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TEACHER PERCEPTION OF MATH FACT FLUENCY AND THE IMPACT ON
STUDENT ACHIEVMENT AND CLASSROOM INSTRUCTION
A Dissertation
Presented to
The Faculty of the Education Department
Carson-Newman University
In Partial Fulfillment
Of the
Requirements for the Degree
Doctor of Education
Andrea Noel Bringard
October 2017
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Copyright © 2017 by Andrea Noel Bringard
All Rights Reserved
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Abstract
With the advent of Common Core Standards and the ever changing environment of
schools, has the perceived importance of math fact fluency diminished in the classroom?
This qualitative study investigated teacher perception of fact fluency on student
achievement and whether or not teachers transferred this perception into their daily
lessons. The research design utilized focus groups, follow up interviews, and
observations of third, fifth and high school math teachers. (a) What are educator’s
perception of the importance of math fact fluency and achievement? (b) Do teachers
perceive a difference in academic achievement based on fact fluency? Why? (c) Do
teachers base instruction on the need for fact fluency? The study revealed teacher
perception of the correlation between student success and math fact fluency. Themes
emerged including conceptualization, memorization, standards, testing, grades, and in
class instruction. The data generated supported the conclusion of the teachers’ perceived
importance of math fact fluency in the classroom.
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Acknowledgment
I firmly believe the only way to get anywhere is to stay positive and surround
yourself with the best people. This dissertation would not have been possible without the
winning team effort of my dissertation committee. Dr. Patricia Murphree, my sincere and
earnest thankfulness for your constant patience and steadfast support to help me see the
light at the end of the tunnel and actually make it there. To Dr. P. Mark Taylor, for
quickly and kindly answering all of my questions throughout the entire process. Not only
the relevant ones, but also the ones that pretty much answered themselves. Dr. Beth
Batson, thank you is not enough for your time and guidance throughout this endeavor.
Dr. Christy Walker, you are the best reader I have ever met and I am thankful you were
always available to proof read. I am more thankful you always listened, calmed me
down, and became my friend.
I am truly indebted for the support provided to me by Dr. Mark Gonzales, Rachel
Ripley, Dr. Earnest Walker, and Dr. Michelle Keaton. Without this group answering
questions and lending emotional support and guidance, I never would have finished
courses, much less the dissertation. I would be remiss if I did not thank Dr. Deborah
Hayes and Dr. Julia Price two of the finest professors at Carson-Newman University.
Joy Bauman and Linda Owen are two of the finest examples of leadership in
education today. It is a great pleasure to thank them for teaching me how to lead by
example with grit, determination, compassion, and most of all a sense of humor.
Great friends are with you through it all and mine also have to be able to proof
read. I want to thank Jacinta Brothers, Amy Cabral, and Michele Vanderslice. Thanks
for all you have listened to, you are the greatest group to talk it all through with.
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Finally, my family has heard it all and listened patiently. To my parents and
siblings for always being available. To my sons, Lawrence and Michael, for
understanding when I decided to go back to school helping whenever they could. To my
husband Larry, for making sure I never quit. I know you will not miss the hysteria that
surrounds deadlines, rewrites, and citing or finding a source. Thankfully, we are done.
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Dedication
I dedicate this dissertation to my family for providing emotional support and guidance
throughout the entire process. To Michael for the grace in which you accepted me going
back to school. Thank you for allowing the time necessary for my school work to infringe
on your time. You are going to do great things because you are a phenomenal person.
To Lawrence for your endless time proof reading and editing papers. I am humbled you
decided to pursue teaching and am not surprised in the least by the exceptional educator
you have become. I treasure the time we spend debating educational topics. Finally, my
husband Larry, you obviously have the patience of a saint. You encouraged me to pursue
this dream in the midst of traveling, job changes, and moving children into colleges with
nothing but faith and confidence in our success. My heart and soul thanks you for all you
have done, I never would have finished without you.
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Table of Contents
Abstract ................................................................................................................. iv
Acknowledgements ................................................................................................ v
Dedication ............................................................................................................ vii
Chapter One: Purpose and Organization............................................................... 1
Statement of Problem ................................................................................ 2
Purpose of the Study ................................................................................. 3
Theoretical Foundation ............................................................................. 4
Research Questions and Hypothesis ......................................................... 4
Limitations ............................................................................................... 5
Definition of Terms................................................................................... 5
Organization of Document ........................................................................ 6
Chapter Two: Review of Literature ...................................................................... 7
History of Math Content ........................................................................... 9
Fluency Factors .......................................................................................... 14
Brain Research ........................................................................................... 17
Constructivism ........................................................................................... 21
Teacher Perception..................................................................................... 23
Math Fact Fluency and Achievement ........................................................ 28
Summary .................................................................................................... 31
Chapter Three: Research Methodology ................................................................. 33
Introduction ................................................................................................ 33
Population .................................................................................................. 33
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Setting ............................................................................................ 35
Description of Instruments ......................................................................... 35
Data Collection Procedures ........................................................................ 36
Summary .................................................................................................... 37
Chapter Four: Analysis of Data ............................................................................. 38
Demographic Results ................................................................................. 39
Research Question One .............................................................................. 41
Research Question Two ............................................................................. 46
Research Question Three ........................................................................... 48
Individual Teacher Data ............................................................................. 50
Third Grade .................................................................................... 50
Seventh Grade ................................................................................. 53
High School ................................................................................... 56
Summary .................................................................................................... 59
Chapter Five: Conclusions, Implications, Recommendations ............................... 60
Summary .................................................................................................... 60
Conclusions and Discussion of the Findings ............................................. 61
Research Question One .............................................................................. 61
Research Question Two ............................................................................. 64
Research Question Three ........................................................................... 65
Implications................................................................................................ 66
Recommendations for Future Studies ........................................................ 67
Conclusion .................................................................................................. 68
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References ........................................................................................................ 70
Appendices ........................................................................................................ 82
A: Focus Group Questionnaire .................................................................. 84
B: Individual Questionnaire ....................................................................... 86
C: Institutional Review Board Approval ................................................... 88
D: District Permission to Research ............................................................ 94
E: Email to Teachers .................................................................................. 96
F: Informed Consent Document................................................................. 98
G: Permission to Reproduce ..................................................................... 101
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Table of Figures, Tables, and Illustrations
Table 3.1 Demographics of Schools in Study ........................................................ 34
Table 4.1 Demographics of Focus Group Participants .......................................... 40
Table 4.2 Conceptualization and Memorization .................................................... 42
Table 4.3 Reponses in Relation to Standards ......................................................... 44
Table 4.4 Teacher Perception of Fact Memorization............................................. 46
Table 4.5 Testing and Grades ................................................................................ 47
Table 4.6 In Class Instruction ................................................................................ 49
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CHAPTER ONE
Purpose and Organization
Learning begins at home before students ever walk into the classroom. Higher
mathematical achievements by the end of elementary school can be seen when students
are able to count before entering kindergarten (Nguyen, et al., 2016). The skills teachers
rank essential for kindergarten included basic communication skills comprised of the
ability to “follow direction,” “not being disruptive,” and the “ability to take turns”
(Cappelloni, 2010, p. 72). These skills are not academic. Academic skills were ranked
lower (less than 10%) and included alphabet and number sense. Also teachers strongly
agreed parents should be reading to students at home. Unlike teachers, parents tend to
believe academic skills such as knowing the alphabet, counting and other academic
abilities were necessary. Teachers believed those skills were teachable when students had
basic foundational skills such as a curiosity towards learning, support from home, and
regular school attendance (Cappelloni, 2010). Unfortunately, most teachers reported
their perceptions did not matter because state standards mandated what was required and
what was taught. These standards in the State of Tennessee include “Fluently multiply
and divide within 100. By the end of grade three know from memory all products of two
one-digit numbers” (TN Department of Education, 2016).
Mathematical fluency and comprehension must both be taught in order for a
student to obtain automaticity in math (Krudwig, 2003). Fluency is an important skill in
both the educational setting and in real life. Studies have shown a positive relationship
between fluency and performance on advanced math tasks and achievement (Krudwig,
2003; Loveless & Coughlan, 2004; Duncan, et al., 2007).
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Statement of Problem
Math fact fluency (addition, subtraction, multiplication, and division) is a third
grade standard in the State of Tennessee, yet it is not always mastered in third grade. The
premise of this study was to determine if teacher perception of math fact fluency reflected
the current standards. Teachers in different grade levels have different perceptions of
what fluency is and whether or not it is necessary. The Tennessee State Standards build
on a foundation of fact knowledge and fluency. Second grade standards focus on base-
ten number sense, followed by multiplication mastery and an introduction to fractions in
third grade. In fourth grade students are exposed to rigorous fraction experience.
Students without mastery of multiplication facts will struggle with mathematics
depending on levels of understanding and fluency (Wallace & Gurganus, 2005). This
struggle could lead to insufficient skills in further mathematics, such as fractions.
According to a study led by Robert Siegler (2012), fractional knowledge and division was
a predictor of success in high school mathematics.
How do mathematics teachers perceive fact fluency? In elementary and middle
schools, mathematics is taught as a spiraling subject. Concepts are taught and reviewed
in a continuous manner. Curricula are organized so topics are revisited yearly, sometimes
several times a year (Snider, 2004). Research conducted by Cole (1999) concluded
students in fifth and sixth grades, instructed using the spiral method, significantly scored
higher on state achievement testing than students not taught using spiral instruction.
Although the intent of a spiral curriculum is progressive depth of each concept as it is re-
introduced at successive grade levels, opponents disagree. Snider (2004) contends
student learning is limited by the spiral method because topics are not covered in depth,
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topics are not introduced in order, learning time is minimized based on the frequency of
topic switching, and an inadequate amount of review. She proposes the integrated strand
curriculum would benefit student achievement. “Once students can perform a skill
without hesitation that skill is integrated into other, more complex mathematical
procedures” (Snider, 2004, p. 37). Many teachers teach and reteach concepts for
understanding, but do they teach facts for mastery, fluency and automaticity?
Purpose of the Study
The purpose of this study was to investigate the perceptions educators have of
mathematics at various levels. Elementary, middle, and secondary educators were
interviewed about their perceptions of fluency, math comprehension skills, and the
correlation to academic success in both math and language arts. Education standards are
taught following a framework presented in a curriculum based on a progression of
concepts in a systematic manner (Rata 2016). This study was designed to determine
whether teachers across grade levels had differing views on fact fluency and academic
success. Studies show early math exposure and abilities, such as block play and
counting, are crucial for not only future math achievement, but also reading and language
art achievement (Duncan et al., 2007; Nguyen et al., 2016; Wolfgang, Stannard, & Jones,
2001). Are teachers aware of this correlation and does it have any impact in their
teaching methods or their perceptions of the necessity of vertical alignment of the
mathematical curriculum? With the exhaustive amount of standards teachers are required
to teach in every class, do they have instructional time to devote to the memorization of
math facts?
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Theoretical Foundation
With the advent of the Common Core State Standards Initiative, the approach
used by many to teach math today is based on the Constructivist Theory. The
Constructivist Theory emphasizes a student’s understanding of concepts (Baroody,
Bajwa, & Eiland, 2009; Woodward, 2006). Memorization of facts is not necessarily
considered a skill. This does not agree with the Information Process Theory. The
Information Process Theory states that fluency in math facts is necessary before a student
can achieve “success in many areas of higher mathematics” (Woodward, 2006, p. 269).
Advocates of the Information Process Theory believe math must be taught in sequence
for conceptual understanding and some concepts must be memorized. Hence, fact
fluency is a building block for higher order math. The two phases of counting strategies
and reasoning strategies must be understood in order for a student to advance to the phase
of mastery in fact fluency (Baroody, 2006).
The conceptual framework for the study was the Social Constructivist View
utilizing the understanding “reality is not absolute, but defined through community
consensus; multiple realities exist that are time and context dependent” (Mertens, 2005,
p. 231). Qualitative methods of research can be utilized in order to obtain in depth
information on the beliefs of teachers relative to their current educational situations.
Because the nature of the research does not deal in absolutes, as information is obtained,
the focus of the research could shift to an alternative direction.
Research Questions and Hypothesis
The research questions were what are educator’s perceptions of the importance on
math fact fluency and understanding? Do teachers perceive a difference in academic
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achievement based on fact fluency? Why do they believe this? Do teachers base
instruction on the need for fact fluency?
