game, running a secret self-study op inside traditional classes, taking talent development seriously while simultaneously hitting his head on every ledge and making every rookie mistake. Ups and downs, lessons learned, with tons of concrete examples.

Timestamps:
00:00:00 - Intro: Willing Things Into Existence
00:11:43 - How Jason & Sandy Willed Math Academy Into Existence
00:36:45 - Fighting The Gravity of Mediocrity
01:02:29 - Case Studies in Educational Dysfunction
01:21:53 - The Birth of Justin’s Self-Study Madness
01:50:48 - Self-Studying on the Sly During School
02:02:41 - The Highs & Lows of High School Research
02:22:38 - Outro: Paving the Path with Math Academy Read more...

Teach Like Your Life Depends On It - Math Academy Podcast #2

0:00 - What Would a Tutor Do, If Their Life Depended On It? (Part 1)
5:47 - Find Your North Star: Why Justin Quit His Data Science Job to do Math Tutoring Full Time
11:23 - Getting “Inside the Trade”
19:31 - What Would a Tutor Do, If Their Life Depended On It? (Part 2)
27:28 - Efficient Learning Techniques are Obvious if You Think About Athletics
33:45 - Enjoyment is a Second-Order Optimization
39:50 - We Need to Stay Hardcore, But Become Less Harsh
51:14 - Math Academy is Like “Yuri’s Gym”
59:06 - Vision for the Future of Math Academy
1:14:23 - Goal Setting/Advising and Communicating Progress
1:24:58 - If All You Show Up With is AP Calculus, You’re Probably Outgunned
1:51:08 - The Meta-Skills that Kids Need to Work Effectively on Math Academy
2:08:54 - How to Help Students Maintain Successful Learning Habits While Working Independently
2:32:29 - Overhelping: A Common Failure Mode of Well-Intentioned Parents/Tutors Read more...

The Long Game: Building Minds and Machines - Math Academy Podcast #1

0:00 - Introduction
4:00 - Applying the MA Way to X Growth
7:40 - Status of the ML Course and its Kick-Ass Coding Projects (Part 1)
25:50 - Jason’s Near-Infinite List of Important Things
34:20 - The ML Course Has Been a Massive Undertaking
42:10 - Breadth-First Development
44:30 - Status of the ML Course and its Kick-Ass Coding Projects (Part 2)
50:15 - Why Math Academy Needs To Do a CS Course
56:45 - The Never-Ending Stream of Confusion
1:00:30 - The Story of Eurisko, the Most Advanced Math/CS Track in the USA
1:24:20 - Intuition Through Repetition: Machine Learning Edition
1:29:40 - The Importance of Spaced Review
1:43:30 - Upcoming Course Roadmap
1:47:40 - Spaced Repetition 2.0: Accounting For and Discouraging Reference Reliance
1:54:45 - Overhelping: A Pathology of the Over-Involved Parent/Tutor
1:59:21 - Yes, You Need to be Automatic on Math Facts (and Yes, Rapid-Fire Training is Coming)
2:04:55 - What Happens When Students Don’t Know Their Math Facts
2:05:50 - The Horror of Attempting to Teach a Class When Students Have Multi-Year Deficits in Fundamental Skills
2:11:30 - Integrating Coding Into the Math Curriculum
2:18:00 - Combining Math and Coding is the Closest Thing to a Real-Life Superpower
2:18:55 - Creating a Full Math Degree and Getting Full College Credit
2:22:15 - The Power of Pre-Learning: The Greatest Educational Life Hack Read more...

What Happens When Middle School is Put to Good Use

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. Read more...

The Missing Middle in Test Prep

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. Read more...

The Metagame Podcast #39: Math Academy and The Science of Learning

The most comprehensive 2h overview of my thoughts on serious upskilling, to date. Not just how to train efficiently, but also how to find your mission. Not just the microstructure, but also the metagame. We covered tons of bases ranging from the micro level (science of learning & training efficiently) to the macro level (broader journey of finding, developing, and exploiting your personal talents).
[~0:30] 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?
[~9:00] Efficient learning feels like exercise. The point is to overcome a challenge that strains you. It is by definition unpleasant.
[~13:30] Knowledge graphs are vital when constructing efficient learning experiences. They allow you to systematically organize a learner’s performance data to identify their edge of mastery (the boundary between what they know and don’t know), what previously learned topics below the edge are in need of review, and what new topics on the edge will maximize the amount of review that’s knocked out implicitly.
[~18:00] None of this efficiency stuff matters if you don’t show up consistently. Progress equals volume times efficiency. If either of those factors are low then you don’t make much progress.
[~21:30] Getting excited about the idea of getting good provides an initial activation energy, but seeing yourself improve is what fuels you to keep playing the long game, and efficiency is vital for that.
[~26:30] Your training doesn’t have to be super efficient at the beginning. You can gradually nudge yourself into higher efficiency training even if you don’t have a whole lot of intrinsic motivation to begin with. However, there’s often a skill barrier you need to break through to really get to the fun part, and it’s advisable to do that in a timely manner so you don’t stall out. But at the same time, don’t rush it and fall off the rails.
[~34:30] A common failure mode: being unwilling to identify, accept, and start at the level you’re at.
[~41:30] Center your identity on a mission that speaks to you, that you can contribute to, and do whatever else is needed to further it, regardless of whether you perceive these other things to be “you” or not. You’ll be surprised what capabilities you develop, that you hadn’t previously perceived to be a part of your identity.
[~48:30] How to find your mission: sample wide to figure out what activities speak to you, then filter down and pick one (or a couple) that you’re willing to seriously invest your time and effort climbing up the skill tree and going on “quests”. You may not understand this early on, but skill trees branch out, and quests beget follow-up quests, and the act of climbing to these branch-points will imbue you with perspective that you can leverage to keep filtering down. If you iterate this process enough, it gradually converges into a single area that you can describe coherently and uniquely. That’s your mission.
[~55:30] Every stage in the journey to your mission is hard work, and the earlier you get to putting in that work, the better off you’re going to be. It’s never too late, but the longer you wait, the rougher it gets. At the same time, don’t make a rash decision, don’t tear the house down and build up a new house that you don’t even like. But don’t underestimate how fast you can progress when your internal motivation is aligned with your external incentives.
[~1:12:00] Focus on what matters. That’s obvious, but it’s so easy to mess up lose focus and not realize it until after you’ve wasted a bunch of time.
[~1:15:30] How to get back on the horse after you’ve fallen off. How to avoid feeling bad when something outside of your control temporarily knocks you off your horse. A good social environment can push you to get back on your horse.
[~1:26:30] If you’re a beginner, don’t feel like you have to be advanced to join a community of learners. You can do this right away. And don’t shy away from posting your progress – it’s not about where you are, it’s about where you’re going and how fast. It’s only people who are insecure who will make fun of you. Most people, especially advanced people, will be supportive.
[~1:31:30] There are numerous cognitive learning strategies that 1) can be used to massively improve learning, 2) have been reproduced so many times they might as well be laws of physics, and 3) connect all the way down to the mechanics of what’s going on in the brain. The biggest levers: active learning (as opposed to passive consumption), direct/explicit instruction (as opposed to discovery learning), the spacing effect, mixed practice (a.k.a. interleaving), retrieval practice (a.k.a. the testing effect). Read more...

How Taxing Work Becomes Fun

“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.” Read more...

Math Academy’s Eurisko Sequence, 5 Years Later: Student Outcomes Emerging From the Most Advanced High School Math/CS Track in the USA

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. Read more...

Scraping Bits Podcast #137 (Round 4): Learning Math is Hard, Proof Writing, Which Order to Learn Math

[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. Read more...

Q&A: But Don’t You Need 10,000 Hours To Learn Math?

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. Read more...

We’re Working On Streaks!

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. Read more...

CS Primer Show #23: MathAcademy and the efficient pursuit of mastery

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. Read more...

Chalk and Talk Podcast #42: Math Academy: Optimizing Student Learning

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. Read more...

Golden Nuggets Podcast #40 (Round 4): How Justin learns, new ML course, the magic of Twitter

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. Read more...

Scraping Bits Podcast #116 (Round 3): Essential Math for Machine Learning, Math Intuition/Creativity, Proof Vs Computation

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. Read more...

Golden Nuggets Podcast #39 (Round 3): MA’s upcoming machine learning course

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. Read more...

ML Courses can Vary Massively in their Coverage

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.” Read more...

Golden Nuggets Podcast #37 (Round 2): Balancing learning with creative output

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. Read more...

Scraping Bits Podcast #107: Proof Writing, Discovering Math, Expert Systems, Learning Math Like a Language

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. Read more...

The Future of Education

To quote a Math Academy student: “The fastest and most rigorous progress will be made by individuals in front of their computers.” Read more...

Five Steps to Becoming a Fully-Fledged Quantitative Software Engineer

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. Read more...

Golden Nuggets Podcast #35: Optimizing learning efficiency at Math Academy

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. Read more...

Scraping Bits Podcast #102: Learning Mathematics Like an Athlete

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. Read more...

The Tip of Math Academy’s Technical Iceberg

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. Read more...
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Cognitive Science

Learning Rate Differences Exist

A recent study measured a 2x learning rate difference between the 25th and 75th percentile – likely an underestimate due to methodological choices. The authors reported it as 1.5x and called it an “astonishing regularity.” Read more...

Recall First, Reason Second

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. Read more...

The Field of Education is Due For a Copernican Revolution

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. Read more...

The Importance of Learning Your Prerequisites

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. Read more...

Learning is Memory

This might feel obvious, but many learners don’t fully grasp the implications, and as a result, end up not actually learning much. Read more...

Experts Perceive Differently

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. Read more...

What Learning Actually Is – at a Concrete, Physical Level in the Brain

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. Read more...

Retrieval Practice is F*cking Obvious

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. Read more...

Complete Individualization: an Often-Forgotten yet Critical Component of True Deliberate Practice

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. Read more...

Road to Reading Podcast #23: Discussing Cognitive Science

[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. Read more...

A White Pill on Cognitive Differences

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. Read more...

Higher Math Textbooks and Classes are Typically Not Aligned with the Cognitive Science of Learning

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. Read more...

Spaced Repetition vs Spiraling

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. Read more...

Leveraging Cognitive Learning Strategies Requires Technology

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. Read more...

Cognitive Science of Learning: Interleaving (Mixed Practice)

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. Read more...

Cognitive Science of Learning: Spaced Repetition (Distributed Practice)

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. Read more...

Layering: Building Structural Integrity in Knowledge

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. Read more...

Cognitive Science of Learning: Minimizing Associative Interference

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. Read more...

