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Strategies for Developing Mathematics Skills Students in Who Use Braille Gaylen Kapperman Toni Heinze Jodi Sticken August, 1997 Research and Development Institute, Inc. 1732 Raintree, Sycamore, IL 60178 HV1672 K142 S82 This manual is part ofthe project Computer-assisted Instruction forLearning the Code of Braille Mathematics which was supported by a grant from the U.S. Department of Education, Rehabilitation Services Administration (Grant No. H246C40001). Strategies for Developing Mathematics Skills Students in Who Use Braille Gaylen Kapperman Toni Heinze Jodi Sticken August, 1997 Research and Development Institute, Inc. 1732 Raintree, Sycamore, IL 60178 Digitized by the Internet Archive 2015 in https://archive.org/details/strategiesfordevOOgayl ACKNOWLEDGMENTS The staff of Research and Development Institute (RDI) wishes to express its sincere appreciation to several important individuals whose generous contributions greatly enhanced the contents of this manual. Dr. Abraham Nemeth, the developer of the Nemeth Code of Braille Mathematics, has generously communicated with the authors and contributed his knowledge and perspective in the form of personal writings; a conference paper which he presented, and MathSpeak, a guide for consistent reading of mathematical symbols by persons reading such materialto blind consumers.These contributions are included in the main text of this manual. Dr. Larry Chang, of the University of California, was also very generous in contributing his Speakeasy, a system he developed for the consistent reading of mathematical symbols, especially in the areas of algebra, geometry, trigonometry and calculus. His Speakeasy is presented in its entirety in Appendix B. Mr. Mario Cortesi, of the Chicago Public Schools, the major author of the tutorial entitled Computer-assisted Instruction for Learning the Code of Braille Mathematics, has contributed numerous creative ideas for involving students in higher mathematical experiences. He has also contributed a set of quick reference sheets which are included in the tutorial. The reference sheets are contained in Appendix C. Innovative ideas and pertinent issues and questions were contributed by numerous teachers who participated in several focus groups and/orwho responded to a national survey regarding several of the topics addressed in this manual.These individuals are named Appendix in F. TABLE OF CONTENTS INTRODUCTION 1 Barriers to Achievement in Mathematics 1 Impact of Restricted Development of Mathematical Skills 2 Factors Critical to Success 2 Curriculum and Standards 6 Assessment 8 TEACHING MATHEMATICAL CONCEPTS 11 Number Sense 11 Basic Concepts 11 One-to-One Correspondence and Counting Skills 16 Place Value 18 Measurement 19 BASIC NUMBER FACTS AND OPERATIONS 22 Fractions and Decimals 26 ADVANCED MATHEMATICS 29 TEACHING NEMETH CODE 38 Approach 38 Strategies for Teaching Symbols and Rules 39 CALCULATION TOOLS AND AIDS 40 Sequence 40 The Braillewriter 41 The Abacus 54 Fingermath 60 Mental Math 61 Talking Calculators 68 TACTILE DISPLAYS AND GRAPHICS 73 Guidelines for Designing Tactile Displays 74 Teaching Students to Use Tactile Displays 80 Materials Used to Develop Tactile Displays and Graphics 86 COLLABORATIVE AND INCLUSIVE STRATEGIES 89 The Transdisciplinary Model 89 SPOKEN MATHEMATICS 93 “MathSpeak” (Abraham Nemeth) 94 APPENDIX MATHEMATICS AND THE BLIND STUDENT 99 A: APPENDIX HANDBOOK FOR SPOKEN MATHEMATICS 103 B: APPENDIX C: NEMETH CODE QUICK REFERENCE SHEETS 166 APPENDIX RESOURCES 226 D: APPENDIX REFERENCES 234 E: APPENDIX SURVEY RESPONDENTS AND FOCUS GROUPS 242 F: iVf. INTRODUCTION Barriers to Achievement in Mathematics With the notable exception of the article reprinted in Appendix A (included for its interesting historical perspective), a thorough review of the literature reveals that achievement in mathematics among blind and severely visually disabled persons is, and always has been, extraordinarily low. There are several reasons for this unfortu- nate situation.The first and perhaps most important reason is that mathematics is very visual in nature.Visual reference isthe basis for much ofthe language of mathematics, with the description of such things as direction, quantity, and shape as fundamental elements. The development of spatial and directional concepts, as well as under standing of the concepts of object permanence and conservation of mass and volume are often delayed in children who are congenitally blind; and the blind student must piece together information