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PROBLEM SOLVING & NUMBER SENSE 1

Problem Solving, Number Sense and Their Interactions

Danielle L. Lanigan

Kennesaw State University

Action Research Project


ABSTRACT

In this mixed-method study the author presents the results of research that investigated the

impact of daily Number Talks instruction on fourth grade students’ problem solving skills. The

literature reviewed in preparation for this study identified the importance of problem solving and

Number Talks in mathematics instruction. A classroom of 25 fourth-grade students from a Title I

school in the Metro-Atlanta area participated in a four week study on problem solving. Students

participated in a daily ten-minute Number Talks session followed by a ten minute problem

solving session. Students’ behaviors were observed by the teacher researcher using a checklist

and qualitative notes. These results were analyzed and indicated that daily Number Talks

positively impacted four out of the five student behaviors. The findings are further discussed in

the study and implications of the research presented.


TABLE OF CONTENTS

Abstract ………………………………………………………..…………….…………… Page 2

Chapter 1: Introduction ……………………………………………………………..……. Page 4

Chapter 2: Literature Review …………………………………………………..………… Page 9

Chapter 3: Methodology ………………………………………………….……………… Page 20

Chapter 4: Results ………………………………………………...……………………… Page 25

Chapter 5: Conclusions …………………………………………………..…………...….. 7

Appendix …………………………………………………………………...…………….. Page 40

References …………………………………………………………………..……………. Page 44


Chapter 1: Introduction

The introduction of the Common Core State Standards for Mathematics has made a

significant change to mathematics instruction (Faulkner, 2013). There is now a clear focus on

number sense, questioning, and a conceptual understanding (Common Core State Standards

Initiative, 2014). These new standards are intended to transform mathematical instruction and

ultimately improve students’ understanding of mathematical concepts (Conference Board of the

Mathematical Sciences, 2010). To accomplish these goals and implement the Common Core

Standards, teachers have turned to resources such as Number Talks by Sherry Parrish, or other

similar activities, to ensure that students are engaging in a deep and meaningful study of numbers

and engaging in discourse about their knowledge (Parrish, 2010). Additionally, teachers spend a

large amount of mathematics instructional time working on problem-solving skills to help

students construct a lasting understanding of the mathematical concepts set forth by the Common

Core State Standards (National Council for Teachers of Mathematics, 2014). It seems fitting,

then, to investigate the impact that daily Number Talks instruction has on students’ problem-

solving skills.

Background

In 2010, when the State Common Core Standards were introduced, I felt very confident

in the implementation of the new mathematics standards, which had a clear focus on student

understanding and rigor of study (Common Core Georgia Performance Standards, 2010). One

key aspect of the new standards was the focus on problem-solving skills for students.

Additionally, students were expected to have a deeper and more meaningful understanding of

numbers and operations (Standards for Mathematical Practice, 2014). Many teachers were at a


loss when determining how this looks in daily classroom instruction. At about the same time, the

book Number Talks, by Sherry Parrish, was introduced into the school system that I work in

(Parrish, 2010). This was a novel idea to many teachers and elicited a lot of thought and change

among teachers in regard to how to approach numeracy instruction. After two years of Common

Core and Number Talks instruction, I began to see some improvement in the ways students

approached various problems in mathematics workshop and in their confidence and

understanding. This prompted me to further study the impact that daily number talks instruction

has had on students’ problem-solving skills.

School Context

The school in which the study is taking place is located in the metro Atlanta area and has

an enrollment of about 550 students (Enrollment, 2014). The school participates in the Title I

program, meaning there is a large percentage of the student body that receives free or reduced

lunch (Report Card Overview, 2007; 2010-2011 AYP Report, 2011). In fact, 88% of the school’s

population receives this service, which is extremely high compared to the district’s average of

47% of students and Georgia’s average of 57% of students receiving free or reduced lunch (2010

– 2011 Report Card, 2011; Compton Elementary School, 2014). In the current year, nearly 60%

of students enrolled at the school are African American with the next largest population groups

being Hispanic, representing 26%, and White, representing 10% (Enrollment, 2014). This is

significantly different than the enrollment in the school district where African American students

make up 32%, Hispanic students make up 19%, and White students make up 42% of the student

population (Enrollment, 2014).


Why Numeracy and Problem Solving?

The areas of number sense and problem solving are a key focus for teachers of all grade

levels as indicated by the Common Core Standards (Standards for Mathematical Practice, 2010;

Faulkner, 2013; Cai & Lester, 2010). Additionally, research has demonstrated that, in terms of

whole number concepts and operations, prospective teachers rely heavily on an algorithmic

understanding and yet lack a firm understanding of why the algorithm works (Browning et al.,

2014). If these prospective teachers lack the content knowledge to teach mathematics effectively,

students may also experience difficulties in these same areas. Finally, a strong number sense in

students is linked to a higher mathematical ability in other areas, even into adulthood (Feigenson,

Libertus, & Halberda, 2013; Fenell, 2008). The proven importance of both number sense and

problem solving regarding students’ mathematical understanding and success makes the study of

these areas of high importance. Currently, little research has been done indicating the impact that

students’ number sense has on their problem solving ability or showing any correlation between

the two concepts. Therefore, a study which investigates the research question “to what extent do

daily number talks impact students’ problem solving abilities?” is warranted.

In regard to test scores, the elementary school involved in this study has scores

significantly below that of the district and state. In mathematics, 84% of students in the state

scored a meets or exceeds level on the CRCT in mathematics. In the district, 87% of students

scored meets or exceeds for mathematics. However, the school in this study had only 70% of

students meet or exceed the standards (Cobb County School District, 2014). In addition to the

financial struggle of students, it is clear that there are academic issues plaguing students as well;

therefore, intense, focused instruction targeting two of the main areas of mathematical practices

could likely improve students’ mathematical abilities (2010 – 2011 Report Card). Additionally,


students could benefit from identification of the effectiveness of practices currently in place at

the school-wide level.

