Reading & Writing Quarterly, 20: 2257229, 2004
Copyright # Taylor & Francis Inc.
ISSN: 1057-3569 print
DOI: 10.1080/10573560490272702
FOCUS ON INCLUSION
RECIPROCAL TEACHING AS A COMPREHENSION
STRATEGY FOR UNDERSTANDING MATHEMATICAL
WORD PROBLEMS
Delinda van Garderen
State University of New York at New Paltz,
New York, USA
Ms. Johnson was concerned about the inconsistent performance of several of
her students in solving mathematical word problems. A number of her students
were one to two grade levels below their grade placement in reading, spoke
English as a second language, and had identified reading disabilities. On
mathematics assignments that required minimal reading, all the students in
Ms Johnson’s class performed adequately. The major state assessment in
mathematics was in four months time but Ms Johnson was worried that these
students might fail because the state assessment invariably contained a significant number of mathematical word problems. Ms. Johnson felt frustrated
by her inability to help her students read and comprehend mathematical word
problems.
Mathematics textbooks and standardized tests contain an increasing
number of word problems that students need to be able to solve. As students progress in their education, word problems increasingly demand
greater reading skills (Miller & Mercer, 1997). While concern has been
expressed that ability in reading comprehension, not mathematics
Address correspondence to Dr. Delinda van Garderen, Department of Educational Studies,
OMB 112, SUNY at New Paltz, 75 South Manheim Blvd., New Paltz, New York 12561.
E-mail: vangardd@newpaltz.edu
Focus on inclusion is edited by Michael E. Skinner. Prospective contributor should send 5
copies of their manuscript to Michael E. Skinner, College Charleston, School of
Education—Special Education, 66 George Street, Charleston, SC 29424-0001, USA
( phone: 803-953-5613; Fax: 803-953-5407; e-mail: skinnerm@cofe.edu).
225
226
D. van Garderen
understanding, is what is being tested (Flick & Lederman, 2002), comprehension is a critical aspect of mathematical word problem solving.
Effective math word problem solvers are able to understand the purpose
of a problem. They demonstrate this by their ability to explain the problem
in their own words (Flick & Lederman, 2002). However, like Ms. Johnson’s
students, many students lack this skill (Geary, 1996). This is especially the
case for many students with learning difficulties (Montague, 1997;
van Garderen & Montague, 2003). Factors such as irrelevant numerical
and linguistic information, mathematical terminology, vocabulary level,
number of ideas presented, and syntactic complexity contained within the
word problem can make the wording particularly difficult to understand
(Miller & Mercer, 1997; Salend, 2001). Therefore, teaching all students
to become competent word problem solvers is a concern for educators.
This article presents guidelines for using and modifying reciprocal teaching
to facilitate the development of comprehension of mathematical word
problems.
RECIPROCAL TEACHING AND MATHEMATICAL WORD
PROBLEM SOLVING
Reciprocal teaching is a structured strategy advocated by many reading
specialists for developing comprehension skills (Palinscar & Brown, 1984;
Pressley, 2002). In reading, reciprocal teaching involves students making
predicitions when reading, questioning themselves about the ideas in
the text, seeking clarification when confused, and summarizing content
(Pressley, 2002).
A modified version of reciprocal teaching can be applied to developing
comprehension of mathematical word problems. The four major components of this modified approach are: clarifying, questioning, summarizing, and planning.
During a reciprocal teaching lesson on mathematical word problems, the
students are divided into small groups, and one student is assigned the role
of leader. The leader instructs the group members to silently read a word
problem. After the entire group has read the problem, the leader asks for
vocabulary or phrases that need to be clarified. Any group member then
supplies the meaning of a word or phrase. Once all words and phrases have
been clarified, the leader uses questions to identify the key parts of
the problem. The group leader then summarizes the purpose of the word
problem. The leader guides the group in devising a plan to solve the
problem. The steps and operations needed to solve the problem are listed.
Once the plan has been checked to ensure that it makes sense, the
mathematical word problem is solved. Solving the problem may be done
Reading Comprehension in Mathematics
227
individually or cooperatively. After the word problem has been solved, a
new leader is selected to facilitate completion of the next problem.
RECIPROCAL TEACHING ACCOMMODATIONS
Reciprocal teaching can be modified to accommodate students with
learning difficulties. If students have difficulty reading the problem, the
problem can be read aloud by a group member. The group can be provided
with a dictionary to look up words unknown by any group member. A math
dictionary developed by the students that contains definitions, examples,
and graphics of mathematical terminology to promote mathematical literacy might also be used to contribute to student understanding (Salend,
2001).
Some students may have difficulty coming up with questions to elicit the
key parts of the word problem. A chart of questions, developed by the class
or teacher, can be provided that can be referred to by the students. For
example, questions such as ‘‘Do we have all the information needed to be
able to solve the problem,’’ ‘‘What do we know,’’ and ‘‘What do we not know’’
can be written on the chart.
