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Running Head: THE MODEL METHOD ON ACHIEVEMENT AND MOTIVATION 1

The Effect of Singapore’s Step-by-Step Model Method on Student Achievement and Attitude in

Third Grade

Cayla B. Williams

Kennesaw State University

THE MODEL METHOD ON ACHIEVEMENT AND MOTIVATION 2

Review of Literature

World problems, also known as story problems, can pose significant challenges for

elementary students due to the difficulties they face in finding a solution (Griffin & Jitendra,

2009). In order for students to become successful problem solvers, they must be able to analyze

and interpret information in a way that makes sense. With mathematical problem solving,

students must use their problem-solving abilities to apply math skills to real-world problems

(Fuchs et al., 2006). Mathematical problem solving skills and the application of strategies occurs

through a variety of methods throughout the United States. With the adoption of the Common

Core Standards [CCS] by 43 states, including Georgia, the CCS math focus has caused renewed

interest in students’ ability to relate mathematics to real-world situations through complex, real-

world problems (Common Core Standards Initiative, 2014; Wilson, 2013).

The CCS require teachers to develop instructional methods that support students’ ability

to tackle and make sense of word problems. Problem solving is at the center of mathematics in

the CCS, which requires research-based strategies that will improve problem-solving

performance for all (Porter, McMacken, Hwang & Yang, 2011). In order for students to be able

to tackle complex word problems, they must have a well-developed protocol. This literature

review will explore what research shows about problem-solving strategies. A thorough review

of Singapore’s Model Method is included in order to determine its effectiveness on students’

world problem achievement and motivation.

Problem Solving

The National Council for Teachers of Mathematics [NCTM] defines problem solving as

students engaging in a task where the solution method is unknown (National Council for


Teachers of Mathematics, 2010). Problem solving is more difficult than one may infer. Yuan

(2013) implies that problem solving requires students to do much more than simply remove

numbers from a word problem and solve the problem with an equation. He states that many

students desire to solve math problems by extracting numbers and quickly finding the answer,

with little understanding of how they solved the problem. Problem solving requires much more

than using basic concepts, it requires the use of in-depth mathematical thinking and reasoning for

problem success (Griffin & Jitendra, 2009). According to Clark (2009), mathematical problem

solving is the center of mathematical learning. He states problem solving requires students to

apply mathematical concepts and skills in a wide range of situations, including unfamiliar, open-

ended, and real-world problems. Problem solving may pose a daunting, undesirable task for

many, but with the right method and strategies in place, problem solving can become a much

easier task.

Methods for Problem Solving

Students receive instruction on various problem-solving strategies to help them tackle

and make sense of word problems. Through my research, I discovered that authors discussed

four main problem-solving strategies: identifying key words (see Definition of Terms),

implementing Pólya’s four-step method, and using schematic drawings. England (2010) states

students traditionally receive instruction on solving word problems through the strategy of

seeking key mathematical words. Teachers can guide students to use the words to determine

which mathematical operation to use when solving a word problem. According to mathematical

researcher John A. Van de Walle (2003), using key words in math problems can be misleading.

He states that the problem with key words is that they do not exist in many word problems. Van

de Walle believes that key words send the wrong message to students about doing math. Other


researchers state that key words ignore the meaning and structure of word problems, and fail to

help students make sense of the problem (Jitendra, Griffin, Deatline-Buchman, & Sczesniak,

2007). Therefore, suggesting that students focus solely on the key words in a mathematical word

problem to find a solution seems to be an ineffective strategy for problem solving.

Another problem solving method is Pólya’s four steps to problem solving. George Pólya

published his four-step method in his 1945 book How to Solve It. Pólya’s steps require students

to understand the problem, devise a plan, carry out the plan, and look back and reflect (Pólya,

2014). Pólya’s first step, understand the problem, requires students to identify the unknown in

the problem and make sense of it. In this step, Pólya suggests that students draw a figure in

order to understand the problem. In the second step, making a plan, Pólya suggests that students

connect the data to the unknown, and create a plan for solving the problem (Pólya, 2014). This

step of the method connects to the schema theory since students access schema of the problem

and problem type (Mahoney, 2012). In the third step, carrying out the plan, students use their

plan to solve the problem, checking their steps along the way. In Pólya’s fourth and final step,

looking back, students review their computation to assure that their answer is correct, and use

other methods to check their work. Clark (2009) states mathematical textbooks use methods and

strategies based on Pólya’s method, which seem to provide a successful method for students’

word problem solving.