The hypothesis was that educators would hold differing views based on the grade
levels they have taught previously or in the present. The researcher believed most, but
not all teachers, would passionately believe fact fluency was critical for further math
success if they had any teaching experience at the elementary level.
Limitations
Limitations are those areas which may not be controlled by the researcher. Many
of the participants taught in a low socioeconomic environment. This was beyond the
researcher’s control and had an impact on research. The participants were selected based
on the convenience of the researcher and the participants. The delimitation of the study
was that the data would only come from one school district.
Definition of Terms
Curriculum. The courses of study offered by a school (Curriculum, 2015).
Fluency. The understanding of both facts and concepts and the ability to apply knowledge
to solve problems (NCTM, 2014).
Learning Standards. The specific skills teachers are expected to instruct and students
should master at specific set times in the education timeline (Learning Standards,
2014).
Memorize. To learn fully, “learn by rote” (Memorize, 2017).
Perception. The way a situation is understood (Perception, n.d.).
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Organization of Document
Chapter One provides an introduction and background, statement of the problem,
purpose of the study, theoretical framework, research questions, limitations, and
definitions of terms. The researcher has included a review of professional literature in
Chapter Two. Chapter Three provides an introduction, population, description of
instruments, data collection procedures, proposed data analysis, and a summary. Chapter
Four provides an analysis of demographic and survey data. An evaluation of data as well
as recommendations for further studies are provided in Chapter Five.
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CHAPTER TWO
Review of Literature
Math is a foundation for everyday life, not just education. “Students benefit from
not just numeracy skills and the confidence to use them, but also the mathematical
thinking skills that they develop. Learning maths is not merely about the content”
(Parker, 2012, p. 1). Most students begin math class before they even enter the classroom.
Number specific items such as clocks, remote controls, and telephones are part of society.
Daily life requires math skills. Telling time, reading a calendar, and money tasks all must
be learned. Early experiences with math and reading shape future success in school.
Many children benefit from the use of technology at an early age. Studies indicated
educational math applications (apps) designed for pre-school elevated student success
(Berkowitz, et al., 2015). These apps had non-distracting, simple designs, aligned with
Common Core Standards, and designed to be used collaboratively between parents and
students. Studies have found students with basic skills in math and language show rapid
growth over time with early math counting ability the strongest early predictor of success
(Aunola, Leskinen, Lerkkanen, & Nurmi, 2004; Duncan et al., 2007). Early math
interventions should be considered based on the prediction of success not only in math,
but also as a powerful predictor of future literacy achievement (Duncan et al., 2007).
Parents who provide mathematical activities and opportunities prior to formal education
can significantly improve future math achievement (Huntsinger, Jose, & Luo, 2016;
Anders, et al., 2012).
Math fact fluency is when basic addition, subtraction, multiplication, and division
facts can be correctly mentally calculated and within a few seconds (Frawley, 2012;
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Poncy, Skinner, & Jaspers, 2007). Linda Gojak, President of the National Council of
Teachers of Mathematics, disagrees (2012). Gojak stated that in order for students to
truly be fluent they must understand what they are doing, not just memorize the answer.
Wallace and Gurganus (2005) agreed, “Fluency with multiplication facts includes the
deeper understanding of concepts and flexible, ready use of computation skills across a
variety of applications” (p. 26). Mastering foundational skills in math, including
memorizing facts, in order to move on to advanced skills, can be compared to mastering
foundational skills for other disciplines. For example, in order to play an entire piece, a
violinist must first learn finger positions or a singer must first master notes before singing
an entire aria (Lin & Kubina, 2005).
In 2009, the Council of Chief State School Officers (CCSSO) and the National
Governors Association Center for Best Practices (NGA Center) created and released
Common Core State Standards (2017). These controversial standards were designed to
ensure all students across the nation were consistently learning the same skills. Common
Core framework could then be used by states for their standards. In these standards,
second grade math is focused on base ten understanding and the introduction to
multiplication. Third grade revolves around multiplication and division within 100,
along with strategies for the student to represent, understand the properties of, and solve
problems related to multiplication and division (Common Core State Standards, 2017).
The State of Tennessee adopted standards based on Common Core strategies such as:
Fluently multiply and divide within 100, using strategies such as the relationship
between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows
40 ÷ 5 = 8) or properties of operations. By the end of Grade three, know from
9
memory all products of two one-digit numbers. (TN Department of Education,
2016)
Along with the memorization, students are also expected to use multiplication and
division to solve word problems, and apply multiplication strategies including equal
groups, arrays, and measurement.
History of Math Instruction
Learning standards, or objectives, are the specific skills teachers are expected to
instruct and students should master at specific set times in the education timeline
(Learning Standards, 2014). Curriculum can be technically defined as just the courses
taught in a school, but in practice, it usually encompasses much more including the
standards, the teaching lessons, assignments, and a teacher’s style (Curriculum, 2015).
Social forces shape curriculum.
Mathematics has been seen as a foundation for the nation’s military and economic
preeminence, and in times of perceived national crisis mathematics curricula have
received significant attention. This was the case before and during both World
Wars, the Cold War (especially the post-Sputnik era, which gave rise to the new
math), and the U.S. economic crises of the 1980s. (Schoenfeld, 2004, p. 256)
Beginning in the 1950s, college professors were concerned over the lack of basic
math knowledge and ability to apply knowledge the students entering college showed
(Woodward, 2004). The Soviet Union launched Sputnik in 1957 and the United States
suddenly perceived herself as behind the world in math and science (Herrera & Owens,
2001). The movement continued towards a stronger math and science curriculum. In
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1955, the College Entrance Examination Board directed a committee to reassess their
entrance examinations. This report additionally impacted the need for curricular change.
As a result, curriculum was changed and by the 1960s the emphasis for the lower
grades was on “new math.” Math evolved to include broad concepts especially in the
younger grades. Drills were unnecessary because understanding would lead to answers.
Max Beberman was integral to the change. He stressed vocabulary, materials, patterns,
and observing which would all lead to understanding (Lagemann, 2000). Elementary
changes to curriculum were slower in implementation than high school and middle school
mainly due to the lack of specific math content knowledge necessary for teachers
(Herrera & Owens, 2001). By the 1970s and 1980s, standardized testing had become
important and curriculum had to be changed in order to directly improve scores
(Woodward, 2004). “By the early 1970s New Math was dead” (Klein, 2003, p. 7). In a
reversal of curriculum, classrooms reverted back to what was considered basic
mathematics. Skills and procedures were the primary focus. Experts cited a decline in the
Scholastic Aptitude Test scores. Because of this shift, students lacked problem-solving
skills.
The National Council for Teaching Mathematics (NCTM) published An Agenda
for Action in 1980 (Schoenfeld, 2004). The publication stated that a focus only on skills
and procedures was incorrect, and that a shift in curricula to problem-solving skills
should be implemented.
It set problem solving as the curricular focus, recommended that the definition of
“basic skills” be broadened to include such mathematical skills as estimation and
11
logical reasoning, and promoted the use of calculators and computers in the
classroom at all grade levels. (Herrera & Owens, 2001, p. 88)
As one of the first statements released by the NCTM, the impact of the report was not
sufficient to impact immediate change. Although the shift had begun, problem-solving
could also be interpreted as word problems (Schoenfeld, 2004). Textbooks made minor
changes therefore implementation in the classroom was not seen.
In 1983, A Nation at Risk was published by the National Commission on
Excellence in Education. The report analyzed math issues in education from elementary
curriculum and assessment to teacher preparatory programs to effect of mathematics
education on businesses (Klein, 2003). The report influenced new policy in math across
all grade levels. Toward the end of the 1980s, problem solving had become an important
factor in math. Unfortunately, the general view of problem solving was simple and not
true problem solving. Constructivism was the theory being pushed into the forefront of
math.
President George H.W. Bush began a push towards national standards in 1989
(Klein, 2005). Goals for these standards included student achievement in subjects
including English, mathematics, science, history and geography with measurable mastery
at the end of 4th, 8th and 12th grades. Late in the 1980s, the NCTM (National Council of
Teachers of Mathematics) issued new math standards (Woodard, 2004). These reformed
standards focused on both content and practice. “Underlying these proposed changes in
content is a central focus on the conceptual versus the merely procedural” (Herrera &
Owens, 2001, p. 89). Mental computation, understanding operation, and problem solving
was emphasized while repetitive paper and pencil practice was discouraged.
12
Constructivist Theory was used as a framework for the standards (Herrera & Owens,
2001). Recommendations for classroom teaching practice were:
Active student involvement in discovering and constructing mathematical
relationships, rather than merely memorizing procedures and following
them by rote.
The use of concrete materials, calculator graphics, tables, or other
representations as a means to help students grasp abstract concepts.
Group work, including students sharing and justifying their ideas.
Student writing (including drawings, diagrams, charts) to encourage
reflection on mathematical ideas, and oral presentation to promote
communication of those ideas.
The use of context, whether imaginary or real world, as a way to capture
student interest in problems as a framework.
Teacher as orchestrator of classroom discourse and facilitator of learning
experiences. (p. 89)
Teacher perception, time restraints, accountability, and the lack of basic computational
skill instruction were widely seen as barriers to implementation of the standards.
Conversely, most states redesigned their standards or framework to align with the NCTM
Standards (Herrera & Owens, 2001). By 1994, the United States Government had issued
the Goals 2000 Act and revised the Title I program, laying the foundation for the No
Child Left Behind Act (Klein, 2005).
NCTM published Principles and Standards for School Mathematics in 2000
which updated the standards accounting for concerns voiced by teachers and
13
administrators. The new standards were arranged into five strands and vertically aligned
for easy understanding (NCTM, 2000). Standards were organized into the following five
content strands: Number & Operations, Algebra, Geometry, Measurement, and Data
Analysis & Probability. Each standard included grade level appropriate examples and an
explanation of the teacher’s role for the express purpose of teaching the skills of problem
solving, reasoning & proof, communication, connections, and representation (NCTM,
2000).
The federal government passed the No Child Left Behind Act (NCLB) in 2002
(Klein, 2005). NCLB mandated state implementation of standards in math and reading as
a requirement for federal funding. The act also required an assessment system for
tracking student performance which was meant to hold schools accountable for
progression towards “universal proficiency as gauged by those standards” (p. 5).
By 2009, Common Core Standards presented a national shift in curriculum
(Porter, McMaken, Hwang, & Yang, 2011). The original intention was to align states
into a more common curriculum. They were to be “more focused than current state
standards” (p. 103).
Battista et al. (2009) analyzed many different methods available for mathematical
and educational research. The inquiry concluded research which leads to policy making
should be high quality and multiple methods should be employed. Education needs
“convincing research evidence about which programs and practices produce superior
student learning” (p. 237).
14
Fluency Factors
There are many reasons why students may not have fluency with math facts by
third grade. Some studies found students not only needed to understand concepts in order
to memorize facts, they also needed to have executive function (EF) in order to solve
math problems (Clements, Sarama & Germeroth, 2016). EF included the ability to self-
control emotions, attention, behavior and thinking. These skills were crucial for solving
math problems because they allowed students to focus, plan ahead, and apply past
experiences to problem solving. A combination of teaching both EF and mathematics was
suggested for optimal student benefit. As a side note to the brain study, researchers
discovered that it could be possible to teach EF with “high-quality mathematics
education” (Clements, Sarama & Germeroth, 2016). It was indicated the student would
have the advantage of learning both skills concurrently. This is a benefit which would be
especially convenient for students who are in need of help. Mazzocco and Kover (2007)
found executive function important for math skills in the early school years (up to third
grade), but no specific correlations as students proceeded. They concluded both long-
term and working memory were important factors in math fluency.
Is there a reason some students have a problem mastering fact fluency? Three
stages were found necessary to master math facts (Baroody, 2006):
Phase 1: Counting Strategies – using object counting (e.g., with blocks,
fingers, marks) or verbal counting to determine an answer
Phase 2: Reasoning Strategies – using known information (e.g., known facts
and relationships) to logically determine (deduce) the answer of an unknown
combination
15
Phase 3: Mastery – efficient (fast and accurate) production of answers (p. 22)
There was a difference between memorizing and understanding basic facts. According to
Baroody (2006), students who had learning disabilities were stuck at stage one. They
never made it to mastery. One side of the issue stated a defect in the learner while the
other stated it was due to a lack of instruction. Specific interventions for students have
also been studied.
