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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________

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Signature of researcher Date _________

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