Definition of Terms

Terms central to this study are defined through research as follows:

Problem refers to a situation in which a solution is not readily available or known in

advance (National Council for Teachers of Mathematics, 2014).

Problem Solving refers to “mathematical tasks that have the potential to provide

intellectual challenges for enhancing students’ mathematical understanding and development” as

defined by Cai and Lester in their 2010 publishing of Why is Teaching With Problem Solving

Important to Student Learning (pg.1).

Number Sense refers to “a person’s general understanding of numbers and operations

along with the ability and inclination to use this understanding in flexible ways to make

mathematical judgments and to develop useful and efficient strategies for managing numerical

situations” (Sengul, 2013)

Number Talks refers to short, mental math sessions intended to strengthen students’

number sense that are pulled from the 2010 book by Sherry Parrish, Number Talks.

Significance

This study on the extent to which daily Number Talks impacts students’ problem solving

ability will likely impact the instructional practices of teachers at my school and within the local

professional community. Research findings support the notion that number sense and problem

solving abilities are critical components of mathematical instruction. It is also clear from my

professional endeavors that this area is a source of struggle for teachers and students alike. Given


this understanding, I am very interested in determining if, and to what extent, these daily Number

Talks sessions have on students problem solving abilities.


Chapter 2: Literature Review

Introduction to Problem Solving and Number Sense

The Common Core State Standards Initiative was adopted by the Georgia State Board of

Education on July 8, 2010 with the intent to provide students and teachers with “relevant content

and application of knowledge through high-order skills” (Common Core Georgia Performance

Standards, 2010). This adoption of standards included an entirely new set of mathematics

standards for all public school grades. These new standards were research-based and intended to

provide coherence and rigor across the United States in educational curriculum. The mathematics

standards were specifically designed to allow for a greater depth of knowledge regarding the

content addressed and to reflect modern understanding on the ways in which students learn

(Mathematics Standards, 2014). The adoption of these standards reflects a change in thinking

regarding how mathematics should be taught and is understood by students.

A key component of the Common Core Standards is problem solving. A problem is

typically thought of as a situation where there is a goal and the problem solver is not

immediately able to reach that goal (Lester, 2013). Recently, the concept of problem solving has

expanded from simply reaching the goal to an activity that can be completed by one or more

people, requiring cognitive actions that are not considered routine, along with previous

experience and intuition (Lester, 2013). The National Council for Teachers of Mathematics

(NCTM) reported in a 2008 Research Brief that problem solving in mathematics specifically is

thought of as a “mathematical tasks that have the potential to provide intellectual challenges for

enhancing students’ mathematical understanding and development” (pg. 1). More recently,

NCTMT has identified problem solving as not only an important component of mathematics, but


integral to the subject itself. They also discuss the importance of problem solving not only in

learning but also in the professional world (National Council of Teachers of Mathematics, 2014).

The Mathematics Common Core Standards place such an emphasis on problem solving

that it is their first of eight Standards for Mathematical Practice (Standards for Mathematical

Practice, 2014). The standard states that students should “make sense of problems and persevere

in solving them” and includes a description discussing metacognition while solving problems as

an important aspect of problem solving. Additionally, students should be able to explain their

thinking and solution as well as understand various solutions to the problem. Beyond problem

solving, students should reason abstractly and quantitatively, construct mathematical arguments,

and critique the reasoning of others (Standards for Mathematical Practice, 2014). These essential

concepts of mathematics as addressed through the eight Standards for Mathematical Practice are

the foundation for which the mathematical content is to be taught and should be valued by all

teachers (Swanson & Parrott, 2013).

With the adoption of the Common Core standards, changes in testing procedures and a

shift in the modern workforce to a collaborative environment, it is fundementally important that

students are masters of the Standards for Mathematical Practice. This means that students should

be able to see multiple solutions to a problem and create reasonable arguments and explanations

to support their solutions. The book Number Talks by Sherry Parrish intends to meet the

requirements of the Standards for Mathematical Practice through short, daily practice with the

concepts of numeracy and solving problems (Parrish, 2010). In the past, mathematics instruction

has been thought of as a set of rules and procedures taught to students devoid of any conceptual

understanding of the mathematical concepts at play (Parrish, 2010). Now we know that a deep

conceptual understanding of mathematics is linked directly to mathematics achievement


(Schneider, Grabner, & Paetsch, 2009). Additionally, students must have a working sense of

numbers, be able to reason mathematically, and assess solutions. Fostering these skills is the

purpose of Number Talks instruction (Parrish, 2010).

A deep understanding of problem solving is critical in modern mathematics instruction

(Cai & Lester, 2010; CCGPS, 2010; Lester, 2013). Also, success with word problem solving can

be an indicator of students’ mathematics ability as well as their working memory capacity

(Jitendra, Sczesniak, & Deatline-Buchman, 2005; Swanson, Moran, Lussier, Fung, 2014). A

conceptual based method of problem solving instruction, including using illustrations to

represent the problem, is one of the most effective methods of instruction and has a positive

impact on students’ problem solving ability, especially in students with mathematics difficulties

(Jitendra et. al., 2005; Xin, Zhang, Tom, Whipple, Si, 2011; Csikos, Szitanyi, Kelemen, 2011).

Researchers studied the method of delivery when instructing students on problem solving and

found that whether through computer or teacher instruction, the quality of the program is more

important than the medium of delivery (Bayazit, 2013). Additionally, research has found that

students tend to rely on procedures and rules to solve real-world problems and disregarded a

realistic approach which relies on the real-world understanding of the problem when determining

a solution. The research also indicated the importance of the procedure of problem solving rather

than the solution and that this should be encouraged in students (Bayazit, 2013).

