Word Problems for Solving Strategies

STEPBY

STEPMATH

Strategies for Solving

Word Problems


Word ProblemsBOOK

DDTeaches six basic strategies for solving word problems successfully

Builds on the step-by-step problem-solving process

Improves problem-solving skills and strengthens mathematical reasoning

Provides practice with extended-response problems

S U P P O R T I N G R E S E A R C H

Strategies for Solving Word ProblemsA Research-based Math Program

TABLE OF CONTENTS

Introduction to the Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

How Does Strategies for Solving Word Problems Help Teachers Prepare to Teach Problem-Solving? . . . . . . . . . . . 4

How Does Strategies for Solving Word Problems Support English-Language Learners? . . . . . . . . . . . . . . . . . . . . . . 5

How Does Research Support the Instructional and Learning Strategies in Strategies for Solving Word Problems?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

How Do These Math Strategies Assist Students with Solving Word Problems?. . . . . . . . . . . . . . . . . . . . 10

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Math Strategy Research Support . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Introduction to the SeriesThe Strategies for Solving Word Problems series is the

second tier in the Step-by-Step series. The Step-by-Step series begins with Understanding and Solving Word Problems which provides explicit instruction on a six-step process for reading and understanding word problems. The six-step process and the research base of this first book are also present in Strategies for Solving Word Problems, but the six-step process is presented in an abbreviated format: PREVIEW AND READ; PLAN AND SOLVE; and CHECK AND REVIEW. The focus of Strategies for Solving Word Problems is on key math strategies to use when solving word problems. Students transfer what they have learned in Understanding and Solving Word Problems, to help them learn and apply the mathematics strategies in Strategies for Solving Word Problems.

What Is the Math Content Covered in the Series?The math content that is covered in this series stems

from the five National Council of Teachers of Mathematics (NCTM) content standards. Each content standard is modeled at least once in the series. Each book contains word problems that cover the five content standards of Number and Operations, Algebra, Geometry, Measurement, and Data Analysis and Probability. Books A and B have 28 problems. Books C through H have 44 problems.

Is There a Need for Word Problem Instruction?The findings in research and assessments state the need for

word-problem instruction. In 2003, American students scored below the international average of 500 on two significant areas of the Program for International Student Assessment (PISA) test. They scored 477 on the problem-solving section of the test and scored 483 on the mathematics literacy section. The international average of industrialized nations was 500 on both sections. The United States placed 24th of 29 nations in each category, indicating a lackluster performance. (Cavanaugh, 2005)

In 1991, researchers Wilson and Sindelar cited the following drawbacks to basal mathematics programs. Their concerns included a lack of adequate provisions for effective practice and review, inadequate sequencing of problems, and an absence of strategy teaching and step-by-step procedures for teaching math problem solving.

“Solving word problems in math

involves a complex web of skills

that require learners to be good

readers and to be proficient at

thinking critically, computing,

and using a process to solve

problems” (Forsten, 2004, p. 21)

◆ DRAW A DIAGRAM ◆ FIND A PATTERN ◆ ACT IT OUT

Six students are scheduling appointments with Mr. Scanlon, the guidance counselor,

to talk about courses that they will take next year. Each appointment lasts 30 minutes.

Mr. Scanlon has 6 appointments available on Friday, starting at 9:00 and ending at

12:00. The students have the following time restrictions:

Andrew can only take the 10:00 appointment.

Belle has tests between 8:50 and 10:15.

Carole has a dentist’s appointment and will not be in school until 11:15.

Diego’s only free time is between 8:50 and 9:30.

The only free time Eric has is between 10:20 and 11:00.

Fred has no restrictions on this day.

How must Mr. Scanlon schedule the appointments so that all 6 students can meet

with him on Friday?

17.

Strategy:

Solution:

Fine Foods prints labels for food packages. Labels are worth 5 points, 12 points,

or 18 points. Students may collect the labels and get school supplies donated by Fine

Foods. The kinds of school supplies depend on the point value of the labels. Carly has

collected labels worth 180 points. What is one possible point combination of labels

that she collected?

18.

Strategy:

Solution:

34

TRY IT!

Strategies for Solving Word Problems–Mixed Practice Problems

STEPBY

STEPMATH


Word ProblemsStrategies for Solving

Word ProblemsBOOK

EETeaches six basic

strategies for solving

word problems successfully

Builds on the step-by-step

problem-solving process

Improves problem-solving

skills and strengthens

mathematical reasoning

Provides practice with

extended-response

problems

Research Paper: Strategies for Solving Word Problems 3

According to the U.S. Department of Education in 2001, language proficiency appears to be a contributing factor in problem solving; student performance on word problems is generally 10–30% below that on comparable problems in a numeric format.

And, for further reinforcement of the connection between effective reading and mathematics success, consider the following conclusion proposed by Jitendra and Xin (1997, page 413), “[s]tudents who have difficulty with reading or math computation, or both, are likely to encounter difficulties with word problem solving.”

How Does Strategies for Solving Word Problems Help Teachers Prepare to Teach Problem-Solving?

The No Child Left Behind Act of 2001 requires that teachers are well-prepared for their classroom. “A prepared teacher knows what to teach, how to teach and has command of the subject matter being taught” (U. S. Department of Education, 2005). Strategies for Solving Word Problems helps teachers be well-prepared through several features that offer math-content support.

1. Lesson Overview—In the teacher guide, each lesson begins with an overview that discusses math vocabulary, prior knowledge, and background math content that is relevant to the lesson’s math strategy and problems.

2. An Introduction to the Program—A quick-reference chart in the student book displays when and how to apply each strategy to a word problem.

3. Math Strategy Research Support—Research support behind each strategy can be found in the teacher guide. The research support provides additional information for a more comprehensive picture of the math strategy.

4. Annotated Solutions to Mixed Problems—In the teacher guide, teachers are provided with possible solutions to the mixed-problem section. This feature demonstrates mathematical reasoning and solutions for the teacher to follow, if needed.

Insert p. 94, Part 2 of

Level C

Researchers Hill, Rowan, and

Loewenber (2005) have found

that teachers’ mathematical

knowledge was significantly

related to student achievement

gains in both first and third

grades.

SummaryStrategies for Solving Word Problems is a step-by-step implementation of problem-solving strategies that aid students in the approach and computation of solving word problems. Strategies for Solving Word Problems continues with the comprehension process learned in its companion book, Understanding and Solving Word Problems. Building upon this process, Strategies for Solving Word Problems introduces research-based mathematics strategies students may use to plan and solve word problems. Additionally, students are taught with research-based teaching and learning strategies. These strategies help students not only solve their word problems using multiple methods, but also to self-assess and monitor their learning. Learners of all abilities and backgrounds will benefit from the gradual accomplishment of reaching solutions independently. The consistent application of scaffolded instruction easily allows for differentiated instruction in the classroom. The challenge of creating open-ended solutions will reinforce students’ mathematical reasoning and problem solving. Cooperative learning during the review and assessment phase of the process also reinforces their mathematical reasoning. Each instructional step in the program makes Strategies for Solving Word Problems a comprehensive learning experience that turns a particularly troublesome math area into one that is approachable and attainable by students.

