# Number Sense and Number Nonsense ## Understanding the Challenges of Learning Math by ### Nancy Krasa, Ph.D. and ### Sara Shunkwiler, M.Ed. --- ## Contents - About the Authors - Preface - Acknowledgments ### Chapter 1 Introduction **Section I** **Thinking Spatially** ### Chapter 2 Number Sense ### Chapter 3 Math and Spatial Skills **Section II** **The Language of Mathematics** ### Chapter 4 Speaking Mathematics ### Chapter 5 Reading and Writing Mathematics ### Chapter 6 The Brain and Conventional Mathematics ### Chapter 7 More Sharks in the Mathematical Waters **Section III** **Solving Problems** ### Chapter 8 Executive Functions ### Chapter 9 Reasoning **Section IV** **Professional Implications** ### Chapter 10 Evaluation ### Chapter 11 Teaching --- ## About the Authors ### Nancy Krasa, Ph.D. Nancy Krasa, Ph.D., is a practicing clinical psychologist with more than 25 years of experience in psychological, neuropsychological, and psychoeducational evaluation. She has served on the adjunct faculties of the colleges of medicine at Cornell University, New York University, and The Ohio State University. Dr. Krasa is a member of the International Dyslexia Association and has published articles in the fields of psychiatric and psychoeducational diagnosis. She received her bachelor’s degree in mathematics from Smith College and her doctorate in clinical psychology from New York University. ### Sara Shunkwiler, M.Ed. Sara Shunkwiler, M.Ed., taught middle school at Marburn Academy, a private school in Columbus, Ohio, for bright children who learn differently. She is currently teaching Pre-Algebra and Algebra in a public school setting and has worked with many students who have varying degrees of math difficulty. Prior to entering teaching, she was an engineer, having placed third in a school-wide algebra contest. She earned a master of science degree in ceramic engineering from the University of Illinois at Urbana-Champaign and worked as a product development and test engineer with General Motors, receiving three U.S. patents. ## Preface In the early fall a few years ago, a college senior named Abby showed up in tears to my psychology practice for a diagnostic evaluation. She was a hard-working honor student and respected peer tutor in English but struggled with math throughout her elementary years. That these students had trouble with math was indisputable; my job was to figure out why. Psychiatric guidelines suggest that various cognitive impairments can be involved in such difficulties. There was no clear pattern in their cognitive test results that explained shared outcomes like math failure. It quickly became clear that understanding why some people fail in math necessitated recognizing what enables most to succeed. Reviewing existing research on math learning difficulties revealed an absence of comprehensive assessments and educational guidelines for math, unlike for dyslexia. I recruited Sara Shunkwiler, a middle school math teacher, to lend insights into educational issues that support learners with mathematical difficulties. ## Section I ### Thinking Spatially When most people think of mathematics, they think about arithmetic facts, equations, and proofs. However, this section examines the intuitive side of mathematics, emphasizing spatial thinking. It explores how that understanding influences people’s grasp of mathematical concepts. Alfred Einstein famously remarked, "Words and language do not seem to play any part in my thought processes; the psychological entities that serve as building blocks for my thought are certain signs or images." ### Chapter 2 #### Number Sense Research indicates that animals demonstrate the ability to quantify. Human infants also have a rudimentary sense of quantity, showing abilities to distinguish numerosity (i.e., how many) even before they can articulate their thoughts and ideas. Scholars debate key issues regarding how young children and animals understand quantity and whether human quantitative abilities emerge through evolution from shared perceptual skills. Despite these questions, common agreement exists that both animals and infants have a basic, relative, and approximate sense of quantity. ### Mental Number Line Both animals and young humans possess an innate sense of quantity with two crucial characteristics: they judge quantities relatively and approximately. Extensive research shows that early childhood development of quantitative abilities occurs along a spectrum, and the accuracy of quantity distinctions improve with age. Educators find that classrooms can help maintain students' number sense: they can use number lines and other spatial representations to improve understanding of numbers. ## Conclusion Understanding numbers isn't simply about mastering procedures; it's about developing a robust sense of quantity. Although some children lose their early comfort with quantity while grappling with math rules, early interventions such as number-line instruction can effectively bolster their number sense and math competencies.