In terms of number sense, multiple studies of research have shown there is a link between

number sense and standardized mathematical performance in very young children and in

elementary school, students use of mental computation strategies is beneficial in helping them

understand numbers and come up with multiple strategies to solve problems (Varrol & Farran,

2007, Feigenson, Libertus, & Halberda, 2013). It has also been shown through research on


students’ use of a mental number line, that a conceptual understanding of math is linked more

closely to achievement than the use of a mental number line (Schneider, et al., 2009). However,

this research differs from the current in that it was not directly linked to students’ word-problem

solving ability and is limited in that it only assessed the impact of student use of a mental number

line. Subsequently, there has not been research completed evaluating the impact of a strong

number sense on students’ problem solving ability. It has been shown that word problem solving

is important and that a conceptual understanding of mathematics is the most important indicator

of success as well as the importance of student instruction on mental computation and number

sense. Therefore, it is necessary to see how these two very important aspects of an effective

mathematical program are related by investigating the impact of Number Talks instruction on

fourth grade student’s problem solving ability.

Problem Solving

The National Council of Teachers of Mathematics released a research brief in 2010

explaining the significance and impact of teaching with problem solving (Cai & Lester, 2010).

The findings of this study indicated that problem solving is in fact a crucial aspect of the

mathematics curriculum and should not be taught in isolation, but integrated into mathematics

instruction. The study also indicated that students can participate in problem solving activities at

a very young age. The council conducted a qualitative study on the research surrounding

mathematics instruction to answer key questions regarding problem solving. From the study, the

researchers claimed teachers should understand that problem solving is a slowly developed skill

that is fostered through a classroom culture in which problem solving is consistent and

challenging.


Jitendra, Sczeniak, and Deatline-Buchman (2005) found that word problem solving

success could be a successful indicator of mathematics proficiency. The study investigated

whether curriculum-based word problem solving success was correlated to student achievement

on a criterion-referenced, statewide assessment. Third grade students were given assessments on

computational fluency and word-problem solving fluency. These were compared to the

Standford-9 and TerraNova standardized criterion-referenced tests in the subtest areas of

Problem Solving and Concepts and Application. Students’ scores indicated that the curriculum-

based problem-solving fluency assessment was moderately correlated with the Problem Solving

subtest of the Stanford-9 and the Concepts and Application subtest of the TerraNova. The study

also indicated that curriculum-based measures of problem solving fluency can provide data

regarding concepts and applications of mathematical knowledge. Therefore, teachers can use

students’ ability to solve curriculum-based problems as an indicator for success on standardized

mathematics testing.

The effect of computer-mediated instruction compared to teacher-mediated instruction on

word problem-solving performance was studied and the findings indicated that there were no

statistically significant differences between the groups on the word problem solving post-test

(Leh & Jitendra, 2012). Students with mathematical difficulties were divided into two groups

receiving either teacher instruction or computer instruction that focused on conceptual modeling

of the problem through using visual representations to represent the problem structure. The

students were given a word-problem solving post-test after completion of the intervention and a

retention test four weeks after the post-test. The lack of statistical significance between groups’

performance supports previous research completed by the researchers indicating that the quality

of problem-solving instruction is more important than the medium of learning.


There have been multiple studies conducted on the various problem solving methods and

their effectiveness. Swanson, Moran, Lussier, and Fung (2014) studied the effect of strategy

training and working memory on students’ word problem-solving accuracy in children at risk of

having significant difficulties in mathematics. They found that working memory was a

conditional variable of achievement, but students with a higher working memory capacity

achieved at a higher level when taught problem solving using complete, generative (or

paraphrasing) strategies. This study divided students into groups using three different

instructional strategies: paraphrasing question propositions, paraphrasing relevant proposition,

and paraphrasing all propositions within a word problem. Students took a pretest and posttest to

measure their word-problem solving abilities. Researchers concluded from this study that

students with a higher working memory capacity would benefit from generative problem solving

instruction but that achievement using this method was tied to prior working memory capacity.

The effect of using drawings and representations when solving problems was studied

with Hungarian students and it was found that students who received instruction including visual

representation made greater gains than those who did not (Csikos et al., 2012). Five classes of

third-grade students were included in the experimental group, receiving instruction on using

visual representations to model the word problem, while six classes were considered control

classes. The control group was studied while the curriculum being taught was problem solving in

mathematics and teachers were told only of the importance of their participation. Both groups of

students were assessed using an arithmetic skill and problem solving pre and posttest. The

researchers concluded from the test results that using realistic content and relations within word

problems and teaching a solution strategy of using drawings helps students with problem solving

success and metacognition.


Similarly to the previous study, concept-based problem solving was discussed in two of

the research studies. In one study, small group tutoring on schema-based instruction (studying

the underlying mathematical structure of the problem) versus standards-based curriculum

(school-provided inquiry-based student-driven approach) was studied regarding its impact on

students’ mathematical achievement and word problem solving (Jitendra et al., 2013). The

findings of this study showed that students with a higher pre-test score benefited more from a

schema-based instruction, while students with a lower pre-test score benefited more from a

standards-based content. Third-grade students were divided into two groups receiving either

standards-based curriculum instruction or schema-based instruction. In this study, schema-based

instruction focused on the structure of the word-problem and standards-based content focused on

place value, addition, and subtraction concepts. Students were assessed on a pre-test,

intervention, post-test, and retention exam. The findings suggest that students with significant

mathematical difficulties would benefit more from a standards-based instructional method while

students with a higher word-problem solving ability would benefit more from a schema-based

instructional method.