Response to Intervention (RTI) was an educational method used to identify
students with learning disabilities in need of intervention (Fuchs & Fuchs, 2006). The
framework of RTI was built in tiers leveling intervention strategies based on student
needs. Frequently three tiers were used with the bulk of the student population receiving
minimal intervention in Tier 1, with increasing intensity in Tier 2 and Tier 3. RTI in Tier
1 and Tier 2 specifically targeted problem-solving. Number sense helped to reduce poor
problem solving skills in third grade (Fuchs, et al., 2006). Many RTI interventions
specifically targeted fact fluency due to the studies which indicated a positive correlation
between math fact fluency and achievement (Codding, Burns & Lukito, 2011). A
combination of student self-study, modeling, and continued practice led to the greatest
positive effect.
When intervention is done, math fact fluency improves (Poncy, Skinner &
Jaspers, 2006; Poncy, Skinner & McCallum, 2012). A study conducted by Poncy,
Skinner, and McCallum (2012) examining fact fluency interventions found both Cover,
Copy, and Compare (CCC) and Taped Problems (TP) interventions effective in
increasing math fact fluency. The study suggested more research would need to be done
to determine if there was an age where intervention would no longer be necessary.
16
The Cover, Copy, and Compare (CCC) intervention was adapted from a method
originally developed for spelling intervention by Skinner in 1989 (Poncy, Skinner, &
Jaspers, 2007). CCC was a method whereby the student received a sheet of selected
equations, studied the equations with answers on the one side of the paper, covered that
side of the paper, wrote the equation and answer on the other side of the paper, uncovered
the answer side of the paper and evaluated his or her solutions. If the student had
incorrect answers, he or she re-wrote the correct ones. Taped-Problems (TP) intervention
was a method where students listened to taped recordings of equations. The student
wrote the solutions before the recorded answer was spoken. When the student wrote the
incorrect answer, the student marked a line through it and wrote the correct one. If the
student did not have time to write the correct answer, the correct answer was written as it
was heard. The researchers discovered both interventions improved student performance
in math fact accuracy and automaticity. “However, when time is taken into account, TP
is clearly more efficient than CCC, taking approximately 29% less time (Poncy, Skinner,
& Jaspers, 2007).
There was significant correlation between learning deficiencies in mathematics
and students who struggled with automaticity/fluency of arithmetic facts (Gersten, Jordan
& Flojo, 2005; Poncy, Skinner, & Jaspers, 2007). Students who participated in the study
for two years and showed an initial low mastery of arithmetic facts, continued to show
deficits. Algebra and advanced math concepts were more challenging for those students
due to the failure to automatically retrieve simple math facts (Gersten, Jordan, & Flojo,
2005). The researchers recommended an individualized computer program which
combined both easy to recall and difficult problems for the student as an intervention
17
strategy. Teaching students several strategies were also recommended along with enough
time to adequately master the arithmetic facts.
Fluency and computational understanding are both skills which must be included
in mathematics instruction in order for a student to understand multiplication. Fluency is
a basic skill which must be mastered in order for students to achieve higher level math
skills (Loveless & Coughlan, 2004; Jarema, 2007). Mathematical skills need to be taught
and learned in a progressive stage in order to be mastered. According to Baroody (2006)
there are three phases students must typically progress through in order to achieve
mastery of single digit fact fluency. These phases begin with the phase of counting
strategies. Phase one includes object and verbal counting. The second phase is called
reasoning. Phase two includes relationships and being able to determine an answer. The
final phase is mastery and is being able to produce answers quickly and efficiently. Some
students, especially those who have learning difficulties or those who are struggling
never make it to stage three.
Research concluded students with learning disabilities and students who were
academically low achieving typically display difficulty with automaticity and
memorization of math facts (Woodward, 2006). Intervention strategies which included
repetition and discussion about math resulted in higher performance along with an
increased positive attitude towards math.
Brain Research
Brain research conducted over the last 20 years has provided the field of
education valuable information about how students learn. Research allowed teachers to
combine current instructional teaching techniques with brain research to increase student
18
achievement (Radin, 2009). Teachers need an understanding of the way a student’s brain
works in order to teach effectively. “Brain-compatible teaching is essential for optimal
learning; educators at all levels, pre-school through higher education, need this
component to round out their conceptual framework” (2009, p. 49). The researchers
discovered six characteristics which are considered instrumental for instruction based in
brain compatibility:
Emotional involvement, from the standpoints of both teacher and student
Physical systems to include movement, room arrangement and homeostasis
Lowered stress and threat levels
Experiences in the classroom, including trial and error, exploration, practice,
creativity, and critical thinking
Challenge, problem-solving, and authentic work, in which the students do the
work of learning and create their own meanings (Radin, 2009, p. 44)
The research recommended teachers become fluent in both brain research and
educational practices as teaching students was about teaching the whole student, not just
brain research.
The aforementioned research was supported by the American Federation of
Teachers (2000) who noted teachers must use scientific means to understand how people
learn. Specific guidelines and framework has been determined to be successful in
classroom lessons as a result of brain research. Willis (2016) expand on this research:
And what has emerged from the neuroscience of learning over the past two
decades is a body of highly suggestive evidence that successful strategies teach
for meaning and understanding, that learning-conductive classroom are low in
19
threat and high in reasonable challenge, and that students who are actively
engaged and motivated devote more brain activity (as measured by metabolic
processes) to learning. (p. 698)
Cognitive development of the brain plays a critical role in the shift from learning
basic math to becoming fluent or achieving automaticity. A Stanford study showed the
hippocampus, the region of the brain which can be associated with new memories, was
initially used for learning math facts (Digitale, 2014). Students who became fluent at
retrieval of facts, and adults who had already formed connections and stored the
information, retrieved information from their neocortex. The research explained the
hippocampus serves as scaffolding while the schema for math fact knowledge is being
created, as the “brain gets better at solving math problems its activity becomes more
consistent” (2014, p. 2).
The brain processes information more effectively and stores long-term when
engaged in non-stressful and engaging activities (Willis, 2007) Multiplication facts could
be memorized more easily if approached in a positive manner with intrinsic rewards and
relevance to student accomplishments. “When teachers use strategies to reduce stress and
build a positive emotional environment, student’s gain emotional resilience and learn
more efficiently and at higher levels of cognition. Brain imaging studies support this
relationship” (2007, p. 5).
According to Bransford (2000), the brain uses a method of progressive steps for
learning information successfully. The first step is initial learning and it must be
successful in order to transfer basic information into detailed knowledge. Mastery of this
skill would be necessary for successful transfer. “All new learning involves transfer
20
based on previous learning, and this fact has important implications for the design of
instruction that helps students learn. Without an adequate level of initial learning, transfer
cannot be expected” (2000, p. 53). During this initial phase, skills such as multiplication
drill and practice are essential.
Price, Mazzocco, and Ansari (2013) embarked on a quantitative study of the
brain. The researchers analyzed brain imaging to determine if the regions of the brain
used during simple mental math (fact retrieval) were related to higher order math skills.
The neuroscience approach was a unique way to conduct research on the topic. The
researchers suggested fluency in math facts does have an impact on the brain “facilitating
the learning of higher level mathematical skills” (p. 161).
“Working memory is a system devoted to short-term storage and processing and
is used in various cognitive tasks, such as reading, reasoning, and mental arithmetic”
(Imbo & Vandierendonck, 2007, p. 1759). Research on mental arithmetic and working
memory showed both simple and complex equations require the brain to use working
memory resources. Simple problems include equations such as 3+2 or 4x5. Complex
problems include items such as 45+78 or 35x76. Both of these types of problem solving
require the brain to use working memory resources. There is a difference in the way the
brain retrieves information, both executive working memory and phonological working
memory is accessed. Both executive working memory and phonological working
memory resources are needed to retrieve multiplication and division solutions.
“Furthermore, the acquisition of addition and subtraction skills and strategies is mainly
based on counting procedures, whereas the acquisition of multiplication and division
skills and strategies is based on the memorization of problem—answer pairs” (p. 1760).
21
Cognitive processing theories indicated it was difficult for the brain to perform
several tasks concurrently unless some of the required processes can be performed
without quickly, easily, automatically and with little or no working memory (Poncy,
Skinner, & Jaspers, 2007). Therefore, memorization and mastery of multiplication facts
is necessary for achievement of and advancement in mathematics when those basic skills
are a component step necessary for the more complex skills.
Constructivism
Many educators agree comprehension skills and rote memory skills are necessary
to achieve true fluency and automaticity in math facts (Baroody, 2006; Krudwig, 2003;
Wallace & Gurganus, 2005). Research has shown students who memorize math facts
free up short-term memory space, can quickly answer problems, make fewer mistakes,
and are more successful at math (Digitale, 2014).
The NCTM Principals and Standards for School Mathematics (2000) advocated
students not only memorizing math facts and algorithms, but understanding mathematics
and solving problems. The study suggested the use of manipulatives and choice of
learning activities. Use of the learning theory of Constructivism while supporting the
learning objective of problem solving, could be seen as conflicting with the need for
students to memorize math facts. Chung’s (2004) research on the effectiveness of the
Constructivism approach versus the Traditionalist approach discovered using
manipulatives during multiplication instruction for a minimal amount of time increased
student achievement. Teacher perception of using the Constructivism approach was
negatively impacted by a lack of classroom control. It was reported, the change of
student routine led to an increase in undesirable student behavior.
22
The Social Constructivism Theory is based on the belief that the student
constructs understanding and knowledge from his or her own personal experiences
(Greaney, 2015). The teacher is not seen as necessary for direct instruction, but rather as
a facilitator during the learning experience. The teacher provides the student
opportunities to learn and does not directly provide knowledge to the student. Greaney
(2015) argues
The problem with relying on a problem-based methods of teaching in primary
school, particularly with regard to the teaching of maths and literacy, is that
young students are still developing number sense (for maths) and decoding skills
(for reading comprehension) but without the automaticity required to attend to the
higher level cognitive processes involved in the understanding of the wider
problem.” (p.3)
Information Process Theory supports the belief math fact fluency is the fundamental
knowledge necessary for student success in advanced mathematics (Woodward, 2006;
Greaney, 2015). Automaticity allows cognitive space for complex problem solving and
is helpful to all students. “The pay-off for those who are able to quickly and effortlessly
recall such basic skills is too important to ignore” (Greaney, 2015, p. 6). Advanced math
skills such as factoring algebraic equations are simplified by multiplication fact
automaticity (Woodward, 2006).
Bailey & Pransky (2005) analyzed the Constructivism Theory in relation to
cultural dynamics. They concluded, Constructivism is likely the most widely used
system in education. It is based on the cultural beliefs, norms, and socio-economic status
of the educators primarily from the local middle class communities. The researchers
23
assert this could cause a conflict in the educational expectations from students. Instead of
all instruction based on the Constructivist approach, some instruction should be taught
directly and scaffolded for better student understanding and achievement.
Karen C. Fuson (2003) stated twentieth-century learning based on computational
fluency developed by rote memorization and drill and kill practice was no longer
sufficient for the twenty-first century learner. Problem-solving and computational skills
were attained by weaving the procedures with the understanding. Starting with the
problem situation returns equal or higher computational fluency.
Thinking strategies have been used to promote multiplication facts in the
classroom (Crespo, Kyriakides, & McGee, 2005). Developing a relationship and forming
patterns with numbers helps students understand and therefore learn the multiplication
facts. Once students begin to develop an understanding of the symbols and operations,
memorization can begin (Columba, 2013). For example: both x and / can be used to
signify multiplication. Literacy and concrete manipulatives can then be used for
understanding of multiplication. Repetitive practice which can be fun is just as important
as practice which was challenging.
Teacher Perception
Studies have shown teachers believe students who are not able to quickly or
correctly recall facts are unable to accurately answer more difficult questions (Frawley,
2012). Recall skills allow students to learn at a higher level of engagement because they
have a basic understanding in the four math processes (addition, subtraction,
multiplication and division), they may have less anxiety, and they can focus on harder
skills. Educators find students develop mathematical problem-solving skills without
24
simultaneously developing accurate fact fluency (Krudwig, 2003). When students lack in
fact fluency, this leads to deficiencies in higher math problem-solving. This is true not
just at the elementary level, but also with older student’s understanding of functions,
calculus, and trigonometry. Brain studies have recently indicated that when students
retrieve basic facts, neural networks in the brain that are associated with higher
mathematical skills are activated (Price, Mazzocco & Ansari, 2013).
