reading and writing to learn in mathematics
TRANSCRIPT
Reading and Writing
to Learn in Mathematics Ten Strategies to Improve
Problem Solving
ACKNOWLEDGMENTS
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Denver, CO September 2005
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Table of Contents
Page Reading and Writing in Mathematics………………………………………….
1
Goals for Students………………………………………………………………
2
Reading and Writing Strategies for Problem Solving………………………..
3
Understand the Steps/Process K-N-W-S………………………………………………………….
5
Five-Step Problem Solving……………………………………..
7
SQRQCQ…………………………………………………………
10
Focus on Important Facts/Approaches QAR……………………………………………………………….
13
Three-Level Guide………………………………………………
15
GIST………………………………………………………………
18
Collaborate Word Problem Roulette…………………………………………
20
Cooperative Group Cards………………………………………
22
Communicate Thinking Process Logs…………………………………………………….
24
RAFT……………………………………………………………...
27
Bibliography………………………………………………………………………
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1
Reading and Writing in Mathematics
Reading and writing in mathematics are of particular interest to educators, because they 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)
Principles and Standards calls for excellence in mathematics for all students, and this requires high expectations for all students. We know that students learn mathematics through building new knowledge from prior knowledge and experience, and students must be challenged and supported in their learning of mathematics. Often teachers know the mathematics content that students need, but they may not be able to help all students learn this content to reach high standards in mathematics. Adding to or fine tuning reading and writing strategies in their collection of teaching strategies can help teachers work with all students to attain these high standards. Students arrive in our classrooms with a wide variety of reading abilities. Many students know how to read but still do not know how to learn from reading. For many students, reading mathematics is particularly difficult, perhaps because they must make sense of the specialized vocabulary, symbols, and diagrams; hence, everything on the page. These students struggle with organizing ideas as they read, making meaningful connections, and persevering through the reading material of mathematics. Often mathematics text is presented at a higher reading level than where most students are reading, especially if one considers the difficulties of reading conditional and hypothetical statements. Mathematics problems can be organized in such a way that what might seem to be trivial information is in fact important to the understanding of the problem. The strategies presented in this publication are designed to help with the difficulties of reading for problem solving. Problem solving in mathematics is a natural place to increase students’ competency in writing. Through the writing process, students develop their communication skills by clarifying their thinking and understanding of mathematics, organizing their ideas, and making coherent conclusions. Communication is an essential part of mathematics and needs to be practiced, with guidance, to help students appreciate the power and precision of mathematical language. The strategies in this publication emphasize the importance of writing to learn in mathematics.
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Goals for Students
As teachers, our goals for students’ reading and writing to learn in mathematics include that they learn the mathematics content—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—gain control of their own learning, ask their own questions, find their own answers, and use critical thinking skills. The reading and writing strategies for problem solving in mathematics are designed to help teachers with these goals. They help teachers teach students how to read word problems, how to communicate in mathematics, and how to reason and think mathematically. The 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; and
• Construct meaning as they read; monitor comprehension, evaluate new
information, connect new information with existing ideas, and organize information in ways that make sense.
These characteristics also describe effective readers of mathematics and mathematics problem solvers. The use of reading and writing strategies for word problems in mathematics gives teachers tools with which to help students become more effective readers and communicators in mathematics; that is, helps them address our goals for all students.
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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 process of mathematical problem solving 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 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.
The reading and writing strategies in this publication are associated with this four-step model. Each strategy includes all of the steps in Polya’s model, but in some strategies the main focus is associated with one or two steps. Therefore, we have organized the presentation of 10 reading and writing strategies as follows:
• Understand the Steps/Process K-N-W-S, Five-Step Problem Solving, and SQRQCQ
• Focus on Important Facts/Approaches
QAR, Three-Level Guide, and GIST
• Collaborate Word Problem Roulette and Cooperative Group Cards
• Communicate Thinking
Process Log and RAFT
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For these reading and writing strategies to become an effective part of a student’s “toolbox,” teachers must provide direct instruction on how and when to use them. But, just as with problem-solving approaches (e.g., guess and check, make a table, work backward, look for a pattern, etc.), this should not happen all at one time. Consider the following teaching suggestions:
• Introduce one strategy at a time, and let students practice it several times while the teacher observes 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.
