Math Problem Solving for Primary Elementary Students with Disabilities
Download this document: Microsoft Word | Adobe PDFAbout the Author
Marjorie Montague, Ph.D. , is a former president of the Division for Research, Council for Exceptional Children, and is currently a professor at the University of Miami focusing on learning disabilities and emotional/behavioral disorders. She previously authored Math Problems Solving for Middle School Students in November, 2004 and for Upper Elementary Grades in January, 2005. In this document, Dr. Montague applies strategies for students in the primary elementary grades.
Math Problem Solving for Primary School Students with Disabilities
This
brief is part of a series on teaching primary, upper elementary, and
middle school students with disabilities how to solve mathematical
word problems. The skills and strategies needed for successful mathematical
problem solving develop from the preschool years, when children acquire
a basic conceptual understanding of the base 10 numerical system. During
these early years, they typically develop the “number sense” needed
to process and manipulate numerical information. In primary school,
children continue to acquire mathematical knowledge and skills and
are exposed to a variety of math problem types requiring addition and
subtraction operations. Students in upper elementary school continue
to apply and refine the skills and strategies necessary to solve real
life mathematics problems. By middle school, students should be able
to apply mathematical problem-solving skills and strategies effectively
and efficiently in school, at home, and in the community.
Most children enter kindergarten with a fairly developed sense of numbers (Case, 1998, as cited in Gersten & Chard, 1999). Children with number sense can invent their own procedures for doing math, they can represent numbers in many different ways, and they can recognize benchmark numbers and number patterns. They also seem to understand the magnitude of numbers and recognize when a number’s order of magnitude does not make sense. They are able to talk about quantities without actually performing operations. Children with well-developed number sense are comfortable with mathematics and enjoy solving math problems.
This brief deals specifically with math problem solving for primary school students. Many students in kindergarten through grade 3, especially students with learning disabilities (LD), have difficulty learning how to solve math word problems because they often do not have the necessary conceptual bases. From kindergarten through third grade, both children who have reading skills and those who must be read to are expected to be able to solve problems. The following are examples of second and third grade problems:
- Charlie has 15 stickers. His friend Tony gave him 6 more stickers. How many stickers does Charlie have altogether?
- Sara has 14 balls. Amy has 5 balls. How many more balls does Sara have?
For students who can read, most textbooks are not very helpful when
it comes to teaching students how to solve math problems. They typically
provide a four-step formula: (a) read the problem, (b) decide what
to do, (3) compute, and (4) check your answer. “Read” the problem for
understanding is the first step. Understanding involves a representation
of the relationship between numbers, words, and symbols in the problem.
This representation provides the basis for deciding what to do to solve
the problem. From early on, most students acquire the skills and strategies
needed to “read the problem” and “decide what to do” to solve it. Many
students with LD or other cognitive impairments, however, do not easily
acquire these skills and strategies. Therefore, they need explicit
instruction in mathematical problem-solving skills and strategies to
solve problems in their math textbooks and in their daily lives.
The
following frequently asked questions provide the framework for this
brief and the other briefs in this series brief.
- What is mathematical problem solving?
- How do good problem solvers solve math problems?
- Why is it so difficult to teach students to be good math problem solvers?
- What is the content of math problem-solving instruction?
- What are effective instructional procedures for teaching math problem solving?
Several validated practices for teaching young children math problem
solving are described, and a sample lesson is provided. Additionally,
specific adaptations and accommodations are provided for students with
other types of cognitive disabilities such as traumatic brain injury
(TBI) and attention deficit hyperactivity disorder (ADHD).
What is mathematical problem solving?
Mathematical problem solving is a complex cognitive activity involving
a number of processes and strategies. Problem solving has two stages:
problem representation and problem execution. Successful problem solving
is not possible without first representing the problem. Appropriate
problem representation indicates the problem solver has understood
the problem, and it guides the student toward the solution plan. Students
who have difficulty representing math problems will have difficulty
solving them.
Students in primary school, kindergarten through grade
3, are acquiring declarative and procedural knowledge (i.e., math facts
and math operations) in addition and subtraction. They are expected
to be able to apply this knowledge in solving “simple” addition and
subtraction word problems, but these “simple” problems are more complicated
than at first glance. There are actually four addition and subtraction
problem types: change problems, compare problems, equalize problems,
and combine problems. Within the change, compare, and equalize problem
types, there are six variations of the position of the unknown quantity.
