All Posts

2024

How to Cultivate Discipline
2024 Oct, ~400 words | Tear down the unproductive habit and build up a counter-habit whose gravity eventually becomes strong enough to completely overtake the original habit.

Love What You Do
2024 Oct, ~300 words | If you don't love it, you'll never be able to keep up with the same volume of effective practice as someone who does have that love. You'll never outwork them.

Complete Individualization: an Often-Forgotten yet Critical Component of True Deliberate Practice
2024 Oct, ~500 words | There are many studies demonstrating a benefit of some component of deliberate practice, but these studies often get mislabeled or misinterpreted as demonstrating the full benefit of true deliberate practice. The field of education is particularly susceptible to this issue because it is impossible for a teacher with a classroom of students to provide a true deliberate practice experience without assistive technology that perfectly emulates the one-on-one pedagogical decisions that an expert tutor would make for each individual student.

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

Just Do The F*cking Work
2024 Oct, ~300 words | At the end of the day you can either waste time debating your coach on the training regimen, or you can use that time to just put your head down and do some f*cking work.

Pictures are Valuable in Math Learning, but They’re Often Overvalued
2024 Sep, ~200 words | Pictures can help build mathematical intuition, but sometimes learners think they should fully visualize every single problem they solve, which actually handicaps their thinking. Math involves generalizing patterns in logically consistent ways, and the generalizations eventually go beyond what you can fully picture in your head.

The Importance of Hardcore Skills
2024 Sep, ~500 words | Hardcore skill development is necessary to do big things, it's one of the greatest social mobility hacks, and it gives you the ability/confidence to take risks knowing that you'll be okay.

Career Hack: Put Pressure on Your Boss to Come Up with More Work For You
2024 Sep, ~300 words | One of the best career hacks -- especially for a junior dev -- is to knock out your work so quickly and so well that you put pressure on your boss to come up with more work for you. Your boss starts giving you work that they themself need to do soon, which is really the exact kind of work that's going to move your career forward.

The Future of Education
2024 Sep, ~1400 words | To quote a Math Academy student: "The fastest and most rigorous progress will be made by individuals in front of their computers."

The “Progress Equals Pressure” Formula
2024 Sep, ~100 words | Making progress is all about putting pressure on a problem: applying the force of your skills to a specific problem area (pressure = force / area).

Five Steps to Becoming a Fully-Fledged Quantitative Software Engineer
2024 Sep, ~900 words | Once you get past steps 1-3, it's hard to find scaffolding. You can't just enroll in a course or pick up a textbook. The scaffolding comes from finding a mentor on a mission that you identify with and are well-suited to contribute to. And it can take a lot of searching to find that person and problem area that's the right fit.

Why Talent Development is Necessary in Math
2024 Sep, ~400 words | When students do the mathematical equivalent of playing kickball during class, and then are expected to do the mathematical equivalent of a backflip at the end of the year, it’s easy to see how struggle and general negative feelings can arise.

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

Competition as a Means of Collaboration
2024 Sep, ~300 words | The whole idea is that you want the other person to raise the bar on competition and pass you up, so that you're motivated to come right back and do the same to them.

Writing is a Skill that Can Be Trained
2024 Sep, ~500 words | Every time you put out a post, get feedback, make improvements, and carry those improvements forward into future posts, that's essentially a "rep" of deliberate practice.

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

A White Pill on Cognitive Differences
2024 Aug, ~1000 words | It's a hard truth that some people have more advantageous cognitive differences than others -- e.g., higher working memory capacity, higher generalization ability, slower forgetting rate. However, there are two sources of hope: 1) automaticity can effectively turn your long-term memory into an extension of your working memory, and 2) many sources of friction in the learning process can be not only remedied but also exploited to increase learning speed beyond the status quo.

Resolving Confusion about Deliberate Practice
2024 Aug, ~1000 words | Doesn't "beyond the edge of one's capabilities" mean that you can't do it? How can you practice it if you can't do it? Also, "performance-improving adjustments on every single repetition" is hard to understand in some realms of performance. For instance, does each step a runner takes involve feedback and improvement?

