**Fast, Correct Answers Do Matter in Mathematics **

2024 Aug, ~1400 words | You gotta develop automaticity on low-level skills in order to free up mental resources for higher-level thinking!

All Posts

## 2024

**The Most Effective Way to Motivate Students to Learn Math **

2024 Aug, ~700 words | ... is to not overwhelm them. In my experience, students naturally enjoy math when it doesn't feel overwhelmingly difficult to learn.

**Intuition Behind Polynomial Numerators in Partial Fractions **

2024 Aug, ~300 words | Each decomposition produces a system of linear equations where the number of unknowns equals the number of equations.

**How to Learn Machine Learning: Top Down or Bottom Up? **

2024 Aug, ~300 words | It can be helpful to take a top-down approach in planning out your overarching learning goals, but the learning itself has to occur bottom-up.

**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.

**The 2 Most Common Ways that People Get Retrieval Practice Wrong **

2024 Aug, ~300 words | 1) The information must have already been written to memory. 2) The information must be retrieved from memory, unassisted.

**Why Extrinsic Motivation Matters **

2024 Jul, ~200 words | I think optimal motivation requires a balance of both intrinsic and extrinsic factors.

**What’s the Best Way to Teach Math: Explicit Instruction or Less Guided Learning? **

2024 Jul, ~2300 words | Nobody who knows the science of learning is actually debating this.

**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.

**Sources of Motivation in Successful Math Learners **

2024 Jul, ~400 words | I can think of 4 possible sources.

**Overcoming the Paradox of Serious Training **

2024 Jul, ~100 words | Here's a trick to feel amazingly capable and confident: periodically look back at stuff you originally found challenging months ago.

**How do you apply math to CS when so many software engineers say that there is not that much math in coding? **

2024 Jul, ~400 words | Write code that makes complicated decisions, often involving some kind of inference.

**What’s the Highest Sustainable Daily XP on Math Academy? **

2024 Jul, ~300 words | Around 50-60 XP/day, that is, 50-60 minutes of serious practice per day. Just like the high-end amount of daily exercise you'd expect from people who keep a consistent exercise routine at the gym.

**I’m Writing a Book on the Science of Learning **

2024 Jul, ~300 words | With the science of learning, it's less about "keeping up" with what's happening, and more about "catching up" with what's already happened.

**How to Know When You are Practicing at the Edge of Your Ability **

2024 Jul, ~400 words | Most people can tell when their practice is too easy, but what about when your tasks are too hard? That's often less obvious.

**The Math Death Spiral: How Knowledge Gaps Lead to Student Failure **

2024 Jul, ~400 words | Accumulating mathematical knowledge gaps can lead students to reach a tipping point where further learning becomes overwhelming, ultimately causing them to abandon math entirely.

**Math is Overpowered When Combined with Other Expertise but Underpowered Alone **

2024 Jul, ~200 words | When you're knowledgeable/skilled enough to grapple with problems in a more directly applicable field, math gives you the superpower of being able to compress those problem representations into an abstract space where they're easier to solve.

**Trick to Check Equality of Expression Containing Subscripts Using a Basic LaTeX Expression Evaluator **

2024 Jul, ~200 words | A silly bug turned genius hack.

**Learning Loss, Grade Inflation, and Radical Constructivism **

2024 Jul, ~1700 words | The only way to argue against the existence of learning loss and grade inflation is to argue against the very idea of measuring learning objectively (i.e., radical constructivism).

**“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, ~1300 words | 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.

**Want to know about how the science of learning is missing from teacher education? **

2024 Jul, ~700 words

**The Pedagogically Optimal Way to Learn Math **

2024 Jul, ~3200 words | The underlying principle that it all boils down to is *deliberate practice*.

**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.

**Record for Most Work Done on Math Academy on a Single Date (as of July 2024) **

2024 Jul, ~700 words | 834 XP = 834 minutes = 14 hours of work in a single day. You're probably wondering, what kind of person does that much math in a day? Time for a little story.

**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.

**Levels of Mathematics **

2024 Jul, ~300 words | Research mathematicians are like professional athletes.

**How to Crush a Standardized Math Test: SAT/ACT, AP/IB, GRE/GMAT, JEE, etc. **

2024 Jul, ~1200 words | First, you need extensive and solid content knowledge. Then, you need to work through tons of practice exams for the specific exam you're taking. This might sound simple, but every year, countless people manage to screw it up.

**The Most Superior Form of Training and the Most Hard-Hitting 2 Sentences in All of Talent Development Research **

2024 Jul, ~600 words | "...[D]eliberate practice requires effort and is not inherently enjoyable. Individuals are motivated to practice because practice improves performance."

**What is learning, at a physical level in the brain? **

2024 Jul, ~200 words | Long-term learning is represented by the creation of strategic electrical wiring between neurons.

**If You Want to Learn Math, You Can’t Shy Away from Computation **

2024 Jul, ~500 words | Learning math with little computation is like learning basketball with little practice on dribbling & ball handling techniques.

**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, ~500 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.

**Why Not Just Learn from a Textbook, MIT OpenCourseWare, Khan Academy, etc.? **

2024 Jul, ~600 words | Some shortcomings in my personal experience self-studying a bunch of math on MIT OpenCourseWare (OCW) when I was in high school, that motivated me to help build Math Academy. These shortcomings are pretty general and would also apply to someone learning from miscellaneous textbooks or Khan Academy.