Math Strategy Research Support

Find a Pattern

A precursor to algebraic thinking, the Find a Pattern strategy is an essential strategy for students to know. “Smith (2003) believed that for students to think algebraically, they must be able to identify, extend, and generalize patterns in order to understand quantitative relationships” (Steele, 2005, p. 142).

Act It Out

This strategy draws students into the math problem by having them actively engage in solving the problem. Students may use manipulatives as a means of acting out a word problem. “. . . [H]ands-on learning helps students to more readily understand concepts and boosts their self-confidence. . . . Harnessing the power of manipulatives has proven invaluable in the teaching of mathematics. Students are better able to visualize math concepts and gain insights into necessary fundamentals when they use rods, cubes, and other tools” (DeGeorge & Santoro, 2004).

Guess and Check

The Guess and Check strategy is a strategy that builds students’ ability to work accurately with numbers. “The guessing and checking that students often do to understand the situation provides practice with basic skills and allows them to wrap their minds about the true conditions of a problem” (Kalman, 2004, p. 179).

Make a Table/Draw a Picture or Draw a Diagram

The strategies Make a Table and Draw a Picture or Draw a Diagram are visualization instruments that allow students to see the relationships and patterns between numbers and variables. “Tables and graphs provide visual means for students to organize and summarize numerical and verbal data” (Krulik & Rudnick, 1996; Sorenson et al., 1996).

Write a Number Sentence or Write an Equation

This is an integral mathematics strategy to learn because it requires students to transfer their mathematics knowledge into another representation. According to Mayer (2002), “[Representing] occurs when a student is able to convert information from one form of representation to another . . . Transfer is the ability to use what was learned to solve new problems, answer new questions, or facilitate learning new subject matter (Mayer & Wittrock, 1996)” (p. 226).

124 Research Paper: Strategies for Solving Word Problems

How Does Strategies for Solving Word Problems Support English-Language Learners?

Strategies for Solving Word Problems continues the use of the comprehension process found in Understanding

and Solving Word Problems. However, the process acts as background support as students concentrate on learning problem-solving strategies. The consistent presence of the comprehension process is important to English-language learners (ELL). Experts from the federally funded organization, ESCORT (1998), recommend the following steps in helping ELL students and general education students comprehend math word problems. 1. Understand the question. Teach students to understand

the problem through elaboration and imagery. Then rewrite the question as a statement.

2. Find the needed information. Help students to use selective attention (e.g., disregard irrelevant data or number distractors to find needed information).

3. Choose a plan. Have students identify the operation and what the problem calls for, then choose a plan (e.g., write a number sentence, identify parts of the problem, work with a peer, make a table, make a list).

4. Solve the problem. Students write out the steps of the problem and solve it, using cooperation to review the steps they have taken.

5. Check the answer. Students use a variety of approaches to verify their answer.

These recommendations are mirrored in Strategies for Solving Word Problems. In the PREVIEW AND READ section of the lesson, students must identify the question and key facts and choose an appropriate problem-solving strategy. In the PLAN AND SOLVE section, students develop a visual and written plan for the chosen strategy. Students then carry out the plan to find the solution. In the CHECK AND REVIEW section, students work individually, or in pairs or small groups to complete the problem-solving process.

• According to the U.S.

Department of Education,

nearly 1 in 12 students

receive special assistance to

learn English.

• The population of English-

language learners has grown

over 86% since 1992, while

general K-12 enrollment has

grown only 11%.

Is ELL Instruction Relevant to

Your Classroom?

Strategy One: Draw a Picture

2

Modeled Practice

STUDY

IT! Study how Sean used the strategy Draw a Picture to solve the model word problem. He used this strategy because the problem gives information that can be drawn. Read why Sean chose Draw a Picture and how he found and checked his solution.

PREVIEW and READ

Mrs. Sanchez is arranging the desks in her classroom. The desks are 2 feet wide.

She leaves 3 feet of space between each desk. Mrs. Sanchez puts 6 desks in a row.

What is the total width of a row of desks?

Explain Why: I chose D P because the problem gives information that I can draw and label: 6 desks; each 2 feet wide; 3 feet between each desk. The question asks for the total width of a row of desks.

PLAN and SOLVE

1. Find the information needed for the picture: row of 6 desks each 2 feet wide, 3 feet between desks

2. Draw and label the picture:

2 ft 2 ft 2 ft 2 ft 2 ft 2 ft↔ ↔ ↔ ↔ ↔ ↔

→

3 ft→

→

3 ft→

→

3 ft→

→

3 ft→

→

3 ft→

3. Use the picture to answer the question: Find the total length (27 ft).

Explain How: I wrote the information I needed for the picture (row of 6 desks, each 2 feet wide, 3 feet between desks). I drew 6 rectangles and labeled each one 2 feet. I labeled the space between each desk 3 feet. Then I added the measurements: 2 ft + 3 ft + 2 ft + 3 ft + 2 ft + 3 ft + 2 ft + 3 ft + 2 ft + 3 ft + 2 ft = 27 ft.

Solution: The total width of a row of desks is 27 feet.

CHECK and REVIEW

I checked my solution by multiplying by 2 the number of labels for 2 feet and multiplying by 3 the number of labels for 3 feet. I then added the products: (6 x 2 ft) + (5 x 3 ft) = 12 ft + 15 ft = 27 ft.

Model Word Problem

Students incorporate the process of solving word problems

with the newly learned problem-solving strategies.


Strategies for Solving Word Problems supports ELL students with the additional feature of cooperative learning. Math students benefit from a variety of instructional settings in the classroom. The teacher must guide students through individual, small-group, and whole-class activities. ELL students receive additional aid through a glossary in the student book. The glossary covers mathematical terms used in the word problems.

Strategies for Solving Word Problems is a useful tool for ELL students to use to solve word problems. The comprehension process that is so critical to ELL students continues to support them as they learn new content. ELL students’ learning experiences are also bolstered through cooperative learning, the variety of activities, and the glossary.

How Does Research Support the Instructional and Learning Strategies in Strategies for Solving Word Problems?

Strategies for Solving Word Problems is integrated with research-based instructional and learning strategies.

Scaffolded InstructionOnce students complete the Introduction to the Program

section they enter the main body of the program, where scaffolded instruction begins. Students progress from modeled instruction, to guided instruction, to independent learning. This gradual release of responsibility from the teacher to the student is particularly helpful to ELL students. “Where before there was a spectator, let there now be a participant” (Bruner, 1960).