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

Ability is Built, Not Unlocked
2024 Aug, ~200 words | Curiosity/interest motivates people to engage in deliberate practice, which is what builds ability.

Different Students Need Different Amounts of Practice
2024 Jul, ~500 words | The amount of practice should be determined on the basis of each student's individual performance on each individual topic. Some students may end up having to do more work, but this ultimately empowers them to learn and continue learning into the future.

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

“Following Along” vs Learning
2024 Jul, ~100 words | You haven't learned unless you're able to consistently reproduce the information you consumed and use it to solve problems.

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

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

Who Needs Worked Examples? You, Eventually.
2024 Jul, ~1600 words | Math gets hard for different students at different levels. If you don't have worked examples to help carry you through once math becomes hard for you, then every problem basically blows up into a "research project" for you. Sometimes people advocate for unguided struggle as a way to improve general problem-solving ability, but this idea lacks empirical support. Worked examples won't prevent you from developing deep understanding (actually, it's the opposite: worked examples can help you quickly layer on more skills, which forces a structural integrity in the lower levels of your knowledge). Even if you decide against using worked examples for now, continually re-evaluate to make sure you're getting enough productive training volume.

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

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

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

Conversational Dialogue is a Fascinating Distraction for AI in Education
2024 Jun, ~400 words | Hard-coding explanations feels tedious, takes a lot of work, and isn't "sexy" like an AI that generates responses from scratch – but at least it's not a pipe dream. It’s a practical solution that lets you move on to other components of the AI that are just as important.

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

Student Bite Size vs Curriculum Portion Size
2024 Jun, ~800 words | Students eat meals of information at similar bite rates when each spoonful fed to them is sized appropriately relative to the size of their mouth. (Note that equal bite rates does not imply equal rates of food volume intake.)

Review Should Feel Challenging
2024 May, ~400 words | It's the act of successfully retrieving fuzzy memory, not clear memory, that extends the memory duration.

The Situation with AI in STEM Education
2024 May, ~2500 words | What are LLMs good for in STEM education? Where do LLMs fall short, and why? What does an educational AI need to do for its students to succeed?

If You Want to Learn Algebra, You Need to Have Automaticity on Basic Arithmetic
2024 May, ~1900 words | Solving equations feels smooth when basic arithmetic is automatic -- it's like moving puzzle pieces around, and you just need to identify how they fit together. But without automaticity on basic arithmetic, each puzzle piece is a heavy weight. You struggle to move them at all, much less figure out where they're supposed to go.

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

What Mathematics Can Teach Us About Human Nature
2024 Apr, ~200 words | It highlights the aversion that people have to doing hard things. People will do unbelievable mental gymnastics to convince themselves that doing an easy, enjoyable thing that is unrelated to their supposed goal somehow moves the needle more than doing a hard, unpleasant thing that is directly related to said goal.

What to Do When Math Gets Too Hard
2024 Mar, ~500 words | In general, when you feel yourself running up against a ceiling in life, the solution is typically to pivot and into a direction where the ceiling is higher.

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

Learning vs Feeling
2024 Mar, ~100 words | The strongest people lift weights heavy enough to make them feel weak.

Recreational Mathematics: Why Focus on Projects Over Puzzles
2024 Feb, ~500 words | There's only so much fun you can have trying to follow another person's footsteps to arrive at a known solution. There's only so much confidence you can build from fighting against a problem that someone else has intentionally set up to be well-posed and elegantly solvable if you think about it the right way.

Intuiting Adversarial Examples in Neural Networks via a Simple Computational Experiment
2024 Feb, ~800 words | The network becomes book-smart in a particular area but not street-smart in general. The training procedure is like a series of exams on material within a tiny subject area (your data subspace). The network refines its knowledge in the subject area to maximize its performance on those exams, but it doesn't refine its knowledge outside that subject area. And that leaves it gullible to adversarial examples using inputs outside the subject area.

