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TEACHERS’ PERCEPTIONS OF MATHEMATICS
An Attitudinal Study of 5th-grade Teachers’ Perceptions about Mathematics and the
Influence on Instruction
Submitted by
Margaret Knight
A Dissertation Presented in Partial Fulfillment
of the Requirements for the Doctorate of Education
in Curriculum and Assessment
Southern Wesleyan University
Central, South Carolina
Date February 2, 2023
TEACHERS’ PERCEPTIONS OF MATHEMATICS
Southern Wesleyan University
An Attitudinal Study of 5th-grade Teachers’ Perceptions about Mathematics and the Influence on
Instruction
by
Margaret Knight
has been approved
Date February 2, 2023
Michael Hylen, Ph.D., Dissertation Chair ____________________________________________
Julie McGaha, Ph.D., Committee Member ___________________________________________
Candice Moore Ph.D., Committee Member__________________________________________
TEACHERS’ PERCEPTIONS OF MATHEMATICS
Abstract
Research suggests that math anxiety correlates directly with individuals' views of mathematics-
related experiences. Research also suggests that math anxiety may begin in early elementary
grades and remain into adulthood. This study examined how teachers' experiences, perceptions,
and mathematics confidence levels influence mathematics instruction. To understand teachers'
perceptions of mathematics, the researcher used a qualitative phenomenological approach to
probe into their earliest memories of mathematics before entering school and their experiences
during formal education. Seven 5th-grade teachers with two to fourteen years of experience
agreed to participate in the study. The participants responded to ten open-ended questions
focused on mathematics instruction and seven interview questions examining participants'
experiences with mathematics. The researcher also observed mathematics instruction in each
teacher's classroom. The results indicated that teachers recalled fond early memories of
mathematics and felt reasonably confident about their math instruction. Some teachers struggled
with mathematics as elementary students, while others excelled. The majority of participants
completed the basic requirements for mathematics in high school. Most teachers did not engage
in a mathematics methods course focused on standards-based elementary mathematics. Teachers'
weak areas in mathematics corresponded with similar studies regarding complex mathematics
topics. Several key components of mathematics instruction were absent during observations.
Future research may need to increase the number of observations and the sample size.
TEACHERS’ PERCEPTIONS OF MATHEMATICS
Dedication
I dedicate this work to my dear brother, Charles, who is watching us all from Heaven.
Not many older teenage brothers read Rudyard Kipling's "Just So" stories and the National
Geographic magazine to their three-year-old sisters. Not many explain that Lassie did not die;
she will be back for the next episode because they could not keep having a show if she did not,
and she is a he, by the way, because female dogs are more temperamental; the wolf man is not
real and not under your bed. Bats do not transform into vampires; the sun will not burn out in
your lifetime, and no volcanoes exist in Louisiana. For these and many other reasons, I dedicate
this work to you, my dearest brother. I wish you were here to read the final copy and provide
commentary. You read through my papers, comps, and dissertation proposal before you left to
complete the final journey. You will always inspire me, and I will believe in myself because you
did.
TEACHERS’ PERCEPTIONS OF MATHEMATICS
Acknowledgments
I deeply respect my committee members: Dr. Michael Hylen, Dr. Julie McGaha, and Dr.
Candice Moore. The support they have provided is over and beyond expectations. Your insight
and advice are invaluable, but your encouragement and words of wisdom will be with me
throughout life. Most of all, you have taught me what it means to be a critical thinker and a lover
of research. Dr. Hylen models what he expects. I have never known anyone with a stronger work
ethic or so devoted to improving education. He is a mover and a shaker. He makes things happen
and always looks for ways to improve. He is prompt with responses and specific with feedback.
He gives his all, and that is what he expects of students. There is no guessing game to be played.
Dr. McGaha is very candid with her feedback. She makes me think more deeply about
the reader’s perspective and the clarity needed to ensure that other researchers can duplicate or
build upon the research. She also encouraged me to explore literature about systems outside of
the United States and to consider the complexity of test data and the importance of examining all
factors involved in the results. The complicacy of international tests is that scores do not reveal
the whole picture. These data include socio-cultural, socio-economic, and internal policies within
the countries that contribute to their scores.
I appreciate Dr. Moore’s willingness to join the committee and the keen insight and
thoughtful feedback she provided. Without any background about the study, she swiftly
completed a first read of the entire dissertation from top to bottom in a compendious way with
very insightful comments, which motivated me to probe deeper into the actual problem and
TEACHERS’ PERCEPTIONS OF MATHEMATICS
purpose of the study. I also want to acknowledge Dr. Lisa Hall-Hyman for her sincere and
heartfelt support before and during the dissertation process.
The ladies of the “Fabulous Five” will always be very near my heart. We have come a
long way, and all had challenges, but we stand far stronger together. I am thankful for all the
support and love you have given throughout the process. I hope we can all walk down the aisle
together in May, and if we cannot, I know we will continue to support each other until we all
make it there. Much love to Candice, Tammy, Tiffany, and Krystal.
I want to thank Shannon for his patience and support throughout these three years. You
are an exceptional example of what love and devotion truly mean. Thank you to my children,
Charles and Carol, for your confidence, guidance, and unconditional love. To my sisters,
Elizabeth and Frances, thank you for always listening to my worries and encouraging me along
the way. To my beloved best friend, Deborah Q., we have been together since we were six, a
beacon of light and inspiration through the darkest and happiest moments of my life. Thank you
to my friend and colleague, Dawn Scott. Your advice, proofreads, subtle hints, and
encouragement will never be forgotten. My project completion would not have been possible if it
had not been for Dr. Sandra Brossard. I am forever thankful for your persuasive ability to make
things happen.
TEACHERS’ PERCEPTIONS OF MATHEMATICS
Table of Contents
Chapter 1: Introduction to the Study................................................................................................1
Problem Statement……...................................................................................................................2
Significance of the Study.................................................................................................................6
Purpose of the Study........................................................................................................................7
Research Questions..........................................................................................................................7
Rationale for Methodology..............................................................................................................8
Theoretical Foundations and Conceptual Framework...................................................................10
Definition of Terms........................................................................................................................12
Summary…………………………………………………............................................................13
Chapter 2: Review of Related Literature.......................................................................................15
Math Anxiety and Neurological Factors .......................................................................................16
Math Anxiety and Math Achievement ..........................................................................................17
Math Self-Perceptions…………....................................................................................................21
Math Content Knowledge……......................................................................................................22
Student Anxiety in Mathematics....................................................................................................26
Teacher Anxiety in Mathematics...................................................................................................29
Summary and Integration…….......................................................................................................31
Chapter 3: Methodology ...............................................................................................................32
Introduction....................................................................................................................................32
Purpose of the Study......................................................................................................................32
TEACHERS’ PERCEPTIONS OF MATHEMATICS
Research Questions………………………....................................................................................33
Research Design…….. ..................................................................................................................33
Sources Of Data …………………………………………………………………………………35
Data Collection .............................................................................................................................35
Data Analysis ................................................................................................................................37
Questionnaire.................................................................................................................................37
Interviews......................................................................................................................................38
Observations..................................................................................................................................40
Study Population and Sample Selection........................................................................................41
Ethical Considerations...................................................................................................................43
Summary........................................................................................................................................44
Chapter 4: Data Analysis and Findings........................................................................................46
Introduction....................................................................................................................................46
Sample............................................................................................................................................47
Data Collection..............................................................................................................................47
Interviews.......................................................................................................................................47
Questionnaire.................................................................................................................................48
Observations .................................................................................................................................48
Data Analysis……………………………………………………………………………….……49
Interviews…………………………………………………………………………………….…..49
Questionnaire……………………………………………………………………………….…....60
Observations……………………………………………………………………………………..67
TEACHERS’ PERCEPTIONS OF MATHEMATICS
Summary........................................................................................................................................69
Chapter 5: Conclusions and Recommendations.............................................................................70
Introduction....................................................................................................................................70
Discussion and Interpretation........................................................................................................71
Limitations.....................................................................................................................................79
Implications for Theory.................................................................................................................80
Implications for Practice and Future Research..............................................................................80
Summary........................................................................................................................................82
References .....................................................................................................................................84
Appendices……………………………………………………………………………………….99
TEACHERS’ PERCEPTIONS OF MATHEMATICS
List of Tables
Table 1. Survey Response Themes...............................................................................................38
Table 2. Interview Response Themes.......................................................................................... 39
Table 3. Observation Themes...................................................................................................... 41
Table 4. Demographics of Participants........................................................................................43
Table 5. Observation Results...................................................................................................... 68
TEACHERS’ PERCEPTIONS OF MATHEMATICS
List of Figures
Figure 1. Mathematics Experiences.............................................................................................. 61
TEACHERS’ PERCEPTIONS OF MATHEMATICS 1
Chapter 1
Introduction
The relationship between mathematics self-perception and performance has traditionally
been explored in terms of "math anxiety" and "math self-concept" (Fitzgerald, 2012). Math
anxiety is derived from a person's self-perception of math skills. One's self-perception of ability
in mathematics may influence math anxiety and performance levels. Mathematics self-perception
refers to a person's confidence level in mathematics abilities and efficacy. People who suffer
from math anxiety often have a low opinion of their abilities, contributing to poor performance
and attitudes about mathematics (2012).
Prior research indicates that math anxiety is a legitimate phobic reaction (Ashcraft, 2002).
A more recent study by Schaeffer et al. (2021) suggested that math anxiety in teachers is linked
to poorer mathematics performance in both male and female students.
Unfortunately, this phenomenon affects so many people in the United States that it is a fixture of
our culture. In Beilock's (2019) article, Americans Need to Get Over Their Fear of Math; he
states that it is socially acceptable for people to admit that they struggle in math in our nation. By
contrast, people in the United States generally do not proclaim that they cannot read (2019).
In 1983, A Nation at Risk supported the belief that teachers played an integral role in
improving student mathematics and science learning (NCEE, 1983). When teachers are anxious
about understanding and teaching mathematics concepts, students' confidence and success may
be impeded (Lewis, 2018). Teachers may unintentionally communicate their attitudes about math
to students, positively or negatively influencing students' perceptions of mathematics (Lewis,
2018). The influence of negative mindsets toward mathematics is well-researched as an indicator
TEACHERS’ PERCEPTIONS OF MATHEMATICS 2
of low mathematics achievement in students of all ages. However, research needs to recognize
teachers' significant role in this issue and how their attitudes may influence instruction (Ramirez
et al., 2018).
Problem Statement
Many American students are not performing in mathematics at the same level as their
counterparts in other countries (Federico, 2016). It is essential to consider how students perform
in mathematics in other areas to compare educational policies and teaching strategies that may be
useful in understanding how teachers' attitudes about mathematics in this country affect
mathematics instruction and inevitably influence student achievement.
The Programme for International Student Assessment (PISA) is a thorough exam and a
reliable indication of students' skills that nations use to assess education policies and practices.
The test is administered to fifteen-year-olds triennially throughout the globe. In research from
Jerrim (2021), the common impression of PISA as the measurement of science, reading, and
mathematics skills of 15-year-olds is somewhat more complex. Some top-performing institutions
in China and other countries have been criticized for omitting a substantial section of their 15-
year-old population from their sample selection for PISA 2015 and PISA 2018. (Loveless, 2014).
It is especially likely to inflate PISA scores in nations where a substantial proportion of 15-year-
olds, mainly lower and middle incomes, are not enrolled in school (Jerrim, 2021).
The United States may want to consider the differences between other countries' policies
and the differences in the size and make-up of the populations when comparing students'
performance on the PISA. Nevertheless, the current state of the United States educational system
TEACHERS’ PERCEPTIONS OF MATHEMATICS 3
needs to be revised; the question is whether political and educational leaders are open to learning
from the rest of the world (Anderson, 2013).
The OECD publishes the test results and an analysis of the state of education worldwide,
providing evidence of the most effective policies and methods for assisting nations in delivering
quality education (OECD, 2020). The most recent report published in 2018 from the OECD
indicated that students in the United States ranked third from the bottom in mathematics (2020).
Also, in the United States, socioeconomically advantaged children outscored disadvantaged
students on the most recent 2018 PISA in mathematics (OECD, 2020). Additional assessments,
such as the National Assessment of Education Progress (NAEP), also reveal students' low
achievement in mathematics. The (NAEP) was given in 2019 to a representative sampling of
fourth and eighth graders in participating districts across the states. The assessment measured
students' mathematics understanding and problem-solving ability. In South Carolina, all fourth-
grade students taking the (NAEP) scored significantly below the national average (2019).
International and national assessments provide information about the state of mathematics
teaching and learning across the country and the similarities with underperforming school
districts in South Carolina.
Evidence of a deficit in content knowledge was revealed in a study that asked preservice
teachers to reason about procedural algorithms using whole numbers (Thanheiser et al., 2014).
Researchers found that only seven out of seventy-one preservice teachers could explain why the
algorithm worked conceptually. There is also evidence that mathematics content knowledge does
not improve with teaching years (Browning et al., 2014). Seasoned teachers may not have a
stronghold on conceptual understanding. Conceptual learning is very present in the South
TEACHERS’ PERCEPTIONS OF MATHEMATICS 4
Carolina College and Career ready mathematics standards. If teachers lack a conceptual
understanding of mathematics standards, an adequate content representation may be lost in
instruction.
The teacher's frustration and lack of understanding could be a factor that contributes to
instruction which may influence students' low achievement. The stress of re-learning a content
area can create a tense classroom environment that is not conducive to learning (Sun, 2017).
Beginning in 2021-2022, all elementary schools' grades, k-5, were self-contained in the
participating district. Departmentalization no longer exists at the elementary level. Many
teachers who have not taught math currently teach all content areas, including mathematics.
The South Carolina College and Career-Ready Assessment (SC READY) is a state
assessment with English Language Arts (ELA) and mathematics components. It is administered
to students from third to eighth grades during the last weeks of school. Educators established
four performance levels for the South Carolina READY assessment (SC READY) to indicate
student mastery and command of the skills and understanding defined in the South Carolina
College and Career Ready Standards (SCCCRS). Most students have some knowledge of
academic standards; nonetheless, performance levels succinctly convey the extent to which
students have demonstrated mastery of the knowledge and skills defined in the SCCCRS. By
outlining the information and skills students must display to reach each level, performance levels
provide meaning and context to scale ratings. The four performance levels include: Does Not
Meet Expectations, Approaches Expectations, Meets Expectations, and Exceeds Expectations for
the SC READY assessment (South Carolina Department of Education, 2021).
TEACHERS’ PERCEPTIONS OF MATHEMATICS 5
A student who fails to satisfy the grade-level content standards for knowledge and
abilities required for the grade level will require significant academic support to prepare for the
next grade level and be on track for college and career readiness. A student who approaches the
grade-level knowledge and skills necessary for this level requires further academic support to
prepare for the next grade level and be college and career-ready. A student who meets the
expectations for grade-level content standards is prepared for the next grade level and on track
for college and career readiness. As described by the grade-level content standards, a student
who excels at this level of learning is well-equipped for success in the next grade level and is
college and career ready (2021).
With the introduction of South Carolina College and Career Ready Standards in 2015,
teachers were confronted with teaching mathematics conceptually to support current
mathematical practices and future learning goals. From 2015 to 2019, the participating district's
5th-grade scores ranged from 26% Meets and Exceeds in 2016-2017 to 35 % Meets and Exceeds
in 2017-2018. In 2019, approximately thirty-three percent of all fifth-grade students in the
participating district scored at Meets or Exceeds on the South Carolina College and Career
Ready Exam in the 2018-2019 school year. Out of the Meets and Exceeds categories, about 16%
of 5th-grade students scored Exceeds, and around 17% scored Met. Thirty-one percent of fifth-
grade students scored Approach's Expectations, and almost 36% of students scored Does Not
Meet Expectations.