I also discovered that schema plays an important role in mathematical problem solving.

Schema, defined by Gick and Holyoak (1983), is two or more examples that students use to

group problems that require similar solutions. Similarly, Jitendra et al. (2007) define schema as

a group of problems that hold similar structures and require similar solutions. The schema

theory, addressed by one of the early schema theorist Frederick Bartlett in 1932, states that


schema builds with experience (Bartlett, 1932 as cited in Mahoney, 2012). According to Jitendra

et al. (2013), the schema theory notes that understanding the schematic structure of the problem

is important for word problem comprehension. According to Fuchs et al. (2004), the broader the

schema, the more likely students will be to make connections among similar problems, and that

greater problem transfer will occur. The three main types of word problem schemata are change,

part-whole, and compare (see Definition of Terms). By recognizing these types of problems,

students can successively organize problems by structure, which allows them to accurately

represent the problem and find a solution (Jitendra et al., 2013). Using schema to solve word

problems is a successful strategy through schema-based instruction [SBI], which incorporates

visual representations (Griffin & Jitendra, 2009).

Schema-Based Instruction and Model Drawing

As seen by many researchers, using SBI with students produced positive results in

students’ problem solving abilities on word problems (Fuchs et al., 2006; Griffin & Jitendra,

2009; Jitendra et al., 2013). SBI guides students to identify the schema of the problem and create

a schematic diagram to help with solving the problem (Griffin & Jitendra, 2009). Most SBI

studies look at the method’s use with striving students or students with mathematical difficulties.

Progress measurement in the studies occurred through pretest and posttest measurement

comparison (Jitendra et al., 2007; Jitendra et al., 2013). In both studies completed by Jitendra et

al. (2007; 2013), students participating in the SBI intervention showed growth, indicating that the

SBI method for solving mathematical word problems is beneficial for students. SBI allows

students to make sense of similar word problems, create visual representations that help them

gain a deep understanding of the problem, and transfer their problem solving strategies among

similar problems (Jitendra et al., 2013). Mahoney (2012) linked this effective design to


Singapore’s valued problem solving method due to their similarities in schema identification,

problem connection, and visual representations. Although both methods approach problems

similarly, the way students visually represent problems is different.

Singapore’s Model Method

In the early 1990s, Singapore and the United States’ mathematics achievement were

comparable, but since then Singapore has made great gains in mathematics (Leinwand &

Ginsburg, 2007). Over the past 20 years, Singapore’s students in fourth and eighth grades

received recognition for their top math scores on the Trends in International Mathematics and

Science Study [TIMSS] comparison, a test taken every four years to compare countries’ math

and science achievement (National Center for Education Statistics, n.d.). Singapore’s fourth and

eighth grade students scored first on the assessments in 1995, 1999, and 2003, and scored within

the top three in 2007 and 2011 (Buckley, 2012; Clark, 2009). Due to Singapore’s success,

mathematics educators across the world have gained interest in discovering what Singapore’s

teachers are doing to foster effective learning. According to Clark (2009), the secret to

Singapore’s mathematical success is their central focus on problem solving, and the ways

students in Singapore receive instruction on problem-solving methods.

Although the CCS in the United States emphasize problem solving similarly to

Singapore’s curriculum, Clark (2009) notes several differences that cause students in the United

States to fall short of equal success. Singapore embeds problem solving through their

mathematics text. It is not a separate activity, but central to every skill and concept. Students in

Singapore work on much more complex one- and two-step word problems than those presented

in American text (Leinwand & Ginsburg, 2007). The main difference between the United States

and Singapore is that Singapore’s students receive instruction on explicit, sequenced problem-


solving strategies beginning in second grade, in order to solve both routine and non-routine

problems (Clark, 2009). The most recognized strategy Singapore’s students receive instruction

on to solve word problems is the Model Method, also called model drawing.

Model drawing, also known in the United States as bar modeling, teaches students

beginning in second grade to use visual models that they can manipulate to deal with complex

word problems (Clark, 2009). Students receive instruction on using rectangles to model the

situation of a problem (Fong, 2003). Students can use the rectangles to represent the specific or

unknown numerical values (Ng & Lee. 2009). This method requires students to think

analytically, providing them an important transition between the concrete and abstract. It

provides students with a powerful strategy that allows them to understand the word problems

they are solving (Forsten, 2010). In order for students to understand and solve word problems,

they must first recognize basic mathematical relationships (Forsten, 2010).

