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## Posts

## Active Learning and Cognitive Load

The goal of active learning is *not* to blow up a student’s cognitive load. It’s actually the opposite – to get students actively retrieving information from memory, while *minimizing* their cognitive load. ** Read more...**

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

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. ** Read more...**

## The Situation with AI in STEM Education

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? ** Read more...**

## If You Want to Learn Algebra, You Need to Have Automaticity on Basic Arithmetic

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. ** Read more...**

## Bloom’s 3 Stages of Talent Development

First, fun and exciting playtime. Then, intense and strenuous skill development. Finally, developing one’s individual style while pushing the boundaries of the field. ** Read more...**

## What Mathematics Can Teach Us About Human Nature

It highlights the aversion that people have to doing hard things. People will do unbelievable mental gymnastics to convince themselves that doing an easy, enjoyable thing that is unrelated to their supposed goal somehow moves the needle more than doing a hard, unpleasant thing that is directly related to said goal. ** Read more...**

## What to Do When Math Gets Too Hard

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. ** Read more...**

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

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. ** Read more...**

## Maximizing Learning vs Other Things

Lots of people in education disagree with the premise of maximizing learning. 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. ** Read more...**

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

No matter what skill is being trained, improving performance is always an effortful process. ** Read more...**

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

Learning math early guards you against numerous academic risks and opens all kinds of doors to career opportunities. ** Read more...**

## Learning vs Feeling

The strongest people lift weights heavy enough to make them feel weak. ** Read more...**

## Recreational Mathematics: Why Focus on Projects Over Puzzles

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. ** Read more...**

## Intuiting Adversarial Examples in Neural Networks via a Simple Computational Experiment

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

## Should Students be Asked to Regurgitate Known Proofs?

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. ** Read more...**

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

Six weeks of pure review and six official practice exams. ** Read more...**

## The Double-Edged Nature of Hierarchical Knowledge

It’s easier to run into roadblocks, but also easier to maintain what you’ve learned. ** Read more...**

## You Know it’s Edutainment When…

Passive consumption. Lack of depth. Lack of rigorous assessments. Failing upwards. Lack of skill development. ** Read more...**

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

Initial parameter range, data sampling range, severity of regularization. ** Read more...**

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

It’s like going to the gym without a solid workout plan in place. ** Read more...**

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

If you know your single-variable calculus, then it’s about 70 hours on Math Academy. ** Read more...**

## The Only Way to Teach a More Sophisticated Technique

… is to present a problem where known simpler techniques fail. ** Read more...**

## How I Got Started with Calisthenics

My training has been scattered and fuzzy until recently. Here’s the whole story. ** Read more...**

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

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

## Tips for Learning Math Effectively

Solving problems, building on top of what you’ve learned, reviewing what you’ve learned, and quality, quantity, and spacing of practice. ** Read more...**

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

An oval () fits inside a rectangle [ ] with the same width and height. ** Read more...**

## A Common Source of Student Mistakes

Many students who pattern-match will tend to prefer solutions requiring fewer and simpler operations, especially if those solutions yield ballpark-reasonable results. ** Read more...**

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

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. ** Read more...**

## My Go-To Math Riddle: How Many Squares are in a 10 x 10 Grid?

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. ** Read more...**

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

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. ** Read more...**

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

Answer: It’s not very useful (not in practice, not in theory). ** Read more...**

## Can You Automate a Math Teacher?

For many (but not all) students, the answer is yes. And for many of those students, automation can unlock life-changing educational outcomes. ** Read more...**

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

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

## When Can You Manipulate Differentials Like Fractions?

In general, you can manipulate total derivatives like fractions, but you can’t do the same with partial derivatives. ** Read more...**

## The Tragedy of the Commons in Education

Why it’s common for students to pass courses despite severely lacking knowledge of the content. ** Read more...**

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

Won first place in a state-level competition by finding and exploiting a loophole in the points scoring logic. ** Read more...**

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

Drawing –> Latex commands –> ChatGPT summary –> Google more info ** Read more...**

## For Most Students, Competition Math is a Waste of Time

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. ** Read more...**

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

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. ** Read more...**

## Business Lessons from Science Fair

The most important things I learned from competing in science fairs had nothing to do with physics or even academics. My main takeaways were actually related to business – in particular, sales and marketing. ** Read more...**

