The Foundation of Number Sense: Introducing subQuan: Seeing Numbers Intro Rev3B
Download as PPTX, PDF0 likes14 views
The Foundation of Number Sense: Introducing subQuan: Seeing Numbers Intro Rev3B
The Foundation of Number Sense: Introducing subQuan: Seeing Numbers Intro Rev3B
- 1. Seeing Numbers Introducing subQuan Back cover cover The Foundation Of Number Sense Seeing Numbers This book has three powerful ideas: • Numbers have shapes • You can organize those shapes as subQuans. • patterns in subQuans reveal concepts in algebra and calculus. Discover the HOT new “subQuan” world and the joy of learning number sense in a new, visual, instinctual way. The book’s goal is to address problems in math education through a very uncommon solution. This solution is uncommon because very few people look for solutions where these were found: In the beginning, at the very foundation of what numbers are. Seize the opportunity of a visual perspective, cutting to the core of understanding numerosity. Along the way you will discover new keys to algebra and calculus. These keys will become so natural to you that you will feel they were always there; you just never looked until now! “We are giving kids the ability to turn abstract symbols into something they can see.” Anna-Marie Robertson, a visual learner “Mathematics has been standardized on base ten. It’s like standardizing solely on blue for art.” Cooper Patterson, Cognitive Engineer “When one learns how to ride a bicycle, they understand why training wheels were invented, but they never desire to use them again. Little green cubes are like that.” Rebecca Patterson, Secondary Math Teacher Danielle Cooper Patterson, Rebecca Lynne Patterson, and Anna-Marie Robertson SubQuan on the front cover: 210435 Read the book to find out why. This book is about discovery! “If there is to be knowledge, there must first be something to be known. In other words, knowledge is essentially discovery, or the finding of what already is.” -H. A. Pritchard, Front Cover Back Cover
- 2. ISBN, copyright, publisher, whatever goes here Seeing Numbers Introducing subQuan By D. Cooper Patterson Rebecca L. Patterson Anna-Marie Robertson Dedication This book is dedicated to those with childlike hearts. May they have eyes to see and ears to hear. “But blessed are your eyes for they see, and your ears for they hear; for assuredly, I say to you that many prophets and righteous men desired to see what you see, and did not see it, and to hear what you hear, and did not hear it.” Matthew 13:16-17 “In short, an innumerate citizen today is as vulnerable as the illiterate peasant of Gutenberg’s time.” Lynn Arthur Steen, Professor of Mathematics at St. Olaf College July 17, 1997 “The laws of nature are but the mathematical thoughts of God.” Euclid, The Father Of Geometry 323-283 B.C. “We cannot solve our problems with the same thinking we used when we created them.” Albert Einstein, The Father Of The Theory Of Relativity 1879-1955 ii i
- 3. (Editor’s preface) Also by authors “Profound Advances in Mathematical Thinking Revealed in a Virtual World: Numbers Unleashed” in Emerging Tools and Applications of Virtual Reality in Education “Cross-Reality Math Visualization: The SubQuan System Dream Realizations” in Immersive Environments, Augmented Realities, and Virtual Worlds iv iii
- 4. Introduction Welcome! I’m Papa. Papa is the name that we called the loving, joyful grandfathers in my family for generations. While I get to introduce subQuan to you, this book isn’t about me, or any other characters from the book, or even mathematics. This really is a story about you! The story develops slowly for some, very quickly for others, but the speed makes little difference. For the joy comes from the discoveries you make, not how fast you make them. The kinds of joy you get depend on who you are, what you have done, what and who you already know, and your experiences with children and friends doing math with you. To get the maximum joy from this book, remember what it is like to be a little kid. I know that sometimes it is hard, but hey -- I’m a grandfather, and I can do it, so maybe you can, too. If you can put aside your “adult ways” and come as little children into this learning experience, the discoveries you make about numbers will have you wondering why you didn’t see them before. You might think that if you teach, you must know all there is to know about the topic being taught. That is simply not possible. The world is a wonderful place with so much to know that no one, not even Papas, can begin to grasp all its splendor. No teacher can be aware of all the new ideas that are being discussed about a topic they teach. And sometimes, the greatest ideas are just now being discovered, or are still waiting to be discovered. Maybe by you or your children! This book will introduce you to several fundamental discoveries about number sense. Moreover, the book will encourage you to recreate these discoveries yourself, and to help children do so. The methods of this book will guide you in helping those