Within the combine problem type, there are two variations. Therefore,
students with mathematical difficulties need explicit instruction in
20 different variations of “simple” primary-level word problems. The
position of the unknown quantity has a major influence on the difficulty
level of the problem. So, in line with good principles of instruction
for students with disabilities, problem types should be introduced
beginning with the easiest problem type, and, after mastery, move to
the next level of difficulty (see García, Jiménez, & Hess, in press,
for a taxonomy of the difficulty levels of the four types of addition
and subtraction problems for students with mathematical difficulties).
Teaching mathematical problem solving
is a challenge for many teachers, many of whom rely almost exclusively
on mathematics textbooks to guide instruction. Unfortunately, most
basal series do not provide the teacher with the tools to teach problem
solving. Instruction for primary children with math learning disabilities
must progress gradually from the concrete to the abstract. That is,
symbolic representation may not be possible for these children without
explicit instruction that begins with manipulatives and other tactile
materials that are concrete in nature (e.g., abacus, base ten blocks,
Cuisenaire rods) that will help students move from a concrete to a
more symbolic, schematic level. In other words, teachers must provide
systematic, progressive, and scaffolded instruction that considers
the students’ prior knowledge and cognitive strengths and weaknesses.
Students
who have difficulty solving math word problems usually do not construct
a representation of the problem that considers the relationships among
the problem components and, as a result, they do not understand the
problem and have no clue about a plan to solve it. So, it is not simply
a matter of “drawing a picture or making a diagram;” rather, it is
the type of picture or diagram that is important. Effective visual
representations, whether with manipulatives, with paper and pencil,
or in one’s imagination, show the relationships among the problem parts.
These are called schematic representations (van Garderen & Montague,
2003). Poor problem solvers tend to make immature representations that
are more pictorial than schematic in nature. The illustration below
shows the difference between a pictorial and a schematic representation
of the mathematical problem presented earlier in the brief.
Other cognitive processes and strategies needed for
successful mathematical problem solving include paraphrasing the problem,
which is a comprehension strategy, hypothesizing or setting a goal
and making a plan to solve the problem, estimating or predicting the
outcome, computing or doing the arithmetic, and checking to make sure
the plan was appropriate and the answer is correct (Montague, 2003;
Montague, Warger, & Morgan,
2000). Mathematical problem solving also requires metacognitive or
self-regulation strategies. Students with LD are notoriously poor self-regulators.
In this developmental period in particular, it is imperative that they
be explicitly taught how to self-instruct (tell themselves what to
do), self-question (ask themselves questions), and self-monitor (check
themselves as they solve the problem).
What do good problem solvers do?
Good problem solvers
use a variety of processes and strategies as they read and represent
the problem before they make a plan to solve it (Montague, 2003). First,
they READ the problem for understanding. As
they read, they use comprehension strategies to translate the linguistic
and numerical information in the problem into mathematical notations.
For example, good problem solvers may read the problem more than once
and may reread parts of the problem as they progress and think through
the problem. They use self-regulation strategies by asking themselves
if they understood the problem. They PARAPHRASE the problem
by putting it into their own words. They identify the important
information and may even underline parts of the problem. Good problem
solvers ask themselves what the question is and what they are looking
for. VISUALIZING or drawing a picture or diagram means
developing a schematic representation of the problem so that the picture
or image reflects the relationships among all the important problem
parts. Using both verbal translation and visual representation, good
problem solvers not only are guided toward understanding the problem,
but they are also guided toward developing a plan to solve the problem.
This is the point at which students decide what to do to solve the
problem. They have represented the problem and they are now ready to
develop a solution path. They HYPOTHESIZE by thinking
about logical solutions and the types of operations and number of steps
needed to solve the problem. They may write the operation symbols as
they decide on the most appropriate solution path and the algorithms
they need to carry out the plan. They ask themselves if the plan makes
sense given the information they have. Good problems solvers usually ESTIMATE
or predict the answer using mental calculations, or may even
quickly use paper and pencil as they round the numbers up and down
to get a “ballpark” idea. They are now ready to COMPUTE. So
they tell themselves to do the arithmetic and then compare their answer
with their estimate. They also ask themselves if the answer makes sense
and if they have used all the necessary symbols and labels such as
dollar signs and decimals. Finally, they CHECK to
make sure they used the correct procedures and that their answer is
correct.
