Mathematics Education for
Students with Severe Cognitive Disabilities
By
Pamela G. Frazier
MS University of Wyoming 2012
Plan B Project
Submitted in partial fulfillment of the requirements
for the degree of Masters in Science in Natural Science/Mathematics
in the Science and Mathematics Teaching Center at the
University of Wyoming, 2012
Laramie, Wyoming
Masters Committee:
Professor Dr.Martin Agran, Chair
Associate Professor Dr. Lynne Ipina
Professor Dr. Jerry Hamann
Mathematics for Students with SCD
Abstract
This literature review covers the current body of research conclusions regarding
mathematics for students with severe cognitive disabilities (SCD). It reviews the historical
background and importance of mathematics education as a whole in the special education system
and for students with SCD. It emphasizes that research in this field is recent, within the past two
decades. Although legislation of No Child Left Behind has been a contributing factor, these
students are reaching higher potential than ever known possible, and this is calling for a body of
research, training, and instructional materials to be available and current. More attention is being
given to mathematics education and this attention is for including all special education students.
In this paper, standards are reviewed and the cognitive foundations of disabilities are discussed.
Curricula and interventions are being developed to meet these needs. Evidence shows an
awareness and commitment to making more mathematics available and meaningful for students
with severe cognitive disabilities.
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Mathematics for Students with SCD
Acknowledgments
Writing this Plan B has been a challenging and exceptional journey. I could truly never
have accomplished this task without the many generous people who have helped me along the
way. Thank you all so very much for your expertise, your patience, and your most gracious time.
First, Dr. Ana Houseal, who took us seven cohorts under her wing these past nine months and
seriously taught us how in the world to fly. Secondly, to my Committee Chair, Dr. Martin Agran,
an amazing research source we are so lucky to have in Wyoming, of all places. He is an
international expert on my subject and I have been humbled and grateful for his time and
guidance as I came to him literally out of the blue. Thank you to my committee members, Dr.
Lynne Ipina and Dr. Jerry Hamann. Lynne is the reason most of us ever begin this program. She
is the greatest encouragement and a great mathematics educator. I hope Jerry and Toni will find
some sense of purpose in being involved in this with me. I am grateful to the University of
Wyoming and the Science and Math Teaching Center for providing access to opportunities for
teachers to pursue more knowledge and help us become masters of our trade. My district, Albany
County Schools and our Mathematics Coordinator, Mike Busch, have supported me and Mike
has been an incredible source of mathematical passion and professional development. And
finally my most precious and amazing family who has helped me believe in myself when doubt
was clearly trying to reign. My parents and two sisters have cheered me on and supported us
through Timmy’s extraordinary life. My children Maggie, Timmy, and Graham have tolerated
mom on the computer way too much and reminded me why this all matters at all. And to my
husband Jeff who has sacrificed his time and infused me with sanity every step of the way. He
has been ever supportive, taking on so much of the family work, and has encouraged my passion
every moment. Thank you to Jesus, my reason for everything.
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Mathematics for Students with SCD
Table of Contents
LIST OF TABLES ............................................................................................................. v
LIST OF FIGURES .......................................................................................................... vi
CHAPTER 1 INTRODUCTION ....................................................................................... 1
Background ............................................................................................................ 1
Statement of Problem ............................................................................................. 3
Limitations ............................................................................................................. 5
Purpose and Research Questions ........................................................................... 6
CHAPTER 2 LITERATURE REVIEW ............................................................................ 7
The Standards......................................................................................................... 8
Five Strands Broader Base ................................................................................... 10
Cognition.............................................................................................................. 15
Instruction ............................................................................................................ 21
Interventions ........................................................................................................ 27
Functional Mathematics ....................................................................................... 31
CHAPTER 3 DISCUSSION ............................................................................................ 32
Implications.......................................................................................................... 32
Conclusion ........................................................................................................... 34
Summary .............................................................................................................. 35
REFERENCES ................................................................................................................ 37
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Mathematics for Students with SCD
List of Tables
Table Page
1. Sample Individual Education Plan Goals……………………………………….. 15
2. Comparison of Intervention Programs for Mathematics…………………………30
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Mathematics for Students with SCD
List of Figures
Figure Page
1 Number of Research Studies .................................................................................. 5
2 Working Memory System .................................................................................... 18
3 Chunking Material ............................................................................................... 27
4 Story-based Math Problem .................................................................................. 27
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Mathematics for Students with SCD
Chapter 1
Introduction
Background
Mathematics is thought to have traditionally begun around 5000 B.C. with the Egyptians
and the onset of writing through scribes who would solve simple mathematical problems. The
Babylonians also had a foundational influence in mathematics. Mathematics education, however,
began much later in the early A.D. centuries, with the Greeks and Romans (Berlinghoff &
Gouvea, 2004). Mathematicians were rare and wealthy and could therefore pursue scholarly
endeavors. “They built an intellectual tradition that continues to impress everyone who comes in
contact with it” (Berlinghoff & Gouvea, 2004, p. 15). Men and women were actually making
profound discoveries long before there were names of theorems, postulates, properties, or rules.
Observation and use of whole numbers, geometry, and ratios was a natural product of the world
around them. Pattern, number, shape, measuring, and record keeping seem intuitive in historical
mathematics. Humans have always wondered what, why, and how number and its cohort shape
made sense in the world (Berlinghoff & Gouvea, 2004). As mathematics evolved over centuries,
discoveries were made that had profound implications and applications. For example, Euclid’s
treatise on geometry, The Elements, and Newton’s laws of Physics, Principia (Berlinghoff &
Gouvea, 2004) had a profound impact on world history and development.
Mathematics education has always played an important role in
development over centuries. However, mathematics education and instruction for
the individuals with extra learning needs (i.e., Mathematical Learning Disability)
has not been emphasized (Gersten, Clarke, & Mazzooco, 2007). Dr. David C.
Geary defines Mathematical Learning Disability (MLD) as the following:
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Mathematics for Students with SCD
A mathematics learning disability would be manifest as a deficit in conceptual or
procedural competencies that define the mathematical domain, and these, in
theory, would be due to underlying deficits in the central executive or in the
information representation or manipulation (i.e. sorting memory) systems of the
language or visuo-spatial domains (Geary, 2004, p. 9).