A qualitative study was conducted on a Standards Based Math Improvement
Program (MIP) implemented in Ohio (Thompson, 2009). The MIP study reported an
overall response from teachers that the program, which had a Constructivist approach,
much like the Common Core, was undermined by a student’s lack of fact fluency.
Teachers opined that the Constructivist-based program, much like Common Core, with
the required amount of problem solving and explanation, was more difficult for the
students to achieve because of basic skills were lacking. Additionally, teachers
responded that the curriculum had a lack of “drill” practice necessary to learn facts.
Teachers acknowledged the approach to learning math was effective, just lacking in
skills.
Lisa Buchholz (2016) found similar issues as the curriculum in the
aforementioned Standard Based Math Improvement Program. The curriculum used in the
researcher’s district required teaching different strategies for understanding facts, but no
application of the strategies. Buchholz discovered that students needed to practice using
facts in order to not only understand them, but to learn them. The change to Common
Core standards caused teachers to change the way math instruction is delivered. Higher-
level thinking was deemed just as important as knowing math facts (Wagganer, 2015).
25
Strategies such as math talks were used and classroom math discussions, math
knowledge, and math fluency increased.
Teacher perception in fact fluency directly affects the way a teacher conducts
lessons (Stipek, Givven, Salmon, & MacGyvers, 2001). If an educator holds the belief
that number skills and memorizing multiplication facts is important, more instructional
time will be spent on counting strategies. Conversely, teachers who believe in
constructing answers and problem solving will budget more instructional time on word
problems.
In 2002, NCLB mandated the implementation of state standards in order for states
to receive federal funding (Klein, 2005). Research evaluated the resulting state standards
based on “the standards clarity, content, and sound mathematical reasoning, and the
absence of negative features” (p. 9). Nationally, the average grade applied to the
standards was a D with 15 states receiving Cs and 29 states receiving Ds and Fs.
California, Indiana and Massachusetts were the only states which received As. Klein
identified nine common issues with the standards:
1. Calculators were shown to be used an excessive amount. There is a role for
calculators, but not when it replaces the student’s ability to think about the solution of a
problem.
2. Memorization of basic numbers was not emphasized enough.
3. There was not enough focus on knowledge of standard arithmetic algorithms.
4. Basic fractional skills were not developed or practiced enough for foundational needs
in advanced math.
5. Patterns were given excessive attention.
26
6. Ineffective or overuse of manipulatives.
7. Ineffective or incorrect use of estimation.
8. Excessive and or incorrect use of probability, data, and statistics especially in
elementary education.
9. Mathematical reasoning and mathematical problem solving incorrectly developed and
scaffolded.
“Curriculum Focal Points” released by the NCTM set out to pin point for teachers
the most important standards which should be covered in each grade level (Cavanagh,
2006). These focal points recommended second grade focus on counting units in
multiples of ones, tens, and hundreds. By the end of second grade, students should also
understand numbers relative to place value and order numbers. By fourth grade, students
should understand multiplication, including “quick recall” of both multiplication and
division facts. Estimation and the use of correct methods for mental calculation of
estimation should be mastered at this level. Upon exiting sixth grade, students should
know the meanings of fractions, multiplication, and division. Students should be able to
multiply and divide fractions and decimals, as well as understand the relationship
between the two. Students should also be able to complete multi-step problems
(Cavanagh, 2006). This framework provided teachers a clearer vertical vision for math
concepts.
A study conducted on fifth through twelfth grade students and their teachers
found teacher perception had direct positive effect on student achievement (Campbell, et
al., 2014). Research showed teacher mastery of subject along with awareness of student’s
mathematical disposition predicted positive student achievement. Teacher perception of
27
the importance of modeling for students how to find solutions to tasks, and teaching skills
incrementally also impacted student achievement.
Stigler & Hiebert (2009) studied the American education system in comparison to
the German and Japanese education systems. Teacher perception emerged as one of the
main differences. In the United States, teachers perceived math lessons as “a set of
procedures for solving problems” (p. 89). The teachers also perceived skill practice was
not fun and felt it was necessary to embellish lessons with music, changing topics, using
real life objects (such as basketballs to measure), or finding ways to be entertaining. This
was not seen in an isolated event, bus as a cultural norm. Whereas the Japanese teachers,
who typically have higher student achievement in math, perceived mathematics as
concepts, rules, facts, and procedures and the relationships between them. “These
relationships are revealed by developing solution methods to problems, studying the
methods, working toward increasingly efficient methods, and talking explicitly about the
relationships of interest” (p. 89). Japanese teachers believe and act as if mathematics is
interesting and do not feel the need to be entertaining. Memorization also plays an
important role in the Japanese classroom.
Teacher perception regarding reform efforts is critical in the classroom
environment (Stigler & Hiebert, 2009). Policy changes and academic reform occur
frequently. Teachers are asked to implement programs, change teaching practices, and
teach new standards. Often there is a gap between policy and practice. Even when
teachers believe changes may benefit students and student achievement, making the
decision on which changes to make and actually implementing change does not always
occur. Classroom practices are directly related to beliefs, therefore teachers filter new
28
skills and learning through current belief systems (Stipek, et al., 2001). Research showed
teacher beliefs did not change much after student teaching and reading about teaching
methods does not impact classroom teaching. Truly changing instruction methods and
reflecting on student outcome does income teacher perception. This research supports
the implementation of Professional Learning Communities for teacher development in
schools.
Math Fact Fluency and Achievement
Is memorization necessary for fluency and automaticity of multiplication facts?
Multiplication is a base level math skill and a scaffolding skill for higher math such as
division, fractions, algebra, and estimation. Students who have not memorized facts could
find themselves behind on skills and experience a loss of confidence (Jarema, 2007).
NCTM stated in 2000, “Knowing the basic number combinations – the single digit
addition and multiplication pairs and their counterparts for subtraction and division – is
essential. Equally essential is computational fluency—having and using efficient and
accurate methods for computing” (p. 32).
The NCTM publication Principals and Standards for School Mathematics (2000)
noted students required a “balance and connection between conceptual understanding and
computational proficiency” (p. 35) in order to achieve true fluency. The main
computational focus points released include:
Computational fluency is an essential goal for school mathematics
The methods that a student uses to compute should be grounded in
understanding
29
Students should know the basic number combinations for addition and
subtraction by the end of grade 2 and those for multiplication and division
by the end of grade 4
Students should be able to compute fluently with whole numbers by the
end of grade 5
Student can achieve computational fluency using a variety of methods and
should, in fact, be comfortable with more than one approach (Russell,
2000, p. 156).
Fluency includes efficiency, accuracy, and flexibility (Russell, 2000). In order to be
fluent, students must be able to correctly find the solution to mathematic problems using
a variety of approaches in a manner which is easily followed and explained.
Studies were conducted to test the correlation between math fact fluency and
achievement. Does overall fluency have any effect on achievement? In 1993, Zentall and
Ferkis found students with fluency in math facts demonstrated higher performance on
problem solving tasks. They also concluded computational speed was a predictor of how
well a student could solve word problems. According to a study conducted by Nelson,
Parker, and Zaslofsky (2016), as students became more successful on single-digit math
fact testing, they also tended to improve on state testing. The study concluded fact
fluency had some relevance to math achievement up to eighth grade. While the afore
mentioned study used basic single-digit fact testing and state standardized testing for
correlation purposes, other studies have also shown a correlation between basic facts and
curriculum based math tests. A study at the University of Minnesota utilized curriculum-
based mathematics measures (M-CBMs) to determine if basic multiplication and division
30
math fact mastery and concept application could predict achievement on middle school
students (Codding, Mercer, Connell, Fiorello & Kleinert, 2016). These findings were
also dependent on grade level, but found both math facts and concept application skills
necessary to some degree for higher order math achievement.
Although the relationship between basic math knowledge and higher achievement
in math is a generally established relationship, the underlying reason was not fully
understood. Research conducted by Cowan et al. (2011) was employed to contribute to
the ongoing correlation between basic fact fluency and math achievement. In this study,
researchers observed similar results to other studies. Basic fact fluency skills were
typically learned between second and third grade. The skills overlap and were somehow
linked with conceptual understanding. The correlation study concluded that students with
higher math fact fluency also used math more both in school and outside of school.
Therefore, just like students who enjoy reading read more, students who understood the
basic concepts of math used math more and thus became more proficient due to the added
use of skills both inside and outside of the classroom. Working memory and executive
functioning were also indicated as necessary for children to master math (Cowan, et al.,
2011; Mazzocco & Kover, 2007).
The US Department of Education, National Mathematics Advisory Panel
(NMAP) (2008) found a lack of achievement in High School Algebra courses for United
States students. Algebra I is considered a gateway for advanced mathematics in high
school. This means completion of Algebra II has a direct correlation with completion of
college and potential workforce earnings. The NMAP determined:
31
Computational proficiency with whole number operations is dependent on
sufficient and appropriate practice to develop automatic recall of addition and
related subtraction facts, and of multiplication and related division facts. It also
requires fluency with the standard algorithms for addition, subtraction,
multiplication, and division. Additionally, it requires a solid understanding of core
concepts, such as the commutative, distributive, and associative properties.
Although the learning of concepts and algorithms reinforce one another, each is
also dependent on different types of experiences, including practice. (2008, p.
xix).
Procedural fluency according to the NCTM (2014) builds the foundation for
mathematical knowledge. In order for a student to be proficient in math, he or she must
be fluent. The memorization of facts should coincide with the conceptual understanding
of addition, subtraction, multiplication, and division. Each concept is important.
“Effective teaching strategies provide experiences that help students to connect
procedures with the underlying concepts and provide students with opportunities to
rehearse or practice strategies and to justify their procedures” (NCTM, 2014).
Summary
The purpose of the study was to research educator’s perceptions of the importance
on math fact fluency and understanding. The research shows memorizing basic facts has
a direct impact on student achievement. Students who do not memorize basic arithmetic
facts will struggle compared to their counterparts who memorize and conceptualize
multiplication in order to obtain fluency (Klein, 2005). Research has shown teachers
believe student automaticity has a direct relationship to student ability to correctly solve
32
increasingly difficult mathematical problems with less anxiety (Frawley, 2012).
Although educators believe achievement is possible without mastery of the facts, they
firmly believe mastery of the facts is beneficial. “The importance of automaticity
becomes apparent when it is absent” (Orefice, 2013, p. 15). The research showed fact
automatization allows the brain to access necessary areas of brain including working
memory for complex mathematical equations and problem solving.
33
CHAPTER THREE
Research Methodology
Introduction
The purpose of the study was to research educators’ perceptions of the necessity
of fluency in math facts. Memorization of math facts is a third grade standard in
Tennessee which students do not always master. This lack of mastery could have lasting
mathematical achievement. The research questions were:
What are educators’ perceptions of the importance of math fact fluency and
understanding? Do teachers perceive a difference in academic achievement based
on fact fluency? Why do they believe this? Do teachers base instruction on the
need for fact fluency?
Chapter Three includes the population and sample of participants in the study and
a description of the instruments the researcher used for the study. Research procedures
are explained. In conclusion, the data analysis methods used to answer the research
questions are identified.
Population
The population for the study was chosen from a small school district in Middle
Tennessee. The district serves a vast population of low socio-economic status families.
There are six elementary schools, three middle schools, three high schools and one
alternative school serving all grades located in the district. Schools were chosen based on
convenience for the researcher. The chart illustrates the demographics of the district.
34
Table 3.1
Demographics of Schools in Study
The researcher utilized three separate focus groups. Focus groups are useful
because they are a group interview about a particular subject or issue, and the researcher
can discover why and how the participant is thinking (Ary, Jacobs, Sorensen, & Walker,
2014). The focus groups were based on convenience sampling. The first group was
comprised of a group of six teachers who taught third grade. The second group was
comprised of a group of five teachers who taught seventh grade. The third group was
comprised of five teachers who were teaching high school mathematics.