Although students were administered the exam in the 2020-2021 school year, 678 5th-
graders within the participating district did not take it. Therefore, the district 2018-2019 data was
compared with the state data from the same school year. It was evident in the SC READY 2018-
TEACHERS’ PERCEPTIONS OF MATHEMATICS 6
2019 mathematics scores that almost sixty-seven percent of 5th-grade students in the
participating school district were not fully grasping mathematics' content. Because students'
understanding of mathematics needs improvement for scores to increase, a close examination of
teachers' understanding of the content and feelings about mathematics is necessary. A thorough
investigation of the literature associated with math anxiety and lower math achievement provided
fundamental background information about how this phenomenon may influence teachers' and
students' confidence and perceptions of mathematics.
Significance of the Study
Teachers, students, and all stakeholders benefit from the qualitative nature of the data
gathered from this study. This study differs from Commodari and La Rosa (2021), who
examined connections between students' overall academic anxiety, math anxiety, and student
achievement. Instead, it investigates past experiences and feelings towards mathematics and the
relationship of those experiences to teachers' current attitudes toward mathematics instruction.
Teachers' narratives and feelings about mathematics and confidence in understanding
mathematics concepts make this study unique. The focus is on the teachers' confidence in
teaching math concepts rather than a connection between student achievement and teacher
anxiety. The data from this study will be used to support teachers' understanding and comfort
level with teaching fifth-grade mathematics concepts. Additionally, the researcher used the data
to make recommendations in Chapter 5 that support mathematics instruction in other school
districts outside the participating district's demographic area.
TEACHERS’ PERCEPTIONS OF MATHEMATICS 7
Purpose of the Study
This study explored the relationship between teachers' mathematical self-perceptions and
their influences on mathematics instruction. The researcher examined contributing factors
between teachers' confidence level with mathematics competency and the quality of instruction.
Exploration of teachers' past experiences in math clarified how or when teachers' notions of
mathematics developed. Examining teachers' self-perceived strengths and weaknesses gave the
researcher insight into which mathematics domains impacted the conceptual understanding of
mathematics and instruction.
Observations of teacher instruction indicated positive and negative influences and
perceptions. Observations also revealed teachers' levels of mathematics competency. The
qualitative design sought to uncover the factors contributing to teachers' attitudes and
perceptions about mathematics instruction. The methods included interviews, classroom
observations of instruction, and an open-ended questionnaire.
Research Questions
Question 1: How do 5th-grade teachers' experiences and self-perceptions of mathematics
influence mathematics instruction?
Question 2: How do 5th-grade teachers' understanding of mathematics concepts influence
instruction?
TEACHERS’ PERCEPTIONS OF MATHEMATICS 8
Rationale for Methodology
The methodology consisted of a qualitative phenomenological design. The design was
chosen to explore teachers' experiences and perceptions of mathematics by collecting data from
interviews, observations, and responses to a questionnaire. A descriptive phenomenological
investigation was appropriate in illustrating a deeper meaning of how teachers' experiences and
attitudes played a significant role in mathematics instruction. Phenomenology allowed for
examining teachers' past and current mathematics experiences that influenced instruction.
Through this approach, the researcher developed a theory based on teachers' interpretations of
their lived mathematics experiences, with phenomenology as the research philosophy.
The qualitative interviews captured in-depth personal information from participants
(Creswell, 2014). Personal experiences of teachers' mathematical journey through early
childhood, school careers, and experiences with teaching mathematics were vital to this study's
foundation, rooted in teachers' confidence and attitudes about mathematics instruction. The
questionnaire delved into teachers' comfort level with mathematics content and which areas of
mathematics were enjoyable or frustrating to teach.
The researcher also sought to understand or explain teachers' experiences and perceptions
by contextualizing the math anxiety phenomenon. The purpose of the qualitative observations
was to determine if teachers found mathematics engaging and exciting content to teach or felt
uncomfortable and anxious about their instruction and depth of knowledge. The teacher's
confidence level affected how the content was taught. Bandura et al. (1977) stated that evidence
of a teacher's belief in their ability to instruct students could explain individual levels of
effectiveness.
TEACHERS’ PERCEPTIONS OF MATHEMATICS 9
Examination of teachers' instructional practices was mutually agreed upon and negotiated
in which the teacher chose the times and dates of the observations to avoid stress. During the
data-gathering process, research subjects felt at ease, which ensured that interactions produced
valid results (Aluwihare-Samaranayake, 2012). Some teachers felt self-conscious about
inexperience with mathematics or mastery of mathematics concepts. However, they readily
admitted to not understanding the mathematics concepts yet felt satisfied with their grasp of
procedural knowledge. This study came at a crucial time when policies changed, leaving some
teachers vulnerable and unprepared to teach 5th-grade mathematics. The researcher ensured that
the stress levels involved in research participation needed to be kept at a minimum during this
disquietful time.
The questions used in the interview promoted metacognitive thought without placing
more stress on the teachers. The researcher hoped to stimulate teachers' reflective thoughts about
their content knowledge and classroom interactions during mathematics instruction. The
interview questions also probed teachers' childhood experiences with mathematics to identify
particular periods of stress or enjoyment of mathematics. Seven 5th-grade teachers were
interviewed to assess their perceptions about teaching mathematics. Additionally, classroom
observations of the same teachers and their responses to an open-ended questionnaire provided
meaningful qualitative data. A thorough examination of the interviews, observations, and
questionnaire data and analysis of developing themes provided a basis for evidence compared
with the literature review and the researcher's suppositions.
TEACHERS’ PERCEPTIONS OF MATHEMATICS 10
Theoretical Foundations and Conceptual Framework
The constructivist approach to mathematics demonstrated that it provides a solid basis for
learning mathematics conceptually while still adhering to the intent of standards (Van de Walle,
2004). The primary idea underlying this research study was grounded in conceptual mathematics
teaching and learning. Piaget claimed that children utilize arrangements like mathematical
structures and patterns to reason about mathematical contexts (Wavering, 2011). These
structures, characterized by propositional logic, focus on children's reasoning and practical logic
(2011). According to Lev Vygotsky, students construct their mathematical understandings as
they learn to explain and defend their reasoning to others (Steele, 2001). This constructivist
theory suggests that acquiring mathematics language and expanding mathematical understanding
occurs through a sequence of connecting concepts to generate additional meaning (2001).
Students need opportunities to grapple with conceptual understanding to connect mathematical
ideas and structures. A confident mathematics teacher that fosters the development of
metacognitive strategies provides students with the means to justify thinking with a command of
mathematical language and deep conceptual understanding.
Dr. John A. Van de Walle, a well-known mathematics leader, states that following
procedural directions without reflective thought provides little to no construction of mathematics
understanding. Student learning becomes limited because of the rules and procedures (Van de
Walle, 2004, as cited in Smith, 2010). The rules and steps followed in mathematics sometimes
blur proper conceptual understanding. "The ineffective practice of teaching procedures in the
absence of conceptual understanding results in a lack of retention and increased errors, rigid
approaches, and inefficient strategies" (Van de Walle et al., 2016, p. 25). Teachers who are not
TEACHERS’ PERCEPTIONS OF MATHEMATICS 11
comfortable with mathematics concepts may revert to strictly procedural teaching. When
students cross-multiply fraction numerators and denominators to determine an equivalency, they
follow a procedure that could just as well be performed using whole numbers. As a result,
students get bogged down trying to remember steps that can cause distress and lead to negative
feelings towards math. The whole concept of equivalence as it relates to fractions is lost.
Therefore, teachers must understand the foundations for effective mathematics instruction
to give students a thorough understanding and lessen their anxieties regarding mathematics
instruction (Smith, 2010). Teachers must also have a deep understanding of mathematics, how it
appears in daily life, and how to effectively move students from concrete to abstract
understanding. The Concrete Representational Abstract (CRA)(2012) is an effective instructional
sequence that prepares students to understand abstract mathematical concepts. When students
begin to think mathematically, they must be supported by concrete material before reaching the
more abstract operations phase of learning. This combination of experiences with concrete
objects eventually leads to the sense-making of more sophisticated mathematical structures.
Piaget (1964) indicates that conceptual mathematical understanding is an experience of
the child's interactions with concrete materials instead of the concrete materials themselves. He
emphasized that mathematics content is heavily abstract, and it is vital for students to begin with
sufficient concrete experiences necessary for understanding the abstractions like the concepts.
(Yıldırım & Yıkmış, 2022, p. 94).
According to research from Concrete Representational Abstract (CRA) (2012), The
initial phase of the CRA sequence is known as the concrete stage. It involves physically
manipulating items to understand math conceptually. The next step is the representational stage,
TEACHERS’ PERCEPTIONS OF MATHEMATICS 12
which bridges the concrete and the abstract. Students can move from hands-on manipulation to
drawings as representations of concrete objects at this stage. The final stage of this method is
known as the abstract stage and involves solving math problems using only numbers and
symbols. CRA is a continuous progression. Each stage is dependent on the previous stage and
must be taught sequentially. However, students' rate of successfully moving through the stages is
not necessarily a linear progression (CRA Assessment, n.d.). Teachers must be confident in their
ability to provide appropriate instruction based on this sequence for students to be successful
with mathematics beyond elementary.
In this study, the researcher will rely on the collected data and literature from similar
studies and professional journals to gain insight from multiple theories and interrelated ideas to
support and inform the research. The core of a structure or an experience is the focus of
phenomenological research. It is a method of inquiry that explores inner feelings and
experiences. Participants' experiences are evaluated and compared to determine the essential
qualities of the phenomenon (Merriam & Grenier, 2019). The goal of this study was to explore
the phenomenon of math anxiety contextually with 5th-grade mathematics elementary teachers
to uncover the implications for teachers' perceptions of mathematics teaching and learning.
Definition of the Terms
Concrete- Representational-Abstract (C-R-A): (Concrete Representational Abstract (CRA),
2012). Concrete Representational Abstract (CRA) is a three-step approach to teaching
mathematical concepts that have proven to be highly effective.
Math Anxiety: An unfavorable response to math and mathematical situations (What Is
Mathematics Anxiety? | Centre for Neuroscience in Education, n.d.).
TEACHERS’ PERCEPTIONS OF MATHEMATICS 13
Organization for Economic Co-operation and Development (OECD): OECD (2020). OECD
is the abbreviation for the Organization for Economic Co-operation and Development, consisting
of 34 countries that discuss and develop economic and social policy. The OECD allows different
countries' governments to solve common problems worldwide.
Programme for International Student Assessment (PISA): PISA - PISA. (2019). OECD.
https://www.oecd.org/pisa/. PISA is the abbreviation for the Programme for International
Student Assessment, which measures 15-year-old students' scholastic performance in
mathematics, science, and reading every three years.
South Carolina College and Career Ready Test (SCCCR or SCReady): South Carolina
Department of Education Test Scores. (2019). South Carolina Department of Education. SC
Ready is the abbreviation for the standardized test administered in late spring to students in
South Carolina from third to eighth grades in mathematics and English language arts.
Survey of Adult Skills: Programme for the International Assessment of Adult
Competencies (PIAAC): PIAAC (2019). This survey measures adults' proficiency in essential
information, processing skills, literacy, numeracy, and problem-solving. It gathers information
and data on how adults use their skills at home, at work, and in the broader community.
Summary
In the United States, aversion to mathematics is a phenomenon often discussed in
research (Beilock, 2019). However, with unsuccessful attempts at reform, an inadequate
understanding of mathematics prevails in our nation. Agasisti and Zoido (2018) referenced a
2012 OECD study comparing the numeracy proficiency of 16- to 65-year-olds in 20 nations;
Americans were among the lowest five in terms of numeracy. Mathematics proficiency levels are
TEACHERS’ PERCEPTIONS OF MATHEMATICS 14
low across age levels. Teacher candidates and veteran teachers fit into the age span in the OECD
study. If elementary teachers are not comfortable with mathematics, will they be comfortable
teaching mathematics? Could instructors' experiences, attitudes, and levels of mathematics
confidence influence how they teach mathematics?
This study explored teachers' earliest memories of mathematics before entering school
and mathematics experiences during formal schooling to learn more about the elements that
influence their perceptions of mathematics. The study also considered the elements impacting
teachers' current attitudes toward math content and instruction. Additionally, the study explored
the possible relationship between the teacher's competency levels in teaching mathematics and
the quality of instruction. In the next chapter, the researcher analyzed the literature associated
with the topic. In Chapter Two, the researcher chose literature similar to the topic but not
necessarily in agreement to provide a wide range of literature that provided insight and
challenges to the study.
TEACHERS’ PERCEPTIONS OF MATHEMATICS 15
Chapter 2
Review of Related Literature
Mathematics is taught at all grade levels throughout a student's education. It is one of the
primary fields of study needed for success in daily life. However, it is an area that people are
afraid of the most and fail the most (Gürbüz & Yıldırım, 2016). This study aimed to examine the
relationship between fifth-grade teachers' experiences and perceptions of mathematics and the
quality of their instruction.
The decreasing numbers of students meeting and exceeding achievement requirements
led to exploring teachers' attitudes towards mathematics and their confidence in mathematics
instruction. In 2019, 2021, and 2022 SC READY scores showed a marked decrease in fifth-grade
mathematics students who scored met and exemplary in the participating district. Overall, close
to 55% of 5th graders in South Carolina landed in the “not met” or “approaches” category on the
SC READY for the 2019 school year, which increased to 73 % in 2021 and decreased by 1 % in
2022. (South Carolina Department of Education, 2022). This research aimed to determine how
common math anxiety was in 5th-grade teachers in the participating district and how that might
have affected fifth-grade math instruction.
In this section, the literature is associated with mathematics anxiety, mathematics content,
perceptions, confidence levels, and the relationship to mathematics instruction. The discussion
examines society's perceptions of mathematics and how attitudes may impact the enjoyment of
teaching mathematics. The literature also includes research focusing on a connection between
math anxiety, neurological activity in the brain, and the ability to perform mathematics tasks. In
addition, the review includes research concerning the conceptual understanding of mathematics
TEACHERS’ PERCEPTIONS OF MATHEMATICS 16
content and its implications on instruction. How math anxiety impacts students’ perceptions and
understanding of mathematics provides a broad spectrum of information. Lastly, the researcher
examined the literature on teachers' math anxiety and confidence in understanding and teaching
mathematical concepts.
Mathematics Anxiety and Neurological Factors
Mathematics anxiety is trait anxiety and differs from test anxiety and state anxiety. "State
anxiety" is a short-term reaction to a traumatic experience, while "trait anxiety" is a more
enduring personality attribute (Saviola et al., 2020). It differs from dyscalculia, a specific
learning disability that affects the development of arithmetic skills. A brain imaging study
focusing on adults and comparisons of brain response between groups with high and low levels
of math anxiety found that high-level math anxiety is linked to more significant activity in brain
areas that process danger and pain (Hartwright et al., 2018). Elevated levels of math anxiety were
also linked to a depletion of working memory during complex mathematics tasks (2018).
Working memory is a brain function that affects how we process, utilize, and remember
information (Child Mind Institute, 2021). Working memory is required to remember a phone
number, directions, and math facts and procedures. Working memory is like a mental file folder
that holds all the information we need to recall quickly (2021). Some anxiety is favorable to
performance, but high levels can undermine thought processes (Hebb, 1955).
