example of problem solving using agonsa in math

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3.1: Use a Problem-Solving Strategy

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Learning Objectives

By the end of this section, you will be able to:

• Approach word problems with a positive attitude
• Use a problem-solving strategy for word problems
• Solve number problems

Before you get started, take this readiness quiz.

• Translate “6 less than twice x ” into an algebraic expression. If you missed this problem, review Exercise 1.3.43 .
• Solve: \(\frac{2}{3}x=24\). If you missed this problem, review Exercise 2.2.10 .
• Solve: \(3x+8=14\). If you missed this problem, review Exercise 2.3.1 .

Approach Word Problems with a Positive Attitude

“If you think you can… or think you can’t… you’re right.”—Henry Ford

The world is full of word problems! Will my income qualify me to rent that apartment? How much punch do I need to make for the party? What size diamond can I afford to buy my girlfriend? Should I fly or drive to my family reunion? How much money do I need to fill the car with gas? How much tip should I leave at a restaurant? How many socks should I pack for vacation? What size turkey do I need to buy for Thanksgiving dinner, and then what time do I need to put it in the oven? If my sister and I buy our mother a present, how much does each of us pay?

Now that we can solve equations, we are ready to apply our new skills to word problems. Do you know anyone who has had negative experiences in the past with word problems? Have you ever had thoughts like the student below (Figure \(\PageIndex{1}\))?

When we feel we have no control, and continue repeating negative thoughts, we set up barriers to success. We need to calm our fears and change our negative feelings.

Start with a fresh slate and begin to think positive thoughts. If we take control and believe we can be successful, we will be able to master word problems! Read the positive thoughts in Figure \(\PageIndex{2}\) and say them out loud.

Think of something, outside of school, that you can do now but couldn’t do 3 years ago. Is it driving a car? Snowboarding? Cooking a gourmet meal? Speaking a new language? Your past experiences with word problems happened when you were younger—now you’re older and ready to succeed!

Use a Problem-Solving Strategy for Word Problems

We have reviewed translating English phrases into algebraic expressions, using some basic mathematical vocabulary and symbols. We have also translated English sentences into algebraic equations and solved some word problems. The word problems applied math to everyday situations. We restated the situation in one sentence, assigned a variable, and then wrote an equation to solve the problem. This method works as long as the situation is familiar and the math is not too complicated.

Now, we’ll expand our strategy so we can use it to successfully solve any word problem. We’ll list the strategy here, and then we’ll use it to solve some problems. We summarize below an effective strategy for problem solving.

USE A PROBLEM-SOLVING STRATEGY TO SOLVE WORD PROBLEMS.

• Read the problem. Make sure all the words and ideas are understood.
• Identify what we are looking for.
• Name what we are looking for. Choose a variable to represent that quantity.
• Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebraic equation.
• Solve the equation using good algebra techniques.
• Check the answer in the problem and make sure it makes sense.
• Answer the question with a complete sentence.

Example \(\PageIndex{1}\)

Pilar bought a purse on sale for \($18\), which is one-half of the original price. What was the original price of the purse?

Step 1. Read the problem. Read the problem two or more times if necessary. Look up any unfamiliar words in a dictionary or on the internet.

Let p = the original price of the purse.

Step 2. Identify what you are looking for. Did you ever go into your bedroom to get something and then forget what you were looking for? It’s hard to find something if you are not sure what it is! Read the problem again and look for words that tell you what you are looking for!

In this problem, the words “what was the original price of the purse” tell us what we need to find.

Step 3. Name what we are looking for. Choose a variable to represent that quantity. We can use any letter for the variable, but choose one that makes it easy to remember what it represents.

Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Translate the English sentence into an algebraic equation.

Reread the problem carefully to see how the given information is related. Often, there is one sentence that gives this information, or it may help to write one sentence with all the important information. Look for clue words to help translate the sentence into algebra. Translate the sentence into an equation.

Step 5. Solve the equation using good algebraic techniques. Even if you know the solution right away, using good algebraic techniques here will better prepare you to solve problems that do not have obvious answers.

Step 6. Check the answer in the problem to make sure it makes sense. We solved the equation and found that \(p=36\),which means “the original price” was \($36\).

If this were a homework exercise, our work might look like this:

Pilar bought a purse on sale for \($18\), which is one-half the original price. What was the original price of the purse?

Step 7. Answer the question with a complete sentence. The problem asked “What was the original price of the purse?”

Try It \(\PageIndex{2}\)

Joaquin bought a bookcase on sale for \($120\), which was two-thirds of the original price. What was the original price of the bookcase?

Try It \(\PageIndex{3}\)

Two-fifths of the songs in Mariel’s playlist are country. If there are \(16\) country songs, what is the total number of songs in the playlist?

Let’s try this approach with another example.

Example \(\PageIndex{4}\)

Ginny and her classmates formed a study group. The number of girls in the study group was three more than twice the number of boys. There were \(11\) girls in the study group. How many boys were in the study group?

Try It \(\PageIndex{5}\)

Guillermo bought textbooks and notebooks at the bookstore. The number of textbooks was \(3\) more than twice the number of notebooks. He bought \(7\) textbooks. How many notebooks did he buy?

Try It \(\PageIndex{6}\)

Gerry worked Sudoku puzzles and crossword puzzles this week. The number of Sudoku puzzles he completed is eight more than twice the number of crossword puzzles. He completed \(22\) Sudoku puzzles. How many crossword puzzles did he do?

Solve Number Problems

Now that we have a problem solving strategy, we will use it on several different types of word problems. The first type we will work on is “number problems.” Number problems give some clues about one or more numbers. We use these clues to write an equation. Number problems don’t usually arise on an everyday basis, but they provide a good introduction to practicing the problem solving strategy outlined above.

Example \(\PageIndex{7}\)

The difference of a number and six is \(13\). Find the number.

Try It \(\PageIndex{8}\)

The difference of a number and eight is \(17\). Find the number.

Try It \(\PageIndex{9}\)

The difference of a number and eleven is \(−7\). Find the number.

Example \(\PageIndex{10}\)

The sum of twice a number and seven is \(15\). Find the number.

Try It \(\PageIndex{11}\)

The sum of four times a number and two is \(14\). Find the number.

Try It \(\PageIndex{12}\)

The sum of three times a number and seven is \(25\). Find the number.

​​​​​​ Some number word problems ask us to find two or more numbers. It may be tempting to name them all with different variables, but so far we have only solved equations with one variable. In order to avoid using more than one variable, we will define the numbers in terms of the same variable. Be sure to read the problem carefully to discover how all the numbers relate to each other.

Example \(\PageIndex{13}\)

One number is five more than another. The sum of the numbers is 21. Find the numbers.

Try It \(\PageIndex{14}\)

One number is six more than another. The sum of the numbers is twenty-four. Find the numbers.

Try It \(\PageIndex{15}\)

The sum of two numbers is fifty-eight. One number is four more than the other. Find the numbers.