A similar study was conducted that compared two common problem-solving strategies:

(a) a conceptual-model based problem solving approach, which focuses on the structure of the

problem, to a (b) general rule-based problem-solving approach which used a “SOLVE” (Search,

Organize, Look, Visualize, Evaluate) rule (Xin et al., 2011). The study found that students with

learning problems benefited from a conceptual model expression of mathematics more than the

general instruction group. Students were randomly divided into two groups: a conceptual-model

based problem solving or general instruction. Students participated in problem-solving learning

activities three days a week for 30 – 45 minutes each session. They were given a pretest then


instructed based on their group’s method as provided by the researcher and were assessed on a

pre-algebra and criterion-referenced problem solving posttest. The researchers found that

students with learning difficulties benefit from a strategy being clearly taught that focuses on a

conceptual understanding and that these students are capable of thinking algebraically through

representation.

In 2013, Bayazit investigated 7th and 8th grade students’ approaches to solving real-world

problems which are those that present problems typical in human life. The research findings

showed that most students did not supply answers that would make sense in daily life. Instead,

students applied factual and procedural knowledge to the problem without regarding its real

world situation and thus gave an answer that would not make sense in terms of the problem. This

mixed-method research focused on student use of their real world understanding of the problem

and application of that knowledge to their problem solving approach. Students were asked to

answer six real-world mathematical problems and their solutions were assessed and categorized

as a realistic answer, non-realistic answer, other answer, or no answer. Additionally, four

students were selected to interview about their responses and still defended their non-realistic

responses. From this study, researchers concluded that it is important for teachers to emphasize

the process by which students come up with solutions more so than their answers.

Number Sense

Number sense and its integral nature in students’ mathematical learning were discussed in

a news bulletin released by the National Council of Teachers of Mathematics (Fennell, 2008). In

this bulletin, Francis Fennell, the president of the organization, discusses the urgent need for

teachers to integrate number sense into the mathematics curriculum. He discusses that students


with number sense are able to reasonably answer mathematical problems and use their sense of

numbers to help them when computing answers. Another benefit of number sense is that students

gain flexibility when solving problems which is very helpful when solving problems. Overall,

the news bulletin indicates the importance of number sense in students and its foundational

nature in mathematics ability and problem solving.

This need was mirrored in the book Number Talks: Helping Children build Mental Math

and Computational Strategies by Sherry Parrish (2010). Parrish discusses the importance of

number sense in students and supports the need for this skill development in mathematics

classrooms and echoes the call for a deep, foundational understanding of numbers and

mathematics that is required according to the Common Core State Standards (pg. xxiii). Parrish

elaborates on how daily number talks support a deeper understanding of numbers by allowing

them to construct their own understanding of operations and properties of numbers through the

exploration and discussion of problems.

The need for a conceptual understanding of mathematics was demonstrated through an

investigation of students’ use of a mental number line and its relationship to mathematical

achievement (Schneider et al., 2009). Researchers conducted three studies involving fifth and

sixth grade students that investigated the distance effect, SNARC effect, and numerical

intelligence as they relate to a conceptual understanding of mathematics. The distance effect

study relates to students more easily comparing numbers when the distance between them on a

mental number line is greater, which was assessed through a computer program. The SNARC

effect is also a behavioral indicator of the use of a mental number line and relates to the concept

of smaller numbers being placed on the left and larger numbers on the right, which was also

assessed through a computer program. Finally, in the third study, numerical intelligence was


assessed through the use of an external number line on a computer game. The results from the

first two studies showed a great importance on conceptual understanding and little reliance on an

internal number line. The results from the third study indicated some relationship between

numerical intelligence and mathematical achievement. The implication drawn from these three

experiments was that the influence of a mental number line is not as important as a conceptual

understanding of the mathematics in this age group.

Students’ number sense and mental computation abilities were also researched in a study

evaluating the connection between mathematics ability and number sense that a “core sense of

approximate number is linked to formal mathematical ability” (Feigenson et al., 2013). The

researchers reviewed previous evidence of this relationship to identify how the approximate

number system (ANS) relates to a person’s mathematical ability. It was found that there is a clear

relationship between approximate number ability and mathematical proficiency. It was also

shown through the review that people with mathematical difficulties have deficits in the

representation of quantities. Additionally, findings regarding individual differences indicate a

causal relationship between ANS and standardized mathematical achievement from an age as

early as three years old. Based on their findings, researchers asserted that there was a need for

this basic human knowledge system to inform our formal knowledge system.

Mental computation, an aspect of number sense, was examined through a literature study

on the mental computation abilities of elementary school students with findings that students

benefited from mental computation instruction (Varo & Farran, 2007). This study investigated

the difference between mental computation and traditional computation using a pen and paper

along with the various mental computation strategies that teachers use when solving problems.

These strategies were named by the researchers and related to the ways subject decomposed the


numbers being added and subtracted. Other key points in the study were the relationship between

conceptual understanding and procedural skills and teaching mental computation. The

researchers concluded that mental computation instruction is important for helping students

understand how numbers work, understanding various solution strategies, and making procedural

decisions.


Chapter 3: Methodology

Design

A qualitative checklist was used throughout the study to measure students’ problem

solving skills. Each student was assessed each day for five key problem solving behaviors. The

behaviors observed included students starting the problem quickly, working with their partner,

justifying thinking, using an appropriate strategy, and correctly answering the problem. The

values noted across four weeks were compiled into an excel spreadsheet that was subsequently

analyzed using quantitative methods. Notes were also taken throughout the study to document

behaviors that were not anticipated on the checklist. These qualitative notes were studied to

determine repetitive behaviors related to students’ problem solving skills.

Participants

Participants in the study included 25 fourth grade students from a Title I elementary

school in the metro-Atlanta area. Within the classroom, there were 14 male students and 11

female students within the age range of 9 - 10 years old. Ten students were receiving services

through the Gifted and Talented Education program at the school and were pulled out of the

classroom for instruction with the gifted education teacher for one full day during the week. Two

students in the room were receiving EIP services in mathematics. These students qualified for

EIP status due to having earned a score below 800 on the 3rd grade Criterion-Referenced

Competency Test. Additionally, three students were considered ESOL students but did not

receive pull-out or push-in services and were instead being monitored throughout the year. The

general ability level of the class was on or above grade level for reading and math.