Working memory, flexibility in thinking, and self-regulating composure are all
controlled in the brain. Problems with cognitive processing can impede focus and control over
emotions. Emotions can cloud thinking and stop processes necessary for carrying out
mathematical procedures. Emotions heighten math anxiety. People with high mathematics
TEACHERS’ PERCEPTIONS OF MATHEMATICS 17
anxiety will be less proficient at computation and less likely to utilize effective strategies or
make connections between mathematical concepts (Omoniyi-Israel, & Olubunmi, 2014).
Ashcraft and Kirk (2001) delved into math anxiety's impact on mathematical thinking and
processing. They found that when regrouping was added to a computational task, it temporarily
overloaded participants' working memory, and their performance dropped dramatically.
Furthermore, lower accessible working memory capacity was linked to higher levels of
arithmetic anxiety, but not permanently, but as a transitory functional loss in processing capacity
(2001).
Lauer et al. investigated math and spatial anxiety in children during elementary school
years (2018). Prior research suggested that general anxiety accounted for the impacts of math
anxiety on math performance in school-aged children (Hill et al., 2016). Lauer et al. found
gender disparities in math and spatial anxiety across the board and domain-specific anxiety as a
distinct predictor of children's performance on arithmetic and spatial tests (2018). Additionally,
the findings demonstrated the need for educational interventions to reduce math and spatial
anxiety, particularly in females, implying that such treatments may be most advantageous if
implemented in the initial years of school (2018). In summary, math anxiety is a phenomenon
that temporarily blocks the ability to think through challenging aspects of mathematics.
Mathematics Anxiety and Mathematics Achievement
High math anxiety is frequently associated with low math achievement. Recent research
primarily examines the potential factors influencing the association between math anxiety and
math achievement. Studies indicate that as America's math phobia increases, achievement and
opportunities for success decrease. Ramirez et al. (2013) conducted a study to assess math
TEACHERS’ PERCEPTIONS OF MATHEMATICS 18
anxiety in 154 first and second-grade students. They found that math anxiety appears in students
as early as first and second grade. More than half reported varying levels of math anxiety during
the assessment. The results revealed a strong correlation between high levels of math anxiety and
poor performance. The higher the anxiety, the lower the score (2013).
Cargnelutti et al. (2017) conducted a study to advance other recent studies associated
with early childhood mathematics anxiety. Cargnelutti et al. examined whether math anxiety
impacts early math proficiency and gathered information from children in Grades 2 and 3 to
identify existing and developing patterns of anxiety (2017). Overall, the findings suggested that a
proactive response to math anxiety in early childhood is crucial to preventing negative
experiences in subsequent grades, possibly due to early learning gaps (2017).
A study conducted by Claessens and Engel (2013) investigated whether beginning
kindergarteners with low math skills predicted subsequent outcomes for students as they
progressed through school. Researchers measured students' kindergarten mathematics
proficiency levels and other markers of school performance in eighth grade and at different
stages throughout primary school. The study revealed that children with poor math skills are
nearly twice as likely to be Black or Hispanic (nearly 40%) compared to only 24% of the overall
population (2013). Data collection and analysis revealed that early mathematics knowledge and
skills are the strongest determinants of later math achievement and achievement in other content
areas. "Interestingly, these results indicate that kindergarten entry math scores on Proficiency
Levels 1 and 2 are more predictive of math and reading achievement in eighth grade than is the
reading test score at kindergarten entry" (2013, pp. 13-14). The results acknowledge that low-
TEACHERS’ PERCEPTIONS OF MATHEMATICS 19
income schools need strong teachers in early childhood years to ensure success in upper
elementary and secondary levels.
Several studies in the United States indicate that ill-prepared teachers are more likely to
work in low-income schools (Tröbst et al., 2018). Research also indicates that American teacher
preparation programs draw from a weaker pool of future mathematics teachers since the
population generally performs poorly on international K-12 mathematics examinations (2018).
"The fact that early mathematics knowledge and skills are the most important predictors not only
for later math achievement but also for achievement in other content areas and grade retention
indicates that math should be a primary area of academic focus during the kindergarten year"
(Claessens and Engel, 2013, p. 23). School districts often focus the majority of professional
learning and support efforts on the "tested" grades rather than concentrating on preemptive
measures in the early childhood years. Confident and well-prepared teachers are vital to
strengthening the foundation for early childhood mathematics. Children with gaps in learning
who are also anxious about mathematics find it exceedingly more difficult each year they spend
in school.
A recent meta-analysis of the relationship between math anxiety and math achievement
was conducted by Barroso et al. (2021) and discovered a strong link between math anxiety and
math achievement, implying that high mathematics anxiety results in poorer academic
achievement. Math anxiety may also impact particular brain functions such as working memory,
flexibility of thought, and self-control (Ashcraft and Kirk, 2001). Issues with neurological and
cognitive processing in the brain may decrease focus and control of emotions, influencing
student achievement outcomes (Beilock and Willingham, 2014). Math anxiety may affect
TEACHERS’ PERCEPTIONS OF MATHEMATICS 20
competence in math-related circumstances. The neurological shutdown may also result in long-
term consequences when people are anxious about math (Beilock, 2019). Low math skills,
course selection, and even occupational choices are associated with mathematics anxiety
(OECD, 2020). Because of its link to math achievement, math anxiety is crucial when improving
math experiences. Students' past and recent test scores demonstrated a need to examine
mathematics instruction and 5th-grade teachers' attitudes toward mathematics.
Vanbinst et al. (2020) suggest that mathematics anxiety results from a complicated
interaction between nature and nurture. In nature, the brain reacts to anxiety by inhibiting the
individual from efficiently attending to a task or assessing numerical information (Corbetta and
Shulman, 2002). Even the anticipation of participating in mathematics activities activates the
same neural centers in the brain that register threats and physical pain (Corbetta and Shulman,
2002). Data from Vanbist et al. pointed to a complicated familial basis for mathematics anxiety.
People with math anxiety take fewer math courses, earn lower grades in the classes they
take, and demonstrate lower math achievement (Ramirez et al., 2018). Some teachers' negative
perceptions about mathematics may be exacerbated by instruction focusing on rote memorization
and procedures over conceptual learning from early childhood experiences in mathematics
(Geist, 2010). The realization that children with negative math perceptions could become
elementary teachers with negative mindsets toward mathematics needs to be considered to create
positive change. In this study, the researcher explored the origins and impacts of math anxiety in
teachers and how their current perceptions of mathematics influence instruction.
The literature portrays a convincing argument that more investigation is needed on the
origin, progression, and mode of action of mathematics anxiety in children and its impact on
TEACHERS’ PERCEPTIONS OF MATHEMATICS 21
achievement. It is also evident that children from lower-income situations may enter
kindergarten without a firm grasp of basic mathematics understandings. This demonstrates the
need for equal access to knowledgeable and confident teachers to avoid the onset of math anxiety
in early childhood.
Self-Perceptions of Mathematics
Poor attitudes toward math have plagued our nation for quite some time (Looney et al.,
2017). According to a Stanford study, a positive attitude toward math enhances the brain's
memory region and positively impacts math achievement even when anxiety is present (Chen et
al., 2018). A negative attitude towards mathematics may cause evasion of classwork, disorderly
behavior, and mathematics anxiety (Dossel, 2016). Avoiding classwork and misbehaving will not
improve students' foundational mathematics knowledge to compete for higher-paying jobs. The
strong connection between mathematics skills and wages is represented in the United States
(OECD, 2020). Students who struggle in mathematics have limited career choices as young
adults. Even students who perform well in mathematics courses are unlikely to enjoy
mathematics into adulthood (Browning et al., 2014). As a result, individuals with math anxiety
continue to struggle with mathematics as adults and reinforce their negative beliefs about their
abilities in mathematics (Jameson, 2014).
Female students' confidence and achievement levels are more likely to be negatively
impacted by teachers with high math anxiety than male students (Beilock et al., 2010).
According to a study conducted by Cvencek et al. (2011), math gender biases occur early in life
and exhibit a distinct impact on boys' and girls' perceptions of mathematics. Researchers found
that elementary school girls demonstrated a weaker association with mathematics through
TEACHERS’ PERCEPTIONS OF MATHEMATICS 22
implicit and self-report measures than boys (math self-concept) (2011). This shows that the
math–gender stereotype emerges early in life.
Robinson-Cimpian et al. (2014) collaborated to explore how teacher perceptions may
influence the relationship between gender and mathematics achievement. The findings suggested
that teachers perceive girls' mathematics abilities equally with similarly achieving boys only if
they believe the girls exhibit more effort and behave better than the boys (2014). More typically,
teachers ranked girls' mathematical prowess lower than boys', which may account for the
widening mathematics gender gap favoring males at the elementary level (2014).
Bafflingly, some college students pursue careers in elementary education, knowing that
they lack mathematics content knowledge and experience mathematics anxiety (Stoehr, 2017).
Stoehr's investigation included three female preservice teachers who expressed worry about
teaching elementary mathematics. They experienced mathematics anxiety about teaching
mathematics to k-12 students and found difficulty with math methods courses required in their
teacher preparatory programs (2017). Unfortunately, gender-specific mathematics anxiety
impacts students who pursue teaching careers, where 90% of the workforce is female (United
States Department of Education, 2017). Elementary teachers, unlike secondary, are held
responsible for acquiring a solid understanding of all content areas they teach. Teachers may
gain knowledge through professional learning opportunities, graduate courses, or research.
Mathematical Content Knowledge
If teachers feel that they know the mathematics content well, they are more confident in
teaching math and imparting a positive attitude (Geist, 2015). The Survey of Adult Skills
(PIAAC) was administered to 5,010 people in 15 countries worldwide (2019). The results
TEACHERS’ PERCEPTIONS OF MATHEMATICS 23
indicate that almost one-third of individuals tested in the United States have a numeracy score
below level two (2019). Level two includes proficiency with whole numbers, simple decimals,
percent, and fraction calculations. It also includes measurements, estimation, simple data
analysis, and probability. A solid grasp of elementary mathematics content is crucial for teachers
of young children.
In the United States, a third of adults scored below a level of mathematics that contained
elementary concepts. This deficit suggests a critical need for knowledgeable mathematics
teachers in early grades and upper elementary. Research in mathematics content knowledge is
limited (Gresham, 2018). According to research conducted by Thanheiser et al., preservice
teachers' knowledge of whole number operations may be inadequate and grounded in knowledge
of standard algorithms alone (2014). A study conducted by Chen et al. suggests that teacher
confidence varies with specific math content knowledge and teaching and assessing mathematics
(2014).
Antonelli (2019) conducted a mixed-methods study including kindergarten through fifth-
grade teachers. Her study investigated teachers' perceptions of their technical abilities,
mathematics content understanding, pedagogy, and readiness to use technology integration in
their classrooms. In the past, expectations for teaching and learning mathematics were
substantially different, according to most of the participating teachers in the study. Their
educational experiences and academic achievements influenced participants' attitudes toward
mathematics. They stated that past experiences impacted their current confidence levels as
mathematics teachers. Although similar in some aspects to Antonelli’s, this study delved into
teachers’ perceptions about math instruction and confidence with the concepts they teach. It
TEACHERS’ PERCEPTIONS OF MATHEMATICS 24
explored the earliest childhood memories of mathematics interactions. It focused on how well-
prepared teachers felt to teach mathematics and what mathematics methods courses they took in
college. Additionally, teachers recalled what experiences influenced their current feelings about
mathematics and how that has impacted their current instruction.
Antonelli's quantitative data analysis revealed that teachers' perceptions of mathematics
content knowledge were strong. However, qualitative data suggested they were primarily
focused on rote skills and procedural actions rather than deep learning of concepts. Participants
believed they had a solid grasp of fundamentals but struggled with problem-solving, conceptual
knowledge, and teaching various math strategies. Similarly, some teachers in the current study
felt confident about teaching mathematics, but observations revealed that the instruction was
primarily procedural in those classrooms.
If teachers have a poor understanding of mathematics, they may teach students discrete
procedures that exclude the mathematical thinking required for conceptual understanding
(Thomas & Hong, 2012, as cited in Antonelli, 2019). Teachers are expected to apply conceptual
and procedural knowledge in various contexts. The new expectations and demands of instructing
and mastering mathematics are significantly demanding, and teachers perceive this as a
significant paradigm shift and steep learning curve for which they are unprepared (Antonelli,
2019).
Teacher education programs must provide continual professional support to mathematics
preservice and in-service teachers and determine specific contexts in which the level of math
anxiety can be decreased (Thanheiser et al., 2014). Sun (2017) analyzed teacher interactions with
others and learning communities' participation to understand more clearly how mathematics
TEACHERS’ PERCEPTIONS OF MATHEMATICS 25
teachers construct their identities. The study aimed to learn more about the relationship between
mathematics teacher identities and professional development involvement, specifically how
teachers' identities influenced how they participated in learning communities and professional
learning opportunities (2017). Sun found that teachers were aware of the need to continuously
develop their content knowledge and instructional strategies to stimulate, analyze, and respond to
student thinking and reasoning (2017). However, Sun found that it was difficult for teachers to
be wholly committed to mathematics professional learning when they disagreed with the school's
vision and did not have a clear and consistent understanding of the goals of the professional
development sessions (2017). For mathematics professional learning communities to thrive,
characteristics must include collaborative participation in professional learning, commitment to
the school's goals, and teacher input into the topics for professional learning (2017).
Teachers must model the thinking process necessary for understanding mathematics
concepts. Students must be taught how to think and persevere through problem-solving and make
logical decisions to solve and represent problems. In a study comparing constructivist and
traditional methods of instruction, Alsup (2005) found evidence of a strong interaction between
content knowledge and instruction. Preservice teachers with math anxiety took a semester-long
course focused on conceptual learning. Teachers that took the course became less anxious about
math content and gained self-efficacy (2005). However, the control-group students took a more
traditional mathematics course. They experienced the steepest decrease in math anxiety of all
participants, indicating that the overall decrease in math anxiety was likely due to the instructor's
clarity and teaching style rather than the course itself (2005). Teachers' success with the course
was primarily due to the instructor's ability to communicate and clarify mathematical ideas, the
TEACHERS’ PERCEPTIONS OF MATHEMATICS 26
emphasis on deep conceptual understanding, and the interconnectedness of mathematical
concepts. The instructor's use of various representations and approaches to problems may have
had the most pronounced effect on students' mathematics anxiety and teaching efficacy (2005).
The math course was necessary for teachers' success because math anxiety has
devastating effects on learning mathematics content. According to research with preservice
teachers, negative experiences in the past potentially lead to mathematics anxiety at some point
in students' academic journey. The major source of occurrences is linked to the behavior of the
teachers from students' past school experiences (Bekdemir, 2010). The consequences of these
negative experiences increasingly worsen as students proceed through school (2010). Teacher
education programs should include proactive support to decrease or prevent the cyclical
perpetuation of anxiety before preservice teachers graduate. Research findings regarding the
continuance of math anxiety in our nation raise the question of the role of teacher education
programs in mitigating negative perceptions of mathematics among preservice teachers (Looney
et al., 2017).
Student Anxiety in Mathematics
Mathematics anxiety is measured through the level of enjoyment associated with items
having to do with making good grades or the level of comfort when doing mathematics work
(OECD, 2010). Research shows that anxiety toward mathematics differs by grade level, and
anxiety toward assessment is higher among middle and secondary levels. Escalera-Chávez et al.