who are first learning to talk with their first number words. As a parent, teacher, or friend, you can help them deal with a foundational difficulty found in many languages when teaching the number words above nine. But this book is not only for those who work with babies and toddlers. There are some discoveries that grown-ups can share with children of all ages. We also hope we may plant some seeds of thought that might spring forth a doctoral thesis in cognitive psychology, mathematics, or education. Between toddlers and academics, there are discoveries for those wishing to know math topics that they did not understand in high school or college. This book is designed for all levels, ages two to a hundred and two, to learn, discover, and connect. And there might even be some who learn something new about themselves when they get to re-live childhood and play, yes play, with numbers. Who knows? That you can play with numbers may be one of your greatest discoveries! Enjoy! --- Papa Acknowledgements We would like to thank Maria Droujkova and Yelena McManaman for believing in us and in our work, for the opportunities DeltaStream provides for math educators, and for their compassionate encouragement throughout this adventure. Thanks to our editor, Carol Cross, for her undaunted spirit. Her inquisitiveness opened our eyes to better explanation and even our own discoveries along the way. We would also like to thank our early reviewers who greatly enriched the readability and the message: Irina, Natalia, Joshua, J. Grab, Sally, Ramona, … This section will be completed just before we go to press! Thank you for joining us. Preface It is normal for individuals throughout history to reinvent new explanations or systems. This is especially true when new discoveries have been made in the field or are fundamental to the field. Numbers and mathematics are no exception, though discoveries or revelations in the foundation of number sense have been very rare. Recently there have been two profound discoveries, one so simple that we pause and wonder how it could have been overlooked for so long. The other discovery required modern technology to confirm that the patterns in our brains were connected to our awareness of the number of things. This book applies the latter to the former. Understanding numeracy is a very powerful and freeing acquisition. This understanding is something everyone is capable of acquiring, but few have. It is similar to acquiring the ability to read, another powerful tool enabling people to share knowledge over space and time. Many people are in the dark when it comes to understanding numeracy. We believe we are now aware of the cause of this darkness. We believe we can make others aware of it and how to bring light to these people. This book is an invitation to open your eyes and see. We don’t know what we don’t know. All of us have acquired some knowledge that is rather unique to us. If the knowledge is helpful we need to share it with one another. Part of the sharing involves simply making others aware that there may be something they do not know. Once they are aware of its existence, then the other part of sharing is letting them know what you have discovered so far. Wish us luck. We are all in this together. May God bless us with understanding and wisdom. vi v
- 5. Virtual Worlds Best Practices Exhibit Boise State University Innovation Pavilion @ EdTech Island The Sushi Bar Live Demo @ EdTech Island Virtual Worlds Best Practices Workshop First Generation subQuan Cooper’s Lab at University of Washington The joys of SL creation! Designing Hello World Cooper Rebecca Alexsis Papa Jason Susie Let us give you a little background about our characters, the authors. Cooper, Rebecca, and Anna-Marie (Alexsis) met in a research class on teaching math in a virtual world online while both Rebecca and Anna-Marie were getting their Master’s degree. Cooper also joined the class so that she could find a better way to show people what she was discovering about numbers. All three of us had to create an avatar to represent ourselves and attend class online, virtually. You will see us throughout this book talking about what we love and teaching deeper meanings and patterns. We are glad to meet you! We really existed in a virtual world. We moved around and spoke to one another through computers just like in real life. In this space, we studied things about numbers that are very difficult and sometimes impossible to see in real life. We were not hindered by gravity, space, and time. Through programming we manipulated thousands of unit cubes into shapes our eyes could see very quickly. This book is the beginning of sharing our own discoveries about how we, as humans, were meant to understand numbers. This foundational start ties into the way we believed God designed us and helps to make math and number symbols more comprehensible. (See Patterson & Robertson research referenced in the Appendix.) Papa and the kids are also avatars that help us teach others about the joys of discovery. If you can put aside your ‘adult ways’ and come as little children into this learning experience, the discoveries you will make about numbers will have you wondering why you didn’t see it before. In the virtual world we actually have an interactive display representing number that we call the ‘Oh my God, I see it!’ board. Many eyes have been opened so far and we are hoping that you will see it, too. Enjoy! About the Characters viii vii
