This book will help you to understand elementary mathematics more deeply, gain facility with creating and using mathematical notation, develop a habit of looking for reasons and creating mathematical explanations, and become more comfortable exploring unfamiliar mathematical situations.

The primary goal of this book is to help you learn to think like a mathematician in some very specific ways. You will:

Make sense of problems and persevere in solving them. You will develop and demonstrate this skill by working on difficult problems, making incremental progress, and revising solutions to problems as you learn more.

Reason abstractly and quantitatively. You will demonstrate this skill by learning to represent situations using mathematical notation (abstraction) as well as creating and testing examples (making situations more concrete).

Construct viable arguments and critique the reasoning of others. You will be expected to create both written and verbal explanations for your solutions to problems. The most important questions in this class are “Why?” and “How do you know you’re right?” Practice asking these questions of yourself, of your professor, and of your fellow students.

Model with mathematics. You will demonstrate this skill by inventing mathematical notation and drawings to represent physical situations and solve problems.

Use appropriate tools strategically. You will be expected to use computers, calculators, measuring devices, and other mathematical tools when they are helpful.

Attend to precision. You will write and express mathematical ideas clearly, using mathematical terms properly, providing clear definitions and descriptions of your ideas, and distinguishing between similar ideas (for example “factor” versus “multiple”.)

Look for and make use of mathematical structure. You will find, describe, and most importantly explain patterns that come up in various situations including problems, tables of numbers, and algebraic expressions.

Look for and express regularity in repeated reasoning. You will demonstrate this skill by recognizing (and expressing) when calculations or ideas are repeated, and how that can be used to draw mathematical conclusions (for example why a decimal must repeat) or develop shortcuts to calculations.

Throughout the book, you will learn how to learn mathematics on you own by reading, working on problems, and making sense of new ideas on your own and in collaboration with other students in the class.


This book was developed at the University of Hawai`i at Mānoa for the Math 111 and 112 (Mathematics for Elementary Teachers I and II) courses. The materials were written by Prof. Michelle Manes with tremendous assistance from lots of people.

I owe a huge debt to Dr. Tristan Holmes, who has taught the courses for years and assisted greatly on the revision and current format of the textbook. I also thank the graduate students who helped to design and develop the original iBook version of these materials: Amy Brandenberg, Jon Brown, Jessica Delgado, Paul Nguyen, Geoff Patterson, and especially Ryan Felix. Thanks to Monique Chyba, PI of the SUPER-M project (NSF grant DGE-0841223), for supporting this work, and to the UH Mānoa College of Natural Sciences and College of Education for their support as well.

Thanks also to the hundreds of Math 111 and 112 students I’ve taught over the past ten years. Your enthusiasm, energy, joy, and humor is what keeps me going.

I am grateful to all of my colleagues and professors, past and present, from whom I have learned so much about mathematics and about education. Special thanks to Dr. Carol Findell and Dr. Suzanne Chapin at Boston University, who gave me an entirely new perspective on mathematics teaching and learning.

I can never thank Dr. Al Cuoco enough for his support and intellectual leadership. I owe him more than I can say.

Unless otherwise noted, images were created by Michelle Manes using LaTeX, Mathematica, or Geometer’s Sketchpad.

Michelle Manes

Honolulu, HI

December, 2017


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Mathematics for Elementary Teachers Copyright © 2018 by Michelle Manes is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License, except where otherwise noted.