Cognitive Science of Learning: Developing Automaticity

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. Read more...

Myths and Realities about Educational Acceleration

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. Read more...

Effective Learning Does Not Emulate the Professional Workplace

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. Read more...

People Differ in Learning Speed, Not Learning Style

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. Read more...

The Story of the Science of Learning

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. Read more...

Cognitive Science of Learning: How the Brain Works

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. Read more...

Optimized, Individualized Spaced Repetition in Hierarchical Knowledge Structures

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). Read more...
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Upskilling

Self-Discovery Is an Effortful Process

Self-discovery doesn’t feel pleasant every step of the way – that’s the point. You discover what you’re good at and love by working hard at various challenges until the signal emerges from the noise. There is no shortcut. Read more...

Optimize Actions, Not Plans

Optimization is best spent on actions, not plans. Take action and then optimize the next rep. No plan survives contact with reality, so there’s no point layering optimizations on scenarios that may not play out. Read more...

How To Get The Most Value Out Of a STEM Degree

Pre-learn your major before college. The compound effect: placing out of intro courses leads to a reputation as the advanced student, which opens research and internship doors immediately and lets you widen the gap throughout college. Read more...

How Learning Debt Starts

Learning debt usually starts with adults letting compensatory hacks slide – not calling out weak fundamentals before they compound. When this happens at scale across many students and schools, it degrades the entire educational system. Read more...

The Cycle of AI

Progress is made in AI, people lose their shit thinking Jarvis-level AGI is just around the corner, the singularity gets canceled, then nothing is AI, until more progress is made. Rinse & repeat. Read more...

Summary of My Own Math Journey

Grinding it like a video game, running a secret self-study op inside traditional classes, taking talent development seriously while making every rookie mistake, and how it unlocked a life I almost certainly never would’ve found otherwise. Read more...

What Happens When Middle School is Put to Good Use

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. Read more...

I Believe You Have Potential

What people tend to need the most yet have the least in their lives is a supportive hard-ass. Not to be confused with an unsupportive hard-ass or a supportive pushover. That’s the gap I aim to fill as best I can with my writing. Read more...

What’s the Point of Being Way Ahead? It Buys You Time.

Time is the #1 killer of dreams and aspirations. When someone gives up on their dream, or gives up on figuring out what that dream is, it’s typically a result of them losing the race against time. That is the point of compressing time, of removing skill bottlenecks early. Read more...

Protect The Habit

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. Read more...

The Metagame Podcast #39: Math Academy and The Science of Learning

The most comprehensive 2h overview of my thoughts on serious upskilling, to date. Not just how to train efficiently, but also how to find your mission. Not just the microstructure, but also the metagame. We covered tons of bases ranging from the micro level (science of learning & training efficiently) to the macro level (broader journey of finding, developing, and exploiting your personal talents).
[~0:30] 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?
[~9:00] Efficient learning feels like exercise. The point is to overcome a challenge that strains you. It is by definition unpleasant.
[~13:30] Knowledge graphs are vital when constructing efficient learning experiences. They allow you to systematically organize a learner’s performance data to identify their edge of mastery (the boundary between what they know and don’t know), what previously learned topics below the edge are in need of review, and what new topics on the edge will maximize the amount of review that’s knocked out implicitly.
[~18:00] None of this efficiency stuff matters if you don’t show up consistently. Progress equals volume times efficiency. If either of those factors are low then you don’t make much progress.
[~21:30] Getting excited about the idea of getting good provides an initial activation energy, but seeing yourself improve is what fuels you to keep playing the long game, and efficiency is vital for that.
[~26:30] Your training doesn’t have to be super efficient at the beginning. You can gradually nudge yourself into higher efficiency training even if you don’t have a whole lot of intrinsic motivation to begin with. However, there’s often a skill barrier you need to break through to really get to the fun part, and it’s advisable to do that in a timely manner so you don’t stall out. But at the same time, don’t rush it and fall off the rails.
[~34:30] A common failure mode: being unwilling to identify, accept, and start at the level you’re at.
[~41:30] Center your identity on a mission that speaks to you, that you can contribute to, and do whatever else is needed to further it, regardless of whether you perceive these other things to be “you” or not. You’ll be surprised what capabilities you develop, that you hadn’t previously perceived to be a part of your identity.
[~48:30] How to find your mission: sample wide to figure out what activities speak to you, then filter down and pick one (or a couple) that you’re willing to seriously invest your time and effort climbing up the skill tree and going on “quests”. You may not understand this early on, but skill trees branch out, and quests beget follow-up quests, and the act of climbing to these branch-points will imbue you with perspective that you can leverage to keep filtering down. If you iterate this process enough, it gradually converges into a single area that you can describe coherently and uniquely. That’s your mission.
[~55:30] Every stage in the journey to your mission is hard work, and the earlier you get to putting in that work, the better off you’re going to be. It’s never too late, but the longer you wait, the rougher it gets. At the same time, don’t make a rash decision, don’t tear the house down and build up a new house that you don’t even like. But don’t underestimate how fast you can progress when your internal motivation is aligned with your external incentives.
[~1:12:00] Focus on what matters. That’s obvious, but it’s so easy to mess up lose focus and not realize it until after you’ve wasted a bunch of time.
[~1:15:30] How to get back on the horse after you’ve fallen off. How to avoid feeling bad when something outside of your control temporarily knocks you off your horse. A good social environment can push you to get back on your horse.
[~1:26:30] If you’re a beginner, don’t feel like you have to be advanced to join a community of learners. You can do this right away. And don’t shy away from posting your progress – it’s not about where you are, it’s about where you’re going and how fast. It’s only people who are insecure who will make fun of you. Most people, especially advanced people, will be supportive.
[~1:31:30] There are numerous cognitive learning strategies that 1) can be used to massively improve learning, 2) have been reproduced so many times they might as well be laws of physics, and 3) connect all the way down to the mechanics of what’s going on in the brain. The biggest levers: active learning (as opposed to passive consumption), direct/explicit instruction (as opposed to discovery learning), the spacing effect, mixed practice (a.k.a. interleaving), retrieval practice (a.k.a. the testing effect). Read more...

How Taxing Work Becomes Fun

“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.” Read more...
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Coding

On Writing Good Code

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. Read more...

Recommended Language, Tools, Path, and Curriculum for Teaching Kids to Code

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). Read more...

Decision Trees

We can algorithmically build classifiers that use a sequence of nested “if-then” decision rules. Read more...

Euler Estimation

Arrays can be used to implement more than just matrices. We can also implement other mathematical procedures like Euler estimation. Read more...

K-Means Clustering

Guess some initial clusters in the data, and then repeatedly update the guesses to make the clusters more cohesive. Read more...

Merge Sort and Quicksort

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. Read more...

Single-Variable Gradient Descent

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. Read more...
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Algebra

Q&A: How to Introduce Sigma Notation

First have students write sums the long way until it gets annoying. Then sigma notation lands as a solution to a problem they’ve personally felt, not needless complexity imposed from above. Read more...

Compositions of Functions

Compositions of functions consist of multiple functions linked together, where the output of one function becomes the input of another function. Read more...

Completing the Square

Completing the square helps us gain a better intuition for quadratic equations and understand where the quadratic formula comes from. Read more...

Linear Systems

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. Read more...
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Q&A

Q&A #1: WM taxation, ML ETA, catching errors, coding tutorials, math vs calisthenics, foundations

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. Read more...

Recreational Mathematics: Why Focus on Projects Over Puzzles

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. Read more...

Recommended Language, Tools, Path, and Curriculum for Teaching Kids to Code

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). Read more...

Q&A: How to Scaffold Epsilon-Delta Limit Proofs

Rather than warning students that epsilon-delta proofs are hard, scaffold the curriculum so the leap isn’t so big. Start with a specific easy limit, play a game with concrete numbers, build up gradually to the general definition. Read more...

Q&A: How to Explain Improper Fractions to a Kid

The heuristic ‘a/b means you want a out of every group of b’ extends naturally to improper fractions: wanting more than available in the group just means wanting more than one whole – so the question becomes how many wholes. Read more...

Q&A: How to Introduce Sigma Notation

First have students write sums the long way until it gets annoying. Then sigma notation lands as a solution to a problem they’ve personally felt, not needless complexity imposed from above. Read more...
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Calculus

Variation of Parameters

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. Read more...

Undetermined Coefficients

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. Read more...

Separation of Variables

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. Read more...

Integration by Parts

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. Read more...

L’Hôpital’s Rule

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. Read more...

Properties of Derivatives

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? Read more...

Chain Rule

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. Read more...

Evaluating Limits

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. Read more...

Intuiting Limits

The limit of a function is the height where it looks like the scribble is going to hit a particular vertical line. Read more...
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Machine Learning

The (Free) Textbook I Wrote While Teaching The Most Advanced High School Math/CS Sequence In The USA

It scaffolded high school students up to doing masters/PhD-level coursework: reproducing academic research papers in artificial intelligence, building everything from scratch in Python. A former student worked through it right before conducting research that won 1st place ($250,000) in the Regeneron Science Talent Search, getting personally recruited by the head of NASA (with a fighter jet ride as a signing bonus), and publishing his results, solo-author, in The Astronomical Journal. Read more...

ML Courses can Vary Massively in their Coverage

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.” Read more...

Intuiting Adversarial Examples in Neural Networks via a Simple Computational Experiment

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. Read more...

Decision Trees

We can algorithmically build classifiers that use a sequence of nested “if-then” decision rules. Read more...

K-Means Clustering

Guess some initial clusters in the data, and then repeatedly update the guesses to make the clusters more cohesive. Read more...

Intuiting Ensemble Methods

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. Read more...

Intuiting Neural Networks

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. Read more...

Intuiting Linear Regression

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). Read more...

Intuiting Naive Bayes

Naive Bayes classification naively assumes that the presence of a particular feature in a class is unrelated to the presence of any other feature. Read more...
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Clips

Summary of My Own Math Journey

Grinding it like a video game, running a secret self-study op inside traditional classes, taking talent development seriously while making every rookie mistake, and how it unlocked a life I almost certainly never would’ve found otherwise. Read more...
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Education

I Love Seeing Hardcore People Moving Into Serious Edtech

We put man on the moon with computers weaker than a digital watch. Why don’t we have efficient learning at scale? We overcame Earth’s gravity half a century ago, but we can’t overcome the gravity of educational mediocrity? Bullshit. That’s why I get so excited seeing hardcore people moving into serious edtech. People who don’t take bullshit for an answer. Read more...

The Missing Middle in Test Prep

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. Read more...

The Importance of Learning Your Prerequisites

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. Read more...

