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.
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Top Selected
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."
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.
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, then you're going to get your ass handed to you.
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.
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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.
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.
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.
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).
Other Selected
Why is the EdTech Industry So Damn Soft?
2024 Jul, ~500 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.
The Pedagogically Optimal Way to Learn Math
2024 Jul, ~3200 words | The underlying principle that it all boils down to is deliberate practice.
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 to Crush a Standardized Math Test: SAT/ACT, AP/IB, GRE/GMAT, JEE, etc.
2024 Jul, ~1100 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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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).
My Experience with Teacher Credentialing and Professional Development
2023 Apr, ~500 words | It's centered around political ideology rather than the science of learning.
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.
Tips for Developing Valuable Models
2022 Oct, ~3000 words | Stuff you don't find in math textbooks.
But WHERE do the Taylor Series and Lagrange Error Bound even come from?!
2019 Dec, ~1300 words | An intuitive derivation.
Other
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.
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.
Want to know about how the science of learning is missing from teacher education?
2024 Jul, ~700 words
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.
Levels of Mathematics
2024 Jul, ~300 words | Research mathematicians are like professional athletes.
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.
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.
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.
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.)
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.
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.
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.
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.
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.
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.
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 Requires Intense Effort
2024 Jan, ~2900 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.
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.
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.
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.
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).
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.
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.
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.
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.
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
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.
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.
Minimalist Strength Training, Phase 2: Gaining Mass
2023 Apr, ~700 words | Minor changes to increase workout intensity and caloric surplus.
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.
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.
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.
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.
Tips for LaTeX Math Formatting
2020 Sep, ~600 words | How to avoid some of the most common pitfalls leading to ugly LaTeX.
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.
Trick to Apply the Chain Rule FAST - Peeling the Onion
2019 Dec, ~1600 words | A simple mnemonic trick for quickly differentiating complicated functions.
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.
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.
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.
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.
A Formula for the Partial Fractions Decomposition of $x^n/(x-a)^k$
2013 Aug, ~600 words | And a proof via double induction.
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.