**Individual Variation in Working Memory Capacity (WMC): a First Step Down the Research Rabbit Hole **

2024 Jul, ~400 words | There are many, many studies that measure variation in WMC vs variation in other metrics.

**The Problem with “Think Really Hard, Struggle for a While, Eventually Solve it or Look Up The Answer” Problems **

2024 Jul, ~500 words | Challenge problems are not a good use of time until you've developed the foundational skills that are necessary to grapple with these problems in a productive and timely fashion.

**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.

**The Problem with Teaching Through “Productive Struggle” **

2024 Jul, ~500 words | Beginners benefit more from direct instruction.

**The Greatest Breakthrough in the Science of Education Over the Last Century **

2024 Jul, ~600 words | If you understand the interplay between working memory and long-term memory, then then you can actually derive – from first principles – the methods of effective teaching.

**Conversational Dialogue is a Fascinating Distraction for Educational AI **

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 us move on to other components of the AI that are just as important.

**Silly Mistakes are Still Mistakes **

2024 Jun, ~200 words | ... and they should be treated as such.

**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.

**Paper Idea: A Theory of Optimal Learning Efficiency in Hierarchical Knowledge Structures **

2024 Jun, ~200 words | An idea for a paper that I don't currently have the bandwidth to write.

**Want to Major in Math at an Elite University? Getting A’s in High School Math is Not Good Enough **

2024 Jun, ~900 words | If all the knowledge you show up with is high school math and AP Calculus, and you're not a genius, then you're going to get your ass handed to you.

**What People Think Maximum-Efficiency Learning Should Feel Like, vs What it Actually Feels Like **

2024 Jun, ~500 words | When you're developing skills at peak efficiency, you are maximizing the difficulty of your training tasks subject to the constraint that you end up successfully overcoming those difficulties in a timely manner.

**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 Vicious Cycle of Forgetting **

2024 May, ~300 words | To transfer information into long-term memory, you need to practice retrieving it without assistance.

**Why 4x8 and 6x8 Are, Perhaps Surprisingly, Some of the Hardest Multiplication Facts for Students to Remember **

2024 May, ~300 words | There's a cognitive principle behind this: associative interference, the phenomenon that conceptually related pieces of knowledge can interfere with each other's recall.

**When Does the Learning Happen? **

2024 May, ~200 words | Learning is the incremental gain in your ability to perform a tangible, reproducible skill.

**Q&A: Does Self-Studying Advanced Math Create Bad Habits? **

2024 May, ~800 words | Sure, accelerating via self-study not as optimal as accelerating within teacher-managed courses, but it's way better than not accelerating at all.

**The Goal of Active Learning is NOT to Increase Cognitive Load **

2024 May, ~500 words | It's actually the opposite -- to get students actively retrieving information from memory, while *minimizing* their cognitive load.

**A Quick Trick for Finding a Matrix Transformation Formula **

2024 May, ~300 words | Perform the desired transformation on identity matrix to get a left-multiplier, and maybe transpose the output.

**Which Cognitive Psychology Findings are Solid, That Can Be Used to Help Students Learn Better? **

2024 May, ~3400 words | There are numerous cognitive learning strategies that 1) can be used to massively improve learning, 2) have been reproduced so many times they might as well be laws of physics, and 3) connect all the way down to the mechanics of what's going on in the brain.

**The 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?

**Q&A: How to Create a Matrix whose Eigenvalues have Specified Algebraic and Geometric Multiplicity **

2024 May, ~600 words

**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.

**Q&A: How is it Possible that People can Learn and Apply University-Level Math Yet Not be Able to Solve Competition Problems? **

2024 Mar, ~700 words

**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.

**Estimating a Visitation Interval: an Exercise in Bivariate Bayesian Statistics **

2024 Mar, ~400 words | Loosely inspired by the German tank problem: several witnesses reported seeing a UFO during the given time intervals, and you want to quantify your certainty regarding when the UFO arrived and when it left.

**Q&A: How to Explain Commutativity of Addition to a Kid **

2024 Mar, ~100 words

**Q&A: How to Grade Partial Credit **

2024 Mar, ~300 words

**Lots of People in Education Disagree with the Premise of Maximizing Learning **

2024 Mar, ~700 words | But in talent development, the optimization problem is clear: an individual's performance is to be maximized, so the methods used during practice are those that most efficiently convert effort into performance improvements.

**There is No Such Thing as Low-Effort Learning **

2024 Mar, ~300 words | No matter what skill is being trained, improving performance is always an effortful process.

**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 you can hone your math skills by working on 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.

**Should Students be Asked to Regurgitate Known Proofs? **

2024 Feb, ~300 words | Imitating without analyzing produces a robot / ape who can't think critically; analyzing without imitating produces a critic who can't act on their own advice.

**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.

**Active Learning: If You’re Active Half the Time, That’s Still Not Enough **

2024 Feb, ~300 words | During practice, the elite skaters were over 6 times more active than passive, while non-competitive skaters were nearly as passive as they were active.

