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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...

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...

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...

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...

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...

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...

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 the opposite of mindless repetition. The amount of deliberate practice has been shown to be one of the most prominent underlying factors responsible for individual differences in performance across numerous fields, even among highly talented elite performers. Deliberate practice demands effort and intensity, and may be discomforting, but its long-term commitment compounds incremental improvements, leading to expertise. Read more...

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

Not everybody can learn every level of math, but most people can learn the basics. In practice, however, few people actually reach their full mathematical potential because they get knocked off course early on by factors such as missing foundations, ineffective practice habits, inability or unwillingness to engage in additional practice, or lack of motivation. Read more...

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...

The Abstraction Ceiling: Why it’s Hard to Teach First-Principles Reasoning

Everyone has some level of abstraction beyond which they are incapable of engaging in first-principles reasoning. That level is different for everyone, and it’s not a hard threshold, but beyond it the time and mental effort required to perform first-principles reasoning skyrockets until first-principles reasoning becomes completely infeasible. 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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Math Education

Why is the EdTech Industry So Damn Soft?

If you depend on a massive base of learners, most of whom are unserious, that puts hard constraints on how you teach. You have to employ ineffective learning strategies that do not repel unserious students. 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 the opposite of mindless repetition. The amount of deliberate practice has been shown to be one of the most prominent underlying factors responsible for individual differences in performance across numerous fields, even among highly talented elite performers. Deliberate practice demands effort and intensity, and may be discomforting, but its long-term commitment compounds incremental improvements, leading to expertise. Read more...

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

Not everybody can learn every level of math, but most people can learn the basics. In practice, however, few people actually reach their full mathematical potential because they get knocked off course early on by factors such as missing foundations, ineffective practice habits, inability or unwillingness to engage in additional practice, or lack of motivation. Read more...

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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Coding

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

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

Decision Trees

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

Euler Estimation

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

K-Means Clustering

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

Merge Sort and Quicksort

Merge sort and quicksort are generally faster than selection, bubble, and insertion sort. And unlike counting sort, they are not susceptible to blowup in the amount of memory required. Read more...

Single-Variable Gradient Descent

We take an initial guess as to what the minimum is, and then repeatedly use the gradient to nudge that guess further and further “downhill” into an actual minimum. Read more...
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Algebra

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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Q&A (Misc)

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Machine Learning

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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Cognitive Science

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...

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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Q&A

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...

The Abstraction Ceiling: Why it’s Hard to Teach First-Principles Reasoning

Everyone has some level of abstraction beyond which they are incapable of engaging in first-principles reasoning. That level is different for everyone, and it’s not a hard threshold, but beyond it the time and mental effort required to perform first-principles reasoning skyrockets until first-principles reasoning becomes completely infeasible. 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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Blog (Tier 2)

Why is the EdTech Industry So Damn Soft?

If you depend on a massive base of learners, most of whom are unserious, that puts hard constraints on how you teach. You have to employ ineffective learning strategies that do not repel unserious students. 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...

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...

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...

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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Graphs

Decision Trees

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

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Graphing Calculator

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Drawing

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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...

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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Stories

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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Blog (Tier 1)

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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Talent Development

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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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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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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Math Academy

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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Neural Networks

Intuiting Adversarial Examples in Neural Networks via a Simple Computational Experiment

The network becomes book-smart in a particular area but not street-smart in general. The training procedure is like a series of exams on material within a tiny subject area (your data subspace). The network refines its knowledge in the subject area to maximize its performance on those exams, but it doesn’t refine its knowledge outside that subject area. And that leaves it gullible to adversarial examples using inputs outside the subject area. Read more...
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Physics

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Integrals

Integration by Parts

We can apply integration by parts whenever an integral would be made simpler by differentiating some expression within the integral, at the cost of anti-differentiating another expression within the integral. Read more...
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Quadratic Equations

Completing the Square

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

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Games

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Notation

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Arithmetic

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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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Education

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...
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Learning

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...
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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 the opposite of mindless repetition. 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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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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Computer Science

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Neuroevolution

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Blondie24

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Quant

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Strength Training

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Proofs

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Persistent Homology

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Videos

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Geometry

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Eurisko

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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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Tips

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Calisthenics

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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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Multivariable Calculus

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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

The Abstraction Ceiling: Why it’s Hard to Teach First-Principles Reasoning

Everyone has some level of abstraction beyond which they are incapable of engaging in first-principles reasoning. That level is different for everyone, and it’s not a hard threshold, but beyond it the time and mental effort required to perform first-principles reasoning skyrockets until first-principles reasoning becomes completely infeasible. Read more...
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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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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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Career

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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 the opposite of mindless repetition. 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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Test Prep

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