Training with Gradients and Backpropagation
A worked explanation of loss, gradient descent, the chain rule, and what backpropagation actually computes.
Read featured guide →Guides that teach the idea, not just the answer.
Read instructor-written explainers for mathematics, CS/AI, science, and competition preparation. For classroom-ready PDFs, use the separate Decks and Practice libraries.
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41 guides and articles
PCA: Reducing Dimensions Without Guessing
An introduction to centering, variance, principal directions, explained variance, and the information a projection loses.
Read More →August 3, 2026
Neural Networks as Composed Functions
A mathematical view of layers, weights, activations, and why nonlinear functions give neural networks expressive power.
Read More →July 29, 2026
Nearest Neighbors and Embedding Evaluation
A careful workflow for testing whether an embedding preserves useful local relationships instead of trusting a visualization.
Read More →July 22, 2026
Embeddings: Turning Objects into Coordinates
A practical guide to feature representations, embedding geometry, and what it means for nearby vectors to preserve a useful relati...
Read More →July 16, 2026
Distance and Cosine Similarity
How Euclidean distance and cosine similarity compare vectors, when their rankings disagree, and how normalization changes the prob...
Read More →July 10, 2026
Modular Arithmetic: Clocks, Remainders, and Large Powers
A COSMO guide to congruences, cycles, divisibility tests, and why remainder thinking turns huge problems into small ones.
Read More →June 22, 2026
Invariants: The Thing That Refuses to Change
A COSMO guide to invariants, coloring arguments, parity, and why some impossible problems can be solved by tracking one stable fea...
Read More →June 22, 2026
Euler Characteristic: Why V - E + F Keeps Showing Up
A long COSMO guide on the Euler characteristic, planar graphs, polyhedra, and why a simple-looking formula contains a serious stru...
Read More →June 20, 2026
The Pigeonhole Principle Without the Fancy Name
A beginner-friendly introduction to the pigeonhole principle using socks, birthdays, remainders, and guaranteed outcomes.
Read More →June 19, 2026
Graph Theory: The Math of Connections
A first COSMO-style lesson on vertices, edges, paths, degrees, and why simple drawings can solve real problems.
Read More →June 18, 2026
Graph Theory: When a Drawing Becomes a Proof
A long-form student guide to vertices, edges, paths, degree, Euler routes, trees, and why graph theory is more than pretty diagram...
Read More →June 17, 2026
Probability Through Games
A COSMO-style lesson note on expected value, fair games, and why probability is more than counting outcomes.
Read More →June 15, 2026
Counting Without Double Counting
A practical introduction to combinatorics through cases, complements, symmetry, and careful organization.
Read More →June 14, 2026
Monoids: One Structure Behind Sums, Strings, and Functions
An accessible abstract algebra guide showing how closure, associativity, and identity connect arithmetic, coding, transformations,...
Read More →June 12, 2026
Sequences and Recursion
A guide to arithmetic, geometric, and recursively defined sequences through patterns students can extend before formalizing.
Read More →June 11, 2026
Proof Writing for Beginners
How to move from examples to explanation using contradiction, induction, and direct proof.
Read More →June 10, 2026
Logic Puzzles Without Guessing
How to turn a logic puzzle into a system of constraints instead of a pile of guesses.
Read More →June 7, 2026
Optimization: Choosing the Best Possible Option
A math circle introduction to optimization through routes, constraints, tradeoffs, and objective functions.
Read More →June 6, 2026
Applied Optimization: What Does Best Actually Mean?
A COSMO applied math guide to objectives, constraints, greedy algorithms, tradeoffs, and why optimization depends on what you choo...
Read More →June 5, 2026
Generating Functions: Turning Counting Problems into Algebra
A long COSMO guide to generating functions, showing how coefficients can store counting information and make combinatorics feel le...
Read More →June 5, 2026
Game Theory: Why Incentives Change Behavior
A beginner-friendly look at strategy, payoffs, equilibrium, and why systems often produce surprising outcomes.
Read More →June 2, 2026
Estimate Before You Calculate
A short guide on teaching students to predict size, direction, and reasonableness before doing exact arithmetic.
Read More →May 30, 2026
When Greedy Works, When Greedy Fails
A long COSMO guide to greedy algorithms, counterexamples, exchange arguments, and why an intuitive local choice still needs proof.
Read More →May 29, 2026
Casework That Does Not Become Chaos
How to split a contest problem into cases that are complete, non-overlapping, and actually useful.
Read More →May 24, 2026
Invariants: The Thing That Does Not Change
A guide to invariant thinking, parity, coloring arguments, games, and impossibility proofs for math circle students.
Read More →May 21, 2026
Expected Value: The Average You Can Use Before Anything Happens
A long COSMO guide to expected value, linearity, games, and why probability can reason about a process before the outcome is known...
Read More →May 18, 2026
Optimization: Choosing the Best Possible Option
A first math circle lesson on optimization using routes, constraints, objective functions, and tradeoffs.
Read More →May 18, 2026
Proof by Induction: The Domino Argument With Teeth
A careful student guide to induction, base cases, inductive assumptions, common mistakes, and why examples are not proof.
Read More →May 9, 2026
How to Run a COSMO Math Circle Session
A practical teaching note for chapter leaders on pacing, discussion, questioning, and keeping students engaged.
Read More →May 9, 2026
Counting Without Double Counting
A long-form COSMO guide to cases, complements, symmetry, bijections, and the bookkeeping habits behind combinatorics.
Read More →April 26, 2026
Monoids: A Structure Hiding in Plain Sight
A COSMO-style explanation of monoids through numbers, strings, functions, and combining data.
Read More →April 24, 2026
When the Greedy Choice Fails
A short problem guide on local choices, global outcomes, and why optimization needs proof.
Read More →March 16, 2026
A Gentle Introduction to Number Theory
Basic ideas in number theory for budding problem solvers: divisibility, primes, modular arithmetic, and why they matter.
Read More →November 2, 2025
Navigating STEM as an Underrepresented Minority
Advice and strategies for minority students to thrive in STEM fields.
Read More →October 25, 2025
Navigating STEM as a First-Generation Low-Income Student
Strategies and advice for first-gen students pursuing STEM careers.
Read More →October 5, 2025
Signaling Theory in Economics and Biology
An introduction to signaling theory, its mechanisms, and applications in different fields.
Read More →September 20, 2025
Base Conversion and Modular Arithmetic
Guide to converting number bases and working with modular arithmetic for competitions.
Read More →September 10, 2025
Exploring the Cosmos: An Introduction to Astrophysics
A beginner-friendly guide to the wonders of astrophysics, from stars and galaxies to black holes and dark matter.
Read More →August 28, 2025
The Art of Problem Solving: AMC Strategies
Learn key strategies and tips for tackling the American Mathematics Competitions (AMC), one of the most challenging high school ma...
Read More →August 25, 2025
The Beauty of Euler's Identity
A deep dive into one of the most profound equations in mathematics, connecting five fundamental constants.
Read More →August 20, 2025