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Section 1: The Problem
Introduction
The competency to solve various mathematics problems is critical to a student’s
educational success. Solving mathematics problems dependsent on a student’s basic
understanding and application of mathematics concepts as well as critical thinking skills
(Wilson, 2009). Mastery of these concepts and the skills are critical to a student’s future
educational success.
In today’s classroom, middle school students are deficient in their ability to solve
grade-appropriate minimum mathematics competencies problems due to weak
problemsolving and critical thinking skills. For these students, the challenges of solving
mathematics problems begin in the early grades and are compounded as they move into
the higher grades (Cotik & Zujlan, 2009). In the early grades, students initially learn to
solve simple mathematics problems through various exploratory instructional
mathematics practices: Children use their senses and manipulatives to count, add,
subtract, and multiply (Robelen, 2012). Teachers may start with a variety of activities
which could include rhymes and songs, riddles and clapping games to introduce the
basics of problem solving so that students are engaged in problem solving (Rapp, 2011).
These activities allow the students to problem solve by using conceptual understanding
techniques. Once students are introduced to these exploratory instructional mathematics
practices, they are then expected to work basic computations of addition, subtraction,
multiplication, and division.
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But in the upper elementary grades and middle school, many teachers strive to
teach mathematical concepts from a theoretical perspective without first engaging
learners or appealing to the sensory aspects of the learning process (Holmstrom, 2010).
An example of this could include teaching students traditional methods for multi-digit
multiplication without showing them conceptual methods such as using an area model.
This shift in instructional practice may leave students questioning their ability to use
critical thinking skills to solve mathematical problems (Holmstrom, 2010). The focus on
traditional, rather than multisensory instructional practices, can reduce students’
confidence in problem solving and performance (Rapp, 2011). For some students, the
result might be poor mathematics performance.
Data obtained from 2011 state standardized tests revealed that some students in a
Colorado urban school district struggle with demonstrating mastery of required
mathematics concepts. An average of 37% of seventh grade and eighth grade students in
this district failed to meet minimum mathematics competencies, as measured by the 2011
state standardized tests (Colorado Department of Education [CDE], 2011). Based on this
result, this project study gathered data from a panel of middle school mathematics
teachers on the instructional practices that could improve student mastery of mathematics
concepts and content related to the middle school National Council of Teachers of
Mathematics (NCTM) Content Standards. By using appropriate problem-solving
strategies for learning mathematics concepts, a student may more easily master the
complex mathematics concepts presented in the middle school mathematics content
standards of the NCTM middle school mathematics content standards. This student
mastery could also impact mathematics achievement scores (Cole, 2010). Educators may
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benefit by learning about instructional mathematics practices that have proven successful
for other educators.
Based on a literature review of instructional practices for middle school
mathematics students, this project study explored the perspectives of a panel of middle
school mathematics teachers on instructional practices for learning the concepts and
content of middle school mathematics. The perspectives were used to inform educators
and they can help plan future mathematics instruction. Additionally, these instructional
matthematics practices may ensure alignment with NCTM content standards of middle
school mathematics.
In the following sections, Definition of the Problem, Rationale, Evidence of the
Problem at the Local Level, Evidence of the Problem from the Professional Literature,
Definitions, Significance, and Guiding/Research Questions, the problem of low levels of
mathematics achievement for middle school students at the local and national level is
discussed. It includes a literature review and a theoretical framework that is related to
mathematics comprehension. Finally, the implications of this project are discussed.
Definition of the Problem
In their initial years of education, teachers are helping students to develop
competencies necessary to solve basic mathematics problems. During this time, students
use manipulatives for a visual and kinesthetic way to learn; manipulatives help them to
retain the concepts (Robelen, 2012). Visual and kinesthetic instructional mathematics
practices are examples of sound, instructional mathematics practices that are used help
students master difficult concepts. Students must learn visual and kinesthetic instructional
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mathematics practices if they are to expand their skills and move forward in the learning
process (Rutherford et al., 2010).
Instructional mathematics practices extend into all mathematics areas as well as
other academic subjects . As students advance through middle school, some students
retain the ability to apply the instructional mathematics practices necessary to solve
complex problems. Others students emerge with a deficit in using what they have learned
from various instructional mathematics practices or transferring them to more complex
problems or problems in different types of mathematics. According to Rutherford et al.
(2010), some students consistently struggle to solve mathematics problems and are often
unable to understand the mathematics concepts necessary for success in various types of
mathematics or in practical applications. These students must navigate complex
mathematics problems using only traditional instructional mathematics practices (Cotik &
Zujlan, 2009). These traditional methods do not help students gain a deep level of
mathematics concepts. Without more appropriate instructional mathematics practices,
students may become disheartened, stymied, and exhausted by what they consider to be a
series of random symbols and variables because they lack the critical thinking skills,
problem-solving skills, or ability to understand the mathematics concpets required to
reach solutions (Erden & Akgül, 2010).
Instructional mathematics practices should be specifically taught. Some middle
school mathematics teachers may need more information about these alternative teaching
approaches that could improve student achievement on mathematics assessments
(Robelen, 2012). Although some mathematics educators are successful in explaining
difficult concepts, these same educators may not know about newer instructional
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practices or they may need practice in helping students grasp the concepts in all areas of
mathematics (Perrit, 2010).
For the majority of students, mathematics achievement at the middle school level
may be part of a nationwide and international issue. While some gains have been made in
recent years, middle school students in the United States are behind at least eight other
countries in mathematics achievement (McKinney & Frazier, 2008). The gap includes
skills in number sense (numbers and operations), algebra, geometry, measurement, data
analysis, and probability. If middle school students in the United States are to compete in
a worldwide economy, the gap needs to narrowed or closed.
Consistent with nationwide mathematics statistics, one Colorado urban school
district is dealing with these same obstacles. Its students are evaluated on their ability to
solve mathematics problems measuring state standards, including all of the types of
mathematics (CDE, 2011). In 2011, 36% of urban Colorado eighth graders scored
Proficient or higher on the Colorado Student Assessment Program (CSAP, CDE, 2011).
Although the 2011 eighth grade mathematics CSAP scores have improved since 2009,
these data still indicate that 64% of urban students tested below the Proficient level.
These scores indicate a need for appropriate instructional mathematics practices that help
all students master abstract and more difficult middle school mathematics concepts. If
these practices were infused successfully into existing curricula, gains could decrease the
achievement gap and could encourage student proficiency in mathematics.
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Rationale
Evidence of the Problem at the Local Level
The CDE outlined the Colorado State Standards, including information all
elementary and secondary students should retain as a product of their learning in a
Colorado public school (CDE, 2011). This framework includes standards that were
created using the NCTM middle school mathematics content standards /framework and
the Core Content standards. Each year, parts of the Colorado Student Assessment
Program (CSAP) are administered to students in Grades 3 through 10, and evaluate a
student’s level of achievement in four subjects: reading, math, science, and writing (CDE,
2011). Outcomes are categorized as unsatisfactory, partially proficient, proficient, or
advanced (CDE, 2011). Partially proficient or unsatisfactory indicates that a student has
not met the minimum expectations (CDE, 2011).
Based on data analysis, some middle school students at a Colorado urban school
district struggle to fully comprehend concepts in mathematics as evidenced through local
test scores from the CSAP (CDE, 2011). For instance, in this specific Colorado urban
school district, data from the mathematics CSAP data in 2011 revealed that an average of
37% of seventh and eighth grade students scored at the Proficient level or higher (CDE,
2011). According to these data, 63% of the middle school students in this district did not
pass the mathematics assessment in 2011. These data indicate a potential gap in student
learning and a failure to meet minimum requirements of the standards.
Even though inner city schools have demonstrated small gains in mathematics
achievement, instructional practice needs to be addressed to make bigger gains
(McKinney & Frazier, 2008). This study solicited feedback from a panel of middle school
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mathematics teachers on which instructional mathematics practices would help students
to better understand middle school mathematic instruction. By means of the panel, I
collected research-based instructional mathematics practices that follow best practice and
align with the concepts taught and Colorado’s mathematics standards, which, ultimately,
may improve middle school students’ mathematics achievement. The results of the study
are expected to help teachers improve students’ understanding of middle school
mathematics concepts.
Evidence of the Problem from the Professional Literature
Obstacles to student mastery of middle school mathematics concepts might be due
to a scarcity in professional development training guides that is not useful for middle
school teachers providing effective instructional mathematics practices (NCTM, 2000).
Obstacles might also be due to an increase in the level of student frustration when
performing more complex mathematics problems (Erden & Akgül, 2010) and the
persistent use of mostly traditional instructional methods to teach students how to solve
mathematics problems (Holmstom, 2010). When these factors are combined, student
mastery of mathematics concepts may be limited.
Student achievement in all mathematics areas requires mathematics teachers to
have a thorough knowledge and the ability to successfully teach the instructional
mathemtaics practices necessary for students to solve complex mathematics problems.
Many of these teachers are unapprised of the significant role these mathematics
instructional strategies play as they relate to student success, ability, and learning (Cave
& Brown, 2010). Mathematics teachers may be unaware of mathematics instructional
strategies and instructional methods that, once demonstrated, would empower some
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students to grasp complex mathematics problems successfully (Erden & Akgül, 2010).
Other teachers may be confronted with pressure to finish their mathematics curricula
within the school calendar, and may not deem mathematics instructional strategies as a
priority in their classrooms (Rapp, 2011).
Combined with day-to-day teaching hurdles, inadequate opportunities for
professional development exist for mathematics teachers who hope to effectively
integrate mathematics instructional strategies into their curricula (Erden & Akgül, 2010).
Perrit (2010) affirmed that it is the obligation of all teachers to inspire and use a variety of
instructional strategies to help students in becoming stronger mathematics students.
Although mathematics teachers may be considered experts in their fields, they may face
struggles in explaining and teaching mathematics content to students as a consequence of
a deficiency of knowledge and time to incorporate effective mathematics instructional
strategies into their curriculum.
The selection and use of effective mathematics instructional strategies in middle
school is lacking, in spite of federal and other programs implemented to enhance
mathematics instruction (Rutherford et al., 2010). Bottge, Rueda, Grant, Stephens, and
Laroque (2010) asserted that as students advance through their elementary school career,
they are progressively exposed to more complex mathematics problems. According to
Robelen (2012), much consideration has been given to the best mathematics instructional
practices that focus on elementary students, however, little has been given to middle
school students struggling with mathematics. Research through the National Mathematics
Advisory Panel (NMAP) demonstrated that as students arrive at their middle and high
school experiences, much elementary mathematics instruction supports are not available
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(NMAP, 2008). These supports can include strategies such as active learning, visual
support, kinesthetic support with manipulatives, and musical mathematics songs to help
students remember information. Many students will have moved on to challenging
disciplinary NCTM middle school mathematics content standards using only traditional
instructional strategies to solve problems.
The capacity to decipher complex mathematics problems is a fundamental
component in most subject areas and students are required to solve problems in many
different strands of mathematics throughout their elementary and secondary school
experiences (Bottge et al., 2010). According to Erden and Akgül (2010), many middle
school students fail to grasp the material or select not to complete the assignments
because they lack essential mathematics strategies or select inappropriate mathematics
instructional strategies needed to solve grade-level mathematics problems. Mathematics
in a variety of strands such as geometry and probability may require its own specialized
mathematics instructional strategies (Kang & Zentall, 2011). Although students may
exhibit proficiency in some strands of mathematics such as number sense (numbers and
operations), their proficiency to solve problems in other strands of mathematics may be
unsuccessful, resulting in gaps in mathematics content area learning.
For some middle school students, comprehending complex mathematics problems
is taxing. Likewise, many of their mathematics teachers may feel unprepared to teach
outside of traditional mathematics instructional strategies. Combined with evidence from
professional literature and standardized test figures from a specific Colorado urban school
district, the problem of trying to solve complex mathematics problems in middle school
mathematics and higher has endured for many years. During this time, some students
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have not reached minimum mathematics competency levels and as an outcome, may have
gaps in their learning.
Definitions
Terms used in this project study are as follows:
Colorado Student Assessment Program (CSAP): According to the Colorado
Department of Education (2011), CSAP tests students in Grades 3-10 in reading, writing,
and mathematics. CSAP also tests students in science in Grades 5 and 8.
Multiple Intelligences (MI): The MI theory includes nine intelligences that could
be identified as strengths in students’ learning. The intelligences are linguistic,
logicalmathematical, musical, spatial, bodily-kinesthetic, naturalistic, interpersonal,
intrapersonal, and existential (Gardner, 2006).
Standards: NCTM middle school mathematics content standard used in this are
from NCTM (2000) and refer to the concepts students should master in mathematics
including algebra, numbers and operations, geometry, measurement, data analysis and
probability and process.
Instructional mathematics practice: Instructional mathematics practices (Alberta
Learning, 2002) are methods teachers use to help students turn out to be strategic,
autonomous students. These practices help students to concentrate, coordinate, and
comprehend information and to evaluate learning.
Significance
The ability to use critical thinking skills in is crucial in order for citizens to make
active contributions to society (LaVenia & Pineau, 2010). Citizens are called upon to
make decisions in data analysis using rational numbers and problem solving skills from
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sources including bank statements and research or marketing data. LaVenia and Pineau
(2010) noted that many occupations require and rely on mathematics critical thinking
skills to compete in a progressively changing work arena. Today’s learners require a
range of mathematics instructional strategies to be able to secure a comfortable position
with employers in the community. Low levels of mathematics skills may also contribute
to elevated unemployment levels, heightened prospect of imprisonment, and lower wages
during an individual’s lifetime (Gifford, Evans, Berlin & Bai, 2011).
If students are not educated in the mathematics instructional strategies needed to
master complex mathematics problems, their grasp of the world may be restricted. Ozgen
and Bindaka (2011) suggested that as students begin to develop better problem solving
skills when working with more complex mathematical problems, they build mathematics
self-efficacy which makes learning new mathematics concepts a positive experience.
Guiding/Research Question
The problem in this study is related to low test scores in state mathematics
assessments. These scores suggest that teachers need to select and use appropriate
instructional mathematics practices that help all students master abstract mathematics
concepts. The difficulty for teachers is the selection and use of the most appropriate
instructional mathematics practice that aligns with the concepts taught. Help choosing the
most appropriate instructional mathematics practices could be a key to solving this
problem.
The problem leads to the guiding question for this project study: In an urban
middle school in Colorado, what are the mathematics teachers’ perspectives on
instructional practices for abstract mathematics concepts and content ? The local school
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district would benefit from clear evidence of instructional practices that lead to successful
outcomes . Such instructional practices could be used to teach mathematics or practice
concepts.