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UNDERSTAND THE STEPS/PROCESS
K – N – W – S Description of K-N-W-S What I Know, What I Want to Learn, What I Learned (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 (K) about a topic, 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 answer 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). Rationale for K-N-W-S strategy 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 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. Individual charts can be
prepared for students to write on, or they can construct their own. 2. Using a word problem, model how the columns are used. Explain how you know
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.
K N W S What facts do I KNOW from the information in the problem?
Which information do I NOT need?
WHAT does the problem ask me to find?
What STRATEGY or operations will I use to solve the problem?
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Example of K-N-W-S Video Heaven rents movies for $3 apiece per night. They also offer a video club plan. The plan costs $100 per year and allows unlimited rentals at $1 per movie per night plus two free rentals per month. How many movies must you rent in a year to make the video club worthwhile?
K N W S What facts do I KNOW from the information in the problem?
Which information do I NOT need?
WHAT does the problem ask me to find?
What STRATEGY or operations will I use to solve the problem?
$3 to rent the movies. Plan charges $100. Each movie under the plan costs $1. There are 2 free movies per month under the plan.
The video club plan allows an unlimited number of movie rentals.
How many movies must be rented in a year to make joining the club worthwhile?
Make a chart to compare the costs of both regular and club plans. Write an inequality to compare the cost of the regular plan to the club plan.
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UNDERSTAND THE STEPS/PROCESS
Five-Step Problem Solving Description of Five-Step Problem Solving This strategy was presented by Braselton and Decker (1994) in an article on “Using Graphic Organizers to Improve the Reading of Mathematics.” They claim that students’ understanding of word problems can be enhanced by teaching them to read the problems as short stories from which students can construct meaning based on prior knowledge and experience. A graphic organizer (see Exhibit 1 on page 8) leads the students through a five-step problem-solving process (similar to Polya’s four-step process). 1. Restate the problem/question. 2. Decide what information is needed to solve the problem. 3. Plan the steps and calculations to be performed. 4. Find an answer by doing the steps or calculations. 5. Evaluate the reasonableness of the solution. The shape of the graphic organizer should suggest to students that they all begin with the same information about the problem, and that they should arrive at the same solution to the problem, if they are successful. Students may use a variety of methods and steps depending upon their reasoning and experience going from the top to the bottom of the shape. Rationale for Five-Step Problem Solving A graphic organizer in a reading strategy lets readers use a graphic representation of material to assist comprehension. The National Reading Panel suggests that the use of graphic organizers as a reading strategy “has a firm scientific basis” (p. 4-42). In this case, the graphic organizer allows students to represent their understanding of a problem, to plan their steps in solving the problem, and to explain their reasoning throughout the process. Guidelines for Use of Five-Step Problem Solving 1. Introduce students to the layout and purpose of the graphic organizer for Five-Step
Problem Solving. This introduction should include comments on starting and ending in the same place, but having many possibilities for the steps in between.
2. Explain each of the steps in Five-Step Problem Solving and show how these are
depicted in the graphic organizer. 3. Present a word or story problem. Read the problem aloud with the class and solicit
comments from students about their prior knowledge of the “story.”
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4. If this is the first time students are using Five-Step Problem Solving, model how to complete the first step in the organizer. You may want to restate the problem or question in several different ways. Continue by modeling how to complete the remaining steps in the graphic organizer.
5. Offer guided practice by selecting another word (story) problem and leading the
class through each step of the process. Ask students to discuss their thinking about the problem and to articulate reasons for responses they give at each step. Point out that there may be different methods or ways to solve the same problem. Also, emphasize that the graphic organizer is a guide, not a formula for solving word problems.
6. Arrange for students to work in groups of two to four to discuss and complete
several word problems using the graphic organizer for Five-Step Problem Solving. 7. Make copies of the graphic organizer available for students to use on other word
problems and suggest they practice this individually over the next several days. Exhibit 1. Graphic Organizer for Five-Step Problem Solving
1. Restate the
problem/question.