Intervention

In the study, students participated in daily number talks lessons pulled from Number

Talks by Sherry Parrish. These lessons focused on a specific mental math goal for students to

apply to a given problem, but they encourage student use of a variety of computation strategies

to determine the correct answer for the problem provided. A sample of strategies and problems

can be found in Appendix A of this paper. Students were given time to solve the problem

mentally, and then asked to explain their answers to their math partner. Finally, students

volunteered to explain the strategy they used and while the rest of the class followed along in an

attempt to understand that student’s work. These mental math problems along with the

collaboration and explanation of solutions were intended to build student understanding of the

four main operations in mathematics and therefore increase their ability to identify these

concepts within a word problem.

Following this activity, students engaged in a problem solving session. Students were

read aloud a problem, which is also into their problem solving journals. A list of the specific

problems used for this study can be found in Appendix B. The students then worked with their

math partners to solve the problem using problem solving strategies previously taught and

reviewed throughout the year. This collaboration was crucial to the intervention, as it intended to

help students understand the problem before attempting a solution. These strategies included

determining the mathematical concept present in the problem though a deep understanding of the

four main operations (addition, subtraction, multiplication, and division) along with strategies

found in Step by Step Model Drawing: Solving Word Problems the Singapore Way (Forsten,

2010). Students combined these skills to read and solve the word problem while working with

their partner to explain their thinking, receive assistance, and provide assistance.


Data Collection

To collect data surrounding students’ problem solving skills, I used a mixed-method

approach. To gather quantitative data surrounding this question, I used a checklist of student

behaviors I determined to be important indicators of students’ problem-solving skills. To

determine these skills, I combined the research surrounding problem-solving in elementary

students with my personal observations of student behavior during our problem solving session.

The skills in the finalized checklist include the following: begins the problem quickly, works

collaboratively with partner, justifies thinking, uses an appropriate strategy, and determines a

correct answer. The specific checklist used in the study can be found in Appendix C with the

names of students involved in the study removed for confidentiality. The first two behaviors

aimed to look at the confidence students have when solving a word problem and their comfort

with solving problems. The final three behaviors were included to assess students’ mathematical

understanding related to the word problem and were most closely related to the research question

The qualitative aspect of the study included anecdotal notes recorded by the teacher.

These notes were logged each day on a section of the checklist and included information about

behaviors not present on the checklist, aid provided by the teacher, or any other interesting

information that needed to be included. These notes, however, were not the foundation of the

study. Additionally, the time requirements of the checklist left little time to record detailed notes

each day. This contributed to the lesser influence these notes make to the overall findings.

Quantitative Data Analysis

To analyze the results of the checklist, student behaviors were entered into an Excel

spreadsheet using either a 1 to represent the occurrence of that behavior for that student or a 0 to


indicate that the student did not exhibit the behavior. These data were recorded each day for each

of the five behaviors. The totals for each day were then averaged together to determine the

percentage of students exhibiting the problem solving behavior. These percentages were

calculated for each day and each behavior throughout the study. To record percentages for

correct student answers, it was important to distinguish between the number of students

answering correctly and the number of students completing the problem; therefore, the

percentage of students correctly solving the problem included only students who completed the

problem with a correct answer out of the number of students attempting the problem. However,

the percentage of students completing the problem was also important to note during the study as

it impacts the percentage of correct answers as well as encompasses its own implications

regarding students’ problem-solving skills.

This set of data was used to create a graph providing a visual representation of the overall

data. A trend line was added to the graphs representing the data collected for the entire class.

These lines help illustrate the overall change in the data during the duration of the study, which is

less visible due to the variations in the data from day to day and the numerous data points. To

determine data for each subgroup, the student scores for the week were averaged together for a

weekly average. Then, these results were averaged using only students included in the particular

subgroup to determine an average for the subgroups’ behaviors during each of the four weeks of

the study. Since there were fewer data points in these graphs, values were added to help clarify

the exact changes taking place over the four-week study. A combination of these representations

was used to interpret the results of the analysis.


Qualitative Data Analysis:

The notes taken daily were reviewed but were not coded since there were very few notes

to review. Instead, common behaviors occurring frequently throughout the study were noted after

looking through all notes. Then, these behaviors were tallied based on occurrence for each day

throughout the study. These behaviors were then reviewed in comparison to student achievement

that day and the word problem used. All of these factors were considered when analyzing the

qualitative data results for the study. This method ensured that the multiple aspects including

problem type and student achievement with the problem were considered when determining

findings.


Chapter 4: Results

After entering the quantitative data from the study, graphs were created to provide an

overview for each of the five behaviors analyzed in the study. These graphs allowed the data to

be analyzed for trends. Additionally, the data were disaggregated by subgroup to analyze. The

findings of the study are explained immediately below.

Student Begins the Problem Quickly

This behavior was included in the checklist to gauge whether students became more

comfortable with solving problems during the time of the study and as a result were less hesitant

when beginning to solve a problem. As indicated in Figure 1, this trend line showed that the

percentage of students beginning the problem quickly increased slightly over the duration of the

study. This increase was likely influenced by the ability of students to work with a partner to

solve the problem and their comfort level in working with their partner.

Figure 1

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b0

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40

60

80

100

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Percentage of Students Beginning Problem Quickly

Begins Problem Quickly PercentageLinear (Begins Problem Quickly Percentage)


When disaggregated into the three main subgroups present within the classroom, it is

clear that the percentage of students identified as gifted who began the problem quickly were

much higher than the other two groups. Additionally, ESOL students began the problem quicker

than general education students throughout most of the study, with a decline occurring in the

fourth week of the study. It is also important to notice that all three of the subgroups maintained

similar increases between weeks one and two, with week three showing some difference in

change. During week four, ESOL students declined sharply while general education students

increased sharply. Gifted students showed a slow decline after the initial jump during week two.