Learning is Memory

This might feel obvious, but many learners don’t fully grasp the implications, and as a result, end up not actually learning much. Read more...

What Learning Actually Is – at a Concrete, Physical Level in the Brain

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. Read more...

How I Would Go About Learning an Arbitrary Subject Where No Full-Fledged Adaptive Learning System is Available

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. Read more...

Student Bite Size vs Curriculum Portion Size

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.) Read more...

Critique of Paper: “An astonishing regularity in student learning rate”

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. Read more...

Q&A: Why Do So Many People Hate Math?

People don’t like being forced to do hard things they find uninteresting. What turns mild dislike into outspoken hatred is social contagion – hatred of math has been normalized, and there’s a vocal group to join. Read more...
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Q&A (Misc)

Q&A: Why are Simpler Proofs Not Given Credit?

Solving an unsolved problem at all is generally much harder than finding a simpler solution to an already-solved one. Credit goes to the original solver. The value of a simpler proof lies in the deeper understanding it reveals. Read more...

Q&A: Why Do So Many People Hate Math?

People don’t like being forced to do hard things they find uninteresting. What turns mild dislike into outspoken hatred is social contagion – hatred of math has been normalized, and there’s a vocal group to join. Read more...

Q&A: Transitioning to One-Arm Chinup

The blocking point for OACs is almost always the biceps, not the back. Weighted chinups don’t adequately load the bicep stabilizers. One-arm dead hangs at the top and bottom positions build the specific strength needed. Read more...

Q&A: Tip for Grading Proof-Based Courses

Review a taxonomy of common proof errors before grading. When you recognize an error quickly, you can provide more targeted feedback and spend less time puzzling over what the student was trying to do. Read more...
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Blog (Tier 1)

Prerequisite Knowledge is Intellectual Capital

It’s easy to feel untalented when you’re really just missing prerequisite skills. A Real Analysis student who thought she might fail her class had simply never gotten much practice with proof-writing – a few sessions filling that gap and she came out with an A. Read more...

The Most Costly Part of Failure

The most costly part of failure is usually the time you spend on the ground before getting back up. A v-shaped recovery is almost always possible – but only if you stop dwelling on the past. Read more...

Why Can’t College Students Do Middle School Math?

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”. Read more...

Summary of My Own Math Journey

Grinding it like a video game, running a secret self-study op inside traditional classes, taking talent development seriously while making every rookie mistake, and how it unlocked a life I almost certainly never would’ve found otherwise. Read more...

What Happens When Middle School is Put to Good Use

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. Read more...

What’s the Point of Being Way Ahead? It Buys You Time.

Time is the #1 killer of dreams and aspirations. When someone gives up on their dream, or gives up on figuring out what that dream is, it’s typically a result of them losing the race against time. That is the point of compressing time, of removing skill bottlenecks early. Read more...

The Field of Education is Due For a Copernican Revolution

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. Read more...

Learning is Memory

This might feel obvious, but many learners don’t fully grasp the implications, and as a result, end up not actually learning much. Read more...

Advice on Upskilling

Advice on consistency, skills, discipline, the grind, the journey, the team, the mission, motivation, learning, and expertise. Read more...
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Motivation

The Cure for Procrastination

The cure for procrastination is often just a tiny dosage of action. Tell yourself you’ll stop after a few minutes. Usually you won’t want to, because you had built the task up in your head to be worse than it is. Read more...

Self-Discovery Is an Effortful Process

Self-discovery doesn’t feel pleasant every step of the way – that’s the point. You discover what you’re good at and love by working hard at various challenges until the signal emerges from the noise. There is no shortcut. Read more...

Optimize Actions, Not Plans

Optimization is best spent on actions, not plans. Take action and then optimize the next rep. No plan survives contact with reality, so there’s no point layering optimizations on scenarios that may not play out. Read more...

The Most Costly Part of Failure

The most costly part of failure is usually the time you spend on the ground before getting back up. A v-shaped recovery is almost always possible – but only if you stop dwelling on the past. Read more...

Protect The Habit

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. Read more...

Advice on Upskilling

Advice on consistency, skills, discipline, the grind, the journey, the team, the mission, motivation, learning, and expertise. Read more...
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Teaching

Recommended Language, Tools, Path, and Curriculum for Teaching Kids to Code

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). Read more...

Q&A: How to Scaffold Epsilon-Delta Limit Proofs

Rather than warning students that epsilon-delta proofs are hard, scaffold the curriculum so the leap isn’t so big. Start with a specific easy limit, play a game with concrete numbers, build up gradually to the general definition. Read more...
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Stories

Prerequisite Knowledge is Intellectual Capital

It’s easy to feel untalented when you’re really just missing prerequisite skills. A Real Analysis student who thought she might fail her class had simply never gotten much practice with proof-writing – a few sessions filling that gap and she came out with an A. Read more...

Summary of My Own Math Journey

Grinding it like a video game, running a secret self-study op inside traditional classes, taking talent development seriously while making every rookie mistake, and how it unlocked a life I almost certainly never would’ve found otherwise. Read more...

Q&A: But Don’t You Need 10,000 Hours To Learn Math?

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. Read more...

Business Lessons from Science Fair

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. Read more...
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Learning

Learning Is a Balancing Act

Figure skating illustrates how effective learning is a balancing act: strong foundations are required before practicing advanced skills, but advanced skills also make your foundations more robust – provided you’ve mastered the basics well enough to get a grapple on the harder moves. Read more...

Prerequisite Knowledge is Intellectual Capital

It’s easy to feel untalented when you’re really just missing prerequisite skills. A Real Analysis student who thought she might fail her class had simply never gotten much practice with proof-writing – a few sessions filling that gap and she came out with an A. Read more...

Math Trauma Isn’t Really About Math

Math trauma tends to be less about math and more about being asked to do advanced maneuvers before you’ve mastered the basics – and then being told to try harder when you inevitably fall. Read more...

Learning Rate Differences Exist

A recent study measured a 2x learning rate difference between the 25th and 75th percentile – likely an underestimate due to methodological choices. The authors reported it as 1.5x and called it an “astonishing regularity.” Read more...

Retrieval Practice is F*cking Obvious

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. Read more...

How I Would Go About Learning an Arbitrary Subject Where No Full-Fledged Adaptive Learning System is Available

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. Read more...

Student Bite Size vs Curriculum Portion Size

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.) Read more...

Critique of Paper: “An astonishing regularity in student learning rate”

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. Read more...
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Linear Algebra

Eigenvalues, Eigenvectors, and Diagonalization

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. Read more...

N-Dimensional Volume Formula

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. Read more...

Span, Subspaces, and Reduction

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. Read more...

Lines and Planes

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. Read more...
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Career Advice

Get On the Right Team

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. Read more...

The Importance of Hardcore Skills

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. Read more...
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Math Academy Podcast

Failure Modes in Teaching - Math Academy Podcast #8, Part 2

What we covered:
– In elementary school, there’s often an intense focus on conceptual understanding, but not enough time spent building real fluency with core skills. And this has left many kids without automaticity on basic things like multiplication facts. Math is extremely hierarchical, and when students don’t have the basic facts at their fingertips, they quickly run into bottlenecks as the material gets more complex.
– Sure, drills can be made more fun, but the bottom line is that they have to get done. In high school and college, most of the class time is spent copying notes from the board – and these notes are often copied by the instructor from a textbook or from other source material. This game of telephone through transcribing is just a performative activity. It’s theater. It’s passive and it does next to nothing for learning and retention.
– In upper level college math courses especially, students may only receive short weekly problem sets, which really aren’t enough to build mastery, even if the problems are really hard, because students just spend most of their time flailing around.
– The bottom line is that students need reps: lots of them, building up scaffolding to the highest, hardest levels that they’re expected to reach. High school assignments tend to be better in that regard, but students frequently don’t receive timely feedback, and often their work isn’t even graded for accuracy. That feedback loop is so critical: without it, students won’t know what they’re doing wrong or how to improve.
– So rather than just pattern matching to how math has traditionally been taught, what actually makes training effective? There’s a few core principles:
1) Maximize the amount of time spent actively learning, interleaving minimum effective doses of explicit guided instruction active practice.
2) Make sure students are consistently working at the edge of their abilities: not bored, but not overwhelmed.
3) Provide frequent, timely feedback so students can adjust and improve.
These principles should be applied to math education and training environments everywhere.

0:00 - Introduction
3:13 - Professors often wing pedagogy
5:37 - Too much class time is spent transcribing notes
7:41 - College problem sets are too short
12:53 - A lot of homework isn’t even graded for accuracy
18:22 - Copying notes in class is performative productivity
22:29 - Alex taught math courses at University College London
25:35 - Teaching is often an annoying obligation for research professors
30:03 - The bar for teaching is on the floor
32:34 - Even football practices often waste players’ time
34:20 - Most training is inefficient because people pattern match to the status quo
34:57 - First principles for effective training
37:13 - Too many models can paralyze and become a crutch for kids
39:24 - Kids can get stuck using training wheels in math forever
42:05 - Non-standard methods are often distracting and inefficient
46:18 - Designing 6th-8th grade courses to align with school curricula
52:30 - Conceptual understanding without ability is useless
55:17 - Skills practice can and should be gamified Read more...

Role of Teachers in the Math Academy Classroom - Math Academy Podcast #8, Part 1

What we covered:
– A lot of schools have recently begun using Math Academy in their classrooms. And one of the biggest benefits of using Math Academy is that it automates all the mechanical parts of teaching, like writing questions, keeping track of what students know and what they don’t know, monitoring student progress, assigning extra practice when needed, grading, all that grindy stuff.
– None of these tasks is enjoyable. They suck. Just ask any teacher. I mean, we grinded through all that back when we were teaching ourselves, and it takes so much effort just to get even a halfway decent approximation of doing it right. And there’s just a limit to how well that you can do it if you’re doing it manually. It’s the whole reason why we built the system.
– And what that system does, what Math Academy we does is it frees up teacher bandwidth to focus on the human elements of teaching: building relationships, connecting what students are working on to their own unique interests. Those kind of things that enhance the learning experience, but that really can’t replace skills practice.
– I mean, in-class projects can be great, but only if students have the prerequisite knowledge to be successful with them. If they don’t, then projects are frustrating, and the students who understand the material will end up doing all the work and carrying everybody else, who will learn next to nothing. It’s inefficient and frustrating all around unless students have their skills in place.
– Ultimately, if students don’t master the math in each class, they’ll be unprepared for the next one. And in a subject as hierarchical as math, these gaps compound quickly. True empowerment isn’t simply telling students they have potential. It’s making sure they actually have the real skills to move forward and realize that potential.