**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.

**What To Do Leading Up to a Standardized Exam Like AP Calculus BC **

2024 Feb, ~600 words | Six weeks of pure review and six official practice exams.

**The Double-Edged Nature of Hierarchical Knowledge **

2024 Feb, ~200 words | It's easier to run into roadblocks, but also easier to maintain what you've learned.

**You Know it’s Edutainment When… **

2024 Feb, ~200 words | Passive consumption. Lack of depth. Lack of rigorous assessments. Failing upwards. Lack of skill development.

**Subtle Things to Watch Out For When Demonstrating Lp-Norm Regularization on a High-Degree Polynomial Regression Model **

2024 Feb, ~800 words | Initial parameter range, data sampling range, severity of regularization.

**Why Poking Around Wikipedia Doesn’t Move The Needle on Math Learning **

2024 Feb, ~300 words | It's like going to the gym without a solid workout plan in place.

**How Much Math Do You Need to Know for Machine Learning? **

2024 Feb, ~300 words | If you know your single-variable calculus, then it's about 70 hours on Math Academy.

**The Only Way to Teach a More Sophisticated Technique **

2024 Feb, ~300 words | ... is to present a problem where known simpler techniques fail.

**How I Got Started with Calisthenics **

2024 Jan, ~1000 words | My training has been scattered and fuzzy until recently. Here's the whole story.

**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.

**Q&A: Sources of Bivariate Data from Various Function Families **

2024 Jan, ~300 words

**The Easiest Way to Remember Closed vs Open Interval Notation **

2024 Jan, ~100 words | An oval () fits inside a rectangle [ ] with the same width and height.

**Q&A: What is the Best Trick for Doing Small-Integer Addition Problems? **

2024 Jan, ~100 words

**Q&A: Is it Ever Okay to Use Dismissive Language During Instruction? **

2024 Jan, ~400 words

**Q&A: Should You Teach Alternative Multiplication Algorithms Like the Lattice Method? **

2024 Jan, ~400 words

**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.

**The Greatest Educational Life Hack: Learning Math Ahead of Time **

2024 Jan, ~2700 words | Learning math early guards you against numerous academic risks and opens all kinds of doors to career opportunities.

**Q&A: Why is Newton’s Method Useful? **

2024 Jan, ~300 words

**Q&A: Which is Faster, Bisection or Newton’s Method? **

2024 Jan, ~500 words

**Effective Learning Requires Intense Effort **

2024 Jan, ~3100 words | Effortful processes like testing, repetition, and computation are essential parts of effective learning, and competition is often helpful.

**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.

**Q&A: Can You Teach a Math Class on the Basis of Projects Only? **

2024 Jan, ~0 words

**Q&A: Can You Teach a Math Class on the Basis of Projects Only? **

2024 Jan, ~300 words

**Q&A: Can You Run a Successful High School Class Using the Moore Method? **

2024 Jan, ~300 words

**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.

**Q&A: Should You Tell Students How Difficult You Think a Problem Is? **

2024 Jan, ~400 words

**Q&A: Should You Tell Students How Difficult You Think a Problem Is? **

2024 Jan, ~0 words

**Talent Development vs Traditional Schooling **

2024 Jan, ~2400 words | Talent development is not only different from schooling, but in many cases completely orthogonal to schooling.

**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

**Q&A: How to Explain Implications Intuitively **

2023 Dec, ~0 words

**Q&A: How to Explain Implications Intuitively **

2023 Dec, ~200 words

**Q&A: How to Learn Effectively On Your Own **

2023 Dec, ~0 words

**Q&A: How to Learn Effectively On Your Own **

2023 Dec, ~700 words

**Q&A: Are Homework Problems at Top Colleges as Hard as Competition Problems? **

2023 Dec, ~400 words

**Q&A: Undefined versus Infinity **

2023 Dec, ~200 words

**Q&A: Undefined versus Infinity **

2023 Dec, ~0 words

**Q&A: Activity for a Probability Class **

2023 Dec, ~0 words

**Q&A: Activity for a Probability Class **

2023 Dec, ~400 words

**Q&A: How to Help Bright Students Articulate Proofs **

2023 Nov, ~500 words

**Critique of Paper: An astonishing regularity in student learning rate **

2023 Nov, ~8900 words | It rests on a critical assumption that the amount of learning that occurs during initial instruction is zero or otherwise negligible, which is not true.

**Q&A: How to Find Open Problems for Math Research? **

2023 Nov, ~400 words

**Q&A: What’s the Appropriate Level of Granularity for a Proof? **

2023 Nov, ~400 words

**Q&A: Is Square Meter a Badly-Phrased Term? **

2023 Nov, ~0 words

**Q&A: Is Square Meter a Badly-Phrased Term? **

2023 Nov, ~200 words

**Q&A: Will You Always Be Behind IMO Toppers? **

2023 Nov, ~500 words

**Q&A: Why is the Order of Operations the Way That It Is? **

2023 Nov, ~200 words

**Q&A: Intuiting Edge Cases of Convergence and Divergence **

2023 Nov, ~100 words

**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.

**Q&A: What’s the Intuition for these Weird Logical Equivalences? **

2023 Nov, ~300 words

**Q&A: What’s the Intuition Behind the Order of Function Transformations? **

2023 Nov, ~300 words

**Study Sessions Should be Short and Frequent as Opposed to Long and Sparse **

2023 Nov, ~400 words | First, you want to form a habit. Second, you want to operate at peak productivity during your session. Third, you want to minimize the amount you forget between sessions.

