ERIC Identifier: ED468579
Publication Date: 2002-07-00
Author: Warger, Cynthia
Source: ERIC Clearinghouse on
Disabilities and Gifted Education Arlington VA. ERIC/OSEP Special Project.
Helping Students with Disabilities Participate in
Standards-Based Mathematics Curriculum. ERIC/OSEP Digest.
The bar on what students with disabilities are expected to learn was raised
by the 1997 Amendments to the Individuals with Disabilities Education Act
(IDEA), which emphasize students' participation and progress in the general
education curriculum. Navigating the general education math curriculum has
become a key to student success.
The mathematics curriculum has changed over the last 20 years due to
educational reforms driven by standards. A significant element driving this
change is the National Council of Teachers of Mathematics (NCTM) Principles and
Standards for School Mathematics (first published in 1989 and revised in 2000),
which focus on conceptual understanding and problem solving rather than
procedural knowledge or rule-driven computation. Most states and districts have
used the NCTM Standards to some degree in revamping their mathematics curricula.
[For more information, visit the NCTM web site at http://standards.nctm.org.]
The challenge for teachers is to provide effective math instruction to
students with disabilities so they can meet the high standards set for what all
students must be able to know and do mathematically. Unfortunately, many
students with disabilities experience difficulties with the reformed math
curriculum. As University of Maryland researchers Paula Maccini and Joe Gagnon
have found, students may have difficulty processing and distinguishing relevant
information, have deficits in computational skills, or lack reasoning and
problem-solving skills. But with the right support, students with disabilities
can succeed in a higher level math curriculum.
For many years, the US Department of Education, Office of Special Education
Programs (OSEP) has supported research to improve mathematics achievement for
students with disabilities. This ERIC/OSEP Digest examines how selected
researchers are informing practice in four areas: enhancing students'
understanding of mathematics, teaching students mathematical problem-solving
strategies, using assistive technology in instruction and assessment, and making
accommodations to support student participation in state and district-wide
ENHANCING STUDENT'S UNDERSTANDING OF MATHEMATICS
Cawley, Professor Emeritus at the University of Connecticut, has found that for
students with disabilities to do better in math, math must be meaningful for
them. Both knowing and doing mathematics must be emphasized to enhance the
quality of mathematics instruction and learning for students with disabilities.
Knowing about mathematics means that the student comprehends the basic
principles of a mathematics problem, knows there is more than one way to explain
the mathematics of the problem, and knows that there is frequently more than one
acceptable answer. This is in contrast to doing mathematics, which means the
student can apply a number of different strategies and mathematics principles to
complete an item. Cawley believes that many of the difficulties students face
with math stem from educators' neglecting the "knowing" and overemphasizing the
Consider this example that highlights the distinction between knowing about
subtraction and being able to do subtraction. Subtraction as a mathematical
topic is much more meaningful than the rote computation take-away approach that
has been advocated for students with disabilities since the 1920s. It is a
process that allows the student to understand and find the difference between
two numbers. The big idea for students to understand is that subtraction
represents a difference. Knowing about subtraction involves reasoning in the
form of proof and explanation. It also involves the ability to demonstrate the
connectedness between one facet of mathematics (e.g., subtraction) and another
(e.g., addition). Cawley has found that understanding subtraction in this way
offers teachers numerous opportunities to stress number sense and skill
development, which can result in improved student understanding and performance.
TEACHING STUDENTS STRATEGIES FOR MATHEMATICAL PROBLEM SOLVING
According to University of Miami researcher Marjorie Montague, a
major focus of the NCTM standards and of reformed math curriculum is problem
solving. Her research has shown that effective mathematical problem solving
depends on the ability to select and apply task-appropriate cognitive and
metacognitive processes and strategies for understanding, representing, and
solving problems. Montague describes cognitive processes as the "to do"
strategies and metacognitive processes as the reflective strategies (e.g., "What
am I doing?" and "What have I done?").
To help teachers understand the knowledge and skills needed for effective and
efficient mathematical problem solving, Montague developed Solve It!, an
approach that incorporates the cognitive processes critical to mathematical
problem solving in each step of the strategy:
* Reading the problem. Students are taught how to read mathematical problems,
including using reading strategies to understand the problem (e.g., focusing on
important information), developing mathematical vocabulary, and recognizing when
they do not understand relationships among mathematical terms and quantitative
concepts expressed in a problem.