Leveraging Cognitive Learning Strategies Requires Technology
2024 Feb, ~4100 words | While there is plenty of room for teachers to make better use of cognitive learning strategies in the classroom, teachers are victims of circumstance in a profession lacking effective accountability and incentive structures, and the end result is that students continue to receive mediocre educational experiences. Given a sufficient degree of accountability and incentives, there is no law of physics preventing a teacher from putting forth the work needed to deliver an optimal learning experience to a single student. However, in the absence of technology, it is impossible for a single human teacher to deliver an optimal learning experience to a classroom of many students with heterogeneous knowledge profiles, each of whom needs to work on different types of problems and receive immediate feedback on each of their attempts. This is why technology is necessary.

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

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

Cognitive Science of Learning: Interleaving (Mixed Practice)
2024 Feb, ~3400 words | Interleaving (or mixed practice) involves spreading minimal effective doses of practice across various skills, in contrast to blocked practice, which involves extensive consecutive repetition of a single skill. Blocked practice can give a false sense of mastery and fluency because it allows students to settle into a robotic rhythm of mindlessly applying one type of solution to one type of problem. Interleaving, on the other hand, creates a "desirable difficulty" that promotes vastly superior retention and generalization, making it a more effective review strategy. But despite its proven efficacy, interleaving faces resistance in classrooms due to a preference for practice that feels easier and appears to produce immediate performance gains, even if those performance gains quickly vanish afterwards and do not carry over to test performance.

Cognitive Science of Learning: Spaced Repetition (Distributed Practice)
2024 Feb, ~5100 words | When reviews are spaced out or distributed over multiple sessions (as opposed to being crammed or massed into a single session), memory is not only restored, but also further consolidated into long-term storage, which slows its decay. This is known as the spacing effect. A profound consequence of the spacing effect is that the more reviews are completed (with appropriate spacing), the longer the memory will be retained, and the longer one can wait until the next review is needed. This observation gives rise to a systematic method for reviewing previously-learned material called spaced repetition (or distributed practice). A repetition is a successful review at the appropriate time.

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

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

Cognitive Science of Learning: Developing Automaticity
2024 Feb, ~4400 words | Automaticity is the ability to perform low-level skills without conscious effort. Analogous to a basketball player effortlessly dribbling while strategizing, automaticity allows individuals to avoid spending limited cognitive resources on low-level tasks and instead devote those cognitive resources to higher-order reasoning. In this way, automaticity is the gateway to expertise, creativity, and general academic success. However, insufficient automaticity, particularly in basic skills, inflates the cognitive load of tasks, making it exceedingly difficult for students to learn and perform.

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

A Brief History of Mastery Learning
2024 Feb, ~1300 words | Mastery learning is a strategy in which students demonstrate proficiency on prerequisites before advancing. While even loose approximations of mastery learning have been shown to produce massive gains in student learning, mastery learning faces limited adoption due to clashing with traditional teaching methods and placing increased demands on educators. True mastery learning at a fully granular level requires fully individualized instruction and is only attainable through one-on-one tutoring.

Deliberate Practice: The Most Effective Form of Active Learning
2024 Feb, ~3800 words | Deliberate practice is the most effective form of active learning. It consists of individualized training activities specially chosen to improve specific aspects of a student's performance through repetition and successive refinement. It is 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.

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

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

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

Recommended Language, Tools, Path, and Curriculum for Teaching Kids to Code
2024 Jan, ~1000 words | I'd start off with some introductory course that covers the very basics of coding in some language that is used by many professional programmers but where the syntax reads almost like plain English and lower-level details like memory management are abstracted away. Then, I'd jump right into building board games and strategic game-playing agents (so a human can play against the computer), starting with simple games (e.g. tic-tac-toe) and working upwards from there (maybe connect 4 next, then checkers, and so on).

Tips for Learning Math Effectively
2024 Jan, ~1300 words | Solving problems, building on top of what you've learned, reviewing what you've learned, and quality, quantity, and spacing of practice.

Your Mathematical Potential Has a Limit, but it’s Likely Higher Than You Think
2024 Jan, ~5800 words | Not everybody can learn every level of math, but most people can learn the basics. In practice, however, few people actually reach their full mathematical potential because they get knocked off course early on by factors such as missing foundations, ineffective practice habits, inability or unwillingness to engage in additional practice, or lack of motivation.

