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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?


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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.

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.

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

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

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.

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.

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.

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

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

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


Other

“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.

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.

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.

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

Student Bite Size vs Curriculum Portion Size
2024 Jun, ~700 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.)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Deliberate Practice: The Most Effective Form of Active Learning
2024 Feb, ~3800 words | Deliberate practice is the most effective form of active learning. It consists of individualized training activities specially chosen to improve specific aspects of a student's performance through repetition and successive refinement. It is the opposite of mindless repetition. The amount of deliberate practice has been shown to be one of the most prominent underlying factors responsible for individual differences in performance across numerous fields, even among highly talented elite performers. Deliberate practice demands effort and intensity, and may be discomforting, but its long-term commitment compounds incremental improvements, leading to expertise.

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

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

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

Your Mathematical Potential Has a Limit, but it’s Likely Higher Than You Think
2024 Jan, ~5700 words | Not everybody can learn every level of math, but most people can learn the basics. In practice, however, few people actually reach their full mathematical potential because they get knocked off course early on by factors such as missing foundations, ineffective practice habits, inability or unwillingness to engage in additional practice, or lack of motivation.

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

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

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

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

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.

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

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

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

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.

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

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

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

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.