The focus group participants with strong perceptions were invited to a follow-up
interview. The interviews allowed the researcher to obtain more information from
teachers who passionately believed fact fluency either strongly influenced or did not
influence overall math achievement. Along with follow-up interviews, in-class
observations were also performed to verify interview information. The observations
School
Total
Students
Total
Teachers
Free/reduced
lunch
Minority
students
SpEd
Students
E05 425 40 60% 13.1% 14.8%
HS10 636 44 55% 6.9% 9.2%
M14 627 49 56% 9.0% 17.0%
E15 393 39 60% 6.8% 14.5%
M18 561 43 38% 3.9% 12.8%
HS19 580 39 31% 5.5% 7.9%
E20 386 31 44% 5.1% 11.3%
E25 267 27 46% 6.3% 13.8%
E30 479 40 33% 4.8% 13.1%
M32 779 53 39% 4.4% 15.4%
HS33 738 50 32% 5.0% 6.2%
E35 349 36 59% 6.0% 16.3%
R40 24 10 0% 8.3% 33.3%
35
enabled the researcher to confirm that the teacher used instructional methods that
reflected his or her beliefs. Triangulation of data served for validation and reliability.
Setting. The school district that participated in this study was a small rural school
system in Middle Tennessee. The district is located in the middle of several of the largest
and wealthiest in the state. The school district is the largest employer in the county. The
county, according to 2015 census information, is made up of 93% white, 2% African
American, .5% American Indian, .5% Asian, and 2.8% Hispanic residents. The median
household income in the county was $52, 138.00. Over twelve percent of the population
of the county lived in poverty (U.S. Department of Commerce, 2015). The school district
serves a population of over 5,250 students and employees over 500 teachers and
administrators. There are three high schools, three middle schools, six elementary
schools, and one alternative school in the district.
Description of Instruments
The purpose of the study was to examine educators’ perceptions of the importance
of math fact fluency and understanding. A qualitative study was designed in order to
determine whether teachers believed math fluency had an impact on academic
achievement. Qualitative research allowed the researcher a broader perspective of what
is considered a complex topic through the use of in-depth interviews and observations
(Best & Kahn, 2003). This approach allowed the researcher to be “open and responsive to
its subject” (p. 76). The researcher used focus groups, follow-up interviews, and
observations to determine the educators’ perception of the importance of math fact
fluency.
36
Open ended questions, Focus Group Questionnaire, (Appendix A) were used with
the focus groups to elicit group discussions based on the research including: what are
educators’ perceptions of the importance of math fact fluency and understanding and do
teachers perceive a difference in academic achievement based on fact fluency? The
questions allowed teachers to elaborate on answers and corroborate viewpoints. These
questions allowed the researcher to determine if any of the participants had strong
viewpoints on the research subject.
Individual semi-structured interview questions, Individual Questionnaire,
(Appendix B) were then posed to three of the participants who expressed an extreme
viewpoint on the importance of math fact fluency and academic achievement. In order
for the researcher to understand individual teacher views, observations of classroom math
lessons were conducted.
Data Collection Procedures
Once the researcher obtained approval for the research from the school district,
the Institutional Review Board (IRB) (Appendix C), and the dissertation committee, an
email was sent or a phone call made to principals in the district. The email included a
request for permission to interview third, seventh, and high school math teachers and
their names. The permission for research letter from the district was provided (Appendix
D). Principals were asked to reply to the email with the names of their third, seventh,
and high school math teachers. The researcher emailed the teachers (Appendix E),
including principals in the email, or made phone calls to teachers to request participation
in the focus groups. The permission for research letter from the district (Appendix D) as
well as the consent letter (Appendix F) were provided to each participant at the focus
37
group. The researcher contacted volunteers, set up focus group times, and met with each
group individually in a neutral location. Participants were selected for each group based
on their current position, willingness to volunteer, and availability for discussion times.
Data were recorded from each session electronically and transcribed by the researcher.
The data collected in qualitative studies should be analyzed during the course of
study for patterns, relationships, connections and impressions (Mertens, 2005). The
research questions used for the interviews were based on teacher perceptions and
analyzed for common patterns. Data from the interviews can be found in Chapter Four.
Summary
The purpose of the study was to investigate teacher perceptions of math fact
fluency on student math achievement. Data were collected using the qualitative method.
Grade specific focus groups, semi-structured individual interviews, and classroom
observations were utilized for data collection. Data were analyzed for trends leading to
relationships between math fact fluency and teacher perceptions of math achievement.
Chapter Three was composed of an introduction, description of population and
instruments, and data collection procedures.
38
CHAPTER FOUR
Analysis of Data
The purpose of the study was to research educators’ perceptions of the necessity
of math fact fluency. The following research questions guided the study. What are
educator’s perception of the importance of math fact fluency and achievement? Do
teachers perceive a difference in academic achievement based on fact fluency? Why? Do
teachers base instruction on the need for fact fluency? This chapter discusses the analysis
of the data collected through focus groups, follow-up interviews with individual
educators and observations of individual educators.
Third grade, seventh grade, and high school math teachers within a rural
Tennessee school district participated in grade level focus groups. Focus groups produce
data from a group setting. Focus groups allow members to share experiences which are
similar or different while engaging in active discussions of experience or memories
(Harding, 2013). The Focus Group Questionnaire (Appendix A) was designed by the
researcher using empirical research and the focus questions of the study. Ten open-ended,
neutral, non-leading questions were used to guide discussions. These questions were
used during each focus group to elicit conversation and answers in the context of the
focus questions. Teachers who felt a strong inclination one way or another regarding fact
fluency were interviewed individually using the Interview Questionnaire (Appendix B).
The chosen educators from each grade level were then observed for instruction in fact
fluency compared to their perception of the importance of the need for fluency in math
facts based on focus group discussion and Interview Questionnaire answers.
39
The procedures of collection validate the data by means of triangulation.
Triangulation involves collection and analyzation of data using different methods or from
different sources for validation purposes (Carlson, 2010; Creswell & Miller, 2000). For
purposes of the study, focus groups, individual interviews, and observations were used.
Member checks were conducted in order to add credibility and reliability to the study.
Member checks involve “taking data and interpretations back to the participants in the
study so that they can confirm the credibility of the information and narrative account”
(Creswell & Miller, 2000, p. 127). Participants were invited to review the data and
narrative included in the study and make comments on the accuracy and validity of the
study. All teachers who provided feedback acknowledged the data were credible and no
adjustments were made.
Demographic Results
Participants were selected from a rural school district in Tennessee. Sixteen
teachers participated in the focus groups. Participants were selected based on
convenience. Fifteen teachers were female and one teacher was male. For the purpose of
anonymity and confidentiality, all participants will be referred to as female. Six educators
participated in the third grade focus group. The seventh and high school groups each had
five participants. All participants were provided with a copy of the consent form to sign
and read prior to participation in the focus group. No participant opted out of the focus
group discussion. All participants understood the possibility of being chosen for a further
interview and observation.
Table 4.1 displays participant demographics including, years teaching, years
teaching grade level, and self-measured familiarity with standards.
40
Table 4.1
Demographics of Focus Group Participants
Data were collected during face-to-face focus group interaction with the
participants at a neutral location within the school district. Each focus group was asked
the same ten questions. Question one asked basic information about the participant.
Questions two through ten asked open-ended questions based on the research questions.
Data were analyzed using the method of constant comparison, an analysis process where
as the data emerge they are compared to existing data (Lewis-Beck, Bryman, & Futing
Liao, 2004; Dye, Schatz, Rosenberg, & Coleman, 2000). Constant comparison is rooted
in the Grounded Theory approach. Using this analysis, the researcher looked for similar
statements in relation to the data whereby categories were adopted and comparisons were
implied. Fram (2013) recognized the importance of using constant comparison analysis
method “to identify patterns in the data and to organize large amounts of data so as to
abstract categories” (p. 20). Open coding allowed data to condense while constant
comparison analysis method allowed the researcher to identify patterns, create categories,
Participant
Focus
Group
Years
Teaching
Years Teaching FG
Grade Level
Familiarity with
Standards
P1 Third ½ ½ Somewhat familiar
P2 Third 20 9 Familiar
P3 Third 15 3 Very familiar
P4 Third 1 1 Somewhat familiar
P5 Third 5 4 Very familiar
P6 Third 4 1 Not very familiar
P7 Seventh 20 6 Very familiar
P8 Seventh 3 1 Familiar
P9 Seventh 7 7 Very familiar
P10 Seventh 18 18 Very familiar
P11 Seventh 15 12 Very familiar
P12 High School 1 1 Somewhat familiar
P13 High School 8 1 Somewhat familiar
P14 High School 4 2 Familiar
P15 High School 2 ½ 2 ½ Familiar
P16 High School 4 4 Somewhat familiar
41
and synthesize data into themes (Anfara, Brown, & Mangione, 2002). During the process
of identifying, coding, and categorizing the data, several themes emerged. Grounded
Theory can be used as an emergent method allowing the researcher to make choices
while analyzing the data resulting in the ability to see and correlate the data with
categories (Charmaz, 2008). Using constant comparison based in this Grounded Theory,
the researcher was able to establish the relationship among the categories in order to
identify themes and link them to the research questions.
Research Question One
The first research question was, “What are educators’ perceptions of the
importance of math fact fluency and understanding?” Once the data were analyzed,
several themes emerged related to this question. Fluency and understanding led the
educators in all three focus groups into a discussion on the importance of
conceptualization, or knowledge of concepts, versus the actual memorization of
mathematical facts. Fluency was also a discussion in the third grade focus group.
Standards and memorization were themes which also emerged in relation to this question.
Although all groups maintained the necessity of both conceptualization
understanding and memorization, 81% or 13 of the 16 educators eventually expressed the
need for memorization of math facts, especially multiplication facts, in order to continue
to efficiently and timely solve advanced mathematical problems. All third grade
participants expressed the need for math fact memorization. Teachers also acknowledged
mastery of multiplication and division facts as part of the current third grade standards,
therefore prominent in their classrooms.
42
Table 4.2
Conceptualization and Memorization
Grade level Participant Comment
Third Strategies are a way of seeing it. I see what it is, a visual, but in the
back of my mind it is still there, the need to memorize.
Third Memorization comes before fluency. Though maybe it is strategies you
memorize, then you are fluent.
Third Because you are memorizing them, you are fluently getting them.
Third They go together, fluency and memorization.
Third They don’t have the number sense they used to have and the flexibility
with numbers enough to master the facts. They are lacking in that area.
Third Lack of number sense to memorize.
Third I agree but for a third grade standard, memorization is more important
they have got to know them (snapping).
Third I want my kids to be fluent – they would need to memorize. It is hard to
be fluent without memorization (agreement).
Third I believe it is important to start with memorization. That is how I was
taught and how I teach, memorize all of the stuff and then learn how we
do it.
Seventh No way to do fractions without understanding your multiplication
tables.
Seventh Can draw out answers conceptually if understand multiplication.
Seventh If good at modeling can find the answer without knowing the facts, but
their speed and the time factor involved…
Seventh If they know multiplication is repetitive addition, they can find the
answer. If they understand repetitive addition they don’t necessarily
need to know their multiplication tables.
Seventh By the time they get to us they have to know the facts to solve what they
are doing. They have to know to do better in multi- step problems.
Seventh Need to know multiplication facts, if they spend too much time trying to
conceptualize, they are losing track of just where they are in the problem
to begin with.
Seventh As long as you have them memorized, the more knowledge you have,
you can work backwards and you can say, oh I see why the number
sense makes sense here but you memorized first.
Seventh If you don’t have number sense, you better memorize them because you
can’t do anything else.
High
School
I think it important to understand the concept, they need to know
multiplication is repeated addition.
High
School
Both are important, At base, facts help build the foundation.
High
School
Both. We need them to memorize for quick recall during work and
because the standards call for non-calculator sections.
43
While this is true, it is one of an overwhelming number of standards which does not
always receive time or attention. During the discussion, third grade teachers expressed
confusion and concern in regard to the definition of the word “fluency”. After much
discussion, one teacher expressed her definition as “Fluency means you have a flexibility
with numbers where you can manipulate them and work through them and memorization
means you don’t necessarily understand or know why, but you can find the answer.”
The teachers agreed this was a suitable definition and acknowledged the necessity of an
advanced ability in number sense in order to achieve fluency with multiplication and
division facts. Every one of the participants in the third grade focus groups agreed their
students were not academically capable of complete fluency in all mathematical facts due
to their lack of number sense. Seventh grade participants also discussed the third grade
standards and the procedures of “pushing” standards down to lower grades which results
in student standards not being age appropriate according to their perception to current
research and recommendations.