(2016) used a unique scale to quantify math anxiety in their research. Test anxiety, anxiety about
temporality, anxiety toward understanding mathematical issues, anxiety about numbers and
mathematical operations, and anxiety toward real-life mathematics situations were among the 24
TEACHERS’ PERCEPTIONS OF MATHEMATICS 27
items on the scale. Escalera-Chávez et al. discovered that math anxiety among high school
students was linked to worry associated with assessment outcomes (2016).
In another study, mathematics anxiety was most common among second graders and
found less often in fifth graders (Sorvo et al., 2017). More recent studies confirm that math
anxiety is linked to math achievement and math self-esteem in early school-age children
(Szczygieł, 2020). Furthermore, findings by Szczygiel also indicate that math anxiety in children
is a distinct type of anxiety, separate from general test anxiety. A study conducted by Foley et al.
(2017) suggested that the better a student is in math, the more intensely their performance will be
diminished by anxiety. Math anxiety and the brain's emotional system interfere with students'
ability to retrieve information during a test, so they perform much worse than they would if they
were not anxious (2017). Furthermore, the study indicated that the relationship between anxiety
and achievement occurs in the United States and worldwide (2017).
According to data from 2018 PISA results, in several countries, including the United
States, students scored higher in reading when they perceived their teacher as more enthusiastic,
especially when their teachers seemed interested in the subject (OECD, 2020). While Finland
excels academically on PISA, it has a low equity ranking (Sahlberg, 2021). In 2015, the
country's equality scores for boys and girls and immigrant students were below the OECD
average (2021). The United States, on the other hand, fared about average in terms of gender
parity among boys and girls and slightly better than average in terms of immigrant students
(2021).
In Finland, only nine compulsory school years are required for students (Sahlberg, 2021).
According to the National Center for Education Statistics (2017), compulsory age limits in the
TEACHERS’ PERCEPTIONS OF MATHEMATICS 28
United States range between sixteen and eighteen. PISA tests children enrolled in a school at
grade 7 or higher between the ages of 15 and 3 months and 16 and 2 months (OECD, 2020). In
Finland, Upper Secondary School students prepare for the Matriculation Test, which determines
whether they will be admitted to a university after three years. This choice is typically based on
their achievements during primary education. The other path for Finnish students is vocational
training for various non-university careers with the option of taking the Matriculation exam after
three years of training. In new research from Pulkkinen & Rautopuro(2022), most, but not all, of
the PISA students in Finland are in the ninth grade, meaning they are at the last grade level of
primary education before Upper Secondary School or vocational training. In America's present
educational system, students between fifteen and sixteen, regardless of achievement or
socioeconomic level, may be eligible for the PISA examination.
Finland's policies work because it is a small country with a relatively homogeneous
population; however, comparable changes may be difficult to implement in large countries with
vast social differences and immigrants or English language learners (Hendrickson, 2012).
Furthermore, Finland's changes go beyond the classroom, with all students receiving free health
care, nutrition, counseling, and further education, removing some variables that negatively affect
academic performance (2012). The United States may want to consider the differences between
the two countries' policies and the differences in the size and make-up of the populations when
comparing students’ performance on the PISA.
An OECD study in 2010 focused on students' perceptions of math teaching and learning
and the connection to performance in mathematics. According to the study, mathematical
competencies are highly connected with confidence in one's strengths in mathematics and a
TEACHERS’ PERCEPTIONS OF MATHEMATICS 29
strong sense of efficacy in meeting challenges in learning tasks (2010).The disciplinary climate
at the student and school levels, as well as total hours per week of homework, stand out as
having the most substantial effects across the majority of countries, with student use of strategies
and student-teacher relations having positive associations with mathematics performance in some
countries but not in others (2010). Research has also shown that students differ tremendously in
their teacher's perceptions (Göllner et al., 2018).
Siebers (2015) found that elementary students considered understanding patterns and
solving problems fun, but students with math anxiety began to avoid mathematical thinking and
problem-solving in upper grades. The same students were frustrated during math discourse and
were found to have low self-efficacy about math (Siebers, 2015). Math anxiety results are
apparent in students who negatively compared themselves to their classmates because they
tended to earn lower test scores. These findings indicated that it is critical to consider students'
confidence levels in mathematics when examining factors associated with achievement (House
& Telease, 2011). Students' anxiety levels can vary across grade levels, but the relationship
between frustration and low levels of confidence is apparent in those students with mathematics
anxiety.
Teacher Anxiety in Mathematics
According to recent studies, it has been found that mathematics teachers who like their
jobs have lower anxiety levels than those who do not like their jobs (Gürbüz & Yıldırım, 2016).
Many factors contribute to the development of mathematics anxiety, such as the quality of
instruction, motivation, peer influences, the method of teaching, lack of opportunities to relate
math to daily life, topics not appropriate for the cognitive level of students, the very nature of
TEACHERS’ PERCEPTIONS OF MATHEMATICS 30
mathematics, students' preconceived negative attitudes against mathematics, inadequate level of
basic mathematics, and the quality of teacher-student relationships (Gürbüz & Yıldırım, 2016).
Research has also revealed that mathematics anxiety has roots in some preservice
teachers' histories, low-performance and weak backgrounds, and a lack of positive experiences in
school. Some negative experiences included embarrassment, humiliation, shame, being dumb or
stupid in front of peers, and being afraid of speaking up for fear of being the only student that did
not understand (Stoehr, 2017). Students may sense teachers' negative attitudes toward the content
they are teaching (Dossel, 2016).
In a recent study, Hardacre et al. (2021) examined possible factors related to minority
teacher candidates' low test-passing rates on the required standardized exams for teacher
certification in California. According to survey findings, students expressed general anxiety
about taking teacher exams, particularly math exams. Respondents did not score well on math-
related multiple-choice and constructed response questions. Key findings included students'
beliefs that math test anxiety was a barrier to passing the examinations and entering the teacher
preparatory program and the teaching profession,
In a study comparing constructivist and traditional methods of instruction, Alsup (2005)
found evidence of a strong interaction between content knowledge and instruction. Teachers that
took the course became less anxious about math content and gained self-efficacy (2005). Math
anxiety was studied because it has a devastating effect on learning mathematics. Research has
revealed that preservice elementary teachers have the most significant level of math anxiety of
any college major (2005). Research findings regarding the continuance of math anxiety in our
TEACHERS’ PERCEPTIONS OF MATHEMATICS 31
nation raise the question of the role of teacher education programs in mitigating negative
perceptions of mathematics among preservice teachers (Looney et al., 2017).
Additionally, teacher preparatory programs should examine prospective teachers'
mathematics skills at the onset of college education to better understand the progression of
prospective teachers' competencies (Samuels, 2015). Elementary teachers must be well-prepared
as preservice teachers to positively affect the future of quality mathematics instruction.
Summary and Integration
Students and teachers suffer from math anxiety. Research shows that the notion of math
anxiety is common and accepted by American society (Ramirez et al., 2018). Many new
teachers, as well as veteran teachers, suffer from math anxiety or dislike mathematics. Teachers
with negative feelings about their teaching content could pass those attitudes on to students.
Students may develop a distaste for mathematics that begins early in their careers and festers as
students progress through school. Research indicates that disciplinary issues in mathematics
classrooms could also play a part in mathematics anxiety (OECD, 2020). The nature of math and
teacher responses to students' needs, teaching methods, and teacher knowledge of content could
affect students' mathematics perceptions. In the next chapter, the researcher discusses the
methodology and research design in detail.
TEACHERS’ PERCEPTIONS OF MATHEMATICS 32
Chapter 3: Methodology
Introduction
In the previous chapter, the literature connected research that suggests many students and
teachers may suffer from math anxiety. Additionally, the research indicates that some elementary
teachers may need more confidence in teaching mathematics because mathematics was a
historically weak area throughout their school years. The literature also reveals that math anxiety
is a common occurrence and is acknowledged by American society. According to Fiss, math
anxiety's history in America persists in the manner of math communication. The current written
high-stakes assessment is still a source of anxiety akin to the previous feeling of stage fright
while writing on the blackboard in front of the class (2020).
In this chapter, the author discussed the purpose of the study, the research questions, the
research methodology, and the study's design. Also included in this chapter is information about
the sample represented in the study. In addition, the author addressed the limitations and ethical
concerns of the research. A brief and concise synopsis was provided to summarize the chapter.
Purpose of the Study
This study aimed to determine how 5th-grade teachers' experiences and perceptions of
mathematics influenced mathematics instruction. The researcher examined the literature to
determine if there was evidence of teacher anxiety towards mathematics and math instruction
and teachers' lack of content knowledge and math achievement. The researcher also considered
other factors that might impact teachers' perceptions of mathematics and included literature
suggesting that other extraneous variables contribute to anxiety in mathematics learning and
instruction. The research questions were crafted to explore the roots of math anxiety and whether
TEACHERS’ PERCEPTIONS OF MATHEMATICS 33
this phenomenon guided teachers' perceptions of mathematics. The questions also aimed to
investigate whether teachers' understanding of mathematics content influenced math instruction.
Research Questions
Question 1: How do 5th-grade teachers' experiences and self-perceptions of mathematics
influence mathematics instruction?
Question 2: How do 5th-grade teachers' understanding of mathematics concepts influence
instruction?
Research Design
By its very nature, qualitative research is exploratory and descriptive. Additionally, it is
utilized to delve deeper into topics and investigate intricacies tied to the topic under
investigation. This type of “research has a long history of living with the criticism that it engages
in some revealing theorizing based on evidence that would otherwise not satisfy traditional
criteria” (Gioia, 2017, p. 455). The qualitative approach assumes that organizational phenomena
were socially built by individuals who understood what they attempted to do and explained their
thoughts, intentions, and behaviors (Gioia, 2017). Gioia et al. (2013) state that a qualitative
researcher seeks a credible, defendable explanation of the how and why of a phenomenon.
For this study, a qualitative phenomenological design was utilized. This approach used a
structured interview protocol, open-ended questionnaire, and observations to answer the research
questions. Phenomenological studies help researchers better understand and describe the impact
of specific experiences and perceptions of individuals. Phenomenology attempts to explain the
TEACHERS’ PERCEPTIONS OF MATHEMATICS 34
meaning of people's experiences. Phenomenological investigations look into what people have
experienced and focus on their feelings about the phenomena (Groenewald, 2004). The
researcher's goal is to describe the phenomenon as precisely as possible, avoiding any prenotions
while remaining factual (Groenewald, 2004).
The researcher focused on a phenomenon in mathematics instruction in which a small
group case study included three sources from which data was accessed (Igbol, 2021). The
qualitative data collection and analysis process included combining the data and identifying links
to the literature review and analysis. The purpose of the design was to collect cogent data from
the interviews, the questionnaire, and observations to bolster the validity of the findings. The
research questions focused on the experiences and perceptions of teachers and required
qualitative data collection to provide a chronicle of authentic embodiments from participants.
The inductive nature of this qualitative approach necessitated vignettes from teachers'
experiences, attitudes, and instructional behaviors associated with elementary mathematics. The
research utilized qualitative surveys, interviews, and observation data to capture the essence of
participants' feelings and actions fully. The qualitative surveys and interviews aligned with the
research questions' purpose. The interview questions probed the participants' experiences in
mathematics, their current feelings toward mathematics instruction, and information about their
previous and teacher preparatory mathematics courses. The survey questions investigated
teachers' current math instruction to include enjoyable and frustrating concepts. The observations
were scripted with teacher actions and practices during instruction to investigate connections
between verbal and written accounts with actions and observable behaviors.
TEACHERS’ PERCEPTIONS OF MATHEMATICS 35
Sources of Data (Qualitative)
The data sources consisted of interview questions written by the researcher, a modified
observation protocol that utilized sound instructional practices found in the Massachusetts
Curriculum Framework (2012), and open-ended survey questions created by the researcher. The
articles, books, and dissertations were obtained by utilizing Southern Wesleyan's library online
resources, "ONE search" and "ProQuest Dissertations and Theses." Other articles and books
were gleaned from Research Gate, the Association for Supervision and Curriculum Development
(ASCD), and Phi Delta Kappan (PDK).
Data Collection
A successful phenomenological study must focus on the various ways information is
extracted from respondents. In a qualitative phenomenological research design, the focus is on
the research questions. The researcher must develop rapport without influence to understand the
participants' experiences thoroughly. (Essential Guide to Coding Qualitative Data, n.d.). In this
phenomenological case study, in-depth research supported the understanding of the group in
their actual situations. The goal of combining several data sources was to learn more about
different elements of the phenomenon (Maxwell, 2013). This broadened the scope of topics
covered. Interviews helped to understand instructors' historical experiences and opinions on
mathematics, while observational data looked at the behavior of teachers and students in the
math classroom setting. The survey included questions about mathematics instruction, which
provided valuable information about particular areas of instruction, content teachers felt
confident about, and weaknesses that needed growth.
TEACHERS’ PERCEPTIONS OF MATHEMATICS 36
Qualitative data was collected by recording responses from the online open-ended survey,
interviews, and observations. Participants received an email containing a detailed explanation of
the study and how data would be collected. A link to surveyplanet.com was provided to
respondents to complete the survey. The online questionnaire contained ten open-ended
questions constructed by the researcher to generate written responses from participants. The
purpose of the questions was to elicit perspectives and experiences from respondents about
mathematics content and teaching mathematics. The open-ended survey was used as a prelude to
the structured interviews. The data gathered from the survey aided in identifying initial themes.
The survey took respondents approximately 20-30 minutes to complete.
The interviews were a structured type of questionnaire conducted verbally. Six of seven
teachers scheduled time during their planning periods to meet via Microsoft Teams for the
interview. Planning times varied from 7:45 a.m. to 1:45 p.m. One teacher completed the
interview through Microsoft Teams after work hours at home. In the structured interviews, the
responder answered a set of predetermined questions. During the interview, the researcher did
not elaborate or ask further questions for clarification from respondents. The constancy of the
interview questions also allowed for easy comparison and analysis of the results. The interviews
took twenty to thirty minutes to complete. Teachers allowed the researcher to record the
interviews using an iPhone but declined the video recording through Microsoft Teams.
The researcher created a protocol for observing mathematics classroom content and
practice using a "look fors" document (see Appendix D) from the Massachusetts Curriculum
Framework (MCF) for guidance. The information from the MCF document was used to craft the
observation protocol that focused on the instruction and assessment domains. The "look fors"
TEACHERS’ PERCEPTIONS OF MATHEMATICS 37
included what the teacher and students did during mathematics instruction. Teachers provided
schedules to the researcher to avoid testing or other situations compromising instruction. Each
teacher was observed once for thirty to sixty minutes during the mathematics block. The
researcher observed from the back of the classroom to minimize disruption to instruction. The
researcher recorded audio from the observation using an iPhone to ensure transcription accuracy.
The observations revealed further information about the teacher's mathematical and conceptual
understanding and instructional practices. The observation data “primarily relied on descriptive
field notes” (Maxwell, 2013, p. 89) and transcriptions from the audio recordings. The data was
gathered and housed within the participating school district on an encrypted external drive.
Data Analysis
The researcher read the collected data from notes and questionnaire responses, then
compared them with transcriptions while listening to audio recordings. Every word, pause, and
stutter was recorded. All filler words and unintelligible utterances were later omitted to provide
more clarity to the reader without losing the integrity of the response passages.