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

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. ** Read more...**

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

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). ** Read more...**

## Minimalist Strength Training, Phase 2: Gaining Mass

Minor changes to increase workout intensity and caloric surplus. ** Read more...**

## My Experience with Teacher Credentialing and Professional Development

It’s centered around political ideology rather than the science of learning. ** Read more...**

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

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. ** Read more...**

## Selecting a Good Problem to Work On

Good problem = intersection between your own interests/talents, the realm of what’s feasible, and the desires of the external world. ** Read more...**

## Minimalist Strength Training, Phase 1: Getting Ripped

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. ** Read more...**

## Quants vs Systems Coders

Two subtypes of coders that I watched students grow into. ** Read more...**

## Tips for Developing Valuable Models

Stuff you don’t find in math textbooks. ** Read more...**

## The 5 Breeds of Quants

… are summarized in the following table. ** Read more...**

## From Procedures to Objects

An aha moment with object-oriented programming. ** Read more...**

## Reimplementing Blondie24: Convolutional Version

Using convolutional layers to create an even better checkers player. ** Read more...**

## Reimplementing Blondie24

Extending Fogel’s tic-tac-toe player to the game of checkers. ** Read more...**

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

Reimplementing the paper that laid the groundwork for Blondie24. ** Read more...**

## Introduction to Blondie24 and Neuroevolution

A method for training neural networks that works even when training feedback is sparse. ** Read more...**

## Reduced Search Depth and Heuristic Evaluation for Connect Four

Combining game-specific human intelligence (heuristics) and generalizable artificial intelligence (minimax on a game tree) ** Read more...**

## Minimax Strategy

Repeatedly choosing the action with the best worst-case scenario. ** Read more...**

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

Building data structures that represent all the possible outcomes of a game. ** Read more...**

## Backpropagation

A convenient technique for computing gradients in neural networks. ** Read more...**

## Introduction to Neural Network Regressors

The deeper or more “hierarchical” a computational graph is, the more complex the model that it represents. ** Read more...**

## Decision Trees

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

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

Computing spatial relationships between nodes when edges no longer represent unit distances. ** Read more...**

## Distance and Shortest Paths in Unweighted Graphs

Using traversals to understand spatial relationships between nodes in graphs. ** Read more...**

## Breadth-First and Depth-First Traversals

Graphs show up all the time in computer science, so it’s important to know how to work with them. ** Read more...**

## Naive Bayes

A simple classification algorithm grounded in Bayesian probability. ** Read more...**

## K-Nearest Neighbors

One of the simplest classifiers. ** Read more...**

## Multiple Regression and Interaction Terms

In many real-life situations, there is more than one input variable that controls the output variable. ** Read more...**

## Regression via Gradient Descent

Gradient descent can help us avoid pitfalls that occur when fitting nonlinear models using the pseudoinverse. ** Read more...**

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

Just because model appears to match closely with points in the data set, does not necessarily mean it is a good model. ** Read more...**

## Power, Exponential, and Logistic Regression via Pseudoinverse

Transforming nonlinear functions so that we can fit them using the pseudoinverse. ** Read more...**

## Regressing a Linear Combination of Nonlinear Functions via Pseudoinverse

Exploring the most general class of functions that can be fit using the pseudoinverse. ** Read more...**

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

Using matrix algebra to fit simple functions to data sets. ** Read more...**

## Simplex Method

A technique for maximizing linear expressions subject to linear constraints. ** Read more...**

## Hash Tables

Under the hood, dictionaries are hash tables. ** Read more...**

## Hodgkin-Huxley Model of Action Potentials in Neurons

Implementing a differential equations model that won the Nobel prize. ** Read more...**

## SIR Model For the Spread of Disease

A simple differential equations model that we can plot using multivariable Euler estimation. ** Read more...**

## Euler Estimation

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

## Tic-Tac-Toe and Connect Four

One of the best ways to get practice with object-oriented programming is implementing games. ** Read more...**