MLD research is increasing. The past 10 years have brought an increase in study,
research, and documentation regarding MLD. An added challenge has been students with
Significant Cognitive Disabilities (SLD), an even more specified population of individuals who,
according to Browder and Spooner (2006):
(a) require substantial modifications, adaptations, or supports to meaningfully
access the grade-level content; (b) require intensive individualized instruction
in order to acquire and generalize knowledge; and (c) are working toward
alternate achievement standards for grade-level content (p. xviii)
As result of a recent study by Browder, Harris, Spooner, and Wakeman, (2008), an
extensive amount of information and recommendations from 68 experiments on teaching
mathematics to individuals with significant cognitive disabilities have recorded information to
guide and enhance the mathematics education of students with SCD. Several excellent textbooks:
Teaching Language Arts, Mathematics, & Science to Students with Significant Cognitive
Disabilities (Browder & Spooner, 2006); Teaching Students with Moderate and Severe
Disabilities (Browder and Spooner, 2011); and Why Is Mathematics So Hard for Some
Children? (Berch & Mazzocco, 2007) are now informing educators, professionals, parents, and
researchers about MLD and mathematics for students with significant cognitive disabilities. The
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Mathematics for Students with SCD
field has definitely advanced beyond its infancy, even if it remains a relatively young field
(Berch & Mazzocco, 2007). Internet sites for different disabilities are providing families and
professionals with outstanding resources to gather, compare, and become knowledgeable and
proactive in educating and raising these distinct children. These professional books, articles, and
sites are now public domain and have been invaluable in writing this review. More and more
information is available. Parents, educators and politicians need to engage in this topic and enter
into its discussion and practice (Siegler, 2004). Students with disabilities and their families have
capacity to thrive and contribute to society. Mathematic knowledge can only enhance and
promote their contributions.
Statement of Problem
Intellectually challenged students have historically been discounted in much of mathematics
education of the past. Only in the last decades has research examined the issues relating to MLD,
and much is still ill defined and unclear as to what constitutes MLD or dyscalculia (Mazzocco,
2007). A good number of students with cognitive disabilities have MLD. Up until the 1990’s,
students with severe intellectual challenges have been virtually disregarded in mathematics
education except for selected functional skills. With the onset of the Individuals with Disabilities
Education Improvement Act (IDEA) 1997, educators have become more assertive and aware of
including all children in general curriculum (Spooner & Browder, 2006). Students with
significant cognitive disabilities represent a population of students with pervasive support and
learning needs. They can include those with Down Syndrome, brain injuries, cerebral palsy,
autism spectrum disorders, and other various genetic and chromosomal disorders such as Fragile
X Syndrome, and Fetal Alcohol Spectrum Disorders.
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Mathematics for Students with SCD
Mathematics curricula and interventions follow the ever changing pendulum swings of
computational versus conceptual; facts versus problem solving; procedure versus application and
meaning; structure versus abstract; and rote mathematics versus making sense of knowledge
(Loveless, 2001). Special education mathematics materials have traditionally tended towards the
procedural (Browder et al., 2006). Test-driven criteria force special education classrooms into
structured curricula and easier data gathering resources, primarily focused on number theory.
Studies are suggesting the students with SCD would actually better thrive on a broader, active
and inquiry based methodology (Browder et al., 2008). This involves the use of programs that
target a broad spectrum of mathematics. While the researchers solve the deepest number
theories, pattern driven data, and scientific quandaries in the universe, there is a great deal of
learning to be gleaned by these students. There are specific special education and intervention
programs and curricula that cover the five strands of mathematics: algebra, data-analysis,
geometry, measurement and number, in a broad, hands-on, concrete, questioning, engaging and
technologically rich way (Browder et al., 2006). Students with SCD can achieve success and
discovery in the mathematical arena. Curricula and interventions need to meet the needs of
students with SCD. The development of effective mathematical teaching procedures for the
student with significant cognitive disability is presently emerging (Browder et al., 2011).
Literature is suggesting and supporting a distinct paradigm for the mathematics students of
today, including students with SCD. Trends that promote a balance between procedural and
conceptual instruction are developing and progressing (Gersten & Chard, 1999), but it is
imperative that correct information is disseminated and professionals are more aware of
misconceptions concerning the students with SCD.
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Mathematics for Students with SCD
Limitations
The research on this particular topic of mathematics education for the students with SCD
th
has been sparse. Students with SCD were primarily institutionalized until the mid 20 century.
Yet, this is not the only reason research has been sparse and slow. Mathematics Learning
Disabilities (MLD) themselves have only recently become a diagnostic term and recognizable
condition. The field was in its infancy in the mid-1990s (Berch & Mazzocco, 2007). There was a
small body of research about MLD that was available and even less so for students with SCD.
One other limitation to the study of mathematics has been the capacious amount of
research and information studied, analyzed, and written about reading. Research on reading
disabilities and its all-encompassing effects on education have stunted that of mathematics
(Gersten et al., 2007). The ratio of reading research studies to mathematical studies was 10:0 and
100:1 from 1956 to 1975. It increased to a mere 36:1, 22:1, and 14:1 from 1976 to 2005 (Gersten
et al., 2007) (see Figure 1).
Figure 1. Number of research studies conducted on reading disability (RD) versus mathematical
learning disability (MLD) and the ration for each decade from 1956-2005 (Gersten et al., 2007).
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Mathematics for Students with SCD
Purpose and Research Questions
The purpose of this literature review was to investigate the past and current information
about teaching mathematics to students with SCD and their abilities as learners. The past 20
years have produced increased research in this area. The fact that literacy research and reading
th
difficulties received most of the attention in the second half of the 20 century is unquestionable
(Gersten et al., 2007). However, the onset of No Child Left behind (NCLB) and the
augmentation of the National Council for the Teaching of Mathematics (NCTM) also helped to
stimulate interest in mathematics education. Educators, especially special education teachers, and
families need to become more informed and competent in mathematics education practice and
help these students improve their competency in mathematics. The following chapter reviews
current research about mathematics instruction for students with disabilities, generally, and with
severe cognitive disabilities, specifically. It will emphasize the what, who, and how we might
approach teaching and learning for the SCD student. Finding materials to support these
conclusions may be the biggest hurdle.
The primary question addressed in this literature review is:
What teaching practices are most effective for the mathematics education of children with
SCD?
Secondary questions include:
1. What is known and documented regarding the mathematic instruction of children
with SCD and particularly those with Down Syndrome?
2. Are interventions available and research based to support the education of these
students with SCD?