Questionnaires
The analysis process began immediately upon completing the surveys with assistance
from two colleagues with research experience. The first step in the qualitative analysis was to
read the survey responses. The research team read all of the survey responses and took note of
similar responses and outliers. The team chose highlighter colors to develop overarching
categories and assign the data to more refined groupings. Next, a diagram was created to
visualize the themes in an organized manner. The visual provided the researcher with a
TEACHERS’ PERCEPTIONS OF MATHEMATICS 38
simplified and truthful account of the responses. An organizational chart in Table 1 demonstrates
categorizing and coding of the survey data.
Table 1
Questionnaire data chart
Interviews
After each interview, the researcher listened to the responses with the corresponding
notes taken to give an accurate and complete description. Next, the qualitative interview data
were transcribed and reread. Notes were taken to gather any missed information from the first
read. The researcher provided a transcription generated from the Google tool "Voice Typing" to
the two supporting colleagues for review and comparison with the audio recordings and notes.
As with the surveys, the research team examined the data to categorize and code the significant
themes. The data was then sifted to reveal unique nuances between responses.
The researcher and colleagues also listened to recordings of the interviews for
comparison with notes to improve the accuracy of transcriptions. The researcher and colleagues
TEACHERS’ PERCEPTIONS OF MATHEMATICS 39
looked for patterns, similarities, differences, and connections from the memos and notes to
categorize the data into meaningful chunks of information. The information was placed in a
graphic organizer to begin the open coding process. Patterns of phrases and words associated
with respondents' experiences and perceptions of mathematics and any outlying data contained in
the results were sorted during the process. The researcher and colleagues determined which
categories fell under areas that provided substantial, theoretical, or organizational evidence
(Maxwell, 2013). Organizational categories provided the big picture and helped drill down
information into meaningful topics (2013). Substantial categories were more descriptive and
aligned closely with respondents' words (2013). Theoretical categories included information
representing similar results from prior studies found in the literature review (2013). Table 2
contains the analysis process of the interview data.
Table 2
Interview Response Themes
Results Categories
Similar Responses
Outlier Responses
Mathematics experiences before school
Counting games
Learning to count from
1-10
Mathematics experiences during school
Struggles with number sense,
positive experiences with teachers
Montessori
experiences, problem-
solving
High School Math Courses
Basic Algebra and Geometry
courses
Calculus, Probability,
Statistics,
Trigonometry
TEACHERS’ PERCEPTIONS OF MATHEMATICS 40
Teacher Preparatory Math Courses
College level mathematics
involving Algebra and Geometry
Manipulatives training,
connecting children’s
literature with math
Subject Area Strengths
Mathematics, Social Studies,
Science
Reading, Phonics
Influences on Mathematics Instruction
District professional development,
mentor teachers
Montessori Training,
previous mathematics
employment
experience outside of
teaching
Mathematics Weaknesses
Fractions, Decimals, Metric
Conversions, deeper understanding
of math concepts, using concrete
models to demonstrate math concepts
Number Lines
Mathematics Strengths
No similarities
Finding gaps in
understanding, pacing,
enthusiasm, teaching
procedures, teaching
the CRA sequence, and
scaffolding
Confidence in Mathematics Instruction
Pretty good, willing to learn more
Very Strongly, I like
when kids make
connections
Observations
The researcher read observational notes and listened to recordings of the observations.
Part of the process involved "writing memos on what was seen and heard in the data to develop
tentative ideas about categories and relationships'' (Maxwell, p. 105, 2013). The observational
data provided comparable information relating to the second research question, which focuses on
teacher content knowledge and how that might influence instruction. In the organization and
analysis of the observation data, the researcher focused on clearly delineated elements from
TEACHERS’ PERCEPTIONS OF MATHEMATICS 41
Massachusetts's Department of Elementary and Secondary Education Observation Protocol
(2012) (see appendix D). Since the protocol focused on specific elements, deductive coding was
used to analyze the observed data. The codes arose from the observation protocol "look fors."
The research team reviewed the data and aligned the observation excerpts with the codes. In
Table 3, an (x) represents the "look fors" observed during an instructional period. This chart
provides a clear picture of the frequency of observances from each theme.
Table 3
Observation Themes
Note: (x)= observed during instructional period
Study Population and Sample Selection
The researcher employed purposeful sampling to ensure the representation aligned with
the research focus. The participating district approved twenty schools for the research. Three of
the schools volunteered to participate in the study. The schools included one school from a rural
area, one Montessori school from a suburban area, and one from an urban area. Eighty-seven
percent of the students in the rural school are African American, five percent are Hispanic, and
TEACHERS’ PERCEPTIONS OF MATHEMATICS 42
two percent are white. One hundred percent of students from that school are from low-income
homes. Only fifteen percent of fifth-grade students in the school passed the state mathematics
exam in 2021 (GreatSchools, n.d.).
The Montessori school’s demographics include seventy-five percent white, five percent
Hispanic, and fifteen percent African American students. One hundred percent qualify for free or
reduced lunch. Seventy-three percent of students passed the state mathematics exam in 2021.
The Montessori school is in a suburban area. Students at this school are making significant
academic gains in mathematics and are outperforming peers at other schools across the state
(GreatSchools, n.d.).
The urban school’s demographics include ninety-four percent African American students,
Four percent with more than one race, one percent Hispanic, and less than one percent white.
One hundred percent of families qualify for free or reduced lunch. Twenty-four percent of
students passed the state mathematics exam in 2021(GreatSchools, n.d.).
The participants in this study included five female and two male certified 5th-grade
teachers. Participating teachers were current instructors of 5th-grade math from the 2021-2022
school year. The teachers ranged from almost two years to 14 years of experience. Forty-three
percent of the participating teachers were African American, and fifty-seven percent were white.
The participating school district population spans all socioeconomic levels. The data was
gathered and housed within the participating school district. Table 4 displays the participant’s
demographic information.
TEACHERS’ PERCEPTIONS OF MATHEMATICS 43
Table 4
Demographics of Participants
Ethnicity
Sex
Highest
Educational
Degree
Years
Teaching
Years
Teaching 5th
grade Math
African American
F
Bachelor’s
3
2
African American
M
Bachelor’s
4.5
2
White
F
Bachelor’s
2
1.5
African American
M
Bachelor’s
2
2
White
F
Master’s
5
3
White
F
Master’s
10
2
White
F
Master’s + 30
14
10
Ethical Considerations
Ethics in a qualitative study required that participants are informed and give voluntary
consent. Teachers' names were not used in the study to protect confidentiality, and no participant
was harmed in any way. Before the researcher approached people to participate in the study, they
were given detailed information about the study's purpose and possible benefits from the results.
The researcher made a list of actions needed to accomplish the research goals. The researcher
also carefully examined options to clarify that they were ethically sound. The researcher then
TEACHERS’ PERCEPTIONS OF MATHEMATICS 44
identified how participants might be affected by any portion of the study to be confident that no
one would be harmed.
For the study to be equitable, the researcher ensured that gender, race, and socioeconomic
levels were fairly represented. There are nearly 30 elementary schools in the participating
district, ranging from rural to urban, with varying socioeconomic levels in each. Fifty-two
percent of teachers are African American, and forty-eight percent are Caucasian. To be equitable,
the researcher strived to be strategic about choosing the teachers' population to represent the
district accurately. The participating teachers in the study included forty-three percent African
American and fifty-seven percent white. There were fewer African American and more white
participants in the study, with almost a ten percent difference from the district percentages.
Summary
In this chapter, the research method and design were disclosed and outlined in this
chapter. The critical points included explaining the population studied, the research design, and
the instrumentation utilized to collect data. The researcher explored characteristics and
experiences that might affect teachers' views of mathematics instruction and content. The goal of
the research questions was to investigate the mathematical memories of teachers and determine if
those recollections had a lasting effect on how teachers currently viewed mathematics instruction
and content. The qualitative design of the study supported the rationale to examine the human
reaction to positive and adverse revelations about mathematics. This design allowed the
researcher to explore different feelings and attitudes about teaching mathematics content.
The population of participants embodied the primary goal of examining different views
and experiences. The participating schools varied in demographic data, excluding income levels
TEACHERS’ PERCEPTIONS OF MATHEMATICS 45
which were the same in each school. Data collection processes enabled the research to gather
comparable details applicable to the research questions. In addition, the research questions were
reiterated and included to allow the reader to connect the questions and research design. The data
was analyzed in Chapter four, and a detailed summary of the findings was presented.
TEACHERS’ PERCEPTIONS OF MATHEMATICS 46
Chapter 4: Data Analysis and Findings
Introduction
Research reveals that some teachers suffer from mathematics anxiety (Stoehr, 2017).
There is limited extensive research on how teachers' math anxiety influences instruction
(Ramirez et al., 2018). This study examined how fifth-grade teachers' experiences, perceptions,
and confidence in mathematics may influence math instruction. The overarching questions
utilized in the study are:
Question 1: How do 5th-grade teachers' experiences and self-perceptions of mathematics
influence mathematics instruction?
Question 2: How do 5th-grade teachers' understanding of mathematics concepts influence
instruction?
Chapter four contains the results of the case study conducted with fifth-grade
mathematics teachers. In this phenomenological approach, the researcher completed the
qualitative data collection by conducting virtual interviews first, gathering the questionnaire data
next, and ending with a face-to-face observation of math instruction.
The organization of chapter four begins with a description of the qualitative data
collection process in the study. The next section of the chapter is devoted to qualitative analysis,
which includes the transcription of interviews, organization, and coding of the data. Also
discussed in the chapter is the process used to analyze transcripts from the interviews to uncover
codes and themes. The researcher and colleagues teamed up to evaluate the teacher questionnaire
and compared results with themes emerging from the interviews to organize the information into
a display. The research team analyzed the observation data and organized it into a table that
TEACHERS’ PERCEPTIONS OF MATHEMATICS 47
displayed the elements of the observation protocol. The findings accompany a summary of the
data and how the results align with the research questions. The chapter also includes samples
from the individual interviews, the questionnaire, and observations that highlight critical areas
and substantiate the analyses' organizational flow.
Sample
As stated previously, seven teachers participated in this study. All seven participants
completed the questionnaire and the subsequent interviews. Participants' names were not used in
the study. Instead, they were represented using the letters A to G. Five teachers in the study were
female, and two were male. The male teachers taught math for less than five years. Two female
teachers were upper-level Montessori teachers, and most of their students were at a fifth-grade
level. The other three female and two male teachers were in a general education setting. The
female teachers ranged in teaching experience from two to fourteen years. Three teachers were
African American, and four were White and non-Hispanic. Four teachers have a bachelor's
degree, and three teachers have a master's degree.
Data Collection
Interviews
The teacher interviews served as the primary source of research data. All interviews were
conducted virtually through Microsoft Teams and scheduled over a two-to-three-week period.
Six teachers completed the interviews within two weeks, and one teacher during the third week.
The teachers opted out of having a video recording through Microsoft Teams. The researcher
used an iPhone to make an audio recording of the interview and took hand-written notes. The
TEACHERS’ PERCEPTIONS OF MATHEMATICS 48
researcher transcribed the audio recording using Google “voice typing.” The researcher
destroyed recordings and transcriptions after the analysis and usage of passages in the paper.
Questionnaire
The teacher survey was distributed to participants via email with a link to the open-ended
questionnaire. The teachers were given three weeks to complete the questionnaire. All teachers
completed the questionnaire within three weeks. The results were listed by the date of
submission. The researcher and colleagues downloaded the results and read through each
document several times, noting similarities and differences between each participant. The
researcher and colleagues used highlighters in the coding process of labeling and organizing to
categorize results to identify themes.
Observations
The observations provided information about the second research question focusing on
mathematics content understanding and instruction. The observations were each scheduled at the
end of the interviews. The observations were completed two to three weeks after the interviews.
The naturalistic observations took place in the teachers' classroom setting. The researcher sat in
the back of the classroom and did not engage with the students or teachers during the
observations to prevent obtrusive interaction. The researcher recorded the sixty-minute
observations with an iPhone and hand-scripted all actions and comments from the teacher and
students. The Google voice typing tool was used to transcribe the observation data. The
researcher repeated the same procedures with the interview data, listening to the audio recording
while the transcribing tool typed the recordings. The researcher and colleagues stopped several
times to ensure the accuracy of the transcription.
TEACHERS’ PERCEPTIONS OF MATHEMATICS 49
Data Analysis
Interviews
To initiate the analysis process, the researcher asked one colleague with qualitative
research experience and one doctoral candidate studying coding and wanted to experience the
process. Both colleagues agreed to assist the researcher with the interviews, survey, and
observation data. The two colleagues are women. One received her Doctor of Philosophy in
educational leadership, and the other pursued her Educational Doctorate in Leadership. The
researcher provided the colleagues with copies of the transcribed data from the Microsoft Teams
interviews. The researcher recorded the interviews with a cell phone rather than utilizing the
recording feature on Microsoft Teams. To preserve confidentiality, the researcher transcribed the
interviews using a Google tool called “voice typing.” The recorded interview played while the
application typed and transcribed the interviews. Although Microsoft Teams was used to
complete the interviews, the district policy requires that Microsoft Teams recordings are
available to staff. They upload automatically to a shared district folder. By using Google
transcription, the researcher protected anonymity.
The colleagues and researcher listened separately to the recordings and verified the
transcriptions for accuracy. The three team members compared notes, separated, and looked for
themes in the transcriptions. The researcher and two colleagues read each transcription several
times while listening to the audio recording to ensure the accuracy of the translation application.
The data included positive and negative experiences. The group reconvened to compare notes
and created the categories and subcategories noted in the data.
TEACHERS’ PERCEPTIONS OF MATHEMATICS 50
The group agreed that "Mathematics Experiences" encompassed the intent of the
questions and the resulting responses. A hierarchical organizational chart displayed the section
headings, subheadings, and responses (see Figure 1). Section headings include; education,
content, and mathematics instruction. Under the section headings, the subheadings broke the
content into smaller, more specific sections, giving readers clarity and making it easier to
understand and compare with other chart sections. There were positive and negative experiences
associated with various mathematics concepts throughout the interviews and survey. The
interview questions and responses related directly to the research questions.
The first two interview questions asked about teachers' experiences in mathematics that
related precisely to the first research question, which asked how 5th-grade teachers' experiences
and perceptions of mathematics influence their mathematics instruction. To understand how
experiences influence instruction, data needed to be collected describing the experiences that
influenced teachers' perceptions about mathematics. The first interview question asked
participants to recall their first informal memories of mathematics before entering formal
schooling. This question also provided comparable data with the second interview question that
asked participants to describe mathematics experiences from their school years. Six of the seven
teachers reported fond memories from childhood. The memories included counting money,
playing card games with siblings, counting real-life objects, and learning to count from 1 to 10.
Samples from the six teachers with positive responses are below:
Teacher A: I remember counting how many plates, forks, and knives we needed to set the
table.
TEACHERS’ PERCEPTIONS OF MATHEMATICS 51
Teacher B: I guess I remember being little and my neighbor had a garden and we used to
go over there and count strawberries. We counted just little things around the house too.
Teacher C: I mean I think coins and time. I just always like to play with coins and Daddy
let me. Daddy was an engineer, so he always encouraged you to know that kind of stuff
like asking me what time it was and playing with coins. Daddy had a collection of other
kinds of coins.
Teacher D: Before school I learned my numbers 1-10.
Teacher E: Before school, hmm, I never thought about that. I guess playing card games
with my sisters. Yeah, playing cards and other games.
Teacher F: My mom taught me colors and had me sort toys by color. She also had
Montessori type toys that we used to count and sort.