Standards were discussed in depth in the third grade and seventh grade focus
groups (Table 4.3). In relation to the first research question, the standards play an
important role. All of third and seventh grade educators agreed there were too many
standards to successfully teach all of them. Math fact fluency is a standard that is
supposed to be mastered in third grade however; 100% of third and seventh grade
teachers agreed this does not occur regularly in classroom practice. While third grade
teachers expressed frustration mainly with the number of standards, seventh grade
teachers also expressed frustration with the “gaps” students displayed because previous
educators did not teach all of the previous standards
44
Table 4.3
Responses in Relation to Standards
Grade level Participant Comment
Third One of the standards is makes sense of problems and persevere in
solving them but we want them to spit out the answers. They can’t
really persevere if they are being timed (snapping).
Third I have seen over the last few years my kids coming in with better
number sense than the previous years so I do feel like they are getting it.
It is getting like they are seeing more of it in Kindergarten and I feel like
you are starting to see a progression with it.
Third Common core is helping with number sense (3 agreements).
Third We are pressured as teachers to move on because of the standards. Not
enough time too many standards.
Third There is not enough time – even though we know that it is the
foundation, the struggle to balance. I, you believe it, but the struggle to
get the standards done.
Third You just practice those facts when you can. There is so much to hit and
so much to cover you just keep on going.
Third I think that is her point. I think it is a brilliant point. Our third math
standards, there are so many and so many to teach if we backed off of
some of those and really focused on multiplication and division it would
set them up for success.
Seventh 7th grade is overwhelmed with standards.
Seventh Along with just the standards, all of the language needs to be taught and
gaps filled.
Seventh I don’t feel like I am overwhelmed with standards, but along with
everything else. It is all of the gaps I have got to fill because they don’t
get their standards done.
Seventh Gaps in standards before me (4 in agreement).
Seventh Finding and filling the gaps in the majority of students while enriching
the about 4 who need it.
Seventh 7th grade is one of the toughest grades, not just the standards, it is the
depth you have to go into.
Seventh 7th grade is the hardest, every single question is an application or a word
problem.
Seventh Knowing the multiplication language in 7th and 8th is just as important as
the computation.
Seventh Only teaching the standards and moving on regardless is what causes the
gaps. That’s only going to hurt that kid in the long run and set them up
for failure later on.
45
Seventh grade educators also had an intensive discussion on the necessity of teaching
only their standards. One teacher expressed concern with only teaching grade level
standards not only for her students, but also for herself as a teacher:
I have sat in meetings with the math coordinator, and he has always said DO NOT
teach anything that is not in your standards and we jumped him. We said OK, one
of our standards is no longer order of operations. That went down to 6th grade. If I
do not teach order of operations, they cannot do some of our standards. And this
is what I told him! I said they teach it in 6th grade, I know they do. But if they
forget the stuff and that is taught way at the beginning of the year if I remember
right and if you think by the time they get to me they still remember order of
operations, and I’m telling you they don’t, I’m in big trouble. I’m telling you
those kids, you can’t teach solving an equation.
Participants then went on to discuss the necessity of sometimes picking and choosing the
importance of standards. One high school teacher made a significant comment on
standards. She responded “younger students think more openly… which is what the
standards are pushing for!”
All three focus groups closely related memorization to fluency. Discussion in all
three groups frequently circled back to the need for memorization of math facts. All of
the teachers agreed students who memorized math facts were more successful than
students who did not memorize math facts (Table 4.4).
46
Table 4.4
Teacher Perception of Fact Memorization
Research Question Two
The second research question was “Do teachers perceive a difference in academic
achievement based on fact fluency? Why?”
Grade level Participant Comment
Third Fluency is why memorize.
Third They need to know it for the ease. It just gets harder and harder,
especially when add long division.
Third Math just gets harder and harder; they just need to know it. They can’t
draw the arrays, they can’t do all that equal groups.
Third Now we have to do the groups and the number line and everything to
get in their brain.
Third If you know your facts you are good to go.
Third Students who memorize facts do better (All 6 teachers agree).
Seventh Timed tests need memorization (agreement).
Seventh You have to have fast recall and pattern recognition quickly.
Seventh You can have a conceptual understanding but you must know your times
tables (snaps fingers).
Seventh Math coordinator disagrees with this.
Seventh I believe you have to at least memorize to 10’s maybe not 12’s but
definitely 10’s.
Seventh Purpose of memorization is to speed up the process in math.
Seventh Purpose of memorization is fluency, to do better on multi-step problems.
Seventh Purpose of memorization is to arrive at an answer quickly to move on to
harder more complicated problems.
High
School
Memorizing helps students with fluency and accuracy in mathematics.
High
School
Memorizing helps with accuracy. Makes answering faster, helps with
linking things in the mind.
High
School
Students memorize for building a solid foundation.
High
School
Memorizing leads to fluency throughout lessons.
High
School
Memorization is for fluency, building blocks.
47
Table 4.5
Testing and Grades
Grade level Participant Comment
Third I would say the ones who struggle are the ones who want to rely on
those crutches, those arrays and repeated addition strategies, they just
don’t want to take the step and memorize them.
Third Even on the test [TCAP] my lowest kids were drawing pictures, I loved
it, but, you don’t have all this time, you have got to get going
(snapping). – If you had been practicing those facts like I told you at
home, you wouldn’t be struggling.
Third Look at my students now – end of the year – the ones with the high
grades know them. If I walk up to one of them and say what is 8x6 –
they know it.
Third Better on achievement tests when they memorize facts. (5 in agreement)
Third It takes less time and effort after 20 minutes they are over it. If know
the answers they are good if not they are over it and they are done.
Third Long division is just a killer when they get to 6th grade and if they don’t
know their facts.
Seventh They don’t necessarily excel, but they are definitely at grade level and
they don’t struggle as much.
Seventh I think they have a better chance of excelling, you are not going to spend
as much time on this and you can actually worry about what is being
taught.
Seventh Kids that don’t know them are already frustrated and they you are
adding these three steps or multi-step equations and they just quit.
Seventh 1st question for struggling students, do you know your multiplication
facts? (3 agreements)
Seventh Not necessarily just multiplication for me, also number sense.
High
School
I think student do perform well when they know and understand the
facts.
High
School
Yes, they do well when they know their facts and their age doesn’t
matter.
High
School
Yes, I think they have to know their math facts to do well on
achievement tests.
High
School
Not necessarily. Students who don’t know the facts may be have strong
number sense in other areas. However, if they have their facts
memorized, they find math to be easier and can feel more confident.
High
School
High grades come from intrinsic motivation, effort, and hard work.
48
Testing and grades were the main themes to emerge from the focus group data in relation
to academic achievement. Table 4.5 highlights participant responses in relation to testing
and grades.
Third grade teachers expressed a perception of the direct correlation between
academic success and memorization of multiplication tables. All participants noticed a
time factor involved with the speed students were able to complete problems, therefore
allowing students to feel successful in not only answering questions, but successfully
finishing within actual imposed or self-imposed time constraints. Seventh grade
educators also noted concern regarding time constraints both in class and during test
taking. Both focus groups mentioned math fact knowledge as one of the first questions
asked when evaluating needs or deficiencies of struggling students. Four of the five high
school educators contended memorization of the math facts was necessary for success.
Over all, high school educators believe student success, by the time students are in high
school, depends mainly on the student and the amount of effort the student is willing to
put forth. Therefore, the high school educators’ focus group discussion frequently circled
back to the assumption if the students in their school had not learned their multiplication
facts by the time they arrived to their class rooms, it was too late.
Research Question Three
The third research question was “Do teachers base instruction on the need for fact
fluency?” This question directly relates to how the educator perceives her own teaching
in the classroom (Table 4.6).
49
Table 4.6
In Class Instruction
During the third grade focus group discussion, 100% of the educators shared one or more
of the strategies used for in class direct instruction of math fact fluency for retention.
Grade level Participant Comment
Third I like if practice is more fun for my kids. It is a center. They get to put their
names on a chart. We just have fun with it and I get a reward. I’s not like I’m
going to be mad because you didn’t get this you didn’t master this number
right.
Third There is a fine line between teaching them the importance of memorization and
terrifying them and making them afraid of math and that is a hard line to walk.
Third They are stressed and we keep them stressed.
Third They are all terrified. Kids today have so many things thrown at them.
Third I do spend class time, but I also have to follow the standards.
Third We do a fact drill every day.
Third Throw the ball and do math facts when you catch the ball. I don’t do timed
drills at all.
Third I do timed drills but I know when I want to stop them but I don’t tell them
when I am going to stop them, that way they don’t know. I just know when I
want to stop them.
Third There is an overall push for kids behind in reading not math.
Third Encouragement is important.
Seventh Take time in class – no.
Seventh Not in my class.
Seventh RTI would be the place to learn multiplication facts.
Seventh By now they need intrinsic motivation to learn the facts.
Seventh We can give them the tools they need now.
Seventh Homework could be an app on the phone, math ninja app or math vs zombies.
As long as they were doing the math multiplication facts. Make it something
the can just do and say I have proof of 15 minutes or you made it to whatever
level instead of just doing 10 problems.
Seventh It is just the hardest to stop and go over multiplication facts.
Seventh You don’t want to embarrass them in front of their peers.
Seventh RTI is a great place for facts.
Seventh Sometimes it takes peer pressure for them to succeed.
Seventh Number talks helps keep them accountable.
High School No Time! Don’t believe they are that important. Too many standards.
High School Unfortunately we can talk about it for a few minutes, but I have to move
forward.
High School I don’t have students memorize math facts. I teach math talks which value
being able to talk about math creatively.
High School I tend to teach the fact that the facts should be memorized, but I know that
there is a good chance that they won’t have all of them memorized.
50
Differing views were obvious, whether educators employed daily timed drill methods or
only infrequently practiced fun games. None of the seventh grade teachers expressed
using direct instruction time for math fact instruction. Seventh grade teachers also
expressed concern at their grade level for student embarrassment when math facts have
not been memorized. Although high school educators still perceive fact fluency as
important, they expressed a resigned acceptance of their notion that the students could
use calculators and they did not have enough time to go all the way back to teaching a
third grade standard.
Individual Teacher Data
Third Grade. Based on the analysis of the data, one educator from each focus
group was chosen for observation. Triangulation is necessary for validity of data. The
use of focus groups to elicit interaction and comments which might not have been made
otherwise, individual interviews for specific information, and classroom observations
provided triangulation. (Carter et al., 2014). During the third grade focus group, one
participant consistently commented memorization was important. She stated the
following:
I see what it is, a visual, but in the back of my mind it is still there, the need to
memorize. They go together, fluency and memorization. Important to start with
memorization. That is how I was taught and how I teach, memorize all of the stuff
and then we learn how to do it. I would say the ones who struggle are the ones
who want to rely on those crutches, those arrays and repeated addition strategies
they don’t want to just take the step and memorize them.
51
Data analysis revealed her perception of math fact fluency and the effect on student
academic success was clearly strong. Because she believed she directly taught lessons on
memorizing mathematical facts she was asked to participate in the individual interview
and observation.
Her responses to the Interview Questionnaire reiterate this teacher’s perceptions
regarding the importance of math fact fluency and the effect on student academic
achievement. In addition, her responses clearly state her belief in her active teaching of
fluency of math facts during classroom instruction time. Question six asks if lessons are
planned or if time is allowed during class for memorization of math facts and for an
explanation of the use of that time. She responded “I do math drills daily. I use flash
cards, math games, fact drills on paper, etc… It is a yearlong commitment.” According
to her answers, she ascertained fluency was necessary for higher-order math skills,
students would struggle without fluency, and their future grades would be impacted. This
educator stated she supported the alignment of standards and believes learning standards
are intended to be scaffolded.
The researcher observed this educator in the classroom on September 6, 2017.
The objectives for the class were: “Interpret the factors and products in whole number
multiplication equations (3.OA.A.1). Fluently multiply and divide within 100 using
strategies such as relationship between multiplication and division or properties of
operations. By the end of 3rd grade, know from memory all products of two one-digit
numbers and related division facts (3.OA.C.7).” (TN Department of Education, 2016, p.
37). During the course of the approximate 80-minute block, both students and the
educator verbally expressed the need to memorize multiplication facts seven times.
52
Eleven multiplication strategies were directly taught or reviewed. This supported her
belief in the importance of memorization of math facts. Students were observed by the
researcher wearing bracelets made to assist with the multiplying by 5’s. Pictures of arrays
were posted around the room and commented on by the students as they entered the
room. The first ten minutes of the 80-minute block was devoted to bell work which was
a review of previous standards, including repeated addition. The next 25 minutes of the
block were spent on review of the multiplication facts focusing on multiplication by five.