## K-Means Clustering

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

## Reduced Row Echelon Form and Applications to Matrix Arithmetic

You can use the RREF algorithm to compute determinants much faster than with the recursive cofactor expansion method. ** Read more...**

## Basic Matrix Arithmetic

We can use arrays to implement matrices and their associated mathematical operations. ** Read more...**

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

In 9 months, these students went from initially not knowing how to write helper functions to building a machine learning library from scratch. ** Read more...**

## Merge Sort and Quicksort

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

## Selection, Bubble, Insertion, and Counting Sort

Some of the simplest methods for sorting items in arrays. ** Read more...**

## Multivariable Gradient Descent

Just like single-variable gradient descent, except that we replace the derivative with the gradient vector. ** Read more...**

## Single-Variable Gradient Descent

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

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

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. ** Read more...**

## Solving Magic Squares via Backtracking

Backtracking can drastically cut down the number of possibilities that must be checked during brute force. ** Read more...**

## Brute Force Search with Linear-Encoding Cryptography

Brute force search involves trying every single possibility. ** Read more...**

## Cartesian Product

Implementing the Cartesian product provides good practice working with arrays. ** Read more...**

## Roulette Wheel Selection

How to sample from a discrete probability distribution. ** Read more...**

## Simulating Coin Flips

Estimating probabilities by simulating a large number of random experiments. ** Read more...**

## Recursive Sequences

Sequences where each term is a function of the previous terms. ** Read more...**

## Converting Between Binary, Decimal, and Hexadecimal

There are other number systems that use more or fewer than ten characters. ** Read more...**

## Some Short Introductory Coding Exercises

It’s assumed that you’ve had some basic exposure to programming. ** Read more...**

## Tips for LaTeX Math Formatting

How to avoid some of the most common pitfalls leading to ugly LaTeX. ** Read more...**

## Path Dependency in Multivariable Limits

The behavior of a multivariable function can be highly specific to the path taken. ** Read more...**

## Thales’ Theorem

Every inscribed triangle whose hypotenuse is a diameter is a right triangle. ** Read more...**

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

An intuitive derivation. ** Read more...**

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

A simple mnemonic trick for quickly differentiating complicated functions. ** Read more...**

## Intuition Behind Completing the Square

Hidden inside of every quadratic, there is a perfect square. ** Read more...**

## Matrix Exponential and Systems of Linear Differential Equations

The matrix exponential can be defined as a power series and used to solve systems of linear differential equations. ** Read more...**

## Generalized Eigenvectors and Jordan Form

Jordan form provides a guaranteed backup plan for exponentiating matrices that are non-diagonalizable. ** Read more...**

## Recursive Sequence Formulas via Diagonalization

Matrix diagonalization can be applied to construct closed-form expressions for recursive sequences. ** Read more...**

## Eigenvalues, Eigenvectors, and Diagonalization

The eigenvectors of a matrix are those vectors that the matrix simply rescales, and the factor by which an eigenvector is rescaled is called its eigenvalue. These concepts can be used to quickly calculate large powers of matrices. ** Read more...**

## Inverse Matrices

The inverse of a matrix is a second matrix which undoes the transformation of the first matrix. ** Read more...**

## Rescaling, Shearing, and the Determinant

Every square matrix can be decomposed into a product of rescalings and shears. ** Read more...**

## Matrix Multiplication

How to multiply a matrix by another matrix. ** Read more...**

## Linear Systems as Transformations of Vectors by Matrices

Matrices are vectors whose components are themselves vectors. ** Read more...**

## Higher-Order Variation of Parameters

Solving linear systems can sometimes be a necessary component of solving nonlinear systems. ** Read more...**

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

Shearing can be used to express the solution of a linear system using ratios of volumes, and also to compute volumes themselves. ** Read more...**

## Volume as the Determinant of a Square Linear System

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. ** Read more...**

## N-Dimensional Volume Formula

N-dimensional volume generalizes the idea of the space occupied by an object. We can think about N-dimensional volume as being enclosed by N-dimensional vectors. ** Read more...**

## Elimination as Vector Reduction

If we interpret linear systems as sets of vectors, then elimination corresponds to vector reduction. ** Read more...**