Teacher G: My mom is also a Montessori educator so growing up even before I went into
school, she told me stories of things that she would do with us at home that were very
Montessori Centered, and I remember being at the grocery store with my I parents
counting the number of items that we had in our cart and a lot of different practical real-
life things with counting
The second interview question asked participants to recall mathematics experiences as
students. Out of the seven teachers, three had negative experiences in school. A sample is
provided from each negative experience to highlight the patterns in responses. The responses
included having difficulty memorizing facts, a hate for math, and a fear of math. Transcriptions
provided the passages from teachers’ interviews.
TEACHERS’ PERCEPTIONS OF MATHEMATICS 52
Teacher A: I loved math until 4th grade. The teacher didn’t explain why or how the math
worked. I was confused from elementary to the present because I am afraid of math. I
pray that I never have to take another math class.
Teacher B: I didn't like math or have strong fact fluency. I didn’t know my math facts and
I believe that's why I'm so hard on my kids about it now because after I've gotten in the
teaching, I learned the skills. I just didn't ever really know my facts, so I didn't like math
because I didn't really have a solid understanding of my number sense at the time.
Teacher D: I hated math in school until college where I had good teachers
In the next section, excerpts from the responses described teachers’ positive experiences.
One participant enjoyed problem-solving in mathematics and the other attended Montessori
school. The responses included strong teachers, self-motivation, and the curriculum. The
responses highlight positive and negative experiences related to the first research question that
asks how experiences and perceptions may impact mathematics instruction. The interview
questions gathered data that excludes other reasons that may influence teachers’ perceptions and
strengthened the validity of the findings. Below are the positive experience responses from the
second interview question:
Teacher C: In 6th grade and I had a teacher, and she had a big old southern accent and
she said she said, “How do you not know your Lawus?” and she kept saying that, so she
ended up keeping me after school. My mother was just like “What!? You know she got
A’s at her old school” She was like “Well she needs to stay after school.” Then it turned
out she was talking about properties! - you know -the law-us was the properties. She
spent time with me. I remember her talking about you know 5 + 4 is 4 + 5 and things like
TEACHERS’ PERCEPTIONS OF MATHEMATICS 53
that. I knew that but I didn't know the terms. I had questions about math, and she didn't
shut me down.
Teacher E: I think using manipulatives got my interest and then also trying to figure out
the unknown. I like a lot of problem solving. It was really what interested me because I
like the challenge. I like trying to figure out things.
Teacher F: I really started enjoying learning about math in third grade. My third-grade
teacher was very hard, but she made math fun. Third grade is when I realized math
doesn't have to be hard or boring all the time.
Teacher G: I was a Montessori student which for me I think gave me a real concrete
understanding of math which I don't think you get if you are in a traditional school all the
time. For me because I was able to really see how the different concepts were connected
at an early age. I've always loved math and I really truly think it has a lot to do with
Montessori because no concept was taught in isolation. It wasn't just a theory or an
abstract concept. Being taught from an early age from elementary on I just really had a
concrete understanding of math which allowed me to work towards abstract.
The third interview question asked teachers what mathematics courses they took in high
school and their teacher preparatory program. The third question aligned with the second
research question, which asked how teachers' understanding of mathematics concepts influenced
their instruction. The interview question revealed the level of mathematics teachers were
exposed to in high school. It also provided information about mathematics training received
during teacher preparatory training. Two teachers took higher-level mathematics courses, and
five took the basic mathematics requirements. All seven teachers had at least one mathematics
TEACHERS’ PERCEPTIONS OF MATHEMATICS 54
methods course during teacher preparation. However, only three teachers indicated that the
course was at the elementary level. Six out of seven teachers had negative comments about the
elementary mathematics course in the teacher preparation program. A sample from the positive
response is below, followed by the negative responses:
Teacher D: I took Algebra 1 & 2. In college we took a math methods course where we
used the lattice method to multiply. It was fun and I used it with my students.
Teacher A: I took general math, business math, and algebra in high school. I didn’t take
geometry until college. I failed miserably at all but the general math. I had a tutor in high
school and in college, I guess I was just lucky. My teacher prep only included one
elementary math course, yet many reading courses. The math course was all about how
to use children’s literature to teach math. It didn’t help at all. It was mostly about
connecting a few stories to math.
Teacher B: I took algebra and geometry. In college all I can remember as far as math
goes is something entitled elementary math, but now it wasn't any elementary math in
it. It was like algebra two or three.
Teacher C: I didn't take calculus in high school, but I took the next highest algebra. In
college I took calculus and math 101. In my teacher preparatory math course, the first
day I walked in there the teacher said, “Don't worry people I hate math too.”
Teacher E: Calculus is all I can kind of remember from high school. In the teacher
preparatory program, we had a methods course, but it was not in depth and not on an
elementary level.
TEACHERS’ PERCEPTIONS OF MATHEMATICS 55
Teacher F: Algebra 1 and 2 and then throughout college we had the basic prerequisites,
but we had to take the class after that which covered how to use manipulatives and I can't
remember what we learned.
Teacher G: In high school I took a lot of math like probability and statistics. I took AP
Calculus. I also took general courses like pre-algebra, algebra stuff like that. In my
college coursework I did some more probability and statistics. I did take an education
Math course, but it is way too hard to explain. We worked with different number systems,
so we weren't working with a number system based on ten.
The fourth interview question asked participants to share experiences that influenced
their perceptions of math instruction. The question aligned with the first research question, which
asked how teachers' experiences and perceptions of mathematics influence instruction because it
divulges information about how past experiences shape teachers' current perceptions of
mathematics. The question also relates to teachers' understanding of mathematics content and
whether the experiences involved professional training. Six teachers related their perceptions of
math instruction to the district or external professional learning sessions, student teaching
experiences, and individual teachers that impacted them. One teacher felt that circumstances
outside of education influenced her math instruction.
Samples from the responses are below:
Teacher A: The most influential experiences that helped me develop understanding of
elementary math instruction came from the state department and district professional
training sessions.
TEACHERS’ PERCEPTIONS OF MATHEMATICS 56
Teacher B: Not knowing the math actually motivated me to do the work. I had a good
relationship with my teacher in student teaching and that experience helped me with
math instruction. I may not be the best at math and I might not be doing it right, but I am
learning and that's the best part about it.
Teacher C: I think a lot of my experience is that I was good at math, and I was in banking
before I was a teacher.
Teacher D: The most impact has been professional development sessions the district
provided.
Teacher E: My Montessori training involved a lot of hands-on experience. A lot of
concrete materials.
Teacher F: I really loved what we had yesterday, so it's on my mind. The external
consultant came yesterday, and I think she's helped me with the way I teach math. She
helps us with multiplying and dividing decimals. I also like talking to the other teachers
on my team and learning about what they do and how I can try to make it my own by
combining the different ways.
Teacher G: I feel like Montessori definitely has shaped a lot of my view on education as a
whole, specifically mathematics. It's even more ingrained and I'm passionate about math.
The Montessori curriculum influenced the way that I teach today because I think that's
where my passion for education is.
The fifth interview question asked what content areas the teachers' strengths were. The
question aligned with the second research question, which asked how teachers' understanding of
mathematics content influences their instruction. Teachers revealed which content areas were
TEACHERS’ PERCEPTIONS OF MATHEMATICS 57
their strongest. Six teachers included mathematics, and three teachers included social studies.
Only one teacher indicated a specific concept in mathematics. One teacher did not include
mathematics and stated that math and science were not favorites.
Teacher A: Social Studies and Reading. I don’t care much for math and science.
Teacher B: Math and Reading
Teacher C: Math and Phonics are my two strongest areas.
Teacher D: Math and Science
Teacher E: Social Studies and Math
Teacher F: Math and Social Studies.
Teacher G: Fractions are my strong suit.
The sixth interview question asked participants to describe their strengths and
weaknesses in mathematics. The interview question aligned with the second research question
that asked how teachers' understanding of mathematics concepts influenced instruction. All
teachers indicated a strength that did not include a specific mathematical domain; however,
specific domains were given as weak areas. Excerpts are shown from the responses to validate
the connection between teachers' perceptions of math content and math instruction.
Teacher A: My strength is that I use a lot of higher order questioning and real-life
examples because I didn’t have that at all when I was in elementary. My weakness is that
I still don’t feel totally comfortable with 5th grade math standards. Geometry and
Algebra standards are difficult for me. Geometry should be easy because it’s everywhere
in nature, but to me it’s very abstract.
TEACHERS’ PERCEPTIONS OF MATHEMATICS 58
Teacher B: I think I'm good at breaking down the steps. I think I can get them pretty
good. Conversions in measurement are hard for me because I don't know them as well as
I should probably be able to. Then I also struggle with fractions. I know the
procedures, but I have trouble teaching the concept.
Teacher C: I have enthusiasm and I think outside the box. I think measurement is my
weakness. I just don’t like it.
Teacher D: My strength is being able to scaffold instruction, being able to understand
students’ struggles having struggled myself. Number lines are my weakness. Especially
fractions and decimals. I struggle with teaching a number line. I do not like them.
Teacher E: I am able to translate the relationship between concrete and abstract. I make
it realistic and applicable to their lives. I think maybe explaining abstract algebraic
concepts is a weakness.
Teacher F: I think I am good at pacing. I think that one of my strong suits and I try to
relate it somehow to their lives. I think knowing when to use certain manipulatives and
which ones to use is my weak area.
Teacher G: I would say that I am really strong when it comes to math and being able to
find gaps in understanding and since I am certified first through sixth grade, I'm able to
revert back to lower grades and be able to help fill those holes for these children. I would
say an area of weakness when it comes to math and teaching math would be geometry.
That's just an area for me where I don't feel as strongly in so I guess it would just be a
weak point.
TEACHERS’ PERCEPTIONS OF MATHEMATICS 59
The final interview questions asked teachers how they felt about their mathematics
instruction. The question also aligned with the second research question, which asked how
teachers' understanding of mathematics concepts influenced instruction. It could also relate to the
first research question, which focuses on perceptions and experiences. Four teachers related their
current confidence in teaching math to the lack of understanding at varying levels in their
educational careers. Three teachers felt optimistic about their instruction.
Teacher A: I feel like I have a lot to learn about mathematics to teach it well. I do not
have confidence and I am afraid to take a college course because I have failed so often, I
just can’t see myself being successful. I need to gain confidence so I can be a better
teacher to my students.
Teacher B: I am a work in progress.
Teacher C: I feel that I am a work in progress. There is always more to learn about math.
Teacher D: I feel strongly about my mathematics instruction.
Teacher E: I feel confident about it and we follow the child so if there's ever a child that
needs to go farther than another, then I know what steps to take, and it is something I feel
pretty good about.
Teacher F: I feel pretty good about it but always open to learn more.
Teacher G: I do a pretty good job. There is always room for improvement.
The interview questions delved into the participants' feelings about mathematics and their
experiences with informal and formal mathematics. The intent was to reveal the teachers'
perceived strengths and weaknesses in mathematics. The questions also examined the level of
mathematics courses taken in high school and during college-level teacher preparatory programs.
TEACHERS’ PERCEPTIONS OF MATHEMATICS 60
The high school courses taken demonstrate the level of mathematics achievement before entering
college. The teachers' preparatory information provided a glimpse into the elementary-level math
courses provided at the college level.
Most teachers related their weaknesses in mathematics to particular mathematics domains
and their strengths to instructional practices or broader areas of mathematics. The interview
questions provided data to determine if there was a connection between math experiences,
perception, content knowledge, and instruction. In the next section, a discussion of survey data
supplies the reader with a clearer perspective of the qualitative data and a source of comparison.
Questionnaire
The research team utilized inductive coding in the open-ended responses from the survey
to discover repetitive patterns and themes (see Figure 1). Each team member read the open-
ended responses several times. The team took notes and created memos. They developed codes
from the themes, placed them into categories, and reached a consensus about unique patterns in
the data. The open-ended questions provided information regarding mathematics instruction (See
Figure 1). Figure 1 represents the data in a hierarchical chart that
combines similar themes from the interview data and the questionnaire.
TEACHERS’ PERCEPTIONS OF MATHEMATICS 61
Figure 1
Mathematics Experiences
The first question asked for teachers' highest academic qualifications. Four teachers had
bachelor's degrees, and three had master's degrees. The question related to the first and second
research questions regarding content understanding and the variety of mathematics experiences.
The second question asked teachers to provide the number of years each would attain by the end
of the school year. Five teachers have between two and five years of experience. Two teachers
will have ten and fourteen years of teaching experience by the end of the year. The question
related to the first research question about mathematics experiences.
TEACHERS’ PERCEPTIONS OF MATHEMATICS 62
The third question asked teachers how prepared they felt to teach elementary
mathematics during student teaching. This question aligned with the first and second research
questions. The question could relate to experiences and confidence in content knowledge.
Teachers used terms such as fairly, mostly, somewhat, very, or unprepared. Two teachers felt
very prepared or mostly prepared. Four teachers felt somewhat or reasonably prepared, and one
did not feel prepared. Excerpts, as seen below, show elaborative individual comments that bring
deeper meaning to vague terms.
Teacher A: During student teaching, I felt fairly prepared to teach elementary
mathematics. I knew the content of what I was teaching, but there were some occasions
that I was unsure of how to explain concepts.
Teacher B: I felt mostly prepared to teach mathematics during student teaching. I learned
a lot about teaching math during my time as a student teacher.
Teacher C: Not very prepared by the program. I loved math already but most of the other
interns did not and the Math Methods teacher started by saying "I hated math when I
became a teacher."
Teacher D: Fairly confident. I have always been strong in Mathematics, so it is the
subject I am naturally drawn to.
Teacher E: I felt somewhat prepared. I did well in math class growing up, so it came
more naturally to me but my college class about teaching math was unhelpful because it
focused on 6th grade.
Teacher F: During student teaching, I felt prepared to teach math. However, that was
mainly because the students I was teaching were prepared to learn the content.
TEACHERS’ PERCEPTIONS OF MATHEMATICS 63
Teacher G: I felt very prepared. I took an extensive elementary math methods course in
undergrad and I student taught with a teacher who only taught math.
Question four asked teachers how long they had taught 5th-grade mathematics. Six
teachers have been teaching 5th-grade math for less than three years. One teacher is in her
eleventh year of teaching 5th-grade math. Question five asked teachers to list the math concepts
they enjoy teaching. Two said they enjoyed teaching the coordinate plane, and three included
teaching fractions, multiplication, addition, subtraction, number sense, and place value. Two
teachers included geometry as enjoyable. Questions four and five are related to research question
one about experiences. Question five could also relate to research question two about
participants’ confidence in teaching concepts.
Question six asked teachers to list the math concepts they found the most frustrating to
teach. Two teachers found fractions frustrating, and one listed decimals and place value. Four
teachers included measurement; one mentioned conversion as the most challenging measurement
concept. One teacher expressed that geometry was the most difficult to teach. Passages from the
questionnaire are below to add clarity to the responses. Question six aligns with both research
questions in that participants' confidence in teaching math content and their experiences with
teaching mathematics.
Teacher A: I find teaching fractions most frustrating because it's a difficult concept for
me and the students to grasp.
Teacher B: Fractions and conversions are two things that can be frustrating to teach.
Teacher C: Measurement
Teacher D: Decimals and place value
TEACHERS’ PERCEPTIONS OF MATHEMATICS 64
Teacher E: Measurement is frustrating because there isn't enough time to teach students
and ensure they have mastery of the standards.