Example \(\PageIndex{16}\)

The sum of two numbers is negative fourteen. One number is four less than the other. Find the numbers.

Try It \(\PageIndex{17}\)

The sum of two numbers is negative twenty-three. One number is seven less than the other. Find the numbers.

Try It \(\PageIndex{18}\)

The sum of two numbers is \(−18\). One number is \(40\) more than the other. Find the numbers.

Example \(\PageIndex{19}\)

One number is ten more than twice another. Their sum is one. Find the numbers.

Try It \(\PageIndex{20}\)

One number is eight more than twice another. Their sum is negative four. Find the numbers.

\(-4,\; 0\)

Try It \(\PageIndex{21}\)

One number is three more than three times another. Their sum is \(−5\). Find the numbers.

\(-3,\; -2\)

Some number problems involve consecutive integers. Consecutive integers are integers that immediately follow each other. Examples of consecutive integers are:

\[\begin{array}{l}{1,2,3,4} \\ {-10,-9,-8,-7} \\ {150,151,152,153}\end{array}\]

Notice that each number is one more than the number preceding it. So if we define the first integer as \(n\), the next consecutive integer is \(n+1\). The one after that is one more than \(n+1\), so it is \(n+1+1\), which is \(n+2\). \[\begin{array}{ll}{n} & {1^{\text { st }} \text { integer }} \\ {n+1} & {2^{\text { nd }} \text { consecutive integer }} \\ {n+2} & {3^{\text { rd }} \text { consecutive integer } \ldots \text { etc. }}\end{array}\]

Example \(\PageIndex{22}\)

The sum of two consecutive integers is \(47\). Find the numbers.

Try It \(\PageIndex{23}\)

The sum of two consecutive integers is 95. Find the numbers.

Try It \(\PageIndex{24}\)

The sum of two consecutive integers is −31. Find the numbers.

Example \(\PageIndex{25}\)

Find three consecutive integers whose sum is −42.

Try It \(\PageIndex{26}\)

Find three consecutive integers whose sum is −96.

-33, -32, -31

Try It \(\PageIndex{27}\)

Find three consecutive integers whose sum is −36.

-13, -12, -11

Now that we have worked with consecutive integers, we will expand our work to include consecutive even integers and consecutive odd integers. Consecutive even integers are even integers that immediately follow one another. Examples of consecutive even integers are:

\[\begin{array}{l}{18,20,22} \\ {64,66,68} \\ {-12,-10,-8}\end{array}\]

Notice each integer is \(2\) more than the number preceding it. If we call the first one \(n\), then the next one is \(n+2\). The next one would be \(n+2+2\) or \(n+4\). \[\begin{array}{cll}{n} & {1^{\text { st }} \text { even integer }} \\ {n+2} & {2^{\text { nd }} \text { consecutive even integer }} \\ {n+4} & {3^{\text { rd }} \text { consecutive even integer } \ldots \text { etc. }}\end{array}\]

Consecutive odd integers are odd integers that immediately follow one another. Consider the consecutive odd integers \(77\), \(79\), and \(81\).

\[\begin{array}{l}{77,79,81} \\ {n, n+2, n+4}\end{array}\]

\[\begin{array}{cll}{n} & {1^{\text { st }} \text {odd integer }} \\ {n+2} & {2^{\text { nd }} \text { consecutive odd integer }} \\ {n+4} & {3^{\text { rd }} \text { consecutive odd integer } \ldots \text { etc. }}\end{array}\]

Does it seem strange to add 2 (an even number) to get from one odd integer to the next? Do you get an odd number or an even number when we add 2 to 3? to 11? to 47?

Whether the problem asks for consecutive even numbers or odd numbers, you don’t have to do anything different. The pattern is still the same—to get from one odd or one even integer to the next, add 2.

Example \(\PageIndex{28}\)

Find three consecutive even integers whose sum is 84.

\[\begin{array}{ll} {\textbf{Step 1. Read} \text{ the problem.}} & {} \\ {\textbf{Step 2. Identify} \text{ what we are looking for.}} & {\text{three consecutive even integers}} \\ {\textbf{Step 3. Name} \text{ the integers.}} & {\text{Let } n = 1^{st} \text{ even integers.}} \\ {} &{n + 2 = 2^{nd} \text{ consecutive even integer}} \\ {} &{n + 4 = 3^{rd} \text{ consecutive even integer}} \\ {\textbf{Step 4. Translate.}} &{} \\ {\text{ Restate as one sentence. }} &{\text{The sum of the three even integers is 84.}} \\ {\text{Translate into an equation.}} &{n + n + 2 + n + 4 = 84} \\ {\textbf{Step 5. Solve} \text{ the equation. }} &{} \\ {\text{Combine like terms.}} &{n + n + 2 + n + 4 = 84} \\ {\text{Subtract 6 from each side.}} &{3n + 6 = 84} \\ {\text{Divide each side by 3.}} &{3n = 78} \\ {} &{n = 26 \space 1^{st} \text{ integer}} \\\\ {} &{n + 2\space 2^{nd} \text{ integer}} \\ {} &{26 + 2} \\ {} &{28} \\\\ {} &{n + 4\space 3^{rd} \text{ integer}} \\ {} &{26 + 4} \\ {} &{30} \\ {\textbf{Step 6. Check.}} &{} \\\\ {26 + 28 + 30 \stackrel{?}{=} 84} &{} \\ {84 = 84 \checkmark} & {} \\ {\textbf{Step 7. Answer} \text{ the question.}} &{\text{The three consecutive integers are 26, 28, and 30.}} \end{array}\]

Try It \(\PageIndex{29}\)

Find three consecutive even integers whose sum is 102.

Try It \(\PageIndex{30}\)

Find three consecutive even integers whose sum is −24.

−10,−8,−6

Example \(\PageIndex{31}\)

A married couple together earns $110,000 a year. The wife earns $16,000 less than twice what her husband earns. What does the husband earn?

Try It \(\PageIndex{32}\)

According to the National Automobile Dealers Association, the average cost of a car in 2014 was $28,500. This was $1,500 less than 6 times the cost in 1975. What was the average cost of a car in 1975?

Try It \(\PageIndex{33}\)

U.S. Census data shows that the median price of new home in the United States in November 2014 was $280,900. This was $10,700 more than 14 times the price in November 1964. What was the median price of a new home in November 1964?

Key Concepts

• Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation.

\[\begin{array}{cc}{n} & {1^{\text { st }} \text { integer }} \\ {n+1} & {2^{\text { nd }} \text {consecutive integer }} \\ {n+2} & {3^{\text { rd }} \text { consecutive integer } \ldots \text { etc. }}\end{array}\]

\[\begin{array}{cc}{n} & {1^{\text { st }} \text { integer }} \\ {n+2} & {2^{\text { nd }} \text { consecutive even integer }} \\ {n+4} & {3^{\text { rd }} \text { consecutive even integer } \ldots \text { etc. }}\end{array}\]

\[\begin{array}{cc}{n} & {1^{\text { st }} \text { integer }} \\ {n+2} & {2^{\text { nd }} \text { consecutive odd integer }} \\ {n+4} & {3^{\text { rd }} \text { consecutive odd integer } \ldots \text { etc. }}\end{array}\]

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MATH QUIZ- AGONSA METHOD

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Problem: How many centimeters of ribbon will Jane need to decorate the sides of a bed sheet with length 100 cm and width of 40 cm?

