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The mathematics of language and learning
Beta
Interactive coursework

Visualizing the Mathematics Behind Machine Learning.

Learn the mathematics behind natural language processing through worked examples, interactive visualizations, and small models you can run in your browser.

Read
Worked derivations

Work through the steps behind each formula and check them against a concrete example.

Explore
Try the calculations

Drag points, adjust sliders, and train small models to see how the calculations respond.

Check
Quick checks

Use the quizzes to check your understanding. Each option includes feedback to help you work through the answer.

LM
Language Models

From counting words to training transformers: probability, gradients, embeddings, attention, and the models built on them.

AC
Advanced Concepts

Alignment, retrieval, translation, recurrent architectures, and speech: topics that build on the core language-model material but stand on their own.

In development

Computational Linguistics
is being built.

This volume will cover how computers analyze language structure and meaning, from tagging words and parsing sentences to tracking references across a passage. Lessons and interactive examples are still in development.

Sequence labeling Constituency parsing Dependency parsing Information extraction Semantic roles Sentiment lexicons Coreference Discourse coherence Conversation

About
this project

MathCore is a guide to the mathematics behind natural language processing and machine learning. The lessons pair explanations with interactive visualizations and small, runnable models, so you can try an idea as you learn it.

Train a small neural network, move a decision boundary, or generate text one token at a time. The derivations explain how the calculations work; the demos let you change the inputs and see what follows.

I built MathCore to make this material easier to study independently. You can work through the lessons in order or explore a topic that interests you. The range of topics draws on Speech and Language Processing by Daniel Jurafsky and James H. Martin; the explanations and interactive examples were developed for this site.

Visual-first Runnable models First principles Open access In-browser

MathCore in the classroom

Learning together

Snapshots from my teaching sessions at V.R.V. Matriculation High School, bringing n-gram models, vocabulary growth, and other machine learning concepts into the classroom.

2025

Tokenization, words and tokens, and minimum edit distance algorithms

Summer 2025 at V.R.V. Matriculation High School.
Introducing students to n-gram models.
Using MathCore to help students explore Herdan’s law and vocabulary growth in machine learning.
Working through vocabulary growth and Herdan’s law with the website’s interactive examples.
Teaching Levenshtein distance at the blackboard.
Working through a Levenshtein distance example with the class.

2026

N-gram models, interpolation, gradient descent, and neural networks

Explaining n-gram models to students.
Discussing n-gram models and how interpolation combines their predictions.
Teaching gradient descent with MathCore’s Gradient Descent Explorer.
Guiding students through the website’s Gradient Descent Explorer.
Contents

In the classroom