Review of the Literature
Database Search
In searching for literature on middle school students’ mathematics achievement, I
used the following databases: ProQuest, ERIC, Education Research Complete, Proquest
Central, and The Teacher Reference Center. In the beginning of the search, I used middle
school mathematics as a place to start. After locating some literature on learning
deficiencies in middle school mathematics, I broadened the search and included
mathematics instructional strategies in other middle school content areas. The following
concepts were used : low math achievement, middle school math, middle school math,
math instructional strategies, mathematics instructional strategies, math achievement
gaps, mathematics achievement gap, math learning theory, mathematics learning theory,
predictors of math achievement, predictors of mathematics achievement, math learning
theories, junior high math, junior high mathematics, primary learning theories,
mathematics secondary learning theories, brain-based learning, brain-based teaching,
brain-based learning strategies, strategic teaching, cognitive strategy, problem-solving
strategies, innovative strategy, literacy strategies, and math strategies.
Strategic Teaching in Mathematics
Some researchers reported that students in content areas, like mathematics, need
to be taught following a process similar to strategic planning used in the business world
(Graeff, 2010). Mathematics instruction could be taught to students in a way that accounts
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for various parts of the overall learning plan. Students can be active rather than passive
beneficiaries of knowledge, and the learning event should go beyond what can be
acquired just from a text book (Graeff, 2010).
Reasons for using strategies. Several reasons exist for using strategies or
strategic teaching in mathematics instruction. Strategies are often active learning which
can be motivational for students (Graeff, 2010). Strategic teaching and learning helps to
move the brain from its comfort zone into a higher working capacity (Halakatti, 2010).
Strategies in teaching and learning also help to meet different student learning styles
(Freeman & Walsh, 2013).
Benefits and obstacles in strategic teaching. Graeff (2010) suggested benefits
for strategic teaching. Strategic teaching is used to combine skills necessary for learning
rather than teaching skills in isolation. Combining skills in strategic teaching allows
teachers to teach more content and skills, and allows students to make connections among
the skills learned. When students are taught using strategic instruction, students learn
more, retain what is learned longer, and apply their knowledge in new situations.
As a second benefit of strategic teaching is the teacher-student relationship
involved in using this delivery model. Students are empowered to be learn side by side
with the teacher, which enables students to communicate what they understand and do not
understand in a risk-free environment (Graeff, 2010). This working relationship may
reduce anxiety in learning new concepts and content for the students, because the teacher
is directly involved in interaction with students and what they learn. In traditional
classrooms where lecture is the predominant method of instruction, instruction is only
provided one-way from the teacher to the student. Lecture may be necessary to provide
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foundational content, but this method of delivery rarely allows for student-teacher
interaction.
A third advantage of strategic teaching is the information the classroom teacher
can access about student learning (Graeff, 2010). When strategic teaching is applied, a
teacher can address any student misunderstandings immediately rather than leaving
students frustrated if they have not grasped the concepts and skills taught for specific
mathematics content. This immediacy in addressing problems in learning the content
lessens the chance of reteaching concepts and skills, thus, increasing the amount of
content and skills to be taught. Ultimately, strategic teaching keeps classroom learning
from being redundant and tied to the learning in textbooks. When planning mathematics
instruction, all of these benefits should be considered in the teaching and learning
process.
Kay and Swanson (2011) noted some of the following obstacles that can cause
problems when introducing strategic teaching to teachers. These researchers identified
teachers’ beliefs or biases, lack of confidence, and lack of sufficient exposure to teaching
strategies as the main obstacles to strategic teaching. Teachers might have beliefs or
biases against some of the strategies. For example, teachers who revert to traditional
teaching methods for mathematics based on how he or she learned this content when they
were students in school. Some teachers might have a lack of confidence in their ability to
implement teaching strategies due to minimal experience or training in strategic
teaching. Pre-service teachers may have little exposure to some teaching strategies
because their college and/or student teaching experiences did not include newer or
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innovative strategies. These issues should be considered when exploring strategic
teaching in mathematics.
Best practices in teaching mathematics. Best practices in teaching mathematics
may include cognitive strategies, problem-solving strategies, innovative strategies,
strategies borrowed from literacy instruction, or strategies very specific to mathematics
instruction (NCTM, 2000). The term best practice (research-based or scientificallybased),
common in evidence based education, is used to describe what works in the classroom.
Teachers are encouraged to use their professional wisdom to determine what works for
their students in the content area.
Cognitive strategies may enhance mathematics achievement in students. Swanson
(2014) studied application of cognitive strategies dependent upon the student’s working
memory capacity (WMC). These strategies can include helping students solve
mathematics word problems using verbal, spatial, or verbal and spatial strategies. The
results of the study showed that students with high WMC’s fared well with the verbal and
spatial strategies such as diagramming. Students with a low WMC did not do well with
some of the cognitive strategies such as verbal (key word location), visual strategies
(placing numbers into diagrams), and verbal and visual (diagramming numbers or
combination of verbal and visual. Based on Swanson’s (2014) study, it cannot be assumed
that all students would benefit from cognitive teaching strategies in mathematics;
however, cognitive strategies should be considered and teachers’ discretion and
professional judgment used should be applied.
Bayazit (2013) studied whether middle school students could successfully solve
real life problems using problem-solving strategies that could include listing strategies,
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application of mathematics models, and drawing pictures. The students were unable to
apply problem-solving strategies to the presented mathematics problems, and Byazit’s
(2013) findings suggested that teachers work on problem-solving strategies with students
to reinforce their critical thinking skills. Problem-solving strategies should be considered
essential to strategic mathematics instruction.
Innovative teaching strategies may also help raise student mathematics
achievement. Sherer and Grunow (2011) studied effective strategies in development
mathematics at the community college level. The study used a 90-day process or cycle to
determine the program efficacy, allowing a quick determination on whether innovative
mathematics strategies were helpful or worthy of time in the classroom. Innovative
strategies should be investigated to determine if they are worthy of inserting into
mathematics education (NCTM, 2000).
Some instructional strategies are appropriate for all content areas and could be
applied specifically to mathematics instruction. Howe, Mundy, Kopczynski, and
Cummins (2012) investigated teacher knowledge, use of, and recommendations regarding
instructional strategies for literacy (e.g. brainstorming, graphic organizers, and
vocabulary cards). The study results indicated that teachers with related graduate courses
were more likely to apply these strategies and that more experienced teachers were more
likely to use or recommend them to others. Although this study focused on teacher
implementation and sharing of instructional strategies in literary courses, these same
instructional strategies may be useful in a mathematics classroom as well.
Mathematics strategies can be used to engage students and to help them to
become more interested in learning mathematics (Ludwig, 2014). Ludwig (2014)
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suggested several mathematics instructional strategies that can be used across all strands
of mathematics. Structured learning, cooperative learning groups, teaching of vocabulary,
using manipulatives, varying assessments, and mathematics journals are beneficial in
learning, regardless of the mathematic content. All of these strategies should be
considered when planning mathematics lessons to better prepare students and improve
student achievement.
Conceptual Framework
The conceptual framework for this study is based on the quest for effective
instructional mathematics practices in mathematics instruction. The best resource to use
as a starting point in this research comes from the NCTM. Principles and Standards for
School Mathematics is one of NCTM’s publications that was created to guide
policymaking related to the improvement of mathematics education (NCTM, 2000).
Within this guide, six principles are examined that may assist in planning mathematics
instructional methods, mathematics learning, and in the creation of top-quality
mathematics programs (NCTM, 2000). These six principles include (NCTM, 2000): (a)
equity, (b) curriculum, (c) teaching, (d) learning, (e) assessment, and (f) technology.
Equity principle. The first principle addressed is equity. Equity includes the
themes of significant expectations and meaningful potentials for all learners (NCTM,
2000). This principle is based on the idea that one’s potential to learn mathematics should
not be lowered due to extenuating circumstances which could include language
deficiencies, socioeconomic status, or disabilities. Instead, this principle advocates that
additional resources be used to help all learners meet high mathematics learning
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expectations. One of the most important resources identified that can help with equity is
to increase professional development for teachers to help them to better understand and
accommodate mathematics instruction for various student needs (NCTM, 2000).
In alignment with the Equity Principle from the NCTM (2000), some research has
been conducted to determine how high expectations in student learning has impacted
student learning in mathematics. The results of this research could help with planning the
best possible mathematics instruction for middle school students.
Research was conducted with students performing at a low level in mathematics to
see if higher teacher expectations had an impact on the students’ achievement (Woolley,
Strutchens, Gilbert, & Martin, 2010). The equity principle looks for high expectations in
mathematics achievement from teachers with for all students regardless of learning
issues, language deficiencies, or socioeconomic status (NCTM, 2000). Several factors
were studied along with high teacher expectations in this study (Woolley et al., 2010).
Many students performed better as a direct result of higher expectations, but for some
students, higher expectations caused heightened anxiety, which lowered assessment
scores. The researchers also suggested that students might need time to adapt to these
heightened expectation, so it was recommended that further longitudinal studies should
be conducted to see if the anxiety decreases as students get used to the idea (Woolley et
al., 2010).
Williams (2010) studied nine minority high school students who succeeded in
higher level mathematics classes despite obstacles. She discovered that there were several
commonalities, and one of the most identified by the students was high expectations by
mathematics teachers. One recommendation from this researcher was related to the
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environment of minority students who are successful in mathematics. In her research, it
was determined that environmental factors helped the students to succeed and access to
rigorous content and high expectations were part of this environment. She recommended
that further research related to these areas would be beneficial to help future students
experience success in high level high school mathematics classes (Williams, 2010).
Curriculum principle. The next principle is curriculum. The idea behind this
theme relates to connecting the different strands of mathematics so that they are not
taught discretely (NCTM, 2000). The mathematics concepts taught should be worthwhile
or have purpose in everyday life, and these connections should be presented in
mathematics instruction. Throughout the years, mathematics concepts should build upon
prior mathematics knowledge and take students deeper in the level of sophistication and
understanding of concepts (NCTM, 2000).
Kelly (2008) studied the effect of a mathematics intervention program on middle
school student achievement that included the use of real-life learning experiences. The
overall increase in mathematics achievement for the test group was not significant, but
there was significant increase in three strands of mathematics based on the state test
scores. The three mathematics strands with the highest impact from this intervention
program included computation and estimation, statistics and probability, and patterns and
relationships. Recommendations for future research included looking at more qualitative
and quantitative research that would help to dig deeper into the impact of this type of
mathematics intervention on the student’s understanding using tools other than just an
achievement test (Kelly, 2008).
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Nehme (2011) studied the impact of real life connections of matrices related to
students’ engagement and motivation. The students were required to research real life
application of matrices and to report their findings and interact on a blog created for the
purpose. The students reported high engagement and motivation at the end of the activity
for a concept that might not be one where students easily form a connection (Nehme,
2011). The idea of real life application of complex mathematics concepts is one that
might engage students in all grade levels.
Teaching principle. Another principle shared is the teaching principle. This
theme centers on the practice of sound teaching (NCTM, 2000). An important part of
sound teaching is creating a thought-provoking yet compassionate teaching setting.
Sound teaching also means that a teacher should continuously be seeking improvement in
teaching practices related to mathematics (NCTM, 2000).
One recent study examined the success of developing leaders out of teachers with
strong mathematics teaching practices. These teachers were used to mentor other
mathematics teachers and lead learning communities which resulted in higher student
achievement in mathematics (Vale et al., 2010). Recommendations from the researchers
include delving further to determine what teaching practices lead to higher levels of
student achievement in mathematics.
Gasser (2011) observed five keys to being able to teach mathematics successfully,
and several correlated to sound teaching practice. One of the ideas was related to using
problem-based instruction in learning activities. Another idea was creating a classroom
climate where students feel comfort in taking risks. Creating fun in the mathematics
learning environment is another idea that Gasser (2011) credited to help students have
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higher levels of mathematics achievement. These ideas could be explored further as
effective instructional mathematics practices are analyzed so that they can be reproduced
in many classrooms.
Learning principle. The learning principle follows the teaching principle. This
principle is based on student learner’s developing deep and flexible understanding of
mathematics concepts (NCTM, 2000). Teachers need to provide experiences that provide
a deeper meaning and learning level for students through appealing activities and
classroom communications (NCTM, 2000).
Delacruz (2011) studied the impact of games on student mastery of mathematics
concepts. The study was conducted with control groups of fourth and fifth grade students.
Attributes of the game including incentives for feedback seeking and detailed rules were
examined to determine if this led to higher levels of understanding. The finding included
the most growth in students with very low pretest scores who were provided with both
detailed rules and feedback seeking incentives. The recommendations from Delacruz
(2011) included including more mathematics games that have mechanisms for students to
have more detailed rules and incentives for seeking feedback. Another recommendation
was to look for the motivating factors in learning mathematics with well-planned games
in future studies.
Xiong (2010) researched in a longitudinal study the causal relationship between
mathematics instructional strategies and student achievement. The results of state
achievement tests for students in Grades 2 through 6 was compared over 3 years after a
new mathematics curriculum was introduced. The data showed varying levels of
improvements in student achievement from year to year. The new mathematics program
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emphasized teaching in an organized system using strategies that included all five senses.
Xiong (2010) recommended that future studies should be conducted to study causal
relationships between mathematics strategies and student achievement.
Schmitz and Perels (2011) studied how self-regulated learning affected
mathematics achievement with eighth grade students. The students used a diary to
selfmonitor mathematics learning throughout a unit of study. The students’ learning was
compared to a group of students who completed the same unit without the selfmonitoring.
The group who used the diary during the unit showed greater growth than the control
group based on pre and post assessment data. Schmitz and Perels (2011) recommended
that this and similar types of self-regulatory strategies should be applied in mathematics
and other subjects to provide students with a learning advantage because of the positive
results from this study.
Assessment principle. Assessment is something that should increase learning.
Using assessment as an instrument is one of the best way to make educational decisions
(NCTM, 2000).
Shaffer (2011) conducted a study to determine if differentiated instruction made a
different in middle school mathematics achievement scores. The differentiation was based
on using pre assessment and formative assessment to influence instructional decisions.
The research results showed a significant increase in mathematics achievement scores as
a result of differentiated instruction. Shaffer (2011) recommended that further studies
could be done to study the implementation of differentiated instructional strategies in
mathematics and other content areas.
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Jackson (2012) studied the experiences of teachers in using data-driven instruction
to help with instructional decisions in middle school mathematics instruction. The
findings were that these teachers with different years of experience and education levels
lacked the training or understanding of how to use data to make instructional decisions in
their mathematics classrooms. Jackson (2012) recommended that future studies could be
conducted in administrator perceptions of teacher data use and in the effects of data
driven instruction on mathematics achievement.