2. Find needed information.
3. Plan what to do.
4. Solve the problem. Find the answer.
Solution/answer
5. Check: Is the answer (solution) reasonable?
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Example of Five-Step Problem Solving
Shane is making eggnog. The recipe calls for 4 eggs, ½ cup sugar, 1 quart (32 ounces) milk, and 1 tsp. vanilla. Shane has only 3 eggs. How much milk and how much sugar should he use?
1. Restate the problem/question.
If you have 3 eggs, how should you alter a recipe that calls for 4 eggs keeping the proportions of other items the same?
2. Find needed information. Recipe “new” Recipe 4 eggs 3 eggs ½ cup sugar ____ sugar 32 oz. milk ____ milk
3. Plan what to do. Consider a full recipe. Divide all the ingredients in half to get ½ of a recipe.
Again, divide all the ingredients in half to get ¼ of a recipe.
½ of a recipe uses 2 eggs, and ¼ of a recipe uses 1 egg.
Add the ingredients in ½ and ¼ of a recipe to find a “recipe” with 3 eggs. 4. Solve the problem. Find the answer. One recipe: 4 eggs ½ recipe: 2 eggs ¼ recipe: 1 egg
½ cup sugar ¼ cup sugar ⅛ cup sugar 32 oz. milk 16 oz. milk 8 oz. milk
Add ½ recipe and ¼ recipe to get ¾ recipe: 3 eggs
⅜cup sugar 24 oz. milk.
Solution/answer ⅜ cup sugar and 24 oz. milk
5. Check: Is the answer (solution) reasonable?
3 out of 4 eggs is the same as 24 out of 32 oz. milk (3 8 = 24 and 4 8 = 32); and is the same as ⅜ out of 4/8 (½) cup sugar.
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UNDERSTAND THE STEPS/PROCESS
SQRQCQ Description of SQRQCQ The six-step study strategy SQRQCQ, initialing Survey, Question, Read, Question (Organize), Compute (Construct), Question, was designed by Fay (1965) and was modeled after SQ3R. The reading strategy SQ3R ( Survey, Question, Read, Recite, Review) was originated by Robinson (1961) as an independent study tool. It has become one of the best known reading strategies and a widely used model for textbook study. The underlying idea of SQ3R is: 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. 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. Rationale for SQRQCQ Some of the difficulties students have with word problems are related to reading difficulties. 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. The strategy involves previewing, setting purpose, and monitoring success, all reading comprehension strategies that the National Reading Panel recommends. Guidelines for Use of SQRQCQ Give students a description of the steps for SQRQCQ. Then model using the strategy with one or two word problems before asking students to practice it with other word problems. 1. Survey.
Scan 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.
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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 or construct a solution to the problem. This might involve 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?”
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Example of SQRQCQ Marcie has 96 counters in three colors, red, blue, and yellow. She has twice as many red counters as blue and five times as many blue as yellow. How many counters of each color does Marcie have? Survey. Scan the problem to get a general idea of what it’s about. Clarify terms.
Counters are in three colors. There are conditions on the 96 counters Marcie has.
Question. What is the problem about and what is the information in the problem?
How many counters of each color does Marcie have?
Read. Identify relationships and facts needed to solve problem.
96 counters (red, blue, and yellow) #red is 2 #blue; #blue is 5 #yellow
#red + #blue + #yellow = 96 #red = 2 #blue #blue = 5 #yellow
Question. What to do? How to solve the problem?
Write equations with variables r, b, and y. Use substitution. Make a table and try numbers, keeping relationships, until the total is 96. #red #blue #yellow total 20 10 2 32
Compute (or construct) Do the calculations or construct a solution.
Algebra: r + b + y = 96 r = 2b and b = 5y 2b + b + y = 96 (sub. r = 2b) 2(5y) + 5y + y = 96 (sub. b = 5y) 10y + 5y + y = 16y = 96 y = 6 and b = 30 and r = 60 Guess and Check: #red #blue #yellow total 20 10 2 32 40 20 4 64 60 30 6 96 Marcie has 60 red counters, 30 blue counters, and 6 yellow counters.