Question #1: What is asked in the problem?

color of ribbon

sides of the bedsheet

Full name of Jane

Centimeters of ribbon needed by Jane to decorate the sides of bedsheet

Question #2: What are the given facts?

100 cm and 40 cm

Jane needs to decorate the sides

Question #3: What operation/s will you use?

Addition and/or Multiplication

Addition and/or Subtraction

Addition and/or Division

Multiplication and/or Division

Question #4: What is the number sentence?

100 cm - 40 cm= N

100 cm +40cm= N

100cm+100cm+40cm+40cm= N

100 x 40 cm = N

Question #5: What is the solution?

100 cm x 40 cm= 4 000 centimeters

100 cm - 40 cm= 60 centimeters

100 cm + 40 cm= 14o centimeters

100 cm + 100 cm + 40 cm + 40 cm = 280 centimeters

Question #6: What is the complete answer?

Therefore, Jane needs 280 centimeters of ribbon to decorate the sides of the bedsheet.

Therefore, Jane needs 140 centimeters of ribbon to decorate the sides of the bedsheet.

Therefore, Jane needs 4, 000 centimeters of ribbon to decorate the sides of the bedsheet.

Therefore, Jane needs 60 centimeters of ribbon to decorate the sides of the bedsheet.

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Solving Word Problems using Schemas: A Review of the Literature

Sarah r. powell.

Vanderbilt University

Solving word problems is a difficult task for students at-risk for or with learning disabilities (LD). One instructional approach that has emerged as a valid method for helping students at-risk for or with LD to become more proficient at word-problem solving is using schemas. A schema is a framework for solving a problem. With a schema, students are taught to recognize problems as falling within word-problem types and to apply a problem solution method that matches that problem type. This review highlights two schema approaches for 2 nd - and 3 rd -grade students at-risk for or with LD: schema-based instruction and schema-broadening instruction. A total of 12 schema studies were reviewed and synthesized. Both types of schema approaches enhanced the word-problem skill of students at-risk for or with LD. Based on the review, suggestions are provided for incorporating word-problem instruction using schemas.

Since Pólya (1945) introduced four steps for solving word problems (understand the question, devise a plan, carry out the plan, and look back and check), teachers have been encouraged to provide more systematic instruction on problem solving in mathematics. Word-problem instruction has become vital for students. High-stakes standardized tests like the National Assessment of Educational Progress (NAEP; National Assessment Governing Board, 2009 ) place heavy emphasis on mathematics word problems and national educational organizations like the National Council of Teachers of Mathematics ( NCTM; 2000 ) heavily value the teaching of problem solving across grades K through 12. Many researchers have investigated methods for teaching problem solving to general-education students ( Marshall, 1995 ; Schoenfeld, 1992 ; Shavelson, Webb, Stasz, & McArthur, 1988 ) and, in more recent years, to students with learning disabilities (LD) (e.g., Case, Harris, & Graham, 1992 ; Mastropieri, Scruggs, & Shiah, 1997 ; Miller & Mercer, 1993 ).

Over the last two decades, a sizeable literature has begun to accumulate with an emphasis on helping students develop schemas to solve word problems in mathematics (e.g., Fuchs, Fuchs, Finelli, Courey, & Hamlett, 2004 ; Fuchs, Seethaler, et al., 2008 ; Griffin & Jitendra, 2009 ; Jitendra & Hoff, 1996 ; Willis & Fuson, 1988 ). A schema is a framework, outline, or plan for solving a problem ( Marshall, 1995 ). In mathematics, students can use schemas to organize information from a word problem in ways that represent the underlying structure of a problem type. Pictures or diagrams, as well as number sentences or equations, can be used to represent schemas.

Often, word problems can be differentiated into types of problems. The problem type is determined by what is happening in the word-problem narrative. For example, students may be given the following information: There are 7 blue birds and 4 red birds sitting on a tree . If, however, students are asked, How many birds are on the tree? , the problem type is combining or totaling (the birds). If students are asked, Five blue birds flew away, how many blue birds are left sitting in the tree? , the problem type is finding the change (in the number of blue birds). Combining or totaling is different from finding a change in that the examples represent two distinct problem types. Two distinct schemas can be used to solve the problems. Once students determine the problem type, they can apply a schema (i.e., diagram, equation, or plan) to assist in solving the word problem. In the elementary grades, most word problems can be sorted into only a few types ( Riley & Greeno, 1988 ). If students know a schema for each type, and understand how to sort problems into the problem types and apply the solution method for each schema, then students should be able to solve most word problems ( Cooper & Sweller, 1987 ).

The first purpose of the present paper was to review and synthesize the literature on schemas within word-problem instruction to determine (a) what schemas were taught to elementary students at-risk for or with LD, (b) how these schemas were taught, and (c) what effects were associated with solving word problems using schemas. The second purpose was to provide suggestions for classroom teachers on how to teach students to use schemas to solve word problems.

Word-Problem Difficulty

Students at-risk for or with LD often struggle with word-problem solving ( Parmar, Cawley, & Frazita, 1996 ). For example, Wilson and Sindelar (1991) worked with second- through fifth-graders with LD. On a test of addition and subtraction word problems, these students performed significantly below third-grade students without LD. At second- and fourth-grade, Englert, Culatta, and Horn (1987) tested 24 students with LD on 16 addition word problems. When compared to grade-level peers, the students with LD demonstrated significantly lower accuracy on word-problem solutions. More recently, Jordan and Hanich (2000) administered 14 addition and subtraction word problems to 20 second-grade students at-risk for or with mathematics LD and 29 second-grade students without LD. Students at-risk for or with LD answered fewer word problems correct than students without LD and employed less efficient strategies. These findings were corroborated with a larger group of second-grade ( Hanich, Jordan, Kaplan, & Dick, 2001 ) and third-grade students ( Jordan & Montani, 1997 ). Moving beyond simple word problems, Fuchs and Fuchs (2002) administered 10 word problems comprising four types (i.e., shopping list, buying bags, half, and pictograph) and 10 multi-step word problems with tables and graphs to fourth-grade students. The performance of 40 students with LD was compared to normative data collected from typical fourth-grade students. On both word problem sets, students with LD scored significantly lower than students in the normative group with effect sizes (ESs) ranging from 0.49 to 1.10 favoring the normative group.