Technology principle. Technology enriches student learning in all mathematics
strands, algebra, geometry, measurement, data analysis, and probability. There are
applications and programs that can help students on an IEP for mathematics or students
who need more visual strategies for learning. Technology, itself, should not be the
mathematics teacher. Teachers should use it in efficient ways to support teaching and
learning. In addition, decisions about teaching mathematics are partly determined by
current technology (NCTM, 2000).
Lewis (2011) conducted research related to the effect of computer assisted
instruction (CAI) on mathematics achievement with a group of fourth grade students. In
the study, there was a control group who received traditional mathematics instruction
only, while the test group was provided with the same traditional mathematics instruction
along with CAI technology integration in lesson. The test group scored significantly
higher on the post assessments provided to both groups. Lewis (2011) recommended that
CAI should be researched in other grade levels to decide if the results could be
generalized within higher grade level mathematics achievement levels.
24
Allison (2012) researched the use of Computer Performance System (CPS), also
known as clickers, along with peer instruction (PI), as an additional learning strategy, in
relation to student mathematics scores. The idea was to determine whether technology
along with sound instructional strategy could raise eighth grade mathematics
achievement. The results showed that technology like CPS along with sound instructional
strategies like PI raised mathematics achievement significantly. Allison (2012)
recommended that CPS should be a focus in future research combined with other
grounded mathematics strategies to see if this combination of technology and
instructional practice has impact on student achievement.
Implications
Based on feedback from a panel of middle school mathematics teachers and a
review of the literature, through an exploratory study using the modified Delphi method, I
explored middle school mathematics teachers’ perspectives regarding instructional
mathematics practices for abstract mathematics concepts and content taught in an urban
middle school in Colorado. After input was collected and analyzed, it was possible to
create a guide that would support any administrator or teachers seeking improved
mathematics scores. Possible projects could be a curriculum or instructional guide that
provides instructional mathematics practices that align with NCTM middle school
mathematics content standards or resources and professional development for new or
struggling mathematics teachers. The impact of these data could be powerful for any
teacher working with struggling mathematics students as they would indicate
instructional mathematics practices of middle school mathematics teachers based on the
NCTM middle school mathematics content standard.
25
An essential element to this project study’s usefulness is that the actual project be
presented with effective instructional mathematics practices from a practitioner’s
perspective in a comprehensible and clear-cut format. If stringent observance of these
recommendations are followed, teachers may be willing to assimilate these instructional
mathematics practices into their everyday lesson plans. As an outcome, student grasp of
complex mathematics concepts may strengthen confidence and lower the feelings of
exasperation by the student and teacher.
Summary
Some middle school students at a Colorado urban school district have had
problems mastering mathematics concepts, as shown from figures obtained from state
standardized tests. The guiding research question for this project study was created to
uncover what instructional mathematics practices should be inserted to the everyday
practice of middle school mathematics teachers in a Colorado urban school district to
improve student achievement in this content area as exhibited on the outcomes of annual
state standardized tests. This project study was aimed at affording middle school
mathematics teachers with a pool of instructional mathematics practices in a professional
development training plan. When used with instructional mathematics practices regularly
implemented in the mathematics classroom, this resource guide could be used to boost
student achievement of middle school mathematics concepts.
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Section 2: The Methodology
Introduction
The purpose of this study was to improve classroom instruction by using a
modified Delphi method to exploring the perspectives of an expert panel of middle school
mathematics teachers on instructional mathematics practices that draw on theories that
are foundational to best practices in mathematics. In this section, the following topics are
covered: the modified Delphi method and rationale, the sample and setting for the study,
the data collection and analysis techniques, measures for protecting participant rights, and
the role of the researcher in this process.
Research Design
The Delphi method, which originated at the RAND Corporation (Dalkey, 1969), is
a method of collecting and synthesizing the opinions of a panel of experts on a topic in
order to make decisions about policy or practice; it can be used across a wide range of
fields (Clark, 2006). In general, in a Delphi study a series of surveys is designed to
generate ideas or to synthesize opinions about a topic. Exploratory research in emerging
areas is an appropriate place to conduct a Delphi study (Päivärinta, Pekkola, & Moe,
2011). This method is also appropriate in cases where the overarching problem might
benefit from the insights of experts (Hsu & Sandford, 2007). These surveys are
administered to experts, often dispersed over a wide geographical area, in three or more
rounds of data collection. This method has been used in a range of educational settings to
gather stakeholder beliefs about instruction and educational policy (Franklin & Hart,
2006; Mahmood, Iqbal, & Saeed, 2009; Williams, Boone, & Kingsley, 2004). Typically,
the first round of data collection consists of open-ended questions and qualitative data
27
(Williams et al., 2004), followed in subsequent rounds by quantitative or both
quantitative and qualitative question types. After each round, the responses are analyzed
and summarized; a feedback survey is developed for the same respondent group. In this
second survey, respondents are asked to rate the responses from the group given in the
first round using a Likert-type scale(Hsu & Sandford, 2007). A final round is often the
chance for the experts to comment on opinions from previous rounds that deviate from
the norm (Hsu & Sandford, 2007).
In this case, the modified Delphi method was used to obtain the opinions of experts
on the types of instructional practices might increase student mastery of middle school
mathematics concepts. The results from the research were critical to selecting a project
for this study. The participants a middle school mathematics teachers who had taught
adolescent learners for at least 5 years, who held an advanced degree in education, and
who had taught mathematics to adolescent learners for at least 3 years. Their responses,
which were collected and analyzed using both quantitative and qualitative techniques,
were used to determine what instructional mathematics practices would be most helpful
to these students.
Data Collection Strategy
This Delphi study may be considered a mixed method sequential exploratory
study because the data were collected in three rounds (based on the surveys) and that the
questionnaires consisted of quantitative and open-ended questions. The first round survey
was used to collect open-ended responses from the middle school mathematics teacher
panel. Subsequent surveys contained a combination of numeric and open-ended questions
that will be created based on the first round results. This method was exploratory because
28
the research process itself created opportunities to learn what areas should be explored
further for this study research.
In a sequential exploratory mixed-methods study, data are collected in rounds
using qualitative and then quantitative methods, and the analysis is investigated and
analyzed for qualitative exploration (Creswell, 2013). In exploratory mixed methods, the
qualitative data are collected and analyzed first. The information in this study was
analyzed using content analysis (Stemler, 2001), and the results from it were used to
create the numeric and open-ended questions for subsequent surveys. Dalkey (1969), the
pioneer of modern day Delphi research methods, described this mixed methods Delphi
process as “conducting the exercise in sequence of rounds between which a summary of
results of the previous round are communicated to the participants” (p. 16). The three
rounds of data collection that comprise this particular modified Delphi study are
described in the next section.
Multiple Forms of Data Collection and Analysis
In the first modified Delphi data collection round, I collected qualitative data by
asking the participants to respond to a series of open-ended internet survey questions in
which they described instructional mathematics practices that can be used in the middle
school mathematics classroom to teach the different NCTM middle school content
standards. The participants were middle school mathematics teachers who meet inclusion
criteria of at least 5 years teaching adolescent learners, an advanced degree in education,
and at least 3 years teaching mathematics to adolescent learners. These open-ended
responses were analyzed using qualitative content analysis to shape the creation of the
mixed-question type surveys to be use in the second and third rounds. Surveys
29
administered in the second and third rounds asked participants to rank the themes that
emerged during the first survey and to explain why the participants believed these
instructional mathematics practices should be ranked in this order.
The internet survey tool, Survey Monkey, was used to collect the survey responses.
The first survey (see Appendix D) asked the participants to identify instructional
mathematics practices and scenarios that might help students solve problems similar to
the mathematics problems listed that match each of the mathematics strands in middle
school mathematics. The participants were asked to provide examples and other details to
help understand how the instructional mathematics practices might be used by students
solving these types of problems. These instructional mathematics practices can later be
analyzed with the goal of deciding how to implement them in curricula to improve
student understanding of middle school concepts. Participant responses were analyzed
using content analysis (Stemler, 1990). Instructional mathematics practices/scenarios
corresponding to each question on the first round survey were summarized and similar
responses were combined.
The second modified Delphi round (see Appendix E) consisted of a combination of
closed- and open-ended questions based on the results of the first round’s results. After
analyzing the list of instructional mathematic practices generated by the middle school
mathematics teachers panel in Round 1 of the data collection, a second set of questions
was presented again in an internet survey. A summary of the instructional mathematics
practices was listed that were collected from the first round, and the panel rated their
opinion of each instructional mathematics practices’ effectiveness in relation to its
possible use in seventh grade mathematics instruction. Each instructional mathematics
30
practice was rated in a 1 through 5 point Likert system with 5 representing the highest
rating. Then, in an open-ended follow-up question, the respondents were asked for
reasons or details regarding why they felt this way about the instructional mathematics
practice. Data from surveys that include close ended questions are quantitative (Creswell,
2013).
Finally, the quantitative internet survey administered in the third round (see
Appendix F) was created based on the analysis of Round 2 quantitative and qualitative
data. The Round 3 information presented to the participants was the compilation of the
Round 2 findings in summary form. The purpose of the Round 3 data collection is for
participants to arrive at a final consensus. In order to achieve final consensus, the Round
3 panel rated the instructional mathematics practices using a Likert scale of 1 to 5, where
A five-point Likert scale was used, with scores ranging from 1 (disagree very much) to 5
(agree very much). The results were tracked for each question by finding the mean and
then ranked from highest to lowest.
Justification for Delphi Method Research Design
This mixed methods study was based on the Delphi technique. A Delphi study is a
method used to build a consensus about a concept or construct when one does not exist
(Yousuf, 2007). It is characterized as a way for facilitating discussion between experts in
the field of study. A Delphi study’s purpose is to create consensus among knowledgeable
individuals to address a complex problem (Yousuf, 2007). In the context of this study, a
group of educators and researchers reached a consensus on the best instructional
mathematics practices to use when teaching middle school mathematics so that the
mathematics objectives can be attained. This study resulted in a collective prioritization of
31
instructional mathematics practices that may be used to improve classroom instruction and
student achievement on state standardized tests.
Since the Delphi process helps to develop consensus, it allows for participation in
the research process by a panel of experts who assume the responsibility of making
judgments about the best responses to the problem. While other mixed methods survey
techniques could have been used to collect expert opinions about the most effective
instructional mathematics practices, the Delphi method includes the expert participants in
the judgment of those responses. Another advantage of the Delphi method over other
qualitative research design methods is the anonymity of the participants with each other.
If the panel were to meet face-to-face, the interpersonal dynamics would differ from those
in an anonymous format. The Delphi method can provide results using more strategic
information than other research methods since the feedback is controlled. This method
allows for less bias due to the fact that all participants’ opinions are presented without
influence of the other participants (Hsu & Sandford, 2007). Finally, proponents of the
Delphi method “recognize human judgment as a legitimate and useful input in generating
forecasts and therefore believe that the use of experts, carefully selected, can lead to
reliable and valid results” (Olds, Streveler, Miller, & Nelson, 2003, p. 2). The collective
judgments of the panel are used to arrive at consensus which can lead to further research
and/or help to make changes to policy or processes.
The Delphi method is also useful because it avoids bias that can occur when a
group of experts meets face-to-face by taking out the communal connections that can
change the way opinions are formed (Yousuf, 2007). The Rand report suggested that the
anonymous feature of Delphi method makes the results more accurate (Dalkey, 1969). As
32
a result of the Delphi method, this study was organized to show prioritization of
instructional mathematics practices following an unprejudiced process.
The exploratory Delphi methodology provides several rounds that include mixed
type survey questions. This methodology was chosen over a strictly qualitative or
quantitative methodology. First, the qualitative input provides a way to solicit different
ideas. The quantitative input using the Likert scale helps to order the ideas from most
effective to least effective. The combination of qualitative and quantitative rounds in
Delphi help to develop a consensus based on the ideas of the entire panel of experts. It
also gives the expert participants more opportunity for fully describing their opinions on
the types of instructional mathematics practices needed in particular scenarios. This
explanation happens when a member of a panel either chooses on the low end, a 1, or on
the high end, a 5, on the Likert scale when rating an instructional mathematics practice.
At that point, the expert is asked to provide more information to explain the rating.
This method works well when data on a given topic are not documented or
existing due to the difficulty of gathering experts together to work on a consensus on the
topic (Yousuf, 2007). Yousuf (2007) stated that the Delphi method is “useful where the
opinions and judgments of experts are needed, but time, distance, and other factors make
it unlikely or impossible for the panel to work together in the same physical location” (p.
80). Bringing the opinions of the experts together in an exploratory Delphi study help to
create a consensus of ideas that might not have been documented due to the constraints
listed.
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Integration of Research Methods
The integration of both qualitative and quantitative methods was evident in the
three rounds of data collection, interpretation, and analysis. This modified Delphi method
incorporates three rounds within the process. The first stage or round used qualitative
data collection methods in an internet survey format. The internet survey included
mathematics problems related to each strand of mathematics taught in middle school. The
middle school mathematics teacher panel was asked to provide instructional mathematics
practices that would help students learn how to solve the different problems. The data/
instructional mathematics practices collected were listed under each scenario and
presented to the panel again as part of Round 2. If instructional mathematics practices
collected from the panel were similar, they were not listed multiple times under a single
scenario. For example, if one middle school mathematics teacher suggested an
instructional mathematics practice of using real life applications in mathematics
instruction, and another teacher from the panel suggest using authentic learning situations
for scenario one, these were combined into one instructional mathematics practice.
In Stage 2, the middle school mathematics teacher panel judged the list of
instructional mathematics practices collected in Round 1 for effectiveness in mathematics
instruction for the specific scenarios in a new internet survey. The panel rated each
instructional mathematics practice on the internet survey using a five-point Likert scale
was used, with scores ranging from 1 (disagree very much) to 5 (agree very much). The
ratings provided quantitative data. Then, the panel was asked for follow-up feedback with
explanations for ratings. The follow-up feedback provided additional qualitative
feedback. The data for each instructional mathematics practice collected was
34
descriptively analyzed including frequencies, means, medians, and modes. Panel
justifications were included for ratings of each instructional mathematics practice. All of
the data from Round 2 was organized into a summary report which was presented in an
email to the middle school mathematics teacher panel for Round 3. Creswell (2013)
stated that quantitative methods include statistical analysis which can be in the form of an
average. The qualitative feedback data were interpreted by the panel later in Round 3.
Finally, Round 3 included only quantitative data collection, interpretation, and
analysis. The middle school mathematics teacher panel was presented the summary report
from Round 2, and using this information, they rated the instructional mathematics
practices one more time on a Likert scale. No qualitative feedback was collected in this
stage. Each instructional mathematics practice was analyzed based on the mean for each
instructional mathematics practice. The instructional mathematics practices were
prioritized based on the quantitative analysis of this average and rank ordered from
highest to lowest based on mean scores. These data were used to inform the final group
consensus of the instructional mathematics practices.