Question. Is the algebra correct? Are the calculations correct? Does the solution make sense?
# red is 60 (Twice as many red as # blue is 30 blue. Five times as many blue # yellow is 6 as yellow.) The total number of counters is 96.
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FOCUS ON IMPORTANT FACTS/APPROACHES
QAR
Description of QAR The question answer relationship (QAR) framework (Raphael, 1982, 1986) complements the problem-solving model recommended by Polya. The QAR strategy is used to analyze the questioning process which correlates to Polya’s first step, “Understand the problem.” Four levels of questions are used in the QAR strategy, but only “Right There” questions and “Think and Search” questions apply to word problems. “Right There” questions use words from the text to pose the question as well as answer them; the answer is literally in the text. “Think and Search” or interpretive questions require students to think about how the pieces of information in the text relate to one another and search the passage to find all the information that applies; the answers may be found in the text but may require the reader to make connections among ideas from the text. Rationale for QAR strategy The National Reading Panel (2000) says that question answering, where students answer questions posed by teachers and receive immediate feedback, is a reading strategy that is effective and most promising (p. 4-42). Students who become skilled with the QAR strategy recognize the relationship between the questions being asked and the corresponding answers. Some questions require exploration of text to find an existing answer; some questions require explanation or evaluation to supply an answer. Students learn where to find the information needed for a correct response. QAR teaches students that answering different kinds of questions requires different thought processes. In both types of questions, students are using higher order thinking skills to formulate responses to questions. Guidelines for Use of QAR strategy 1. Introduce students to each question-answer relationship. 2. Construct word problems that can be used for answering “Right There” or “Think and
Search” questions. Using a multiple choice approach for the questions coincides with formats used on many district and state assessments.
3. Assign problems to students. Point out the differences between each question and
the kind of answer it requires. 4. After students demonstrate that they understand the difference among the types of
QAR questions, assign several more problems to students. Discuss what type of question is being asked.
5. Eventually, students should generate QAR questions on their own to present to the
rest of the class for identification and answers.
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Example of QAR “Right There” Question The temperature is rising at a constant rate of 3 degrees each hour. In how many hours will the temperature be at 21 degrees if it is now 6 degrees?
Which questions does the problem want us to answer? What is the temperature at the end of 3 hours? How many hours will it be until the temperature reaches 21 degrees? How many hours will it be until the temperature goes up 21 degrees? How many degrees does the temperature change in 1 hour?
“Think and Search” Question
Ron and his family went to see the doubleheader at the ballpark. The first game lasted exactly 2 hours and 30 minutes. During the intermission between games, Ron had a hot dog and a cold drink. The second game ended at 6:15 p.m. At what time did the second game start?
Which pieces of information are necessary to answer the question? The first game started at 1 p.m. Hot dogs and cold drinks cost $3.00 each. The intermission lasted 30 minutes. They left home for the stadium at noon.
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FOCUS ON IMPORTANT FACTS/APPROACHES
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 for students to solve and get a deeper understanding of word problems. 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, recommended by Davis and Gerber (1994), are modeled after the 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. Rationale for the Three-Level Guide strategy Creating a three-level guide for a word problem asks teachers first to 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 expected student difficulties. The questions or statements in a guide help students to develop and appreciate three levels of understanding—given and relevant information identified explicitly in the problem, relationships or inferences implicit in solving the problem, and conclusions or applications from solving the problem. Guidelines for Use of the Three-Level Guide strategy Construct guides for problems with three parts (levels), as in the example. Part I should include a set of true or false facts suggested by the information given in the problem. Part II should have mathematics concepts, ideas, or rules that might apply to the problem. Part III should include possible methods (e.g., calculations) to use in getting 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, state 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.
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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 a guide to determine both its validity and its usefulness in solving the problem.
4. With advanced students, you might select word problems for which they write three-
level guides to share with the class or to exchange with a partner. Exhibit 2. Three-Level Guide Template
Read the problem. Follow the directions given for each part.