3

Instruct students to work alone, cooperatively in pairs, or collaboratively in groups. Have a range of manipulatives available, and tell students that they may use any that they find useful. Because students must be able to choose the appropriate manipulative when working independently, none is specified in the student book.

In this section, students need only write and complete the steps of the strategy and arrive at a solution. They are to show all of their work. Tell students the level of detail that you wish them to provide in their responses. Ensure that students check their solutions, and then have them compare their solutions with a partner, in small groups, or as a class. Have students explain why they chose any particular strategy.

How can I meet individual needs with this program?

Because of the range of languages, cultures, and learning in any classroom, ensure that all students comprehend what is written in the student book and what they are being asked to do. Because math-language instruction is especially important for English-language learners, you will need to spend additional time defining math terms and related vocabulary, and clarifying the usage of those words which have an everyday meaning as well as a math-related definition. Also, by probing students for prior knowledge about the context of the problem and watching for cultural assumptions about the problem, you can ensure students’ understanding. You might rephrase a problem and question, give examples to highlight vocabulary, and explain cultural activities, such as games or sports cited in the problems. You may choose to adapt any part of a strategy lesson, as necessary, for particular students. (See Research Summary, page 8.) In Mixed Practice, you may wish to suggest a strategy for each problem to those students who need support. (See the annotated solutions for possible strategies, page 32.)

How much time is needed to complete Strategies for Solving Word Problems?

The chart on page 7 shows a suggested schedule for completing this program in either 23 days or 28 days, depending upon the length of your class period and the progress of your students.

How does Strategies for Solving Word Problems build on the foundation of Understanding and Solving Word Problems?

The Strategies for Solving Word Problems series builds on the problem-solving process taught in the first Step-by-Step Math series, Understanding and Solving Word Problems. The purpose of the first series is to teach students a problem-solving process that they can apply to solve any word problem. The process consists of six steps: preview, read, plan, solve, check, review, each with its own strategies. Some of these strategies include identifying the question and key facts in a word problem, visualizing the problem, drawing a sketch of the problem, thinking aloud through the problem, noting each step leading to a solution, and writing a clear solution statement. Because the process is new to students, each step and its accompanying strategies are first introduced and practiced one step at a time, and then reinforced and practiced as a whole. After students complete Understanding and Solving Word Problems, they have mastered the problem-solving process and can apply it independently.

In Strategies for Solving Word Problems, the problem-solving process is maintained as a framework for learning and applying the six math strategies: Draw a Diagram (or Draw a Picture), Find a Pattern, Act It Out, Make a Table, Write an Equation (or Write a Number Sentence), Guess and Check. For the purpose of this series, the six steps are combined in logical pairs: preview and read, plan and solve, check and review. At each stage of the process, the focus is on the new math strategy. For example, when students preview and read a word problem, they evaluate the question and important information to determine which strategy would be appropriate for solving the problem. Through writing, students explain why the strategy is an appropriate one. When students plan and solve, they write a plan for using the strategy and then execute the plan to find the solution. Here students show their work and explain how they used the strategy to solve the problem. When students check and review, they choose a method to check their solution, reviewing their work with the strategy. Again students write about their work. Class discussion follows independent work.

Strategies for Solving Word Problems provides teachers

with instructional suggestions for ELL students.

99

These recommendations are mirrored in Strategies for Solving Word Problems. In the preview and read section of the lesson, students must identify the question and key facts and choose an appropriate problem-solving strategy. In the plan and solve section, students develop a visual and written plan for the chosen strategy. Students then carry out the plan to find the solution. In the check and review section, students work individually, or in pairs or small groups to complete the problem-solving process.

Strategies for Solving Word Problems supports ELL students with the additional feature of cooperative learning. Math students benefit from a variety of instructional settings in the classroom. The teacher must guide students through individual, small-group, and whole-class activities. ELL students receive additional aid through a glossary in the student book. The glossary covers mathematical terms used in the word problems.

Strategies for Solving Word Problems is a useful tool for ELL students to use to solve word problems. The comprehension process that is so critical to ELL students continues to support them as they learn new content. ELL students’ learning experiences are also bolstered through cooperative learning, the variety of activities, and the glossary.

How Does Research Support the Instructional and Learning Strategies in Strategies for Solving Word Problems?

Instructional and Learning Strategies in Strategies for Solving Word Problems

Scaffolded Instruction

Modeled Guided Independent

Prior-Knowledge ActivationMath-Language Instruction

Think-Aloud Instruction

Scaffolded InstructionStrategies for Solving Word Problems is integrated with research-based instructional and learning strategies. Once students complete the Introduction to the Program section they enter the main body of the program, where scaffolded instruction begins. Students progress from modeled instruction, to guided instruction, to independent learning. This gradual release of responsibility from the teacher to the student is particularly helpful to ELL students. “Where before there was a spectator, let there now be a participant” (Bruner, 1960).

Modeled Practice“Many students, particularly low performing students, learn more quickly from a clear, concise explanation of what to do and how to do it” (Marchand-Martella, Slocum, & Martella, 2004, p. 211). Students’ first exposure to the problem-solving strategies is in the Modeled Practice section. Students read and learn why and how a strategy works for a given problem. “Direct modeling is a highly sensible approach” (Kilpatrick, Swafford, & Findell, 2001, p. 186). Students then apply the knowledge learned from the model problem to a practice problem in the Try It activity that accompanies the model.

Guided Practice Guided instruction occurs when there is a balance of responsibility between the teacher and the student for the learning in the lesson. According to Bransford, Brown, and Cocking (2000) in How People Learn, teachers using guided instruction benefit because this is a time for teachers to obtain students’ domain knowledge and ideas, and to correct any misconceptions. In Guided Practice, students are provided with partially detailed clues to complete each of the strategy steps. Students are responsible for deciding what the problem is about, how to solve the problem with the available strategy, and proving that the problem was solved using a logical method.

Independent Practice and Mixed PracticeStudents apply what they have learned about the problem-solving strategies from the Modeled Practice and Guided Practice sections and apply their new knowledge in the Independent Practice and Mixed Practice sections. Once more, scaffolding plays a role in aiding students to become independent learners. In the Independent Practice section, students apply the lesson’s specified strategy to solve two word problems (one word problem in books A and B). At this point in their learning, students may choose to refer back to the Modeled or Guided Practice sections for assistance or they may choose to work without assistance. In the Mixed Practice section, students are completely responsible for their strategies and for assessing their application of the strategies. “Individual work settings ensure that all students process lessons at their own rate of learning” (Buchanan & Helman, 2000).

The instructional design of Strategies for Solving Word Problems is further supported by additional research-based instructional and learning strategies. These strategies include: prior-knowledge activation, math language instruction, and a think-aloud learning strategy.