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Mathematics for Students with SCD
Chapter 2
Literature Review
Before the 1990s, the bulk of mathematic instruction for children with moderate to severe
cognitive disabilities (SCD) focused on number skills and was often limited to money and telling
time. Mathematics can be challenging to teach and learn, and educators have often limited their
instruction for these students to the basics of computation (Browder & Spooner, 2006). The
number of research studies in this area for these children is sparse. Although the Individuals with
Disabilities Education Act (IDEA), previously known as the Education for All Handicapped
Children Act (EHA), was established in the 1970s, it has taken time for the NCTM’s principle of
equity to filter into the field of mathematics for students with SCD. This past decade, 2000-2011,
brought much discussion, research, literature and instructional information regarding
mathematics instruction for students with disabilities. Fields of “developmental psychology,
cognitive science, mathematics education, special education, and even law have given rise to the
multidisciplinary field of Math Learning Disabilities (MLD) research and practice that exists
today” (Gersten, et al., 2007, p. 7). Specific mathematical intervention and instruction for the
students with SCD was vague if not non-existent prior to recent research. Because of the
reauthorization of the IDEA in 1997 and 2004 that state all students with disabilities are required
to have access to the general curriculum and to be included in state and district large-scale
assessments and the reinforced inclusion of students with disabilities in standards-based reform
of No Child Left Behind (NCLB), 2001, all students need to be included in state accountability
and assessment systems. “Although federal policy continues to evolve, the changes created by
NCLB had a lasting impact on services for the students with severe disabilities (Browder,
Spooner, & Wakeman, 2011, p. 40).
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Mathematics for Students with SCD
The Standards
An important turning point in the assimilation and dissemination of better mathematic
education for all students came through the implementation of the National Council of Teachers
of Mathematics (NCTM). Established in 1920, the NCTM has become a highly respected and
influential voice for mathematics. This organization is by definition “a public voice of
mathematics education, supporting teachers to ensure equitable mathematics learning of the
highest quality for all students through vision, leadership, professional development, and
research (NCTM, 2011, p. 1). The International community has also influenced these standards.
NCTM has provided professional guidance through mathematic teachers and others
knowledgeable and experienced in mathematics. Prior to the 1980s, curricular content came
primarily from textbook publishers and standardized test publishers (NCTM, 2011). They
published three landmark documents: Curriculum and Evaluation Standards for School
Mathematics (1989), Professional Standards for Teaching Mathematics (1981), and Assessment
Standards for School Mathematics (1995). NCTM was the first to introduce “standards” in 1989.
In addition, states each created their own Standards documents. In the late 90’s, other
organizations (e.g. The America Diploma Project, Performance Standards, and College Board
Standards for College Success) began producing Standards documents as well (NCSM, 2011).
Using the input of mathematicians, teachers, and researchers, the Standards 2000 Project
came to guide the teaching of mathematics. Standards were identified that centered on following
six principles: (a) Equity, (b) Curriculum, (c) Teaching, (d) Learning, (e) Assessment, and (d)
Technology. Additional standards were developed that included five content areas: (a) Number
and Operations, (b) Algebra, (c) Geometry, (d) Measurement, and (e) Data Analysis and
Probability. Also, five process areas were identified: (a) Problem Solving, (b) Reasoning and
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Mathematics for Students with SCD
Proof, (c) Communication, (d) Connections, and (e) Representation (NCTM, 2011). In 2009, the
National Governors Association and Council of Chief State School Officers decided that it was
time to address the incoherence of each state having their own standards. Subsequently, they
commissioned Achieve, a bipartisan, non-profit education reform organization, the College
Board, and ACT to develop a set of college and career readiness standards that would be
available for states to adopt. Jason Zimba, William McCallum, and Phil Daro were mathematic
experts who also had enormous input into these refined standards. Daro then chose other
renowned education researchers, educators, and policy makers with specialty areas to help lead
the standards development process (M. Busch, personal communication, December 6, 2011).
More specific grade level standards were developed based on The College and Career
Readiness Standards. These were voluntary national standards. National organizations such as
NCTM and NCSM and experts provided feedback for revisions. As of 2010, states were
systematically adopting this unifying set of standards called the Common Core State Standards.
While 44 states have adopted these standards, 48 had input in the writing process (M. Busch,
personal communication, December 6, 2011).
Mathematics education will likely eventually be guided by these CCSS. NCLB has
legally mandated that students with disabilities be included in state testing and be given access to
general curriculum (NCLB, 2002). State testing will be driven by the Common Core Standards.
This, however, is where it gets tricky in terms of the inclusion of special education students.
Special educators have long been challenged to teach students with extremely varied needs and
provide an individualized education for each of them according to their legally binding
Individual Education Plans (IEPs). In addition, this is one of the most transient sectors of
education. Special Education teachers are in high demand and transition often (McLeskey, Tyler,
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Mathematics for Students with SCD
& Flippen, 2004). Experts in special education are difficult to find (Smith et al., 2011). Because
of this, the binding standards of NCTM and IDEA are helpful and difficult. Special educators
and special education systems are held to a high standard, yet a serious shortage of qualified and
knowledgeable professionals exacerbates the problem of quality education for students with
SCD. Mathematics education needs to be broadened to the previously mentioned content areas,
principles, and processes, and this is particularly important to address in special education and
for the students with SCD. IEP teams rarely know what opportunities a student may have as an
adult. To limit or restrict a student’s education based on assumptions about the student’s
disability would be unfortunate (Browder et al., 2011).
One reason for the lack of research on mathematic disability (Gersten et al., 2007) was
the assumption that students with SCD lacked the ability to learn mathematical concepts
(Browder, Spooner, & Trela, 2011). This misconception has been challenged in recent years.
Although students with SCD have a wide variation in their abilities to learn, retain, and use
knowledge, including mathematics, they need the opportunity to discover their capabilities like
all other students. These students deserve as equitable an education as possible. In mathematics,
this includes the opportunity to learn and reason through all the content areas and all the process
areas.
Five Strands Broader Base
These 2001 standards emphasize five content areas including: (a) algebra, the study of
patterns, relations, and functions; (b) geometry, the study of spatial organization; (c) data
analysis, the study of organizing and interpreting facts and data; (d) measurement, the study of
defining attributes in the standard format; and (e) numbers and operations, or the study of
quantity and number (Browder et al., 2011). Students with SCD may be able to master more
complex mathematical concepts than once considered possible (Jimenez et al., 2008). These
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Mathematics for Students with SCD
skills will then prepare them for more educational, job, and life opportunities in their futures.
More and more SCD students have opportunities to go to collegiate programs such as “Think
College” and “Learning into Future Environments.” Mathematical skill creates better problem
solvers, communicators, and thoughtful adults. Discovering a broad and useful facet of each of
these content strands is not only a part of NCTM’s standards but a balanced and relevant
mathematical instructional program for all students including those with SCD.