These processes and strategies are developmental in nature
and reach maturation at different stages. For example, visualization
matures in most learners sometime between the ages of 8 and 11. Primary
school students with math disabilities will need cues and prompts and
other ongoing supports as they learn how to solve problems and also
during practice sessions. Typically, these children have memory and
self-regulation problems, so instruction must provide compensatory
supports to mediate the cognitive problems that interfere with problem
solving. Many children will require cues and prompts not only during
instruction but also following instruction to maintain the problem-solving
skills and strategies they have learned.
Why is it so difficult
to teach students to solve math problems?
Students who are
poor mathematical problem solvers, as most students with LD are, do
not process problem information effectively or efficiently. They lack
or do not apply the resources needed to complete this complex cognitive
activity. Generally, these students also lack metacognitive or self-regulation
strategies that help successful students understand, analyze, solve,
and evaluate problems. To help these students become good problem solvers,
teachers must understand and teach the cognitive processes and metacognitive
strategies that good problem solvers use. This is the CONTENT of
math problem-solving instruction. Teachers must also use instructional PROCEDURES that
are research-based and have proven effective. These procedures are
the basis of COGNITIVE STRATEGY INSTRUCTION, which
has been demonstrated to be one of the most powerful interventions
for students with LD (Swanson, 1999).
What is the content
of math problem-solving instruction?
The previous sections
described the content of math problem-solving instruction as the cognitive
processes and metacognitive strategies that good problem solvers use
to solve mathematical problems. Students learn how to use these processes
and strategies not only effectively, but efficiently as well. For primary
school students, teaching needs to be systematic. The focus should
be on reading, paraphrasing, and visualizing. Manipulatives are essential.
A representation using manipulatives is developed with the support
of the teacher. Based on the three-dimensional representation, a schematic
representation using paper and pencil is formed. Eventually the representation
is transformed into a symbolic mathematical representation using mathematical
notation.
Jitendra, Griffin, Haria, Leh, Adams, and Kaduvetoor (2005) have developed a procedure called schema-based strategy instruction for teaching primary school students how to solve change, group, and compare problems. They use a four-step process.
- Find the problem type.
- Organize the information in the problem using the diagram.
- Plan to solve the problem.
- Solve the problem.
For each problem, a diagram and self-regulation checklist was provided. See the below figure for an example of the diagram and checklist for a change problem.
What are effective instructional procedures for teaching math problem solving?
Explicit Instruction
Explicit Instruction,
the basis of cognitive strategy instruction, incorporates research-based
practices and instructional procedures such as cueing, modeling, verbal
rehearsal, and feedback. The lessons are highly organized and structured.
Appropriate cues and prompts are built in as students learn and practice
the cognitive and metacognitive processes and strategies. Each student
is provided with immediate, corrective, and positive feedback on performance.
Overlearning, mastery, and automaticity are the goals of instruction.
Explicit instruction allows students to be active participants as they
learn and practice math problem-solving processes and strategies. This
approach emphasizes interaction among students and teachers.
Through
an extensive and statistical review of the intervention studies conducted
over 20 years with students with LD, Swanson (1999) identified the following
eight components of effective strategy instruction. They are described
as they would be used in teaching mathematical problem solving.
Sequencing
and Segmenting
Sequencing and segmenting
means breaking the task into component subparts, providing short activities,
and synthesizing the parts into a whole. For example, each cognitive
process/self-regulation strategy routine is taught consecutively, beginning
with reading the problem as a necessary first step for solving the
problem. Students are taught to read the problem and then to ask themselves
if they understood it. They are then taught to go back and reread it
or read parts until they decide they understand it. At the beginning
of instruction, the teacher models the process and provides plenty
of step-by-step cues and prompts as students practice. Eventually these
cues and prompts are phased out. After students know what to do when
they read math problems, they learn how to paraphrase or retell problems.
Students learn the paraphrasing routine, which is then added to the
reading routine. Subsequently, students have mastered a sequence of
two important processes for solving mathematical problems.
Drill-repetition and Practice-review
This component
includes progress checks to measure skill mastery, sequenced review,
repeated practice, distributed review and practice using the same or
similar practice problems, and ongoing and positive feedback. For example,
the paraphrasing routine is taught and then students practice on their
own or with peers.
PARAPHRASE (your own words)
Say: Underline the important information. Put the problem in my own words.