Problem:
Part I (Literal Comprehension) Directions: Read the statements. Check column A if the statement is true according to the information in the problem. Check column B if the information will help solve the problem. Part II (Interpretive Comprehension) Directions: Read the statements. Check the ones that state mathematics ideas useful for the problem. Part III (Applied Comprehension) Directions: Check the calculations that can be used in solving this problem.
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Example of a Three-Level Guide Read the problem. Follow the directions given for each part.
Problem: The markup rate on electronics at Ed’s Electronics Mart is 35%. Ed, the store owner, bought 25 Boom speaker systems for a cost of $70 each. What is the selling price of the Boom speaker system at Ed’s Electronics Mart?
Part I (Literal Comprehension) Directions: Read the statements. Check column A if the statement is true according to the information in the problem. Check column B if the information will help solve the problem. A (true or false?) B (helpful?) __________ _________ Ed’s markup rate is 35% __________ _________ Ed bought 25 Boom speaker systems. ___________ __________ The markup rate is less than ¼. __________ _________ The cost to Ed for a Boom speaker
system is $70. __________ _________ The selling price of a Boom speaker
system is more than $70. Part II (Interpretive Comprehension) Directions: Read the statements. Check the ones that state mathematics ideas useful for the problem. ___________ Selling price is greater than cost. __________ Markup equals cost times markup rate. __________ Selling price equals cost plus markup. ___________ Markup divided by cost equals markup rate. __________ The total cost of the systems is the cost of one
system times the number of systems. Part III (Applied Comprehension) Directions: Check the calculations that can be used in solving this problem. _________ 0.35 × $70 _________ 25 × $70 _________ 25 × 35 _________ ($70 × 35) ÷ 100 _________ 1.35 × $70 _________ $70 + (7/20 × $70)
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FOCUS ON IMPORTANT FACTS/APPROACHES
GIST Description of GIST GIST is an acronym for Generating Interactions between Schemata and Texts (Cunningham, 1982). The reading strategy helps students distinguish between significant and supporting details of a passage as they organize a concise, paraphrased summary of the reading. In a mathematics class, students comprehend what a word problem is asking them to do before attempting to solve the problem. The task is to condense the word problem into fewer words than in the original problem. The strategy forces students to discard extraneous details. Rationale for GIST strategy Synthesizing is a culmination of reading comprehension strategies because it draws upon making connections, questioning, visualizing, inferring, and determining importance. One key component of synthesizing is summarizing. The National Reading Panel (2000) includes summarization, where readers integrate ideas and generalize information, as a reading comprehension strategy that appears to be effective for classroom instruction (p. 4-42). Students need to be given the opportunity to reflect and monitor their understanding of passages and to participate in summarizing activities that identify pertinent information, delete trivial information, and generalize in their own words; major strategies necessary for comprehension. GIST is used to generate summarizing sentences of the word problem. In order to perform this task, students must analyze the problem for the essential information needed to solve the problem. Guidelines for Use of GIST 1. Compose or select a word problem that has an abundance of information, either
essential or trivial. A chart with squares representing the maximum number of words can be drawn on the board or chart paper. Students may need to be provided with a prepared chart in order to record their words.
2. Teachers need to model the process with the students, explaining their reasoning as
they proceed through the problem. 3. Students read the first sentence of the problem and summarize the content using
some predetermined number of words; e.g., 20 words or less. 4. The next sentence is read. A summary of the sentence along with the first sentence
is conducted in the predetermined number of words. 5. The process continues until the entire word problem is summarized into the
predetermined number of words. 6. Students can work in groups or individually to complete other problems using the
GIST strategy.
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Example of GIST
Write the following problem in less than 40 words. When Jenny helps her dad run his café in the summer, she makes drinks for the customers. Her favorite drink to make is a fruit smoothie. She uses strawberries, cherries, and bananas and blends them together in a blender. She uses 3 more strawberries than cherries and 2 more cherries than bananas. She has 6 bananas but 2 are rotten, so she decides to use only the good ones. There are at least 12 good cherries and strawberries all together that she can use to make smoothies. How many smoothies can Jenny make for the customers and how much of each fruit will she use? Summary of first sentence: Jenny makes drinks for her dad’s customers in the summer.