Algebra. The study of Algebra is often portrayed as a high level area involving only
complex functions, variables, and slope. However, the underlying principle of Algebra is the
discovery and recognition of pattern and its relationships. Recognizing the sequence of units of
pattern such as blue, blue, and red or 3, 6, 9, and 12 are all foundational to algebra. Being able to
determine the next unit in a sequence can lead to all types of observations in nature. For
example, in machinery, poetry, and music the patterns involved can be exquisite. Changes
involved in our world such as temperature, finances, time, and goals are the essence of the
graphing involved in slope and algebraic understanding. Being able to recognize algebraic slope
in the real world does not necessarily involve knowing or manipulating the formula. Students
with SCD can certainly gain skills and relevance in algebraic problems and observations in the
real world (Bird & Buckley, 2001).
Geometry. Geometry involves recognition of shape and spatial relationship as well as
coordinate planes, directional terms, and maps. From the foundational terms of point, line, curve,
and shape, to the complex geometry of Euclid, there is a great deal of geometry to be gleaned at
all developmental levels. Concepts of perimeter and area are very simple and useful. Students
with SCD will relish the four quadrant coordinate plane as they plot points and discover points of
intersection. Mapping skills, grids, and directions across town are also geometric in nature. Some
11
Mathematics for Students with SCD
key job opportunities require knowledge of this concept. Geometry skills provide descriptions of
objects and moving through space (Browder et al., 2011).
Measurement. Being able to use money and tell time are such basic life skills that they
are a regular addition to most IEPs. However, measurement as a mathematical topic is so much
more. Terms such as large, slow, and deep are foundational to daily life. Tools that measure
length, weight, and capacity are important in classrooms and homes. Deigns that involve angles
and curves are a part of the world around us. Many students, including those with disabilities are
curious and naturally want to measure their world (Ginsburg & Pappas, 2007).
Data analysis. Data analysis is about organizing information and answering questions
about data. Students with SCD have long been involved in managing their behavior and
academics and creating graphs or charts to do this (Browder et al., 2006). Though maybe not
taught as data analysis, these children and adults are already adept in graphing and charting
skills. Students can learn to apply these skills elsewhere and make great sense of and interpret
data. Charts, tables, and pictures are excellent ways of modifying math to be better understood.
Graphing and charting information gives mathematics a visual aspect and refreshing application
of quantitative data in a variety of subjects such as comparing sales, monitoring change over
years, or knowing statistics of products on the market.
Number and operations. Numbers and operations are the most obvious aspects of
mathematics. Representing and computing numbers is seen every day as we make purchases and
move from one place to another. Relationships among numbers and numbers systems literally
run our world. From speed signs to percentage sales and taxes or from government to the mother
buying diapers, numbers are as intrinsic in our lives as letters. In early years of education,
particularly for the students with SCD, teachers and students often got bogged down with money
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Mathematics for Students with SCD
recognition and making change. These are actually higher level thinking skills for all students
and many kindergarten and first grade students have great difficulty with them. There is so much
more to number than representation, recognition, counting, and operation work. Many students
with SCD are capable of identifying and understanding numbers and combinations of numbers,
number correspondence, place value and numerical order, even fractions. Students can learn to
apply entry-level number sense in increasingly complex contexts across the grade-level content
of mathematics. They can progress in sophisticated number use across the grade levels.
Computer software, calculators, and other materials may be used to manage difficult facts and
larger numbers (Browder et al., 2006).
Number sense. Number sense is a different skill set than number and operations.
Number sense refers to a student’s fluidity and flexibility with numbers. The NCTM process
standards flow out of the foundation of number sense, making sense of numbers, what they
mean, mental math, and real world application. Constructive, conceptual mathematical thinking
is founded on this number sense. Phonemic awareness is the building block to reading, and
number sense is the same to mathematics (Gersten, 1999). Developing such a sense is difficult
for the student with SCD but not impossible. “The most important outcome students can receive
from mathematics instruction is to learn to solve problems” (Browder et al., 2011 p. 175). We
must direct our attention towards developing this sense.
Focus on a broader base of mathematics rather than a deep, detailed understanding of the
intricacies of math is proving to be a valid approach, specifically for students with SCD.
Additional research is being conducted and supporting instruction that covers all strands of math.
“There is widespread agreement that children should be exposed to a broad range of
mathematical content” (Ginsburg & Pappas, 2007, p. 434). Conclusive research from 40 editors
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Mathematics for Students with SCD
of Why is Math so Hard for Some Children indicates that one major goal of “mathematics
education is to promote mathematical thinking and communication, not just the mastery of skills
and concepts and this kind of intervention may be especially useful for the students with MLD
(p. 435)” and SCD. Studies suggest that math instruction needs to:
1. Focus on the big ideas-generalizable concepts rather than individual details,
2. Teach conspicuous strategies, neither too broad nor too specific for conducting math
operations and solving problems,
3. Make efficient use of time in prioritized objectives,
4. Communicate strategies in a clear and explicit manner, and
5. Provide practice and review to promote retention (Carnine, 1997).
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Mathematics for Students with SCD
Table 1
Sample Individual Education Plan Goals
Strand IEP Goal
When given two or more shapes, numbers, colors or designs, student will be
Algebra
able to complete the pattern identifying the next shape in the pattern,
increasing from 20% to 80% as measured by teacher created assessments
Geometry/ When given a triangle, rectangle or square, student will be able to measure the
Measurement sides of the shape and use the measurements to find the perimeter of the shape
using the appropriate formula. Student will also find the measurements of
perimeter using various units of measurement (centimeters, inches, and feet)
up to twenty improving from 0 out of 10 correct to 7 out of 10 correct as
measured by data collection and teacher observation.
Data Analysis When given various sets of data to be analyzed and compared, student will
graph the information using various types of graphs (line graph, pie charts
and bar graphs), improving from 20% to 80% accuracy as measured by
teacher observation and data collection.
Number and When given a fourth grade level addition, subtraction, division, and/or
Operations multiplication problem, student will input the digits into a calculator and
solve the equation, improving from 20% to 80% accuracy as measured by
data collection and teacher observation
Note. Sample IEP goals that cover all strands of mathematics
Cognition
Genetic, traumatic, and cognitive obstacles combine to cause significant delays for
students with SCD. Students with SCD often have similar deficits and strengths (Barnes,
Fletcher, & Cobbs, 2007). Cognitive processing involves the mental operations we do to evoke
meaning or solve a problem whether it is word related or number related. Cognition is how we
acquire, organize, represent and process knowledge visually, intuitively, selectively and through
reasoning. It is our thinking and our remembering (Pellegrino, Chudowsky, & Glaser, 2001).
Whether a student has traumatic brain injury or is born with cognitive disability, it
manifests itself in a delay of processing information and difficulty remembering certain aspects
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Mathematics for Students with SCD
of number theory and spatial operation. This review will focus on students with SCD cognitive
processing problems in learning of mathematics, particularly those with Down Syndrome.