**Educational resources commonly address slant asymptotes. Why not general polynomial asymptotes? **

2023 Oct, ~400 words | Answer: It's not very useful (not in practice, not in theory).

**Q&A: How Can You Reconcile Cognitive Activation with Cognitive Load Minimization? **

2023 Oct, ~500 words

**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.

**Q&A: Are Math Ability and Language Ability Inversely Related? **

2023 Oct, ~100 words

**Q&A: What are the Sides on the Argument about Lowering Math Requirements? **

2023 Oct, ~600 words

**Q&A: What Does it Mean to Say that a Proof is Trigonometric or Calculus-Based? **

2023 Oct, ~200 words

**Q&A: What is the Easiest Way to Memorize Absolute Value Inequalities? **

2023 Oct, ~300 words

**Q&A: What is the Easiest Way to Memorize Absolute Value Inequalities? **

2023 Oct, ~0 words

**Q&A: Why Do Students Skip Concepts Before Examples? **

2023 Oct, ~600 words

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

2023 Oct, ~600 words | 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.

**Q&A: Why Do Students Skip Concepts Before Examples? **

2023 Oct, ~0 words

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

2023 Oct, ~400 words

**Q&A: How To Explain Why Two Numbers with Arbitrarily Small Distance Are Equal **

2023 Oct, ~200 words

**When Can You Manipulate Differentials Like Fractions? **

2023 Oct, ~300 words | In general, you can manipulate total derivatives like fractions, but you can't do the same with partial derivatives.

**Q&A: Real-Life Application Involving a Non-Trivial Limit **

2023 Oct, ~300 words

**The Tragedy of the Commons in Education **

2023 Oct, ~500 words | Why it's common for students to pass courses despite severely lacking knowledge of the content.

**Q&A: What to Do When Students in Advanced STEM Courses Cannot Do Math **

2023 Oct, ~300 words

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

2023 Oct, ~200 words

**Q&A: What To Major In If There Is No Data Science Major **

2023 Oct, ~200 words

**Precision Spaced Repetition in Hierarchical Knowledge Structures **

2023 Oct, ~0 words

**Optimized, Individualized Spaced Repetition in Hierarchical Knowledge Structures **

2023 Oct, ~4800 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).

**Q&A: Why can Any Set of Data Points with Different Inputs be Fit Perfectly by a Polynomial of Sufficient Degree? **

2023 Oct, ~400 words

**Q&A: Why can Any Set of Data Points with Different Inputs be Fit Perfectly by a Polynomial of Sufficient Degree? **

2023 Oct, ~0 words

**Q&A: What are some Common Errors and Misconceptions about the Pythagorean Theorem? **

2023 Oct, ~0 words

**Q&A: What are some Common Errors and Misconceptions about the Pythagorean Theorem? **

2023 Oct, ~200 words

**Q&A: Why are the Segment Addition Postulate and the Partition Postulate Two Different Things? **

2023 Oct, ~300 words

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

2023 Oct, ~600 words

**Q&A: Has the Topology of Student Errors in Algebra Been Studied Systematically? **

2023 Oct, ~500 words

**How I Won a Heat Capacitor Competition Without a Heat Capacitor **

2023 Oct, ~700 words | Won first place in a state-level competition by finding and exploiting a loophole in the points scoring logic.

**How to Look Up the Meaning of an Unknown Math Symbol or Expression **

2023 Sep, ~100 words | Drawing --> Latex commands --> ChatGPT summary --> Google more info

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

2023 Sep, ~200 words

**Q&A: How to Define Common Knowledge for Online Course Discussions **

2023 Sep, ~100 words

**Q&A: How Can Geometric Approaches be Used When Teaching Algebra? **

2023 Sep, ~400 words

**Q&A: Relationship Between Determinant and Number of Solutions **

2023 Sep, ~200 words

**Q&A: Why Must You Know How to Handle Data Manually Before Doing it Automatically? **

2023 Sep, ~300 words

**Q&A: Why Must You Know How to Handle Data Manually Before Doing it Automatically? **

2023 Sep, ~0 words

**Q&A: What is the Easiest Way to Graph a Linear Equation? **

2023 Sep, ~0 words

**Q&A: How to Introduce Sigma Notation **

2023 Sep, ~300 words

**Q&A: What is the Easiest Way to Graph a Linear Equation? **

2023 Sep, ~300 words

**Q&A: Transitioning to One-Arm Chinup **

2023 Sep, ~300 words

**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.

**According to Feynman himself, his classes were a failure for 90% of his students. **

2023 Sep, ~500 words | While some may view Feynman-style pedagogy as supporting inclusive learning for all students across varying levels of ability, Feynman himself acknowledged that his methods only worked for the top 10% of his students.