Word problems may pose a challenge for students at-risk for or with LD because numerous steps and skills are necessary to solve a word problem ( Parmar et al., 1996 ). Additionally, students may struggle with comprehension of the text of the word problem ( Cummins, Kintsch, Reusser, & Weimer, 1988 ). Many students with LD struggle with mathematics and reading difficulty; therefore, embedding mathematics within a linguistic context may challenge students who also have reading deficits ( Fuchs, Fuchs, Stuebing, et al., 2008 ). To solve a word problem, students must use the text to identify missing information, derive a plan for solving for the missing information, and perform a calculation to find the missing information. Even when complex calculations are not required, students with LD struggle with problem solving compared to their average-performing peers ( Pellegrino & Goldman, 1987 ).

Instruction for Students with LD

To help students at-risk for or with LD become more efficient and accurate word-problem solvers, explicit word-problem instruction may be warranted ( Parmar et al., 1996 ). For example, Kroesbergen, Van Luit, and Maas (2004) randomly assigned 265 8- to 11-year-old students with LD or behavior disorders to receive constructivist multiplication instruction, explicit multiplication instruction, or regular classroom instruction (control). Students in the constructivist and explicit conditions received 30 lessons over 4 to 5 months. Intervention focused on multiplication automaticity and problem solving. In the constructivist condition, students were encouraged to discuss different approaches to solving a multiplication problem and then determine whether they could use one of these approaches to solve the problem. In the explicit condition, students were told how a problem should be solved and were provided examples of good problem-solving strategies. Students were always told which strategies to use. In the control condition, students followed the school's regular mathematics curriculum which included instruction on multiplication. At posttest, explicit instruction students significantly outperformed constructivist and control students on a computation multiplication measure and a measure of multiplication word problems. Kroesbergen et al. concluded that explicit or direct mathematics instruction, but not discovery or constructivist learning, may benefit lower-performing students.

Several explicit approaches exist for teaching students at-risk for or with LD to solve word problems ( Jitendra & Xin, 1997 ). These include diagramming or drawing the word problem ( van Garderen, 2007 ); identifying key words in a word problem and solving the problem based on the key word; utilizing computer-assisted instruction with explicit step-by-step work ( Mastropieri et al., 1997 ); using a mnemonic device to guide word-problem solving ( Miller & Mercer, 1993 ); learning metacognitive strategies to monitor word problem-solving progress ( Case et al., 1992 ); and using a checklist of steps to solve word problems along with monitoring work with metacognitive strategies ( Montague, Warger, & Morgan, 2000 ). An additional approach to teaching word-problem solving to students at-risk for or with LD, which has been developed over the last 20 years, is using schemas to solve word problems (e.g., Fuchs, Fuchs, Finelli, et al., 2004 ; Jitendra & Hoff, 1996 ). Word-problem instruction using schemas differs from typical word-problem instruction (e.g., key words, checklist of steps) because students first identify a word problem as belonging to a problem type and then use a specific problem-type schema to solve the problem. In conventional word-problem instruction, students may organize word-problem information or follow a mnemonic device to work step-by-step through the problem; however, students are not taught to determine a problem type and solve word problems according to a problem-type schema. In this paper, the research conducted to evaluate using schemas in word-problem solving was reviewed and synthesized.

Literature Search

The studies selected for this literature review met four criteria. First, the implemented treatments incorporated explicit instruction on solving a word problem though a schema. Second, studies needed to include, but not necessarily be limited to, students at-risk for or with LD. Third, study participants comprised students in second or third grade. These were the target grades because this is often when identification of students with LD occurs ( Fletcher, Lyon, Fuchs, & Barnes, 2006 ) and when written word-problem solving is a major focus of the curriculum as opposed to less formal, oral problems presented in kindergarten and first grade. Fourth, studies needed to be published in a peer-reviewed journal. I conducted searches in electronic databases including ERIC, PsycInfo, and ProQuest using the following terms: schema, word problem, story problem , and problem solving . Then, I read the titles and abstracts of articles to identify studies that fit the four criteria resulting in 12 word-problem solving studies. In all 12 studies, instruction focused on addition and subtraction word problems, which are the two operations most commonly found in second- and third-grade instruction and on standardized tests ( Hudson & Miller, 2006 ).

Overview of Included Studies

Each schema study reviewed in this paper is outlined in Table 1 . Study publication dates ranged from 1996 to 2009. Across studies, almost 4000 students were included. Of these students, 411 were at-risk for LD and 173 were identified as receiving special education services. Jitendra and colleagues tended to work with students with LD whereas Fuchs and colleagues generally worked with students at-risk for LD. The researchers utilized a variety of experimental designs: single subject (1), group teaching without assignment to treatment conditions (1), student random assignment (4), matched pairs random assignment (2), and classroom random assignment (5). Instruction occurred during school hours in individual settings (3), in small groups (2), and in large groups (8). Assessments for determining instructional effects were experimenter-designed in all 12 studies. Two of the studies included standardized assessments as well as experimenter-designed measures in the testing battery. In ten of the studies, more than one assessment was administered.

Two Approaches to Schema Instruction

In this section, two approaches to schema instruction are discussed. The first, referred to as schema-based instruction , teaches students to use schematic diagrams to solve addition and subtraction word problems ( Jitendra, Griffin, Deatline-Buchman, & Sczesniak, 2007 ; Jitendra & Hoff, 1996 ). The student reads a word problem, selects a schema diagram into which the word problem fits, and uses the structure of the diagram to solve the problem. In more recent studies, students are taught to use a mathematical equation (i.e., 4 + ? = 7), after filling in a schematic diagram, to solve the problem ( Griffin & Jitendra, 2009 ). The work by Jitendra and colleagues uses schema-based instruction. By contrast, Fuchs et al. (2003) uses a second approach to schema instruction, schema-broadening instruction . Schema-broadening instruction is similar to schema-based instruction in that students read the word problem and select a schema (from the taught schema) to solve word problems. Schema-broadening instruction differs from schema-based instruction because students are taught to transfer their knowledge of problem types to recognize problems with novel features (e.g., different format, additional question, irrelevant information, unfamiliar vocabulary, or information presented in charts, graphs, or pictures) as belonging to a problem type for which they know a solution. As with Jitendra and colleagues, Fuchs and colleagues also teach students to set up and solve mathematical equations (e.g., X – 3 = 7) representing the structure of problem types ( Fuchs et al., 2009 ).

In terms of schema instruction, the schema-based instruction of Jitendra and colleagues differs from the schema-broadening instruction of Fuchs and colleagues in one primary way. With schema-broadening (but not schema-based) instruction, students receive explicit instruction on transfer to novel problems. The schemas that Jitendra and colleagues used rely on diagrams for organizing word-problem work. (See Figure 1 for an example.) Fuchs and colleagues, in contrast, teach students to organize word-problem information in sections or in mathematical equations. (See Figures 2 and ​ and3 3 for examples.)

Schema-Based Instruction

To understand how schema-based instruction may benefit students with LD, Jitendra and Hoff (1996) worked with three third- and fourth-grade students with LD. During 13 to 16 days of intervention, students learned to recognize defining features of addition and subtraction word-problem types, classify problems in terms of problem types, map the word problem information onto the schema's diagram, and use the diagram to solve the problem. Jitendra and Hoff taught three schemas: change, group, and compare. All three students demonstrated positive growth as the study progressed and maintained skills 2 to 3 weeks after the final intervention session, with only a slight decline in scores. Through this multiple-baseline, single-subject design, Jitendra and Hoff demonstrated the possible benefit of using schemas for teaching word-problem solving to students with LD.