Unfortunately, “professionals have not given a great deal of thought to how students with severe
cognitive disabilities think” (Kleinert, Browder, & Reeves, 2009, p. 305). The focus has been on
behavior modification and analysis that lead to more independent living skills. Applying
cognitive theories to education of students with SCD has been ineffective primarily because they
focus on the deficit or what these students lacked instead of a “capacity building-model
approach” (p. 305), which emphasizes what students could do. Without a cognitive framework
such as Piaget’s 1978 model or more recent constructs of James Pellegrino’s cognition vertex
(Pellegrino et al., 2001), understanding how these students think and construct knowledge is
“especially difficult and great care must be taken in making inferences about these students
cognition” (Kleinert et al., 2009, p. 306). A model developed by Pellegrino (2001) suggests “four
perspectives for understanding the nature of human learning and knowing. Each of these
perspectives has important implications for the assessment of students with SCD” (Kleinert et
al., 2009, p. 306).
Pellegrino’s (2001) differential perspective emphasizes products, learning about how
much we know. Assessment scores play a huge role here and have been primarily based on
mental age versus grade level production. Students with SCD have scored poorly over the years
and have at times even been excluded as untestable. This perspective severely limits the
understanding of what these students are capable of performing. The behaviorist perspective
regards task-analysis and stimulus-response associations. Its focus is systematic assessment
particular to measurable and observable targeted skills, a small bit of what we know. This has
been the most influential perspective regarding education of the students with SCD but does not
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Mathematics for Students with SCD
focus on “how students construct, organize and/or use the knowledge they attain” (Kleinert et al.,
2009, p. 307). The cognitive perspective focuses on this how. It emphasizes strategies to connect
knowledge to quality, organization, and meaning. It is more about growth over time, a much
more essential goal for students with SCD. “A one-time snapshot of what they know might not
capture the significant gains in how they have learned to represent their knowledge over time”
(Kleinert et al., 2009, p. 308).
The fourth and final model of learning and knowing according to Pellegrino (2001) is the
situative, or socio-cultural, perspective, one’s place in a community of learners. Here,
controversial yet supported research practices such as inclusion and least restrictive environment
issues arise. Students use their skills for “real work of community”. Inclusive settings “promote
not only social benefits but attainment of educational goals” (Kleinert et al., 2009, p. 308).
Students with SCD may have trouble transferring skills to contribute to the real world and should
be allowed practice and effective strategies to develop application skills (Kleinert et al., 2009).
Finally, given the current emphasis on technical competence, which draws on high levels of
mathematics, mathematic ability and achievement, seem critical (Swanson, 2007).
There has been considerable research on the neurological factors associated with
intellectually disabilities. Suffice it to say, little conclusion can be drawn as to the
neurobiological causes of Mathematical Learning Disability (MLD). Much research is ongoing
but conclusions are unsubstantiated at this point (Geary et al., 2007). Memory deficits, however,
represent a significant problem. “It has been well established that many children with MLD do
not perform as well as their same-age peers on a variety of working memory tasks”(Geary et al.,
2007). Memory issues in students with SCD should be recognized in order to develop an overall
17
Mathematics for Students with SCD
understanding and plan for their teaching and learning. Working memory and long term memory
are the fundamental elements of cognition and the mind’s cognitive
architecture (Pellegrino et al. 2001).
Working memory. Working memory is sometimes referred to as short-term memory. It
is what people use to access and process information right in front of them. Its key component is
capacity (Pellegrino, Chudowsky, & Glaser, 2001). Working memory is ability to hold a mental
representation in mind while simultaneously engaged in other mental processes (Geary et al.,
2007). It is a “highly limited system” (Pellegrino et al., 2001). Alan Baddeley’s model of
working memory (1986) proposes three looped components of memory. These include:
1. Central executive- component controlling attention and processing mental
information
2. Visual-spatial short-term memory, holds images, pictures, and locations; and
3. Verbal short-term memory or phonological loop, holds verbal words and numbers
typically speech based (Broadley, MacDonald, & Buckley, 1995) (see Figure 2).
Figure 2. Working Memory System (Buckley, 2008)
Typically, the older we get, the better our working memory because of the faster loop and
the more capability. Students with SCD need learning strategies to chunk or code information, as
18
Mathematics for Students with SCD
their working memory capacity is weaker and they have trouble with multistep directions
(Kleinert et al., 2009, p. 309). Working memory certainly affects long-term memory but “what
matters most in most situations is how well one can evoke the knowledge stored in long-term
memory and use it to reason efficiently about current information and problems” (Pellegrino et
al., 2001, p. 3).
A “virtually limitless store of knowledge” (Pellegrino et al., 2001, p. 3), long-term
memory stores two types of information: semantic, the way the world is, and procedural, how
things are done. Memory skills are particularly crucial for students with SCD (Kleinert et al.,
2009). Students must connect the ways of the world to the procedures of the world, especially in
regard to mathematical information. “Procedural knowledge can be gained only by sustained,
explicit practice across the range of exemplars or situations in which the student would be
expected to learn the skill” (Kleinert et al., 2009, p. 309). Difficulties in number procedures and
basic fact memorizing by students with SCD stem from difficulty storing and retrieving this
information in long-term memory. These foundational skills inhibit other sequential
mathematical skills. The deficits in memory are to be acknowledged and recognized, and
“identification of the underlying processes that might account for these deficits could provide a
clue to either remediation or alternative teaching and learning strategies” (Broadley et al., 1995,
p. 3).
Finally, Pellegrino (2001) also makes a distinction between concepts of development and
learning. Some types of knowledge, particularly regarding mathematics, come in the course of
normal development while other types are learned through deliberate teaching. Geary (2007)
calls these “biologically primary cognitive abilities” and “secondary cognitive abilities”. The
first are fairly innate in our development, the second require effort. “Yet for students with SCD,
19
Mathematics for Students with SCD
these primary cognitive forms of learning are often not acquired incidentally, but rather require
very intentional and focused instruction as well”(Kleinert et al., 2009, p. 311).
Visual, auditory, or kinesthetic (VAK) learners. Students with SCD have difficulty
acquiring information through auditory means (Horstmeier, 2004). “Verbal short-term memory
tasks typically involve auditory presentation of the to-be-remembered items, and clearly if one
has difficulty in hearing and identifying these items then one's memory for them is likely to be
severely reduced” (Jarrod & Baddeley, 2001, p. 18). Hearing difficulties are known to be
relatively common in SCD students such as those with Down Syndrome (Jarrod & Baddeley,
2001). Students with SCD do have “higher incidence of sensory or physical disabilities then
students with less severe disabilities” (Kleinert et al., 2009, p. 311), and this affects their
kinesthetic or motor abilities and learning through imitative or exploratory hands-on methods.
