Table Of ContentStrategies 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 quan-
titative 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 neces-
sary for entrance into such fields.
Factors Critical to Success
Professional development
One solution to this problem is bettertrained teachers and rehabilitation special-
ists. 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