- 6. How this book is organized: Objects Needed & Play Preparation – Discoveries – Action – Please take the time to read through the instructions and enjoy the play. Example – Extra Instructions – Reality Check – Informational – Kid Fun – Research – This banner indicates a new set of discoveries. I’m Susie. I’m in third grade. I’m good at helping others so I get to show you my examples for the different Discoveries so you can see if you’re understanding it like me. Outlined boxes introduce what is needed for discoveries in the chapter. You can always substitute other objects for the ones listed, but you may need to modify the discoveries accordingly: i.e. make a new measuring device. I’m Susie’s big brother Jason. I get to add my own thoughts most of the time. I’ll tell you things that might be fun or nice to know. I am Cooper. I’m the engineer that discovered subQuan. My goal is to bring light into darkness. I am Papa. I’ll sometimes give you an opportunity to take an activity further. I like to use the wisdom I’ve gained through the years to help explain why or what you’re doing. I find that helps a lot of adults and my grandkids. I am Rebecca. I enjoy keeping up with any research related to seeing numbers and mathematics. Research bubbles show documentation supporting our findings and take a look at possible avenues of new research to be done. I will help you find clarity and make connections as you work through the discoveries in the lessons. I will encourage you to think deeper, to feel what you know, and to apply your thoughts and feelings to your new discoveries. Table of Contents: Chapter 1: Place Shapes – The concept of place and its value proved to be a leap in man’s ability to understand and manipulate numbers. However, the concept of Place Shapes provides the foundation for number sense that opens the eyes to see! Discover for yourself how place value and place came about. Page 1-1 Chapter 2: subQuan – When unit cubes are organized into shapes, and then the groups of shapes are ordered by size, and then coded symbolically, taking notice of any shapes that are missing, the result is extremely eye-opening. But referring to this as ordered-organized-shapes-converted-into-symbols-to-represent-a-number is awkward. Call them subQuans. SubQuans gives us a far-reaching perspective on the patterns in numbers that surpasses what we see from only looking at numbers in the traditional way. SubQuans are a superset of traditional numbers. Page 2-1 Chapter 3: Missing Negatives – Often real-life organizes ‘things’ into rectangular patterns. You will decompose rectangular patterns into place shapes to obtain subQuans. This opens your eyes to seeing subQuan patterns in tiles, bricks, fabric, and other everyday objects. Turning our eyes to not quite perfect rectangles opens our minds to negatives! Page 3-1 Chapter 4: Measurement – The ruler is the first number tool and helps you in two ways. First to build the place shapes and second to determine a fully composed subQuan. Page 4-1 Chapter 5: subQuan Metapatterns – A metapattern is a pattern of patterns. These patterns can be discovered from place shapes, measurements, or rectangular patterns. The metapattern of subQuans unlocks one of the powers of numbers: the power to predict! Algebra teaches this power symbolically, but with subQuan we discover the power visually which lays a solid foundation for understanding algebra. This is big. Page 5-1 Chapter 6: Change– Simply taken the change between each pair of numbers in a sequence of data can help determine the metapattern. This powerful analytic tool becomes understandable with a foundation of subQuan. Page 6-1 Conclusion – Numbers do make sense. Seeing numbers in a new visual way improves everyone’s number sense! Read it at the end of Chapter 6. Appendix 1: Digit-Speak – This appendix introduces the reader to digit-speak: A straight-forward solution to the conflict between how our brains understand simple number patterns and how some of our languages contradict those patterns. The magnitude of the conflict is very dependent on the reader’s language. Digit-speak is used throughout the book. Appendix 2: Subitizing – Subitizing is an ability recently discovered as instinctual, even in other mammals, birds, and possibly fish and amphibians. Infants one day old can distinguish between zero, one, two, and three objects, movements, and sounds. Learning to subitize removes the crutch of counting and allows you see the number of shapes in each place more quickly. Subitizing usage is preferred, but not required in this book. Appendix 3: Resources, References, Answers. x ix