With OSEP support, researchers
are developing approaches that enhance students' math performance by focusing
on both the knowing and the doing aspects of mathematics. The following
examples show that effective math instruction must address the students'
deeper understanding of mathematics.
ENHANCING THE QUALITY OF MATHEMATICS FOR STUDENTS WITH DISABILITIES
"If we want students with disabilities to do better in math, teachers
have to make it more accessible and meaningful to them," asserts John
Cawley, Professor Emeritus at the University of Connecticut.
Cawley began his career in mathematics in 1953 and since that time has
focused many of his academic pursuits on helping students with disabilities
achieve mathematically. For him, both knowing and doing mathematics must
be emphasized if we are to enhance the quality of mathematics instruction
and learning for students with disabilities.
"Knowing about mathematics means the student comprehends the basic principles
of the mathematics and knows there is more than one way to explain the
mathematics and that there is frequently more than one acceptable answer,"
Cawley explains. "This is contrasted with doing mathematics, which means
the student can apply a number of different strategies and mathematics
principles to complete an item. Today, many of the problems students face
with math stem from educators neglecting the knowing and overemphasizing
the doing."
With OSEP support, Cawley has conducted many investigations that underscore
the importance of helping students with disabilities know mathematics.
Cawley shares an 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," Cawley points out. "The take away view of subtraction
is limited and fails to assist students in achieving higher standards
of mathematics knowledge and competence." Cawley maintains that subtraction
is a process that allows you to understand and find the difference between
two numbers. "The big idea for students to understand is that subtraction
represents a difference," Cawley states. "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. "In
developing an understanding of subtraction as a difference, students might
analyze and discuss situations, such as determining what number must be
added to another to make them the same, how much larger or smaller one
number is in comparison to another, or what remains of a number after
part of it has been taken away," Cawley describes. "Teachers also may
explore the reason there is a difference between two numbers; namely,
that the sets of numbers lack onetoone correspondence. Or, they may
help students understand that subtraction takes place in only one column,
whether the column be 1's, 10's, 100's and so forth."
BASIC PRINCIPLES FOR DOING SUBTRACTION
Students should be able to:
 Determine the differences
between numbers.
 Use facts accurately and
rapidly.
 Use multiple algorithms
when subtraction involves two or more digits.
 Represent subtraction
of two or more digits with expanded notation and use expanded notation
to explain the transition from manipulative to traditional symbolic
representations.
 Demonstrate knowledge
that subtraction is not cumulative and that changing the order of numbers
will change the answer.

This way of thinking also leads teachers to consider different approaches
to subtraction, including troubleshooting problems students may be having.
"Students with disabilities often have difficulty borrowing and renaming
when doing subtraction," Cawley says. "If we address the mathematics of
this difficulty, we see that we can teach students to subtract without
having to borrow."
THE INTEGRATED AND FLEXIBLE UNDERSTANDING OF NUMBERS
Each year, teachers in the middle grades are faced with a range of student
abilitiessome students do not know their multiplication tables, some have
difficulty with long division, others may not be able to complete subtraction
problems efficiently. John Woodward, researcher at the
University of Puget Sound, explains the issue of skill deficits in this
way: "Most of the difficulties teachers experience stem from the way they
look at their students' math performance. Their remediation strategies are
largely based on a traditional conception of arithmetic and mastery learning.
Rather, teachers' goals should be to increase their students' flexible understanding
of numbers."
With OSEPfunding, Woodward and his colleague Juliet Baxter, have been
studying how students might use their knowledge of math facts as they
learn more conceptual algorithms for basic operations. In Woodward's approach,
facts extend into the critical numeric skills of mental computation and
approximation. "In our classrooms, mathematical topics have been reorganized
to help students achieve a greater flexibility with numbers and provide
them with an adequate foundation on basic operations," Woodward asserts.
"This enables them to move on to more complex topics such as decimals,
geometry, and negative numbers."