which is perceived as a whole, in its entirety, by the sighted student. For a person who has no sight orvery little useful sight, the study of mathematics is difficult. It requires considerably more effort on the part of the blind student than does the study of fields which are more verbal in nature. Generally, in orderto achieve at reasonably high levels in mathematics, blind students must possess greateraptitude for the subject than their sighted counterparts. Since mathematics is difficult for blind demand more emphasis on the subject. persons to learn, students are unlikely to Younger students, of course, have no knowledge of the fact that they are not receiving sufficient training. A second major reason for poor achievement in mathematics among blind students is thatteachers of visually impaired students and rehabilitation specialists often lack skill and knowledge in the area of mathematics instruction. Many have had inadequate preparation in the Nemeth Code. While it is logical for personnel preparation programs to concentrate on the literary braille code, instruction in the Nemeth Code is often relegated to a subordinate position in the array of skills and knowledge. Consequently, many teachers lack confidence in their ability to teach the reading and writing of braille symbols in mathematics; this becomes a neglected area of instruction. A third reason for inadequate mathematics instruction is that only a small minority of individuals who join the field of special education and rehabilitation have technical backgrounds in which mathematics was a major portion of their study. Many teachers and rehabilitation specialists have concerns about their personal level of mastery of the field. There is a natural tendency in the art and practice of teaching to place emphasis on areas which are of interest to the instructor, Introduction 1 areas in which the instructor has expertise. Mathematics, therefore, usually is not the focus of instruction. Impact of Restricted Development of Mathematical Skills Insufficient preparation in mathematics has a profound effect upon blind persons. The disadvantages in daily life are numerous, including difficulties and unnecessary limitations in normal tasks of daily life as well as educational and occupational opportunities. While it is commonly agreed that the level of mathematics achievement required for daily living is not high, many blind persons, in reality, are functionally illiterate in mathematics. For example, it is not uncommon to find blind or severely visually disabled persons who cannot determine the correct change which should be received when making a purchase, calculate the amount of interest one might pay on a loan, or add one-half cup and three-fourths cup to accurately measure ingredients in a recipe. Given real life” tasks, many cannot choose the basic mathematics operation or combination of operations to solve a common problem. Inability to achieve in mathematics has a deleterious effect upon educational opportunities afforded to blind persons. Most entrance examinations contain a quantitative subtest. Scores from that subtest are included in the calculation of the overall score, representing the examinee’s performance and aptitude. fundamental mathe- If matics skills are poorly developed, then the general aptitude score will be depressed, impeding entrance into higher education or professional programs. In a modern technological society, well-honed quantitative skills are very important. Because most blind persons have poorly developed mathematics skills, they tend to avoid technical areas of employment. In fact, many technical areas are closed to them because they are unable to demonstrate the fundamental level of mathematics competence necessary for entrance into such fields. Factors Critical to Success Professional development One solution to this problem is bettertrained teachers and rehabilitation specialists. If these individuals have confidence and the technical expertise to provide the necessary instruction, many more blind persons in the future will experience success and competence in the essential and critical area of mathematics. Given the paucity of resources designed to help teachers remain current in methods and materials for instruction in mathematics for blind students, the following are sources of information which may be helpful. Catalogs of non-profit as well as for-profit organizations which produce materials and devices for use by blind students 2 Strategies for Developing Mathematics Skills in Students Who Use Braille