Tags

Math Education

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...

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...

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...

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 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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Blog (Tier 3)

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...

Writing is a Skill that Can Be Trained

Every time you put out a post, get feedback, make improvements, and carry those improvements forward into future posts, that’s essentially a “rep” of deliberate practice. 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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Talent Development

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

During its operation from 2020-23, Eurisko was the most advanced high school math/CS sequence in the USA. It culminated in high school students doing masters/PhD-level coursework (reproducing academic research papers in artificial intelligence, building everything from scratch in Python). It’s still early and the first cohort hasn’t even graduated from college yet, but there have already been some amazing student outcomes in terms of college admissions, accelerated graduate degrees, research publications, and science fairs. 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...

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...

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...

Writing is a Skill that Can Be Trained

Every time you put out a post, get feedback, make improvements, and carry those improvements forward into future posts, that’s essentially a “rep” of deliberate practice. 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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Cognitive Science

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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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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Blog (Tier 2)

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...

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...

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...

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...

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 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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Algebra

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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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

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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Math Academy

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 Sequence in the USA

During its operation from 2020-23, Eurisko was the most advanced high school math/CS sequence in the USA. It culminated in high school students doing masters/PhD-level coursework (reproducing academic research papers in artificial intelligence, building everything from scratch in Python). It’s still early and the first cohort hasn’t even graduated from college yet, but there have already been some amazing student outcomes in terms of college admissions, accelerated graduate degrees, research publications, and science fairs. 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...

Demonstration of Setting Encompassing Weights

Encompassing weights control how much spaced repetition credit is propagated backwards from a more advanced topic to a simpler prerequisite topic when a student does a spaced repetition on the more advanced topic. Setting them is tedious, and it sucks, but it’s completely necessary. That’s sometimes what you’ve got to do when you want to build a solution that actually solves a problem. You have to put in the hard work. 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 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 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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Q&A (Misc)

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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...
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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...
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Education

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 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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Motivation

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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Podcast

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...

Demonstration of Setting Encompassing Weights

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

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...

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 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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Blog (Tier 1)

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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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Stories

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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Upskilling

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...

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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Applications

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Graphing Calculator

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Drawing

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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 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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Learning

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 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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Regression

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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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Blog (Pinned)

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...

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

During its operation from 2020-23, Eurisko was the most advanced high school math/CS sequence in the USA. It culminated in high school students doing masters/PhD-level coursework (reproducing academic research papers in artificial intelligence, building everything from scratch in Python). It’s still early and the first cohort hasn’t even graduated from college yet, but there have already been some amazing student outcomes in terms of college admissions, accelerated graduate degrees, research publications, and science fairs. 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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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

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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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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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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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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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Limits

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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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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Eurisko

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

During its operation from 2020-23, Eurisko was the most advanced high school math/CS sequence in the USA. It culminated in high school students doing masters/PhD-level coursework (reproducing academic research papers in artificial intelligence, building everything from scratch in Python). It’s still early and the first cohort hasn’t even graduated from college yet, but there have already been some amazing student outcomes in terms of college admissions, accelerated graduate degrees, research publications, and science fairs. Read more...
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Computer Science

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Neuroevolution

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Blondie24

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Quant

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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

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Calisthenics

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Proofs

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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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Persistent Homology

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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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Expository

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Category Theory

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Functions

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Sequences

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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

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Intelligence

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Absolute Value

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AI

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Logic

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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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Test Prep

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Journal

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Advice

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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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Probability

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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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Working Memory

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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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Startups

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Edtech

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Podcast (Pinned)

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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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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