To understand Woodward's approach, consider the case of Tina, a sixth
grade student with mild disabilities who doesn't know her multiplication
tables. In Woodward's approach, the easy multiplication facts (e.g., X1,
X2) are identified as one distinct set. Tina passes the pretest, so she
skips on to the harder facts. She also is shown strategies for learning
facts, such as doubling. For example, Tina found it easier to learn 6X4
by starting with 6X2 and then doubling it. Another strategy Tina found
useful involved helping or near facts. When Tina encountered a difficult
fact, such as 6X7, she used a helping or near fact to solve it (e.g.,
Tina knew 6X6 =36, so she added 6 to 6X7 to find the answer).
"Two related topics that are intimately related to a flexible understanding
of numbers are the skills of mental computation and approximation," Woodward
says. Students are shown how they can apply their knowledge of basic facts
to approximate the answers to problems with large numbers. For example,
a problem such as 12,654  5,788 can be converted to approximate numbers.
A beginning strategy might be to simply round up to 13,000  6,000 and
use the fact 13  6 to yield an answer of 7,000. As students become more
comfortable with mental calculations and approximations, other strategies
for rounding numbers may be introduced (e.g., 654 and 788 are close in
quantity and thus, it may be just as sensible to round to 12,000  5,000).
"Such strategies help students learn to see other ways of looking at numbers,"
Woodward summarizes.
Woodward and his colleagues have created a variety of curriculum materials
that exemplify his approach. You can find a selection of them on his web
site at http:olsoncs.wou.edu/transmath/.
Thinking About Math Differently— Teacher's Experience
For Cal Young Middle School (Oregon) special education teacher Linda
Vie, getting her students to think about math required some personal
changes. "To use Woodward's approach, I had to let go of some things I
thought were importantlike teaching long division without calculators,
which, by the way, I am very good at! I realized that the students needed
to understand the concept of division, and that once they understood it,
they could do it on the calculator."
Vie has integrated discussion about numbers into all aspects of her mathematics
teaching. "When I started with the students, they could do some basic
rote math facts okay, but they did not understand numbers." Vie now pairs
the methodical teaching of math facts with discussions about the meaning
of numbers. "When we are doing addition, we talk about the meaning of
the numbers," Vie explains. "We also are careful to refer to numbers correctly."
She cites the following example: When presented with an addition fact,
9 + 3, you often will hear the statement, "Carry the 1." Yet, it is not
a "one" that is being carried but a ten. Vie notes the importance of breaking
down numbers with students to help them understand what they are doing.
"Thinking about what the numbers mean lays the foundation for other skills—
as looking at an answer and asking, Does my answer make sense?"
Vie has carried these discussions into students' work with math word
problems. "Before we start solving the problem, we think about what the
problem is asking. We look for necessary information, and sometimes even
rewrite what the problem is asking in our own words," Vie describes. "We
then go through every sentence, asking if the information is necessary
to solving the problem. Before students ever attempt to work the problem,
they must know what is needed to work the problem."
"Too often, special education
math instruction focuses on...repeated practice with limited opportunities
for students to explain verbally their reasoning and receive feedback
on their evolving knowledge of concepts and strategies....special education
mathematics instruction continues to focus on computation rather than
mathematical understanding."
Russell Gersten and David Chard
Researchers 
Mathematical Problem Solving Instruction
Tiki and Dale are going to the movies. They have $12 between them.
Tickets cost $4.50 each. How much will they have left over for snacks?
Ms. Anderson needs to purchase enough paint to cover the walls and
ceiling in an 8' by 8' room. One gallon of paint covers 200 square feet.
How much paint does she need?
The Bailey family is planning a vacation to the city. The map shows
the mileage to be 180 miles. If the family drives 55 miles per hour
75 percent of the way and 35 miles per hour the rest of the distance,
how much time will it take to get there?
Each day, we are presented with problems which require math. How can
we help students to be better math problem solvers? "We can start by understanding
that students who have problems with mathematics typically do not know
how to decide what to do," points out University of Miami researcher Marjorie
Montague. "Teaching students how to decide what to do is a cognitive
process critical to instruction in mathematical problem solving."
More than a decade ago, Montague received OSEPfunding to begin studying
the problems students with disabilities have solving math word problems.
Since then, her research has shown that effective and efficient mathematical
problem solving depends on the ability to select and apply taskappropriate
cognitive and metacognitive processes and strategies for understanding,
representing, and solving problems. Montague describes cognitive processes
as the "to do" strategies (see sidebar, Cognitive Processes Associated
with Problem Solving), and metacognitive processes as the reflective strategies
(e.g., "What am I doing?" and "What have I done?").