Although visual-spatial acquisition of information may be weaker for the child with SCD
than a traditional child, research suggests visual and kinesthetic teaching materials are still a
stronger place to implement learning than auditory methods (Horstmeier, 2004). Special
instruction and practice is needed and, fortunately, information and practical help are emerging.
Research studies consistently show that children such as those with Down Syndrome
have characteristically poor short-term memory. Children and adults with Syndrome show poor
auditory memory compared with visual memory and recognition memory (Broadley et al., 1995).
The Down Syndrome child has difficulty with the abstract ideas of math. Arithmetic and number
skills are areas of particular difficulty for individuals with Down Syndrome. “Children with
Down Syndrome show severe difficulties in mastering basic number skills as assessed by tasks
that include size and numerosity judgments, counting and simple arithmetic” (Brigstocke et al.,
2008, p. 75). Number vocabulary is difficult, as it is with many children since it used so much
20
Mathematics for Students with SCD
less frequently than general vocabulary. Repetition breeds memory and repetition takes a
concerted effort to learn math vocabulary (Graves, 2009).
In summary, students with SCD are helped by teaching methods that take account of this
research into their strengths and weaknesses in their: (a) motor delays make manipulating small
items, drawing and writing difficult. (b) speech and language delays lead to their understanding
being underestimated. (c) auditory processing and working memory difficulties make learning
from listening difficult. (d) strengths in social understanding and enjoyment in learning from
social interaction with peers and adults. (e) relative strengths in visual processing and visual
memory make learning from seeing important and effective and they are visual learners, and (f)
strengths in using gestures to communicate and in showing their understanding by pointing or
choosing an answer (Horstmeier, 2004).
Instruction
The most frequently mentioned instructional technique to support the students with SCD
is that of systematic instruction (Spooner et al., 2011). Systematic instruction (SI) entails these
five basic components:
1. Socially meaningful skills;
2. Target skills, observable and measurable;
3. Using data for results;
4. Behavioral principles, reinforcement, prompting and fading, error correction; and
5. Behavior change that can carry over to other contexts, skills and materials (Spooner et al.
2011).
Systematic instruction is, as it suggests, a very specific system of steps that allow for
careful planning and record keeping monitoring progress. It defines skills, methods, and
21
Mathematics for Students with SCD
frequency of practice and assessment and records data emphatically (Spooner, Browder, &
Mims, 2011). To plan SI teachers follow four steps: (1) define skills to be acquired, (2) define
methods to use, (3) implement the SI plan and frequency, and (4) review student progress to
modify instruction with charts, lists, and graphs. SI is the overarching instructional package with
strong evidence of effectiveness.
This “evidence-based” practice (Spooner et al., 2011, p. 93) of instruction is encouraged
for use, especially for students with SCD. However, in this emerging field of mathematics,
evidence-based criteria are more difficult to establish for several reasons. Identifying evidence-
based intervention and practice is difficult if there were not enough studies or participants to
make conclusions. “Researchers are just beginning to discover what works” (Spooner et al.,
2011). In applying evidence-based practices professionals need to be careful to pay attention to
prescribed procedures and adapt and evaluate procedures for the individual student. Assessing
progress is key. The individual student is the unit of analysis and visual inspection of graphed
data is the primary method used to analyze change (Spooner et al., 2011).
Instructional intervention programs should be research-based and proven effective.
However, there are intervention programs that do not have the stamp of approval of the
“research- based” criteria, yet can provide especially beneficial support. Several research
specialists have suggested principles for teaching numeracy and the broader array of mathematic
areas to students with SCD. They all corroborate one another and support the overarching, sound
bedrock of metacognition, mathematical thinking skills. These features also uphold the premises
of SI. Ginsburg, Pappas, and Griffin support 10 insights:
Guided integration of what children bring to the task, informal knowledge
Wisdom of culture, what is taught in school, or at least should be
22
Mathematics for Students with SCD
General models that immerse children in the world of number
Draws on their prior knowledge
Attempts to transform that knowledge into a more mature and formal form
Align instruction with the natural developmental progression of mathematical thinking
Provide hands-on games and activities that encourage children to construct meaning
Encourage communication in spoken language and writing
Ensure that activities capture children’s emotions and imaginations, that is they enjoy
mathematical learning (emphasis added)
Create activities appropriate for children from different cultural and social backgrounds
(Ginsburg & Pappas, 2007).
Allsopp suggests 13 research-supported strategies to look for in programs and instruction
of students with SCD:
Building meaningful student connections
Continuous monitoring/charting of student performance
Dynamic assessment for mathematics
Explicit teacher modeling
Instructional games
Planned discovery activities
Self-correcting materials
Scaffolding instruction
Structured cooperative learning groups
Structured language experiences
Structured peer tutoring
23
Mathematics for Students with SCD
Teaching concepts/skills within authentic context
Teach using big math ideas in all five strands (Allsop, 2007).
He also identified four universal features of meaningful mathematics instruction for
struggling learners. These features are: (a) understanding and teaching the big ideas in
mathematics, (b) understanding learning characteristics of and barriers for struggling learners, (c)
making mathematics accessible through responsive teaching, and (d) continuously assessing
learning to make informed instructional decisions (Allsopp, 2007). Horstmeier (2004) promotes
nine general principles for teaching students with SCD:
Teach in the same stages in learning as other children
Provide a wide range of progress levels and good teaching helps
Interactive, social learning situations
Consider and accommodate speech production difficulties
Create vocabulary lists that expose and give clear context
Visual supports that link specific images with the spoken word
Multi-sensory materials that teach through play and support recall (Horstmeier, 2004).
Three of these strategies stand out as they repeat themselves in all three resources: (a)
hands-on instructional and interactive games and activities that help construct meaning, (b)
spoken and written language experiences, and (c) developmentally progressive big ideas of
mathematical stages and all five strands. Dice, board, and card games, puzzles and teacher
created activities such as money, numeral, and bingo (or ‘fringo’ for the fractions) boards and
visual pictures and variety of color of objects will all bring interaction and meaning to
mathematics. Dice dots turn into real counting and combining skills for use in sums and products
24
Mathematics for Students with SCD
in everyday living. 10 frames, Base Ten Blocks, Cuisenaire rods, Fraction Bars and real Hersey
bars or other small food items bring number sense and are enjoyable to children. Cutting apart
charts, bars, or pictures brings an added dimension to the learning experience. Communicating
their steps and expressing what they are doing and why verbally, can bring meaning and
vocabulary development (Bley & Thornton, 2001).