The educator engaged students in direct instruction using a clock, pattern recognition,
repetition of both facts and answers. Students used flashcards and were sent to the board
in teams to play a review game which required quick recall of facts. The educator
engaged students in group discussions on strategies for finding answers if memorization
had not occurred yet using seven different methods prior to taking a paper based fact
drill. Before taking a teacher-timed fact drill on paper, teacher and students mentioned
the need to memorize facts three times. After completing and correcting the fact drill, the
teacher announced to the class “We want to memorize, that is where we are headed.”
The remaining block time was spent on arrays. The lesson included an engaging activity
where the students were asked to divide objects into groups, then everyone wrote onto a
whiteboard the following: a repeated addition sentence, multiplication equation, reversed
equation, and all skip counting patterns. Two students were then asked to take the
objects and form them into an array. After a discussion, another group of objects was
analyzed. During this activity, the students or teacher expressed the need to memorize
math facts three times, for a total of seven times during the entire lesson. Once the
students finished this activity, they were given an independent activity on arrays to
53
complete. The teacher then ended the lesson by wrapping up the discussion on arrays and
reminding students of their homework assignment due Tuesday. As noted in the focus
group and individual interview, fluency in math facts is not only important for third
grade, it is a standard that is supposed to be mastered.
Seventh Grade. The seventh grade focus group had one participant who
consistently responded, in contrast to the majority of the focus group, that memorization
or fluency of math facts was not necessary. This participant cited there were other factors
more important for student success. Those responses in the focus group included:
“If they know multiplication is repetitive addition, they can find the answer. If
they understand repetitive addition they don’t necessarily need to know their
multiplication tables. If you have number sense, you can figure the out through
other ways around it. First question for struggling students – whether or not they
can use rational numbers, fractions, decimals, can you put stuff on a number line?
It is not necessarily just multiplication for me. High grades if standards based,
number sense and reading skills. I use number talks to keep them accountable.”
Research showed significant correlation between learning deficiencies in math and
students who struggled in automaticity/fluency of arithmetic facts (Gersten, Jordan, &
Flojo, 2005; Poncy, Skinner, & Jasper, 2007). This educator had strong perceptions of
mathematical fluency which diverged from the focus group data and current research.
She participated in the individual interviews and in the observations.
The first question on the Interview Questionnaire related to the importance of
math fact fluency for student success. In answering this question, the educator reiterated
the perception that a student could be successful without fluency if he or she had number
54
sense. The only reason given that a student would be more successful having fluency
was when a student without fluency lost his or her place and had to re-read a problem
multiple times in order to complete it. The educator did not believe fluency was
necessary for success. In contrast, when asked about grades, the educator believed many
students that have lower math grades struggle with math facts. This was due to time
issues. The seventh grade focus group discussed the overwhelming amount of standards
that they were required to cover as well as the amount of “gaps” students seemed to have
because other teachers did not finish all of the previous standards. This educator, when
asked about standards and academic success, perceived it was necessary to teach to the
student, not necessarily the standards. “I feel we need to get to know our kids and how
they think and tailor our teaching to them. I have found that kids often do better when
they set the pace of my teaching.” When asked about class time for direct instruction on
math fact fluency, the educator responded,
I use some class time, but not for memorization. I am currently a huge fan of
“math talks” and feel that this helps students create a sense of number sense on
their own turf. They get to find what works for them.
Memorization is not emphasized, but the participant perceived she used class time to
teach strategies which led to fluency and number sense.
This educator was observed by the researcher during a seventh grade math class
which had a 45-minute block on August 31, 2017. The objective for the class was
students will be able to apply the rules for adding and subtracting integers to negative and
positive numbers with 95% accuracy. I can: add and subtract integers. This lesson
required the student to be fluent in mathematical facts in order to be successful. Warm
55
up questions were posted on the Promethean board for students to answer. A general
discussion was held on the procedure for the activity. The educator utilized a set of cards
for each student, red (disagree with answer), green (agree with answer), and yellow (have
questions about answer/discussion please). Think-pair-share activity was utilized with a
minimal amount of time given for each step. Discussion for each question was held,
begun by one student being chosen (name pulled from a bucket) to go to the whiteboard,
solve the problem, and explain how he arrived at the answer. The teacher facilitated the
entire process for each question. Cards were held up for student engagement. The third
problem was computational problem 102 – 3. The student went to the board and set the
problem up as a traditional subtraction algorithm. The educator led a discussion on the
ways to determine the answer. This discussion included using the method of counting on
fingers. The fourth problem was eventually reduced to 20-11. The educator had to
explain to a student the difference between a minus sign and a plus sign. The teacher
stated, “plus signs and minus signs and how well you read the problems will be your
nemesis all year.” Once the initial warm up problems were finished, the teacher moved
to word problems. The first problem was a rate problem. Part of the problem required
the multiplication of 25 x 10. The researcher observed approximately 60% of the class
knew 25 x 10 immediately with fluency and accuracy. Students understood to add a zero
to the end of 25 and arrive at the answer 250. The researcher observed 40% of the class
used alternative methods to determine the answer. One of the students explained he
knew 25 x 4 was like 4 quarters which equals one dollar or 100, so 25 x 8 would be 200
and then he added 50 to that to get 250. This was only a portion of the problem. The
teacher spent at least five minutes discussing the strategy of memorizing any number
56
times ten add a zero to the end of the number, 100 two zeros, etc. This is in contrast to
the educator’s perception of not allowing class time for memorization of multiplication
facts as this is clearly a memorization technique. The class continued with two more
word problems, one completed and one not completed before class ended. The educator
quickly reviewed negative and positive numbers and reminded students to clean up
before leaving. The researcher observed the teacher spending time in class helping
students find answers to problems which could easily be solved if students were fluent in
mathematical facts.
High School. The high school focus group discussed the necessity of fluency of
math facts, but they agreed by the time students reached high school, it was a common
occurrence to just hand them a calculator and not worry if they were fluent in math facts
or not. One high school teacher responded she didn’t believe students struggled or
excelled based on fact fluency, but they struggled because they didn’t give any effort, had
a fixed mindset, or were genetically inclined to struggle. She also ascertained
memorization was an important base for other concepts, but there was no time to teach
fluency in class. Because of these conflicting views, she was asked to participate in the
individual interview and the observation.
In answering the individual interview questions, the educator continued to assert
students did not need to be fluent in math fact to be successful, but fluency did shorten
the time necessary to complete assessments and assignments. She also stated they should
be more successful than others who did not have fluency because of the ability to “make
connections and process more quickly.” When asked individually about students who
struggle, the educator opined students who struggle do not know math facts. She stated,
57
They are not as confident in their answers and constantly rely on the calculator.
They also arrive at an answer slower or cannot grasp a standard as fast because
they are having to take time to figure out the multiplication at hand in the
problem. This does result in a lower grade because we cannot spend extra time in
the classroom on a standard that involves multiplication. They miss key pieces of
the concept because they are still grappling with the basic math.
Even though the educator understood the fundamental necessity of fluency of math facts,
she stated no lessons are ever planned for math fact memorization. High school math
standards are too far removed from rote memorization. During the focus groups, seventh
grade and high school participants discussed and agreed on the need to always be
teaching only the standards assigned to their grade levels. The high school educator did
state she would use class time if a lesson ended early to play a computer based game like
Kahoot to practice math facts. She stated since it was won with speed and accuracy her
students tended to be more engaged. Students were also offered after school tutoring and
access to a class website where links were posted for videos, games, and practice
problems specific to mathematic facts.
The researcher observed the educator during a 45-minute Math One, Freshman
class on September 14, 2017. The “I can” statement was listed for the students as I can
graph equations (ST4). The students entered the classroom, were seated, and found the
words Number Talk projected on the front board. As the students entered, the educator
reminded the students it was Thursday, Number Talk Day. She then reminded them they
had two minutes of silent think time to come up with as many solutions as they could for
the problem on the board. Mental math only. The problem on the board was revealed to
58
be 67 + 28. The students were given two minutes, then answers were given and put on
the board. One student answered 95, one answered 91. One student answered 85, but
quickly said they knew why and the answer was not put on the board. One student said
“I accidently multiplied.” The teacher then asked the students to explain how they
arrived at their answers. Many responses were given including the standard algorithm of
stacking the numbers, adding the tens then the ones place then adding them together, and
several different ways of rounding then modifying the answer. All of the ten methods
explained by either the teacher or the students required the student to have fluency with
math facts. The students who arrived at incorrect answers made simple computation
mistakes. Although the high school teachers said as a group they did not spend class time
on fluency of math facts, if they are all participating in “math talks,” they are all spending
some time in class with fact fluency. This portion of the class block took approximately
10 minutes. The rest of the class block was spent on word problems which could be
solved using either an equation or a table. Although no direct instruction time was spent
during any more of the block teaching fact fluency, the lesson required the students to
multiply and divide in order to work all of the equations. The first word problem was an
equation which could be written 10 + 3x = 15 + 2x. The purpose was determining the
number of movies necessary to rent in order to decide between joining two video
memberships. As the educator stated in both the focus group and the individual
interview, fact fluency is necessary in order to quickly solve the problems presented to
them in high school. Without it, students take too much time and cannot deduce the
correct answer.
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Summary
Chapter Four provided an analysis of the demographic and survey data. The
purpose of the study was to investigate teacher perceptions of math fact fluency on
student math achievement. In this study data were collected from third, seventh, and high
school educators in a rural Tennessee school district. The research questions were: What
are educator’s perception of the importance of math fact fluency and achievement? Do
teachers perceive a difference in academic achievement based on fact fluency? Why? Do
teachers base instruction on the need for fact fluency? Data were collected using focus
groups, individual interviews, and classroom lesson observations. Through open coding
of data and analyzation using constant comparison, themes emerged from the data
relevant to the study research questions.
Chapter Four included demographic information. Additionally, data were
presented from focus groups, individual interviews, and classroom lesson observations.
Chapter Five contains a summary of the study, discussion of research findings based on
the data, and recommendations for further research.
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CHAPTER FIVE
Conclusions, Implications, Recommendations
The purpose of the study was to research educators’ perceptions of the necessity
of math fact fluency. The study investigated teacher perception of fact fluency on student
achievement and whether or not teachers transferred their perceptions into classroom
lessons. Educators located in a rural Tennessee school district participated in three grade
level focus groups. Third grade, seventh grade, and high school educators were selected
by convenience sampling to respond to questions using the Focus Group Questionnaire
(Appendix A). Participants with strong views from each focus group were selected to
participate in individual interviews and observations. The educators chosen responded to
the Individual Questionnaire (Appendix B) and were observed teaching an exemplary
class lesson by the researcher. The data were analyzed using open coding and constant
comparison method. The following research questions guided the study:
What are educator’s perception of the importance of math fact fluency and
achievement? Do teachers perceive a difference in academic achievement based
on fact fluency? Why? Do teachers base instruction on the need for fact fluency?
The purpose of this chapter is to provide a summary of the findings, offer conclusions,
discuss implications, and make recommendations for further studies.
Summary
The participants for this study included sixteen educators from a rural district in
Tennessee. At the time of the study, six of the educators were teaching in third grade
math, five were teaching seventh grade math, and five were teaching high school math.
Fifteen of the educators were female and one was male. All were referred to as female to
61
protect anonymity. The third grade educators had approximately 46 years total teaching
experience with an average of approximately three years teaching experience in third
grade. The seventh grade educators had approximately 63 years total teaching experience
with an average of nine years teaching experience in seventh grade. The high school
teachers had approximately 20 years of total teaching experience with an average of
approximately two years teaching experience in high school mathematics.
Conclusions and Discussion of the Findings
Focus group data allow the researcher insight and information on how people
“think, feel, or act regarding a specific topic” (Freitas, Olivera, Jenkins Popjoy, 1998, p.
1). The research questions driving this study revolved around teacher perception of
student achievement based on math fact fluency. Data collected, identified, coded and
analyzed using open coding and constant comparison method, revealed categories
relevant to the research questions. Using the thought perception is reality, the researcher
triangulated the data by conducting individual interviews with participants within each
focus group and then conducted observations necessary to obtain data to correlate
whether or not teacher perception of necessity of mathematical fluency was actually
being implemented into actual lesson plans.