Ask: Have I underlined the important information? What is the question?
What am I looking for?
Check: That the information goes with the question.
After the teacher models the routine and guides the students as they
go through the routine, they are provided with practice until the routine
becomes automatic. As they learn how to paraphrase math word problems,
they can evaluate themselves using a checklist, and plot their improvement
on a graph.
Directed Questioning and Responses
Cognitive strategy
instruction uses a guided discussion technique to promote active teaching
and learning. Students are engaged from the very beginning through
an initial discussion of the importance of mathematical problem solving.
With the teacher, they set individual performance goals and make a
commitment to becoming a better problem solver. Teachers ask both “process-related”
and “content-related” questions. Students are directed by the teacher
to ask questions. Students are also taught when and how to ask for
help.
Control Difficulty or Processing Demands of the Task
Arrange
tasks from easy to difficult. The teacher provides simplified demonstrations,
necessary assistance, appropriate cues and prompts, and guided discussion.
For primary school students, math problem-solving instruction should
start with one-step change addition problems in which the “ending”
is unknown (see Figure 2). When students have mastered the problem
solving routine with problems of this type, they can progress to one-step
change problems in which the change is unknown. They can then progress
to change problems in which the beginning is unknown, and so forth.
Technology
Technology extends beyond calculators
and computers to include structured text, diagrams, flow charts, structured
curricula, scripted lessons, and video demonstrations. Students who
are learning to be better math problem solvers should be taught how
to use calculators to facilitate computation after their understanding
of math facts for addition and subtraction have been mastered.
Group
Instruction
Students with LD who have math problem
solving difficulties should be taught in small groups (5-8 students) to
maximize teacher and student interaction. Interaction between teachers
and students and among peers is the cornerstone of cognitive strategy instruction.
Cognitive strategy instruction is intensive and time-limited.
Supplements
to Teacher and Peer Involvement
Children with second and third
grade reading skills can be given cue cards to study for homework as they
memorize and learn the various problem-solving processes and self-regulation
strategies. For example, using the Jitendra, et al., strategy for change
problems, students first learn to “find the problem” type by reading the
problem, retelling the problem, and asking themselves if it is a change
problem. They check the box on the self-monitoring checklist as they complete
the task. When they have mastered how to “read” and retell a math word
problem, they advance to the organization step. Each step is added successively
until they have learned and applied the entire routine. After small group
instruction or homework, students are expected to return to the general
education math class and use what they have learned about solving math
problems. General education teachers must be made aware of the instruction
that students are receiving and supplement and support this instruction
in the general education math classes.
To do this, it is essential that
general and special education teachers communicate regularly about the
children and the instruction and coordinate what is taught in the general
education class with what is taught in the special education resource class
and vice versa. Continuity across general and special education is essential
for student success. General education teachers must reinforce what students
have learned to ensure that they apply appropriately, and also maintain,
acquired skills and strategies.
See page 17 for an example of how a general education teacher and a special educator
collaborate on instruction in math problem solving for their second grade students.
Strategy
Cues
Students who have a second or third grade reading level are given
reminders and prompts such as individual Student Cue Cards to carry with them
for home and class use, Master Class Charts on the classroom walls, problem
type diagrams and checklists, think-aloud protocols, and discussion about
the benefits of using strategies.
The next section presents several instructional procedures that are the
foundation of cognitive strategy instruction. These include verbal rehearsal,
process modeling, visualization, role reversal, performance feedback, distributed
practice, and mastery learning.
Verbal Rehearsal
Before students actually solve problems,
they must first learn the steps and memorize them by using verbal rehearsal.
This is a memory strategy that enables students to recall automatically
the math problem-solving processes and strategies. Students in primary
school can learn a SAY, ASK, CHECK routine similar to Montague’s (2003)
or Jitendra’s (2005) as they are learning how to represent problems. Frequently,
acronyms are created to help students remember as they verbally rehearse
and internalize the labels and definitions for the processes and strategies.
For Jitendra’s math problem-solving routine, the acronym FOPS was
created (F = Find the problem type, O =
Organize the information using the change diagram, P =
Plan to solve the problem, and S = Solve the problem.
For non-reading students visual prompts can be used as substitutes for
letters or along with letters. Cues and prompts are used to help students
as they memorize the processes and their definitions. When students have
memorized the math problem-solving routine, they can cue other students
and the teacher during practice sessions.