**Q&A: Why is Zero Considered Even? **

2023 Aug, ~100 words

**Q&A: Tip for Grading Proof-Based Courses **

2023 Aug, ~100 words

**Q&A: When (and How) to Correct Student Notation **

2023 Aug, ~200 words

**Q&A: How to Leverage Self-Studied Math in College Applications **

2023 Aug, ~300 words

**Q&A: Suggestions for Getting Started with a Personal Website **

2023 Aug, ~300 words

**Q&A: Translating Between Tensor Indices **

2023 Aug, ~500 words

**Q&A: Why are Eigenvalues and Singular Values Important? **

2023 Aug, ~300 words

**Q&A: Why are Eigenvalues and Singular Values Important? **

2023 Aug, ~0 words

**Q&A: Lower Body Calisthenics for Gaining Muscle Mass **

2023 Jul, ~0 words

**Q&A: Lower Body Calisthenics for Gaining Muscle Mass **

2023 Jul, ~200 words

**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.

**How to Remember Type I, II, and III Regions in Multivariable Calculus **

2023 May, ~400 words | Type I pairs with the variable that runs vertically in the usual representation of the coordinate system. The remaining types are paired with the rest of the variables in ascending order.

**The Story of Math Academy’s Eurisko Sequence: the Most Advanced High School Math/CS Sequence in the USA **

2023 May, ~1100 words | During its operation from 2020 to 2023, Eurisko was the most advanced high school math/CS sequence in the USA. It culminated in high school students doing masters/PhD-level coursework (reproducing academic research papers in artificial intelligence, building everything from scratch in Python).

**Minimalist Strength Training, Phase 2: Gaining Mass **

2023 Apr, ~700 words | Minor changes to increase workout intensity and caloric surplus.

**My Experience with Teacher Credentialing and Professional Development **

2023 Apr, ~500 words | It's centered around political ideology rather than the science of learning.

**Q&A: Resources and Advice for Gifted Student **

2023 Mar, ~300 words

**Why I Don’t Worship at the Altar of Neural Nets **

2023 Jan, ~500 words | In order to justify using a more complex model, the increase in performance has to be worth the cost of integrating and maintaining the complexity.

**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

**Minimalist Strength Training, Phase 1: Getting Ripped **

2022 Oct, ~1100 words | Daily 20-30 minute bedroom workout with gymnastic rings hanging from pull-up bar -- just as much challenge as weights, but inexpensive and easily portable.

**Quants vs Systems Coders **

2022 Oct, ~700 words | Two subtypes of coders that I watched students grow into.

**Tips for Developing Valuable Models **

2022 Oct, ~3000 words | Stuff you don't find in math textbooks.

**The Counterintuitive Nature of Effective Learning Strategies **

2022 Sep, ~200 words | Effective learning strategies sometimes go against our human instincts about conversation.

**Memory vs Time Graphs **

2022 Sep, ~200 words | A way to visualize some cognitive learning strategies.

**The 5 Breeds of Quants **

2022 Aug, ~0 words | ... are summarized in the following table.

**From Procedures to Objects **

2022 Aug, ~600 words | An aha moment with object-oriented programming.

**Reimplementing Blondie24: Convolutional Version **

2022 Mar, ~500 words | Using convolutional layers to create an even better checkers player.

**Reimplementing Blondie24 **

2022 Mar, ~800 words | Extending Fogel's tic-tac-toe player to the game of checkers.

**Reimplementing Fogel’s Tic-Tac-Toe Paper **

2022 Mar, ~1400 words | Reimplementing the paper that laid the groundwork for Blondie24.

**Introduction to Blondie24 and Neuroevolution **

2022 Mar, ~1000 words | A method for training neural networks that works even when training feedback is sparse.

**Reduced Search Depth and Heuristic Evaluation for Connect Four **

2022 Mar, ~900 words | Combining game-specific human intelligence (heuristics) and generalizable artificial intelligence (minimax on a game tree)

**Minimax Strategy **

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

**Canonical and Reduced Game Trees for Tic-Tac-Toe **

2022 Mar, ~500 words | Building data structures that represent all the possible outcomes of a game.

**Backpropagation **

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

**Introduction to Neural Network Regressors **

2022 Feb, ~1000 words | The deeper or more "hierarchical" a computational graph is, the more complex the model that it represents.

**Decision Trees **

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

**Dijkstra’s Algorithm for Distance and Shortest Paths in Weighted Graphs **

2022 Feb, ~800 words | Computing spatial relationships between nodes when edges no longer represent unit distances.

**Distance and Shortest Paths in Unweighted Graphs **

2022 Feb, ~400 words | Using traversals to understand spatial relationships between nodes in graphs.

**Breadth-First and Depth-First Traversals **

2022 Feb, ~900 words | Graphs show up all the time in computer science, so it's important to know how to work with them.

**Naive Bayes **

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

**K-Nearest Neighbors **

2022 Jan, ~1500 words | One of the simplest classifiers.

**Multiple Regression and Interaction Terms **

2022 Jan, ~1200 words | In many real-life situations, there is more than one input variable that controls the output variable.

**Regression via Gradient Descent **

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

**Overfitting, Underfitting, Cross-Validation, and the Bias-Variance Tradeoff **

2022 Jan, ~2200 words | Just because model appears to match closely with points in the data set, does not necessarily mean it is a good model.