Modeled Practice“Many students, particularly low performing students,

learn more quickly from a clear, concise explanation of what to do and how to do it” (Marchand-Martella, Slocum, & Martella, 2004, p. 211). Students’ first exposure to the problem-solving strategies is in the Modeled Practice section. Students read and learn why and how a strategy works for a given problem. “Direct modeling is a highly sensible approach” (Kilpatrick, Swafford, & Findell, 2001, p. 186). Students then apply the knowledge learned from the model problem to a practice problem in the Try It activity that accompanies the model.

Guided Practice Guided instruction occurs when there is a balance

of responsibility between the teacher and the student for the learning in the lesson. According to Bransford, Brown, and Cocking (2000) in How People Learn, teachers using guided instruction benefit because this is a time for teachers to obtain students’ domain knowledge and ideas, and to correct any misconceptions. In Guided Practice, students are provided with partially detailed clues to complete each of the strategy steps. Students are responsible for deciding what the problem is about, how to solve the problem with the available strategy, and proving that the problem was solved using a logical method.

Independent Practice and Mixed PracticeStudents apply what they have learned about the problem-

solving strategies from the Modeled Practice and Guided Practice sections and apply their new knowledge in the Independent Practice and Mixed Practice sections. Once more, scaffolding plays a role in aiding students to become independent learners. In the Independent Practice section, students apply the lesson’s specified strategy to solve two word problems (one word problem in books A and B). At this point in their learning, students may choose to refer back to the Modeled or Guided Practice sections for assistance or they may choose to work without assistance. In the Mixed Practice section, students are completely responsible for their strategies and for assessing their application of the strategies. “Individual work settings ensure that all students process lessons at their own rate of learning” (Buchanan & Helman, 2000).


2

Modeled Practice

STUDY

IT! Study how Sean used the strategy Draw a Picture to solve the model word problem. He used this strategy because the problem gives information that can be drawn. Read why Sean chose Draw a Picture and how he found and checked his solution.

PREVIEW and READ

Mrs. Sanchez is arranging the desks in her classroom. The desks are 2 feet wide.

She leaves 3 feet of space between each desk. Mrs. Sanchez puts 6 desks in a row.

What is the total width of a row of desks?

Explain Why: I chose D P because the problem gives information that I can draw and label: 6 desks; each 2 feet wide; 3 feet between each desk. The question asks for the total width of a row of desks.

PLAN and SOLVE

1. Find the information needed for the picture: row of 6 desks each 2 feet wide, 3 feet between desks

2. Draw and label the picture:

2 ft 2 ft 2 ft 2 ft 2 ft 2 ft↔ ↔ ↔ ↔ ↔ ↔

→

3 ft→

→

3 ft→

→

3 ft→

→

3 ft→

→

3 ft→

3. Use the picture to answer the question: Find the total length (27 ft).

Explain How: I wrote the information I needed for the picture (row of 6 desks, each 2 feet wide, 3 feet between desks). I drew 6 rectangles and labeled each one 2 feet. I labeled the space between each desk 3 feet. Then I added the measurements: 2 ft + 3 ft + 2 ft + 3 ft + 2 ft + 3 ft + 2 ft + 3 ft + 2 ft + 3 ft + 2 ft = 27 ft.

Solution: The total width of a row of desks is 27 feet.

CHECK and REVIEW

I checked my solution by multiplying by 2 the number of labels for 2 feet and multiplying by 3 the number of labels for 3 feet. I then added the products: (6 x 2 ft) + (5 x 3 ft) = 12 ft + 15 ft = 27 ft.

Model Word Problem

Strategy One: Modeled Practice

Strategy One: Guided Practice

Strategy One: Independent Practice

The problem gives information that can be shown in a picture.

The question asks for a solution that can be found in the picture.

Check your solution by solving the problem another way.

1. Find the information needed for the picture.

2. Draw and label the picture with information from the problem.

3. Use the picture to answer the question.

3

Read the practice word problem. Explain why Draw a Picture can be used to solve the problem. Then solve the problem. Check your solution.

PREVIEW and READ

Harry has 4 cards. Each card has a number: 1, 2, 3, or 4. He picks

two cards. How many different pairs of numbers could Harry pick?

Explain Why:

PLAN and SOLVE

Explain How:

Solution:

CHECK and REVIEW

Practice Word Problem

TRY IT!

HINTS


Guided Practice

TRY IT!

4

Use the strategy Draw a Picture to solve the word problem.Show your work. Then check your solution.

1. Find the information needed for the picture.2. Draw and label the picture with information

from the problem.3. Use the picture to answer the question.

PREVIEW and READ

Amy wants to use the fewest number of colors possible

to color this design. Parts that touch must be a different

color. What is the fewest number of colors that Amy can use?

PLAN and SOLVE

Explain How:

Solution:

CHECK and REVIEW

REMEMBER

5


Independent Practice

Use the strategy Draw a Picture to solve each word problem. Show your work. Then check your solution.

In a game, cards are placed in 3 rows of 4 cards. One third of the game cards

are blue. How many game cards are blue?

Solution:

1.

Wade is making a hot plate with 16 tiles. Each tile is a square with 2-inch sides.

He puts the tiles in rows of 4. Wade leaves a 1-inch space between the tiles.

What is the length of one side of the hot plate?

Solution:

2.

TRY IT!

Students receive modeled, guided, and independent practice

for each of the six strategies.


The instructional design of Strategies for Solving Word Problems is further supported by additional research-based instructional and learning strategies. These strategies include: prior-knowledge activation, math language instruction, and a think-aloud learning strategy.

Prior-Knowledge ActivationThis learning strategy is predominant in Understanding

and Solving Word Problems and in Strategies for Solving Word Problems. Activating prior knowledge helps readers relate their existing knowledge to the concepts in a text. Prior knowledge allows students to make unconscious inferences during reading. Students also try to figure out how the text they are reading relates to their personal prior knowledge (Pressley, 2002). In Strategies for Solving Word Problems, teachers are provided with key mathematics terms and concepts that are needed to complete each lesson. Students engage these terms and concepts as they complete the PREVIEW AND READ section of the lesson. “Several studies of second-language speakers and reading comprehension indicate that prior and existing cultural experiences are extremely important in comprehending text” (Steffensen, Joag-Dev, & Anderson, 1979). When students PLAN AND SOLVE, they apply their prior knowledge. During CHECK AND REVIEW, students reinforce their prior-knowledge activation by discussing their responses with a peer.

Math Language InstructionLearning and understanding mathematical language is

challenging to all students. Mathematical vocabulary can be difficult to learn and remember because of how the words are used in everyday English, which often contrasts with how the words are used in mathematics. Many educators agree with Miller (1993) that “without an understanding of the vocabulary that is used routinely in mathematics instruction, textbooks, and word problems, students are handicapped in their efforts to learn mathematics” (Monroe & Orme, 2002, p. 140).