0:00 - Introduction
2:56 - What is the teacher’s role alongside Math Academy?
5:37 - Math Academy frees up teachers to do the human parts of teaching
7:03 - Projects are great if students have the prerequisite skills
7:42 - Drills without context are boring
8:43 - Games without skills are inefficient
11:14 - Build fun activities on top of a solid foundation of skills
12:15 - Teachers can tailor the class to the students’ preferences
13:28 - Implementing mastery learning is too much work for a single teacher
15:27 - Doing projects without prerequisites is frustrating
16:57 - True empowerment is giving kids the skills they need to succeed
19:30 - Missing skills compound in hierarchical skill trees
24:06 - Lack of automaticity in lower level skills slows down higher level tasks
27:14 - The MA team builds and improves courses through experience
29:21 - The MA team targets tasks with low pass rates for additional scaffolding
31:03 - Alex built knowledge graph intuition through years of experience
37:40 - Social media enforces hyper-accountability
39:19 - Differential equations courses are often a hodgepodge of disjointed techniques
43:20 - Math Academy university courses are a superset of elite university content
45:18 - Differential equations is a highly branching subject
49:21 - The breadth of Differential Equations makes it often poorly taught Read more...

Earning the Right to Scale - Math Academy Podcast #7, Part 2

What we covered:
– As Math Academy has grown over the past year, we’re getting a better sense of general do’s and don’ts when scaling a startup. We’ve learned hard lessons about overloading the database, the task processor, and our team, requiring numerous infrastructure and process updates.
– Schools have been using the system and we’ve built plenty of additional features to, among other things, accommodate unique billing schemes and make it easy for teachers to manage classes on the system.
– We’ve intentionally grown organically and were self-funded, which forced us to do things manually at the beginning. Years ago, we taught math classes in person and Jason onboarded our first online users on hundreds of hour-long individuals and calls. These were crucial experiences to learn who our customers are, what they want from the product, and common failure modes.
– In our experience, doing things manually at the beginning ensures that you 1) build a product that customers actually get value from, and 2) you don’t clutter your product with unnecessary bells and whistles that don’t add value. In other words, you have to do the manual work to earn the right to scale.

0:00 - Introduction
2:18 - Building infrastructure to handle increasing load
3:41 - Bringing on AWS expertise to robustify the backend
4:22 - An overloaded database enters a new realm of physics
5:50 - Prioritizing execution over perfection in start-ups
6:33 - Paying the bill for accumulated infrastructure debt
7:53 - Improving job prioritization of the task processor
9:52 - Benefits of scaling organically
11:42 - Wisdom is the result of failures
12:18 - There is no substitute for experience
13:17 - Focusing on solving problems, not advertising
14:48 - Upgrading with surgical precision
15:35 - The pain-point compass
17:04 - Managing finite time and resources
18:27 - Development of the gravity feature
20:42 - Gravity is a suggestion, not a hard override
22:25 - Limiting gravity to avoid cognitive overload
28:29 - Balancing customization and customer confusion
31:28 - The feature sandbox
33:58 - Increasing volume of customer support emails
35:22 - Additional infrastructure requirements for schools
36:18 - Learning about the customer through direct interaction
38:14 - Step 1: Manually added schools using spreadsheets
40:22 - Step 2: Developed tools to handle specialized school requests
41:23 - Step 3: Goal is 100% self-service sign-ups for schools
42:32 - Solve the problem manually first, then automate it
43:44 - Why focus on schools?
46:15 - Math Academy goes to college
49:37 - You can’t anticipate every edge case
52:14 - Letting user behavior build the product roadmap
58:54 - Becoming successful means working harder
1:00:24 - The customer support hurdle
1:03:27 - How Justin’s expanding roles drove growth (both personal & company)
1:09:03 - Teaching as market research for Math Academy
1:10:52 - The value of having been inside the trade Read more...

2025 in Review: Content Production - Math Academy Podcast #7, Part 1

This is a Math Academy “Wrapped” for 2025, focusing on the content side of things. In summary, here’s the good:
– We released a Discrete Mathematics course.
– We added hundreds of “missing middle” topics to our SAT Math Fundamentals course to bridge the chasm between what’s in standard school curricula versus what’s tested on the SAT.
– We soft-launched a SAT Math Prep course that students automatically promote into after finishing the fundamentals course, where they see their estimated SAT score instead of a progress percentage, and they do even more SAT-specific training such as taking frequent mock SAT practice exams and doing rapid-fire problem practice to build up speed and comfort with all the slight variations in the ways that questions could be phrased on the test.
– We added tens of thousands of free response questions throughout our middle school and high school courses.
– We developed all the content including coding projects for our first machine learning course, to be released once the coding interface is ready. (If you’re waiting on that course and absolutely must start your ML journey right this moment, note that there’s a freely available 400+ page textbook that I wrote while teaching this stuff manually in the school program – it’s called “Introduction to Algorithms and Machine Learning.”)

Of course, we’re under no illusion that we need to ramp up our rate of course production and transition from a workshop to a factory. We started pursuing that goal last year, and while there has been much pain from hitting our heads on basically every ledge possible, we’ve learned a lot and have just recently, in the past couple weeks, hit an inflection point where the factory transition is finally coming together. As Alex summarized in a recent post on X: we’re working tirelessly to upgrade our course development pipeline, building new tools and processes to help us manage a higher volume of courses so we can increase output while maintaining the quality you’ve all come to expect. In particular, we’re using our nearly-finished Differential Equations course as a guinea pig to test-drive some of our new tools and processes. This is the year that Math Academy comes out of the basement and onto the factory floor.

0:00 - Introduction
3:57 - Added 115 “Missing Middle” topics to SAT Prep
6:06 - Integrating the SAT Missing Middle topics into other courses
9:42 - Added tens of thousands of free response questions
10:34 - Free response questions are useful because they don’t prime you
13:33 - When to use free response vs. multiple choice questions
14:54 - Too many free response questions taxes learners
16:39 - Limiting the length of free response answers
18:08 - Building infrastructure for free response questions was a beast
20:42 - SAT test prep course
22:22 - Machine Learning has been the hardest course to develop so far.
23:12 - People who know machine learning, math, and how to teach them are rare
25:06 - The Eurisko book was the best resource for developing the Machine Learning course
28:51 - Balancing repetition and computational load in Machine Learning problems
29:43 - Designing minimum viable problems for Machine Learning
33:53 - Building the infrastructure for dynamic select questions was a nightmare
36:12 - Dynamic select questions are good for proofs and university-level math
38:03 - The Differential Equations course is almost finished
40:23 - Iterating on course development to make better courses
42:00 - 2026 is the year of scaling up course production
43:03 - How to scale up the team without sacrificing course quality
44:39 - Learning the hard way about hiring too quickly
46:20 - Challenges of managing a fully remote, geographically dispersed team
48:54 - Building tools to measure company output
50:06 - Optimizing content writer performance is like optimizing student learning
52:31 - Incentivizing content creation to improve output
56:36 - Courses planned for the longer term
58:01 - You need to learn concrete computations before abstract proofs
59:32 - Why we separate university-level courses into computational vs proof-based
1:01:07 - The best textbooks for beginners are NOT the most complex
1:02:37 - Teaching proofs and computations at the same time overloads most students
1:04:16 - Intuition through repetition
1:04:49 - Wisdom is the abstract compression of lived experiences
1:07:39 - Mastering details before abstracting Read more...

Learning Debt and Skill Insolvency - Math Academy Podcast #6, Part 3

What we covered:
– The dangers of accumulating learning debt: the gap between what you can do and what you need to be able to do.
– If you miss building up your foundational skills in school or sports, you can get by for a while. You develop some compensatory strategies, like favoring your forehand over your backhand, or using ChatGPT to write all your school essays.
– But learning debt is like any other kind of debt: it accrues interest and eventually comes due. Over time, the workarounds become more complex. The cognitive load increases. You start avoiding situations that expose the gap, and this is where you hit your ceiling. You can’t pursue an engineering degree if you can’t do algebra. You can’t be competitive in tennis if you can’t hit with your backhand.
– Learning debt often begins because of a lack of oversight by adults. Parents, teachers, and even coaches sometimes think they’re being nice not telling you that you need to work on your weaker side, or you need to stop using a calculator on your math problems. It feels like nagging, and it can create conflict between adults and learners. So they let it slide.
– But this failure to hold the line early on inhibits students’ future potential. And when it occurs across many students across many schools, it degrades the whole educational system – leading to the current situation in which many students are totally unprepared for the rigors of college.

0:00 - Introduction
2:04 - Course phases: instruction, final review, final exam, remediation if needed
5:25 - Generating full-length SAT exams for our prep course
6:53 - Loosening up the gravity throttle for high-performing students
14:59 - Aptitude is measured by accuracy rate
18:07 - Accuracy correlates first with aptitude, second with conscientiousness
21:35 - Assessment vs. non-assessment accuracies
23:43 - Propagating accuracy through the knowledge graph
24:27 - Hidden skill gaps force bad compensations
25:27 - Sports make skill deficits and bad compensations obvious
33:38 - The Math Academy system holds you accountable for every skill
34:18 - Completing the square: a common skill deficit with temporary workarounds
36:15 - Reliance on Desmos undermines students’ ability to graph functions
37:38 - You need to know your multiplication facts for factoring
38:13 - Foundational deficits are usually caused by lack of adult oversight
38:52 - Shoring up foundations is effortful but has huge ROI
40:40 - Filling in missing foundations makes kids so much more confident
41:12 - Missing foundations stall learning and drive cheating
42:12 - Faking competence backfires downstream
45:33 - The truth hurts but is the kindest thing in the long run
46:26 - Learning debt eventually comes due, with students paying the biggest price
47:12 - Kicking the can down the road in education
49:46 - The cost of a broken education system Read more...

On The Rails and Out Of Scope - Math Academy Podcast #6, Part 2

What we covered:
– The benefits of short problems. Math Academy problems typically take only a minute or two. This way, students can stay on the rails with lots of reps, successfully building up complexity instead of getting crushed by it from the start.
– What goes wrong in college math classes: they tend not to scaffold content very well, forcing students to build their own bridges across knowledge & skill gaps. Weekly problem sets often consist of a handful of hour-long problems that instructors hope students will “self-scaffold” up to. In reality, what happens more often is that students fall off the rails.
– Founders of growing start-ups cannot be hands-off. “Things falling off the rails” is the most realistic and most dangerous failure mode, not micromanaging. Founders of small, scaling companies need to be in “founder mode,” not the “manager mode” that CEOs of huge, well-established companies are in.
– Within teams, it’s important to let conversations flow out of scope. Every innovation, every solved problem, requires relevant background context, and you often don’t know what the full context is beforehand. It’s easy to let conversations flow out of scope when you like who you’re working with and what you’re working on.