Effective Learning Does Not Emulate the Professional Workplace
2024 Jan, ~3900 words | The most effective learning techniques require substantial cognitive effort from students and typically do not emulate what experts do in the professional workplace. Direct instruction is necessary to maximize student learning, whereas unguided instruction and group projects are typically very inefficient.

People Differ in Learning Speed, Not Learning Style
2024 Jan, ~4400 words | Different people generally have different working memory capacities and learn at different rates, but people do not actually learn better in their preferred "learning style." Instead, different people need the same form of practice but in different amounts.

Accountability and Incentives are Necessary but Absent in Education
2024 Jan, ~5400 words | Students and teachers are often not aligned with the goal of maximizing learning, which means that in the absence of accountability and incentives, classrooms are pulled towards a state of mediocrity. Accountability and incentives are typically absent in education, which leads to a "tragedy of the commons" situation where students pass courses (often with high grades) despite severely lacking knowledge of the content.

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

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

Bloom’s Two-Sigma Problem
2024 Jan, ~300 words | The average tutored student performed better than 98% of students in the traditional class.

A Common Source of Student Mistakes
2024 Jan, ~400 words | Many students who pattern-match will tend to prefer solutions requiring fewer and simpler operations, especially if those solutions yield ballpark-reasonable results.

2023

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

My Go-To Math Riddle: How Many Squares are in a 10 x 10 Grid?
2023 Nov, ~400 words | Q: Draw a 10 x 10 square grid. How many squares are there in total? Not just 1 x 1 squares, but also 2 x 2 squares, 3 x 3 squares, and so on. A: The total number of square shapes is the total sum of square numbers 1 + 4 + 9 + 16 + ... + 100.

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

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

For Most Students, Competition Math is a Waste of Time
2023 Sep, ~500 words | If you look at the kinds of math that most quantitative professionals use on a daily basis, competition math tricks don't show up anywhere. But what does show up everywhere is university-level math subjects.

Business Lessons from Science Fair
2023 Jun, ~400 words | The most important things I learned from competing in science fairs had nothing to do with physics or even academics. My main takeaways were actually related to business -- in particular, sales and marketing.

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

2022

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

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

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

Naive Bayes
2022 Jan, ~1100 words | A simple classification algorithm grounded in Bayesian probability.

Regression via Gradient Descent
2022 Jan, ~1300 words | Gradient descent can help us avoid pitfalls that occur when fitting nonlinear models using the pseudoinverse.

2021

Simplex Method
2021 Mar, ~3000 words | A technique for maximizing linear expressions subject to linear constraints.

Hash Tables
2021 Mar, ~600 words | Under the hood, dictionaries are hash tables.

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

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

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

Basic Matrix Arithmetic
2021 Mar, ~600 words | We can use arrays to implement matrices and their associated mathematical operations.

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

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

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

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

Simulating Coin Flips
2021 Jan, ~500 words | Estimating probabilities by simulating a large number of random experiments.

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

2020

2019

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

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

Inverse Matrices
2019 Aug, ~2400 words | The inverse of a matrix is a second matrix which undoes the transformation of the first matrix.

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

Span, Subspaces, and Reduction
2019 Jun, ~2600 words | The span of a set of vectors consists of all vectors that can be made by adding multiples of vectors in the set. We can often reduce a set of vectors to a simpler set with the same span.

Lines and Planes
2019 Jun, ~1700 words | A line starts at an initial point and proceeds straight in a constant direction. A plane is a flat sheet that makes a right angle with some particular vector.

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

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

What is AI?
2019 May, ~500 words | Turing test, games, hype, narrow vs general AI.

Introductory Python: Functions
2019 May, ~700 words | Rather than duplicating such code each time we want to use it, it is more efficient to store the code in a function.

Solving Differential Equations with Taylor Series
2019 Apr, ~700 words | Many differential equations don't have solutions that can be expressed in terms of finite combinations of familiar functions. However, we can often solve for the Taylor series of the solution.