Process Modeling
Process modeling is thinking aloud while
demonstrating an activity. For mathematical problem solving, this means
that the problem solver says everything she or he is thinking and doing
while solving a problem. When students are first learning how to apply
the processes and strategies, the teacher demonstrates and models what
good problem solvers do as they solve problems. Students have the opportunity
to observe and hear how to solve mathematical problems. Both correct and
incorrect problem-solving behaviors are modeled. Modeling of correct behaviors
helps students understand how good problem solvers use the processes and
strategies appropriately. Modeling of incorrect behaviors allows students
to learn how to use self-regulation strategies to monitor their performance
and locate and correct errors. Self-regulation strategies are learned and
practiced in the actual context of problem solving. When students learn
the modeling routine, they then can exchange places with the teacher and
become models for their peers. Initially, students will need plenty of
prompting and reinforcement as they become more comfortable with the problem-solving
routine. However, they soon become proficient and independent in demonstrating
how good problem solvers solve math problems. One of the instructional
goals is to gradually move students from overt to covert verbalization.
As students become more effective problem solvers, they will begin to verbalize
covertly and then internally. In this way, they not only become more effective
problem solvers, but they also become more efficient problem solvers.
Visualization
Visualization is critical to problem representation.
It allows students to construct an image of the problem on paper or mentally.
Students must be shown how to select the important information in the problem
and develop a schematic representation. To do this, teachers model how
to use manipulatives to represent a problem, and then how to draw a picture
or make a diagram that shows the relationships among the problem parts
using both the linguistic and numerical information in the problem. These
three-dimensional and two-dimensional visual representations can take many
forms and will vary from student to student. As students become better
problem solvers, they will use a variety of visual representations including
manipulatives, pictures, tables, graphs, or other types of displays. Initially,
students must be shown how to use the manipulatives and also how to translate
the results of their manipulations with concrete objects to more symbolic
representations using paper and pencil, e.g., the problem type diagrams.
Later, as students become more proficient, they will progress to mental
images. Interestingly, if the problem is novel or challenging, they frequently
return to conscious application of processes and strategies, which is typical
of good problem solvers.
Role Reversal
Role reversal is an important instructional
activity that promotes independent learners. As students become familiar
with the math problem-solving routine, they can take on the role of teacher
as model and actually change places with the teacher. They may use an overhead
projector just as the teacher did and engage in process modeling to demonstrate
that they can apply effectively the cognitive and metacognitive processes
and strategies they have learned. Other students can prompt or ask questions
for clarification. In this way, students learn to think about, explain,
and justify their visual representations and their solution paths. Teachers
may also take the role of the student, guiding the “student as teacher”
through the process.
Performance Feedback
Performance feedback is critical
to the success of the program. Progress checks are given throughout instruction
to determine mastery of the routine. Teachers and parents can assist students
with graphing their progress and visually displaying their performance,
which is very reinforcing for them. Teachers carefully analyze performance
during practice sessions and on mastery checks and provide each student
with immediate, corrective feedback. Appropriate use of processes and strategies
is reinforced continuously until students become proficient. Students need
to know the specific behaviors for which they are praised so they can repeat
these behaviors. Praise should be honest. Students should be taught how
to give and receive reinforcement and to reinforce themselves, and should
have plenty of opportunities to practice doing it. The goal is to teach
students to monitor, evaluate, and reinforce themselves as problem solvers.
Distributed
Practice
Distributed practice is the cornerstone for ensuring
that students maintain what they have learned. To become good math problem
solvers, students learn to use the processes and strategies that successful
problem solvers use. As a result, their math problem-solving skills and
performance levels improve. However, to achieve high performance, students
must be given ample opportunity to practice initially as they learn the
math problem-solving routine and, then, to maintain high performance, they
must continue to practice intermittently over time. They may practice individually
or in teams or small groups. They should be involved in solving a range
of problems from textbook-type problems to problems encountered in real
life.
Mastery Learning
Prior to instruction, a pretest is given
to determine baseline performance levels of individual students. During
instruction, periodic mastery checks are given to monitor student progress
over time and to determine effectiveness of the program. If students are
not making sufficient progress, modifications to the program to ensure
success must be made. Following instruction, periodic maintenance checks
are provided. If students do not meet criterion on maintenance checks,
booster sessions must be provided to improve performance levels to mastery.