**Power, Exponential, and Logistic Regression via Pseudoinverse **

2022 Jan, ~2000 words | Transforming nonlinear functions so that we can fit them using the pseudoinverse.

**Regressing a Linear Combination of Nonlinear Functions via Pseudoinverse **

2022 Jan, ~1500 words | Exploring the most general class of functions that can be fit using the pseudoinverse.

**Linear, Polynomial, and Multiple Linear Regression via Pseudoinverse **

2022 Jan, ~2200 words | Using matrix algebra to fit simple functions to data sets.

## 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.

**Hodgkin-Huxley Model of Action Potentials in Neurons **

2021 Mar, ~700 words | Implementing a differential equations model that won the Nobel prize.

**SIR Model For the Spread of Disease **

2021 Mar, ~400 words | A simple differential equations model that we can plot using multivariable Euler estimation.

**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.

**Reduced Row Echelon Form and Applications to Matrix Arithmetic **

2021 Mar, ~500 words | You can use the RREF algorithm to compute determinants much faster than with the recursive cofactor expansion method.

**Basic Matrix Arithmetic **

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

**The Ultimate High School Computer Science Sequence: 9 Months In **

2021 Feb, ~3000 words | In 9 months, these students went from initially not knowing how to write helper functions to building a machine learning library from scratch.

**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.

**Selection, Bubble, Insertion, and Counting Sort **

2021 Feb, ~1500 words | Some of the simplest methods for sorting items in arrays.

**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.

**Estimating Roots via Bisection Search and Newton-Raphson Method **

2021 Feb, ~800 words | Bisection search involves repeatedly moving one bound halfway to the other. The Newton-Raphson method involves repeatedly moving our guess to the root of the tangent line.

**Solving Magic Squares via Backtracking **

2021 Feb, ~600 words | Backtracking can drastically cut down the number of possibilities that must be checked during brute force.

**Brute Force Search with Linear-Encoding Cryptography **

2021 Feb, ~700 words | Brute force search involves trying every single possibility.

**Cartesian Product **

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

**Roulette Wheel Selection **

2021 Jan, ~500 words | How to sample from a discrete probability distribution.

**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.

**Converting Between Binary, Decimal, and Hexadecimal **

2021 Jan, ~1200 words | There are other number systems that use more or fewer than ten characters.

**Some Short Introductory Coding Exercises **

2021 Jan, ~1000 words | It's assumed that you've had some basic exposure to programming.

## 2020

**Tips for LaTeX Math Formatting **

2020 Sep, ~600 words | How to avoid some of the most common pitfalls leading to ugly LaTeX.

## 2019

**Path Dependency in Multivariable Limits **

2019 Dec, ~900 words | The behavior of a multivariable function can be highly specific to the path taken.

**Thales’ Theorem **

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

**But WHERE do the Taylor Series and Lagrange Error Bound even come from?! **

2019 Dec, ~1300 words | An intuitive derivation.

**Trick to Apply the Chain Rule FAST - Peeling the Onion **

2019 Dec, ~1600 words | A simple mnemonic trick for quickly differentiating complicated functions.

**Intuition Behind Completing the Square **

2019 Dec, ~400 words | Hidden inside of every quadratic, there is a perfect square.

**Matrix Exponential and Systems of Linear Differential Equations **

2019 Sep, ~2600 words | The matrix exponential can be defined as a power series and used to solve systems of linear differential equations.

**Generalized Eigenvectors and Jordan Form **

2019 Sep, ~6100 words | Jordan form provides a guaranteed backup plan for exponentiating matrices that are non-diagonalizable.

**Recursive Sequence Formulas via Diagonalization **

2019 Sep, ~1600 words | Matrix diagonalization can be applied to construct closed-form expressions for recursive sequences.

**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.

**Rescaling, Shearing, and the Determinant **

2019 Aug, ~2900 words | Every square matrix can be decomposed into a product of rescalings and shears.

**Matrix Multiplication **

2019 Aug, ~2200 words | How to multiply a matrix by another matrix.

**Linear Systems as Transformations of Vectors by Matrices **

2019 Aug, ~2500 words | Matrices are vectors whose components are themselves vectors.

**Higher-Order Variation of Parameters **

2019 Jul, ~1800 words | Solving linear systems can sometimes be a necessary component of solving nonlinear systems.

**Shearing, Cramer’s Rule, and Volume by Reduction **

2019 Jul, ~1600 words | Shearing can be used to express the solution of a linear system using ratios of volumes, and also to compute volumes themselves.

**Volume as the Determinant of a Square Linear System **

2019 Jul, ~1600 words | Rich intuition about why the number of solutions to a square linear system is governed by the volume of the parallelepiped formed by the coefficient vectors.

**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.

**Elimination as Vector Reduction **

2019 Jun, ~800 words | If we interpret linear systems as sets of vectors, then elimination corresponds to vector reduction.

**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.

**Dot Product and Cross Product **

2019 Jun, ~1600 words | What does it mean to multiply a vector by another vector?

**N-Dimensional Space **

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

**CheckMySteps: A Web App to Help Students Fix their Algebraic Mistakes **

2019 May, ~3000 words | A prototype web app to automatically assist students in self-correcting small errors and minor misconceptions.

**Solving Tower of Hanoi with General Problem Solver **

2019 May, ~2200 words | A walkthrough of solving Tower of Hanoi using the approach of one of the earliest AI systems.

**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.

**The Third Wave of AI: Computation Power and Neural Networks **

2019 May, ~600 words | As computation power increased, neural networks began to take center stage in AI.

**The Second Wave of AI: Expert Systems **

2019 May, ~300 words | Expert systems stored "if-then" rules derived from the knowledge of experts.

**The First Wave of AI: Reasoning as Search **

2019 May, ~500 words | Framing reasoning as searching through a maze of actions for a sequence that achieves the desired end goal.