2020

Strategy ThreeLesson Overview

Act It Out

The strategy Act It Out can be used to solve word problems that give information that can be acted out with manipulatives. Before presenting the lesson, do a pantomime of going up stairs, and have students determine what you are doing. Have volunteers pantomime the playing of a particular sport and the taking of a mode of transportation, and ask the rest of the class to guess what is being enacted. Explain to students that they will be acting out math problems.

Vocabulary: See page 36 in the student book.perimeter the distance around a figurerectangle a flat shape with 4 sides and

4 right anglessquare a rectangle with 4 sides of equal lengthalternate to arrange by turns, first one and then

the otherequal is the same value as

Prior Knowledge:perimeter the distance around a figure, found by

adding the measures of all the sides

Background for the Strategy:In this lesson, students will learn how to determine when to use the strategy Act It Out by previewing and reading a problem and asking themselves, “Does the problem give information that can be acted out? Does the question ask for a solution that can be found by acting something out?” Students will also learn how to use the strategy by following three steps when planning and solving a problem:1. Find the information that can be acted out.2. Act out the information in as many ways

as possible.3. Choose the way that answers the question.

Modeled PracticeStrategy Three: Act It Out

STUDY

IT!

Page 10Introduce the strategy Act It Out by completing this page as a classroom activity. The page presents students with why and how this strategy can be used to solve a word problem.

Manipulatives: math tiles (optional)

PREVIEW and READ

Read the introduction aloud as students follow along in their book.

Alan has 12 square tiles. Alan wants to arrange

the tiles in equal rows to make a rectangle. He

wants to make the rectangle with the smallest

perimeter possible. How should Alan arrange

the tiles?

Model Word Problem

Have students read the problem aloud, and then help them preview the problem by determining what it is asking (How should Alan arrange the tiles?). Discuss what information in the problem is needed to answer the question (12 square tiles arranged in equal rows to make a rectangle with the smallest perimeter possible).

Explain Why: Lead students to see why Ramona chose the strategy Act It Out. She found some information (an arrangement of square tiles in a rectangle with the smallest perimeter possible) that she could act out with math tiles in order to solve the problem.

PLAN and SOLVE

Discuss with students how Ramona applied the strategy Act It Out to the word problem. She made a plan of what she could act out: an arrangement of 12 square tiles in equal rows in a rectangle. She found the solution by acting out 2 rectangles with 12 math tiles and choosing the one with the smallest perimeter.

Strategy Three: Act It Out

10

Modeled Practice

STUDY

IT! Study how Ramona used the strategy Act It Out to solve the model word problem. She used this strategy because the problem gives information that can be acted out. Read why Ramona chose Act It Out and how she found and checked her solution.

PREVIEW and READ

Alan has 12 square tiles. Alan wants to arrange the tiles in equal rows to make

a rectangle. He wants the rectangle with the smallest perimeter possible.

How should Alan arrange the tiles?

Explain Why: I chose A I O because the problem gives information that I can act out with math tiles. The question asks for a possible arrangement. I can find that information by trying different arrangements with the tiles.

PLAN and SOLVE

1. Find the information that can be acted out: arrangements of 12 tiles in equal rows made into rectangles

2. Act out the information:1 2 3 4 5 6

16 7

15 814 13 12 11 10 9 Perimeter = 16

1 2 3 414 5

13 6

12 711 10 9 8 Perimeter = 14

3. Choose: the 4 x 3 rectangle

Explain How: I acted out the information and found two ways to make a rectangle: 6 x 2 and 4 x 3. I counted the number of sides of the tiles around the outside of each rectangle to find the perimeter. I got 16 and 14. The 4 x 3 rectangle has the smaller perimeter.

Solution: Alan should arrange the tiles in a 4 by 3 rectangle.

CHECK and REVIEW

I checked my solution by adding together the measures of the sides of each rectangle: 6 + 6 + 2 + 2 = 16; 4 + 4 + 3 + 3 = 14.

Model Word Problem

Lesson Overview in Teacher Guide

Teachers engage students by activating student’s prior

knowledge of math vocabulary and concepts.

36

alternate to arrange by turns, first one and then the other

data information collected

equal is the same value as

geometric shapes flat shapes, such as squares, rectangles, triangles, and circles

number sentence two or more expressions set equal to each other, such as 5 + 6 = 11 and 20 – 3 = 17

operation addition, subtraction, multiplication, or division

pattern a repeated sequence or design, such as 2, 4, 6, 8 or ■ ◆ ▼ ■ ◆ ▼

perimeter the distance around a figure

product the answer to a multiplication problem

reasonable based on common sense or sound thinking; logical; not extreme

rectangle a flat shape with 4 sides and 4 right angles:

Glossary

37

row a line of people or things that go across

rule a statement of what to do

set a group whose members have things in common, such as the set of even numbers

square a rectangle with 4 sides of equal length and 4 angles:

table a listing of items or numbers that are alike in some way, arranged in rows and columns:

rows

columns➞ ➞

➞ Homework Time to Complete

➞ Math 20 minutes

➞ Reading 30 minutes

➞ Spelling 15 minutes

triangle a flat shape with 3 sides and 3 angles:

unknown a quantity whose value is not identified

A glossary with graphics assists students in their comprehension

of math vocabulary.


According to Rubenstein and Thompson (2002), the following are some of the ways in which mathematical language can be difficult for students.

1. Some words are used in both everyday English and in math, but have different meanings in each context. Example: right angle versus right answer

2. Some mathematical words are only found in a mathematical context. Example: quotient, isosceles

3. Some mathematical words have more than one meaning. Example: a circle is round versus to round a number to the

tenths place 4. Some mathematical words are homonyms to everyday

English words. Example: sum versus some, pi versus pie 5. Some mathematical concepts are verbalized in more than

one way. Example: one-quarter versus one-fourth

Strategies for Solving Word Problems provides math-language instruction with the aid of a glossary in the student book. Teachers are also provided with a list of vocabulary words that they can review before the lesson begins and discuss during the lesson.

Think-Aloud Learning StrategyStrategies for Solving Word Problems continues to employ

the metacognitive learning strategy—using Think-Alouds—to build students’ understanding of mathematics. This strategy, a secondary strategy in Strategies for Solving Word Problems, encourages students to demonstrate their problem-solving process through monitoring, adapting, and self-assessing. The Think-Aloud activities are concrete representations of their thought processes. Students express their thought processes in the PREVIEW AND READ section, the PLAN AND SOLVE section, and the CHECK AND REVIEW section. These three activities provide insight into students’ thought processes, for both the teacher and the students. “Teachers can gain valuable insights into students’ ways of interpreting and thinking about mathematics by looking at their representations” (NCTM, 1989). “Teachers benefit children most when they encourage them to share their thinking process and justify their answers out loud or in writing as they perform math operations. More and more of the standardized tests are geared toward this type of response (through long- and short-response questions)” (Berman, 2003, p. 172).

