Caveat Academy course · Grades 8 to 12

The History of Mathematics

Five thousand years of mathematics, worked out by hand: tally bones, Egyptian doubling and Babylonian base 60, Greek proof from Pythagoras to Archimedes, zero, numerals and algebra from India, China and Baghdad, symbols, logarithms, coordinates and chance, calculus with Euler and Gauss, and the strange geometries and open problems of today. Six modules at your own pace, with a check in every lesson and a certificate at the end.

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$19one-time · every device you sign in on
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Included with the Caveat Academy pass and Whole-school licenses. Or get it in the STEM explorations set, $49 for 9 items.

6 modules · 18 lessonsabout 12 hours$19

Six modules, eighteen lessons

Each module ends with a practice set that tracks mastery, an explain-it-back prompt, a short written response and a module exam. The final opens when every module is proficient; passing it earns a certificate with a verification code.

  1. Module 1: The first numbers

    Compare tally, Egyptian and Babylonian number systems, and compute with Egyptian doubling, unit fractions and base 60.

    • Counting before writing
    • Egyptian numbers and fractions
    • Babylonian base 60 and place value
    about 2.1 h

  2. Module 2: Greek mathematics and the idea of proof

    Explain how Greek mathematicians introduced proof, and reproduce arguments of Pythagoras, Euclid, Archimedes and Eratosthenes.

    • Pythagoras and the first proofs
    • Euclid's Elements
    • Archimedes and Eratosthenes
    about 2.1 h

  3. Module 3: Zero, numerals and algebra

    Trace the development of zero, place value numerals and algebra in India, China and the Islamic world, and solve problems with their methods.

    • India: place value and zero
    • China: counting rods and the Nine Chapters
    • The House of Wisdom and the road to Europe
    about 2.1 h

  4. Module 4: New tools: symbols, logarithms, coordinates and chance

    Explain how symbols, logarithms, coordinates and probability changed mathematics, and calculate with each.

    • Writing algebra with symbols
    • Logarithms
    • Coordinates and probability
    about 2.1 h

  5. Module 5: Calculus and the age of Euler and Gauss

    Explain the ideas of calculus and the contributions of Newton, Leibniz, Euler and Gauss, and apply Euler's network rule and Gauss's sum.

    • Newton, Leibniz and calculus
    • Leonhard Euler
    • Carl Friedrich Gauss
    about 2.1 h

  6. Module 6: Modern mathematics

    Evaluate non-Euclidean geometry, computer proofs and open problems, and distinguish evidence from proof.

    • New geometries
    • Mathematicians who changed the field
    • Computers and unsolved problems
    about 2.1 h

Try it before you start

This is how every lesson works: step through a worked example at your own pace, then answer a check and get the reason right away. Nothing here is saved; the course keeps your progress.

Module 3 · The House of Wisdom and the road to Europe

Worked example: completing the square, al-Khwarizmi's way

  1. His famous example is x² + 10x = 39.

A lesson check from each module

The checks need JavaScript turned on. The course itself works the same way.

What you get

Enrichment · STEM Explorations$19 · included with the Academy pass and Whole-school licenses

What you finish with
A finished course in the history of mathematics: you can multiply the Egyptian way, read Babylonian base-60 numbers, follow the proofs of Pythagoras and Euclid and the measurements of Archimedes and Eratosthenes, explain how zero, our numerals and algebra traveled from India and Baghdad to Europe, multiply with logarithms and split a stopped game by probability, sum a series like Gauss and test a network like Euler, and explain why checking examples is not proof. Passing the final earns a Caveat certificate of completion.
Who it is for
Students in grades 8 to 12 who like math, history or puzzles, students who want to see where school mathematics came from, and homeschool families who want a complete enrichment unit. The math is arithmetic, fractions and basic algebra; no calculus is required.
Format
An online self-paced course: 6 modules and 18 lessons with worked examples you reveal step by step, 150 practice and exam items (multiple choice, numeric, fill-in and ordering), an explain-it-back prompt and a rubric-checked short response in every module, module exams with one retake, a 24-question final and a certificate with a verification code. About 12 hours in all; works on a phone or a computer.
Access
$19 one-time, yours on every device you sign in on; the course opens once you are signed in with it. Included with the Caveat Academy pass and Whole-school licenses. Also in the STEM explorations set ($49) and the Enrichment Suite pass ($119), and 1 more bundle. Progress is saved in this browser on this device, and the certificate page lets you download your record.
Where it lives
The STEM Explorations space in Caveat Enrichment.
Support
Access, a receipt, a refund or a correction to the course: team@caveatacademic.com.

How it works

  1. Open the course and start Module 1. Each lesson takes about 25 to 35 minutes; two or three a week finishes the course in about a semester.
  2. Answer the checks as you read. Every answer shows the reason, and a miss links you back to the lesson that covers it.
  3. Finish each module with its practice set and exam. Proficient on every module opens the final; 70 percent passes it.
  4. Print or download your certificate and record from the certificate page.

Start the course

Sources and accuracy

The lessons are original writing by Caveat Academic Services. Facts were checked against these free, open references, which are also good places to read further.

  1. MacTutor History of Mathematics archive, University of St Andrews: mathshistory.st-andrews.ac.uk
  2. Yale Babylonian Collection, YBC 7289: peabody.yale.edu
  3. The British Museum, the Rhind Mathematical Papyrus: britishmuseum.org
  4. Clay Mathematics Institute, the Millennium Prize Problems: claymath.org
  5. Euclid's Elements, online edition by David E. Joyce, Clark University: mathcs.clarku.edu

The History of Mathematics is a self-paced enrichment course from Caveat Academic Services. It does not replace a school mathematics course and awards no credit. Dates for ancient objects and people are approximate and labeled as such; where historians disagree, as about the Ishango bone or famous anecdotes, the lessons say so. Support: team@caveatacademic.com.