Presenting a variety of strategies for concepts helps provide differentiated instruction
throughout the development of mathematical ideas and makes students think about different
ways of working out problems and they in turn are thinking mathematically, with metacognition.
“If teachers teach students how to break problems into small steps, relate symbols to words
throughout the school experience, and preview vocabulary when needed, students can become
successful problem solvers” (Bley & Thornton, 2001, p. 41).
Though these students often lack systematic approaches to identifying and solving
problems, there are problem-solving strategies that can be directly taught to students with SCD
(Agran, Blanchard, & Wehmeyer, 2000). Many students with Down Syndrome and similar
cognitive disability typically reach what Piaget (Piaget, 1950) describes as the concrete
operational learning stage. Typical children reach this stage about 7-12 years old:
A child is able to solve problems through mental thought according to logical
rules. However, this problem solving can only be done when the child is dealing
with concrete information that he can perceive directly. If only the abstract
numbers are given, he may not be able to set up the problem. These students need
hands-on materials to learn math concepts. (Horstmeier, 2004, p. 5)
25
Mathematics for Students with SCD
Students with SCD need to have distinct smaller steps and much more practice before
them to organize their thoughts and keep learning in manageable chunks (Horstmeier, 2004) (see
Figure 3). Instruction that involves a variety of instructional approaches and a meaningful
application is critical. These approaches may be different and more explicit for the SCD children.
Some examples include embedding mathematical problems in stories with in the natural schema
of life (Browder et al., 2011) (see Figure 4).
Contextualizing math problems provides meaning and bridges the abstract qualities of
mathematics that sometimes stump the students with SCD. Music can also make connections for
students and be highly motivational (Browder et al., 2006). Calculator instruction and use are
uniquely beneficial for the students with SCD to help them get past their calculation or mental
math difficulties likely due to their working memory problems (Horstmeier, 2004). Other
assistive technology such as computers, iPads, and smart boards are rich sources of support for
these students who tend to be more visual learners. Technology sources are usually interactive,
colorful, driven by application and engaging (Horstmeier, 2004).
If the expectation is for students to access the many environments of our world
through travel, the Internet, the media, and their jobs, then students need the
opportunity to comprehend mathematical concepts. The ongoing challenge is to
be sure that as mathematics is learned, students also are taught to make
connections to their daily lives so that these skills have meaning and utility.
(Browder et al., 2006 p. 193)
26
Mathematics for Students with SCD
Figure 3. Examples of chunking material into smaller steps and pieces (Horstmeier, 2004).
Figure 4. Story-based math problem examples (http://mast.ecu.edu/modules/sscd_mc/concept/)
Interventions
Mathematic materials for instruction come in many different forms. Complete curricula
of each school are important to consider and support. Several examples of common curricula for
the general mathematic classes presently are Investigations, Everyday Math, MATH
27
Mathematics for Students with SCD
Connections, Teaching to Standards: MATH, and Bridges. “When planning the use of alternative
materials, it is important to use the general education class materials whenever possible”
(Browder et al., 2006 p. 188). Many computer- assisted programs are available for a wide
spectrum of support, particularly memory training and capacity building.
Information below distinctly compares Intervention programs (see Table 1). In special
education classrooms it is generally an intervention program that forms the basis to the
instruction for students with SCD (Griffin, 2007). “Until recently, these programs were produced
almost exclusively by commercial publishing companies” (p. 374). These past 20 years have
brought education and cognitive science psychologist experts in to help in this arena to try to
develop intervention programs that would be more effective. The following is a comparison of
the most commonly mentioned in all of my research and the attributes of such that support my
findings of better instructional practices. The seven most mentioned intervention programs in my
research were the following.
Do the Math was created by renowned American Marilyn Burns. It is a number- based
intervention program that focuses on fluency with whole numbers and fluency with fractions.
The program scaffolds these topics and addresses these research- based practices: Explicit
Instruction, Multiple Strategies, Gradual Release routines, Student Interaction, Meaningful
Practice, Assessment & Differentiation, and Vocabulary and Language
(http://teacher.scholastic.com/products/dothemath/po.htm).
Voyager or VMath provides instruction that balances conceptual development,
computational fluency, and problem solving. It covers all the strands of mathematics and
provides hands-on lessons to help teachers present important math concepts using common
manipulatives (http://www.voyagerlearning.com/vmath/curriculum.jsp).
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Mathematics for Students with SCD
ADD Vantage/Math Recovery touts the phrase Learning Framework In Number (LFIN),
and provides participants with an understanding of how children learn mathematics. It helps
teachers assess their students accurately and custom-fit their instruction to enable students to
progress in the area of mathematics (http://www.mathrecovery.org/).
Number Worlds builds foundational skills and concepts, and makes learning fun. It
builds foundational math skills and prepares younger children to understand more complex
concepts later. It looks to be the most comprehensive program that helps children structure and
understand the world.
Kumon and Numicon are both United Kingdom programs that are often used
supportively. Kumon is strictly numeracy drill and practice that improves automaticity and has
been found to be very effective for this and research supports growth in computation. Numicon
then provides more unique multi-sensory manipulatives especially for counting, ordering, and
subitizing and provides a combination of action, imagery and conversation. Both are numeracy-
based programs (http://www.kumon.co.uk/) (http://www.numicon.com/Index.aspx).
Math Perspectives is the newest intervention program and it has the most variety of
excellent numeracy materials and an exceptional computerized assessment system making it very
efficient and easier for teachers. Exceptional variation and innovative analysis are its strength.
29
Mathematics for Students with SCD
Table 2
Comparison of Intervention Programs for mathematics
Evidence/Research Primarily Systematic Hands–On/
Intervention Based 5 Strands Numeracy Instruction Concrete
Do the Math X X X X
Voyager X X X
ADD X X X
Vantage/Math
Recovery
Number Worlds X X X X
Kumon X X
Math Perspectives
X X X X
Numicon X X X X
Note. This chart compares the components of Seven Mathematic Intervention Programs
30
Mathematics for Students with SCD
The intervention program cannot be the only means of instruction or drive the teaching of
the students with MLD or SCD. The complete repertoire of strategies previously mentioned
should be considered and implemented according to individual needs. All of these interventions
programs promote strong number sense, but more intervention programs need to be created that
support all the strands and provide systematic instructional and research-base to all the strands. It
is clearly hard work (Bicard, Bicard, Casey, & Nichols, 2008).
Functional Mathematics
Although the focus of this review is on academic mathematics, it is safe to say that most
students with SCD are provided instruction in functional mathematics, too. Thus, it is appropriate
that this area is addressed to some extent. Functional mathematics involves skills that prepare
students for life such as grocery shopping, preparing a meal, cleaning, negotiating transportation,
and paying bills. Preparing for jobs and work are an ultimate goal (Browder and Spooner, 2006).