Teachers preview vocabulary words before each lesson.

1414

Strategy OneLesson Overview

Draw a Picture

The strategy Draw a Picture can be used to solve word problems that give information that can be drawn. Before presenting the lesson, ask students to describe graphical signs without words, that they have seen. If necessary, prompt students by asking how they know if a hospital is nearby, what building is a school, or if part of a roadway is reserved for bicyclists. Have volunteers draw such common signs and any international signs, or allow them to create their own graphical representations for representing particular buildings, places, or warnings. Lead students to see that they may use such to help them solve word problems. Mention to students that, for the strategy Draw a Picture, their pictures need be only accurate enough to aid their finding the solution to the problem.

Vocabulary: See page 36 in the student book.row a line of people or things that go acrossproduct the answer to a multiplication problemrectangle a flat shape with 4 sides and 4 right anglessquare a rectangle with 4 sides of equal length

Prior Knowledge:fractions parts of a whole, such as

12,

13,

and 14

geometric shapes plane shapes, such as squares, rectangles, triangles, and circles

multiplication facts products of 1-digit factors

Background for the Strategy:In this lesson, students will learn how to determine when to use the strategy Draw a Picture by previewing and reading a problem and asking themselves, “Does the problem give information that can be shown in a picture? Does the question ask for a solution that can be found in the picture?” Students will also learn how to use the strategy by following three steps when planning and solving a problem:1. Find the information needed for the picture.2. Draw and label the picture with information

from the problem.3. Use the picture to answer the question.

Modeled PracticeStrategy One: Draw a Picture

STUDY

IT!

Page 2Introduce the strategy Draw a Picture by completing this page as a classroom activity. The page presents students with why and how this strategy can be used to solve a word problem.

PREVIEW and READ

Read the introduction aloud as students follow along in their book.

Mrs. Sanchez is arranging the desks in her

classroom. The desks are 2 feet wide. She

leaves 3 feet of space between each desk.

Mrs. Sanchez puts 6 desks in a row. What

is the total width of a row of desks?

Model Word Problem

Have students read the problem aloud, and then help them preview the problem by determining what it is asking (What is the total width of a row of desks?). Discuss what information in the problem is needed to answer the question (6 desks, each 2 feet wide; 3 feet between each desk).

Explain Why: Lead students to see why Sean chose the strategy Draw a Picture. He found some information that he could represent in a drawing (6 2-foot-wide desks with 3 feet between them) in order to solve the problem.

PLAN and SOLVE

Discuss with students how Sean applied the strategy Draw a Picture to the word problem. He made a plan by writing the information that he needed: the number of desks, the width of the desks, the amount of space between the desks. He found the solution by drawing a picture of the desks with the measurements labeled. He added the measurements to find the total width of the row.


10

Modeled Practice

STUDY


PREVIEW and READ





PLAN and SOLVE



16 7

15 814 13 12 11 10 9 Perimeter = 16

1 2 3 414 5

13 6

12 711 10 9 8 Perimeter = 14




CHECK and REVIEW


Model Word Problem

Students use the think-aloud strategy to make their problem-solving thoughts visible as they

apply each strategy.


How Do These Math Strategies Assist Students with Solving Word Problems?

Strategies for Solving Word Problems assists students with solving word problems by making the abstract nature of mathematics more visual and therefore more attainable. Students learn about multiple representations of math concepts and procedures and also develop multiple solutions. Moreover, Strategies for Solving Word Problems supports the development of self-autonomy. By providing students with multiple problem-solving tools, students will have a repertoire of solution strategies to apply to a word problem. With these tools, students, who might become overwhelmed and stagnated when working with word problems, become actively engaged in solving a word problem. “Problem-solving ability is enhanced when students have opportunities to solve problems themselves and to see problems being solved” (National Research Council, 2001, p. 420).

VisualizationVisualization is defined by Keene and Zimmermann

(1997) as a comprehension strategy that enables readers to make the words on a page real and concrete. Visualization in mathematics is providing evidence or proof of understanding of the mathematical solution process. In Strategies for Solving Word Problems, students make their visualizations apparent by applying the mathematics strategy learned in the lesson. This step occurs during the PLAN AND SOLVE section of the lesson. For example, the ACT IT OUT strategy entails students using manipulatives or live action to work out a solution to a word problem. Students use concrete tools such as counters, pieces of paper, and math tiles to express their mental mathematics. This modification of the visualization allows students to see in another medium how well their solution is working.

Using another medium to demonstrate their mathematical reasoning is helpful to ELL students. Visualization allows them to communicate their mathematical understanding when language may be a barrier. “Manipulatives, pictures, diagrams, kinesthetic tools, and body language can help these students demonstrate what they know about facts, concepts, and algorithms” (Lee & Sikjung, 2004, p. 270).


Word Problems


10

Modeled Practice

STUDY


PREVIEW and READ





PLAN and SOLVE



16 7

15 814 13 12 11 10 9 Perimeter = 16

1 2 3 414 5

13 6

12 711 10 9 8 Perimeter = 14




CHECK and REVIEW


Model Word Problem

Abstract

Visual

Students use visualization to make their problem-solving

apparent to peers and teachers.


“The . . . strategies may provide students with different ways to ‘see’ word problems and better grasp the concepts of what is being asked and how to work toward a solution” (Shellard, 2004, p. 42).

Development of Multiple Representations of Math Concepts

“. . . [A]ll mathematical ideas require representations, and their usefulness is enhanced through multiple representations. Because each representation has its advantages and disadvantages, one must be able to choose and translate among representations” (National Research Council, 2001, p. 110). Strategies for Solving Word Problems focuses on the development of physical representations of numbers, such as symbols, actions, pictures, and words. The editors of Adding

It Up: Helping Children Learn Mathematics (2001) explain that “[p]hysical representations serve as tools for mathematical communication, thought, and calculation, allowing personal mathematical ideas to be externalized, shared, and preserved. They help clarify ideas in ways that support reasoning and build understanding. These representations also support the development of efficient algorithms for the basic operations” (p. 94). Strategies for Solving Word Problems develops students’ repertoire of representations through the strategies: DRAW A PICTURE (or DRAW A DIAGRAM), FIND A PATTERN, ACT IT OUT, MAKE A TABLE, WRITE A NUMBER SENTENCE (or WRITE AN EQUATION), AND GUESS AND CHECK.

Open-Ended Solutions/Solution AutonomyOpen-ended solutions or solution autonomy is the

practice of allowing students to investigate and explore their own problem-solving solutions in a reasonable and logical manner. This flexibility provides students with the opportunity to explore different solution methods rather than a prescribed solution. Solution autonomy reinforces the mathematical concept that there is more than one solution method for a problem. According to the 2000 Nation’s Report

Card for Mathematics, “Students at all three grades in 2000 who disagreed with the statements that math was mostly memorizing facts and that there was only one way to solve a mathematics problem scored higher, on average,

Students working together or in small groups can effectively share their problem-solving strategies and processes through multiple

representations.