## Span, Subspaces, and Reduction

The span of a set of vectors consists of all vectors that can be made by adding multiples of vectors in the set. We can often reduce a set of vectors to a simpler set with the same span. ** Read more...**

## Lines and Planes

A line starts at an initial point and proceeds straight in a constant direction. A plane is a flat sheet that makes a right angle with some particular vector. ** Read more...**

## Dot Product and Cross Product

What does it mean to multiply a vector by another vector? ** Read more...**

## N-Dimensional Space

N-dimensional space consists of points that have N components. ** Read more...**

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

A prototype web app to automatically assist students in self-correcting small errors and minor misconceptions. ** Read more...**

## Solving Tower of Hanoi with General Problem Solver

A walkthrough of solving Tower of Hanoi using the approach of one of the earliest AI systems. ** Read more...**

## Cutting Through the Hype of AI

Media outlets often make the mistake of anthropomorphizing or attributing human-like characteristics to computer programs. ** Read more...**

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

As computation power increased, neural networks began to take center stage in AI. ** Read more...**

## The Second Wave of AI: Expert Systems

Expert systems stored “if-then” rules derived from the knowledge of experts. ** Read more...**

## The First Wave of AI: Reasoning as Search

Framing reasoning as searching through a maze of actions for a sequence that achieves the desired end goal. ** Read more...**

## What is AI?

Turing test, games, hype, narrow vs general AI. ** Read more...**

## Introductory Python: Functions

Rather than duplicating such code each time we want to use it, it is more efficient to store the code in a function. ** Read more...**

## Introductory Python: If, While, and For

We often wish to tell the computer instructions involving the words “if,” “while,” and “for.” ** Read more...**

## Introductory Python: Lists, Dictionaries, and Arrays

We can store many related pieces of data within a single variable called a data structure. ** Read more...**

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

We can store and manipulate data in the form of variables. ** Read more...**

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

Equations involving compositions of trigonometric functions can create wild patterns in the plane. ** Read more...**

## Graphing Calculator Drawing: Lissajous Curves

Lissajous curves use sine functions to create interesting patterns in the plane. ** Read more...**

## Graphing Calculator Drawing: Rotation

Absolute value graphs can be rotated to draw stars. ** Read more...**

## Graphing Calculator Drawing: Non-Euclidean Ellipses

Non-euclidean ellipses can be used to draw starry-eye sunglasses. ** Read more...**

## Graphing Calculator Drawing: Euclidean Ellipses

Euclidean ellipses can be combined with sine wave shading to form three-dimensional shells. ** Read more...**

## Graphing Calculator Drawing: Shading with Sine

High-frequency sine waves can be used to draw shaded regions. ** Read more...**

## Graphing Calculator Drawing: Roots

Roots can be used to draw deer. ** Read more...**

## Graphing Calculator Drawing: Sine Waves

Sine waves can be used to draw scales on a fish. ** Read more...**

## Graphing Calculator Drawing: Parabolas

Parabolas can be used to draw a fish. ** Read more...**

## Graphing Calculator Drawing: Absolute Value

Absolute value can be used to draw a person. ** Read more...**

## Graphing Calculator Drawing: Slanted Lines

Slanted lines can be used to draw a spider web. ** Read more...**

## Graphing Calculator Drawing: Horizontal and Vertical Lines

Horizontal and vertical lines can be used to draw a castle. ** Read more...**

## Solving Differential Equations with Taylor Series

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. ** Read more...**

## Manipulating Taylor Series

To find the Taylor series of complicated functions, it’s often easiest to manipulate the Taylor series of simpler functions. ** Read more...**

## Taylor Series

Many non-polynomial functions can be represented by infinite polynomials. ** Read more...**

## Tests for Convergence

Various tricks for determining whether a series converges or diverges. ** Read more...**

## Geometric Series

A geometric series is a sum where each term is some constant times the previous term. ** Read more...**

## Variation of Parameters

When we know the solutions of a linear differential equation with constant coefficients and right hand side equal to zero, we can use variation of parameters to find a solution when the right hand side is not equal to zero. ** Read more...**

## Integrating Factors

Integrating factors can be used to solve first-order differential equations with non-constant coefficients. ** Read more...**

