Reading and Writing to Learn in Mathematics: Strategies to Improve Problem Solving

by Clare Heidema

Reading and Writing in Mathematics

Reading and writing in mathematics are of particular interest to educators because these processes are essential to both problem solving and concept development in mathematics.  Martinez and Martinez (2001) discuss what happens when children read and write mathematics:

For starters, their learning incorporates some key ideas in the National Council of Teachers of Mathematics new Principles and Standards for School Mathematics (NCTM, 2000).  They learn to use language to focus on and work through problems, to communicate ideas coherently and clearly, to organize ideas and structure arguments, to extend their thinking and knowledge to encompass other perspectives and experiences, to understand their own problem-solving and thinking processes as well as those of others, and to develop flexibility in representing and interpreting ideas. (p. 5)

Reading research clearly points to several characteristics of effective readers.  They can:

• Locate key information
• Distinguish between main ideas and supporting details
• Modify their reading behaviors when faced with difficulty
• Ask questions before, during, and after reading
• Construct meaning as they read by monitoring comprehension, evaluating new information, connecting new information with existing ideas, and organizing information in ways that make sense

These characteristics also describe effective readers of mathematics and skills needed to be mathematics problem solvers. 

Reading and Writing Strategies for Problem Solving

Mathematics is about problem solving, and reading comprehension is an important component, especially for word problems.  Writing, too, is a critical component, because students should monitor and reflect on the problem-solving process as well as communicate their thinking during problem solving.

Problem solving in mathematics often is viewed with a conceptual model proposed by George Polya (1957).  Polya’s model has four steps:

1. Understand the problem. Determine what information is given, what is the unknown, what information is needed or not needed, and what is the context or conditions of the problem.  Restate the problem to make sure terminology and facts are understood.
2. Devise a plan. Consider how to go about solving the problem and what strategies would help in finding a solution.  This may be as simple as selecting the numbers and operations demanded by the problem.  It might include examining different ways to approach the problem, for example, comparing it to problems solved previously, or finding related problems, or making and checking predictions.
3. Carry out the plan. Use the plan as devised, and check or prove that each action taken is correct.
4. Look back (and forward). Examine the result or solution to make sure it is reasonable and solves the problem.  Ask if there could be other solutions or if there are other ways to get a solution.  Perhaps extend or generalize the problem.

Here is a quick preview of the strategies discussed in this article and how you might use them. The strategies are associated with Polya’s four-step model; i.e., each strategy includes the steps in Polya’s model.  

• K-N-W-S and SQRQCQ. These are especially useful for helping students understand the steps in problem solving
• Three-level guide. This is a good choice for focusing on important facts or approaches.
• Word problem roulette. This strategy lends itself nicely to collaboration.
• Process log and RAFT. These two strategies are well suited to help students communicate their thinking.

For these reading and writing strategies to become an effective part of a student’s “toolbox,” teachers must provide instruction on how and when to use them.  When providing instruction, consider the following teaching suggestions:

• Introduce one strategy at a time, and let students apply it several times while you observe what they are doing and where they may need help.
• Model and explain the use of a strategy in an activity that lets students see how and why to use it.
• Practice a strategy as a whole class before asking students to use it independently.  During the whole-class activity, solicit and compare various responses on how the strategy can work.
  
  
  

K–N–W–S

Description of K-N-W-S

K-W-L (Ogle, 1986) is an active reading tool to help students build content knowledge by focusing on the topic and setting the purpose for the upcoming reading.  During K-W-L, students list what they know about a topic (K), note questions they have and what they want to learn (W), and summarize what they learned (L).  In a similar pattern, K-N-W-S allows students to use word problems to determine what facts they know (K), what information is not relevant (N), what the problem wants them to find out (W), and what strategy can be used to solve the problem (S).

In reading, the K-W-L strategy helps all students, no matter the age or achievement level, activate their prior knowledge, develop a purpose for the reading, and make connections between new information and familiar ideas.  The K-N-W-S strategy does this as well. In addition, the K-N-W-S strategy allows students to plan, organize, and analyze how to solve word problems, while teachers can evaluate students’ understanding and possible misconceptions about word problems.  The strategy allows students to focus on what effective students do when assigned word problems.