It is possible that ESOL students showed higher rates of early participation in problem solving

due to a learned behavior of working with others to understand English and the task at hand.

Additionally, gifted students are aware of their gifted status and often express a high confidence

level regarding their mathematical ability. This may have contributed to their willingness to

begin the problem quickly. The declines shown by both ESOL and Gifted students may be

related to the difficulty of the problem.

Figure 1.1

Week 1 Week 2 Week 3 Week 40

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Percentage of Students Beginning the Problem Quickly by Subgroup

GiftedESOLGeneral Ed.


Student Works Collaboratively with their Partner

This behavior was added to the checklist to assess students’ engagement with their peers

when solving word problems. This is important to students’ overall problem solving ability

because it indicates their comfort level with discussing the problem along with their solution.

This skill lends itself to students explaining their thinking and reasoning when solving problems.

This skill was practiced daily during Number Talks instruction as students worked with their

math partner to explain their thinking surrounding a computational problem. The trend line for

this skill, shown in Figure 2, indicated a greater amount of growth than the percentage of

students beginning the problem quickly, but is fairly weak overall. The growth as a class could

also be explained by the comfort level students had with their partner over the time of the study.

This trend line is similar to the percentage of students beginning the problem quickly and is also

related in that both assess students working with their math partner and their comfort level when

beginning and discussing the problem. Therefore, it is reasonable that students would experience

similar changes for each behavior.

Figure 2

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Percentage of Students Working Collaboratively with their Partner

Works Collaboratively With Partner PercentageLinear (Works Collaboratively With Partner Percentage)


When assessing the changes in the data by subgroup, it is interesting to note that the

subgroups converged at week three as shown in Figure 4. Gifted and general education students

experienced somewhat significant drops between weeks one and two, rose during week three,

and then separated at week four with a higher percentage of gifted students working

collaboratively with their partners. General education students and ESOL students exhibited a

slight decline from week three to week four. ESOL students showed little change from week’s

two to four after an initial drop in the percentage of students working collaboratively with their

partners. The lack of change shown by ESOL students could again be due, in part, to their

tendency to work with others to gain an understanding of the task at hand since this is a skill

taught when learning English. The changes shown by the Gifted and general education students

indicate that the gap in percentage decreased slightly but overall remained during the study. The

reason for this may be related, to an unknown degree, the students’ conceptual mathematical

knowledge.

Figure 2.1


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Percentage of Students Working Collabo-ratively with their Partner by Subgroup



Student Justifies Thinking

Justifying thinking is extremely important to students’ overall mathematical

understanding and is an essential part of the Common Core Standards. Additionally, when

students justify their thinking using the word problem or their mathematical knowledge, they are

applying their learning to construct a new understanding of problem solving. As such, this

behavior was of major importance to the study and is strongly related to the intervention of daily

Number Talks. The trend line for this behavior (Figure 5) shows a noticeable change from the

start to the end of the study. This indicates that a higher percentage of students were able to

justify their thinking at the end of the study. It can then be concluded, regarding the present

study, that the daily practice with explaining reasoning during Number Talks did positively

impact students’ ability to explain their thinking when solving problems as shown in Figure 3.

Figure 3

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Percentage of Students Justifying their Think-ing

Justifies Thinking PercentageLinear (Justifies Thinking Per-centage)


When disaggregated by subgroup, there is a visible increase among both general

education and ESOL students (Figure 6). While ESOL students initially had a higher percentage

of students justifying their thinking, the percentage decreased to nearly the same amount as the

general education students and rose at a similar rate during weeks three and four. The percentage

of gifted students justifying their thinking increased between weeks one and two, and then

decreased to the same percentage as general education students during week three. However, the

same group increased to a percentage above both ESOL and general education students during

week four. These fluctuations in the percentage of gifted students justifying their thinking from

week to week are puzzling but may relate to the type of problem studied over the week. Further

research regarding this phenomenon appears warranted.

Figure 3.1


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Percentage of Students Justifying their Thinking Verbally by Subgroup


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Student Uses an Appropriate Strategy

The behavior of using an appropriate strategy for the problem is closely related to student

understanding of the concept present in the word problem. The struggle for students to accurately

model the mathematical operation(s) within the problem is the basis for many of the studies

reviewed prior to beginning this study. This behavior was important in determining the effect of

daily number talks on students’ problem solving skills. The data for this behavior (Figure 7)

indicate an overall decline in students modeling the correct strategy. This decline is interesting

when compared to the increase in students justifying their thinking. This decline may be related

to some degree to a lack of instruction from the teacher as the study progressed. It is possible

students became more comfortable with working with their partner and less concerned with their

representations; therefore students may not have recorded their thinking to the degree they had at

the beginning of the study when students were less comfortable with their partner.

Figure 4

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Percentage of Students Using an Appropriate Strategy

Uses an Appropriate Strategy PercentageLinear (Uses an Appropriate Strategy Percentage)


When looking at the data by subgroup shown in Figure 8, each of the three subgroups

declined in the percentage of students using an appropriate strategy. Each group increased and

decreased between weeks one and four, but began the study and ended the study at nearly the

same percentage. The changes in the percent of students using an appropriate strategy for each

group from week to week could also be due to a decrease in the students’ use of representation

when solving the problem as they began to rely on their partner.

Figure 4.1


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Percentage of Students Using an Appropriate Strategy by Subgroup


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Students Determining the Correct Answer

While it is of upmost importance that students determine the correct answer when solving

word problems, this skill has been deemphasized through the Common Core Standards.