Teacher F: Measurement
Teacher G: At times, Geometry seems the most difficult to teach. It often feels as though I
let it slip to the backburner.
Question seven asked teachers what is needed to be successful in mathematics. Four
teachers suggested that number sense builds a strong foundation. Three teachers included basic
facts, and two included place value. One teacher indicated that finding patterns in mathematics
was integral to success. Excerpts from responses show more detailed responses to illuminate
meaning.
Teacher A: That mathematics is made of patterns connecting all things in our world
together.
Teacher B: They need knowledge of place value and number sense. They also need to be
fluent in basic number operations.
Teacher C: They need to know that everything surrounds 10. You need 10 ones to make a
ten, ten tens to make one hundred. Students need to know the addition and subtraction
and multiplication.
Teacher D: Students need a strong, or at least f; oneal, number sense in order to be able
to learn any math concepts.
Teacher E: Students need to know their basic math facts (adding, subtracting,
multiplying, and dividing), as well as math terms. They should also know basic strategies
that they can use to solve math problems.
TEACHERS’ PERCEPTIONS OF MATHEMATICS 65
Teacher F: Strong foundation of number sense
Teacher G: In Montessori, students must have a strong foundational understanding of
number sense and place value. All Montessori mathematics builds on these foundational
understandings.
Question eight asked participants to describe their most recent memory of teaching
mathematics. Each teacher described a different topic. The broad topics included place value,
numbers and operations, fraction concepts, multiplication, and geometry. Four related the
memory to the topics they taught, and three included students' reactions to the instruction.
Examples of responses show more detail for clarity.
Teacher A: A recent memory is teaching regrouping and borrowing using base 10 blocks.
A student said, “this is finally making sense to me”.
Teacher B: My most recent memory of math is the order of operations lesson I just
completed.
Teacher C: The “aha” moment when a student learns that multiplication is groups of.
Pure Joy.
Teacher D: Today, I taught students the skills needed to round numbers to the nearest
tenth, hundredth, and thousandths.
Teacher E: One of my students that has been working with finding common denominators
with fractional materials came to the realization that she was finding multiples.
Teacher F: My most recent memory about teaching mathematics was in my classroom
yesterday. We are focusing on place value and powers of 10, so we are looking for
decimals and the pattern that happens when we multiply.
TEACHERS’ PERCEPTIONS OF MATHEMATICS 66
Teacher G: My most recent memory is the geometry lesson I gave this afternoon
introducing degrees, the Montessori protractor and how to use a protractor.
Question nine asked teachers to name the instructional strategies they found most
effective in teaching mathematics. Several different responses included games, real-world
examples, problem-solving, fluency practice, and using manipulatives. Excerpts from responses
are included below:
Teacher A: I find using manipulatives and showing photo/real-world examples to be
effective in my classroom. I also think using strategic questioning is effective. For
example, asking "how did you solve that?"
Teacher B: Using manipulative and creating/ playing maybe games that get kids
competitive and eager to win. It helps them stay engaged and want to actually learn the
concepts.
Teacher C: Using manipulatives in my classroom. This is the best way for students to
learn the conceptual understanding of mathematics.
Teacher D: The most effective strategy I have found is consistent number sense/fluency
practice.
Teacher E: Presenting content at the child's level and allowing them to problem-solve on
their own. Knowing when to interject and stand back is vital in a child's learning
development.
Teacher F: No certain one. The cool thing about math is that there are many ways to
answer and different strategies work more efficiently for different students.
TEACHERS’ PERCEPTIONS OF MATHEMATICS 67
Teacher G: The most effective strategy is to use the Montessori materials with fidelity and
truly assess where a child is and what misconceptions he/she has.
The final question asked teachers how they knew when a student had mastered a concept.
Three teachers indicated that having a student teach the concept to them or another student
demonstrated mastery. Two teachers related mastery to consistently completing work without
assistance. One suggested that student discussion about the topic demonstrated mastery, and one
defined mastery as students being able to transfer the knowledge to other mathematical concepts.
The open-ended questions in the survey complemented the interview feedback. The
survey items involved experiences and perceptions of mathematics related to the first research
question. The responses provided information regarding teachers' preferences and frustrations
about mathematics and instruction. Also, teachers’ self-efficacy about math content related to the
second research question focusing on content and its influence on instruction.
Observations
Each participant was observed for one sixty-minute mathematics period of instruction.
During the focused observations, the researcher ignored entities considered to be insignificant
such as classroom cleanliness, temperature, and student behavior. The researcher used an iPhone
to preserve an audio record of the observation for accurate transcription. Although the
observation protocol (see Appendix D) contains specific “look fors,” the researcher grasped all
teachers’ actions, commentary, and nuances. The researcher checked the notes and transcription
for accuracy when the notes were compared with the audio recording.
The use of an online application was the only observable form of instruction in two
classrooms and limited the gathering of information related to the protocol. No classrooms used
TEACHERS’ PERCEPTIONS OF MATHEMATICS 68
mathematics exemplars during instruction, and one out of seven administered an assessment
during the observation. All teachers provided feedback related to behavior or performance. One
teacher addressed misconceptions. Three teachers completed a sequence of instruction. Two
teachers used concrete objects or illustrations in the sequence of instruction. Two teachers were
observed using precise mathematics vocabulary during instruction, and expectations were
verbalized in six of the classrooms. Table 5 includes teachers’ commentary and the researcher’s
observations as they aligned with the observation protocol.
Table 5
Observation Results
TEACHERS’ PERCEPTIONS OF MATHEMATICS 69
Summary
The qualitative data provided connections and relevance to the research questions. The
purposeful interview questions aligned with the study's intent and the research questions. The
interview responses presented information about teachers’ feelings toward mathematics and the
experiences that accompanied their perceptions. The survey data provided comparable responses
from the interviews and questionnaire that will be discussed further in Chapter five. The
observation data enhanced and illuminated the information gathered from the interviews and
questionnaire responses. The observation data also accommodated the second research question
by adding coherence and more detail.
In Chapter Five, the researcher will expound upon the results and make inferences based
on the findings. Additionally, in the next chapter, the researcher acknowledges the study's
limitations in data collection, analysis, and sample size of participants. Further research is
suggested, and what impact it may have on future studies. Finally, the researcher will address
how the findings may benefit the participating school district and serve as a foundation for a
deeper investigation of teachers’ mathematical strengths and weaknesses.
TEACHERS’ PERCEPTIONS OF MATHEMATICS 70
Chapter 5: Discussion and Conclusions
Introduction
According to research, mathematics anxiety poses difficulty for many educators, particularly
primary school teachers, who often have difficulty with mathematics content knowledge. Depending on
their educational paths, elementary teachers may choose or shy away from higher-level mathematics
courses before teaching math in the classroom. Since fearfulness of mathematics is associated with math
eschewal (Jaggernauth & Jameson-Charles, 2015), educators not confident about math may avoid math
specialization throughout their education (Porsch, 2017). Those who want to become elementary school
teachers rarely do so because they want to teach mathematics (Porsch, 2017). Elementary school
significantly influences a student's math skills and attitudes. Elementary school teachers are role models
for their students during the learning process. Therefore, students may encounter primary school teachers
with a negative attitude toward mathematics or suffer increased arithmetic anxiety. It is particularly
challenging for teachers in elementary school systems where the expectation and certification include all
content areas.
Through exploration, the author collected and examined the attitudes and experiences of
mathematics teachers and how these intellections presented themselves in their mathematics instruction.
Each participant was responsible for teaching the 5th-grade mathematics academic content in the
participating school system. Collected data revealed teachers' self-perceptions about mathematics
instruction and their understanding of content. The ultimate goals of this study were to explore teachers'
self-perceptions and understanding of mathematics and mathematics instruction through the following
research questions:
Question 1: How do 5th-grade teachers' experiences and self-perceptions of mathematics influence
mathematics instruction?
Question 2: How do 5th-grade teachers' understanding of mathematics concepts influence instruction?
TEACHERS’ PERCEPTIONS OF MATHEMATICS 71
The first research question delved into teachers' lived mathematics memories from early
childhood into their professional lives as mathematics teachers. The second research question considered
how teachers' understanding of mathematics content influenced their mathematics instruction. The
findings depict a montage of attitudes and personal accounts in connection with mathematics content and
pedagogy.
Discussion and Interpretation
The participating fifth-grade teachers' diverse backgrounds shaped their experiences and other
characteristics, such as professional training and years of experience. Gender, training, and experience
allowed for various responses regarding content knowledge and attitudes about mathematics. The
qualitative phenomenological research design included three types of data to be gathered. Interviews,
questionnaires, and observations supported the research questions with expressive authenticity.
The results from the interviews showed that before attending school, participants were involved in
counting activities initiated by family members or friends. Counting activities included currency,
household objects, and food. Other math-like activities included learning colors, sorting objects by color,
and playing card games. Participants' experiences after entering school included positive and negative
accounts. Five participants recalled fond memories of mathematics in elementary school, while only two
reported negative memories from their elementary years. The negative memories stemmed from a lack of
automaticity with numeracy or a lack of clarity from teachers.
The survey revealed some parallel responses with the interviews. In the interviews, six of the
seven participants said that math was their strongest subject. Three of the six also chose social studies as a
strength. One teacher included science and math, and another indicated that phonics and math were
strengths. One teacher chose social studies and reading. Their weak areas were geometry, abstract
concepts in algebra, measurement, how to choose specific manipulatives for concepts with fractions,
TEACHERS’ PERCEPTIONS OF MATHEMATICS 72
algebra, and decimals, and teaching number lines with fractions and decimals. When the researcher asked
teachers about their strengths as mathematics teachers, no one mentioned a particular "math topic."
All teachers spoke of instructional strategies such as moving students from the concrete to the abstract,
finding the gaps in learning and addressing them, pacing of instruction, enthusiasm for the content,
teaching procedures, and using higher-level questioning. However, during the survey, teachers were more
specific about particular mathematics content when asked which concepts they enjoyed teaching. The
content they enjoyed teaching included multiplication, addition, subtraction, fractions, and the coordinate
plane. The math content they did not enjoy teaching paralleled most of the weak areas mentioned in the
interview. Teachers found fractions, decimals, place value, general measurement, and specifically metric
conversions frustrating to teach. These same areas were also identified as weak by the participants.
Other similarities between the interview responses and the questionnaire included the math
courses completed before teaching and how prepared teachers felt when entering the field. Fifty-seven
percent of the participants graduated with a bachelor's degree in elementary education. Two had master's
degrees, and one participant completed thirty hours above a master's degree. Two participants took
calculus in high school and college and felt prepared to teach when they entered the classroom. Both
teachers received additional Montessori training and have master's degrees in education. However, they
did not engage in an elementary standards-based math methods course during teacher preparation.
The other five participants took general math courses in high school, such as algebra and
geometry. Teacher preparatory math courses were unique to each person. All participants reported taking
one elementary math methods course during the teacher preparatory program. On the first day of class,
one participant's teacher told the students not to worry and that she hated math too. The course focused on
higher math, not elementary math. Another said that the only activity remembered from the course was
lattice multiplication. Lattice multiplication is not part of a standard or mentioned in the South Carolina
supporting documents as a viable strategy. An introduction to manipulatives was the focus of one course;
TEACHERS’ PERCEPTIONS OF MATHEMATICS 73
however, the participant could not recall details from the course. One participant commented that there
was no elementary math involved in the methods course, only college algebra. According to one
participant, the math methods instructor's new book contained literature used to teach mathematics and
involved no elementary mathematics pedagogy of mathematics content instruction.
The findings demonstrate the range of mathematics requirements and expectations in the
participants' teacher preparatory programs. A study by Scheiner et al. (2017) focused on determining
recommendations for courses preparing elementary (K-8) teachers. It sought to clarify and explore what
makes mathematics knowledge specialized compared to other content areas and what mathematics
knowledge signifies in the context of teaching. Mathematics teachers need to understand content
differently than mathematicians.
Within the mathematical knowledge domain, common content knowledge refers to the
mathematical knowledge and skill possessed by any well-educated adult and by all mathematicians used
in contexts other than instruction. Specialized content knowledge is mathematical knowledge adapted to
the specialized applications unique to the teaching profession. It is described as used by teachers in their
work but not held by well-educated adults and not typically utilized for other purposes (2017).
Mathematics expertise is not an intuitive advantage to teaching mathematics. It requires a different type of
knowledge than teachers of other content areas, such as English language arts or social studies.
Mathematics teachers must also show proficiency in mathematics pedagogy and deftly organize and
understand the progression of learning to sequence concepts and instruction logically. Content knowledge
and attitudes towards mathematics noted during instruction attested to individual self-portrayals of
pedagogy and command of the subject.
The observations indicated that approximately 30% of the participants used precise mathematics
vocabulary, assessed students, or addressed misconceptions during the observation. Approximately 43%
of the participants showed high expectations, were involved in direct instruction and completed a logical
TEACHERS’ PERCEPTIONS OF MATHEMATICS 74
instructional sequence. In four out of seven observations, students used an online application to complete
mathematics tasks, and one class used an online formative assessment. In two classes, precise
mathematics vocabulary was used to describe the content during instruction. In two classes, the teachers
explicitly conveyed high expectations regarding the criteria for success. The use of a Concrete-
Representational-Abstract (C-R-A) teaching sequence, which included students manipulating objects to
solve problems with fractions and multiplication, was observed in the two Montessori classes.
Additionally, findings from interviews revealed that most participants' math strengths
were not specifically mathematics. One Montessori teacher felt confident about moving students
from a concrete understanding to a more symbolic understanding of abstract concepts, uniquely
mathematics instruction. However, the other six responses exhibited characteristics of practical
teaching in any content area.
"I can see holes in understanding, scaffolding instruction, breaking down steps, my enthusiasm,
pacing of instruction, and questioning."
Antonelli (2019) conducted a mixed-methods study involving kindergarten through fifth-
grade teachers in an urban setting. She aimed to examine and explore primary school teachers'
perceptions of their technical knowledge, content understanding, pedagogy, and readiness to
adopt technology integration in mathematics education. The findings below reveal a similar
disposition about content knowledge related to the current study.
"Although quantitative data analysis revealed that participants rated Mathematics
Content Knowledge as the second highest mean, the qualitative portion of the study
revealed that participants were only referring to the basic skills of mathematics at the
grade level they instructed The survey data from the quantitative phase showed that
TEACHERS’ PERCEPTIONS OF MATHEMATICS 75
teachers felt strongly about their mathematics content knowledge. However, this data was
divergent from the qualitative findings where teachers expressed that they were only
comfortable with basic rote mathematics skills with a single path of inquiry" ( pp. 135,
138).
In the current study, interview responses suggested that teachers' dispositions about their
instruction were primarily positive, with responses ranging from "pretty good" to "very
strongly." However, when asked about strengths in mathematics, teachers responded with
general pedagogical structures in any content area. The questionnaire asked teachers which math
concepts they enjoyed teaching. The majority of responses included teaching coordinate planes
and reviewing basic math facts. Two teachers included fractions. Except for fractions and the
coordinate plane, the concepts teachers enjoyed teaching involved rote procedural skills. South
Carolina 5th grade standards (2020) for whole number operations are procedural, and the
coordinate plane involves only the first quadrant. The geometry standards covering the
coordinate plane are listed below.
● 5.G.1 Define a coordinate system.
a. The x- and y- axes are perpendicular number lines that intersect at 0 (the
origin);
b. Any point on the coordinate plane can be represented by its coordinates;
c. The first number in an ordered pair is the x-coordinate and represents the
horizontal distance from the origin;
d. The second number in an ordered pair is the y-coordinate and represents the
vertical distance from the origin.