Setting and Sample
The setting for the project was Colorado urban school district. The demographics in
this district included 33% English Llanguage learners, 38% speaking Spanish as first
language, and 72.5% qualifying for free or reduced lunch (DPS, 2013).
The sample for this research study included a middle school mathematics panel
with at least 5 years of experience teaching, an advanced degree or higher in an
education-related area, and at least 3 years teaching mathematics to adolescents. These
education specialists included mathematics teachers and/or administrators with
35
mathematics expertise. Some members of the middle school mathematics panel were also
instructors at universities. The names of the participants were obtained from
administrators. These candidates were sent an invitation via email that explained the
study and asked them to respond if they were interested in participating (see Appendix
B). Included in the email invitation was a consent form which participants were asked to
return if they agreed to participate in the project study.
The total sample solicited for participation was 15-20 educators who met the
eligibility criteria set forth in this section. As Hsu and Sandford wrote (2007), “Delphi
subjects should be highly trained and competent within the specialized area of knowledge
related to the target issue” (p. 3). Although Grisham (2009) noted that a minimum of 15
participants are necessary to conduct a valid Delphi study, Hsu and Sanford (2007)
emphasized that there has not been an agreement on the ideal number of experts that
should be surveyed in a Delphi study.
The sampling method used to choose the population was based on purposeful
selection due to the nature of Delphi methodology. If more than 20 participants met the
minimum qualifications, then random selection would have been used. The candidates
would have been placed in a jar, and 20 names would have been selected Creswell (2013)
indicated that purposeful research is common to qualitative research because this type of
selection determined participants who have extensive knowledge of the problem and
research question. Purposeful sampling was used to identify study participants for this
research study (Neuman, 2003). Because the study’s intent was to identify successful
instructional mathematics practices from a middle school mathematics teacher panel,
36
using confirming or disconfirming sampling was a purposeful instructional mathematics
practice that could be used to test or explore further studies (Creswell, 2013). The sample
invited to participate needed to have knowledge based on experience in teaching
mathematics. Upon initial selection to participate, a selection letter was sent to each
participant (see Appendix C).
Sequential Data Collection Strategy
In this section, I discuss the data collection sequences, including both qualitative
and quantitative sequences.
Qualitative Sequence
Gaining access to participants. Potential teachers for the middle school
mathematics teacher panel were solicited from educational communities with members
meeting the panel criteria. The panel also included teachers who were instructors at
colleges. The middle school mathematics teachers for the panel were solicited by
contacting the administrator at the school district or dean or department head at the
college. Candidates received an emailed description of the study and an invitation to
participate. The email included an introductory paragraph explaining the purpose of the
study and their role should they choose to participate in it. Additional information
regarding the study was provided to the middle school mathematics teacher panel upon
request via email or personal phone call.
Name/type of instrument and number of rounds. The instrument used for the
qualitative sequence in Round 1 was a questionnaire containing a question for each of the
mathematics strands (see Appendix D). These questions requested participants to list
instructional best practices that might be useful for students to master and comprehend
37
mathematics concepts. The instrument used for Round 2 was a 2-part questionnaire used
to collect both quantitative and qualitative data as the respondents were provided pre
choices and then justify these selections with personal commentary (see Appendix E).
The instrument used for Round 3 was a strictly quantitative questionnaire that gathered a
final rating for each of the instructional best practice from each teacher on the panel (see
Appendix F).
Researcher/participant relationship. I developed a working relationship with
the panel through regular email and/or phone contact. In this communication, I shared
timelines, provided directions for each round of data collection, and answered questions.
Data triangulation. Data triangulation is not built into Delphi methodology
because each round of data collection is separate (Hsu & Sandford, 2007). Each round
was analyzed and a new data collection tool was created based on the previous round of
data. This process further narrowed the focus on best instructional mathematics practices
to use in mathematics instruction. All of the analysis was completed within the current
round of data.
Role of the researcher. I was responsible for data collection and data analysis for
each round of the data collection. I have been a mathematics teacher at either a high
school or college for over 17 years; I have taught full time in a local high school and
adjunct part time at two different colleges. I have been with my current full time
employer for 2.5 years and my colleges for 6 years. Some of the participating middle
school mathematics teachers were current or previous colleagues either from a school
district or university where I have worked. The teachers on the panel, however, were not
subordinates, in my reporting line, or subject to my authority. Therefore, my relationship
38
with the participants did not affect data collection. The only connections to participants
was through professional working relationships at the school district or university where
we might both be employed. I encompass a few biases/experiences to my role as the
researcher. These biases included experience and training in methodologies in multiple
intelligences and brain-based learning theories. My experiences with applying these
learning theories could have affected my perspective when evaluating data.
Quantitative Sequence
Name/type of instrument and number of rounds. The instrument used for the
quantitative sequence was a questionnaire based on the information collected during the
Round 1 and 2 of the modified Delphi process. The I nstructional best practices collected
from Round 1 were rated using A five-point Likert scale was used, with scores ranging
from 1 (disagree very much) to 5 (agree very much).The participants were deciding
whether they agreed\ that the instructional best practice selected is one that would help
the students to master each mathematics concept according to the NCTM middle school
mathematics content standards. Qualitative data were presented in a summary report to
the participants, which shows the explanations provided by the middle school
mathematics teacher panel for rating choices. The panel rated the same list of
instructional best practices one last time in Round 3 using the same Likert scale as in
Round 2.
Concept measured by instrument. The instrument in Rounds 2 and 3 listed all
of the instructional best practices collected from Round 1. Each participant rated each
instructional best practice with the provided Likert scale based on its effectiveness in
teaching the focus concept.
39
How ratings are calculated. During Rounds 2 and 3, each instructional best
practice collected during Round 1 was presented to the middle school mathematics
teacher panel so its effectiveness could be evaluated using a five-point Likert scale with
scores ranging from 1 (disagree very much) to 5 (agree very much). The mean and mode
for each instructional best practice were calculated. The data were used to analyze which
instructional best practices the panel deemed effective for specific mathematics concepts.
The instructional best practices were then rank-ordered based on the panel’s opinions.
Processes for assessment of reliability and content validity of the instrument(s).
The quantitative instruments were checked for reliability and content validity in a few
ways. Some of the methods for this included field testing and member checks. Field
testing includes giving the survey to others outside of the study to determine if the
questions make sense and are asking for the information desired. Member checking
includes checking with participants to be sure that the any possible interpretations are
accurate (Creswell, 2013).
Data Analysis and Validity
Validity and reliability for both the qualitative and quantitative processes are
critical components of a research study. When considering quantitative data, validity
involves looking for exactitude in results using processes designed for this purpose
(Creswell, 2013; Cone & Foster, 2006). This process enables others to apply findings
from a study knowing that the findings came from an accurate research process.
Quantitative reliability refers to the ability to use results from the instrument to make
informative suppositions (Creswell, 2013). Reliability is not the same concept as validity,
and it refers to the process of checking the “consistency of responses” (Creswell, 2013, p.
40
190). Using methods to ensure reliability and validity ensure that results from a study are
viewed as trustworthy. Trustworthiness is the qualitative equivalent of validity and
involves credibility, transferability, dependability, and confirmability.
Credibility
When analyzing data collected in this study, credibility is the relative confidence
in the truth of the findings. Credibility was achieved by using the following procedures.
Credibility was addressed is through member checks (Cohen & Crabtree, 2006) that were
employed when the data are sorted from the first round was presented to the middle
school mathematics teacher panel. This process occurs when participants are asked to
check for correctness when information collected has been analyzed and restated in
another way. Most information or collected data were presented as is; however, if
instructional mathematics practices were very similar, they were combined into a single
instructional mathematics practice to avoid repetition. The sample questions used in the
survey for the first round represented concepts found in each strand of middle school
mathematics. The panel’s opinions were solicited as to whether they feel the
categorization of their Round 1 qualitative feedback was accurate. If the members of the
panel are in consensus that there is a problem with the combining of data, their input was
employed to make changes in the summary. Input from the panel was submitted to me on
an individual basis.
Qualitative Validity
Validity issues in research originated from the researcher and/or middle school
mathematics teacher panel showing bias in relation to the topic (Creswell, 2013). In
qualitative research this issue is referred to as confirmability. To provide confirmability, I
41
allowed the panel to provide feedback based on the research topic without offering my
opinion or giving advice so that I did not influence the panel’s feedback in any way. The
panel member’s comments remained anonymous.
Quantitative Validity
Content validity involves being able to make consequential conclusions from scores
on the questionnaires (Creswell, 2013). Content validity was checked in this study to
determine whether questions on the survey accurately measure the content that was
intended to be measured. Each survey was field tested with 2 to 3 of the middle school
mathematics teacher panel to ensure content validity. The panel was asked to give
feedback as to whether each question on the survey measures the intended content.
Feedback from the panel was used to make changes to survey questions throughout the
process.
Sample size can also be a validity issue in regards to quantitative validity
(Creswell, 2013). The number of experts should be a valid sample size which was
previously identified as in the range of 15 to 20 participants in a Delphi study (Grisham,
2009; Hsu & Sandford, 2007). I contacted as many qualified middle school mathematics
teachers in this field as possible based on the criteria described earlier in this section and
randomly selected members to participate in the panel.
Reliability
Creswell (2013) articulated that qualitative reliability can be observed in different
processes of research including transcription. Transcription issues could occur when
categorizing the qualitative feedback from Round 1. The middle school mathematics
teacher panel was asked to give input in the process to ensure they agreed with the list of
42
summaries collect from Round 1 before the Round 2 rating instrument is created. Also, I
included well documented information concerning the procedure used to transcribe this
Round 1 feedback into a list of instructional mathematics practices. Creswell (2013)
indicated that well documented qualitative procedures are crucial to reliability within a
study. Coding was carefully completed to protect the identity of the various member of
the panel. Each teacher was assigned a letter to identify his or her responses rather than
using names.
Transferability
Generalizability occurs when other researchers try to generalize or replicate
results from this study to another similar study (Creswell, 2013). The instructional
mathematics practices chosen by the experts might be changed based on future research,
but the modified Delphi method processes followed in this study could be used to
research similar problems with mastery of difficult concepts in mathematics and other
content areas.
Confirmability
Using the validation techniques discussed above, both the quantitative and
qualitative data collected in this study using the modified Delphi method were valid and
trustworthy. “Proponents of the Delphi method recognize human judgment as a legitimate
and useful input in generating forecasts and therefore believe that the use of experts,
carefully selected, can lead to reliable and valid results” (Olds et al., 2003, p. 2). Because
Delphi has been recommended when creating educational policy (Olds et al., 2003), the
solicitation of the experts’ opinions on finding the most effective instructional
43
mathematics practices used for middle school concepts was a valid and trustworthy
process.
Assumptions, Limitations, Scope, and Delimitations
In any study, the assumptions, limitations, scope and delimitations need to be
acknowledged.
Assumptions
In this study, the following assumptions were made. Consistent with the
constructs of the modified Delphi methodology used for this study, the first assumption
was that using a team of middle school mathematics teachers as participants in the study
was the best way to explore effective instructional mathematics practices because these
panel members are best equipped to explore this subject area. The second assumption was
that all participants in this study answered all questions accurately and honestly. The third
assumption that the middle school mathematics teachers used for this study actively
participated in this study from the beginning of data collection through the end of data
collection.
Limitations
The study was subject to the following three limitations :
1 Some of the participants worked together at the same school or in the same
district.If participants talked to one another about the questions before
providing their own responses, the value of their input could be reduced. At
the beginning of each round of data collection, participants were told not to
talk to other participants about their responses, either before or after
responding.
44
2 A second potential limitation was that a participant or participants may wish to
withdraw from the study for personal or professional reasons and may have
done so either by voicing this choice or by no longer providing responses to
the research instrument questions. To mitigate this potential limitation, the
study sought to recruit enough middle school mathematics teachers to act as
participants. As part of the recruitment process, the expected timeframe for
participation was provided along with the importance and expectation of each
participant completing the study.
3 A third limitation of the study was the research method. Using the modified
Delphi method is a limitation because it does not allow the researcher to use a
larger group. The method also locks the researcher into a method when there
could be another methodology that might be a better choice.
Scope and Delimitations
The scope of this study included a single Colorado urban school district. The
study was delimited to the middle school mathematics teachers who met the following
criteria: (a) at least 5 years of experience with adolescent students, (b) a graduate degree
in an education-related field, and (c) at least 3 years of experience teaching mathematics
to adolescent learners
Using only information from this group may have excluded other perspectives.
These outside resources, however, would not have met the criteria for the panel necessary
for use of the Delphi method (Yousuf, 2007).
45
Protection of Participants’ Rights
I secured permission to conduct the study from Walden’s Institutional Review
Board (Approval No. 05-21-14-0093230). If participants from school districts were
solicited, I contacted their school district administrator. If participants were solicited from
universities, their university IRB process would need to have been followed. Data were
not collected until IRB approval and participating institutions approval was granted; then
the participants were supplied information including the steps they are expected to follow
in the study.
The participants were notified first via an introductory email that contained a
consent form. In the form, the participants were notified of the right to privacy and the
right to choose to participate. The email also included information about the role and
requirement for participants. The participant was invited to ask clarifying questions on a
phone conversation and/or via email to help make the decision about whether they
wanted to be a part of the study. If the participant decided to participate, they returned the
signed consent form via email within a week. Participants were notified by email to
confirm his or her role and to advise him or her of the next steps in the process. These
steps helped to avoid ethical issues in the process.
The anonymity factor of the modified Delphi process also protected participants.
Since, the questionnaires were completed via Survey Monkey™, the participants did not
meet face-to-face. Participants’ identities remained confidential and were not be shared
with anyone involved in this study. A unique coding identifier was put into place to
ensure participant confidentiality. Each participant was assigned an alphanumeric code
such as A01, B02, and so on. No harm came to the participants as a result of this research
46
because their identities were protected. The data will be stored in my home safe for a
period of 5 years after which it will be destroyed.
Results
The following section details the data collection results for the three rounds of this
modified Delphi study. Data were collected from middle school mathematics teachers to
answer the following research question: What are middle school mathematics teachers’
perspectives of instructional mathematics practices for abstract mathematics concepts and
content taught in an urban middle school in Colorado? After soliciting participation from
professional mathematics educator resources, seven middle school mathematics teachers
were willing and eligible to participate in this study. Table 1 details the criteria and
participant responses to the qualifying questions prior to participation in the study.