Students apply a different solution method to check

the reasoning of their first solution method.

Strategy Four: Make a Table

14

Modeled Practice

Study how Jess used the strategy Make a Table to solve the model word problem. She used this strategy because the problem gives more than one set of data and asks for the data to be continued. Read why Jess chose Make a Table and how she found and checked her solution.

PREVIEW and READ

Sandra began earning $8 a week for helping her parents with the housework.

She is saving all the money to buy a jacket that costs $50. After how many weeks

of saving will Sandra have enough money for the jacket?

Explain Why: I chose M T because the problem gives more than one set of data that needs to be continued: the number of weeks and the amount saved each week. The question asks for the number of weeks of saving for Sandra to have enough money for a $50 jacket. I can find that information by making a table.

PLAN and SOLVE

1. List the sets of data: weeks; amount saved2. Make a table of the data. Circle the information needed to answer

the question:

Weeks 1 2 3 4 5 6 7Amount Saved $8 $16 $24 $32 $40 $48 $56

Explain How: There are two sets of data (weeks and amount saved each week), so I made a table with two rows. In the first row, I wrote the number of weeks. In the next row, I put how much money Sandra will have saved each week. I multiplied the number of weeks by $8 to get the amount saved. Then I found the week when Sandra will have at least $50.

Solution: After 7 weeks, Sandra will have enough money to buy the jacket.

CHECK and REVIEW

To check my solution, I used repeated addition with $8: $8 + $8 + $8 + $8 + $8 + $8 + $8 = $56.

STUDY

IT!

Model Word Problem

The problem gives more than one set of data.

The question asks for the data to be continued.

Use another operation to check your solution.

1. List the sets of data.

2. Make a table of the data. Circle the information needed to answer the question.

15

Read the practice word problem. Explain why Make a Table can be used to solve the problem. Then solve the problem. Check your solution.

PREVIEW and READ

Darell likes to learn new words. Each day, he studies and learns 3 new

words. After how many days will Darell have learned 25 new words?

Explain Why:

PLAN and SOLVE

Explain How:

Solution:

CHECK and REVIEW

Practice Word Problem

TRY IT!

HINTS


than those who agreed” (p. 21). In Strategies for Solving Word Problems, students are taught multiple strategies to solve each problem using sensible and logical problem-solving methods. Supplemental multiple solutions are applied when students use a second solution method during the CHECK AND REVIEW section. At this juncture, students think about and modify their solution methods to correct any mistakes. “Competent learners and problem solvers monitor and regulate their own processing and change their strategies as necessary” (Bransford, Brown, & Cocking, 2000, p. 238).

Finally, collaborative work among peers allows further investigation of multiple-solution strategies. Students work in pairs or small groups to exchange papers and examine one another’s solution processes. This exchange of ideas and communication of mathematical ideas is an effective instructional method for students of all abilities, including ELL students. Strategies for Solving Word Problems provides students with ample practice in exploring and investigating multiple solutions. From this exploration, students reflect upon their solutions and make any adjustments so that the solution processes are logical.

Engaged LearningStrategies for Solving Word Problems is a vehicle that

promotes active and engaged learning. Students must actively apply the strategy in each lesson. After every CHECK AND REVIEW step, students discuss as a group or in pairs their solutions. “The discussions that follow every problem review many concepts and skills as students present multiple methods. Because students are actively trying to solve problems instead of passively listening to a lecture or reading material, their minds become fully involved. They feel that they are in charge of their own learning . . . . They learn to think mathematically” (Kalman, 2004, p. 179).

Students who explain their solutions to peers are actively engaged in the lesson.


SummaryStrategies for Solving Word Problems is a step-by-step

implementation of problem-solving strategies that aid students in the approach and computation of solving word problems. Strategies for Solving Word Problems continues with the comprehension process learned in its companion book, Understanding and Solving Word Problems. Building upon this process, Strategies for Solving Word Problems introduces research-based mathematics strategies students may use to plan and solve word problems. Additionally, students are taught with research-based teaching and learning strategies. These strategies help students not only to solve their word problems using multiple methods, but also to self-assess and monitor their learning. Learners of all abilities and backgrounds will benefit from the gradual accomplishment of reaching solutions independently. The consistent application of scaffolded instruction easily allows for differentiated instruction in the classroom. The challenge of creating open-ended solutions will reinforce students’ mathematical reasoning and problem solving. Cooperative learning during the review and assessment phase of the process also reinforces their mathematical reasoning. Each instructional step in the program makes Strategies for Solving Word Problems a comprehensive learning experience that turns a particularly troublesome math area into one that is approachable and attainable by students.

Research Paper: STARS Series 13

STEPBY

STEPMATH



Word ProblemsBOOK

AATeaches six basic






skills and stregthens



extended-response

problems

STEPBY

STEPMATH


Word Problems


Word ProblemsBOOK

BBTeaches six basic strategies for solving word problems successfully


Improves problem-solving skills and stregthens mathematical reasoning


STEPBY

STEPMATH



Word ProblemsBOOK

CCTeaches six basic









extended-response

problems

STEPBY

STEPMATH


Word Problems


Word ProblemsBOOK

DDTeaches six basic strategies for solving word problems successfully




STEPBY

STEPMATH



Word ProblemsBOOK

EETeaches six basic






skills and strengthens



extended-response

problems

STEPBY

STEPMATH


Word Problems


Word ProblemsBOOK

FFTeaches six basic strategies for solving word problems successfully




STEPBY

STEPMATH



Word ProblemsBOOK

GGTeaches six basic









extended-response

problems

STEPBY

STEPMATH


Word Problems


Word ProblemsBOOK

HHTeaches six basic strategies for solving word problems successfully




Research Paper: Strategies for Solving Word Problems


FIND A PATTERN

A precursor to algebraic thinking, the FIND A PATTERN strategy is an essential strategy for students to know. “Smith (2003) believed that for students to think algebraically, they must be able to identify, extend, and generalize patterns in order to understand quantitative relationships” (Steele, 2005, p. 142).

ACT IT OUT

This strategy draws students into the math problem by having them actively engage in solving the problem. Students may use manipulatives as a means of acting out a word problem. “. . . [H]ands-on learning helps students to more readily understand concepts and boosts their self-confidence. . . . Harnessing the power of manipulatives has proven invaluable in the teaching of mathematics. Students are better able to visualize math concepts and gain insights into necessary fundamentals when they use rods, cubes, and other tools” (DeGeorge & Santoro, 2004).

GUESS AND CHECK

The GUESS AND CHECK strategy is a strategy that builds students’ ability to work accurately with numbers. “The guessing and checking that students often do to understand the situation provides practice with basic skills and allows them to wrap their minds about the true conditions of a problem” (Kalman, 2004, p. 179).