COGNITIVE PROCESSES ASSOCIATED WITH PROBLEM SOLVING
 Comprehending linguistic
and numerical information in the problem.
 Translating and transforming
that information into mathematical notations, algorithms, and equations.
 Observing relationships among the elements of the problem.
 Formulating a plan to
solve the problem.
 Predicting the outcome.
 Regulating the solution
path as it is executed.
 Detecting and correcting
errors during problem solution.
Source: Montague,
M. (2002). Mathematical problem solving instruction: Components, procedures,
and materials. In M. Montague, & C. Warger (Eds.), Afterschool
extensions: Including students with disabilities in afterschool programs.
Reston, VA: Exceptional Innovations. 
"To be good problem solvers, teachers need to know what good problem
solvers do," Montague says. To help teachers understand the knowledge
and skills needed to be effective and efficient mathematical problem solvers,
Montague developed the Solve It! approach. Solve
It! is a researchbased instructional program in which teachers
explicitly teach the processes and strategies that underlie mathematical
problem solving. It 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 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 algorithms.
 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 path.
 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.
"Metacognitive strategies
help problem solvers gain access to strategic knowledge, provide guidance
in applying the cognitive strategy, and regulate the use of the cognitive
strategies and overall performance," Montague adds.
A Look at Solve It!
in the Classroom
According to Gretchen
Daniel, it was easy getting her Dublin City (Ohio) Middle School students
with learning disabilities hooked on Solve It! "When you
hear sixth grade boys call math fun, you know something is working right!"
Daniel has been implementing
the Solve It! approach in pullout settings with students
who receive resource room support for math difficulties. "I use explicit
instruction to teach the strategy—I
model each cognitive process, I
have the students verbalize the cognitive activities until they become
automatic, I provide opportunities for students to practice with their
peers, and I continually monitor their progress and provide feedback."
Although Daniel stresses
the importance of replicating Montague's approach as it was researched—"if
you want the program to work, you need to implement it as it was written"—
added a few minor strategies to meet the individual needs of her students.
"My sixth graders wanted to create a mnemonic for the cognitive steps
(read, paraphrase, visualize, hypothesize, estimate, compute, check),"
Daniel reports. "The students came up with rhinos play vicious hyenas,
except competitive camels— they used until they internalized
the steps." In addition, Daniel addressed a motivational issue. "Many
of my students had previously failed at math and needed help developing
a can do attitude," Daniel explained. The solution came in the
form of affirmations that students internalized. "We made posters that
stated Strategy Use = Success! Students also quickly adopted
the phrase, I will use the strategy and do well."
Overall, Daniel found that
it took three to four sessions for students to master the strategy. "The
key is to thoroughly understand the strategy yourself so that you can
teach it and model it."
RESOURCE
READING
AND WRITING THE BRAILLE CODE OF MATHEMATICS
Imagine you are unable to
read or write the symbols that comprise mathematics. You would be forced
to learn concepts and perform calculations entirely in your head, limiting
your ability to master the intricacies of mathematics. Unfortunately,
many students who are blind find themselves in this situation.
To address this need, OSEP
has supported researchers Gaylen Kapperman and Jodi
Sticken of Northern Illinois University in developing an interactive
software tutorial that can be used by students who are blind. The software
helps them to study the Nemeth Code (the Braille code for mathematics).
The software is installed in a Braille Litea small, portable Braille
notetaker that is equipped with synthetic speech and a refreshable Brailled
display. The speech and tactile Braille features enable students to study
the Nemeth Code independently or under the direction of a teacher. A tutorial
on the Nemeth Code for sighted individuals also is available.
The software program may be
downloaded from Freedom Scientific at www.freedomscientific.com/fs_downloads/notenemeth.asp.
The tutorial for sighted individuals is available from the Texas School
for the Blind and Visually Impaired web site at www.tsbvi.edu/math/mathresources.htm#Download.
For more information, contact Kapperman [gkapper@niu.edu]
and Sticken [jsticken@niu.edu]
at the Department of Teaching and Learning, Northern Illinois University,
DeKalb, IL 60115. 