Table 1
Criteria for Participants in Project Study
Participation Criteria
Yes
No
Teaching adolescent learners, > 5 years
7
0
Earned advanced degree in education, > Masters
7
0
Teaching mathematics to adolescent learners , > 3 years Total
Eligible Participants
7
0
7
0
Each of these middle school mathematics teachers participated in all three modified
Delphi rounds of data collection; each round provided opportunity to gather data to
47
determine perspectives on appropriate instructional mathematics practices for middle
school students. Teachers provided responses to open ended, Likert scaled, and
rankordered questions via an electronic Survey Monkey link. The responses to the Round
1 Survey were compiled to create the Round 2 Survey. The Round 2 Survey also provided
the foundation for the Round 3 Survey. The findings for Rounds 1, 2, and 3, including
sample questions from the survey instruments, are presented sequentially in order to
answer the research question.
Round 1: Initial Modified Delphi Round
In Round 1 of the data collection, seven middle school mathematics teachers were
asked to provide recommended instructional mathematics practices for helping middle
school students meet the NCTM middle school mathematics content standards. These
standards include (a) numbers and operations, (b) algebra, (c) geometry, (d) measurement,
and (e) data analysis and probability. The district and school mathematics objectives for
common core standards adopted by the state of Colorado are aligned with these five
NCTM middle school mathematics content standards. The Round 1 Survey includes the
five NCTM middle school mathematics content standards, and a corresponding sample
mathematics problem. Panel members responded to four questions designed to elicit
effective instructional mathematics practices for teaching the sample problem. The four
questions listed for each content standard/sample problem are listed below.
• Please provide a detailed description of what you would do to facilitate
student understanding for this sample problem.
• Please share exemplar or relevant problem/contexts.
48
• Which instructional strategies would you identify as most helpful from your
own experience.
• Share your rationale about why/how these strategies work.
Numbers and Operations Content Standard. The first item for the panel was
focused on the Numbers and Operations NCTM middle school mathematics content
standard. The sample problem related to this standard and given to the panel was “A car
travels 140 miles on 10 gallons of fuel. How far can it go on a tankful of gas if the tank
holds 15 gallons?” The panel was provided with the four bulleted questions listed in the
description of Round 1 as question 1a, 1b, 1c, and 1d.
Table 2 displays the data collected from Round 1, Numbers and Operations
Content Standard. The responses for each question were examined to locate instructional
mathematics practices. The practices that were located are listed in the table (see Table 2)
along with the corresponding question and the participant. Many times, multiple
participants agreed upon the same practices. For example, five of the participants listed
inquiry learning/student led instruction in their responses. The only instructional
mathematics practices that were not suggested by multiple participants were using colors
to help track steps or patterns and using different numbers to solve similar problems
Numbers and Operations Content Standard outcomes. Eleven instructional
mathematics practices were identified in the panel’s Round 1 responses. These practices
included (a) real world application, (b) small group collaboration and discussion, (c)
vocabulary, (d) template/model, (e) using colors to help track steps or patterns, (f)
connections to similar concept strategies/scaffolding, (g) independent practice, (h) use
different numbers to solve similar problems, (i) use graphic organizers, charts, and tables,
49
(j) inquiry learning/student led instruction, and (k) pictures and visuals. All practices were
identified by the entire panel: one practice was identified by five participants; one
practice was identified by four participants; five practices were identified by three
participants; two practices were identified by two participants; and two practices were
identified by one participant.
The mathematics instructional practices that were the most frequently identified
were inquiry learning/student led instruction and pictures and visuals. Both of these
practices were mentioned in three of the questions for the Numbers and Operation
Content Standard. Inquiry learning/student led instruction was mentioned by five
participants while pictures and visuals were shared by four participants.
Three participants identified real world applications, small group collaboration
and discussion, vocabulary, template/model and graphic organizers, charts, and tables as
instructional practices useful for solving the Numbers and Operations sample problem.
The practice mentioned in all questions was small group collaboration. Real world
application, template/model, and graphic organizers, charts, tables were shared in two
questions. Vocabulary was shared in just one question.
Other mathematics instructional practices in the responses included independent
practice and connections to similar concept strategies/scaffolding. Each practice was
shared by two participants. Independent practice was seen in two questions. Connections
to similar concept strategies/scaffolding appeared in two questions.
The mathematics instructional practices mentioned by the least number of
participants were colors to help track steps or patterns and use of different numbers to
solve similar problems. Both of these practices were shared by only one participant.
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Colors to help track steps or patterns was seen in three questions. The use of different
numbers to solve similar problems was shared in only one question.
51
Table 2
Round 1: Numbers and Operations Content Standard
Instructional Mathematics Practice
Question
Participants
Real world application
1a, 1d
1A, 5E, 7G
Small group collaboration and discussion
1a, 1b, 1c, 1d
4D, 5E, 7G
Vocabulary
1a
2B, 5E, 6F
Template/model
1a, 1c,
2B, 4D, 6F
Colors to help track steps or patterns
1a, 1c, 1d
2B
Connections to similar concept strategies/scaffolding
1a, 1b
1A, 2B
Independent practice
1b, 1c, 1d
4D, 6F
Use different numbers to solve similar problems
1a
3C
Graphic organizers, charts, tables
1a, 1c
3C, 4D, 7G
Inquiry learning/student led instruction
1a, 1b, 1d
1A, 3C, 4D, 6F,7G
Pictures and visuals
1a, 1c, 1d
4D, 5E, 6F, 7G
Numbers and Operations Content Standard rationale. In Round 1 of data
collection, panel members were asked to provide rationales for the practices they
identified for the numbers and operations content standard. Four panel members
identified similar rationales for the practices they selected. Panels members 1A and 7G
recommended real world application because this practice would be useful to assist
students in understanding numbers and operations through inquiry learning in real life
situations. Panel members 4D and 6F explained that vocabulary is important because
terms students know can spark their interest.
52
Algebra Content Standard. The Algebra NCTM middle school mathematics
content standard was the second item on the survey. The panel was presented with the
following Algebra Content Standard sample problem: “ABC Phones sells monthly cell
service for $0.50 per minute for the first 30 minutes but only $0.10 a minute for each
minute after. Graph the rate of change for this plan.” The panel was provided with the
four bulleted questions listed in the description of Round 1 as question 2a, 2b, 2c, and 2d.
The data collected from Round 1, Algebra Content Standard, are included in Table
3. The responses for each question were examined to locate instructional mathematics
practices. All practices were identified by the entire panel; two practices were identified
by five participants; three practices were identified by three participants, one practice was
identified by two participants; and two practices
Algebra Content Standard outcomes. Nine instructional mathematics practices
were identified in the panel’s Round 1 responses. These practices included (a) real world
application, (b) small group collaboration and discussion, (c) vocabulary, (d)
template/model, (e) connections to similar concept strategies/scaffolding, (f) independent
practice, (g) use graphic organizers, charts, and tables, (h) inquiry learning/student led
instruction, and (i) pictures and visuals. Some of these practices were identified by just
one participant and others were identified by up to five participants each (see Table 3).
None of the practices were mentioned by all seven of the participants in this round. The
practices that were most frequently recommended were inquiry learning/student led
instruction and pictures and visuals. Pictures and visuals were listed in four of the
questions, and inquiry learning/student led instruction was listed in three of the questions.
Both of these instructional mathematics practices were listed by five participants.
53
Several mathematics instructional practices were mentioned by three participants.
The practices included (a) small group collaboration and discussion, (b) vocabulary, and
(c) graphic organizers, charts and tables. All three practices were listed in two questions.
A few practices were mentioned by two participants. The practices included real world
applications and connections to similar concept strategies/scaffolding. Both of the
practices appeared in two questions for each.
The mathematics instructional practices mentioned by the least number of
participants were template/model and independent practice. Both of these strategies were
shared by only one participant. Template/model was seen in 2 questions. Independent
practice was shared in only one question.
Table 3
Round 1: Algebra Content Standard
Instructional Mathematics Practice
Question
Participants
Real world application
2b, 2d
1A, 3C
Small group collaboration and discussion
2a, 2c,
1A. 4D, 5E
Vocabulary
2a, 2b,
1A. 5E, 6F
Template/model
2c, 2d
3C
Connections to similar concept strategies/scaffolding
2a, 2b
2B. 7G
Independent practice
2d
4D
Graphic organizers/charts/tables
2a, 2c
2B, 3C, 4D
Inquiry learning/student led instruction
2a, 2b, 2c
1A, 4D, 5E, 6F, 7G
Pictures and visuals
2a, 2b, 2c, 2d
1A, 2B, 4D, 5E, 6F
54
Algebra Content Standard rationale. Panel members were asked to provide
rationales for the practices they identified for the algebra content standard in Round 1 of
data collection. Two panel members provided similar rationales for a mathematics
practice they identified for this content standard. Panel members 1A and 2B use visuals
and pictures to provide a graphic representation of equations to help students apply
algebra to real world situations. These panel members also suggested that using visuals
could help create a connection to the theories of number sense.
Geometry Content Standard. Geometry NCTM middle school mathematics
content standard was the third item on the survey. The panel was presented with the
following Algebra Content Standard sample problem: “List a triangle that is similar to the
one with measurements 4, 4 and 7. Draw the new model and explain how you knew it
was similar to the original.” The panel was provided with the four bulleted questions
listed in the description of Round 1 as question 3a, 3b, 3c, and 3d.
The data collected from Round 1, Geometry Content Standard, are included in
Table 4. The responses for each question were examined to locate instructional
mathematics practices. The practices that were located are listed along with the
corresponding question and the participant (see Table 4). Four participants suggested the
same practices such as inquiry learning/student led instruction. Three instructional
practices, template model, using colors to help track steps or patterns, and independent
practice, were not suggested by any participants but were listed under other content
standards included.
Geometry Content Standard outcomes. Eight instructional mathematics
practices were identified in the panel’s Round 1 responses: (a) real world application, (b)
55
use of technology, (c) small group collaboration and discussion, (d) vocabulary, (e)
connections to similar concept strategies/scaffolding, (f) graphic organizers, charts,
tables, (g) inquiry learning/student led instruction, and (h) pictures and visuals. All
practices were identified by the entire panel; one practice was identified by four
participants; three practices were identified by three participants; two participants were
identified by two participants; and one practice was identified by one participant.
The practices that were rated as the most effective were inquiry learning/student
led instruction and pictures and visuals Inquiry learning/student led instruction was
mentioned by five participants in two of the questions for the Measurement Content
Standard. Pictures and visuals were shared by four participants in three of the questions.
Real world application, small group collaboration and discussion and connections to
similar concept strategies/scaffolding were practices mentioned by four participants and
all three of these practices were included in all four questions. Vocabulary and graphic
organizers/charts/tables were shared by two of the participants. Vocabulary was included
in two of the questions while graphic organizers/charts/tables was included in one of the
questions. The mathematics instructional practice, technology, was mentioned by the least
number of participants. This practice was shared by only one participant and appeared in
just one of the questions.
56
Table 4
Round 1: Geometry Content Standard
Instructional Mathematics Practice
Question
Participants
Real world application
3a, 3b, 3c, 3d
1A, 2B, 3C
Use of technology
3a
2B
Small group collaboration and discussion
3a, 3b, 3c, 3d
4D, 6F, 7G
Vocabulary
3a
3C, 6F
Connections to similar concept strategies/scaffolding
3a, 3b, 3c, 3d
1A, 2B, 7G
Graphic organizers, charts, tables
3a, 3c
4D, 6F
Inquiry learning/student led instruction
3a, 3b
1A, 3C, 4D, 6F, 7G
Pictures and visuals
3a, 3b, 3c, 3d
2B, 3C, 4D, 6F
Geometry Content Standard rationale. In Round 1 of data collection, panel
members were asked to provide rationales for the practices they identified for the
geometry content standard. Four panel members provided similar rationales for practices
they identified. Panel members 4D and 7G both recommended using the practice of small
group collaboration and discussion to bolster other practices such as scaffolding and the
use of graphic organizers. Panel members 3C and 6F both discussed the importance of
having the students, not the teacher, draw pictures of the geometry concepts and problems
they are working on as a way of developing and demonstrating an understanding through
the use of visuals.
57
Some panel members described the teaching of geometry in a few ways. A few
described it as a visual learning. Because of the visual focus, a few talked about drawing
as a strategy for students to work through geometry problems.
Measurement Content Standard. The measurement NCTM middle school
mathematics content standard was the fourth item on the survey. The panel was presented
with the following Measurement Content Standard sample problem: “Scale factor: 1 inch
= 300 miles. If the distance from Denver, CO to Salina, UT is 1.5 inches on the map, how
far is the distance between the two cities in miles?” The panel was provided with the
four bulleted questions listed in the description of Round 1 as question 4a, 4b, 4c, and 4d.
The data collected from Round 1, Measurement Content Standard, is included in
Table 5. The responses for each question were examined to locate mathematics
instructional practices (see Table 5). All practices were identified by the entire panel;
three practices were identified by four participants; two practices were identified by four
participants; and one practice was identified by one participant.
Measurement Content Standard outcomes. Nine instructional mathematics
practices were identified in the panel’s Round 1 responses. These practices included (a)
real world application, (b) use of technology, (c) small group collaboration and
discussion, (d) template/model, (e) connections to similar concept strategies/scaffolding,
(f) independent practice, (g) graphic organizers, charts, tables, (h) inquiry
learning/student led instruction, and (i) pictures and visuals. Real world experiences,
graphic organizers, charts, tables, inquiry learning/student led instruction, and pictures
and visuals were most frequently recommended practices by the participants. Real world
application was shared in all four questions. All of these practices were recommended by
58
three participants. Graphic organizers, charts, tables were each included in three
questions. Inquiry learning/student led instruction was mentioned in two of the questions.
Use of technology, small group collaboration and discussion, template/model,
and connections to similar concept strategies/scaffolding were mentioned by two or fewer
participants. The use of technology was shared in three questions. All of the other
practices were shared in one question each. Independent practice was mentioned by just
one participant in two of the questions.
59
Table 5
Round 1: Measurement Content Standard
Instructional Mathematics Practice
Question
Participants
Real world application
4a, 4b, 4c, 4d
1A, 2B, 6F
Use of technology
4a, 4c, 4d
2B, 4D
Small group collaboration and discussion
4d
3C, 7G
Template/model
4a
2B, 7G
Connections to similar concept strategies/scaffolding
4a
1A, 7G
Independent practice
4c, 4d
4D
Graphic organizers, charts, tables
4b, 4c, 4d
4D, 6F, 7G
Inquiry learning/student led instruction
4a, 4b
1A, 4D, 6F
Pictures and visuals
4a, 4c, 4d
3C, 4D, 6F
Measurement Content Standard rationale. Panel members were asked to
provide rationales for the practices they identified for the measurement content standard
in Round 1 of data collection. Three panel members provided similar rationales for two of
the practices identified for this content standard. Panel members 2B and 6F suggested that
the use of real world examples including itmes they are familiar with will help with this
kind of problem. These panel emembers suggest using comparisons between small and
large animals or the movie, Honey I Shrunk the Kids to help studnets understand
measurement concepts and proportion. Panel members 3C and 6F recommended the use
of visuals to help students gain an understanding of the concept of small to large, large to
small, and similar ratios., because these visuals will help the students properly set up the
problem, which is key to being able to understand and solve the problem.