## Undetermined Coefficients

Undetermined coefficients can help us find a solution to a linear differential equation with constant coefficients when the right hand side is not equal to zero. ** Read more...**

## Characteristic Polynomial of a Differential Equation

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. ** Read more...**

## Solving Differential Equations by Substitution

Non-separable differential equations can be sometimes converted into separable differential equations by way of substitution. ** Read more...**

## Slope Fields and Euler Approximation

When faced with a differential equation that we don’t know how to solve, we can sometimes still approximate the solution. ** Read more...**

## Separation of Variables

The simplest differential equations can be solved by separation of variables, in which we move the derivative to one side of the equation and take the antiderivative. ** Read more...**

## Improper Integrals

Improper integrals have bounds or function values that extend to positive or negative infinity. ** Read more...**

## Integration by Parts

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

## Integration by Substitution

Substitution involves condensing an expression of into a single new variable, and then expressing the integral in terms of that new variable. ** Read more...**

## Finding Area Using Integrals

To evaluate a definite integral, we find the antiderivative, evaluate it at the indicated bounds, and then take the difference. ** Read more...**

## Antiderivatives

The antiderivative of a function is a second function whose derivative is the first function. ** Read more...**

## L’Hôpital’s Rule

When a limit takes the indeterminate form of zero divided by zero or infinity divided by infinity, we can differentiate the numerator and denominator separately without changing the actual value of the limit. ** Read more...**

## Differentials and Approximation

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. ** Read more...**

## Finding Extrema

Derivatives can be used to find a function’s local extreme values, its peaks and valleys. ** Read more...**

## Derivatives of Non-Polynomial Functions

There are convenient rules the derivatives of exponential, logarithmic, trigonometric, and inverse trigonometric functions. ** Read more...**

## Properties of Derivatives

Given a sum, we can differentiate each term individually. But why are we able to do this? Does multiplication work the same way? What about division? ** Read more...**

## Chain Rule

When taking derivatives of compositions of functions, we can ignore the inside of a function as long as we multiply by the derivative of the inside afterwards. ** Read more...**

## Power Rule for Derivatives

There are some patterns that allow us to compute derivatives without having to compute the limit of the difference quotient. ** Read more...**

## Derivatives and the Difference Quotient

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. ** Read more...**

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

Various tricks for evaluating tricky limits. ** Read more...**

## Evaluating Limits

The limit of a function, as the input approaches some value, is the output we would expect if we saw only the surrounding portion of the graph. ** Read more...**

## Compositions of Functions

Compositions of functions consist of multiple functions linked together, where the output of one function becomes the input of another function. ** Read more...**

## Inverse Functions

Inverting a function entails reversing the outputs and inputs of the function. ** Read more...**

## Reflections of Functions

When a function is reflected, it flips across one of the axes to become its mirror image. ** Read more...**

## Rescalings of Functions

When a function is rescaled, it is stretched or compressed along one of the axes, like a slinky. ** Read more...**

## Shifts of Functions

When a function is shifted, all of its points move vertically and/or horizontally by the same amount. ** Read more...**

## Piecewise Functions

A piecewise function is pieced together from multiple different functions. ** Read more...**

## Trigonometric Functions

Trigonometric functions represent the relationship between sides and angles in right triangles. ** Read more...**

## Absolute Value

Absolute value represents the magnitude of a number, i.e. its distance from zero. ** Read more...**

## Exponential and Logarithmic Functions

Exponential functions have variables as exponents. Logarithms cancel out exponentiation. ** Read more...**

## Radical Functions

Radical functions involve roots: square roots, cube roots, or any kind of fractional exponent in general. ** Read more...**

## Graphing Rational Functions with Slant and Polynomial Asymptotes

A slant asymptote is a slanted line that arises from a linear term in the proper form of a rational function. ** Read more...**

## Graphing Rational Functions with Horizontal and Vertical Asymptotes

If we choose one input on each side of an asymptote, we can tell which section of the plane the function will occupy. ** Read more...**

## Vertical Asymptotes of Rational Functions

Vertical asymptotes are vertical lines that a function approaches but never quite reaches. ** Read more...**