Research Question One
The first research question was, “What are educator’s perceptions of the
importance of math fact fluency and understanding?” Analysis of focus group data in all
three grade levels clearly led to the majority of educators expressing the belief of the
importance of fact fluency and understanding impacting grades, in class ability, and
achievement. Themes emerged from data including conceptualization versus
62
memorization, in class participation and abilities, standards, and fluency. All focus
groups discussed the impact of memorization and 81% or 13 of the 16 educators
eventually expressed the need for memorization. Fluency was discussed and definitions
agreed upon. Based on the data, the participants would add memorization into the
definition of fluency. All of the focus groups eventually discussed their perception of the
necessity of memorization. The data showed third grade teachers taught memorization of
facts, according to their standards, along with concepts using methods such as arrays,
groups, and word problems. All of the third grade participants agreed their students were
not capable of complete fluency due to their ages. The teachers who taught the older
students expressed a similar desire for students to have memorized facts, but stated they
would not spend class time for rote memorization. Seventh grade and high school
teachers were more concerned with student memorization based on their conceptual
understanding and experience with working problems. This was evidenced by the older
grades use of “number talks.”
Seventh grade participants also discussed age and agreed standards were being
“pushed back.” The teachers agreed students were not capable of mastering standards
due to standards not being age appropriate. Third grade teachers expressed their concern
with an overwhelming amount of standards they were required to cover. Seventh grade
teachers believed they not only had too many standards but their students had “gaps” they
were required to locate and fill where teachers before them did not cover standards or
students did not retain the information. High school teachers did not express concern
over standards, they were resigned to the fact students either knew material or did not.
63
Individual interviews and observations were conducted with an educator in each
focus group. The participant from the third grade group answered repeatedly during the
focus group and during the individual interview her strong belief regarding the necessity
of the importance of math fact fluency and understanding. Her observation directly
correlated with her beliefs. During class time, the students or teacher were observed
seven times repeating math facts needed to be memorized and the educator reviewed or
introduced 11 strategies for memorizing or multiplying during the observation. She
confirmed the first research question and is practicing it within her classroom instruction.
One seventh grade teacher expressed a different view from the rest of the focus
group. Although the educator believed fluency was important, she did not feel it was
necessary and expressed knowledge of number sense was sufficient for a student to
succeed. During the individual interview, she maintained this position. The researcher
noted during the observation, although her perception was fact fluency was not vital, she
allowed for approximately 5-7 minutes of a 45-minute class block to explain to students
the “easiest” way to arrive at the answers to multiples of ten equations such as 10x26,
100x26, etc. This is in direct contrast to her previous statement that fact memorization
was not important. Most students were figuring out the answer using various methods
which she previously stated was acceptable. However, during instruction, those methods
were time consuming.
The high school participants acknowledged resigned acceptance that by the time
students reach them if they do not know their facts, they use calculators. The research
questions asked do teachers perceive fact fluency is important and the majority agreed
fluency was important. This was important because students would be expected to take
64
assessments where they would not be allowed to use calculators. Although the observed
participant stated she did not have time to focus on math fact fluency because of time
constraints in the classroom, she did include “math talks” in her classroom routine. The
researcher observed a “math talk.” The talk included discussion on how to mentally find
the answer to a two digit plus two-digit addition problem. This type of discussion took
approximately 10 minutes of a 45-minute block focused on math fact fluency and used
approximately 20% of her class time. This is a considerable amount of time and is in
opposition to what she perceived she is practicing during her class time.
All participants asserted math fact fluency was important. Although seventh grade
and high school teachers did not feel they had time in class to spend on math fact fluency,
both observed teachers used direct instruction time for math fact fluency.
Research Question Two
The second research question was, “Do teachers perceive a difference in academic
achievement based on fact fluency? Why?” According to the literature review, students
who were academically low-achieving, typically display difficulty with automaticity and
memorization of math facts (Woodward, 2006). The research question correlated with the
data themes of testing and grades. All of the third grade participants agreed that students
who were higher achieving and had higher grades had memorized math facts. During the
observation of the third grade participant, the teacher spent the majority of the class time
reviewing strategies, practicing repetitive facts, and introducing new memorization
strategies. Additionally, traditional written fact drills were completed in order for student
preparation for assessment. Seventh grade and high school participants did not express
the necessity of fact fluency for academic achievement as strongly as third grade
65
participants. The researcher questions if this gap is due to the fact students at these grade
levels have learned coping skills necessary for achievement. For example, when asked to
find the product of 25x10, seventh grade students worked out the answer breaking the 10
into 4 and eventually arriving to the answer of 250. Both seventh grade and high school
teachers expressed concern in the time and confusion factor involved when students did
not master facts. Teachers also cited concern for students having possible issues during
standardized testing.
Research Question Three
The third research question was, “Do teachers base instruction on the need for fact
fluency?” This question was the most interesting. Perception is reality, a widely held
view, should lead to teachers’ intentional preparation of lessons and activities in line with
their perceptions. The researcher observed only one teacher actually basing instruction
intentionally on the need for fact fluency. The third grade focus group participants
agreed one of their standards was mastery of math facts. Standard 3.OA.C.7 requires that
students “Fluently multiply and divide within 100 using strategies such as relationship
between multiplication and division or properties of operations. By the end of 3rd grade,
know from memory all products of two one-digit numbers and related division facts” (TN
Department of Education, 2017, p. 37). The third grade focus group also agreed it is a
standard which is extremely time consuming and class time is not spent on mastery. The
majority of the participants agreed students are expected to memorize the facts at home.
Teachers expressed playing games as sufficient coverage of the standard. The third grade
teacher who was observed used this standard as the basis for her lesson plans. The data
she provided in the focus group, individual interview, and classroom observation
66
correlated. Her perception drives her instruction and aligns with her lesson planning.
The seventh grade and high school focus groups including the individual interview
participants did not see the value in including math fact fluency instruction in class.
Comments mentioned that this should be completed outside of class, during RTI, or as
homework. Other comments indicate that there is no way due to lack of time. In direct
contrast to the perception of both focus groups, both teachers observed spent considerable
time in class directly instructing students on fluency of math facts. After attending all
focus groups, interviewing individual teachers, observing individual teachers, and
analyzing data, the research indicated teachers innately believe math fact fluency is
necessary for student achievement and growth; therefore, when presented with a learning
opportunity in the classroom, teachers will take the time to facilitate direct instruction of
math strategies for fact retention and memorization regardless of the standards being
taught or grade level.
Implications
Findings based on the data indicated the number and scope of standards required
by the state for each grade level to master is perceived to be exceptionally high by all age
groups in the study. Third grade educators maintained the amount of standards to be
covered was overwhelming. Seventh grade educators expressed not only an
overwhelming amount of standards, but also “gaps” to fill due to the inability of previous
grade levels to teach all standards. The high school teachers expressed a general attitude
that it is “too late” by the time students reach high school to worry about what has not
been learned.
67
Based on the research, all teachers should devote some class time to focus on
review and retention of math facts. Math fact fluency is necessary for student
achievement. Hoping students learn math facts at home or assuming the next teacher will
fill in the “gaps” is not beneficial to the students. Math fact fluency is necessary for
students to accurately, efficiently, and timely complete higher order math problem
solving and computation.
All groups expressed lack of memorization or fluency in mathematical facts to
cause many students to struggle. The frustrations expressed were described as a lack of
speed, accuracy, “getting lost” in a problem, and struggling on timed achievement testing.
Information-Process theory supports the necessity of automaticity in mathematical facts
in order for students to achieve success in higher order mathematics (Woodward, 2006).
The research indicates all educators should understand the necessity for fluency of math
facts. Care should be taken in the classroom to distinguish between rote memorization
and memorization in order to build fluency. For example, connecting conceptual
knowledge and linking realistic problem solving with learning multiplication facts
promotes deeper understanding for students resulting in higher performance on
standardized testing (Wallace & Gurganus, 2005). Using “math talks” to target specific
areas of need within the classroom in order to build memorization for fluency, allowing
students to work problems together and discuss answers would be a productive use of
class time for all students.
Recommendations for Future Studies
This study was conducted with third, seventh, and high school public school
teachers from a rural area in Tennessee. Further studies should include educators from
68
schools which are not teaching only from Tennessee State Standards. This could include
but not be limited to private, charter, or home school settings.
Future research could be conducted to include various grade levels.
Further research could be conducted to include teacher perception of other
concepts such as literacy.
Further investigation could be conducted to include student perception or
administration perception.
The availability or usefulness of professional development for teachers addressing
time management or managing of standards in relation to mathematics could be a topic
for further research.
Quantitative data could be introduced into the study using surveys and the survey
population could be expanded to include more educators.
Conclusion
The study was significant because prior research showed a correlation between
fluency and student success. “Fluency involves much more than learning a skill; it is an
integral part of learning with depth and rigor about numbers and operations” (Russell,
2000, p. 158). This study concluded teachers perceived the importance of math fact
fluency for all students. However, teachers of middle school and high school students
were unaware of the actual time devoted during the classroom to math fact memorization
or fluency. Therefore, perception may drive instruction subconsciously. The researcher
initially hypothesized most, but not all teachers, would passionately believe fact fluency
was critical for further math success if they had any teaching experience at the
elementary level. Although all teachers supported the need on some level for math fact
69
fluency, the only teachers who truly verbalized the passionate need for fact fluency were
the elementary teachers. With a push for standards based grading, these beliefs would
coincide with standards taught. Middle and high school teachers would found ways to
supplement “gaps” where previously taught standards were not mastered, including math
facts. This could include the use of calculators. The application of “math talks” and
“math tasks” in the classroom setting also allowed the teacher to incorporate
memorization through conceptual understanding and discussion.
70
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Appendices
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Appendix A
Focus Group Questionnaire
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Focus Group Questionnaire
1. What is your name? How many years of experience do you have teaching math in (third,
seventh or high school)? How familiar are you with the math standards?
2. What is the purpose of memorizing math facts?
3. What are the reasons for a student having high grades? Low grades?
4. What are your perception for the reasons a student excels? Struggles?
5. Why do you believe some teachers spend time with multiplication facts and some don’t?
6. Why do some students know their facts and some don’t?
7. What really happens in the classroom when they don’t know their facts?
8. Could you explain if it is more important to understand the concept or memorize the facts?
9. Do you perceive the way you teach is based on your current belief of the importance of the
math facts? For example, if you believe the facts are very important for students to memorize,
do you devote class time to rote memorization?
10. Do you perceive students who know their facts do better on achievement tests? Does the
student age have anything to do with this?
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Appendix B
Individual Questionnaire
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Individual Questionnaire
1. What is your perception of the importance of math fact fluency for successful students?
2. Does academic success, in your opinion, depend on the systematic mastery of standards?
Must they be mastered in the order the state has presented them?
3. Do you feel students who have not mastered the math facts struggle because of it? Have
lower grades? How do you handle that?
4. Are you aware research shows early math exposure and ability has an effect of future math
and literacy achievement? Does this impact your lesson planning or teaching? Do you believe it
is true? Why or why not?
5. Does the vertical alignment of standards effect the way you teach? Do you agree with and
understand the alignment of the standards? Do you agree with the mastery of standards?
6. Do you allow class time or plan lessons for memorization of math facts? Please explain.
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Appendix C
Institutional Review Board Approval
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Appendix D
District Permission to Research
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Appendix E
Email to Teachers
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Hello Everyone!!
I know it is late on Friday and everyone has had a long week! I am
sending all of you this email to ask you to participate in a focus group for teachers who have or are teaching third grade math. As some of
you already know, I am working on my doctorate at Carson-Newman University and I am in the process of collecting data for my
dissertation on Teacher Perception of Math Fact Fluency, Student Achievement and Classroom Practices. I have received permission
from Stacy Brinkley to do some research with third, seventh and high school teachers.
This focus group would just require you coming together as a group
and answering some questions about your perceptions on third grade
standards and how you implement them in the classroom. I can come out to Kingston Spring and meet with you and it would only take about
an hour.
Please email me back and let me know if you would be willing to participate. I know the next few weeks are busy for all of you, but I
would like to come out the week of May 15-19th. If there is a day in that week from 3:30-4:30 that absolutely does not work for you, if you
could let me know, maybe we could come up with a day that is convenient for everyone.
Thanks so much for your help in advance! I look forward to working
with all of you!
Andrea Bringard
Dean of Students
Cheatham Middle School This email may contain privileged, confidential, or other legally protected information. If you are not the intended recipient (even if the email address above is yours), you may not use, copy, or re-transmit it. If you have received this by mistake please notify us by return email, then delete.
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Appendix F
Informed Consent Document
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Appendix G
Permission to Reproduce
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