Mathematics for Elementary Educators

Number systems, the four operations, number theory, and fractions

Author

Emil Sargsyan

Published

September 7, 2026

Preface

This book is written for students preparing to teach mathematics in elementary school. It is not a review of the arithmetic you already know how to do. It asks a harder question: why does it work, and how would you explain it to a child meeting it for the first time?

That difference shapes everything here. We spend a chapter on number systems the world has since abandoned — Roman, Egyptian, Babylonian, Mayan, Gumatj — not out of antiquarian interest, but because working in an unfamiliar base puts you back in the position of your students. Place value is invisible to an adult who has used base ten since childhood. It becomes visible again in base five.

The four binary operations get the same treatment. Each is developed through its models and interpretations before any algorithm appears, because the algorithms are the end of the story rather than the beginning. A teacher who knows only the procedure can show a child what to do, but cannot help when the child asks why.

What is here

  • Philosophy of mathematics education — the pedagogical stance the rest of the book takes, including Liping Ma’s work and the California Common Core standards.
  • Numerals and number systems — place value, seen from outside base ten.
  • The four binary operations — models, interpretations, properties, and only then the classic algorithms.
  • Bar diagrams — forming and solving word problems.
  • Integers, exponents, and order of operations.
  • Number theory.
  • Fractions — through division of fractions and the reciprocal.

Each chapter ends with exercises.

A note on this edition

The text was written and taught first as a course document. This web edition is the working copy from here on, and it is being revised as it is taught. Chapters will change under your feet.

Licence

Copyright 2024 Emil Sargsyan for Los Angeles Mission College. This book is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International Licence.