Working with a larger number of students, Jitendra et al. (1998) recruited 34 second- to fifth-grade students who performed below the 60 th percentile on a word-problem measure. Students were randomly assigned to receive small-group schema instruction or small-group traditional instruction during 17 to 20 sessions. Schema instruction focused on change, group, and compare problems. Students learned how to identify the schema for a word problem and to use a schema diagram to organize the problem's information. The traditional instruction followed a basal mathematics program focused on general mathematics skills and was implemented to control for tutoring time. At posttest, students participating in schema tutoring outperformed students in the traditional tutoring on experimenter-designed measures of word problems. A delayed posttest, administered one week after tutoring commenced, continued to favor schema students over traditional students. Jitendra et al. also recruited 24 average-performing third graders to serve as a normative sample. At posttest, the schema-tutoring students performed comparably to students in the normative sample, whereas traditional-tutoring students did not. These results favoring schema instruction led Jitendra et al. to conclude that word-problem instruction using schemas is more advantageous to students at-risk for LD than traditional word-problem instruction.

In the next phase of this research program, Jitendra moved from small-group schema intervention to whole-class schema-based instruction. Jitendra, Griffin, Deatline-Buchman, et al. (2007) provided schema-based instruction similar to Jitendra and Hoff (1996) with students receiving instruction on using schematic diagrams to solve change, combine or group, and compare problems. Students were taught to fill word-problem information into a problem type's corresponding schematic diagram and then generate a mathematical equation (i.e., a number sentence with missing information) to help solve the problem. A question mark was used to mark the missing information (i.e., ? + 5 = 10). Across three classrooms, 38 lower-performing third-grade students, 9 of whom were identified with LD, received schema-based instruction. Instruction lasted 15 weeks with three 30-min sessions per week. On two experimenter-designed word-problem posttests, students in the three classrooms demonstrated improvement from pretest although the improvement was not significant. Jitendra, Griffin, Deatline-Buchman, et al. concluded that lower-performing students and students with LD need and benefit from explicit word problem-solving instruction focused around schemas. With the absence of control classrooms for comparison purposes or significant growth from pre- to posttest, Jitendra, Griffin, Deatline-Buchman, et al. indicated, but did not verify, that schema instruction may be beneficial for students at-risk for or with LD.

Comparing schema-based instruction to another word-problem solving approach, Jitendra, Griffin, Haria, et al. (2007) randomly assigned 88 third-grade students to two conditions: schema-based instruction and general-strategy instruction. Four of the 88 participants were identified with LD. Schema-based instruction focused on the change, combine, and compare problem types as in Jitendra, Griffin, Deatline-Buchman, et al., (2007) whereas students receiving general-strategy instruction were taught four steps to solve a word problem (i.e., read and understand, plan, solve, and check) along with four strategies to assist in solving a word problem (i.e., use manipulatives, act it out or draw a diagram, write a number sentence, and use information from a graph). Similar to Jitendra, Griffin, Deatline-Buchman, et al., students receiving schema instruction learned to identify the schema of a word problem, fill word-problem information into a schematic diagram, and then generate an equation to help solve the word problem. Students used different schematic diagrams for each of the three problem types, and the use of schematic diagrams was faded toward the end of instruction on each problem type. Many students, however, continued to draw schematic diagrams independently. After all three problem types were introduced, tutors taught the students to solve two-step problems that combined two schemas. All students received 41 lessons, each lasting approximately 25 min. From pre- to posttest, students in the schema-based condition outperformed students in the general-strategy condition on an experimenter-designed word-problem measure, with an ES of 0.52. The same measure, administered six weeks after posttest, again showed students in the schema-based condition outperforming general-strategy condition students (ES = 0.69). The number of students with LD was small ( n = 4), so results for students with disabilities were not presented by Jitendra, Griffin, Deatline-Buchman, et al. separate from the main analysis. Therefore, conclusions about the benefit of schema-based instruction for students with LD could not be inferred.

Interestingly, Griffin and Jitendra (2009) also compared schema-based instruction to general-strategy instruction with third-grade students but did not replicate the results from Jitendra, Griffin, Haria, et al. (2007) . Students from three classrooms ( n = 60; 5 with LD) were matched based on performance on a standardized mathematics test and then the pairs were randomly assigned to schema-based or general-strategy instruction. Schema-based and general-strategy instruction were similar to that provided in Jitendra, Griffin, Haria, et al., except that instruction was provided in 20 lessons lasting 100 min each. Schema instruction included completing schematic diagrams and generating equations. The final four lessons comprised instruction on two-step problems where tutors taught students to solve problems using two schemas. On an experimenter-designed word-problem measure, there were no significant differences between the two groups at posttest or at 12-week maintenance (even though both groups demonstrated growth from pretest to posttest to maintenance). On a measure of word-problem solving fluency administered three times throughout instruction, there were significant differences favoring schema-based instruction at the beginning of treatment. These effects, however, faded over the course of the study: At posttest, schema-based and general-strategy groups performed similarly. Griffin and Jitendra attributed the inconsistency of this finding to the fact that instruction was provided in 100-min sessions and once a week rather than shorter sessions occurring several times a week.

Jitendra and colleagues' program of research on schema-based instruction is impressive and demonstrates that students at-risk for or with LD may benefit from explicit schema instruction. These researchers taught students to use three schemas (i.e., change, combine or group, and compare) on different types of word problems with two operations (i.e., addition and subtraction). Even though the specific nature of schema-based instruction varied in small ways from study to study, the majority of students benefitted from learning about different schema and applying the schema to solve word problems. Across studies, two instructional design features were consistently incorporated within schema-based instruction. First, interventions were of long duration (13 to 45 lessons), and second, explicit instruction focused on recognizing a problem's schema, using a diagram based on the schema, and solving the problem. The research by Jitendra and colleagues offers a solid foundation for future schema-based investigations and provides strategies that teachers can use to enhance the performance of their students with LD on word problems.

Schema-Broadening Instruction

As in Jitendra and colleagues' schema-based instruction, schema-broadening instruction relies on schemas for conceptualizing word problems. Some of Fuchs and colleagues' schema-broadening instruction comprises problem types (i.e., shopping list, half, buying bags, pictograph) that are notably different from the problem types used by Jitendra and colleagues. Other schema-broadening problem types of Fuchs and colleagues (i.e., total, difference, and change) are similar to the combine, compare, and change problem types of Jitendra and colleagues. Schema-broadening instruction includes a focus on transfer features to help students expand their conceptualization of the schema. Thus, schema-broadening instruction helps students recognize a novel problem (with unfamiliar problem features such as different format, additional question, irrelevant information, unfamiliar vocabulary, or information presented in charts, graphs, or pictures) as belonging to the schema for which they know a problem solution strategy.

