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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...

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...

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...

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...

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...

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...

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...

Decision Trees

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

K-Means Clustering

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

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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 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 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 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...

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...

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...

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...