Guidelines for Use of K-N-W-S

1. Draw a four-column chart on the board or chart paper.  Hand out individual charts to students, or have students construct their own.
2. Using a word problem, model how the columns are used.  Explain how you know which pieces of information belong in each area of the chart.
3. Students can work in groups or individually to complete K-N-W-S sheets for other word problems.  Students can also be asked to write their reasoning for the placement of items in each column.
   
   
   

Example of K-N-W-S

SQRQCQ

Description of SQRQCQ

The reading strategy SQ3R (survey, question, read, recite, review) was originated by Robinson (1961) as an independent study tool.  SQ3R is based on the idea that if students are to recall and comprehend difficult text, they need to make a conscious effort and be engaged at each stage of the reading process.  That idea is also an essential element of SQRQCQ, a six-step study strategy—survey, question, read, question, compute (construct), question—that was designed by Fay (1965) and was modeled after SQ3R.  SQRQCQ is intended to assist students in reading and learning mathematics, in particular, solving word problems in mathematics.  It allows students to organize in a logical order the steps necessary to solve a word problem. 

This strategy can help students focus on a process to decide what a problem is asking, what information is needed, and what approach to use in solving the problem.  It also asks students to reflect on what they are doing to solve the problem, on their understanding, and on the reasonableness of a solution.

Guidelines for Use of SQRQCQ

Give students a description of the steps for SQRQCQ.  Then model the strategy with one or two word problems before asking students to practice it with other word problems. 

1. Survey. Skim the problem to get an idea or general understanding of the nature of the problem.

2. Question. Ask what the problem is about; what information does it require?  Change the wording of the problem into a question, or restate the problem.

3. Read. Read the problem carefully (may read aloud) to identify important information, facts, relationships, and details needed to solve the problem.  Highlight important information.

4. Question. Ask what must be done to solve the problem; for example, “What operations need to be performed, with what numbers, and in what order?”  Or “What strategies are needed?  What is given, and what is unknown?  What are the units?”

5. Compute (or construct). Do the computation to solve the problem, or construct a solution by drawing a diagram, making a table, or setting up and solving an equation.

6. Question. Ask if the method of solution seems to be correct and the answer reasonable.  For example, “Were the calculations done correctly?  Were the facts in the problem used correctly?  Does the solution make sense?  Are the units correct?”
   
   
   

Example of SQRQCQ

Three-Level Guide

Description of the Three- Level Guide Strategy

A three-level guide is a teacher-constructed graphic organizer that serves as the basis of a strategy that students can use to solve and get a deeper understanding of the word problem-solving process.  With the organizer, students must evaluate facts, concepts, rules, mathematical ideas, and possible approaches to solving a given word problem.

Three-level guides for word problems, suggested by Davis and Gerber (1994), are modeled after a reading strategy using study guides devised by Herber (1978) and further developed by Morris and Stewart-Dore (1984) to help students think through the information in texts.  Questions or statements in a study guide address three levels of comprehension: literal, interpretive, and applied.

To create a three-level guide for a word problem, teachers first do a process and content analysis.  They think through their objectives for the problem, namely, what students should learn from the problem and what students should know and be able to do.  Then they identify in the problem the essential information, the mathematical concepts and relationships, inferences from the relationships, and potential student difficulties.  The questions or statements in the guide should help students to develop and appreciate three levels of understanding: the given and relevant information identified explicitly in the problem, the relationships or inferences implicit in solving the problem, and the conclusions or applications involved in solving the problem.

Guidelines for Use of the Three-Level Guide Strategy

Construct guides for problems with three parts (levels).  Part I includes a set of true or false facts suggested by the information given in the problem.  Part II has mathematics concepts, ideas, or rules that might apply to the problem.  Part III includes possible methods (e.g., calculations, creating a table or graph) to use to find a solution to the problem.

1. Introduce students to the three-level guide, and explain the kind of statements that are included in each part.

Part I:  Students analyze each fact to decide if it is true or false given the information in the problem, and they decide whether this fact can help them to solve the problem.