**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.

**Introductory Python: If, While, and For **

2019 May, ~1200 words | We often wish to tell the computer instructions involving the words "if," "while," and "for."

**Introductory Python: Lists, Dictionaries, and Arrays **

2019 May, ~2100 words | We can store many related pieces of data within a single variable called a data structure.

**Introductory Python: Strings, Ints, Floats, and Booleans **

2019 May, ~1900 words | We can store and manipulate data in the form of variables.

**Graphing Calculator Drawing: Composition Waves and Implicit Trig Patterns **

2019 May, ~100 words | Equations involving compositions of trigonometric functions can create wild patterns in the plane.

**Graphing Calculator Drawing: Lissajous Curves **

2019 May, ~200 words | Lissajous curves use sine functions to create interesting patterns in the plane.

**Graphing Calculator Drawing: Rotation **

2019 May, ~200 words | Absolute value graphs can be rotated to draw stars.

**Graphing Calculator Drawing: Non-Euclidean Ellipses **

2019 May, ~300 words | Non-euclidean ellipses can be used to draw starry-eye sunglasses.

**Graphing Calculator Drawing: Euclidean Ellipses **

2019 May, ~200 words | Euclidean ellipses can be combined with sine wave shading to form three-dimensional shells.

**Graphing Calculator Drawing: Shading with Sine **

2019 May, ~300 words | High-frequency sine waves can be used to draw shaded regions.

**Graphing Calculator Drawing: Roots **

2019 May, ~300 words | Roots can be used to draw deer.

**Graphing Calculator Drawing: Sine Waves **

2019 May, ~400 words | Sine waves can be used to draw scales on a fish.

**Graphing Calculator Drawing: Parabolas **

2019 May, ~200 words | Parabolas can be used to draw a fish.

**Graphing Calculator Drawing: Absolute Value **

2019 May, ~200 words | Absolute value can be used to draw a person.

**Graphing Calculator Drawing: Slanted Lines **

2019 May, ~300 words | Slanted lines can be used to draw a spider web.

**Graphing Calculator Drawing: Horizontal and Vertical Lines **

2019 May, ~200 words | Horizontal and vertical lines can be used to draw a castle.

**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.

**Characteristic Polynomial of a Differential Equation **

2019 Mar, ~1200 words | Given a linear differential equation with constant coefficients and a right hand side of zero, the roots of the characteristic polynomial correspond to solutions of the equation.

**Solving Differential Equations by Substitution **

2019 Mar, ~500 words | Non-separable differential equations can be sometimes converted into separable differential equations by way of substitution.

**Slope Fields and Euler Approximation **

2019 Mar, ~800 words | When faced with a differential equation that we don't know how to solve, we can sometimes still approximate the solution.

**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.

**Derivatives of Non-Polynomial Functions **

2019 Jan, ~2000 words | There are convenient rules the derivatives of exponential, logarithmic, trigonometric, and inverse trigonometric functions.

**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.

**Limits by Logarithms, Squeeze Theorem, and Euler’s Constant **

2019 Jan, ~1000 words | Various tricks for evaluating tricky limits.

**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.

**Exponential and Logarithmic Functions **

2018 Jun, ~1100 words | Exponential functions have variables as exponents. Logarithms cancel out exponentiation.

**Radical Functions **

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

**Graphing Rational Functions with Slant and Polynomial Asymptotes **

2018 May, ~300 words | A slant asymptote is a slanted line that arises from a linear term in the proper form of a rational function.

**Graphing Rational Functions with Horizontal and Vertical Asymptotes **

2018 May, ~200 words | If we choose one input on each side of an asymptote, we can tell which section of the plane the function will occupy.

**Vertical Asymptotes of Rational Functions **

2018 May, ~600 words | Vertical asymptotes are vertical lines that a function approaches but never quite reaches.

**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.

**Sketching Graphs of Polynomials **

2018 Apr, ~400 words | We can sketch the graph of a polynomial using its end behavior and zeros.

**Rational Roots and Synthetic Division **

2018 Apr, ~900 words | The rational roots theorem can help us find zeros of polynomials without blindly guessing.

**Zeros of Polynomials **

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

**Standard Form and End Behavior of Polynomials **

2018 Apr, ~1300 words | The end behavior of a polynomial refers to the type of output that is produced when we input extremely large positive or negative values.

**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.

**Quadratic Inequalities **

2018 Mar, ~400 words | Quadratic inequalities are best visualized in the plane.

**Linear Inequalities in the Plane **

2018 Mar, ~600 words | When a linear equation has two variables, the solution covers a section of the coordinate plane.

**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.

**Factoring Quadratic Equations **

2018 Feb, ~1500 words | Factoring is a method for solving quadratic equations.

**Standard Form of a Quadratic Equation **

2018 Feb, ~700 words | Quadratic equations are similar to linear equations, except that they contain squares of a single variable.