0:00 - Introduction
1:32 - Why Math Academy problems are short by design
9:48 - Long problems dilute reps on the skill that actually matters
11:00 - Isolate the new skill first, then recombine into full problems
14:10 - Typical undergrad math classes: too few problems, too complex from the start
18:07 - The proof skills gap: often assumed and not taught
29:32 - Alignment decay: teams naturally drift out of sync unless continually aligned
35:04 - Small misalignments compound fast
38:28 - Founder mode: stay in the weeds to stay in sync
49:07 - Early, frequent parent communication avoids end-of-term blowups
50:48 - High-trust collaboration requires relentless communication
57:42 - Out-of-scope conversation enables context sharing
59:14 - Over-scoping kills context sharing
1:00:51 - Enjoyment & trust fuel context sharing
1:06:13 - Missing context produces confidently wrong outcomes
1:10:01 - LLMs fail when context is missing
1:11:38 - Humans fail when context is missing
1:14:19 - Online discourse fails when context is missing Read more...

Why Can’t College Students Do Middle School Math? - Math Academy Podcast #6, Part 1

What we covered:
– A recent report from the University of California San Diego revealed that 1 in 12 incoming freshmen were not proficient in middle school math – basically, anything above arithmetic with fractions. Their existing remedial math course was too advanced for these students, so they had to design even lower remedial remedial math courses. Even crazier, over a quarter of these students had a perfect 4.0 GPA in their high school math courses.
– It’s not just UCSD. This is everywhere. A similar thing happened at Harvard, too, having to add remedial support to their entry-level calculus courses. It’s like that movie Olympus Has Fallen, except this time it’s Harvard. It’s a catastrophe.
– How did things get this bad? Teachers and administrators face relentless pressure to inflate grades, and during the pandemic many universities went test-optional, removing the only signal that reliably correlated with actual math readiness. That decision simultaneously elevated high school grades to the sole gatekeeping metric, intensifying incentives to inflate them.
– This has all coincided with the advent of LLMs, which make it increasingly easy for students to cheat. The result was predictable: grades became untethered from real competence, and multiple cohorts of students entered college without ever having to demonstrate foundational math skills.
– Teachers have to play both good cop and bad cop, and there is no avoiding the latter. If you refuse to play bad cop at all, you eventually end up playing it constantly. The best teachers are strict from the start and ease up later, once students understand that hard, honest work is non-negotiable.

Timestamps:
0:00 - Introduction
2:11 - Freshmen math collapse: 1 in 12 UCSD freshmen don’t know middle school math
6:45 - Remedial remedial math: UCSD created remediation for remedial math
8:40 - Inflated grades: 25% of remedial-remedial students had perfect GPA in HS math
10:06 - Test-optional admissions removed the last objective metric
12:13 - Pandemic inflation: GPAs skyrocketed
14:37 - Removing tests pressures teachers to inflate grades
16:52 - Grade-grubbing: endless negotiating, complaining, accusations
19:01 - Then vs. now: parents, tests, accountability
27:38 - Crisis opportunism: “Never let an emergency go to waste”
29:33 - No tests = no knowledge requirements
33:28 - Elite collapse: Harvard has the same problem
36:31 - No enforcement means no standards
37:40 - LLM cheating is trivially easy
38:25 - Catching a cheater and turning him around
48:46 - Cheating is like taking mob money. Now you’re in, you’re never out.
50:41 - Assessments must be done in person
55:06 - LLM cheating is often obvious yet hard to prove
57:17 - How to prevent cheating on long papers
58:28 - Start hardcore, then lighten up gradually
1:01:37 - Good teachers play bad cop when needed Read more...

Getting Kids To Do Hard Things - Math Academy Podcast #5, Part 2

What we covered:
– Most kids are not intrinsically motivated to do the hard things: practice their soccer drills, do their math homework, eat their broccoli. Getting them to do the hard things often requires gamification and/or incentives.
– A little gamification goes a long way. Jason gamified drills for his kids’ soccer team to get the most out of each practice (e.g., “zombie attack”), and it was unreasonably effective. The XP leaderboards on Math Academy are also unreasonably effective.
– A good incentive can change kids’ behavior overnight. The incentive doesn’t need to be big; it just needs to be something the kid really cares about. Find the thing the kid would rather be doing, and use it to motivate them to do what they’re supposed to be doing. They won’t need the incentive forever; as the kid gets used to the feeling of a new behavior, it gradually turns into a habit that they can maintain on their own.
– Even when you’re doing what you love, there will be grindy phases. But kids typically don’t understand this. They might get interested in a talent domain and want to become good enough to build a life around it, while simultaneously resisting doing the hard work to make that happen (i.e., stage 2 in Bloom’s talent development process). It’s often up to parents, who can see the long game, to push their kids through the difficult parts in paths that they find rewarding.
– For instance, the most mathematically gifted student I ever worked with, who was drawn into math by his own intrinsic interest, still needed to be pushed to learn calculus. Now he’s having the time of his life working on physics-y, calculus-heavy research-level math problems in high school. Even after finding something he loves and is good at, he still needed to be pushed to do the hard work to unlock more of it.

Timestamps:
00:00:00 - Most kids are not intrinsically motivated to do hard things – homework, drills, practice. They usually need incentives to get through.
00:08:16 - A little gamification goes a long way. Jason gamified drills for his kids’ soccer team to get the most out of each practice (e.g., “zombie attack”).
00:14:05 - A good incentive can change behavior overnight. It doesn’t need to be big, just something the kid really cares about, and they won’t need it forever. It’s about building a habit until they can maintain it on their own.
00:54:16 - The most mathematically gifted student Justin ever worked with needed to be pushed to learn calculus, and now he’s having the time of his life working on calculus-heavy research-level math problems.
01:11:54 - Even when you’re doing what you love, there will be grindy phases. It’s important for parents to help kids push through those grindy phases so that they can unlock more of what they love. Read more...

Building Without Bloat - Math Academy Podcast #5, Part 1

What we covered:
– Any successful endeavor requires a great team: capable people, who like and trust each other, and have complementary skillsets and ways of thinking. Some modes of thinking cannot be performed at the same time within a single brain.
– Accountability requires control. You can’t hold someone responsible for outcomes unless you also give them control over the system that produces those outcomes (though you can set reasonable operational boundaries).
– Solve today’s problems today. Smart people can invent endless hypotheticals and build giant solutions to fake problems. Not only does this waste time, but it also burdens the system with complexity that becomes a future straitjacket. Everything you build must be carried forward, so focus on what’s present in front of you, not on imagined futures five steps away.
– In a scaling system, the sheer volume of interactions will expose a long tail of bizarre scenarios, almost like rare diseases you’d never anticipate. Users will often try to repurpose software beyond its design, like hauling a trailer with a motorcycle.

Timestamps:
00:00 - Introduction
03:48 - The importance of finding your complements
24:07 - The origin story of Math Academy’s content team
43:36 - No meta-work; just solve the problems in front of you
54:26 - Jason time vs real time (real time is longer)
59:00 - The long tail of rare edge cases and unexpected user behavior Read more...

Knowledge Graph Engineering: Mental Models & War Stories - Math Academy Podcast #4, Part 2

What we covered:
– Building a knowledge graph is like city planning & road construction. Too many prerequisites leading into a single topic creates a cognitive traffic jam.
– Elegantly rewiring a live knowledge graph: the evolution of our tooling and automatic validations. How to avoid staging servers & migrations and NOT have it blow up in your face.
– UI work takes time and adds complexity, so we spend it on the customer. Internal tools are almost entirely command-line; clickable buttons are for customers.
– Justin’s transition from research coding to real-time systems. He started with mathy, notebook-driven quant code and had to learn production engineering the hard way. Once he did, it was a massive level-up.
– Alex’s plan for dealing with “content papercuts” - small issues that pile up. Inspired by Amazon’s “papercuts team.”
– Our upcoming differential equations course, the last course in the core undergrad engineering math sequence.

Timestamps:
00:00:00 - Building a production-grade knowledge graph is like city planning and road construction
00:07:26 - Elegantly rewiring a live knowledge graph: the evolution of our tooling and automatic validations
00:24:47 - Justin’s transition from research coding to real-time systems
00:44:51 - Alex’s plan for dealing with “content papercuts” - small issues that pile up
00:58:02 - Our upcoming differential equations course Read more...

The Unreasonable Effectiveness of the Knowledge Graph - Math Academy Podcast #4, Part 1

What we covered:
– Why “problem solving” is often just a vague label people use when they haven’t explicitly enumerated the underlying skills, and how those skills can in fact be exhaustively mapped in a knowledge graph.
– How to approach research problems: Alex’s PhD journey, top-down familiarity vs bottom-up mastery.
– If you have natural talent, use it, but not as a crutch, otherwise you’ll stunt your long-term development. Don’t turn your blessing into a curse.
– The story behind building our SAT prep curriculum: realizing that the standard school curriculum leaves a massive “missing middle” unaddressed; identifying 115+ missing topics to bridge the gap between textbook math and the hardest SAT questions.
– Watching the manifold hypothesis play out in test prep: the SAT may appear to allow an astronomical space of possible problem types, but in reality the actual problems live on a compact, highly structured manifold that can be fully enumerated and scaffolded in a knowledge graph

Timestamps:
00:00:00 - Intro: “problem solving” is what you call it when you don’t really know what it is (i.e. you haven’t explicitly enumerated the skills)
00:04:11 - How to approach research problems: Alex’s PhD journey, top-down familiarity vs bottom-up mastery
00:20:28 - If you have natural talent, don’t use it as a crutch. Don’t turn your blessing into a curse.
00:29:06 - SAT prep, iteration 1: Realizing that the standard school curriculum leaves a massive “missing middle” unaddressed
00:33:45 - SAT prep, iteration 2: Covering the “missing middle” problems
00:53:38 - SAT prep, iteration 3: Building the “missing middle” knowledge graph
01:08:11 - Watching the manifold hypothesis play out in SAT prep
01:16:42 - The unreasonable effectiveness of the knowledge graph Read more...

Waging War on Mediocrity: Tales From the Trenches - Math Academy Podcast #3

What we covered:
– How bureaucracies instinctively reject new ideas like an immune system attacking a foreign organ, and what it takes to keep your project from being “spit out.” Concrete example: how Jason & Sandy muscled past institutional resistance to get 8th graders passing AP Calc BC.
– Every system inevitably decays into mediocrity unless someone fights to keep the standards high. The way you keep people, systems, and projects moving is by “horsing” them forward. Concrete example: how Justin kept 8th graders passing AP Calc BC, and what it looks like when a school succumbs to the gravity of mediocrity.
– Justin’s math self-study journey in high school: grinding math like a video game, running a secret self-study op inside traditional classes, taking talent development seriously while simultaneously hitting his head on every ledge and making every rookie mistake. Ups and downs, lessons learned, with tons of concrete examples.

