krasa00FM.i_xvi

Number Sense and Number Nonsense

Understanding the Challenges of Learning Math

by

Nancy Krasa, Ph.D.

and

Sara Shunkwiler, M.Ed.

Contents

Chapter 1 Introduction

Section I Thinking Spatially

Section II The Language of Mathematics

Section III Solving Problems

Section IV Professional Implications

About the Authors

Nancy Krasa, Ph.D.

Nancy Krasa, Ph.D., is a practicing clinical psychologist with more than 25 years’ 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 of arts degree in mathematics from Smith College and her doctorate in clinical psychology from New York University. She and her husband live in Columbus, Ohio, and have three grown children.

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, a career she chose when she placed third in a schoolwide algebra contest and discovered she was "good at math." Her team won the state competition, and the three women on that team went on to graduate at the top of their engineering classes at The Ohio State University. Ms. Shunkwiler also earned a master of science degree in ceramic engineering from the University of Illinois at Urbana-Champaign. For 12 years, she worked as a product development and test engineer with General Motors and received three United States patents during that time. She left engineering to share her love of mathematics and science with students in the pivotal middle school years, and she earned a master of education degree in middle childhood mathematics and science education from The Ohio State University. She and her husband, also a ceramic engineer, live in Frederick, Maryland, with their two teenage sons.

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 there was a good chance that she would not graduate. Why? To receive a degree from her college, she was required to pass one class of pre-calculus level mathematics. Abby had struggled with math throughout her elementary years and, even though she did well in her other subjects, was barely able to earn enough math credits to graduate from high school. An exam during her college orientation had placed her into a non-credit remedial math course. After four failed attempts to pass that class, the dean referred her for an evaluation. She was frantic by the time she arrived at my office. What was I to make of Abby? Her predicament raised many questions...

Section I

Thinking Spatially

When most people think of mathematics, they think about arithmetic facts, equations, proofs, and the like. This math requires an ordered list of counting words and matching written numerals. It also requires other language that tells us what to do with the numbers, such as multiply or take the square root, or that describes how shapes relate to each other, such as parallel or congruent. Furthermore, it includes a long list of rules that explain how they all work together. This is the math we study in school. But what if mathematical language and symbols did not exist? Could people think about quantity at all? Section I examines the intuitive side of mathematics and how that intuition influences people’s grasp of the concepts that the conventional symbols represent.

Chapter 2

Number Sense

Science has demonstrated that humans are not alone in the ability to quantify. Animals, which do not have words and language, use numbers every day for survival. Across the phylogenetic spectrum from insects to chimpanzees, survival of both the individual and the species depends on quantitative skills to communicate, forage, evaluate threat, track offspring, optimize breeding, and conserve energy. Animals’ remarkable quantitative abilities include a sense of how many (e.g., eggs in the nest) and how much (e.g., distance from predators); some animals have even been trained to determine which one in a series (e.g., the third tunnel in a rat’s maze). Animals make these judgments based on information obtained through all of their senses.

The Mental Number Line

Researchers do agree, however, that both animals and young humans have a rudimentary sense of quantity. These primitive quantitative notions have two striking qualities: They are relative and approximate. Without being able to count, both animals and human infants judge quantities in relation to other quantities...

Conclusion

Long before most children see the inside of a formal classroom, they know something about number. They know that three cookies are more than two cookies and that if someone takes one cookie away, there will be fewer. They have vague ideas about a lot and a little that become sharper as they get older. As children learn to count and gain experience with numbers, they develop a mental image of how quantities relate to each other: a mental number line on which each number has its place, like inches on a ruler. Indeed, quantities are coded in the region of the brain that specializes in spatial functions; knowledge of number is intimately tied to that spatial sense. Many children develop an easy familiarity with quantity—number sense—as they gain mathematical experience. With a reliable mental number line, they have a cognitive map that keeps them oriented as they wander through the unfamiliar terrain of school mathematics.

Notes