Booster sessions are brief lessons to review and refresh what students
have previously learned and mastered.
Teaching Math Problem Solving
to Primary School Students with LD Consider Ms. Gerber’s second
grade class. She has 24 students in her class. Four have identified learning
disabilities and receive resource room support. These students have considerable
difficulty solving math word problems. Ms. Gerber notices that another
six students are also having difficulty solving math word problems. She
decides to seek help from the special education resource teacher, Mr. Rafferty.
They decide to work together on improving students’ math problem-solving
skills. The resource teacher will teach the four students with LD during
their resource time. Ms. Gerber will provide small group instruction for
the six students from her class who are also having difficulty during independent
seatwork time. This coincides with the resource period for the students
with LD. They decide to use the routine developed by Jitendra and colleagues.
Together, they develop structured lessons and create the necessary cues
and prompts. They make a plan to meet every other day for about 20 minutes
to identify what is working and what needs improvement in their lessons
and also to appraise the progress of individual students. Some of the students
are very poor readers. They know that they need to read each problem aloud
with the students before having them solve it.
Ms. Gerber and Mr. Rafferty have already progressed through the change problem with a missing ending. They are now ready to start the change problem with a missing change. Their meetings have been very helpful. They have talked through a number of concerns and brainstormed about techniques that will help some of the students who are having some difficulty. All of the students are making progress, although some need additional assistance, which is provided primarily by peers during peer coaching time at the end of the day. Ms. Gerber oversees and monitors the peer coaching. She provides support for poor readers by pairing them with good readers. Below is a structured lesson designed to teach students how to solve the change problem with a missing ending. Ms. Gerber is modeling how good problem solvers solve a change problem with a missing ending. It may be necessary and important to demonstrate solving the problem using an oversized, laminated Change Problem Diagram and manipulatives before demonstrating at this more symbolic level. She places a transparency of the math problem on the projector.
Ms. Gerber: Watch me say everything I am thinking and
doing as I solve this problem.
Sara had 5 flowers. Andy gave her some more flowers. Now Sara has 12
flowers. How many flowers did Andy give her?
I am going to start with Step 1 – Find the problem type.
Did I read and retell the problem? No, I need to do that. (She
reads the problem aloud). Now I will retell the problem. One
girl has 5 flowers and her friend gives her more so she has
12 altogether. How many does her friend give her? Okay, I read
the problem and said it in my own words. I can check that box.
Next, did I ask if it is a change problem? Did I look at the
beginning, change, and ending just like the diagram? Do they
all describe the same thing? Yes, they do. Flowers.
Okay, on
to Step 2 – Organize the information using the change diagram.
Did I underline the label that describes the beginning, change,
and ending and write in the label in the diagram? I will underline
flowers. That is the label. I will write flowers in the beginning
and ending circles and in the change box. Did I underline the
important information, circle the numbers, and write the numbers
in the diagram? I will circle 5 and 12. I will write 5 in the
beginning circle and 12 in the ending. I need to find the change.
Did I write a “?” for what must be solved? I will write a question
mark in the change box. Did I find my question sentence? Yes,
the question is, how many does her friend give her?
Now to
Step 3 –Plan to solve the problem.
Do I add or subtract? I need to subtract 5 from 12 to find
how many flowers her friend gave her. Did I write the math
sentence? The math sentence is 12 minus 5.
Now, the last step.
Step 4 – Solve the problem. Did I solve the math sentence?
12 minus 5 is 7. Did I write the complete answer? The complete
answer is 7 flowers. Did I check if the answer makes sense?
Yes, the answer makes sense. I am done.
Students then review
the Change Problem Diagram and Checklist. As a review, they
verbally rehearse the steps and corresponding self-monitoring
questions. Then they are given a problem and Ms. Gerber guides
them as a group while they think aloud and solve the next problem.
A student is then selected to model the process with assistance
from Ms. Gerber. Instruction is systematic, sequenced, slow
but intense, and structured to ensure mastery as students learn
the routine. Students have now learned to solve two variations
of the change problem: the missing ending, which requires addition,
and the missing change, which requires subtraction. Although
Ms. Gerber and Mr. Rafferty were successful in their collaborative
effort to help students become better problem solvers and they
know the program was effective for their students, they also
know that sometimes collaboration can be difficult.
