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Introductory · Text-based, self-paced · Open access

What Mathematicians Do

Tour twelve branches of mathematics: the problem behind each, how it reasons, and what is still unsolved.

Knowledge InSightVetting in progress

5 courses 12 modules 24 lessons 37h 2m of material

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About this program

Mathematics is not one subject. It is a family of fields, each built to answer a different kind of question: how numbers behave, what stays the same when a shape is stretched, how things change, how likely an outcome is, and what can be proved or computed at all. Most people meet only a slice of it in school, and never see how the slices fit together or why anyone pursues them.

This program gives you a guided tour of twelve branches. For each one, you'll see the problem that brought it into being, the way it reasons, what it is used for today, and what remains unsolved. You'll try a small problem in each branch's own style, using nothing beyond high-school algebra, and you'll meet open questions, some with a million-dollar prize attached.

The aim is orientation, not training. By the end you'll know what each branch is for, how it connects to work and technology you already know, and which one you'd most like to study next.

What you will be able to do

  • Describe what each of twelve branches of mathematics studies and the problem that gave rise to it
  • Work a short problem in the characteristic reasoning style of each branch
  • Explain where each branch is used in science, technology, and everyday life
  • Identify current open problems in each branch and choose one to explore further

How to progress through a program

Every lesson has the same shape, so you can settle into it and spend your attention on the material.

  1. A motivating readingThe problem that started a branch, or a use you have met without knowing it, in three minutes.
  2. Four concept readingsThe ideas themselves, one at a time, each put into words before any symbol appears.
  3. A worked example or walkthroughOne problem or application, followed step by step.
  4. A Guided ConversationA coached problem or a reflective conversation you run in your own AI account.
  5. An optional problem set or activityShort problems that need only high-school algebra, with worked solutions.
  6. A knowledge checkFive questions, ungraded, retry as often as you like.

The last lesson of each module adds a graded quiz with a target score of 80 percent.

Courses

  1. Course 1 Number and Structure This course covers three branches that study numbers and the patterns they form. You'll see how number theory grew from questions about primes into the mathematics that secures online payments, how… 3 modules 6 lessons 8h 58m 0%
  2. Course 2 Space and Shape This course covers two branches that study space. You'll see how geometry became the first field built on axioms and proof, how a doubt about one of Euclid's assumptions led to new geometries that… 2 modules 4 lessons 6h 22m 0%
  3. Course 3 Change and Motion This course covers the mathematics of change. You'll see how calculus answered two ancient puzzles, instantaneous speed and the area under a curve, and how its methods were later made rigorous… 2 modules 4 lessons 6h 22m 0%
  4. Course 4 Chance and Data This course covers the mathematics of uncertainty. You'll see how a gambling dispute led Pascal and Fermat to put numbers on chance, and how probability grew into the basis of insurance, genetics… 2 modules 4 lessons 6h 22m 0%
  5. Course 5 Logic, Proof, and Computation This course covers the branches that study reasoning itself. You'll see how logic and set theory gave mathematics its foundations and then found limits on what those foundations can prove. You'll… 3 modules 6 lessons 8h 58m 0%

Who this is for

Curious adults who stopped formal math after high school, and professionals who meet mathematical fields at work (data, machine learning, finance, research) and want to know what those fields are. You can work with variables, solve a linear equation, and read a simple formula or the graph of a line; nothing beyond that is assumed.

Knowledge InSightVetting in progress

Help us vet this program

You are reading it closely anyway. If something looks wrong, please tell us to help improve the program.

  • Check claims against the linked sources. Every reading names where its ideas come from. If a passage misstates a source, or attributes something to the wrong thinker, say so and point to the item.
  • Report broken or wrong links. Free editions move and paywalls change. A dead link is worth reporting even when the text around it is fine.
  • Flag quiz questions that do not hold up. An answer you can defend that is marked wrong, an option with no good answer, or feedback that misses the point.
  • Suggest improvements. A passage that reads as muddled, an example that would land better, a primary text we should have pointed you to.

Report an issue or suggest an improvement