To pinpoint the effects of explicit transfer instruction within schema-broadening instruction, Fuchs et al. (2003) randomly assigned 24 third-grade classrooms ( n = 375) to four conditions: problem-solution instruction, partial-problem-solution-with-transfer instruction (to control for instructional time), full problem-solution-with-transfer-instruction, or control, business-as-usual instruction with a 6-lesson introductory general-problem solving unit that all 24 classrooms received. Students receiving special education services ( n = 23) were distributed across the four conditions. After this introductory unit, problem-solution instruction was presented over the next 20 lessons, in which students were explicitly taught to understand and recognize four schema (i.e., shopping list, half, buying bags, and pictograph) and to apply rules for solving problems for each schema. Students in the partial-problem-solution-plus-transfer condition received only 10 solution lessons but also received 10 transfer lessons. The transfer lessons included explicit instruction on the meaning of transfer and instruction to broaden schema to address problems with different formats, unfamiliar vocabulary, additional questions, and broader problem-solving contexts. Students in the full-problem-solution-plus-transfer condition received all 20 solution lessons and all 10 transfer lessons. In terms of classroom performance from pre- to posttest, students in the problem-solution, partial-problem-solution-with-transfer, and full-problem-solution-with-transfer classrooms outperformed control classrooms on an experimenter-designed immediate-transfer measure (ESs = 2.61, 2.15, and 1.82, respectively). On a far-transfer measure, students who received the partial- or full-solution-plus-transfer-instruction significantly outperformed control classrooms. Additionally, classrooms that received the full-solution-plus-transfer instruction improved more than classrooms that received the solution instruction alone. For students with disabilities, however, the results were not as promising. In the partial-problem-solution condition, 60-80% of the students were unresponsive to treatment. Students in the problem-solution and full-problem-solution-with-transfer conditions demonstrated greater levels of response. This study, as well as a similar study conducted with 24 classrooms of 366 students by Fuchs, Fuchs, Prentice, et al. (2004) , demonstrated the added value of schema instruction with an explicit focus on transfer schemas. Interestingly, in Fuchs, Fuchs, Prentice, et al. students in special education demonstrated significant gains over control students with ESs of 0.87 to 1.96.

To further extend this research program on schema-broadening instruction, Fuchs, Fuchs, Finelli, et al. (2004) randomly assigned 24 classrooms, with 351 students, to three conditions: schema-broadening instruction that addressed three transfer features, schema-broadening instruction that addressed six transfer features, and business-as-usual control. Twenty-nine students received special education services. All classrooms received six sessions about generic word problem-solving steps. Schema-broadening classrooms also received 28 lessons focused on the four schemas taught in Fuchs et al. (2003) . The schema-broadening instruction condition addressed three transfer features (i.e., different format, different question, or different vocabulary). The six-feature schema-broadening instruction condition addressed different format, different question, different vocabulary, irrelevant information, combined problem types, and mixing of transfer features. On experimenter-designed measures with the shortest transfer distance (unfamiliar problems but without novel features), students participating in both schema-broadening instruction conditions performed comparably but significantly better than control (ESs = 3.69 and 3.72, respectively). On measures assessing word problems with medium transfer distance (i.e., different format, question, or vocabulary transfer features), again there were no significant differences between the two schema-broadening instruction conditions, which outperformed the control group (ESs = 1.98 and 2.71, respectively). However, on the measure assessing the greatest transfer distance (i.e., involving all six transfer features), students in the schema-broadening instruction condition that incorporated all six transfer features demonstrated a significant advantage with an ES of 2.71 over control students and an ES of 0.72 over students in the narrower schema-broadening instruction treatment. Students with disabilities demonstrated similar gains to those of students without disabilities. Fuchs, Fuchs, Finelli, et al. demonstrated that students benefit from explicit schema-broadening instruction focused on a wide variety of transfer features.

In an expansion of Fuchs, Fuchs, Finelli, et al. (2004) , Fuchs and colleagues tested how real-life problem-solving skills might provide added benefit to schema-broadening instruction ( Fuchs et al., 2006 ). From 30 classrooms, 445 third-grade students (34 of whom received special education services) were randomly assigned by classroom to schema-broadening instruction, schema-broadening and real-life instruction, or business-as-usual control. All 30 classrooms received six 40-min sessions on general problem-solving strategies. Both schema-broadening treatments received an additional 30 sessions on four problem types. Additionally, schema-broadening plus real-life instruction classrooms received explicit instruction via video on real-life problem solving skills (i.e., review the problem, determine extra steps necessary for solving the problem, find important information without number, figure out important information not provided within the problem, reread, and ignore irrelevant information). On experimenter-designed measures of immediate and medium word-problem transfer, both schema-broadening treatments outperformed control classrooms with ESs ranging from 3.59 to 6.84. On a far transfer task, the added benefit of explicit real-life problem solving emerged on an open-ended question about what the student could buy. Students could use information from a pictograph, a price chart, or their own experiences to answer the question. On this question, the schema-broadening plus real-life students outperformed schema-broadening students (ES = 1.83). In this way, Fuchs et al. (2006) demonstrated how the combination of schema-broadening and real-life problem-solving instruction is beneficial for solving word problems. Results for students with disabilities, however, were not disaggregated from the entire sample, thus it was unclear if these students performed in a similar manner.

To investigate the effect of schema-broadening instruction for students at-risk for LD, Fuchs, Fuchs, Craddock, et al. (2008) randomly assigned 119 classrooms to receive schema-broadening instruction or to participate in a business-as-usual control group. Then, within each whole-class condition, 243 students at-risk for or with LD were randomly assigned to receive small-group schema-broadening tutoring or to remain in their whole-class condition without tutoring. In this way, 28 students received business-as-usual whole-class instruction and no schema-broadening tutoring, 51 students received whole-class schema-broadening instruction but no schema-broadening tutoring, 56 students received business-as-usual whole-class instruction with schema-broadening tutoring, and 108 students received whole-class schema-broadening instruction plus schema-broadening tutoring. The schema-broadening instruction at the classroom level provided explicit instruction on solving the four problem types (i.e., shopping list, half, buying bags, and pictograph) over 16 weeks. Tutoring occurred 3 times a week for 13 weeks following completion of three weeks of whole-class instruction. Tutoring sessions lasted 20 to 30 min in small groups of two to four students. For students who received whole-class schema-broadening instruction, tutored students outperformed students who did not receive tutoring on experimenter-designed measures (ES = 1.13). In a similar way, for students in business-as-usual classrooms, tutored students outperformed students who did not receive tutoring (ES = 1.34). Importantly, students who received two tiers of schema-broadening instruction (whole class and small-group tutoring) significantly outperformed students who received schema-broadening tutoring without whole-class schema-broadening instruction. This finding suggests that the combination of whole-class instruction and small-group tutoring provided the best outcome for students struggling with word problems. Whole-class instruction was beneficial alone as was small-group tutoring; however, the combination proved better than one or the other.