TEACHERS’ PERCEPTIONS OF MATHEMATICS 76
● 5.G.2 Plot and interpret points in the first quadrant of the coordinate plane to represent
real-world and mathematical situations.
Although the notion of a plane in geometry is quite abstract, the standards only ask
students to follow steps to define, plot, and interpret points in the first quadrant. Similarly, the
whole number multiplication and division standards are primarily procedural in 5th grade.
● 5. NSBT.5 Fluently multiply multi-digit whole numbers using strategies to include a
standard algorithm.
● 5. NSBT.6 Divide up to a four-digit dividend by a two-digit divisor, using strategies based
on place value, the properties of operations, and the relationship between multiplication and
division.
Teaching these standards does not involve the level of conceptual knowledge needed to
teach to the intent of the fraction or decimal standards. Donovan and Bransford (2004) indicated
that frequent student misunderstandings with fractions illustrate the rational number challenges
students face. The perpetrator appears to be the persistent taught application of whole-number
thinking in contexts where it is inappropriate (2004). Although the research was conducted
eighteen years ago, teachers still use whole numbers procedures to teach fraction computation.
The weaknesses and frustrations teachers specifically named were mathematical topics such as
fractions, decimals, teaching with number lines, metric conversions, general measurement,
geometry, and using concrete models to demonstrate math concepts.
Research suggests similar findings in mathematics weaknesses. As stated in the literature
review, the Survey of Adult Skills (PIAAC) was administered to 5,010 individuals in 15
countries worldwide (2019). Results suggest that over one-third of those tested in the United
TEACHERS’ PERCEPTIONS OF MATHEMATICS 77
States scored below level two in numeracy (2019). Calculations, including whole numbers,
simple decimals, percentages, and fractions, are mastered at level two. Measurements,
estimation, elementary data analysis, and probability are all included.
According to Copur-Gencturk (2021), many of the studies on teachers demonstrate that
they need help comprehending fraction ideas, mainly conceptual understanding of fraction
computation. The research aligns with the responses from the interviews and questionnaire
questions asking teachers which math content they felt was a weakness and which math concepts
they found frustrating to teach. An example of the fraction and decimal standards expectation
necessitates a deep understanding of the concepts beyond procedures. The support document also
includes a note in the fraction standard to emphasize that the focus is on using various models.
● 5. NSBT.7 Add, subtract, multiply, and divide decimal numbers to hundredths using
concrete area models and drawings.
● 5. NSF.1 Add and subtract fractions with unlike denominators (including mixed
numbers) using a variety of models, including an area model and a number line.
○ This standard should focus on using various models instead of "tricks," such as
the "butterfly method," which does not contribute to students' fraction
understanding.
These examples are directly related to the constructivist theories of Jean Piaget, Lev
Vygotsky, and John Van de Walle. Constructivist mathematics teaching provides a solid
foundation for conceptual learning while meeting standards-based criteria (Van de Walle, 2004).
Piaget believed children use mathematical structures and patterns to reason mathematically
(Wavering, 2011). Lev Vygotsky believed that students learn math by explaining and defending
TEACHERS’ PERCEPTIONS OF MATHEMATICS 78
their thinking (Steele, 2001). This constructivist idea claims that connecting concepts to create
new meaning helps students learn arithmetic language and understanding (2001). A confident
math instructor who can organize the progression of learning to develop students' conceptual
understanding must be able to explain how mathematical procedures work. Knowing the
sequence of steps in calculations does not clarify mathematics for children.
Van de Walle (2004) indicated that following procedural directions without critical
thought does not build arithmetic understanding. Thus, teachers must grasp the fundamentals of
good mathematics instruction to help students become flexible with strategies and make
connections between concepts. Teachers must also grasp mathematics, its use in daily life, and
how to help students move from concrete to abstract comprehension.
Before studying abstract operations, students need experience with concrete materials to
reason mathematically. Piaget (1964) suggested that conceptual mathematics comprehension
comes from the child's encounters with tangible materials rather than the materials themselves.
He stressed that mathematics is abstract; thus, students need tangible experiences to understand
concepts. (Yıldırım & Yıkmış, 2022). Teachers must also know how to determine which
manipulative objects are appropriate for each concept and how to use them. For children to
succeed in mathematics beyond elementary school, teachers must be confident in teaching this
instructional sequence.
The examples also demonstrate the need for teaching with the CRA sequence to allow
children the opportunity to experience mathematics fully. The teacher's primary responsibility in
a constructivist classroom is facilitating learning by providing various authentic experiences.
Teachers encourage students to explore and experiment while providing opportunities for
TEACHERS’ PERCEPTIONS OF MATHEMATICS 79
collaboration. Social interaction and language skills enable children to build on understanding as
they develop and acquire knowledge.
The data suggested that most teachers recalled positive experiences and felt confident
about math instruction. The weak areas of mathematics were similar to findings from research.
Most teachers did not reveal anxious perceptions of mathematics through the interviews and
questionnaires. The observation data left some areas open for further investigation. The small
sample size and limited data collection time made it difficult to relate the influence of
perceptions of mathematics to teachers’ instruction.
Limitations and Assumptions
The limitations of this study involved the small sample size of participants and the large
size of the participating district. This research design collected data from interviews, a
questionnaire, and observations to gather evidence accurately answering the research questions.
Further research was needed to fortify the validity and reliability of the findings. These
limitations would have been avoidable if the author had researched school district policies and
procedures for research earlier in the process. In future studies, the researcher will seek districts
with fewer access restrictions and approval time limits.
Through the research approval process, limitations appeared and continued to create
extraneous variables and other issues associated with policies and procedures that limited time
and access to a more diverse grouping of schools and individual participants. The researcher
considered the impact on the sample representation of the participating district's diverse
population.The researcher supposed that permission to access school administrators and teachers
would be granted soon after the research was approved. However, there was a six-month waiting
TEACHERS’ PERCEPTIONS OF MATHEMATICS 80
period between submitting the research proposal and contacting schools. Approximately one-
tenth of the total number of principals voluntarily agreed for teachers to participate in the study,
which affected the sample's diversity and size. The researcher purported that participants would
be candid with interview and questionnaire responses. The researcher acknowledged that bias
may occur and collaborated with colleagues during the analysis and interpretation process to
ensure the data's integrity.
Implications for Theory
The findings in this study have contributed a sampling of information to the phenomenon
linked to math instruction, math content understanding, and how past experiences and current
perceptions influence teachers’ mathematics instruction. This study yielded findings that indicate
teachers feel confident about general instructional strategies and mathematical procedures. The
findings concur with constructivist exploration theories with concrete manipulation and inquiry.
The weak areas revealed by participants demonstrate the need for teacher exploration with
concrete materials, representation, and understanding of how it all relates to abstract notions and
symbolic representation. Although interpretations of data collected from a small sample show
that teachers were pretty confident and did not show signs of anxiety during instruction,
refinements to the prevalence of procedural instruction could enhance and expedite student
learning according to the theoretical framework followed in this study.
Implications for Practice and Implications for Future Research
The participating district could benefit from a future study that allows a larger pool of
teachers to participate in a follow-up study to determine the course of action regarding
professional learning opportunities in mathematics. Moreover, the research suggests immediate
TEACHERS’ PERCEPTIONS OF MATHEMATICS 81
actions to ensure that teachers have opportunities to participate in standards-based courses and
workshops focused on the primary topic areas of weakness in this study and similar research
studies found in the literature review. The researcher should make revisions and additions to the
interview and survey questions for future studies to gain more specific information in participant
responses.
Future research recommendations include a deeper examination of teacher preparatory
program requirements for elementary math pedagogical courses. Also, a larger sample size, more
explicit questioning about instructional practices, and at least two observations per teacher may
provide more compelling data. One observation did not demonstrate the totality of a teacher’s
command of content or pedagogy. Having Montessori teachers in the sample was not anticipated
at the beginning of this study. However, the interviews, surveys, and observation responses
provided insight into differences between the general education teachers’ attitudes about
mathematics and the Montessori philosophy of mathematics learning.
Recommendations for the field of elementary education include reflective conversations
about the current teacher preparatory elementary mathematics curriculum and methods courses—
considering teachers’ lack of conceptual understanding of elementary curriculum design. The
researcher also recommends forming a knowledgeable and committed group of stakeholders to
create a plan for designing, implementing, and maintaining high-quality math curricula and
instruction in school districts. The Montessori methods of CRA observed were unique, with
special concrete materials. The interview and survey responses showed that the teaching
philosophy is more closely related to the constructivist view of learning. The researcher suggests
that representatives from the Montessori schools would bring expertise and ideas for training
TEACHERS’ PERCEPTIONS OF MATHEMATICS 82
general education teachers to use the CRA approach to mathematics learning. The overarching
focus for future research should address the question: What do elementary teachers need to know
about mathematics to be exceptional teachers?
Summary
This research aimed to examine and analyze the relationships among 5th-grade teachers’
attitudes about mathematics and their understanding of 5th-grade mathematics content to
investigate the influence on instructional practices in mathematics. Overall, the data collected in
this study indicate that teachers felt confident about teaching mathematics, and most teachers’
past experiences were positive. The data revealed that the teachers' strengths generally were
more related to strategies inclusive of all content areas, not exclusively mathematical.
Conflicting results were found when analyzing and comparing the observational data
with the interviews and survey results. Most teachers reported enjoyment in teaching more
procedural concepts than those requiring conceptual understanding and concrete application.
This was also supported by some of the responses about strengths, i.e., teaching steps and
procedures, multiplication, addition, and subtraction of whole numbers.
Teachers did not provide exemplars during instruction, and most did not utilize quick
formative assessments of students’ learning during the observation. Online activities also
prevented the observer from seeing enough direct instruction in some cases. The direct
instruction observed was short and primarily procedural. However, when Montessori teachers
reported their mathematical experiences, training, and teaching strengths, they included
mathematics instruction moving from concrete to abstract sequence. The observation of the CRA
sequence of instruction was prevalent in the Montessori classes. More classroom observational
TEACHERS’ PERCEPTIONS OF MATHEMATICS 83
visits would give the researcher a more in-depth view of teachers' understanding of content and
instruction. A larger sample size of participants from various school districts may provide more
compelling results for future studies. Deeper observations of teachers involved in direct
instruction and some adjustments to interview and questionnaire items could further explain
findings regarding the relationship between mathematics perceptions, content understanding, and
mathematics instruction.
TEACHERS’ PERCEPTIONS OF MATHEMATICS 84
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Appendix A
Open-Ended Survey Questionnaire
Question 1: What is your highest academic qualification?
Question 2: By the end of this school year, how many years will you have been teaching
altogether?
Question 3: How well prepared did you feel to teach elementary mathematics during student
teaching?
Question 4: How long have you been teaching 5th-grade mathematics?
Question 5: What math concepts do you enjoy teaching?
Question 6: What math concepts do you find the most frustrating to teach?
Question 7: To succeed in mathematics, what do students need to know?
Question 8: What is your most recent memory about teaching mathematics?
Question 9: What instructional strategies do you find the most effective in teaching
mathematics?
Question 10: How will you know when students have mastered a concept?
TEACHERS’ PERCEPTIONS OF MATHEMATICS 100
Appendix B
Interview Questions
Question 1: What were your first informal memories of mathematics before you entered school?
Question 2: What experiences from school (elementary, middle, high) most impacted your
perceptions or attitudes about mathematics?
Question 3: What mathematics courses did you take in high school and your teacher preparatory
program?
Question 4: What mathematics experiences have influenced how you currently teach
mathematics?
Question 5: What content areas are your strengths as an elementary teacher?
Question 6: What are your strengths as a math teacher? Weaknesses?
Question 7: How do you feel about your mathematics instruction?
TEACHERS’ PERCEPTIONS OF MATHEMATICS 101
Appendix C
Initial Email to Prospective Participants
Date:
Dear ___________,
My name is Margaret Knight, and I am a candidate in the doctoral program at Southern
Wesleyan University. Thank you for your interest in participating in a research study exploring
the influence of 5th-grade teachers' perceptions of mathematics on instruction. The purpose of
this study is to examine how 5th-grade teachers' experiences, perceptions, and mathematics
confidence levels influence mathematics instruction. The study utilizes a qualitative approach to
probe into the earliest memories of mathematics before entering school and mathematics
experiences during formal education. The study also considers factors directly related to teachers'
current perceptions of math content and instruction. Exploration of teachers' past and present
perceptions of mathematics will elucidate the influences of math perception on instruction. If you
agree to participate in this study, please respond to this email by simply stating "yes." I will
contact you to schedule an interview face-to-face or through Microsoft Teams. Thank you for
taking the time to consider participating in this study.
Sincerely,
Margaret W. Knight
TEACHERS’ PERCEPTIONS OF MATHEMATICS 102
Appendix D
Observation Protocol
“Look fors” of Instructional Practices in a 5th-grade mathematics classroom
The Massachusetts Department of Elementary and Secondary Education (2012) suggests that
effective mathematics teachers:
● use instructional practices that convey high expectations for content, effort, and work
quality.
● use precise mathematics vocabulary during instruction.
● use concrete objects, illustrations, and abstract expressions to teach mathematical concepts
and relationships.
● provide pupils with actionable and specific feedback on their mathematics work.
● assess students’ mathematical understanding using a variety of formative measures during
the observation.
● provide accurate examples during instruction that demonstrate mathematical reasoning
and understanding.
● plan and deliver mathematics instruction in a logical sequence.
● Identify and quickly address students’ mathematics misunderstandings.
TEACHERS’ PERCEPTIONS OF MATHEMATICS 103
Appendix E
Consent Form
An Attitudinal Study of 5th-grade Teachers’ Perceptions about Mathematics and the
Influence on Instruction
Consent to take part in research
I _____________________________voluntarily agree to participate in this research study. I
understand that even if I agree to participate now, I can withdraw at any time or refuse to
answer any question without any consequences of any kind. I understand that I can withdraw
permission to use data from my interview within two weeks after the interview, in which case
the material will be deleted. I have had the purpose and nature of the study explained to me
in writing, and I have had the opportunity to ask questions about the study. I understand that
participation involves participation in an interview, completion of a survey, and one
observation of instruction. I understand that I will not benefit directly from participating in
this research.
I agree to my interview being audio-recorded.
I disagree with my interview being audio-recorded.
I understand that all information I provide for this study will be treated confidentially. I
understand that in any report on the results of this research, my identity will remain anonymous.
This will be done by changing my name and disguising any details of my interview which may
reveal my identity or the identity of people I speak about. I understand that disguised extracts
TEACHERS’ PERCEPTIONS OF MATHEMATICS 104
from my interview may be quoted in the dissertation, final defense presentation, and published
papers. I understand that if I inform the researcher that I or someone else is at risk of harm, they
may have to report this to the relevant authorities - they will discuss this with me first but may be
required to report with or without my permission. I understand that signed consent forms and
original audio recordings will be retained in an encrypted external hard drive that only the
researcher has access to data until the final dissertation defense committee confirms the results of
their dissertation. I understand that a transcript of my interview in which all identifying
information has been removed will be retained two years from the date of the final defense. I
understand that under freedom of information legalization, I am entitled to access the
information I have provided at any time while it is in storage as specified above. I understand
that I am free to contact any of the people involved in the research to seek further clarification
and information, including names, degrees, affiliations, and contact details of researchers (and
academic supervisors when relevant).
Signature of research participant Date________
____________________________
Signature of researcher Date _________
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