Summary of first and second sentence: Jenny likes to make smoothies for her dad’s customers in the summer.
Summary of first, second, and third sentence: Jenny makes fruit smoothies for her dad’s customers by blending strawberries, cherries, and bananas.
Summary of all sentences in word problem: Jenny makes smoothies with 3 more strawberries than cherries and 2 more cherries than bananas. She has 4 bananas and at least 12 cherries and strawberries. How many drinks can she make? How much of each fruit will she use?
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COLLABORATE
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. 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 their solution to the problem both orally and in writing. Rationale for Word Problem Roulette 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. Word problem roulette is cooperative learning and has the advantages that the National Reading Panel and group dynamics experts tell us about cooperative learning. 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 3 or 4 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 verbally. They talk to each
other 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 group agrees on a method and the steps for how they will solve the problem.
3. When a group believes they have an agreed-upon solution to the problem, members
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.
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4. After all the groups have finished writing their solution papers, choose one group at a time to present their 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.
Examples of problems for Word Problem Roulette Directions: Read the problem, and discuss a solution with your group. Do not do any writing during the discussion (no paper or pencils).
When the group agrees on a solution and method to get the solution, write a group report explaining the solution. Each person writes one sentence or step and then passes the paper to another person to do the same. Use only words (no symbols) in the report. Everyone contributes in writing the report.
Problem: A family of three adults and four children goes to an amusement park.
Adult admission is twice as much as a child’s admission. The family spends $80 on admissions. How much is an adult
admission? How much is a child’s admission? Problem: Mr. Hanson had a party for the 23 students in his class. He brought
5 dozen cookies. Every boy in the class ate 3 cookies, and every girl ate 2 cookies. All the cookies were eaten and Mr. Hanson ate none. How many boys and how many girls are in the class?
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COLLABORATE
Collaborative Group Cards
Description of Collaborative Group Cards Erickson (1989) with project EQUALS at the Lawrence Hall of Science (University of California at Berkeley) wrote a collection of problems for groups of students to solve cooperatively. All the information needed to solve a problem is written on cards. However, different pieces of information are on different cards. Some of the group card problems might have more than one answer or different questions might need to be answered. Rationale for Collaborative Group Cards strategy The National Reading Panel (2000) claims that cooperative learning, where students learn together, as a reading strategy does improve comprehension in normal readers (p. 4-42). A classroom becomes an effective learning group when it features high expectations, shared leadership, and communication among students. Cooperation in the mathematics classroom helps to advance several goals of equitable education: complex problems, persistence to succeed, and autonomous learners. Through the Collaborative Group Cards strategy, students learn to work together, use mathematical language to communicate clearly and concisely, and successfully assess when they have solved complex problems correctly. Guidelines for Use of Collaborative Group Cards 1. Develop problems that have useful information distributed on four to six cards.
Some of the cards can be assigned as a check after the group has solved the problem. Place the cards into an envelope for each group. Arrange the students into randomly selected groups.
2. Each person in the group gets a card. If some of the cards are to be used as a
check after solving the problem, then put those cards aside. 3. Each student may only look at their own card (clue). They may share the clue by
telling others what is on the card, but the card may not be shown to anyone else. 4. Debrief the Collaborative Group Cards strategy with the entire class. Did everyone
get a chance to talk? What did you learn from other group members? What strategies did you use to solve the problems? How did you decide you had the answer to the question(s)?
5. Reflect on the Collaborative Group Cards strategy: What occurred while you were
watching the class? What are the strengths of the class? How did students perform the mathematics? Were any individuals surprisingly dominant or competent in the groups?
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Example of Collaborative Group Cards
Each child living in the Bittersweet cul-de-sac neighborhood rides just one of these cycles: unicycle, bicycle, or tricycle. How many bicycles are in the neighborhood now?
There are 8 children living in the Bittersweet cul-de-sac neighborhood. How many bicycles are in the neighborhood now?
Next year Betty will be old enough to ride a unicycle instead of a bicycle. Then there will be a total of 15 wheels when she rides the unicycle. How many bicycles are in the neighborhood now?