Timestamps:
00:00:00 - Intro: Willing Things Into Existence
00:11:43 - How Jason & Sandy Willed Math Academy Into Existence
00:36:45 - Fighting The Gravity of Mediocrity
01:02:29 - Case Studies in Educational Dysfunction
01:21:53 - The Birth of Justin’s Self-Study Madness
01:50:48 - Self-Studying on the Sly During School
02:02:41 - The Highs & Lows of High School Research
02:22:38 - Outro: Paving the Path with Math Academy Read more...

Teach Like Your Life Depends On It - Math Academy Podcast #2

0:00 - What Would a Tutor Do, If Their Life Depended On It? (Part 1)
5:47 - Find Your North Star: Why Justin Quit His Data Science Job to do Math Tutoring Full Time
11:23 - Getting “Inside the Trade”
19:31 - What Would a Tutor Do, If Their Life Depended On It? (Part 2)
27:28 - Efficient Learning Techniques are Obvious if You Think About Athletics
33:45 - Enjoyment is a Second-Order Optimization
39:50 - We Need to Stay Hardcore, But Become Less Harsh
51:14 - Math Academy is Like “Yuri’s Gym”
59:06 - Vision for the Future of Math Academy
1:14:23 - Goal Setting/Advising and Communicating Progress
1:24:58 - If All You Show Up With is AP Calculus, You’re Probably Outgunned
1:51:08 - The Meta-Skills that Kids Need to Work Effectively on Math Academy
2:08:54 - How to Help Students Maintain Successful Learning Habits While Working Independently
2:32:29 - Overhelping: A Common Failure Mode of Well-Intentioned Parents/Tutors Read more...

The Long Game: Building Minds and Machines - Math Academy Podcast #1

0:00 - Introduction
4:00 - Applying the MA Way to X Growth
7:40 - Status of the ML Course and its Kick-Ass Coding Projects (Part 1)
25:50 - Jason’s Near-Infinite List of Important Things
34:20 - The ML Course Has Been a Massive Undertaking
42:10 - Breadth-First Development
44:30 - Status of the ML Course and its Kick-Ass Coding Projects (Part 2)
50:15 - Why Math Academy Needs To Do a CS Course
56:45 - The Never-Ending Stream of Confusion
1:00:30 - The Story of Eurisko, the Most Advanced Math/CS Track in the USA
1:24:20 - Intuition Through Repetition: Machine Learning Edition
1:29:40 - The Importance of Spaced Review
1:43:30 - Upcoming Course Roadmap
1:47:40 - Spaced Repetition 2.0: Accounting For and Discouraging Reference Reliance
1:54:45 - Overhelping: A Pathology of the Over-Involved Parent/Tutor
1:59:21 - Yes, You Need to be Automatic on Math Facts (and Yes, Rapid-Fire Training is Coming)
2:04:55 - What Happens When Students Don’t Know Their Math Facts
2:05:50 - The Horror of Attempting to Teach a Class When Students Have Multi-Year Deficits in Fundamental Skills
2:11:30 - Integrating Coding Into the Math Curriculum
2:18:00 - Combining Math and Coding is the Closest Thing to a Real-Life Superpower
2:18:55 - Creating a Full Math Degree and Getting Full College Credit
2:22:15 - The Power of Pre-Learning: The Greatest Educational Life Hack Read more...
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Eurisko

The (Free) Textbook I Wrote While Teaching The Most Advanced High School Math/CS Sequence In The USA

It scaffolded high school students up to doing masters/PhD-level coursework: reproducing academic research papers in artificial intelligence, building everything from scratch in Python. A former student worked through it right before conducting research that won 1st place ($250,000) in the Regeneron Science Talent Search, getting personally recruited by the head of NASA (with a fighter jet ride as a signing bonus), and publishing his results, solo-author, in The Astronomical Journal. Read more...

What Happens When Middle School is Put to Good Use

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. Read more...

Math Academy’s Eurisko Sequence, 5 Years Later: Student Outcomes Emerging From the Most Advanced High School Math/CS Track in the USA

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. Read more...

Your Mathematical Potential Has a Limit, but it’s Likely Higher Than You Think

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. Read more...
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Graphs

Decision Trees

We can algorithmically build classifiers that use a sequence of nested “if-then” decision rules. Read more...
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Applications

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Graphing Calculator

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Drawing

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Podcast

Deep Learning With Yacine Podcast #1: The Science of Learning Math (and Anything Else)

[2:10] My background: growing up in a non-technical family and finding math on my own.
[5:45] Self-studying 3,000 hours of college math in high school: starting with calculus the summer after 10th grade and continuing through undergraduate-level math for the rest of high school.
[16:10] Whether the same ground could have been covered more efficiently – and how being responsible for other people’s learning eventually crystallized the underlying principles.
[29:55] How having math foundations in place paid off in research: getting into Fermilab and CERN research projects at university labs.
[43:10] What the Math Academy learning system looks like: adaptive diagnostic, custom knowledge graph, minimum effective doses of instruction followed immediately by problem-solving, mastery before advancing.
[47:34] How we built the knowledge graph: years of manual work by domain experts, refined with analytics for nearly a decade.
[1:10:46] How the FIRE (Fractional Implicit REpetition) algorithm works: solving a harder problem implicitly reviews the sub-skills it encompasses, compressing the review pile significantly.
[1:35:50] Math and sport. Cognitive science principles – mastery before advancing, spaced practice, interleaving – are often easier to see in sport than in math.
[1:42:00] Does doing math well require different skills than teaching it well?
[1:56:25] Automaticity as a prerequisite for deeper understanding.
[2:05:35] The anatomy of “aha” moments.
[2:14:11] Learning math as an adult: the amount of work doesn’t change, only your free time does. Math Academy’s Mathematical Foundations sequence covers the prerequisite stack for university math in roughly 15,000 minutes.
[2:24:10] Balancing fundamentals and exploration: exploration pays off most at the frontier of a subject.
[2:33:55] Is it ever too late?
[2:46:00] Bottom-up versus top-down learning.
[2:56:30] Students with ADHD often feel the effects of inefficient pedagogy more strongly. Interleaving minimum effective doses of guided instruction and active problem-solving is better for everyone.
[3:06:20] AI tools as a multiplier on existing ability: the more you know, the more useful they are; the less you know, the harder it is to detect when they’ve gone wrong.
[3:14:37] What I’m most focused on right now: taking Math Academy from workshop to factory – producing courses at scale without sacrificing quality. Read more...

OpenEd Podcast: The Real Reason Your Kid Hates Math (It’s Not Them)

Most kids hate math because it’s taught inefficiently, through one-size-fits-all lectures, where they’re constantly asked to learn new things despite not having mastered the prerequisites. Students can learn far faster and with less stress when instruction is individualized, mastery-based, interleaved, and reinforced through spaced repetition. Math becomes frustrating when students are pushed ahead with gaps in foundational skills or cognitively overloaded; it becomes motivating when they experience small, frequent wins at the edge of their “knowledge frontier.” Math Academy operationalizes this through an adaptive diagnostic that detects missing prerequisites and constructs a custom course to cover them, short alternating bursts of instruction and practice to ensure that students master material before moving on, and cumulative spaced review & quizzes to prevent forgetting, creating an individualized “glove fit” to each individual student. The broader vision is a shift away from lockstep classrooms toward individualized, coach-like learning. Read more...

Scraping Bits Podcast #137 (Round 4): Learning Math is Hard, Proof Writing, Which Order to Learn Math

[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. Read more...

CS Primer Show #23: MathAcademy and the efficient pursuit of mastery

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. Read more...

Golden Nuggets Podcast #40 (Round 4): How Justin learns, new ML course, the magic of Twitter

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. Read more...

Scraping Bits Podcast #116 (Round 3): Essential Math for Machine Learning, Math Intuition/Creativity, Proof Vs Computation

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. Read more...

Golden Nuggets Podcast #39 (Round 3): MA’s upcoming machine learning course

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. Read more...

Golden Nuggets Podcast #37 (Round 2): Balancing learning with creative output

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. Read more...

Scraping Bits Podcast #107: Proof Writing, Discovering Math, Expert Systems, Learning Math Like a Language

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. Read more...

Golden Nuggets Podcast #35: Optimizing learning efficiency at Math Academy

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. Read more...

Scraping Bits Podcast #102: Learning Mathematics Like an Athlete

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. Read more...

Road to Reading Podcast #23: Discussing Cognitive Science

[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. Read more...
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Algorithms

Intuiting Ensemble Methods

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. Read more...

Intuiting Neural Networks

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. Read more...

Intuiting Linear Regression

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). Read more...

Intuiting Naive Bayes

Naive Bayes classification naively assumes that the presence of a particular feature in a class is unrelated to the presence of any other feature. Read more...
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Research

Student Bite Size vs Curriculum Portion Size

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.) Read more...

Critique of Paper: “An astonishing regularity in student learning rate”

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. Read more...

Optimized, Individualized Spaced Repetition in Hierarchical Knowledge Structures

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). Read more...
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Limits and Derivatives

L’Hôpital’s Rule

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. Read more...

Properties of Derivatives

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? Read more...

Chain Rule

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. Read more...

Evaluating Limits

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. Read more...
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Objects

Euler Estimation

Arrays can be used to implement more than just matrices. We can also implement other mathematical procedures like Euler estimation. Read more...

K-Means Clustering

Guess some initial clusters in the data, and then repeatedly update the guesses to make the clusters more cohesive. Read more...
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Shorts

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Regression

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Blog (Pinned)

The (Free) Textbook I Wrote While Teaching The Most Advanced High School Math/CS Sequence In The USA

It scaffolded high school students up to doing masters/PhD-level coursework: reproducing academic research papers in artificial intelligence, building everything from scratch in Python. A former student worked through it right before conducting research that won 1st place ($250,000) in the Regeneron Science Talent Search, getting personally recruited by the head of NASA (with a fighter jet ride as a signing bonus), and publishing his results, solo-author, in The Astronomical Journal. Read more...

Math Academy’s Eurisko Sequence, 5 Years Later: Student Outcomes Emerging From the Most Advanced High School Math/CS Track in the USA

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. Read more...

Why is the EdTech Industry So Damn Soft?

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. Read more...