Two other studies in the Fuchs's program of research ( Fuchs et al., 2009 ; Fuchs, Seethaler, et al., 2008 ) rely on schema-broadening instruction but with problem types (i.e., change, total, and difference) that parallel those used by Jitendra and colleagues (i.e., change, combine, and compare). In these tutoring studies (conducted on a one-to-one basis), however, students were also explicitly taught to set up and solve mathematical equations that represent the underlying schema of the word problems similar to Griffin and Jitendra (2009) , Jitendra, Griffin, Deatline-Buchman, et al. (2007) , and Jitendra, Griffin, Haria, et al. (2009). In a pilot study, Fuchs, Seethaler, et al. (2008) randomly assigned 35 third-grade students at-risk for or with LD to two conditions: schema-broadening instruction tutoring with mathematical equations or no-tutoring control. All students performed below the 26 th percentile on global math and reading tests. Students in the schema-broadening condition received individual instruction over 12 weeks with sessions conducted 3 times a week, 30 min per session. Instruction focused on the three problem types with three transfer features (irrelevant information, important information embedded within charts, graphs, or pictures, and double-digit numbers). First, students learned to understand and identify the three schemas (i.e., problem types), to set up an equation to represent each schema (i.e., 3 + X = 9), and to solve equations. Then, explicit instruction to broaden schema to the three transfer features occurred. Students receiving schema-broadening tutoring demonstrated significantly better growth than control students on an experimenter-designed test of word problems (ES = 1.80) and on a test of word problems designed by a research team not affiliated with the study (ES = 0.69). On a standardized test of problem solving, however, there were no significant differences.

Expanding the pilot study to focus on the effects of treatment as a function of difficulty subtype (i.e., students at-risk for or with mathematics LD alone versus students at-risk for or with mathematics and reading LD) and controlling for tutoring time with a contrasting math tutoring condition, Fuchs et al. (2009) randomly assigned 133 third-grade students, blocking by difficulty subtype and by site (i.e., Nashville vs. Houston) to three conditions: number combinations tutoring, schema-broadening word-problem tutoring, or no-tutoring control. Students in the two tutoring conditions received individual tutoring on word problems or on number combinations 3 times a week for 15 weeks, each time for 20 to 30 min. Word-problem tutoring relied on schema-broadening instruction with mathematical equations similar to Fuchs, Seethaler, et al. (2008) . Growth from pre- to posttest on an experimenter-designed word-problem measure, including problems that required transfer, indicated that students in word-problem tutoring significantly outperformed students in number-combinations tutoring and in the control group (ESs = 0.83 and 0.79, respectively). On a standardized test of problem solving, students in word-problem tutoring significantly outperformed students in the control group (ES = 0.28). Additionally, difficulty subtype did not moderate the effect of schema-broadening instruction with equations. That is, students at-risk for or with mathematics and reading LD and students at-risk for mathematics without reading LD responded comparably well to the treatments.

Expanding beyond whole-class, schema-broadening instruction to incorporate mathematical equations, the research conducted by Fuchs et al. (2009) and Fuchs, Seethaler, et al. (2008) revealed how students at-risk for or with LD may benefit from tutoring that combines schema-broadening instruction with instruction on setting up and solving addition and subtraction mathematical equations. Because students did not receive concurrent whole-class instruction and individual word-problem tutoring as in Fuchs, Fuchs, Craddock, et al. (2008) , future research may investigate the added value of such a combination with schema-broadening plus mathematical equations instruction provided at the whole-class and small-group or individual tutoring levels.

A Framework for Teaching Word Problems in the Primary Grades

Across the two lines of work applying schema theory to word-problem solving, Jitendra and colleagues and Fuchs and colleagues provide evidence that students, including those at-risk for or with LD, may benefit from this explicit approach to word-problem instruction at the classroom and tutoring levels. In the schema-based instruction of Jitendra and colleagues, students learned to use schematic diagrams to solve word problems. In Jitendra's more recent research, students also learned to set up and solve an equation to find the word-problem answer after filling in a schematic diagram. The schema-broadening instruction of Fuchs and colleagues incorporated explicit schema instruction about word problem transfer features so students could learn how to recognize novel problems as belonging to the schemas they learned, but without reliance on schematic diagrams. Additionally, in the schema-broadening instruction of Fuchs et al. (2009) and Fuchs, Seethaler, et al. (2008) , students learned to use mathematical equations to represent the structure of a word problem.

Limitations

Before proceeding, it is important to discuss a few limitations across the two lines of schema work. First, and perhaps most importantly, many of the measures used to determine treatment effects were designed by the experimenters conducting the research. Some measures included word problems almost identical to those presented to students during instruction, perhaps raising questions about the generalizability of the word-problem instruction. When standardized tests of problem solving were administered, effects were either not significant or not as large as on the experimenter-designed measures. Second, a few of the studies, especially those by Fuchs and colleagues, included students at-risk for LD and not necessarily students with identified LD. Some of these students at-risk for LD received special education services; most did not. Within these studies for students at-risk for LD, the results for students with disabilities were not disaggregated from the primary sample. The same holds true for some of the other studies conducted by both Fuchs and colleagues and Jitendra and colleagues ( Fuchs et al., 2006 ; Fuchs et al., 2009 ; Fuchs, Fuchs, Craddock, et al., 2008 ; Fuchs, Seethaler, et al., 2008 ; Griffin & Jitendra, 2009 ). Therefore, it remains unclear if the interventions benefit students at-risk for LD, students with LD, or both.

Implications for Practice

Other schema investigations (e.g., Jitendra, DiPipi, & Perron-Jones, 2002 ; Jitendra, Hoff, & Beck, 1999 ; Xin, Jitendra, & Deatline-Buchman, 2005 ; Xin & Zhang, 2009 ) suggest using schema to solve word problems in the intermediate grades. That aside, the focus of the present literature review was on the primary grades, where the literature provides the basis for conceptualizing a framework for teaching word problems to students at-risk for or with LD that comprises the following features. First, instruction should be explicit. Across the two lines of schema work, schema were introduced in an explicit manner, and teachers or tutors often modeled or provided worked examples of word problems using each schema. It is not surprising that students at-risk for or with LD benefitted from explicit instruction, given that other mathematics researchers not focused on schema instruction (e.g., Kroesbergen et al., 2004 ; Mercer, Jordan, & Miller, 1996 ) have demonstrated the benefits of explicit instruction for students at-risk for or with LD. Across all the schema studies at the primary grades, students learned one word-problem schema at a time and had adequate practice (i.e., for days or weeks) on the schema before learning another schema.

Next, word-problem instruction should be organized. Students with LD profited from organizing word problems via schemas and having an explicit method for conceptualizing their solutions for each schema. This solution method could be a schematic diagram ( Jitendra et al., 1998 ), a mathematical equation ( Fuchs et al., 2009 ; Griffin & Jitendra, 2009 ), or a way of organizing information ( Fuchs, Fuchs, Finelli, et al., 2004 ). Because methods that work for one student may not work for another, it is important that teachers familiarize themselves with various explicit methods for helping students learn schema approaches to word problems so they can best help their students.