The number of bicycles in the Bittersweet cul-de-sac neighborhood is a multiple of the number of tricycles. How many bicycles are in the neighborhood now?
The number of unicycles and tricycles in the Bittersweet cul-de-sac neighborhood is the same. How many bicycles are in the neighborhood now?
7/8 of the total number of wheels are on bicycles and tricycles. How many bicycles are in the neighborhood now?
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COMMUNICATE THINKING
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 (see Exhibit 3 on page 25) 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. Rationale for Process Logs 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. Teachers can include additional question prompts as they consider other ideas they would like in a student’s log. Give students 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 about how many steps are involved. 5. Be sure to describe any special difficulties with the problem. 6. Explain your problem-solving process step by step.
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Exhibit 3. Writing-Math Worksheet
Problem Activity
Extra Challenge: Writing about Problem Solving How many steps are involved in the problem?
What mathematics operations will you use?
Does the problem have special difficulties, things you have to watch out for?
What do you do first?
Then what…?
Then what…?
Then what…?
What do you do to check your work?
How about the extra challenge?
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Example of Process Log
Problem Activity Christy works for 2 hours and 15 minutes each week doing yard work. She gets paid $3 an hour for this work. How many weeks will it take Christy to earn enough money to buy a jacket that costs $36. Extra Challenge: How much would Christy need to get paid per hour if she wanted to buy the jacket in four weeks? Writing about Problem Solving
How many steps are involved in the problem?
There are two steps: I need to find how many hours Christy must work and how many weeks of work give her this number of hours.
What mathematics operations will you use?
Division (36 ÷ 3) and multiplication. Some number × 2 hours 15 minutes.
Does the problem have special difficulties, things you have to watch out for?
I have to think about the time. 15 minutes is ¼ hour. This will make the multiplication problem easier.
What do you do first?
$36 is the goal for Christy. At $3 per hour she has to work 12 hours (36 ÷ 3 = 12).
Then what…?
Each week she works 2¼ hours. I need to find a number (# weeks) to multiply times 2¼ so the result is equal or more than 12.
Then what…?
I’ll try 5. 5 × 2¼ = 11¼. 5 is too small.
Then what…?
It must be 6. 6 × 2¼ = 13½. In 13½ hours of work, Christy will make more than $36, enough to buy the jacket.
What do you do to check your work?
I’ll find how much Christy makes each week, and then see how much she makes in 6 weeks. Each week: $3 × 2¼ = $6.75. 6weeks: 6 × $6.75 = $40.50 (more than $36) 5 weeks: 5 × $6.75 = $33.75 (not enough)
How about the extra challenge?
To make $36 in four weeks, she would have to earn $9 each week. At $4 per hour, she would earn $9 each week ( 4 × 2¼ = 9).
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COMMUNICATE THINKING
RAFT
Description of RAFT The RAFT strategy (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 reading. The RAFT strategy is an excellent way to involve students in explaining what they know about a topic. RAFT can be used after students have studied a concept as a culminating activity or assessment. Options for the students are provided in each of the four areas in order for them to demonstrate their understanding in a nontraditional format. Rationale for RAFT strategy 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 are formulated in a different mode and to possibly do further investigations. The RAFT strategy provides a method for students to synthesize all 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 contexts 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); and 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.
Teachers may assign the same role to all students or let students choose from several different roles.
3. Provide class time for the work. Teachers 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.
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Example of RAFT
Choose a role to communicate a mathematics topic to someone else. Be as thorough as possible in explaining the topic in the given format. If you want to change the format and audience, please ask. The roles and topic will stay the same. Roles Audience Format Topic Zero Percent Prime Number Container
Whole numbers Customers Jury Self
Campaign speech Advertisement Instructions Diary
Importance of 0 Mental ways to calculate Rules of divisibility Volume measurements
Possible combinations are:
“Zero” is campaigning to the whole numbers about the importance of the 0. “Percent” is preparing an advertisement for customers about mental ways to calculate percents. “Prime number” is giving instructions to the jury as to the rules of divisibility. “Container” is writing in its diary about the volume measurements.
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