60
Data Analysis and Probability Content Standard. The data analysis and
probability NCTM middle school mathematics content standard was the fifth item on the
survey. The panel was presented with the following Data Analysis and Probability
Content Standard sample problem: “Collect data from newspaper weather/temperature
charts about the temperatures in selected cities within a region of the United States.
Calculate central measures and determine which city is warmest. Analyze the data to
make conjectures about the warmest city and determine if different central measures yield
different results.” The panel was provided with the four bulleted questions listed in the
description of Round 1 as question 5a, 5b, 5c, and 5d.
The data collected from Round 1, Data Analysis and Probability Content Standard
is included in Table 6. The responses for each question were examined to locate
instructional mathematics practices (see Table 6).
Data Analysis and Probability Content Standard outcomes. Eleven
instructional mathematics practices were identified in the panel’s Round 1 responses.
These practices included (a) real world application, (b) use of technology, (c) small group
collaboration and discussion, (d) vocabulary, (e) template/model, (f) connections to
similar concept strategies, (g) independent practice, (h) use different numbers to solve
similar problems, (i) graphic organizers, charts, tables, (j) inquiry learning/student led
instruction, and (k) pictures and visuals. All practices were identified by the entire panel;
one practice was identified by six participants; two practices were identifited by four
participants; 1 practice was identified by three participants; two practices were identified
by two participants; and five practices were identified by five participants.
The practice that was most frequently recommended was real life application.
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This practice was mentioned in all four of the questions in the Data Analysis and
Probability Content Standard and was shared by six of the participants. Small group
collaboration and discussion and inquiry learning/student led instruction were mentioned
by four participants. Inquiry learning/student led instruction was shared in all four
questions, and small group collaboration and discussion was mentioned in two questions.
Graphic organizers, charts, and tables was shared by three participants was practice was
in three questions. Template/model and independent practice were mentioned by two
participants, and both of these practices were included in two questions each. The least
number of participants included (a) use of technology, (b) vocabulary, (c) connection to
similar concept strategy/scaffolding, (d) use different numbers to solve similar problems,
in two questions, while the rest of these practices were shared only in one question each.
Table 6
Round 1: Data Analysis and Probability Content Standard
Instructional Mathematics Practice
Question
Participants
Real world application
5a, 5b, 5c, 5d
1A, 2B, 3C, 5E, 6F, 7G
Use of technology
5a, 5c
5E
Small group collaboration and discussion
5a, 5b
2B, 5E, 6F, 7G
Vocabulary
5b
6F
Template/model
5a, 5b
4D, 7G
Connections to similar concept strategies/scaffolding
5a
1A
Independent practice
5b, 5d
5E, 7G
Use different numbers to solve similar problems
5a
6F
Graphic organizers, charts, tables
5a, 5c, 5d
2B, 3C, 4D
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Inquiry learning/student led instruction
5a, 5b, 5c, 5d
1A, 2B, 4D, 6F
Pictures and visuals
5a
4D
Data Analysis and Probability Content Standard rationale. In Round 1 of data
collection, panel members were asked to provide rationale for the practices they
identified for the data analysis and probability content standard. Four panel members
provided similar rationales for two of the practices identified for this content area. Panel
members 2B, 3C, 5E, and 6F recommended real world application. These panel members
suggested that allowing students to perform real life data collection, especially if it is part
of their every day lives instead of just within the classroom, motivates students to want to
gain a better understanding of this challenging content standard. Panel members 2B and
3C recommended the practice of using graphic organizers, charts, and tables as this will
help students to more easily organize their data during data collection and to make it
easier to analyze.
Round 2: Second Modified Delphi Round
Each of the seven participants on the panel received a survey with the five content
standards from Round 1. The instructional mathematics practices collected from the panel
in Round 1 were listed under each question, and the panel was asked to rate each
instructional mathematics practice’s effectiveness using a Likert scale and provide a
rationale for each instructional practice. A practice rated as a was judged to be not
effective, a 2 meant minimally effective, a 3 was somewhat effective, a 4 was effective,
and a 5 was very effective. Each content standard in Round 2 had 8 to 11 instructional
mathematics practices that were identified in Round 1: (a) the Numbers and Operations
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Content Standard had 11 instructional mathematics practices, (b) the Algebra Content
standard had 9 instructional mathematics practices, (c) the Geometry Content Standard
had 8 s instructional mathematics practices, (d) the Measurement Content Standard had 9
instructional mathematics practices, and (e) the Data Analysis and Probability Content
Standard had 11 instructional mathematics practices
Numbers and Operations Content Standard Quantitative outcomes. The data
collected from Round 2, Number and Operatotions Content Standard is included in Table
7. The responses for each question were examined to locate trends and patterns regarding
instructional mathematics practices (see Table 7).
In Round 2, for teaching numbers and operations, the panel rated demonstrate real
world application as the most effective instructional mathematics practice (m = 4.14), also
taking mean and mode into account. The panel also rated explore the vocabulary as the
least effective instructional mathematics practice (m = 2.43) for teaching numbers and
operations.
The most effective instructional mathematics practices for the Numbers and
Operations Content Standard shared a common characteristic of having students make
connections in their learning to their environment, previous learning, and to others
through the demonstrate real life applications practice. Students connect numbers and
operations content to students’ prior knowledge and new knowledge through the connect
learning to similar concepts practice. Students are provided a setting to share and connect
the learning of numbers and operations content through interaction with one another via
the use small group collaboration and discussion.
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The least effective instructional mathematics practices for this content standard
were instructional practices that did not require the students to interact as much with
others, the environment, or to previous learning The least effective instructional
mathematics practices included provide a template/model, use colors to help track steps
or patterns, and explore the vocabulary. Although these practices may be effective
instructional practices in learning, the panel did not rate these instructional mathematics
practices as effective in numbers and operations instruction.
Table 7
Round 2: Numbers and Operations Content Standard
Instructional Mathematics Practices
m
Median
Mode
Demonstrate real world application
4.14
5
5
Connect learning to similar concepts (scaffolding)
4.29
4
4
Use small group collaboration and discussion
4.00
4
4
Provide examples (use different numbers to solve
different problems)
3.86
4
4
Use graphic organizers, charts, and tables
3.29
4
4
Provide pictures and visuals
3.29
4
4
Work independently (practice alone)
3.29
3
2, 3, 4
Use inquiry learning (student-led instruction)
3.14
3
3
Provide a template/model
3.00
3
3
Use colors to help track steps or patterns
2.43
3
3
Explore the vocabulary
2.43
2
2
Note: Scores reported ranged from 1 (least effective) to 5 (most effective).
65
Numbers and Operations Content Standard qualitative outcomes. Panel
members were asked to provide a rationale for their rating of each practice in the
Numbers and Operations Content Standard in Round 2 of data collection. A comment on
the rationale for the rating of each practice was required before the panel member could
proceed to the next practice. Rationales for the practices with a mean rating of 4.0 or
above and below 3.0 are presented here.
Panel members, 2B and 4D, rated the practice of demonstrating real world
application a 5 on the 1-5 Likert scale, but both panel members agreed that while the
practice was helpful for buy in, students might still have problems applying to and
solving individual problems without the use of other practices, too. Panel members, 2B
and 5E, gave similar rationales for their rating of the practice of connecting learning to
similar concepts (scaffolding), recommending that connecting the learning to prior
knowledge makes it more real and easier to apply for the student. For the practice of
using small group collaboration and discussion, panel members, 1A and 5E, suggested
that this practice will allow students to use their social learning skills to develop higher
thinking skills through group collaboration. Panel members, 1A and 4D, provided a low
rating for the practice of using colors to help track steps or patterns, suggesting that this
practice is too basic and would move students away from the larger concept. Panel
members 3C, 6F, and 7G all rated the practice of using vocabulary low for numbers and
operations, identifying that the practice might have some level of importance but would
not help the students get to the right answers.
Algebra Content Standard quantitative outcomes. The data collected
from Round 2, Algebra Content Standard is included in Table 8. The responses for each
66
question were examined to locate trends and patterns regarding instructional mathematics
practices (see Table 8).
In Round 2, for teaching algebra, the panel rated use graphics and visual organizers as the
most effective instructional mathematics practice (m = 4.43). The most effective two
instructional mathematics practices are ones that require students to use visual tools to
help with learning, The most effective practice was graphics, visual organizers, and
charts, and the second most effective practice was provide pictures and visuals.
The panel also rated explore the vocabulary as the least effective effective
mathematics practice (m = 2.71) for teaching algebra. This practice could be helpful with
algebra content; however, the panel did not rate it as being the most effective of the
practices suggested for this content area.
The practices rated as the least effective for teaching algebra were (a) working
independently and (b) exploring the vocabulary. Similarly, the least effective numbers and
operations instructional mathematics practices was use colors to help track steps or
patterns and explore the vocabulary. These mathematics instructional practices allow
students to learn without the support of a small group or partner. These practices allow
the student to work indepenedently
Table 8
Round 2: Algebra Content Standard
Instructional Mathematics Practices
m
Median
Mode
Use graphic organizers, charts, and tables
4.43
5
5
Provide pictures and visuals
4.14
4
4, 5
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Connect learning to similar concepts (scaffolding)
4.14
4
4
Use small group collaboration and discussion
4.00
4
4
Demonstrate real world application
3.86
4
5
Provide a template/model
3.43
4
4
Use inquiry learning (student-led instruction)
3.29
3
3
Work independently (practice alone)
3.00
3
3
Explore the vocabulary
2.71
3
2, 3, 4
Note: Scores reported ranged from 1 (least effective) to 5 (most effective).
Algebra Content Standard qualitative outcomes. Panel members were asked to
provide a rationale for their rating of each practice in the Algebra Content Standard in
Round 2 of data collection. A comment on the rationale for the rating of each practice was
required before the panel member could proceed to the next practice. Rationales for the
practices with a mean rating of 4.0 or above and below 3.0 are presented here.
Panel members, 2B and 4D, provided similar rationales for their rating of 5 on the
using graphic organizers, charts, and tables practice, suggesting that this practice would
help students extrapolate the formula and see the relationship between the formula and
the problem they are solving. Panel members, 4D and 6F, provided similar rationales for
their high ratings for the providing pictures and visuals practice, suggesting that this
practice helps to build understanding, problem solving skills, and transferability. For the
connecting learning to similar concepts (scaffolding) practice, panel members 2B and 6F
recommend this practice for the purpose of helping students see the progression to build
and sustain knowledge of this content standard. Panel members, 2B and 6F, provided
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rationales for the using small group collaboration and discussion practice that suggest this
practice will help students develop content mastery by owning their work as a team and
later applying as individuals. Panel members, 3C and 6F, suggested that the exploring the
vocabulary practice may be helpful but is not essential for this content standard.
Geometry Content Standard quantitative outcomes. The data collected from
Round 2, Geometry Content Standard is included in Table 9. The responses for each
question were examined to locate trends and patterns regarding instructional mathematics
practices (see Table 9).
In Round 2, for teaching geometry, the panel rated provide pictures and visuals as
the most effective instructional mathematics practice (m = 4.71). The panel also rated use
graphic organizers, charts, and tables as the least effective instructional mathematics
practice (m = 3.43) for teaching geometry.
The most effective instructional mathematics practice for the geometry NCTM
middle school mathematics content standard was provide pictures and visuals (m = 4.71),
but the next two practices rated by effectiveness were not rated as effective in the
previous two standards. Use inquiry learning (m = 3.86) was an instructional mathematics
practice rated as effective for the Geometry Content Standard. This practice was not rated
as highly effective in the other content standards. Explore the vocabulary (m = 3.71) is
another instructional mathematics practice the panel felt was more effective to use during
geometry instruction.
Demonstrate real life applications (m = 3.57) and use graphic organizers, charts and
tables (m = 3.43) were rated least effective of the mathematics instructional practices
shared by the panel when teaching geometry. All of the instructional mathematics
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practices can be effective when teaching mathematics, but the panel felt that visuals and
pictures were more effective than real life application and graphic organizers when
teaching geometry.
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Table 9
Round 2: Geometry Content Standard
Instructional Mathematics Practices
m
Median
Mode
Provide pictures and visuals
4.71
5
5
Use inquiry learning (student-led instruction)
3.86
4
5
Explore the vocabulary
3.71
4
4
Connect learning to similar concepts (scaffolding)
3.71
4
4
Use small group collaboration and discussion
3.71
4
3.4
Solve/Demonstrate using technology
3.57
2
2
Demonstrate real world application
3.57
4
2, 4, 5
Use graphic organizers, charts, and tables
3.43
4
4
Note: Scores reported ranged from 1 (least effective) to 5 (most effective).
Geometry Content Standard qualitative outcomes. Panel members were asked
to provide a rationale for their rating of each practice in the Geometry Content Standard.
A comment on the rationale for the rating of each practice was required before the panel
member could proceed to the next practice. Rationales for the practice with a mean
rating of 4.0 or above is presented here. Panel members 1A, 4D, and 6F provided similar
rationales for the providing pictures and visuals practice. These panel members suggested
the practice of providing pictures and visuals is essential for students to gain an
understanding of the geometry content standard; thus allowing students to solve the
problems.
Measurement Content Standard quantitative outcome. The data collected
from Round 2, Measurement Content Standard is included in Table 10. The responses for
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each question were examined to locate trends and patterns regarding instructional
mathematics practices (see Table 10).
In Round 2, for teaching measurement, the middle school mathematics teacher panel
rated provide pictures and visuals as the most effective instructional mathematics practice
(m = 4.14). The panel also rated provide a template/model as the least effective
instructional mathematics practice (m = 3.29), taking mode into account, for teaching
measurement.
The most effective mathematics practice for teaching measurement, provide
pictures and visuals (m = 4.14), appeals to visual learners. Measurement requires the use
of tools and visual representations to help students to see and apply the measurement
process and usually includes pictures and some type of visual. The least effective rated
instructional mathematics practices for teaching measurement were solve/demonstrate
using technology and provide a template/model. These instructional mathematics
practices are helpful, but the panel did not rate them as effective as the other practices.
Table 10
Round 2: Measurement Content Standard
Instructional Mathematics Practices
m
Median
Mode
Provide pictures and visuals
4.14
4
4
Use small group collaboration and discussion
3.86
4
3
Connect learning to similar concepts (scaffolding)
3.71
4
4
Work independently (practice alone)
3.43
4
4
Use graphic organizers, charts, and tables
3.43
4
4
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Demonstrate real world application
3.57
4
2, 4, 5
Use inquiry learning (student-led instruction)
3.43
4
3
Solve/Demonstrate using technology
3.29
4
4
Provide a template/model
3.29
4
2, 4
Note: Scores reported ranged from 1 (least effective) to 5 (most effective).