## Horizontal Asymptotes of Rational Functions

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. ** Read more...**

## Polynomial Long Division

Polynomial long division works the same way as the long division algorithm that’s familiar from simple arithmetic. ** Read more...**

## Sketching Graphs of Polynomials

We can sketch the graph of a polynomial using its end behavior and zeros. ** Read more...**

## Rational Roots and Synthetic Division

The rational roots theorem can help us find zeros of polynomials without blindly guessing. ** Read more...**

## Zeros of Polynomials

The zeros of a polynomial are the inputs that cause it to evaluate to zero. ** Read more...**

## Standard Form and End Behavior of Polynomials

The end behavior of a polynomial refers to the type of output that is produced when we input extremely large positive or negative values. ** Read more...**

## Systems of Inequalities

To solve a system of inequalities, we need to solve each individual inequality and find where all their solutions overlap. ** Read more...**

## Quadratic Inequalities

Quadratic inequalities are best visualized in the plane. ** Read more...**

## Linear Inequalities in the Plane

When a linear equation has two variables, the solution covers a section of the coordinate plane. ** Read more...**

## Linear Inequalities in the Number Line

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. ** Read more...**

## Quadratic Systems

Systems of quadratic equations can be solved via substitution. ** Read more...**

## Vertex Form

To easily graph a quadratic equation, we can convert it to vertex form. ** Read more...**

## Completing the Square

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

## Quadratic Formula

To solve hard-to-factor quadratic equations, it’s easiest to use the quadratic formula. ** Read more...**

## Factoring Quadratic Equations

Factoring is a method for solving quadratic equations. ** Read more...**

## Standard Form of a Quadratic Equation

Quadratic equations are similar to linear equations, except that they contain squares of a single variable. ** Read more...**

## Linear Systems

A linear system consists of multiple linear equations, and the solution of a linear system consists of the pairs that satisfy all of the equations. ** Read more...**

## Standard Form of a Line

Standard form makes it easy to see the intercepts of a line. ** Read more...**

## Point-Slope Form

An easy way to write the equation of a line if we know the slope and a point on a line. ** Read more...**

## Slope-Intercept Form

Introducing linear equations in two variables. ** Read more...**

## Solving Linear Equations

Loosely speaking, a linear equation is an equality statement containing only addition, subtraction, multiplication, and division. ** Read more...**

## Intuiting Ensemble Methods

The type of ensemble model that wins most data science competitions is the stacked model, which consists of an ensemble of entirely different species of models together with some combiner algorithm. ** Read more...**

## Intuiting Decision Trees

Decision trees are able to model nonlinear data while remaining interpretable. ** Read more...**

## Intuiting Neural Networks

NNs are similar to SVMs in that they project the data to a higher-dimensional space and fit a hyperplane to the data in the projected space. However, whereas SVMs use a predetermined kernel to project the data, NNs automatically construct their own projection. ** Read more...**

## Intuiting Support Vector Machines

A Support Vector Machine (SVM) computes the “best” separation between classes as the maximum-margin hyperplane. ** Read more...**

## Intuiting Linear Regression

In linear regression, we model the target as a random variable whose expected value depends on a linear combination of the predictors (including a bias term). ** Read more...**

## Intuiting Maximum a Posteriori and Maximum Likelihood Estimation

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. ** Read more...**

## Intuiting Naive Bayes

Naive Bayes classification naively assumes that the presence of a particular feature in a class is unrelated to the presence of any other feature. ** Read more...**

## Applications of Calculus: Calculating the Horsepower of an Offensive Lineman

It comes out to roughly a fortieth of that of a truck. ** Read more...**

## Applications of Calculus: Derivatives in String Art

String art works because the strings are tangent lines to a curve. ** Read more...**

## Applications of Calculus: A Failure of Intuition

Calculus can show us how our intuition can fail us, a common theme in philosophy. ** Read more...**

## History of Calculus: The Newton-Leibniz Controversy

Nobody came out of the dispute well. ** Read more...**

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

When Joseph Fourier first introduced Fourier series, they gave mathematicians nightmares. ** Read more...**

## Applications of Calculus: Continuously Compounded Interest

Deriving the “Pert” formula. ** Read more...**

## Applications of Calculus: Maximizing Profit

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. ** Read more...**

## Applications of Calculus: Optimization via Gradient Descent

Calculus can be used to find the parameters that minimize a function. ** Read more...**