Model drawing requires students to have a solid foundation of part-whole relationships.

According to Forsten (2009), students must be able to understand and manipulate number

bounds, also known as fact families. Once students have a solid foundation of number bounds,

they can easily use them in simple addition and subtraction problems. After students master

basic part-whole relationship understanding, they can move on to complex word problems. It is

essential for students to understand parts and wholes in addition, subtraction, multiplication,

division, fractions, ratios, and percent, so they can easily use their understanding to create and

manipulate model drawings (Leinwand & Ginsburg, 2007).

Through model drawing, students are able to move from using manipulatives to drawing

pictorial representations while solving problems (England, 2010). The drawings allow students

to use a single powerful model to solve mathematical problems that incorporate several other


strategies, whereas United States mathematics programs urge students to use various

manipulatives to solve different types of problems (Leinwand & Ginsberg, 2007). Using various

manipulatives and strategies makes it increasingly difficult to make sense of what to use and

when. The Model Method allows students to use a consistent tool, the rectangular bar, to solve

various problems successfully. When using the Model Method, students use one tool that

encompasses several other problem-solving strategies and methods.

In comparing Singapore’s Model Method to other problem-solving strategies, it seems

that the Model Method encompasses a variety of problem solving strategies. The Model Method

requires students to make part-whole connections to link schema and create visual

representations (Mahoney, 2012). Singapore also created their problem solving strategy with

special emphasis on the first and second steps of Pólya’s four-step model method, understanding

the problem and reflecting on the solution (Clark, 2009). Singapore’s curriculum builds on

Pólya’s work by teaching students specific strategies for problem solving, and emphasizes the

use of the most effective strategy for a particular problem (Clark, 2009). Although students use

model drawings to solve problems, they must determine which types of model, part-whole,

change, or comparison, best solves the problem (Hoven & Garelick, 2007). With the support of

the additional strategies that Singapore’s Model Method contains, many researchers suggest that

Singapore’s model drawing is an effective method for students’ use in solving word problems.

Char Forsten (2010) searched further into the topic of Singapore’s Model Method and

expanded on the method by adding a step-by-step component (see Appendix B for the list of

steps). Forsten states that brain research shows breaking down new information when it is

learned can be beneficial. Therefore, she created a systematic approach to teach students upon

beginning the use of the Model Method. The steps help students organize and manage their


thoughts when they come across daunting problem-solving tasks. Forsten suggests guiding

students through learning how to use the steps before having them practice the problem solving

steps independently, to assure students’ appropriate use of the steps. Forsten bases the step-by-

step Model Method approach on knowledge she gained from learning about the brain and

working with students in Singapore and the United States, but little research is available

supporting the use of the systematic method.

Model Method Results

There is little data on the use of the Model Method with a step-by-step approach, but

researchers Csikos, Szitanyi, and Kelemen (2011) found that visual representations are important

for students learning to solve word problems beginning in third grade. Other researchers state

that the use of schematic, organized representations may be effective interventions for students at

risk for falling behind in mathematics, or students who have mathematical difficulties (Griffin

and Jitendra 2009; Jitendra et al. 2007). Although the Model Method includes the use of visual

representations and schematic drawings, there are few studies available on the effectiveness of

the specific method on student achievement. Research completed by England (2010) suggests

that students gained a better understanding of how to solve word problems efficiently by using

the Model Method. Mahoney (2012) completed similar research by teaching students the Model

Method as an intervention, and noticed that after students received the intervention they were

able to answer more word problems successfully. In both studies, students who received

instruction on Singapore’s Model Method, showed improvement on their overall problem

solving, as noted by pretest and posttest data comparison. This data, along with information from

the TIMSS test, indicates that efficient instruction and implementation of Singapore’s Model

Method may increase students’ abilities to solve mathematical word problems


Another avenue that I examined was determining if the use of Singapore’s Model Method

affects students’ attitude toward and outlook on tackling and completing word problems. I found

no information on the use of the Model Method in relation to students’ attitude toward word

problems, but I found information on students’ outlook on the use of the Model Method. In

Mahoney’s (2012) study, children indicated that using the Model Method was enjoyable, useful,

and that its use would benefit other students. Similarly, in Fong’s (2004) study, students

indicated that by using the Model Method, they were better able to visualize problems. Many

students in the study stated that the use of the Model Method helped them visualize and

understand the problem. These studies suggest that students find the Model Method to be a

useful problem-solving strategy, that has the potential of helping other students. However, the

research provides little information on how the method affects students’ attitude toward word

problems.