Optimized, Individualized Spaced Repetition in Hierarchical Knowledge Structures

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). Read more...
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Topological Data Analysis

Mapper Use-Cases at Ayasdi

Ayasdi developed commercial Mapper software and sells a subscription service to clients who wish to create topological network visualizations of their data. Read more...
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History

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Differential Equations

Variation of Parameters

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. Read more...

Undetermined Coefficients

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. Read more...

Separation of Variables

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. Read more...
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Artificial Intelligence

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Neural Networks

Intuiting Adversarial Examples in Neural Networks via a Simple Computational Experiment

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. Read more...
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Physics

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Integrals

Integration by Parts

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. Read more...
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Quadratic Equations

Completing the Square

Completing the square helps us gain a better intuition for quadratic equations and understand where the quadratic formula comes from. Read more...
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Series

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Games

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Notation

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Arithmetic

Q&A: How to Explain Improper Fractions to a Kid

The heuristic ‘a/b means you want a out of every group of b’ extends naturally to improper fractions: wanting more than available in the group just means wanting more than one whole – so the question becomes how many wholes. Read more...
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Writing

I Believe You Have Potential

What people tend to need the most yet have the least in their lives is a supportive hard-ass. Not to be confused with an unsupportive hard-ass or a supportive pushover. That’s the gap I aim to fill as best I can with my writing. Read more...
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Mapper

Mapper Use-Cases at Ayasdi

Ayasdi developed commercial Mapper software and sells a subscription service to clients who wish to create topological network visualizations of their data. Read more...
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Limits

Q&A: How to Scaffold Epsilon-Delta Limit Proofs

Rather than warning students that epsilon-delta proofs are hard, scaffold the curriculum so the leap isn’t so big. Start with a specific easy limit, play a game with concrete numbers, build up gradually to the general definition. Read more...

Intuiting Limits

The limit of a function is the height where it looks like the scribble is going to hit a particular vertical line. Read more...
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Linear Equations and Systems

Linear Systems

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. Read more...
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Rational Functions

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Non-Polynomial Functions

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Transformations of Functions

Compositions of Functions

Compositions of functions consist of multiple functions linked together, where the output of one function becomes the input of another function. Read more...
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Vectors

Span, Subspaces, and Reduction

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. Read more...

Lines and Planes

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. Read more...
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Matrices

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Simulation

Euler Estimation

Arrays can be used to implement more than just matrices. We can also implement other mathematical procedures like Euler estimation. Read more...
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Searching

Single-Variable Gradient Descent

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. Read more...
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Quant

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Active Learning

Deliberate Practice: The Most Effective Form of Active Learning

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. Read more...
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Productivity

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Podcast (Solo)

Q&A #1: WM taxation, ML ETA, catching errors, coding tutorials, math vs calisthenics, foundations

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. Read more...
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Neuroscience

Cognitive Science of Learning: How the Brain Works

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. Read more...

The Brain in One Sentence

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. Read more...
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Inequalities

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Polynomials

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Python

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Volume

N-Dimensional Volume Formula

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. Read more...
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Eigenspace

Eigenvalues, Eigenvectors, and Diagonalization

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. Read more...
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Hello World

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Computer Science

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Neuroevolution

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Blondie24

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Tips

Recreational Mathematics: Why Focus on Projects Over Puzzles

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. Read more...
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Strength Training

Q&A: Transitioning to One-Arm Chinup

The blocking point for OACs is almost always the biceps, not the back. Weighted chinups don’t adequately load the bicep stabilizers. One-arm dead hangs at the top and bottom positions build the specific strength needed. Read more...
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Calisthenics

Q&A: Transitioning to One-Arm Chinup

The blocking point for OACs is almost always the biceps, not the back. Weighted chinups don’t adequately load the bicep stabilizers. One-arm dead hangs at the top and bottom positions build the specific strength needed. Read more...
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Proofs

Q&A: Why are Simpler Proofs Not Given Credit?

Solving an unsolved problem at all is generally much harder than finding a simpler solution to an already-solved one. Credit goes to the original solver. The value of a simpler proof lies in the deeper understanding it reveals. Read more...
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Career

Career Hack: Put Pressure on Your Boss to Come Up with More Work For You

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. Read more...
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Startups

Founder Involvement Is What Keeps Everything On The Rails

As Paul Graham has explained, conventional startup wisdom says to hire good people and let them work, but experienced founders know this is a recipe for hiring professional fakers and letting them derail the company. Founders need to stay in the weeds. Here’s an example of what that looked like at Math Academy. Read more...
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Personal Finance

How To Balance Investing Vs. Spending

Two rules of thumb: don’t upgrade your lifestyle until you can sustain it at 3-4% of your net worth annually. And once an upgrade becomes a rounding error, force yourself to do it – but only if it solves a real pain point. Read more...
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Persistent Homology

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Sequences

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Videos

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Geometry

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Multivariable Calculus

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Classification

Decision Trees

We can algorithmically build classifiers that use a sequence of nested “if-then” decision rules. Read more...
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Student Errors

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AI

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Test Prep

The Missing Middle in Test Prep

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. Read more...
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Podcast (Pinned)

American Enterprise Institute’s “The Report Card” Podcast: Math Academy

You know the kind of trainer an actor gets to prep them for the superhero role in a Marvel film? We’re that, for math. 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 about an hour.
Here’s a summary of what we covered:
[1:34] The big problem in math education is a lack of individualized instruction. In a classroom with one teacher teaching the same thing to all the students, it’s way too easy for the top half of the class and way too hard for the bottom class. What we do is pinpoint the exact problem that each student should be working on right at this moment to make maximum progress in their math learning.
[4:46] So much difficulty in math learning can be traced back to missing prerequisite knowledge. That’s why it’s important to start each student off with a diagnostic that combs through many years of prerequisite knowledge that they need to know to succeed in their chosen course. If we find any knowledge gaps, we fill them in before asking the student to learn any more advanced material that depends on it.
[6:50] We get a very high-resolution picture of the student’s knowledge profile by overlaying every question/answer event onto a structure called a “knowledge graph”. The knowledge graph encodes all the dependency relationships between mathematical topics. We leverage it to squeeze a ton of information out of every single question that we ask the student – not just figuring out what they know and don’t know, but also figuring out exactly what learning tasks they should be working on to maximize their learning efficiency every step of the way.
[8:44] Elsewhere, lots of students struggle with calculus due to gaps in prerequisite knowledge. Good teachers know this, and try to fill those gaps, but there’s a limit to how well the teacher can do this because all the students have knowledge gaps in different places and the teacher can only teach one thing at a time to all the students. But we can target these gaps precisely, backfill them, and move on based on what each individual student knows – fully individualized instruction for all students in parallel, delivering exactly what they need to work on, focusing on those weaknesses and not wasting time on things they already know cold.
[12:05] If you have more talent/aptitude, then you’re going to get more bang for your buck out of practice. You’re going to require fewer reps before getting solid enough to move on, and you’re going to generalize more naturally. However, of the students who get all the way up to calculus before struggle really sets in, the biggest roadblock is typically not talent/aptitude but rather gaps/weaknesses in prerequisite knowledge, an issue that can be resolved with fully individualized instruction.
[14:50] Math Academy origin story: Jason & Sandy coached their son’s 4th grade math field day team. That turned into a pull-out class 3 days per week the following year. The superintendent came by and was shocked to see 5th graders doing trigonometry, advanced algebra, even a little bit of calculus. So he asked Jason & Sandy to create a pilot school program in the Pasadena Unified School District.
In the program, which came to be called Math Academy, students learned all of high school math (prealgebra through precalculus) during 6th/7th grade and took the AP Calculus BC exam in 8th. Students were invited to the program by scoring in the top 7-8% of a middle school math placement that all students in the district took at the end of 5th grade. Keep in mind that about two-thirds of students in the district were on free or reduced lunch, and also, nearly half of Pasadena K-12 students are educated in private schools, compared to the California average of ~10%. In other words, generally speaking, these were not smartest kids in California, and their parents were not Caltech professors.
Jason developed software to automate the process of assigning/grading homework, and together during the pandemic we upgraded it to figure out what each individual student should work on and teach it to them directly without any human intervention. We worked like maniacs to get it ready before school went fully remote the next year, and once we put the school program on it, educational outcomes (including AP Calculus BC scores) skyrocketed. Because of the software, our students experienced a massive learning GAIN, not a loss, during the pandemic. Naturally, it only made sense to keep the school program using the software even after class returned in-person.
[21:55] We have spent thousands and thousands of hours over the years building and fine-tuning our knowledge graph. It’s not off-the-shelf, it’s not automatically generated. It’s the hard work from domain experts, primarily our director of content Alex Smith for the forwards graph (what are all the prerequisites you need to learn in order to unlock a topic) and myself for the backwards graph (when you practice a topic, what component sub-skills are implicitly getting practiced and to what extent).
[25:04] We analyze our knowledge graph by overlaying a big heatmap of where students are doing well or struggling at various parts in the graph. It’s almost like traffic intersections in a city – which ones are where most accidents happen? Let’s go make those safer. We’ve been building and refining the knowledge graph for nearly a decade now with all these analytics.
[27:22] We have a wide variety of user segments. We can help anyone who seriously wants to learn math. Basically, anyone in any sort of educational situation, kids, adults, public school, private school, charter school, homeschool, grade school, high school, college, students who are accelerating, students who are just trying to keep up, “math team” people, people who don’t yet think of themselves as “math people”, adults changing careers to a math-ier field or pursuing a math-ier subfield within their current career, the list goes on and on.
[29:31] The best predictor of how long someone will use the system and how much math they’ll learn is what kind of habit structure they have in place. Students who are consistent, as opposed to sporadic, go much further. It’s that simple.
[34:13] The only math learner persona we can’t help is the crammer – the student who has an exam in a week, is nowhere near prepared, and wants a “quick fix”. We are like a gym, and there’s always people who walk in the gym and think they’re going to work out really hard for a week and look like Thor by the weekend. There is no way to make that happen in a week, no matter how hard you work out. If you show up consistently, like 3-6 times per week for a 30-90 minute session, and then you keep that up for months, then you’re going to come out looking like the mathematical equivalent of a Greek god. But if you are looking for some kind of easy, “how can I change my life in one week,” then I’m sorry, I don’t know what to tell you.
[37:16] We alternate between minimum effective doses of text-based guided instruction followed by active problem-solving. It’s the mathematical equivalent of a tennis instructor showing a quick demonstration of how to hit a ball, just for a minute, and then students practice hitting the ball with that technique until they’re solid enough to move onto the next technique.
[40:16] Real-time reactions and hot takes: Jason on collaborating with school districts, my thoughts on the edtech industry, Jason founding a company with his wife, my experience interacting/growing on X, Jason’s impression of Waymo, my impression of math textbooks, Jason’s thoughts on the “move fast and break things” ethos, Justin’s thoughts on people’s screen time concerns.
[52:10] People say, “just give me the intuition.” But intuition comes through repetition. That’s how you get the automaticity, the natural feel, and that’s what intuition is.
At the same time, it’s important to be efficient. Don’t work 100 problems of the same type in one day. Maybe do 10 to start, then 5 the next day, another 5 a week later, and so on, while you’re filling the empty space with practice on a ton of other skills. You have to get your reps, but you also have to distribute them out over time. That’s how you learn efficiently and build long-term retention.
When people want their math learning to be less skill-heavy and more concept-oriented, what they’re often really saying is that they want a fast overview of a subject without going into the details, without really getting your reps on everything. A video that explains all of calculus in an hour, or how neural networks work in 20 minutes.
But what we’re focused on is building up a true level of mastery. Not surface-level, not shallow. The optimization problem we’re solving is NOT “how fast can we imbue you with a shallow level of understanding, enough that you can tell your friend something cool or that you think you have opinions about it.” What we’re focused on is how quickly we can get you to operating mathematically almost like a professional musician plays their instrument, or a professional athlete plays their sport.
[55:51] As a rule of thumb, if it wouldn’t work in sports, it’s not going to work in math.
[57:31] Students on our system typically learn about 3-4x as fast as a normal class. That’s why, in our school program, the students could go from pre-algebra through AP Calculus BC in 3 years, from 6th-8th grade. When that first happened, and the Washington Post wrote articles about it, lots of people couldn’t believe it. Which is why we had them take the AP Calculus BC exam so we actually have results.
[58:49] We hear all the time about students who are behind in their school class, and then use our system to catch up, and then start crushing their class, and then go well beyond their school class – as well as the resulting change in the student’s level of confidence. In just one year or less, just months, a student can go from thinking “I’m not a math person, I’ll never be good at it” to “I’m crushing my school class, it’s so easy.” That change in the student’s experience does wonders for their confidence.
[1:01:28] Is there an upper limit to how much math you can do per day and have it carry over into real learning results? Think about it like going to the gym. If you work out for 45 minutes, 5-6 days per week, you’ll get in incredible shape. You can do more if you want, but there is a point where you hit diminishing returns. Whether it’s Math Academy or the gym, it really comes down to how long you can sustain a productive full-intensity effort. It’s hard to keep that up for multiple hours, though you might be able to get better mileage by splitting up a multi-hour session into a shorter morning session and evening session. But every person is kind of different in their breaking point, how much they can stay focused and work intensely on the system. In general, one hour per weekday is what we’ve found to be the upper end of a sustainable approach for most students.
[1:03:38] We make students do review problems indefinitely into the future, but with expanding intervals – spaced repetition. It’s the optimal way to keep your knowledge base fresh enough to keep building on it without constantly having to go back and re-learn things. But we make this review process as efficient as possible by tracking all the subskills that are implicitly reviewed when you do a review problem, and we’re always trying to select tasks that kill many birds with one stone by exercising many subskills in need of review.
[1:08:41] Lots of people mistakenly think that students need a million different explanations of the same thing, and that one of those explanations is going to stick, and it’s different for each student. But really, all you need is one really good explanation that’s been battle-tested across a large number of students, and the students need to come into that explanation with all their prerequisite knowledge in place.
If you do that then you can get students learning the skills really well – students pass our lessons over 95% of the time on the first attempt, and over 99% of the time within two attempts, without any additional remediation (because enough knowledge has consolidated into their brain from the first attempt that it makes it cognitively easier for them to get over the hump the second time around).
That’s often surprising to people who think that every student needs a different explanation, but typically what they’re seeing is a symptom of the student lacking prerequisite knowledge, and you’re trying to come up with some explanation that allows them to grasp “enough” of the topic (not the whole thing) while at the same time not requiring too much in the way of prerequisite knowledge they’re missing.
[1:11:08] What makes math hard is the same thing that makes climbing a mountain hard: the steepness of the gradient. What we do is break every steep section of math into smaller steps. If you break things into small enough steps, anyone can learn. And that’s what we do with our analytics: where are the congestion points? Where are students struggling? It’s always where we’re trying to do too much at one time, so we break it up into more steps.
[1:12:25] It’s so important to have a reliable source of truth about what a student really knows, and grades are no longer a good source of truth. You remove test scores from the admissions process, the last objective metric and the last Jenga block, and you get bad situations like at UCSD where 8% of students were not proficient in middle school math. So many issues in education stem from a student having a piece of paper that says they’ve learned something when they actually haven’t. Read more...