Some
Realities of Teaching and Collaborating
Modifying Math Problem-solving Instruction for Students with Other Types of Disabilities
Students with other types of disabilities frequently display cognitive characteristics that resemble those of students with LD. However, their cognitive deficits may be more or less severe or may vary in some unique way from those of students with LD. In many cases, though, there seem to be more similarities than differences. For example, students with spina bifida have long-term memory, visual-spatial, and self-regulation problems that adversely affect their ability to comprehend text and do mathematics (Mesler, 2004). Children with chronic illnesses (such as those surviving cancer) who have undergone intrusive medical treatments often display attention difficulties, short-term memory problems, and other cognitive problems that interfere with school success (Bessell, 2001). Characteristically, students with ADHD have serious self-regulation problems. Students with traumatic brain injury and Asperger’s Syndrome also have cognitive deficits that are similar to students with LD. Because of these similarities, it seems reasonable to assume that instruction effective for students with LD may, with modifications, be effective for students with other types of cognitive disabilities.
Conclusion
A systematic, research-based math problem-solving program makes mathematical problem solving easy to teach. Students are provided with the processes and strategies that make math problem solving easy to learn, and they become successful and efficient problem solvers. They also gain a better attitude toward problem solving when they are successful and develop the confidence to persevere. Moving from textbook problems to real life math situations creates a challenge for students, and they begin to understand why they need to be good problem solvers. Cognitive strategy instruction in mathematical problem solving gives students the resources to solve authentic, complex mathematical problems they encounter in everyday life. Teachers who are knowledgeable about the research underlying effective instruction will be able to justify the instructional time spent on small group instruction in math problem solving. They will also be able to explain how the supplemental instruction complements and builds on the mathematics curriculum.
References
Bessell, A. (2001). Children surviving cancer: Psychosocial adjustment, quality of life, and school experiences. Exceptional Children, 67, 345-359.Case, R. (1998, April). A psychological model of number sense and its development. Paper presented at the annual meeting of the American Educational Research Association, San Diego.
García, A.I., Jiménez, J.E., & Hess, S. (in press). Solving arithmetic word problems: An analysis of classification as a function of difficulty in children with and without arithmetic learning disabilities. Journal of Learning Disabilities.
Gersten, R., & Chard, D. (1999). Number sense: Rethinking arithmetic instruction for students with mathematical disabilities. Learning Disabilities Research & Practice, 33, 18-28.
J itendra, A., Griffin, C., Haria, P., Leh, J., Adams, A., & Kaduvetoor, A. (2005, February) A comparison of schema-based instruction and general strategy instruction on third grade students’ mathematical problem solving. Paper presented at the annual meeting of the Pacific Coast Research Conference, San Diego.
Mesler, J.L. (2004). The effects of cognitive strategy instruction on the mathematical problem solving of students with spina bifida. Unpublished doctoral dissertation, University of Miami, Coral Gables, Florida.
Montague, M. (2003). Solve It! A practical approach to teaching mathematical problem solving skills. Reston, VA: Exceptional Innovations.
Montague, M., Warger, C.L., & Morgan, H. (2000). Solve It! Strategy instruction to improve mathematical problem solving. Learning Disabilities Research and Practice, 15, 110-116.
Swanson, H.L. (1999). Instructional components that predict treatment outcomes for students with learning disabilities: Support for a combined strategy and direct instruction model. Learning Disabilities & Research, 14 , 129-140.
Van Garderen, D., & Montague, M. (2003). Visual-spatial representations and mathematical problem solving. Learning Disabilities & Research, 18, 246-254.
For additional information on this or other topics,
please contact
The Access Center at center@air.org.
The Access Center: Improving Outcomes for All Students K-8
The Access Center is a cooperative agreement (H326K020003) funded by the U.S. Department of Education, Office of Special Education Programs, awarded to
the American Institutes for Research
1000 Thomas Jefferson St. NW,
Washington, DC 20007
Ph: 202-403-5000 | TTY: 877-334-3499 |
Fax: 202-403-5001
|
e-mail: center@air.org website: www.k8accesscenter.org
This report was produced under U.S. Department of Education Cooperative Agreement H326K020003 with the American Institutes for Research. Jane Hauser served as the project officer. The views expressed herein do not necessarily represent the positions or policies of the Department of Education.
No official endorsement by the U.S. Department of Education of any product, commodity, service or enterprise mentioned in this publication is intended or should be inferred.