Summary

Overall, it seems that problem solving poses significant difficulties with students as they

approach word problems. Several strategies seem to show success in a variety of studies

completed with students having difficulties in mathematics. TIMSS assessment data points to

the significance of Singapore’s problem-solving strategies, along with minimal data from

research studies. With the increased focus on mathematical problem solving in the United States

due to the implementation of the CCS, I find it vital to seek and utilize strategies that will

structure and build my students’ problem solving abilities, especially on word problems. Studies

indicate that students need to make connections between similar problems and use an organized

method for solving them. Therefore, through my study, I will seek to add information to the

little data on Singapore’s Model Method in regard to word problem achievement and students’


attitudes toward word problems. I will explore more information about the topic through the

following questions: Does the use of Singapore’s step-by-step Model Method increase student

achievement on mathematical word problems? Does the use of Singapore’s step-by-step Model

Method increase students’ positive attitudes towards completing mathematical word problems?


Definition of Terms

Change Problem- In a change problem, students use information in a word problem to

increase or decrease the initial quantity given in order to find a new quantity (Griffin & Jitendra,

2009).

Comparison Problem- In a comparison problem, students compare two or more separate

sets of information and the relation between the sets (Hoven & Garelick, 2007).

Key word- A key word is a specific word in a word problem that helps a student

determine what type of operation to use to solve the problem. For example, in all, combined,

altogether, and total are key words that guide students to use the addition operation. The words

less than, take away, change, and fewer guide students to use the subtraction operation.

Model Method- The Model Method is the teaching strategy developed and utilized in

Singapore for illustrating mathematical word problems (Mahoney, 2012). Researchers and

educators also refer to The Model Method as model drawing, or bar modeling in the United

States. The method consists of first identifying a word problem by problem type, part-whole,

change, or comparison. Then, a model drawing is created using various configurations of

rectangles, or bar models, to form schematic representations of the word problems’ known and

unknown quantities (see Appendix A for a close look at the use of the Model Method with the

three problem types).

Part-whole Problem- In a part-whole problem, students use their understanding of fact

families, also known as number bonds, to represent simple addition and subtraction word

problems (Hoven & Garelick, 2007). In these problems, students use the known information in a

world problem to find the unknown by manipulating number bonds.


Step-by-step model drawing- Step-by-step model drawing is a systematic way of teaching

students to use the Model Method. With step-by-step model drawing, students receive

instruction on the steps to take when encountering a word problem (see Appendix B for the list

of steps). The steps provide a thorough guide for the Model Method implementation that allows

students to organize their thoughts and the information provided in a word problem.


References

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PIRLS 2011 and TIMSS 2011. Retrieved from

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Clark, A. (2009). Problem solving in Singapore Math. Math in Focus: A Singapore Approach.

Retrieved from http://www.scribd.com/doc/36990278/Math-in-Focus-Problem-Solving

by-Andy-Clark

Common Core State Standards Initiative. (2014). Common core state standards initiative:

Preparing America’s students for college and career. Retrieved from

http://www.corestandards.org/standards-in-your-state/

Csikos, C., Szitanyi, J. & Kelemen, R. (2011). The effects of using drawings in developing

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Appendix A

Model Method Examples

Below are basic examples of the three problem types, part-whole, change, and

comparison. The models are adapted from Char Forsten’s text Step-by-Step Model Drawing

(2009). Students can use each problem type with all operations, but the examples displayed

below demonstrate each problems use in addition and subtraction.

Part-Whole Problem

Change Problem

Comparison Problem

One bag of had 20 marbles. Another bag had 15 marbles. What is the total amount of marbles in both bags?

1 bag another bag

Number of Marbles 20 15 ?

20 marbles + 15 marbles = 35 marbles

The total number of marbles in the two bags is ____35___.

35

Tom has $35. Ben has $18. How much more money does Tom have than Ben?

moreTom’s Money $35

?

Ben’s Money $18

$35-$18$35+$2=$37$18+$2=$20$37-$20=$17

$17

There were 11 birds sitting on a tree branch. Four of the birds flew away. How many birds were left?

? F

L

Birds 11

There were __7__ birds left.

7


Appendix B

Model Drawing Steps

Tom has $35. Ben has $18. How much more money does Tom have than Ben?

moreTom’s Money $35

?

Ben’s Money $18

$35-$18$35+$2=$37$18+$2=$20$37-$20=$17


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