Using Math Games and Word Problems to Increase the Math Maturity of K-8 Students David Moursund Robert Albrecht ABOUT THE AUTHORS Dr. David Moursund After completing his undergraduate work at the University of Oregon, Dr. Moursund earned his doctorate in mathematics from the University of Wisconsin-Madison. He taught in the Mathematics Department and Computing Center at Michigan State University for four years, before joining the faculty at the University of Oregon. There he had appointments in the Math Department and Computing Center, served six years as the first head of the Computer Science Department, and spent more than 20 years working in the Teacher Education component of the College of Education. A few highlights of his professional career include founding the International Society for Technology in Education (ISTE), serving as its executive officer for 19 years, establishing ISTE’s flagship publication, Learning and Leading with Technology, serving as the Editor in Chief for more than 25 years, and working as major professor or co-major professor for 75 doctoral students. Dr. Moursund has authored or coauthored more than 50 academic books and hundreds of articles. Many of these materials are now available free on his Website. He has presented several hundred keynote speeches, talks, and workshops around the world. More recently, he founded Information Age Education (IAE), a non-profit organization dedicated to improving teaching and learning by people of all ages and throughout the world. IAE currently provides free educational materials through its Wiki, a free newsletter published twice a month, and a blog. Robert Albrecht A pioneer in the field of computers in education and use of games in education, Robert Albrecht has been a life-long supporter of computers for everyone. He was instrumental in helping bring about a public-domain version of BASIC (called Tiny BASIC) for early microcomputers. Joining forces with George Firedrake and Dennis Allison, he co-founded People’s Computer Company (PCC) in 1972, and also produced and edited People's Computer Company, a periodical devoted to computer education, computer games, BASIC programming, and personal use of computers. Albrecht has authored or coauthored over 30 books and more than 150 articles, including many books about BASIC and educational games. Along with Dennis Allison, he established Dr. Dobb’s Journal, a professional journal of software tools for advanced computer programmers. He was involved in establishing organizations, publications, and events such as Portola Institute, ComputerTown USA, Calculators/Computers Magazine, and the Learning Fair at Peninsula School in Menlo Park, California (now called the Peninsula School Spring Fair). Albrecht's current adventures include writing and posting instructional materials on the Internet for free use, tutoring high school and college students in math and physics, and running HurkleQuest play-by-email games for Oregon teachers and their students. This book is available for purchase through the Math Learning Center: The Math Learning Center P.O. Box 12929 Salem, OR 97309-0929 Phone: 503-370-8130 Toll Free: 800-575-8130 Fax: 503-370-7961 http://www.mathlearningcenter.org Copyright © 2011 David Moursund and Robert Albrecht Table of Contents Preface and Introduction ..................................................................... 1 Chapter 1: WordsWorth Games ......................................................... 7 Chapter 2: Introduction to Math Maturity ...................................... 17 Chapter 3: Introduction to Math Intelligence .................................. 35 Chapter 4: Math Cognitive Development ......................................... 49 Chapter 5: The Language of Mathematics ....................................... 61 Chapter 6: Some Learning Theory ................................................... 71 Chapter 7: Math Word Problems ..................................................... 81 Chapter 8: Math Games and Puzzles ................................................ 95 Chapter 9: Dice, Coins, and Chance ............................................... 109 Chapter 10: Place Value Games ...................................................... 123 Chapter 11: Word Problems Using Dominoes ............................... 143 Chapter 12: Factor Monster ............................................................ 155 Chapter 13: The Game of Pig .......................................................... 165 Chapter 14: More Games and Puzzles ............................................ 179 Chapter 15: Final Remarks .............................................................. 193 Appendix 1: Make It & Take It, and Blackline Masters ............... 197 Appendix 2: Some Free Resources .................................................. 205 Appendix 3: Some Not-Free Resources ........................................... 209 Bibliography ...................................................................................... 211 Index ................................................................................................... 217 Preface and Introduction This book is mainly intended for preservice and inservice teachers of math at the PreK-8 levels, and parents and other caregivers of such students. The goal of this book is to help improve the informal and formal math education of PreK-8 students. The emphasis is on providing students with learning environments that help to increase their levels of math maturity. The learning environments stressed in this book include an emphasis on communication in the language of mathematics, the use of math-oriented games, and the use of math word problems. The next paragraph is a short definition of a mathematically mature adult. The level of math maturity described comes from years of appropriate informal and formal education and math- related experiences. Later parts of the book will provide a more detailed definition of math maturity and more detail about possible roads leading to an increased level of math maturity. Mathematically mature adults have the math knowledge, skills, attitudes, perseverance, and experience to be responsible adult citizens in dealing with the types of math-related situations, problems, and tasks that occur in the societies and cultures in which they live. In addition, a mathematically mature adult knows when and how to ask for and make appropriate use of help from other people, from books, and from tools such as computer systems and the Internet. Scattered throughout this book you will find short Math Maturity Food for Thought subsections such as the one given below. Each asks you to reflect on a particular idea or presents you with some problems that you and/or your students might explore. Such reflection, introspection, and problem-solving challenges are an important aid to learning and to increasing one’s level of math maturity. If you are using this book in a course, these subsections can be used in small group and/or large group discussions and sharing. This can be done in a face-to-face environment or via use of telecommunications systems. Math Maturity Food for Thought. It’s A-OK to have one’s income taxes prepared by an expert or for a person to make use of income tax preparation software. The income tax system and tax law in the United States are frightfully complex and include substantial changes from year to year. It is not possible for a person to gain and maintain a high level of personal expertise in every type of problem area that adults must routinely deal with. Thus, knowing when and how to ask for math-related help (from a person or from a machine) —and how to make effective use of such help—is an important aspect of math maturity. Think about the math that you do for yourself in your everyday life, and the math that you do with the help of other people and/or with the help of calculators, computers, GPS, and so on. Do you consider yourself to be a mathematically mature adult? What could you do to increase your level of math maturity? Math is a vast and steadily growing discipline. Moreover, math is an important component of science, technology, engineering, and many non-science disciplines. As an example, think about 1 the complexities involved in identifying, understanding, and attempting to deal with various aspects of global sustainability. These immensely difficult problems not only involve science, technology, engineering, and math (STEM), they also involve governments and politics, businesses and economies, and the lives of the people and other species on earth. Common Core State Standards Initiative In March 2010 the Common Core State Standards Initiative released a draft of its proposed standards, and this set of standards has been widely adopted. See http://www.corestandards.org/. Quoting from the proposed math standards (with bold face added to highlight emphasis on mathematical maturity): The draft Common Core State Standards for Mathematics endeavor to follow such a design, not only by stressing conceptual understanding of the key ideas, but also by continually returning to organizing principles such as place value or the laws of arithmetic to structure those ideas. The standards in this draft document define what students should understand and be able to do. Asking a student to understand something means asking a teacher to assess whether the student has understood it. But what does mathematical understanding look like? One hallmark of mathematical understanding is the ability to justify, in