Part II:  Students indicate which statements apply to solving the problem, that is, identify concepts or rules that are useful for this problem.

Part III:  Students decide which calculations (or methods) might help them in solving the problem.

2. Model for students the use of a three-level guide in solving a problem.
3. Present students with a three-level guide for another problem, and direct them to complete the guide on their own or with a partner.  In this step, students analyze information you have included in the guide to determine both its validity and its usefulness in solving the problem.
4. With advanced students, you might select word problems for which the students write three-level guides to share with the class or to exchange with a partner. 
   
   
   

Example of a Three-Level Guide

Word Problem Roulette

Description of Word Problem Roulette

The word problem roulette strategy comes from Davis and Gerber’s (1994) discussion of content-area strategies for the secondary mathematics classroom.  They suggest that students should discuss and write about the content of word problems and their solutions.  The word problem roulette strategy is designed to give students an opportunity to collaborate on solving a word problem and then to communicate as a group the thought processes that went into finding a solution to the problem.  The group presents its solution to the problem both orally and in writing.

The strategy involves students in a group problem-solving activity.  They read a problem and, as a group, decide what the problem is about and what they might do to solve the problem.  Students benefit from communicating their own thinking and from hearing how other students think about a problem.  They have a chance to try out different ideas and to come to an agreement on a suitable method to solve a problem.

Guidelines for Use of Word Problem Roulette

Choose word problems that are well suited to collaborative work.

1. Organize the class into cooperative groups of three or four students per group.  Provide each group member with a copy of the word problem for the group.  Explain to students that they are to solve this problem as a group.
2. First the group discusses how to solve the word problem.  The group members talk to one another about what the problem is asking and their ideas for solving the problem, but they do this without writing or drawing on paper.  During this step, the members of the group agree on a solution method and the steps for how they will solve the problem.
3. When the group members have agreed on a solution to the problem, they take turns writing the steps to the solution in words rather than mathematics symbols.  Each group member writes one step or sentence and then passes the group solution paper to the next group member to add the next step or sentence.  Group members may confer on what individual members write, but the solution paper should have contributions from everyone in the group.
4. After all the groups have finished writing their solution papers, choose one group at a time to present its solution to the class.  One member of a group reads the solution steps as they are written on the paper, and another group member writes the symbolic representation of this solution on the board.
5. After all the groups who have the same problem have presented their solutions, compare the methods and results of the different groups.  If groups have different problems, volunteers from other groups may give an immediate review of a group’s solution.
   
   
   

Example of Problem for Word Problem Roulette

Process Logs

Description of Process Logs

Martinez and Martinez (2001) suggest what they call a process log for word problems.  This strategy uses a writing-math worksheet in which students explain the word problem and the steps they will use to solve it.  The worksheet guides the student with question prompts that lead them through the problem-solving process without dictating a method or the steps students use to solve the problem.  Students are asked to use ordinary language as well as mathematical language in their explanations.

Using a process log, students write about their thinking during the problem-solving process.  The questions they answer create a dialogue between what they know and what they are learning by doing the problem.  In this way, students clarify their thinking about the problem and the mathematics involved, they translate mathematical ideas and procedures into ordinary language, and they practice communicating about mathematics and reasoning in problem solving.  Process logs are a type of learning log and have the advantage of reinforcing strategic learning processes that we associate with learning logs.

Guidelines for Use of Process Logs

Prepare a writing-math worksheet for students with a word problem activity and an extra challenge.  You can include additional question prompts as you consider other ideas you would like in a student’s log.  You might give students these directions on how to use this worksheet:

1. Use the writing-math worksheet as you think about and solve this problem.  That is, “think aloud” on paper about the problem activity.

2. Try to personalize your explanations by writing in the first person; use I, my, me, and so on.

3. Use ordinary language as well as mathematical language.

4. Explain the problem itself as you write the steps involved.

5. Be sure to describe any special difficulties with the problem.

6. Explain your problem-solving process step by step.
   
   
   

Example of Process Log

RAFT

Description of RAFT

RAFT (Santa, 1988) is an acronym for the four critical pieces of writing: role, audience, format, and topic.  It is an extended writing activity that expands the topics students have been studying. The RAFT strategy is an excellent way to involve students in explaining what they know about a topic.  RAFT can be used as a culminating activity or assessment after students have studied a concept.  Options for the students are provided in each of the four areas in order for the students to demonstrate their understanding in a nontraditional format. 

