Post on 31-Jan-2018
PROBLEM SOLVING & NUMBER SENSE 1
Problem Solving, Number Sense and Their Interactions
Danielle L. Lanigan
Kennesaw State University
Action Research Project
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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.
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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 …………………………………………………..…………...….. Page 37
Appendix …………………………………………………………………...…………….. Page 40
References …………………………………………………………………..……………. Page 44
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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
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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).
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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,
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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
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this understanding, I am very interested in determining if, and to what extent, these daily Number
Talks sessions have on students problem solving abilities.
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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
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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
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(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
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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.
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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.
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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.
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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
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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
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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
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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
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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.
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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.
PROBLEM SOLVING & NUMBER SENSE 21
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.
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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
PROBLEM SOLVING & NUMBER SENSE 23
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.
PROBLEM SOLVING & NUMBER SENSE 24
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.
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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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Percentage of Students Beginning Problem Quickly
Begins Problem Quickly PercentageLinear (Begins Problem Quickly Percentage)
PROBLEM SOLVING & NUMBER SENSE 26
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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9298 95 94
85 90 908379 84 82 86
Percentage of Students Beginning the Problem Quickly by Subgroup
GiftedESOLGeneral Ed.
PROBLEM SOLVING & NUMBER SENSE 27
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)
PROBLEM SOLVING & NUMBER SENSE 28
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
Week 1 Week 2 Week 3 Week 40
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10094
73
859295
83 85 817768
8983
Percentage of Students Working Collabo-ratively with their Partner by Subgroup
GiftedESOLGeneral Ed.
PROBLEM SOLVING & NUMBER SENSE 29
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)
PROBLEM SOLVING & NUMBER SENSE 30
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
Week 1 Week 2 Week 3 Week 40
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Percentage of Students Justifying their Thinking Verbally by Subgroup
GiftedESOLGeneral Ed.
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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)
PROBLEM SOLVING & NUMBER SENSE 32
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
Week 1 Week 2 Week 3 Week 40
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Percentage of Students Using an Appropriate Strategy by Subgroup
GiftedESOLGeneral Ed.
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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
PROBLEM SOLVING & NUMBER SENSE 33
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
PROBLEM SOLVING & NUMBER SENSE 34
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.
PROBLEM SOLVING & NUMBER SENSE 35
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
PROBLEM SOLVING & NUMBER SENSE 36
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.
PROBLEM SOLVING & NUMBER SENSE 37
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
PROBLEM SOLVING & NUMBER SENSE 38
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.
PROBLEM SOLVING & NUMBER SENSE 39
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.
PROBLEM SOLVING & NUMBER SENSE 40
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 = ________________
PROBLEM SOLVING & NUMBER SENSE 41
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?
PROBLEM SOLVING & NUMBER SENSE 42
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?
PROBLEM SOLVING & NUMBER SENSE 43
APPENDIX C
Date: _____________________________________
Student Namesbegins
problem quickly
works collaboratively
with partner
justifies thinking
uses an appropriate
strategy
correct answer
123456789
101112131415161718192021222324252627
Observational Notes
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