Quantitative Reasoning & Computation
Our math progression offers two or three options for each year as a core curriculum. The Art of Problem Solving sequence, beginning with Beast Academy in the elementary is the most rigorous, but it is not the best fit for all students. There are other math programs that, when paired with engaged teaching, can provide a truly solid foundation in math. The Supplementary Materials section includes books, games, and other resources that cover programming, logic, and math topics beyond the core curriculum. In the Other Resources section, you will find our recommendations for fiction and nonfiction texts that address math, computer science, and related topics. We will edit and add to these list as we come across resources we think are worth sharing.
Early Primary (years 1 & 2)
- Option 1: Miquon Math: A wonderful, hands-on early math curriculum that uses cuisenaire rods to introduce and teach concepts.
- Option 2: Math Mammoth 1-2: A more traditional, but very solid math curriculum. The text is written to the student, and there is ample review for those who need it.
- Option 3: Beast Academy Grade 2: A discovery-based math curriculum. The comic-book guides and cartoon characters belie the often intensely challenging content.
- Mobius Noodles: Creative, hands-on math activities that introduce advanced concepts to young children
- Family Math: Math-related games and activities
- Hands-On Equations: Introduces children to algebraic concepts in an easy-to-grasp intuitive way. To get started, you would need the basic set and the verbal problems book.
- Lollypop Logic and Primarily Logic: A gentle introduction to formal logic.
- Hands-on Logic Games such as Rush Hour Jr., Gravity Maze, and others
- Gattegno Math: a series of short texts that focuses on mathematical processes; uses cuisenaire rods to represent abstract concepts in tangible ways.
- Primary Grade Challenge Math: Supplementary word problems
- Base-10 Blocks: Invaluable for teaching place value, carrying, and borrowing
- Cuisenaire Rods: Our favorite math manipulative (and building toy)
- Robot Turtles: An introduction to programming game
- Bed Time Math: Short, math-related bedtime stories.
- The Adventures of Penrose the Mathematical Cat by Theoni Pappas
- Annos Counting Book by Mitsumasa Anno
- The Librarian Who Measured the Earth by Kathryn Lasky
- One Grain Of Rice: A Mathematical Folktale by Demi
- The History of Counting by Denise Schmandt-Besserat
- Grandfather Tang’s Story by Anne Tompert (and a book of Tangram puzzles!)
Late Primary (years 3 & 4)
- Option 1: Math Mammoth 3-4: A more traditional, but very solid math curriculum. The text is written to the student, and there is ample review for those who need it.
- Option 2: Beast Academy 3-4: A discovery-based math curriculum. The comic-book guides and cartoon characters bely the often intensely challenging content.
- Hands-On Equations: Work though the various levels in the verbal problems book
- Logic Liftoff Series: A decent introduction to formal logic, including analogies, syllogisms, sequences, deductive reasoning, inference, truth-values, and logical notation.
- This is Not A Maths Book: Wonderful book that introduces advanced math concepts through art. If your kids enjoy it, you can check out the second volume.
- Primary Grade Challenge Math or Upper Elementary Challenge Math: Supplementary word problem
- Amazing Math Projects You Can Build Yourself
- Code on the Brink
- Coding Games in Scratch or Coding Projects in Scratch: Easy to use introduction to Scratch programming.
- Building Thinking Skills: Verbal and Building Thinking Skills: Figural: There are three different levels of these books, but the middle/high school level linked here is suitable for bright elementary students.
- Upper Elementary Challenge Math
- Logic Games such as Architecto and Equilibrio, Quorridor, Cathedral, and others
- Murderous Maths
- Sir Cumference books
- Mathematicians Are People, Too: Stories from the Lives of Great Mathematicians by Luetta and Wilbert Reimer
- What’s the Point of Math
Early Intermediate (years 5 & 6)
- Option 1: Math Mammoth: Levels 5 & 6
- Option 2: Art of Problem Solving: Beast Academy 5 and Pre-Algebra: The Pre-Algebra book is a big step up from the Beast Academy series in terms of format and approach, but not in terms of content. If students completed all levels of Beast Academy, they should be well prepared to work at this level. Keep in mind that even very strong math students may take more than one academic year to complete the book if they do every problem and exercise.
- Option 3: Arbor Algebra Sequence: Jousting Armadillos: A pre-algebra book based on Harold Jacobs’s well-respected algebra text. It is not as challenging as the AOPS texts, but still offers a solid introduction to algebraic concepts and may appeal to students who are more inclined toward the humanities.