Manipulating Taylor Series
2019 Apr, ~1300 words | To find the Taylor series of complicated functions, it's often easiest to manipulate the Taylor series of simpler functions.

Taylor Series
2019 Apr, ~1100 words | Many non-polynomial functions can be represented by infinite polynomials.

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

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

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

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

Undetermined Coefficients
2019 Mar, ~1400 words | Undetermined coefficients can help us find a solution to a linear differential equation with constant coefficients when the right hand side is not equal to zero.

Separation of Variables
2019 Mar, ~700 words | The simplest differential equations can be solved by separation of variables, in which we move the derivative to one side of the equation and take the antiderivative.

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

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

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

Finding Area Using Integrals
2019 Feb, ~1700 words | To evaluate a definite integral, we find the antiderivative, evaluate it at the indicated bounds, and then take the difference.

Antiderivatives
2019 Feb, ~1400 words | The antiderivative of a function is a second function whose derivative is the first function.

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

Differentials and Approximation
2019 Jan, ~800 words | We can interpret the derivative as an approximation for how a function's output changes, when the function input is changed by a small amount.

Finding Extrema
2019 Jan, ~2000 words | Derivatives can be used to find a function's local extreme values, its peaks and valleys.

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

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

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

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

Evaluating Limits
2019 Jan, ~1800 words | The limit of a function, as the input approaches some value, is the output we would expect if we saw only the surrounding portion of the graph.

2018

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

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

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

Rescalings of Functions
2018 Jul, ~700 words | When a function is rescaled, it is stretched or compressed along one of the axes, like a slinky.

Shifts of Functions
2018 Jul, ~400 words | When a function is shifted, all of its points move vertically and/or horizontally by the same amount.

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

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

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

Radical Functions
2018 Jun, ~1000 words | Radical functions involve roots: square roots, cube roots, or any kind of fractional exponent in general.

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

Polynomial Long Division
2018 May, ~600 words | Polynomial long division works the same way as the long division algorithm that's familiar from simple arithmetic.

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

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

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

Quadratic Systems
2018 Feb, ~400 words | Systems of quadratic equations can be solved via substitution.

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

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

Quadratic Formula
2018 Feb, ~700 words | To solve hard-to-factor quadratic equations, it's easiest to use the quadratic formula.

Linear Systems
2018 Feb, ~1200 words | A linear system consists of multiple linear equations, and the solution of a linear system consists of the pairs that satisfy all of the equations.

Point-Slope Form
2018 Feb, ~800 words | An easy way to write the equation of a line if we know the slope and a point on a line.

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

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

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

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

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

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

Intuiting Naive Bayes
2018 Jan, ~300 words | Naive Bayes classification naively assumes that the presence of a particular feature in a class is unrelated to the presence of any other feature.

2017

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

Intuiting Derivatives
2017 Oct, ~500 words | The derivative tells the steepness of a function at a given point, kind of like a carpenter's level.

Intuiting Limits
2017 Oct, ~700 words | The limit of a function is the height where it looks like the scribble is going to hit a particular vertical line.

Intuiting Functions
2017 Oct, ~200 words | A function is a scribble that crosses each vertical line only once.

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

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

2016

2015

The Brain in One Sentence
2015 Nov, ~2500 words | The brain is a neuronal network integrating specialized subsystems that use local competition and thresholding to sparsify input, spike-timing dependent plasticity to learn inference, and layering to implement hierarchical predictive learning.

On the Contrasting Educations and Outcomes of Ben Franklin and Montaigne
2015 May, ~1000 words | Montaigne's education, strictly dictated by his parents and university studies, resulted in an isolative work with scholarly impact but limited public reach. Conversely, Benjamin Franklin's goal-oriented self-teaching led to influential creations and roles benefiting his community and nation.

2014

2013

2012

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

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

Evidence for the Existence of Dark Matter
2012 Dec, ~1800 words | Mass discrepancies in galaxies and clusters, cosmic background radiation, the structure of the universe, and big bang nucleosynthesis's impact on baryon density.