However, it is still critical that students find the correct answer to a problem. Additionally, if

students are improving in their problem-solving ability, they should also improve in their

accuracy when solving problems. As shown in Figure 9, the trend line indicates a clear increase

in the percentage of students obtaining the correct answer when solving a word problem. This


finding is further strengthened by the quality and difficulty of the word problems chosen

throughout the study. Given this, it can be reasoned that the increase in correct answers relates to

the intervention in place during the study.

Figure 5

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Percentage of Students Determining the Correct Answer

Correct Answer Percent of ClassLinear (Correct Answer Percent of Class)

The data by subgroup shown in Figure 10 indicate an overall increase for each subgroup

with Gifted and ESOL students showing a sharp increase in week two of the study. From this

point, both groups decreased slightly over weeks three and four. General education students

showed a steady increase in the percent of students with the correct answer over the study period.

A possibility for this steady increase would be that this group benefited from the continual,

routine practice over each week and gradually increased their problem solving skills. The sharp

increase in the percent of Gifted and ESOL students with the correct answer may be related to

the type of problems used for week two and the difficulty of the word problems used during

week three as noted in the qualitative findings. The difference in data trends could also be linked


to the reasoning skills of both Gifted and ESOL students. This area would require further

research to explain the results. Both groups tend to rely heavily on verbal explanations, which

would have explained the initial increase. The subsequent decline could be a result of a similar

decline shown by students in these subgroups regarding their use of an appropriate strategy.

Figure 5.1


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Students Determining the Correct Answer by Subgroup


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Students Completing the Problem

One additional aspect studied related to students’ ability to solve the word problem in the

given amount of time. This percentage was calculated for the entire group and indicates an

increase in the percentage of students completing the problem in the allotted time. This trend

matches the increase in students justifying their thinking and ascertaining the correct answer.

This similarity could result from an increase in problem solving skills resulting in students

spending less time struggling over the problem and therefore solving the problem faster over the

duration of the study.


Figure 5.2:

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Percent of Students Obtaining an Answer Before the End of the Problem Solving Session

Percent of CompletedLinear (Percent of Completed)

Qualitative Data Analysis

During the length of the study, the volume of notes collected by the teacher was minimal

and therefore a coding scheme was not used to analyze the recordings. Instead, the notes were

reviewed and a few patterns emerged. The first behavior, which was noted on day one and

continued throughout the study, was in the form of students commenting that the problem was

hard. This behavior occurred on January 26th, January 28th, February 13th, and February 27th. All

of the problems, with the exception of the problem from February 13th, had more words than the

average of 33 words. This may have contributed to the comments on the difficulty of the

problem. The problem given on February 13th, though short, was worded poorly and was difficult

for students to understand. What is evident from these behaviors is that the wording of the

problem impacted students’ perception of the difficulty of the word problem. Interestingly, the

percentage of students with the correct answer was not the lowest on these days when compared

to the percentages recorded during that week of the study. This hints that student perception of


the difficulty of the problem may not relate to student performance. More research in this area

would be necessary to investigate this thought.

Another note that occurred throughout the study was that I frequently had to provide

some instruction to the class regarding the mathematical concept present in the problem after

noticing students incorrectly solving the problem. These clarifications occurred eight times

during the twenty-day study. The clarifications all related to the concept present in the problem

and ranged from explaining what the question was asking of students to explaining what is

occurring in each part of the problem. These clarifications are an important part of teaching and

are necessary to scaffold student understanding. They were also less frequent during the end of

the study than at the beginning. It is likely that as students were more accurate with their

solutions, I found less need to clarify the problem to ensure most students were able to determine

a solution. This would indicate that students were becoming more successful and therefore more

independent with problem solving during the study.


Chapter 5: Conclusions

The literature surrounding problem solving generally agreed that it was most important

for students to understand the concept present in the problem in order to be successful with

problem solving. The National Council for Teachers of Mathematics and the Common Core

Standards also indicated a distinct and pressing need in education for students to be successful

problem solvers. Additionally, previous studies indicated that students’ number sense could

predict their formal math ability during school. However, despite studies presented on mental

math, problem solving, and number sense, I was unable to identify research that studied the

interaction of these two ideas. This prompted the current research project, which indicates that

daily Number Talks instruction does impact students’ problem solving skills.

This conclusion is supported by the increase in students justifying their thinking as well

as the increase in students attaining the correct answer. These two behaviors are closely related

to the benefits of daily Number Talks instruction and provide support to the finding that the

increases are related to the daily Number Talks instruction students received. Behaviors

concerning student confidence when solving word problems were also studied. These findings

indicate an increase in both the quickness with which students begin the problem and their

engagement with peers when solving the problems. These behaviors are indicators of student

confidence with word problems and their comfort level when solving problems. The increase in

both behaviors, while slight, indicates an increase in student assurance when solving word

problems.

The success of students shown through the study is likely related to the time dedicated to

discourse among students related to mathematical problems. The literature found that daily


practice with problem solving was very important to student success. This is mirrored in the

current study indicating students were more successful and slightly more confident with solving

problems after daily practice with problem solving. Even more intriguing is the increase in

students justifying their thinking. This is likely due to the daily interaction between students

encouraging them to justify and explain their actions to their peers. Prior to the intervention,

students participated in a number talks and problem solving session similar to the current study.

However, the quality of problem, structure of time, and amount of collaboration between

students were increased for the purpose of this study. These changes relate to the increase in

students’ problem-solving behaviors found in the study. From the expectations surrounding

problem solving and importance of collaborative problem solving, it is likely that the increase in

collaboration relates to the increase in success students’ showed with problem solving.

There were findings at the completion of the study that contradict this conclusion.