**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.

**Standard Form of a Line **

2018 Feb, ~500 words | Standard form makes it easy to see the intercepts of a line.

**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.

**Slope-Intercept Form **

2018 Feb, ~1400 words | Introducing linear equations in two variables.

**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 Maximum a Posteriori and Maximum Likelihood Estimation **

2018 Jan, ~300 words | To visualize the relationship between the MAP and MLE estimations, one can imagine starting at the MLE estimation, and then obtaining the MAP estimation by drifting a bit towards higher density in the prior distribution.

**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: Calculating the Horsepower of an Offensive Lineman **

2017 Nov, ~600 words | It comes out to roughly a fortieth of that of a truck.

**Applications of Calculus: Derivatives in String Art **

2017 Nov, ~100 words | String art works because the strings are tangent lines to a curve.

**Applications of Calculus: A Failure of Intuition **

2017 Nov, ~500 words | Calculus can show us how our intuition can fail us, a common theme in philosophy.

**History of Calculus: The Newton-Leibniz Controversy **

2017 Nov, ~500 words | Nobody came out of the dispute well.

**History of Calculus: The Man who “Broke” Math **

2017 Nov, ~300 words | When Joseph Fourier first introduced Fourier series, they gave mathematicians nightmares.

**Applications of Calculus: Continuously Compounded Interest **

2017 Nov, ~500 words | Deriving the "Pert" formula.

**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.

**Applications of Calculus: Optimization via Gradient Descent **

2017 Nov, ~300 words | Calculus can be used to find the parameters that minimize a function.

**Applications of Calculus: Physics Engines in Video Games **

2017 Nov, ~300 words | Physics engines use calculus to periodically updates the locations of objects.

**Applications of Calculus: Rendering 3D Computer Graphics **

2017 Nov, ~300 words | Introducing Kajiya's rendering equation.

**Applications of Calculus: Rocket Propulsion **

2017 Nov, ~500 words | Deriving the ideal rocket equation.

**Applications of Calculus: Modeling Tumor Growth **

2017 Nov, ~300 words | Deriving the Gompertz function.

**Applications of Calculus: Understanding Plaque Buildup **

2017 Nov, ~300 words | Understanding why even slight narrowing of arteries can pose such a big problem to blood flow.

**Applications of Calculus: Cardiac Output **

2017 Nov, ~300 words | Measuring volume of blood the heart pumps out into the aorta per unit time.

**Intuiting Series **

2017 Oct, ~300 words | A series is the sum of a sequence.

**Intuiting Sequences **

2017 Oct, ~400 words | A sequence is a list of numbers that has some pattern.

**Intuiting Integrals **

2017 Oct, ~300 words | Integrals give the area under a portion of a function.

**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.

**The Data Scientist’s Guide to Topological Data Analysis: Preamble **

2017 Sep, ~500 words | Bridging the communication gap between academia and industry in the field of TDA.

**Persistent Homology Software: Demonstration of TDA **

2017 Sep, ~300 words | Demonstrating an open-source implementation of persistent homology techniques in the TDA package for R.

**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 Aunalytics **

2017 Sep, ~400 words | At Aunalytics, Mapper outperformed hierarchical clustering in providing granular insights.

**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.

**Mapper Software: Demonstration of TDAmapper **

2017 Sep, ~700 words | Demonstrating an open-source implementation of Mapper in the TDAmapper package for R.

**Intuiting the Mapper Algorithm **

2017 Sep, ~900 words | Representing a data space's topology by converting it into a network.

## 2016

**A Game-Theoretic Analysis of Social Distancing During Epidemics **

2016 May, ~1700 words | In a simplified problem framing, we investigate the (game-theoretical) usefulness of limiting the number of social connections per person.

**Making Indirect Interactions Explicit in Networks **

2016 Mar, ~1000 words | Category theory provides a language for explicitly describing indirect relationships in graphs.

**Book Summary: Memory Evolutive Systems **

2016 Mar, ~4200 words | Framing complex systems in the language of category theory.

## 2015

**Introduction to Computers **

2015 Nov, ~2200 words | The main ideas behind computers can be understood by anyone.

**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.

**Shaping STDP Neural Networks with Periodic Stimulation: a Theoretical Analysis for the Case of Tree Networks **

2015 Aug, ~1400 words | We solve a special case of how to periodically stimulate a biological neural network to obtain a desired connectivity (in theory).

**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.

**A Brief Overview of Spike-Timing Dependent Plasticity (STDP) Learning During Neural Simulation **

2015 Feb, ~800 words | Implementation notes for STDP learning in a network of Hodgkin-Huxley simulated neurons.

**A Visual, Inductive Proof of Sharkovsky’s Theorem **

2015 Jan, ~3400 words | Many existing proofs are not accessible to young mathematicians or those without experience in the realm of dynamic systems.

## 2014

**Building an Iron Man Suit: A Physics Workbook **

2014 Jul, ~1300 words | A workbook I created to explain the math and physics behind an Iron Man suit to a student who was interested in the comics / movies.

**The Physics Behind an Egg Drop: A Lively Story **

2014 Jul, ~4400 words | A workbook I created to explain the math and physics behind an egg drop experiment to a student who was interested in Lord of the Rings and Star Wars.

## 2013

**A Formula for the Partial Fractions Decomposition of $x^n/(x-a)^k$ **

2013 Aug, ~600 words | And a proof via double induction.

## 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.