Focus+on+Inclusion

Reading & Writing Quarterly , 20: 225 7 229, 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 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 ** <span style="font-family: Calibri; font-size: 12pt; mso-bidi-font-family: AdvP2AA1; msobidifontfamily: AdvP2AA1;"> <span style="font-family: Calibri; font-size: 12pt; mso-bidi-font-family: AdvPS7F80; msobidifontfamily: AdvPS7F80;">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 predictions 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. <span style="font-family: Calibri; font-size: 12pt; mso-bidi-font-family: AdvPS7F80; msobidifontfamily: AdvPS7F80;"> 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 individually or cooperatively. After the word problem has been solved, a new leader is selected to facilitate completion of the next problem. <span style="font-family: Calibri; font-size: 12pt; mso-bidi-font-family: AdvPS7F80; msobidifontfamily: AdvPS7F80;"> <span style="font-family: Calibri; font-size: 12pt; mso-bidi-font-family: AdvPS7F80; msobidifontfamily: AdvPS7F80;"> 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).
 * <span style="font-family: Calibri; font-size: 12pt; mso-bidi-font-family: AdvP2AA1; msobidifontfamily: AdvP2AA1;">RECIPROCAL TEACHING ACCOMMODATIONS **

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. <span style="font-family: Calibri; font-size: 12pt; mso-bidi-font-family: AdvP2AA1; msobidifontfamily: AdvP2AA1;"> <span style="font-family: Calibri; font-size: 12pt; mso-bidi-font-family: AdvPS7F80; msobidifontfamily: AdvPS7F80;"> Although reciprocal teaching is effective for developing comprehension skills, it has some limitations. For example, it should not be assumed that 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: <span style="font-family: Symbol; font-size: 12pt; mso-bidi-font-family: Symbol; msobidifontfamily: Symbol; mso-fareast-font-family: Symbol; msofareastfontfamily: Symbol; msolist: Ignore;"> <span style="font-family: Calibri; font-size: 12pt; mso-bidi-font-family: AdvPS7F80; msobidifontfamily: AdvPS7F80;">
 * <span style="font-family: Calibri; font-size: 12pt; mso-bidi-font-family: AdvP2AA1; msobidifontfamily: AdvP2AA1;">FURTHER CONSIDERATIONS WHEN USING RECIPROCAL TEACHING **
 * <span style="font-family: Calibri; font-size: 12pt; mso-bidi-font-family: AdvPS7F80; msobidifontfamily: AdvPS7F80;">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).

<span style="font-family: Calibri; font-size: 12pt; mso-bidi-font-family: AdvP2AA1; msobidifontfamily: AdvP2AA1;"> <span style="font-family: Calibri; font-size: 12pt; mso-bidi-font-family: AdvPS7F80; msobidifontfamily: AdvPS7F80;">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. <span style="line-height: 115%; font-family: 'Calibri', 'sans-serif'; font-size: 12pt; mso-bidi-font-family: AdvP2AA1; mso-ascii-theme-font: minor-latin; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin; mso-ansi-language: EN-US; mso-fareast-language: EN-US; mso-bidi-language: AR-SA;"> <span style="line-height: 115%; font-family: Calibri; font-size: 12pt; mso-bidi-font-family: AdvP2AA1; msobidifontfamily: AdvP2AA1;"> <span style="font-family: Calibri; font-size: 12pt; mso-bidi-font-family: AdvP2AA1; msobidifontfamily: AdvP2AA1;"> <span style="font-family: Calibri; font-size: 12pt; mso-bidi-font-family: AdvPS7F80; msobidifontfamily: AdvPS7F80;">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. 77 7 89). Reston, VI: National Council of Teachers of Mathematics. <span style="font-family: Calibri; font-size: 12pt; mso-bidi-font-family: AdvPS7F80; msobidifontfamily: AdvPS7F80;"> 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), 105 7 106. <span style="font-family: Calibri; font-size: 12pt; mso-bidi-font-family: AdvPS7F80; msobidifontfamily: AdvPS7F80;"> Geary, D. C. (1996). // Children’s mathematical development: Research and practical applications //. <span style="font-family: Calibri; font-size: 12pt; mso-bidi-font-family: AdvPS7F80; msobidifontfamily: AdvPS7F80;">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), 169 7 174. <span style="font-family: Calibri; font-size: 12pt; mso-bidi-font-family: AdvPS7F80; msobidifontfamily: AdvPS7F80;"> 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. 81 7 96). Austin, TX: Pro-Ed. <span style="font-family: Calibri; font-size: 12pt; mso-bidi-font-family: AdvPS7F80; msobidifontfamily: AdvPS7F80;"> 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: 177 7 200). Austin, TX: Pro-Ed. <span style="font-family: Calibri; font-size: 12pt; mso-bidi-font-family: AdvPS7F80; msobidifontfamily: AdvPS7F80;"> Muth, D. K. (1997). Using cooperative learning to improve reading and writing in mathematical problem solving. // Reading & Writing Quarterly //, 13 (1), 71 7 83. <span style="font-family: Calibri; font-size: 12pt; mso-bidi-font-family: AdvPS7F80; msobidifontfamily: AdvPS7F80;"> Palinscar, A. S., & Brown, A. L. (1984). Reciprocal teaching of comprehension-fostering and monitoring acitivities. // Cognition and Instruction //, 1 , 11 7 175. <span style="font-family: Calibri; font-size: 12pt; mso-bidi-font-family: AdvPS7F80; msobidifontfamily: AdvPS7F80;"> Pressley, M. (2002). // Reading instruction that works: The case for balanced teaching // <span style="font-family: Calibri; font-size: 12pt; mso-bidi-font-family: AdvPS7F80; msobidifontfamily: AdvPS7F80;">(2nd ed.). New York, NY: The Guilford Press. Salend, S. J. (2001). // Creating inclusive classrooms: Effective and reflective practices //. <span style="font-family: Calibri; font-size: 12pt; mso-bidi-font-family: AdvPS7F80; msobidifontfamily: AdvPS7F80;">(4th ed.). Upper Saddle River, NJ: Merrill Prentice Hall.
 * <span style="font-family: Calibri; font-size: 12pt; mso-bidi-font-family: AdvP2AA1; msobidifontfamily: AdvP2AA1;">SUMMARY **
 * <span style="font-family: Calibri; font-size: 12pt; mso-bidi-font-family: AdvP2AA1; msobidifontfamily: AdvP2AA1;">REFERENCES **

van Garderen, D., & Montague, M. (2003). Visual-spatial-representation, mathematical problem solving, and students of varying abilities. // Learning Disabilities Research and Practice //, 18 (4), 246 7 254.