* Paraphrasing. Students are taught how to put the problem into their own
words and convey meaning.
* Visualizing. Students are taught to draw a representation on paper or to
make a mental image of the problem.
* Hypothesizing about problem solutions. Students are taught how to decide
the number of operations that are needed to solve the problem, select and order
the operations, and then to transform the information into correct equations and
* Estimating the answer. Students are taught how to stay focused on the type
of outcome (e.g., number of yards rather than feet), and then how to predict the
answer by using the information in the problem and their projected solution
* Computing. Students are taught how to recall the correct procedures for
working through the algorithms and the necessary math facts for accuracy.
* Checking the problem. Students are taught how to check the mathematical
problem solving process to ensure that they have understood the problem,
accurately represented the problem, selected an appropriate solution path, and
solved the problem correctly.
In the Solve It! approach, students also learn a metacognitive strategy that
they apply at each step. The strategy includes the following steps:
* Say aloud or to themselves what the problem is asking them to do.
* Ask themselves if they understand the problem.
* Check their progress.
USING ASSISTIVE TECHNOLOGY FOR INSTRUCTION AND
IDEA provides that assistive technology will be considered for
students with disabilities as part of their individualized education program
(IEP) planning. Researchers have made significant advancements in providing
technology tools to support mathematics achievement. Examples include the
* Interactive software for students who are blind. Many students who are
blind are unable to read or write the symbols that comprise mathematics, and
thus, must learn concepts and perform calculations entirely in their heads,
limiting their ability to master the intricacies of mathematics. To address this
need, Gaylen Kapperman and Jodi Sticken of Northern Illinois University
developed an interactive software tutorial that helps them to study the Nemeth
Code (the Braille code of mathematics). The software is installed in a Braille
Lite-a small, portable Braille note taker that is equipped with synthetic speech
and a refreshable Brailled display.
* A CD-ROM for students who use American Sign Language (ASL) to communicate.
Using multimedia, Jean Andrews and Donald Jordan at Lamar University developed
the Meet the Math Wiz CD-ROM series that helps students focus on math word
problems over six grades of math difficulty using multicultural names, stories,
and themes. The program features Chris Kurtz, a math teacher who is deaf. He
welcomes users to his castle, where he describes, among other things, a
four-point plan for solving math word problems. He leads users into eight
demonstrations per CD, giving them an ASL translation of the problem, an
animation hint, and an explanation of how to solve the problem in ASL.
MAKING ACCOMMODATIONS TO SUPPORT STUDENT PARTICIPATION IN STATE AND DISTRICT ASSESSMENTS
Most State and district-wide assessments tap
mathematical knowledge and skills. Given the controversy surrounding the use of
accommodations as evidenced by state policy analysis, it is important to know
what the research currently indicates in order to help make appropriate
To help practitioners access emerging research that addresses accommodations
for students with disabilities, Martha Thurlow at the National Center on
Educational Outcomes has created an online, searchable data base of
accommodations (http://education.umn.edu/NCEO/AccomStudies.htm). The database
allows users to search empirical research studies on the effects of various
testing accommodations for students with disabilities.
Andrews, J. & Jordan, D. (Undated). Meet the
Math Wiz. (CD-ROM). Curriculum Publications Clearinghouse, Horrabin Hall 46,
Western Illinois University, 1 University Circle, Macomb, IL 61455,
Cawley, J. (2002). Mathematics interventions and students with high incidence
disabilities. Remedial and Special Education, 23(1), 2-6.
Freedom Scientific, Inc. Nemeth Code Self-Study Instructional Material.
Download from http://www.freedomscientific.com/fs_downloads/notenemeth.asp%20(July
Gagnon, J., & Maccini, P. (2001). Preparing students with disabilities
for algebra. Teaching Exceptional Children, 34(1), 8-15.
Montague, M., Applegate, B., & Marquard, K. (1993). Cognitive strategy
instruction and mathematical problem-solving performance of students with
learning disabilities. Learning Disabilities Research and Practice, 8(4),
223-232. For more information, contact Montague at outreach@Miami.edu.