The Metagame Podcast #39: Math Academy and The Science of Learning

The most comprehensive 2h overview of my thoughts on serious upskilling, to date. Not just how to train efficiently, but also how to find your mission. Not just the microstructure, but also the metagame. We covered tons of bases ranging from the micro level (science of learning & training efficiently) to the macro level (broader journey of finding, developing, and exploiting your personal talents).
[~0:30] 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?
[~9:00] Efficient learning feels like exercise. The point is to overcome a challenge that strains you. It is by definition unpleasant.
[~13:30] Knowledge graphs are vital when constructing efficient learning experiences. They allow you to systematically organize a learner’s performance data to identify their edge of mastery (the boundary between what they know and don’t know), what previously learned topics below the edge are in need of review, and what new topics on the edge will maximize the amount of review that’s knocked out implicitly.
[~18:00] None of this efficiency stuff matters if you don’t show up consistently. Progress equals volume times efficiency. If either of those factors are low then you don’t make much progress.
[~21:30] Getting excited about the idea of getting good provides an initial activation energy, but seeing yourself improve is what fuels you to keep playing the long game, and efficiency is vital for that.
[~26:30] Your training doesn’t have to be super efficient at the beginning. You can gradually nudge yourself into higher efficiency training even if you don’t have a whole lot of intrinsic motivation to begin with. However, there’s often a skill barrier you need to break through to really get to the fun part, and it’s advisable to do that in a timely manner so you don’t stall out. But at the same time, don’t rush it and fall off the rails.
[~34:30] A common failure mode: being unwilling to identify, accept, and start at the level you’re at.
[~41:30] Center your identity on a mission that speaks to you, that you can contribute to, and do whatever else is needed to further it, regardless of whether you perceive these other things to be “you” or not. You’ll be surprised what capabilities you develop, that you hadn’t previously perceived to be a part of your identity.
[~48:30] How to find your mission: sample wide to figure out what activities speak to you, then filter down and pick one (or a couple) that you’re willing to seriously invest your time and effort climbing up the skill tree and going on “quests”. You may not understand this early on, but skill trees branch out, and quests beget follow-up quests, and the act of climbing to these branch-points will imbue you with perspective that you can leverage to keep filtering down. If you iterate this process enough, it gradually converges into a single area that you can describe coherently and uniquely. That’s your mission.
[~55:30] Every stage in the journey to your mission is hard work, and the earlier you get to putting in that work, the better off you’re going to be. It’s never too late, but the longer you wait, the rougher it gets. At the same time, don’t make a rash decision, don’t tear the house down and build up a new house that you don’t even like. But don’t underestimate how fast you can progress when your internal motivation is aligned with your external incentives.
[~1:12:00] Focus on what matters. That’s obvious, but it’s so easy to mess up lose focus and not realize it until after you’ve wasted a bunch of time.
[~1:15:30] How to get back on the horse after you’ve fallen off. How to avoid feeling bad when something outside of your control temporarily knocks you off your horse. A good social environment can push you to get back on your horse.
[~1:26:30] If you’re a beginner, don’t feel like you have to be advanced to join a community of learners. You can do this right away. And don’t shy away from posting your progress – it’s not about where you are, it’s about where you’re going and how fast. It’s only people who are insecure who will make fun of you. Most people, especially advanced people, will be supportive.
[~1:31:30] There are numerous cognitive learning strategies that 1) can be used to massively improve learning, 2) have been reproduced so many times they might as well be laws of physics, and 3) connect all the way down to the mechanics of what’s going on in the brain. The biggest levers: active learning (as opposed to passive consumption), direct/explicit instruction (as opposed to discovery learning), the spacing effect, mixed practice (a.k.a. interleaving), retrieval practice (a.k.a. the testing effect). Read more...

Chalk and Talk Podcast #42: Math Academy: Optimizing Student Learning

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. Read more...
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Expository

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Category Theory

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Functions

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Sorting

Merge Sort and Quicksort

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. Read more...
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Game Trees

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Gymnastic Rings

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Gifted Students

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Educational Acceleration

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Science Fair

Business Lessons from Science Fair

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. Read more...
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Grading

Q&A: Tip for Grading Proof-Based Courses

Review a taxonomy of common proof errors before grading. When you recognize an error quickly, you can provide more targeted feedback and spend less time puzzling over what the student was trying to do. Read more...
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Intelligence

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Absolute Value

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Logic

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Probability

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Technical Diary

Intuiting Adversarial Examples in Neural Networks via a Simple Computational Experiment

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. Read more...
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Working Memory

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Journal

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Advice

Advice on Upskilling

Advice on consistency, skills, discipline, the grind, the journey, the team, the mission, motivation, learning, and expertise. Read more...
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Incentives

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Humanities

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Computers

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Book Summaries

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Game Theory

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Homology

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Derivatives

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LaTeX

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Linear Programming

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Archetypes

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Tensors

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Personal Website

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College Applications

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Syllabus

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Differentials

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Terminology

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Education Policy

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Math Wars

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Learning Strategies

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Riddles

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Competition Math

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Moore Method

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Project-Based Learning

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Datasets

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Automaticity

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Deliberate Practice

Deliberate Practice: The Most Effective Form of Active Learning

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. Read more...
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Mastery Learning

A Brief History of Mastery Learning

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. Read more...
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Bayesian Statistics

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Math Competitions

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Self-Study

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Learning Efficiency

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Book Reviews

Book Review: Developing Talent in Young People by Benjamin Bloom

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. Read more...
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Mathematics

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Edtech

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LLMs

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Habit

Protect The Habit

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. Read more...
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Education Technology

I Love Seeing Hardcore People Moving Into Serious Edtech

We put man on the moon with computers weaker than a digital watch. Why don’t we have efficient learning at scale? We overcame Earth’s gravity half a century ago, but we can’t overcome the gravity of educational mediocrity? Bullshit. That’s why I get so excited seeing hardcore people moving into serious edtech. People who don’t take bullshit for an answer. Read more...
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Analysis

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Gamification

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Growth

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