## Applications of Calculus: Physics Engines in Video Games

Physics engines use calculus to periodically updates the locations of objects. ** Read more...**

## Applications of Calculus: Rendering 3D Computer Graphics

Introducing Kajiya’s rendering equation. ** Read more...**

## Applications of Calculus: Rocket Propulsion

Deriving the ideal rocket equation. ** Read more...**

## Applications of Calculus: Modeling Tumor Growth

Deriving the Gompertz function. ** Read more...**

## Applications of Calculus: Understanding Plaque Buildup

Understanding why even slight narrowing of arteries can pose such a big problem to blood flow. ** Read more...**

## Applications of Calculus: Cardiac Output

Measuring volume of blood the heart pumps out into the aorta per unit time. ** Read more...**

## Intuiting Series

A series is the sum of a sequence. ** Read more...**

## Intuiting Sequences

A sequence is a list of numbers that has some pattern. ** Read more...**

## Intuiting Integrals

Integrals give the area under a portion of a function. ** Read more...**

## Intuiting Derivatives

The derivative tells the steepness of a function at a given point, kind of like a carpenter’s level. ** Read more...**

## Intuiting Limits

The limit of a function is the height where it looks like the scribble is going to hit a particular vertical line. ** Read more...**

## Intuiting Functions

A function is a scribble that crosses each vertical line only once. ** Read more...**

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

Bridging the communication gap between academia and industry in the field of TDA. ** Read more...**

## Persistent Homology Software: Demonstration of TDA

Demonstrating an open-source implementation of persistent homology techniques in the TDA package for R. ** Read more...**

## Intuiting Persistent Homology

Persistent homology provides a way to quantify the topological features that persist over our a data set’s full range of scale. ** Read more...**

## Mapper Use-Cases at Aunalytics

At Aunalytics, Mapper outperformed hierarchical clustering in providing granular insights. ** Read more...**

## Mapper Use-Cases at Ayasdi

Ayasdi developed commercial Mapper software and sells a subscription service to clients who wish to create topological network visualizations of their data. ** Read more...**

## Mapper Software: Demonstration of TDAmapper

Demonstrating an open-source implementation of Mapper in the TDAmapper package for R. ** Read more...**

## Intuiting the Mapper Algorithm

Representing a data space’s topology by converting it into a network. ** Read more...**

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

In a simplified problem framing, we investigate the (game-theoretical) usefulness of limiting the number of social connections per person. ** Read more...**

## Making Indirect Interactions Explicit in Networks

Category theory provides a language for explicitly describing indirect relationships in graphs. ** Read more...**

## Book Summary: Memory Evolutive Systems

Framing complex systems in the language of category theory. ** Read more...**

## Introduction to Computers

The main ideas behind computers can be understood by anyone. ** Read more...**

## The Brain in One Sentence

The brain is a neuronal network integrating specialized subsystems that use local competition and thresholding to sparsify input, spike-timing dependent plasticity to learn inference, and layering to implement hierarchical predictive learning. ** Read more...**

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

We solve a special case of how to periodically stimulate a biological neural network to obtain a desired connectivity (in theory). ** Read more...**

## On the Contrasting Educations and Outcomes of Ben Franklin and Montaigne

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. ** Read more...**

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

Implementation notes for STDP learning in a network of Hodgkin-Huxley simulated neurons. ** Read more...**

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

Many existing proofs are not accessible to young mathematicians or those without experience in the realm of dynamic systems. ** Read more...**

## Building an Iron Man Suit: A Physics Workbook

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. ** Read more...**

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

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. ** Read more...**

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

And a proof via double induction. ** Read more...**

## Sound Waves

A brief overview of sound waves and how they interact with things. ** Read more...**

## Detecting Dark Matter

A brief overview of the experimental search for dark matter (XENON, CDMS, PICASSO, COUPP). ** Read more...**

## Evidence for the Existence of Dark Matter

Mass discrepancies in galaxies and clusters, cosmic background radiation, the structure of the universe, and big bang nucleosynthesis’s impact on baryon density. ** Read more...**