Measurement Content Standard qualitative outcomes. Panel members were
asked to provide a rationale for their rating of each practice in the Measurement Content
Standard. A comment on the rationale for the rating of each practice was required before
the panel member could proceed to the next practice. Only rationales for the practice
with a mean rating of 4.0 or above are presented here. Panel members, 3C, 6F, and 7G,
provided similar rationales for the providing pictures and visuals practice. These panel
members recommend this practice because the maps, scales, and other examples that
teachers and students can provide through real life pictures helps students demonstrate
what they are thinking, which leads to a deeper understanding.
Data Analysis and Probability Content Standard quantitative outcomes. The
data collected from Round 2, Data Analysis and Probability is included in Table 11. The
responses for each question were examined to locate trends and patterns regarding
instructional mathematics practices (see Table 11).
In Round 2, for teaching data and probability, the panel rated use small group
collaboration and discussion as the most useful instructional mathematics practice (m =
4.43). The panel also rated work independently as the least useful instructional
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mathematics practice (m = 2.86), taking mode into account, for teaching data and
probability.
The most effective instructional mathematics practice for the data and probability
NCTM middle school mathematics content standard connects student learning to the use
of small group discussion, and allows them to collect and/or practice manipulating data
with support. Small group learning helps to give the students supported practice before
moving to independent practice.
The least effective instructional mathematics practice for the data and probability
NCTM middle school mathematics content standard refers to the process of students
assimilating the data and personally applying and integrating the mathematics content
independently. This practice requires students to complete classroom activities without
support or interaction with classmates.
Table 11
Round 2: Data and Probability Content Standard
Instructional Mathematics Practices
m
Median
Mode
Use small group collaboration and discussion
4.43
5
5
Demonstrate real world application
4.29
5
5
Use graphic organizers, charts, and tables
4.14
4
4
Explore the vocabulary
4
4
4
Solve/Demonstrate using technology
3.71
4
4
Connect learning to similar concepts (scaffolding)
3.71
4
4
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Provide examples (use different numbers to solve similar
problems)
3.43
4
4
Use inquiry learning (student-led instruction)
3.57
3
3
Provide pictures and visuals
3.29
3
3
Provide a template/model
2.71
3
4
Work independently (practice alone)
2.86
3
2, 3
Note. Scores reported ranged from 1 (least effective) to 5 (most effective).
Data Analysis and Probability Content Standard qualitative outcomes. Panel
members were asked to provide a rationale for their rating of each practice in the Algebra
Content Standard. A comment on the rationale for the rating of each practice was required
before the panel member could proceed to the next practice. Rationales for the practices
with a mean rating of 4.0 or above and below 3.0 are presented here.
With regard to the using small group collaboration and discussion practice, panel
members, 3C and 5E, suggested that having students work together to gather data and
come up with a solution will help them develop a stronger understanding than by working
alone. Panel members, 1A and 2B, provided rationales emphasizing the importance of
demonstrating real world application, suggesting that many of the concepts within this
content standard can best be understood through a real world context. Panel members, 1A
and 7G, recommended using graphic organizers, charts, and tables as a good way to teach
this content standard because it helps with the visualization process. Panel members, 2B,
5E, and 7G, suggested that exploring the vocabulary practice is necessary for this content
standard for students to gain the understanding they need to be able to solve problems.
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Panel members, 1A and 2B, suggested that the practice of providing a template/model
would not be a good practice to use in this content standard. One reason suggested for
this is that that data from different situations would not fit neatly into a template/model.
Panel members, 1A and 3C, suggested using use small groups and collaborative learning
would be a more effective practice versus working independently.
Round 3: Final Modified Delphi Round
Each of the seven middle school mathematics teachers on the panel received a
survey with the same five content standards from Round 1 and Round 2. As in Round 2,
each question (that correlates to a NCTM middle school mathematics content standard)
had the 12 instructional mathematics practices listed under it. However, this time, each
instructional mathematics practice listed included the mean, median, and mode from
Round 2 along with all of the comments provided by each panel member from Round 2.
After reviewing the Round 2 data, each participant was asked one last time to rank the
priority of the instructional mathematics practice for teaching the NCTM middle school
mathematics content standard by marking the 5-point Likert scale.
The 12 instructional mathematics practices collected from the panel in Round 1
were listed under each question along with the mean, mode, and median data and all
comments collected from the panel. The panel was asked to rate each instructional
mathematics practice’s effectiveness a last time using a Likert scale. This time, the panel
was not required to provide an explanation for each rating.
Round 2 data provided the rating and explanations from the group of seven middle
school mathematics teacher on the panel for each of the recommended instructional
mathematics practices best suited to meet the NCTM middle school mathematics content
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standard represented by survey content standards 1-5. The purpose of the Round 3 data
collection was to allow the panel members view how others on the panel valued each of
the instructional mathematics practices generated for the five content standards based on
the NCTM middle school mathematics content standards. In other words, each panel
member was given the opportunity to rethink his or her own rating for each instructional
mathematics practice for each content standard after looking at the instructional
mathematics practice ratings including mean, median, mode, and explanations collected
from Round 2. With this information, the panel rated everything with a 5-point Likert
scale to indicate how essential each instructional mathematics practice would be for
instructing the indicated NCTM middle school mathematics content standard.
Each of the seven middle school mathematics teachers on the panel received a
survey with the same five content standards from Round 2. This time each content
standard had the instructional mathematics practice listed under it ranked in order based
on the mean, median and mode based on the ratings from Round 2. Each instructional
mathematics practice had a list of explanations collected from the panel during Round 2
sharing the reasoning behind each rating. Each participant was asked to rank the priority
of the instructional mathematics practice for teaching the NCTM middle school
mathematics content standard by marking the 5-point Likert scale on last time with no
explanation. The instructional mathematics practices collected from the panel in Round 1
were listed under each content standard, and the panel was asked to rate each
instructional mathematics practices effectiveness using a 5-point Likert scale.
Numbers and Operation Content Standard. The data collected from Round 3,
Question 1, which was related to the numbers and operations NCTM middle school
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mathematics content standard is displayed in Table 12. The instructional mathematics
practice are ranked from most effective to least effectvie.
In Round 3, for teaching numbers and operations, the panel rated, demonstrate
real world application and connect learning practices to similar concepts, as the most
effective instructional mathematics practice (m = 4.71). The panel also rated explore the
vocabulary as the least effective instructional mathematics practice (m = 2.29) for
teaching numbers and operations.
The top instructional mathematics practices for the numbers and operations
standard included: (a) demonstrate real world application, (b) connect learning to similar
concepts (scaffolding), (c) use small group collaborations and discussion, and (d) provide
examples (use different number to solve different problems). Their means ranged from
4.71 down to 4.43. The next instructional practice dropped down to 3.14.
Table 12
Summary of Round 3: Number and Operations Content Standard
Instructional Mathematics Practices
m
Median
Mode
Demonstrate real world application
4.71
5
5
Connect learning to similar concepts (scaffolding)
4.71
5
4
Use small group collaboration and discussion
4.57
5
5
Provide examples (use different numbers to solve
different problems)
4.43
5
5
Use graphic organizers, charts, and tables
3.14
3
4
Provide pictures and visuals
3.14
3
3
Work independently (practice alone)
3.00
3
2, 3, 4
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Use inquiry learning (student-led instruction)
3.00
3
3
Provide a template/model
2.86
3
3
Use colors to help track steps or patterns
2.57
3
3
Explore the vocabulary
2.29
2
2
Note: Scores reported ranged from 1 (least effective) to 5 (most effective).
Algebra Content Standard. The data collected from Round 3, Question 2, which
was related to the algebra NCTM middle school mathematics content standard is
displayed in Table 13. The instructional mathematics practices are ranked from most
effective to least effective.
In Round 3, for teaching algebra, the panel rated use graphic organizers, charts,
and tables and provide pictures and visuals as the most effective instructional
mathematics practice (m = 4.71). The panel also rated work independently and explore
the vocabulary as the least effective instructional mathematics practice (m = 2.57) for
teaching algebra.
The top instructional mathematics practices for the algebra standard included: (a)
use graphic organizers, charts, and tables; (b) provide pictures and visuals; (c) connect
learning to similar concepts (scaffolding); and (d) use small group collaboration and
discussion. Their means ranged from 4.71 down to 4.43. The next instructional practice
dropped down to 3.14.
Table 13
Summary of Round 3: Algebra Content Standard
Instructional Mathematics Practices
m
Median
Mode
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Use graphic organizers, charts, and tables
4.71
5
5
Provide pictures and visuals
4.71
5
5
Connect learning to similar concepts (scaffolding)
4.57
5
5
Use small group collaboration and discussion
4.43
5
5
Demonstrate real world application
3.14
3
3
Provide a template/model
3.00
3
2
Use inquiry learning (student-led instruction)
2.86
3
3
Work independently (practice alone)
2.57
3
3
Explore the vocabulary
2.57
2
2
Note: Scores reported ranged from 1 (least effective to 5 (most effective).
Geometry Content Standard. The data collected from Round 3, Question 3,
which was related to the geometry NCTM middle school mathematics content standard is
displayed in Table 14. The instructional mathematics practices are ranked from most
effective to least effective.
In Round 3, for teaching geometry, the panel rated provide pictures and visuals as
the most effective instructional mathematics practice (m = 5.00). The panel also rated use
graphic organizers, charts, and tables as the least effective instructional mathematics
practice (m = 3.14) for teaching geometry.
The top instructional mathematics practices for the geometry standard included:
(a) provide pictures and visuals and (b) use inquiry learning (student-led instruction),
Their means ranged from 5.00 down to 4.43. The next instructional practice dropped
down to 3.43.
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Table 14
Summary of Round 3: Geometry Content Standard
Instructional Mathematics Practices
m
Median
Mode
Provide pictures and visuals
5.00
5
5
Use inquiry learning (student-led instruction)
4.43
5
5
Explore the vocabulary
3.43
3
3
Connect learning to similar concepts (scaffolding)
3.43
3
3
Use small group collaboration and discussion
3.29
3
3, 4
Solve/Demonstrate using technology
3.29
3
3
Demonstrate real world application
3.29
3
2, 3, 4
Use graphic organizers, charts, and tables
3.14
3
3
Note: Scores reported ranged from 1 (least effective) to 5 (most effective).
Measurement Content Standard. The data collected for Round 3, Question 4,
which was related to the measurement NCTM middle school mathematics content
standard is displayed in Table 15. The instructional mathematics practices are ranked
from most effective to least effective.
In Round 3, for teaching measurement, the panel rated provide pictures and
visuals as the most effective instructional mathematics practice (m = 5.00). The panel also
rated provide a template/model as the least useful instructional mathematics practice
(m = 2.71) for teaching measurement.
The top instructional mathematics practices for the measurement standard
included: (a) provide pictures and visuals, (b) use small group collaboration and
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discussion, (c) connect learning to similar concepts. Their means ranged from 5.00 down
to 4.43. The next instructional practice dropped down to 3.29.
Table 15
Summary of Round 3: Measurement Content Standard
Instructional Mathematics Practices
m
Median
Mode
Provide pictures and visuals
5.00
5
5
Use small group collaboration and discussion
4.43
5
5
Connect learning to similar concepts (scaffolding)
4.43
4
4
Work independently (practice alone)
3.29
3
3
Use graphic organizers, charts, and tables
3.29
3
3, 4
Demonstrate real world application
3.29
4
2, 4
Use inquiry learning (student-led instruction)
3.14
3
3
Solve/Demonstrate using technology
3
3
2
Provide a template/model
2.71
3
2, 3, 4
Note: Scores reported ranged from 1 (least effective) to 5 (most effective).
Data Analysis and Probability Content Standard. The data collected for Round
3, Question 5, which was related to the data analysis and probability NCTM middle
school mathematics content standard is displayed in Table 16. The instructional
mathematics practices are ranked from most effective to least effective.
In Round 3, according to the panel’s ratings, for teaching data analysis and
probability, use small group collaborations and discussion and demonstrate real world
application as the most effective instructional mathematics practice (m = 4.71). The panel
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also rated work independently as the least effective instructional mathematics practice (m
= 2.43) for teaching data analysis and probability.
The top instructional mathematics practices for the data analysis and probability
standard included: (a) use small group collaboration and discussion; (b) demonstrate real
world application; (c) use graphic organizers, charts, and tables; and (d) explore the
vocabulary. Their means ranged from 4.71 down to 4.57. The next instructional practice
dropped down to 3.14.
Table 16
Summary of Round 3: Data Analysis and Probability Content Standard
Instructional Mathematics Practices
m
Median
Mode
Use small group collaboration and discussion
4.71
5
5
Demonstrate real world application
4.71
5
5
Use graphic organizers, charts, and tables
4.57
5
5
Explore the vocabulary
4.57
5
5
Solve/Demonstrate using technology
3.14
3
3
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Connect learning to similar concepts (scaffolding)
3.14
3
3
Provide examples (use different numbers to solve
similar problems)
3.14
3
2
Use inquiry learning (student-led instruction)
3.00
3
2
Provide pictures and visuals
3.00
3
2, 4
Provide a template/model
2.71
3
2, 3
Work independently (practice alone)
2.43
3
3
Note: Scores reported ranged from 1 (least important) to 5 (most important).
Synposis. Round 3 was different from Round 2 because it was generated by the
middle school mathematics teacher panel with knowledge of one another’s thoughts or
trends in the group through the sharing of mean, median, mode, and explanations from
Round 2. The three most effective instructional mathematics practices identified for each
question/ NCTM middle school mathematics content standard were never exactly the
same as another question/ NCTM middle school mathematics content standard, but they
stayed in the same ranking order for each content standard from the Round 2 results. For
just about every question and instructional mathematics practice, the data represented by
the mean in the three most effective instructional mathematics practices strengthened. For
example, for the Numbers and Operations Content Standard, the mean for the highest
ranked instructional mathematics practice, apply real world strategies, increased from
4.14 to 4.71. The mean for the second most effective instructional mathematics practice,
connect learning to similar concepts (scaffolding), increased from 4.29 to 4.71, and the
mean for the third most effective instructional mathematics practice, use small group
collaboration and discussion, increased from 4.00 to 4.57. The only mean of a three most
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effective instructional mathematics practices that decreased was for the Geometry
Content Standard. The mean for the third most effective instructional mathematics
practice for both Round 2 and Round 3, explore the vocabulary, decreased from 3.71 to
3.43. The three most effective mathematics practices and rankings for each question did
not change from Round 2 to Round 3. The strengthening of the averages appears to
demonstrate that the panel was more confident from the previous rankings after being
able to view the mean, median, mode, and explanations from all middle school
mathematics teachers on the panel from Round 2.