Money skills such as purchasing, comparing, computing, and budgeting will develop all the
numeracy work. The “Next Dollar Strategy” involves excellent number sense and focuses on
estimating to the nearest next or lower dollar rather than dealing with confusing coins and
making change. Calculator instruction and practice are essential here (Browder & Spooner,
2011). Many students with SCD are meticulous about order so record keeping, charts, planning,
and banking skills can be taught well. Time management, schedules, and pictures are all still a
good place for mathematical instruction and teaching meaning. Mathematics embeds easily into
these skills and it can continue to do so as the standards based learning is also advanced and
broadened. It might be helpful if there were no distinctions between general and functional
mathematics goals. However, daily living skills may not be taught well in the general curriculum
and should be specified in IEP goals (Horstmeier, 2004).
31
Mathematics for Students with SCD
Chapter 3
Discussion
Implications
Three critical limitations seem distinctive throughout my research. First, there is a limited
amount of research available for understanding and implementing quality educational strategies
and curricula for students with severe cognitive disabilities. Two decades of research are
proportionally lacking compared to over a half century of reading research. “Mathematics has
always been an afterthought in the learning disabilities field” (Gersten & Chard, 1999, p. 25).
There are few to no studies that focus on longitudinal development in more complex numeracy,
data, and algebraic principles (Kleinert et al., 2009). We “envision” (Gersten & Chard, 1999)
research that parallels that of reading and effectively develops and assesses number sense,
mathematical reasoning, and problem solving even for the students with SCD. Related research
in the fields of development, neuroscience, genetics, and diagnostic instruments needs to be
conducted.
th
Second, special education itself is also a product of the second half of the 20 century.
National and state emphasis on education, groups such as NCTM and NCSM, and shortages in
qualified educators have impacted the education community. New general curricula are
developing that offer broader opportunities in the five strands of mathematics and problem -
based learning. Classroom teachers and special educators need training, resources, experience,
and practice (Kleinert et al., 2009). “Limited research on academic content instruction for this
population creates a challenge for practitioners seeking examples from the literature” (Browder,
2003, p. 161). It is critical that intervention programs continue to be developed. Future IEPs for
32
Mathematics for Students with SCD
students with SCD “will undoubtedly contain a blend of academic and functional objectives”
(Browder, 2003, p. 161). They will benefit from whole group and individual instruction.
Third, special educators are presented with the enormous task of being knowledgeable
about so many areas of student disability. They must become experts in all the diagnosed
disabilities such as learning disabilities, severe physical and cognitive disabilities, emotional
disabilities, and the ever-increasing disorders such as operational defiance, attention deficit
disorders, obsessive compulsive, among others. NCLB is beneficial for the students with SCD
but their teachers have been stretched beyond measure (Bicard et al., 2008). These special
educators are then also often put in charge of three to 10 adults to help support special education
students. They are then put in the position of training adults in an already taxing job with
incredible variance of need. Many special education training degrees provide only the tip of the
iceberg in learning best instructional practices, especially in academic areas. We teachers learn
best through experience. Sadly, special education teachers are limited and often changing
positions or leaving altogether (exit attrition), and this makes it difficult to become experts
(McLeskey et al., 2004).
Why can’t general education teachers fill some of these gaps in the great shortage of
special educators? General education teachers could do the job of many of the para-educators
that the system uses to help these special education classrooms and children in them. General
education teachers could share their strength in academics with the special educator’s expertise
in disability, and there could be much accomplished if we worked with one another instead of the
conflict and separation often seen. Most recent data indicates that about 86% of teachers were
prepared for each available position in special education, while more than twice as many
teachers were produced for each available elementary position (Bicard et al., 2008). We need to
33
Mathematics for Students with SCD
come up with strategies that address this immediate shortage with research, recruitment,
retention, diversity, and policy (McLeskey et al., 2004).
It is intriguing that much research in mathematics has been conducted in Europe.
Hopefully, specialists and researchers will be communicating more around the world with the
onset of technological sources to share information and findings. All will benefit. This field of
mathematical instruction for students with MLD and SCD should be increasing in the coming
years. If the national mandate to include these children in general curricula is receded, will there
be enough highly qualified teacher and knowledgeable professionals? Hopefully, teachers at all
levels and across all areas of expertise will become more aware of the potential academic
abilities of students with SCD and act accordingly.
Conclusion
Dr. Elizabetta Martinez (1999), mathematician and researcher at University at Padua in
Italy, has documented essential conclusions of over 20 years of research on progress of students
with Down Syndrome. She and other specialists suggested that these students progress through
the same stages of development as typical children. She argues that the ability to do arithmetic is
not essential to understanding other areas of math such as geometry, problem solving, data,
algebra, and measurement. “They follow the same path as their typical classmates at a slower
rate, with more steps and with individual teaching” (Bird & Buckley, 2001 p. 17). Martinez
argues that the ability to learn arithmetic facts and mental math is not essential to understanding
other areas of math such as geometry, problem solving, data, algebra, and measurement. Italian
schools have been fully including their students with disabilities for over 20 years. The United
Kingdom (UK), the Netherlands, Finland, and Australia also have done much research in this
area. If teachers, parents, and case managers could cooperate in meaningful mathematic IEP
34
Mathematics for Students with SCD
goals for students with SCD, the discrepancy between content standards and functional skills
could be minimized and a full and dynamic mathematic instruction could become the new
normal. As Martinez (1999) notes:
Students with learning disabilities can succeed in academic programs, where even
typical students may have difficulties, and can enjoy studying these programs. If
we believe the academic culture is precious and pleasing for us, why should we
not share it with people with difficulties? If it helps us, why should it not help
them? I think the right path might be a fair balance between academic programs
and training for autonomy” (p. 16).
Summary
Mathematics education is changing in many ways. New, dynamic, and balanced
standards have weathered the storm of ‘old math’ and ‘new math’. Conceptual and procedural
knowledge are interactive and influence and trigger one another (Gersten & Chard, 1999). These
standards will hopefully bring a strategic, broadened, and meaningful knowledge to mathematics
teachers and students. Cooperation instead of competition should reign. Better understanding of
cognitive disabilities should provide critical strategies in instruction and allow the recently
acknowledged potential of students with SCD to expand and flourish. Technology and
communication capabilities are providing exceptional resources to be created and implemented
among the mathematic and special education sectors of education. Mathematics instruction will
be enhanced if the community of teachers will continue to be learners and become more
knowledgeable and master this teaching.
35
Mathematics for Students with SCD
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