To facilitate the summarization of the problem, the students can be
encouraged to highlight or underline the relevant information in the problem and cross out irrelevant information (Salend, 2001). If students have
difficulty orally summarizing the problem, they can be encouraged to use a
diagram. It is important to ensure that students understand that a diagram
is not a picture or drawing but rather a representation that shows the parts
of the math problem. Furthermore, numerous opportunities should be
provided for students to practice generating diagrams and using them as
tools for problem solving (Diezmann & English, 2001).
To aid students in their planning, they can be taught to look for cue
words to indicate the operation to be used to solve the problem (eg, the
words ‘‘all together,’’ ‘‘in all,’’ and ‘‘sum’’ suggest that the problem involves
addition) (Miller & Mercer, 1993; Salend, 2001). However, students need to
understand that key words do not always cue the appropriate operation
and may lead to operational errors (Miller & Mercer, 1993). Additionally,
being able to identify key words does not necessarily mean the student
comprehends the problem.
FURTHER CONSIDERATIONS WHEN USING RECIPROCAL
TEACHING
Although reciprocal teaching is effective for developing comprehension
skills, it has some limitations. For example, it should not be assumed that
228
D. van Garderen
all students will internalize the use to the four strategies practiced in the
group. Further, the transfer of responsibilities of using the four strategies
from the teacher to the students may result in long pauses during the
lesson (Pressley, 2002). The effectiveness of reciprocal teaching for comprehending mathematical word problems can be enhanced by use of
specific instructional actions, such as:
Identifying the purpose of reciprocal teaching and why each strategy is
important.
Providing explicit instruction about what each strategy is and how to
carry out each of the strategies.
Modeling the use of the strategies by the teacher.
Providing repeated opportunities to practice the use to the strategies
with the teacher’s guidance and assistance, provided on an as-needed
basis.
Having the student’s model and explain the use of each strategy to the
teacher and each other.
Highlighting to the students when and where the strategies can be
applied and making apparent how different students might apply the
strategies in different ways to the same content (Pressley, 2002).
SUMMARY
Solving mathematical word problems is often hindered by the student’s
failure to comprehend the problem. Educators can use reciprocal teaching
for improving comprehension. Reciprocal teaching is a non-threatening
approach that allows students to work cooperatively to support each
other’s learning as they work towards a shared academic goal (Muth, 1997).
Accommodations during a reciprocal teaching lesson, such as drawing
diagrams, providing a dictionary, and underlining key phrases, can further
facilitate the mathematical problem solving performance of students.
REFERENCES
Diezmann, C. M., & English, L. D. (2001). Promoting the use of diagrams as tools for thinking.
In A.A. Cuoco and F.R. Curico (Eds.), The roles of representation in school mathematics:
2001 yearbook (pp. 77789). Reston, VI: National Council of Teachers of Mathematics.
Flick, L. B., & Lederman, N. G. (2002). The value of teaching reading in the context of science
and mathematics, School Science and Mathematics, 102(3), 1057106.
Geary, D. C. (1996). Children’s mathematical development: Research and practical applications. Washington, DC: American Psychological Association.
Miller, S. P., & Mercer, C. D. (1993). Using a graduated word problem sequence to promote
problem-solving skills. Learning Disabilities Research and Practice, 8(3), 1697174.
Reading Comprehension in Mathematics
229
Miller, S. P., & Mercer, C. D. (1997). Educational aspects of mathematics disabilities. In D. P.
Rivera (Ed.), Mathematics education for students with learning disabilities: Theory to
practice (pp. 81796). Austin, TX: Pro-Ed.
Montague, M. (1997). Cognitive strategy instruction in mathematics for students with learning
disabilities. In D. P. Rivera (Ed.), Mathematics education for students with learning
disabilities: Theory to practice (pp: 1777200). Austin, TX: Pro-Ed.
Muth, D. K. (1997). Using cooperative learning to improve reading and writing in mathematical
problem solving. Reading & Writing Quarterly, 13(1), 71783.
Palinscar, A. S., & Brown, A. L. (1984). Reciprocal teaching of comprehension-fostering and
monitoring acitivities. Cognition and Instruction, 1, 117175.
Pressley, M. (2002). Reading instruction that works: The case for balanced teaching
(2nd ed.). New York, NY: The Guilford Press.
Salend, S. J. (2001). Creating inclusive classrooms: Effective and reflective practices.
(4th ed.). Upper Saddle River, NJ: Merrill Prentice Hall.
van Garderen, D., & Montague, M. (2003). Visual-spatial-representation, mathematical problem solving, and students of varying abilities. Learning Disabilities Research and
Practice, 18(4), 2467254.