Despite increases in four out of the five behavior areas studied and a decrease in the support

provided by the teacher, the percentage of students using an appropriate strategy declined during

the study. There are many possible reasons for this result, although further research is needed to

identify if one or more, or any, exert a role in this contradictory finding. It is possible that

students relied more heavily on the verbal conversations they had with their partners to solve the

problem than they did on creating a model to fit the problem. This result also may relate to the

increased independence students showed during the study. Additionally, the analysis of notes

taken during the study indicates that the wording of the problem impacts students’ perception of

the difficulty. It is possible that the wording of the problem also impacts students’ ability to

determine the concept present and therefore their ability to accurately depict their thinking.


Finally, it is possible that teacher bias played a role in the decreased results. This is

because as the teacher and researcher, my expectations for appropriate models of the word

problem may have changed or increased as the duration of the study went on. This possibility

cannot be ruled out for any of the behaviors and should be considered especially when analyzing

the results of this study and specifically this behavior.

Implications

Given the findings of previous studies surrounding problem solving, it is clear that a

student’s problem solving success is tied to an understanding of the mathematical concepts

present in the word problem. The findings of this study indicate that while number talks

instruction increases students confidence with solving problems, the frequency with which

students justify their thinking, and their success with the problems, it does not impact their

ability to accurately represent the problem.

A possible future study could be conducted to determine what problem solving

instructional methods similar to those in the previous studies reviewed positively impact

students’ ability to represent the mathematical concept occurring in a problem. Problem solving

and number sense literature indicated that this was of upmost importance when solving

problems. Therefore, it would be beneficial to determine the effectiveness of instructional

methods already in place as well as those which have yet to be used in the classroom.


APPENDIX A

Making Friendly Numbers

1. 98+ 47 = ______________ 2. 126 + 124 = _________________

Place Value

4. 292 + 139 = _________________ 5. 518 + 256 = _________________

Breaking Factors into Smaller Factors

7. 2 x 2 x 12 = _________________ 8. 9 x 4 = __________________

Making Landmark/Friendly Numbers

10. 8 x 50 = __________________ 11. 5 x 19 = ___________________

Partial Products

13. 8 x 57 = __________________ 14. 4 x 36 = ___________________

Partial Quotients

16. 96 ÷ 4 = ___________________ 17. 92 ÷ 3 = __________________

Keeping a Constant Difference

19. 109 – 51 = ____________________ 20. 171 – 136 = ________________


APPENDIX B

Week 1:

1/26 - Dan played 3 games of marbles. In the 1st game, he lost 1/2 of his marbles. In the 2nd game he won 4 marbles. In the 3rd game, he won the same number of marbles as he had at the end of the 2nd game. He finished with 32 marbles. How many marbles did Dan start with?

1/27 - Belinda read 7/9 of her book. If she read 63 pages, how many pages are in her book?

1/28 - The elementary-school students are participating in a fund raiser. The third graders raised a total of $565.15. The fourth graders raised 38.90 more than the third graders. How much did the two grades raise altogether?

1/29 - The butcher sold 63.8 pounds of meat on Monday. He sold 25.3 pounds more than that on Tuesday. On Wednesday, he sold 14.5 pounds more than he did on Tuesday. How many pounds did he sell over the 3 days?

1/30 - Thomas has $36.25. Joey has $235.15 more than Thomas. Malcom has $49.83 more than Thomas. How much money do they have altogether?

Week 2:

2/2 - The Boy Scouts and Girl Scouts are planning a camping trip. They need $300 in order to pay for the trip. The Boy Scouts have $175.50. The Girl Scouts have $53.95 more than the Boy Scouts. How much money do they have altogether?

2/3 - Out of all the butterflies in a garden, 3/7 were red. If there were 63 butterflies, how many were red?

2/4 - The employees at Frank’s Pizza used 198.5 pounds of dough on Friday. They used 65.9 pounds more than that on Saturday. How many pounds did they use over the 2 days?

2/5 - At the zoo, the tigers ate 6.5 pounds of meat. The lions ate 4 times as much as the tigers. How much meat did the tigers and lions eat altogether?

2/6 - At the drama show, cookies sold for $0.25. If Joanne bought 9 cookies, how much money did she spend?


APPENDIX B CONTINUED

Week 3:

2/9 - In a group of children, 2/5 of the children wear glasses. If there are 10 children who wear glasses, how many children do not wear glasses?

2/10 - Bobby had 36 rocks. He threw 6 rocks into the river. What fraction of his rocks did he throw into the river?

2/11 - The Mustang used 6/7 of its gasoline on a road trip. If the tank holds 21 gallons of gasoline, how much gasoline did the Mustang use?

2/12 - A movie ticket costs $9 and a snack costs $3. How much does it cost to buy 4 movie tickets and 4 snacks?

2/13 - Rick earns $6 for each dog he bathes. He also earns $8 each week for doing chores at home. What will Rick earn if he washes 1, 2, or 3 dogs?

Week 4:

2/18 - Kiko bought a sandwich for $4, a juice drink for $2, and an ice cream cone for $2. He gave the clerk $10. How much change did he receive?

2/19 - One morning, the temperature was 58 ˚ F. By noon, the temperature was 71˚F. Then it rose 5˚ before the end of the day. Find the change in temperature from morning to the end of the day.

2/20 - Franklin school has a total of 226 students and teachers in the middle grades. If there are 10 teachers and there are 27 students in each class, how many classes are there?

…

2/23 - Tyrone went running for 2 hours and 45 minutes on Saturday. On Sunday he ran 45 minutes longer than on Saturday. How long did he run in all on Saturday and Sunday?

2/27- Lindsay wants to purchase a bicycle for $109. She sets up a lemonade stand for 1 week. During the first 3 days, she earns a total of $48. If she earns the same amount of money every day for the entire week, how much money does she make? Does she make enough to purchase the bicycle?


APPENDIX C

Date: _____________________________________

Student Namesbegins

problem quickly

works collaboratively

with partner

justifies thinking

uses an appropriate

strategy

correct answer

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Observational Notes

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