- Patty Paper Geometry: A hands-on introduction to geometric constructions.
- The Moscow Puzzles: 359 Mathematical Recreations by Boris A. Kordemsky: The puzzles here are of a different type than those found in the AOPS books. They require logic, math, and out-of-the-box thinking.
- Upper Elementary Challenge Math: A useful supplement for students who need extra practice with word problems.
- Code Academy: Free resources for learning to code.
- Arduino starter kit (such as this one) and a guide using the Arduino such as Arduino Cookbook
- Coding Games in Python or Invent Your Own Computer Games in Python
- Become a Problem Solving Genius: A handbook of math problem solving strategies
- The History of Counting by Denise Schmandt-Besserat
- The Number Devil: A Mathematical Adventure by Hans Magnus Enzensberger
- Scammed By Statistics by Edward Zaccaro
Late Intermediate (years 7 & 8)
- Option 1: Math Mammoth 7 followed by Foerster’s Algebra: Level 7 is the last book in the Math Mammoth sequence, after which, the student should be prepared for high school level Algebra. Maria Miller, the author of the Math Mammoth series, has a very useful article about Algebra 1 options here.
- Option 2: Art of Problem Solving Introduction to Algebra and Introduction to Geometry: Do not be surprised if even strong math students take more than one academic year to complete these texts, particularly if they diligently and carefully work through all of the problems.
- Option 3: Arbor Algebra Sequence: Coconuts and Crocodiles and Chuckles the Rocket Dog: An engaging, text-heavy, solid two-year sequence that covers Algebra 1 and part of Algebra 2. They may be most appealing to students who are more inclined toward the humanities.
- James Madison Critical Thinking Course: An introduction to formal logic and critical thinking that uses crime-solving scenarios to develop skills.
- Real World Algebra: Understanding the power of Mathematics by Ed Zaccaro
- Art of Problem Solving: Introduction to Counting and Probability
- Art of Problem Solving: Introduction to Number Theory
- Raspberry Pi starter kit such as this one and a guide such as Raspberry Pi: A Comprehensive Beginner’s Guide
- Python Programming for the Absolute Beginner
- Conned Again Watson: Cautionary Tales of Logic Math and Probability! by Colin Bruce
- Winning Ways for Your Mathematical Plays: Volume 1 by John Conway and Richard Guy (if you like it, you can check out Volume 2 and Volume 3 as well.)
- Crimes and Mathdemeanors by Leith Hathout
- The Code Book: The Science of Secrecy from Ancient Egypt to Quantum Cryptography by Simon Singh
Secondary Years (9 – 12)
- Art of Problem Solving Introduction to Geometry
- Geometry: Seeing, Doing, Understanding by Harold Jacobs A solid, engaging geometry course for students who do not need or enjoy the challenge of the AOPS books.
- Option 1: Art of Problem Solving Intermediate Algebra
- Option 2: Foerster’s Algebra and Trigonometry: Functions and Applications: A solid second-year algebra textbook.
- Art of Problem Solving Intermediate Counting and Probability
- Art of Problem Solving Pre-Calculus
- Art of Problem Solving Calculus
- Formal Logic text such as Simple Formal Logic with Common-Sense Symbolic Techniques by Arnold vander Nat or An Introduction to Formal Logic by Peter Smith
- The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems by Martin Gardiner
- The Book of Numbers by John Conway and Richard Guy
- The Symmetries of Things by John Conway, Heidi Burgiel, and Chaim Goodman-Strauss
- Fermat’s Enigma: The Epic Quest to Solve the World’s Greatest Mathematical Problem by Simon Singh
- The Calculus Gallery: Masterpieces from Newton to Lebesgue by William Dunham
- Four Colors Suffice: How the Map Problem Was Solved by Robin Wilson
- Problem Solving Through Recreational Mathematics Bonnie Averbach and Orin Chein
- Mathematics: The Science of Patterns: The Search for Order in Life, Mind and the Universe by Keith Devlin
- Secrets and Lies: Digital Security in a Networked World by Bruce Schneier
- The Annotated Turing: A Guided Tour Through Alan Turing’s Historic Paper on Computability and the Turing Machine by Charles Petzold
- Things to Make and Do in the Fourth Dimension by Matt Parker
- Theory of Games and Economic Behavior by John Von Neumann
- How to Solve It: A New Aspect of Mathematical Method by G. Polya
- In Pursuit of the Unknown: 17 Equations That Changed the World by Ian Stewart