A schematic diagram may help some students organize their word-problem work, as in Figure 1 . With a schematic diagram, students fill in the relevant numbers from the word problem. The area of the schematic diagram that represents the question of the word problem (i.e., the word-problem solution) is left blank or filled in with a question mark. Students then learn how to solve for the blank space in the diagram to solve the word problem by calculating the answer. For example, the following is a typical elementary-school word problem: A classroom has 15 students. If 6 of the students are boys, how many students are girls? Using the schemas employed by Jitendra et al. (1998) , this word problem falls under the “group” schema because there is a larger set (i.e., the classroom) with smaller sets (i.e., boys and girls) within the larger set. The larger set (15) and one of the smaller sets (6) are defined within the text of the word problem. The other smaller set is the missing information needed to answer the word-problem question. After a student selects the word-problem schema (i.e., group), they fill in a schematic diagram. The schematic diagrams assist in organizing the word-problem information in pictorial fashion, which, as Jitendra and colleagues have demonstrated, may be beneficial for students at-risk for or with LD.

Another approach that may assist students in organizing word-problem work is to decide on a problem's schema and then use a mathematical equation to represent the underlying structure of the schema. Working with the word problem just discussed: A classroom has 15 students. If 6 of the students are boys, how many students are girls? , Fuchs et al. (2009) categorized this problem as falling within a “total” schema. In a “total” schema, parts are put together for a total. Instead of using a schematic diagram, students are taught a mathematical equation (i.e., P1 + P2 = T) that represents the two parts (i.e., P1, P2) put together for a total (i.e., T). After students decide the word problem's schema, they write the mathematical equation to help organize their word-problem work. (See Figure 2 for a worked example.) Students fill in the relevant numbers from the word problem and write an X for the missing part of the equation. Students then solve for X (i.e., solve the word-problem question). As demonstrated by Fuchs and Jitendra, using an equation to represent the schema is also an effective approach for strengthening the word-problem skill of students at-risk for or with LD.

For more difficult word problems, Fuchs and colleagues (e.g., Fuchs, Fuchs, Finelli, et al., 2004 ) have demonstrated how students can organize their word-problem work without a schematic diagram or mathematical equation, while relying on their knowledge of the problem schema. The word problem, Maya wants to buy 2 bags of pencils for $3 each, 4 notebooks for $2 each, and 6 folders for $1 each. How much will Maya spend? , would fall under the “shopping list” schema because multiple items of various prices are purchased. Instead of using a schematic diagram or equation for a “shopping list” problem, students draw vertical lines on their paper to organize their work. (See Figure 3 for a worked example.) Students calculate one part or step of the shopping list problem (i.e., pencils, notebooks, folders) in each section and calculate the overall cost of items in the right-most section. The vertical lines assist students in organizing the word-problem information and their work, but drawing the lines is not a necessity. That is, students could use their knowledge of schemas to solve the word problem with or without the lines.

Several other dimensions of a word-problem teaching framework using schema theory also emerge across the two lines of schema. These include practice in sorting word problems into schema, many instructional sessions, and multiple settings (i.e., whole class, small group, individual) for schema instruction to occur. With regard to sorting word problems into schema, both Jitendra and colleagues and Fuchs and colleagues made a point of mixing word-problem types so students had to differentiate what problems belonged to which schema. Some of this practice was explicit via flash cards ( Fuchs, Seethaler, et al., 2008 ). Some of the schema identification practice was embedded within the lesson, whereby teachers and tutors presented students with word problems, students had to decide which schema represented the word problem, and then use the structure of the schema to solve the word problem.

Across schema studies, the number of instructional sessions varied from 13 to 45 sessions. For all studies but one, students were taught or tutored multiple times each week, and students demonstrated significant gains from the schema instruction. Only the results of Griffin and Jitendra (2009) proved disappointing, with the absence of significant differences between the schema-based instruction and a comparison group. The authors attributed this lack of significance to the fact that instruction was provided once a week instead of multiple times each week. The significant results from the other 11 studies highlighted in this review suggest that instruction should be of sufficient duration (i.e., weeks and months, not days) and occur multiple times each week.

Finally, in the body of work reviewed in this paper, schema instruction occurred in whole-class, small-group tutoring, and individual tutoring settings. Students at-risk for or with LD benefitted from the schema instruction in all three of these settings. One study ( Fuchs, Fuchs, Craddock, et al., 2008 ) isolated the effects of conducted schema instruction provided in whole-class arrangement versus small-group tutoring settings. They concluded that the combination of both whole-class teaching and small-group tutoring may optimally enhance outcomes for students at risk for LD and that tutoring was essential for promoting strong outcomes. Teachers, therefore, should be mindful that whole-class instruction may not be enough for students with or at risk for LD, and additional tutoring (i.e., Tier 2 or 3 within a Response to Intervention framework) may be necessary to improve the word-problem outcomes of students with LD.

The linking features of these two schema approaches require students to (a) read a word problem, (b) recognize the underlying structure of the word problem as belonging to a specific schema, and (c) solve the word problem using a solution method that represents a schema. Whether students use schematic diagrams, mathematical equations, or another method to help them apply their knowledge of the word-problem schema, the research conducted by Jitendra and colleagues and Fuchs and colleagues demonstrates that students at-risk for or with LD may benefit from explicit word-problem instruction that incorporates schemas.

Acknowledgments

This research was supported by Award Number R01HD059179 from the Eunice Kennedy Shriver National Institute of Child Health & Human Development to Vanderbilt University. The content is solely the responsibility of the authors and does not necessarily represent the official views of the Eunice Kennedy Shriver National Institute Of Child Health & Human Development or the National Institutes of Health.

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Order of Operations (MDAS)

In this section, you are going to learn to use parentheses in your calculations. The reason to use parentheses is to change the order of operations. But what is the order of operations?

The order of operations decides in which order you do your calculations. At this stage, I call the order of operations MDAS (the initials of M ultiplication , D ivision , A ddition and S ubtraction) .

Want to watch animated videos and solve interactive exercises about MDAS? Click here to try the Video Crash Course called “What Is the Order of Operations?”!

First, do all M ultiplication and D ivision.

Then, do all A ddition and S ubtraction.

Let’s take a look at some examples:

Calculate 6 + 3 ⋅ 2 .

Now, think MDAS. First you multiply and divide, then you add and subtract. Here’s how you do it:

Calculate 4 − 2 ⋅ 6 + 3 .

Think MDAS. First you multiply and divide, then you add and subtract. Here’s how you do it:

Calculate 1 2 ÷ 4 − 6 + 8 ⋅ 5 .

Think MDAS. First you multiply and divide, then you add and subtract. Here is how you do it:

Calculate 7 2 ÷ 8 − 6 ⋅ 4 + 1 6 + 3 2 ÷ 8 .

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