MAKE A TABLE/DRAW A PICTURE or DRAW A DIAGRAM

The strategies MAKE A TABLE and DRAW A PICTURE or DRAW A DIAGRAM are visualization instruments that allow students to see the relationships and patterns between numbers and variables. “Tables and graphs provide visual means for students to organize and summarize numerical and verbal data” (Krulik & Rudnick, 1996; Sorenson et al., 1996).

WRITE A NUMBER SENTENCE or WRITE AN EQUATION

This is an integral mathematics strategy to learn because it requires students to transfer their mathematics knowledge into another representation. According to Mayer (2002), “[Representing] occurs when a student is able to convert information from one form of representation to another . . . Transfer is the ability to use what was learned to solve new problems, answer new questions, or facilitate learning new subject matter (Mayer & Wittrock, 1996)” (p. 226).



ReferencesBerman, B. T. (2003). Math anxiety: Overcoming

a major obstacle to the improvement of student math performance. (Review of Research). Childhood Education, 79(3), 170–174.

Bransford, J. D., Brown, A. L., & Cocking, R. R. (2000). How people learn: Brain, mind, experience, and school. Washington, D.C.: National Academy Press.

Bruner, J. S. (1960). The process of education. New York: Random House.

Buchanan, K., & Helman, M. (2000). Reforming mathematics instruction for ESL literacy students. (Washington, D.C.: U.S. Department of Education, 1997). Available online: http://www.ericfacility.net/databases/ERIC_Digests/ed414769.html.

Cavanaugh, S. (2005). Poor math scores on world stage trouble U.S. Education Week, 24 (16), 1&18.

DeGeorge, B., & Santoro, A. M. (2004). Manipulatives: A hands-on approach to math. Principal, 84(2), 28.

Eastern Stream Center on Resources and Training (ESCORT). (1998). Promoting Mathematics (By Any and All Means). In Help! They Don’t Speak English Starter Kit for Primary Teachers: A resource guide for educators of limited English proficient migrant students, grades Pre-K–6 (3rd ed.). Available at: http://www.ael.org/page2.htm?&index=498&pd =1&pv=x

Forsten, C. (2004). The problem with word problems. Principal, 84(2), 21–23.

Hill, H. C., Rowan, B., & Loewenber, D. (2005, Summer). Effects of teachers’ mathematical knowledge for teaching on student achievement. American Educational Research Journal. 42(2), pp. 371–406.

Jitendra, A., & Xin, Y. P. (1997). Mathematical word problem-solving instruction for students with mild disabilities and students at risk for math failure: A research synthesis. Journal of Special Education, 30(4), 412–439.

Kalman, R. (2004). The value of multiple solutions. Mathematics Teaching in the Middle School, 10(4), 174–179.

Keene, E. O., & Zimmermann, S. (1997). Mosaic of thought: Teaching comprehension in a reader’s workshop. Portsmouth, NH: Heinemann.

Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001). Adding it up: Helping children learn mathematics. Washington, D.C.: National Academy Press.

Krulik, S., & Rudnick, J. A. (1996). The new sourcebook for teaching reasoning and problem solving in junior and senior high school. Boston: Allyn and Bacon.

Lee, H., & Sikjung, W. (2004). Limited-English-proficient (LEP) students: Mathematical understanding. Teaching Mathematics in the Middle School, 9(5), 269–272.

Marchand-Martella, N. E., Slocum, T. A., & Martella, R. C. (2004). Introduction to direct instruction. Boston: Pearson Education, Inc.

Mayer, R. E. (2002, Autumn). Rote versus meaningful learning. Theory into Practice, 41(4), 226–232.

Mayer, R. E., & Wittrock, M. C. (1996). Problem-solving transfer. In D. C. Berliner & R. C. Calfee (Eds.), Handbook of educational psychology (pp. 47-62). New York: Macmillan.

Miller, D. L. (1993). Making the connection with language. Arithmetic Teacher, 40(6), 311–316.

Monroe, E. E., & Orme, M. P. (2002, Spring). Developing mathematical vocabulary. Preventing School Failure, 46(3), 139–142.

National Council of Teachers of Mathematics. (1989). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.

National Research Council. (2001). Adding it up: Helping children learn mathematics. J. Kilpatrick, J. Swafford, and B. Findell (Eds.). Mathematics Learning Study Committee for Education, Division of Behavioral and Social Sciences and Education. Washington, D.C.: National Academy Press.

Pressley, M. (2002). Metacognition and self- regulated comprehension. In A. E. Farstrup & S. J. Samuels (Eds.), What research has to say about reading instruction. (pp. 294–295). Newark, DE: International Reading Association.

Rubenstein, R. N., & Thompson, D. R. (2002). Understanding and Supporting Children’s Mathematical Vocabulary Development. Teaching Children Mathematics, 9(2), 107–112.


http://www.ericfacility.net

http://www.ericfacility.net

Shellard, E. G. (2004). Helping students struggling with math. Principal, 84(2), 40–43.

Smith, E. (2003). Stasis and change: Integrating pattern, functions, and algebra throughout the K–12 curriculum. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 136–150). Reston, VA: National Council of Teachers of Mathematics.

Sorenson, J. S., Busckmaster, L. R., Francis, M. K., and Knauf, K. M. (1996). The power of problem-solving: practical ideas and teaching strategies for any k–8 subject area. Boston: Allyn and Bacon.

Steele, D. (2005). Using writing to access students’ schemata knowledge of algebraic thinking. School Science and Mathematics, 105(3), 142–154.

Steffensen, M., Joag-Dev, C., & Anderson, R. (1979). A cross-cultural perspective on reading comprehension. Reading Research Quarterly, 15(1), 10–29.

U.S. Department of Education. No Child Left Behind. Proven methods: The facts about good teachers. Accessed February 24, 2005, from http://www.ed.gov/nclb/methods/teacherteachers.html

U.S. Department of Education. National Center for Education Statistics. Impact of Selected Background Variables on Students’ NAEP Math Performance, NCES 2001-11, by Jamal Abedi, Carol Lord, and Carolyn Hofstetter. Arnold A. Goldstein, project officer. Washington, D.C.: 2001.

U.S. Department of Education. Office of Educational Research and Improvement. National Center for Education Statistics. The Nation’s Report Card: Mathematics 2000, NCES 2001–517, by J. S. Braswell, A. D. Lutkus, W. S. Grigg, S. L. Santapau, B. Tay-Lim, and M. Johnson. Washington, D.C.: 2001.

Wilson, C. L., & Sindelar, P. T. (1991). Direct instruction in math word problems: students with learning disabilities. Exceptional Children, 57(6), 512–519.


http://www.ed.gov/nclb/methods/teachers/

http://www.ed.gov/nclb/methods/teachers/

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