Running Head: THE MODEL METHOD ON ACHIEVEMENT AND MOTIVATION
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The Effect of Singapore’s Step-by-Step Model Method on Student Achievement and Attitude in
Third Grade
Cayla B. Williams
Kennesaw State University
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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, realworld 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
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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, openended, 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
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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
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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
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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-
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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
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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
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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-bystep 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
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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’
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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?
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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.
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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.
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References
Buckley, J. (2012, December). National Center for Education Statistics [NCES] Statement on
PIRLS 2011 and TIMSS 2011. Retrieved from
http://nces.ed.gov/whatsnew/commissioner/remarks2012/12_11_2012.asp
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
young children’s mathematical word problem solving: A design experiment with third
grade Hungarian students. Educational Studies in Mathematics, 81(1), 47-65. doi:
10.1007/s10649-001-9360-z
England, L. (2010). Raise the bar on problem solving. Teaching Children Mathematics, 17(3),
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Fong, N. S. (2004). Developing algebraic thinking in early grades: Case study of the Singapore
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Peterborough, NH: Crystal Springs Books.
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Fuchs, L. S., Fuchs, D., Finelli, R., Courey, S. J., Hamlett, C. L., Sones, E. M., & Hope, S. K.
(2006). Teaching third graders about real-life mathematical problem-solving: A
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Jitendra, A. K., Dupuis, D. N., Rodriguez, M. C., Zaslofsky, A. F., Slater, S., Cozine-Corroy,
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Mahoney, K. (2012). Effects of Singapore's model method on elementary student problem
solving performance: single case research (Doctoral dissertation, Northeastern
University). Retrieved from http://hdl.handle.net/2047/d20002962
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National Center for Education Statistics. (n.d.) Trends in International Mathematics and Science
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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
One bag of had 20 marbles. Another bag had 15 marbles. What is the total amount of
marbles in both bags?
1 bag
Number of Marbles
another bag
20
15
?
35
20 marbles + 15 marbles = 35 marbles
The total number of marbles in the two bags is ____35___.
Change Problem
There were 11 birds sitting on a tree branch. Four of the birds flew away. How many
birds were left?
7
?
F
L
Birds
11
There were __7__ birds left.
THE MODEL METHOD ON ACHIEVEMENT AND MOTIVATION
Comparison Problem
Tom has $35. Ben has $18. How much more money does Tom have than Ben?
more
Tom’s Money
$35
?
Ben’s Money
$17
$18
$35-$18
$35+$2=$37
$18+$2=$20
$37-$20=$17
Tom has $__17__ more than Ben.
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Appendix B
Model Drawing Steps
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