Reading research states that students need and appreciate choices in their learning.  The choices in the RAFT strategy affect the vocabulary, style, and focus of the writing and address the various learning styles.  Students are encouraged to think creatively about the concepts they have been learning.  They make connections and internalize ideas when they formulate concepts in a different mode and use them to do further investigations.

The RAFT strategy provides a method for students to synthesize information into a writing-to-learn episode.  The National Council of Teachers of Mathematics encourages activities to interconnect mathematical ideas in order to produce a coherent whole.  The RAFT strategy allows students to apply mathematics concepts in a fun way to explain what they have learned about a topic.

Guidelines for Use of RAFT

1. Develop a list or brainstorm with students choices for:

• Role of the writer (reporter, observer, eyewitness)

• Audience for the writing (teacher, other students, parent, someone in the community)

• Format to present the writing (letter, article, poem, diary, journal, instructions, advertisement, speech)

• Topic for the writing (application of a procedure, reaction to an event, explanation of a mathematics concept)

2. Each student chooses a role, audience, format, and topic from the generated list.  You may assign the same role to all students or let students choose from several different roles. 
3. Provide class time for the work.  You can conference with students and keep track of their progress on their RAFT choices.
4. Sharing the writing is important.  Students can read to the entire class or read to smaller groups, if time is limited.
   
   
   

Example of RAFT

Our Goals for Students

Principles and Standards calls for excellence in mathematics for all students, and this requires high expectations for all students.  Adding to or fine-tuning reading and writing strategies in our collection of teaching strategies can help us work with all students to attain these high expectations.

As mathematics teachers, our major goal for students is to learn the mathematics content: to retain important information, develop a deep understanding of concepts and topics, and apply and demonstrate knowledge.  In addition, we want students to become independent learners: to gain control of their own learning, ask their own questions, find their own answers, and use critical thinking skills.  The use of reading and writing strategies for word problems in mathematics gives us tools with which to help students become more effective readers and communicators in mathematics—or to put it another way, these strategies help us address our goals for all students.

Bibliography

Davis, S. J., & Gerber, R. (1994, September). Open to suggestion: Content area strategies in secondary mathematics classrooms. Journal of Reading, 38(1), 55–57.

Fay, L. (1965). Reading study skills: Math and science. In J. A. Figural (Ed.), Reading and inquiry (pp. 93–94). Newark, DE: International Reading Association.

Herber, H. (1978). Teaching reading in the content areas (2nd ed.). Englewood Cliffs, NJ: Prentice Hall.

Martinez, J. G. R., & Martinez, N. C. (2001). Reading and writing to learn mathematics: A guide and a resource book. Boston: Allyn & Bacon.

Morris, A., & Stewart-Dore, N. (1984). Learning to learn from text: Effective reading in content areas. New South Wales: Addison-Wesley.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

National Reading Panel. (2000). Teaching children to read: An evidence-based assessment of the scientific research literature on reading and its implications for reading instruction. Washington, DC: National Institutes of Health.

Ogle, D. (1986, February). The K-W-L: A teaching model that develops active reading of expository text. The Reading Teacher, 39, 564–570.

Polya, G. (1957). How to solve it: A new aspect of mathematical method (2nd ed.). Princeton, NJ: Princeton University Press.

Robinson, F. (1961). Effective study. New York: Harper & Row.

Santa, C. M. (1988). Content reading including study systems. Dubuque, IA: Kendall-Hunt.

Clare Heidema is a senior research associate at RMC Research Corporation in Denver, Colorado. She has teaching experience at all levels from elementary school to graduate school, has extensive background in professional development, and has written curriculum materials such as the Comprehensive School Mathematics Program. She is coauthor of Teaching Reading in Mathematics: A Supplement to Teaching Reading in the Content Areas Teacher's Manual.

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