Data Analysis of Complete Project
The data collected in the survey indicated that the panel in the field of middle
school mathematics identified 12 instructional mathematics practices that were present in
all NCTM middle school mathematics content standards presented at this level. Each of
the five questions was connected to one of the NCTM middle school mathematics content
standards, but the mathematics instructional practices that were generated and those that
were rated most effective by the panel were different for each question/standard. Because
of this difference, when searching for the best instructional mathematics practices to use
when teaching a question based on an NCTM middle school mathematics content
standard, the data support that the instructional mathematics practices chosen to use in
instruction should change depending upon which NCTM middle school mathematics
content standard is being taught. The results from the data support sharing the
instructional mathematics practices with educators separately for each NCTM middle
school mathematics content standard because the instructional mathematics practices
were ranked differently depending upon the question.
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After the panel members viewed the explanations from others when generating
the Round 3 data results, the top instructional mathematics practices list stayed the same.
The notable connection for this round and the previous round was that the mean data
strengthened between Round 2 and Round 3. In other words, the data from Round 3
supports the idea that the panel were more confident that the top instructional
mathematics practices were the best ones to use for each of the five questions based on
the 5 NCTM instructional mathematics practices. The strong ratings of the top
instructional mathematics practices for each question from the panel suggests that
teachers could use them to strengthen teaching and learning for each question.
Results Summary
In summary, in this study I collected data from a middle school mathematics teacher
panel over three rounds. In Round 1, the panel provided instructional mathematics
practices that could be used to help students solve five problems. Each problem matched
a NCTM middle school mathematics content standard for middle school mathematics.
The instructional mathematics practices were collected for all five questions and
coded/grouped into a total of 12 instructional mathematics practice. In Round 2, the
instructional mathematics practices collected from Round 1 for each question. The panel
was asked to rate the effectiveness of each instructional mathematics practice for each
question, and the data was analyzed using mean, median, and mode. The panel also
provided reasoning for each instructional mathematics practice rated. For Round 3, the
data collected from Round 2 was presented to the panel for each question including the
reasoning. The panel rated each instructional mathematics practice’s effectiveness for
teaching each NCTM content standard one more time after reflecting upon the reasoning
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provided from the rest of the panel in Round 2. The data collected were shared in Tables
1 through 11 with different instructional mathematics practices rated in the top three for
each question.
The mathematics instructional strategies that the panel generated and rated during
Round 3 related directly to the problem and research question from the study. The
problem was that due to very low state eight grade mathematics scores, a need exists for
appropriate learning and critical thinking instructional mathematics practices that help all
students master abstract and more difficult middle school mathematics concepts. The
Round 3 outcomes in this modified Delphi study included instructional mathematics
practices generated and rated by middle school mathematics teacher panel. The panel
members believed the instructional strategies generated and rated would help students
master mathematic concepts in each of the NCTM mathematics content standards.
The research question also relate to the outcomes from the study. The research
question asked “What are middle school mathematics teachers’ perspectives of
instructional mathematics practices for abstract mathematics concepts and content taught
in an urban middle school in Colorado?” The Round 3 outcomes in this modified Delphi
study included instructional mathematics practices generated and rated by middle school
mathematics teacher panel. The instructional practices were generated for each NCTM
content standard based on the perspectives from the panel. and then were rated according
to effectiveness in the Round 3 outcomes.
The conceptual framework for this study relate to many of the outcomes. The
NCTM Principles are six principles that are examined that may assist in planning
mathematics instructional methods, mathematics learning, and in the creation of
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topquality mathematics programs (NCTM, 2000). These six principles (NCTM, 2000)
include: (a) equity, (b) curriculum, (c) teaching, (d) learning, (e) assessment, and (f)
technology.
Instructional strategies generated by the panel in the Round 3 outcomes relate to
the equity principle. This principle is based on the idea that one’s potential to learn
mathematics should not be lowered due to extenuating circumstances which could
include language deficiencies, socioeconomic status, or disabilities. Instead, this principle
asserts that additional resources be used to help all learners meet high mathematics
learning expectations. Through the outcomes in Round 3, there were instructional
practices generated and ranked at a top level that could meet the equity principle because
the practices provide additional resources to help all learners. Provide examples (use
different numbers to solve different problems) gives all learners additional resources to
help students to be successful in NCTM content standards such as numbers and
operations. Use graphic organizers, charts, and tables was an instructional practice
providing extra resources to help students especially in the algebra content standard and
in data analysis and probability. Provide pictures and visuals also includes a focus on
additional resources, and the panel thought it is especially effective when teaching
geometry and measurement content standards.
Instructional strategies generated by the panel in the Round 3 outcomes relate to
the curriculum principle. This principle is based on the idea that mathematics concepts
taught should be worthwhile or have purpose in everyday life, and these connections
should be presented in mathematics instruction. Through the outcomes in Round 3, there
was an instructional practice generated and ranked at a top level that could help provide
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connections to students’ everyday live. Demonstrate real world application is an
instructional practice generated from the outcomes that gives students a chance to make
these connections in NCTM content standards such as numbers and operations and data
analysis and probability content standards
Instructional strategies generated by the panel in the Round 3 outcomes relate to
the teaching principle. An important part of sound teaching that falls under this principle
is the creation thought-provoking yet compassionate teaching setting. Through the
outcomes in Round 3, a few instructional strategies ranked at a top level can help create a
thought-provoking yet compassionate teaching setting. Use small group collaboration and
discussion was an instructional practice generated from the outcomes that gives students
a chance to make these connections in NCTM content standards such as numbers and
operation, algebra, measurement, and data analysis and probability content standards. Use
inquiry learning (student led instruction) was another instructional practice generated
from the outcomes that does the same thing. This practice was ranked at a high level in
the geometry NCTM content standard.
Instructional strategies generated by the panel in the Round 3 outcomes relate to
the learning principle. An important part of sound teaching that falls under this principle
is that teachers need to provide experiences that provide a deeper meaning and learning
level for students through appealing activities and classroom communications (NCTM,
2000). Through the outcomes in Round 3, a few instructional practices ranked at a top
level can help create experiences where students attain deeper meaning with appealing
activities and communication. These same instructional practices connected to the
previous principle discussed, the teaching principle. Use small group collaboration and
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discussion was an instructional practice generated from the outcomes that gives students
a chance to make these connections in NCTM content standards such as numbers and
operation, algebra, measurement, and data analysis and probability content standards. Use
inquiry learning (student led instruction) was another instructional practice generated
from the outcomes that does the same thing. This practice was ranked at a high level in
the geometry NCTM content standard.
All instructional strategies generated by the panel in the Round 3 outcomes relate
to the assessment principle. An important part of sound teaching that falls under this
principle is that the use of assessment as an instrument is one of the best way to make
educational decisions (NCTM, 2000). Through the outcomes in Round 3, all instructional
practices fit this principle. The first way to apply the practices is to assess students using
an instructional practice to gather assessment data. Any of the practices can be assessed
informally or formally. The second part of this principle is to make educational decisions
based on the assessment data collected. Various instructional practices could be applied to
future learning activities as a result of analyzing assessment data. Using assessment as an
instrument is one of the best way to make educational decisions (NCTM, 2000).
One instructional strategy generated by the panel in the Round 3 outcomes relate
to the technology principle. The technology principle states that teachers should use it in
efficient ways to support teaching and learning. In the beginning of the modified Delphi
process, the teacher panel generated the instructional practice of solve, demonstrate using
technology for three of the NCTM content standards, geometry, measurement, and data
analysis and probability. In subsequent rounds, including Round 3 which was used to
analyze the outcomes, the technology instructional practice was not ranked at a high level
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by the teacher panel. Solve, demonstrate using technology was still listed as a possible
instructional practice for teaching and learning geometry, measurement, and data analysis
and probability in the Round 3 results.
Conclusion
In this section, the research process has been outlined including detailed
descriptions of the data collection and data analysis procedures, description of the
exploratory, modified Delphi method, reliability and validity processes in the study, data
analysis and validation procedures, and participants’ rights. The end result of the research
included a list of research instructional mathematics practices that the panel selected that
are effective in instructing mathematics concepts. The outcomes from Round 3 guided the
development of a resource guide that teachers can use to help students learn mathematics
by using recommended instructional mathematics practices. The top ranked results for
each of the NCTM content standards in Round 3 were used to create the resource guide in
a Training Plan project deliverable. This project deliverable provides teachers with a
resource where they can find out what these top instructional practices are for each of the
NCTM content standards and then apply them in learning activities with students.
In Section 3, the project is described along with the review of literature related to
the project. Section 4 of the study includes the researcher’s reflections related to the
research study and conclusions related to the project study. This section provides a list of
implications, applications, and direction for further research.
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Section 3: The Project
Introduction
This section includes a brief description of my project (see Appendix A). I chose a
professional development training plan for the project format. First, a training plan could
help teachers assess their own use of best practice teaching strategies based on the
outcomes of this study. Then, a plan could help teachers implement these strategies into
their own instruction. Teachers who are using the training plan to improve their
mathematics instruction would learn about the instructional mathematics practices
suggested by the middle school mathematics teacher panel in the modified Delphi study.
Teachers would then use the training plan to help them set goals and create learning
activities that incorporate these instructional mathematic practices into classroom
learning.
Description and Goals
The training plan was created to provide guidance to teachers but also to allow
teachers to be able to choose the practices that work best for their needs and the needs of
their students. The training plan includes: (a) purpose; (b) learning outcomes; (c) intended
audience: (d) components and a suggested timeline; (e) materials, activities, and trainer
notes designed to help teachers learn about using different instructional mathematics
practices; and (f) plans and materials for an evaluation plan.
The goals of the training plan are designed to help teachers address the problem of
low middle school mathematics achievement by using the instructional
mathematicspractices collected from the panel in the modified Delphi study. The goals
are: (a) to enable teachers to integrate best practices for middle school mathematics
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instruction into daily mathematics lessons, and (b) to provide appropriate mathematics
instruction to help teachers facilitate improvement in mathematics student achievement.
Rationale
I chose the training plan to address the problem of low middle school achievement
on mathematics standardized tests. The data collected from the middle school
mathematics teacher panel in the study focused on finding the best instructional
mathematics practices to use when teaching, in accordance with NCTM middle school
mathematics content standards. The training plan provides a way to share a procedure by
which teachers can assess their own instructional methods, integrate best practices, and
customize their instruction and assessments (Beswick, 2014).
The training plan includes the purpose of the training plan along with an
explanation of the importance of using instructional mathematics practices to help
students achieve NCTM middle school mathematics content standards. I designed the
training plan as a type of professional development to help teachers achieve teacher self-
awareness in relation to the problem of the study. The problem was that, due to very low
mathematics scores in the eighth grade state test , mathematics teachers need appropriate
instructional mathematics practices that help all students master challenging middle
school concepts. The teachers using the training plan are able tocan take the information
from the outcomes related to the highest-rated instructional mathematics practices for
each of the NCTM standards and learn how to apply them in their mathematics
instruction. Thus, students will become a part of instructional activities that can help them
to master challenging mathematics concepts.
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A novice or experienced teacher can use the training plan by following a few easy
steps. Teachers should: (a) read the training plan and consider the highest rated
instructional practices for each of the NCTM content standards, (b) apply the training
plan by using the checklist to check his or her current use of the instructional
mathematics practices shared, and (c) use the information shared in the plan about
implementation of highly rated instructional practice instructional practices in his or her
lesson planning. This application of highly rated instructional practices benefits students
by providing them with different paths to master challenging mathematics topics.
Review of the Literature
The training plan genre is appropriate to the problem from the research. The
problem was that due to very low state eight grade mathematics scores, a need exists for
appropriate learning and critical thinking instructional mathematics practices that help all
students master abstract and more difficult middle school mathematics concepts. The use
of best practices or proven instructional mathematics practices collected from the
outcomes of the modified Delphi methodology in the study can help to improve student
achievement scores. The outcomes from the final round of the study included the highest
rated instructional practices from each of the NCTM standards. These highest rated
instructional practices are the practices highlighted in the training plan so that teachers
can find ways to include them in their instructional practice.
The literature review for professional development included a search to saturate
current research regarding professional development of which a training plan is a
subcategory. The Walden library databases were searched and included the following
terms:
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training plan, professional development, professional development and student
achievement, math professional development, professional development research, reading
professional development, science professional development, social studies professional
development. All recent articles were considered for the literature review.
Training Plan
A training plan is a form of professional development for teachers and might
assist professionals better understand best practices related to the profession (Cox, 2015).
The best professional development is based on or backward planned from well-defined
goals (Guskey, 2014). The goals for this training project are based on the problem of low
achievement and the notion of sharing best practices The goals should be the ultimate aim
should be to reverse/eliminate/reduce low mathematics achievement. The primary goal,
as shared earlier, is to enable teachers to integrate best practices for middle school
instruction into daily lessons to facilitate improvement in mathematics student
achievement.. Grusky (2014) also observed the benefits of using learner outcomes to
prepare professional development training. The project study started with learner
outcomes in the problem to guide the methodology and data collection process, which, in
turn, led to the decision to use a training plan to help deepen the teacher knowledge of
recommended instructional mathematics practices.
Once teachers are made aware of the recommended strategies, professional
learning or development works best when the teachers identify his or her needs in relation
to the material (Beswick, 2014). This project includes self-evaluation in the training plan
that allows teachers to assess their learning needs and to use this to help understand
where to grow from the information presented. This pre-assessment or self-assessment
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benefits teacher and students alike. After teachers learn to effectively apply the
recommended instructional mathematics practices in the classroom, the students can
benefit from these new instructional mathematics practices to achieve at higher levels.
Benefits
Finding research-based recommended mathematics instructional practices and
sharing them through professional development with teachers, helps teachers to broaden
their own states of application and knowledge that can help broaden their students’
content knowledge. A training plan for this study fits into this philosophy because it helps
teachers to broaden practice through self-assessment and application of new instructional
mathematics practices shared by the middle school mathematics teacher panel in the
modified Delphi study. In the end, higher student achievement should help students to
master challenging mathematics concepts when research-based instructional mathematics
practices are shared effectively through professional development. According to Shaha
and Ellsworth (2013), schools with solid professional development plans performed
higher in many areas including the area of student achievement.
Mathematics achievement. A study on teaching mathematics at the elementary
level provided results showing that teachers who are strong in pedagogy in mathematics
have students that perform at higher achievement levels in students’ mathematics
assessments (Erskine, 2010). The content knowledge of teachers could be enhanced in
several ways with one being targeted professional development. Higher student
mathematics achievement could be attained more easily when teachers read and complete
the training plan in this project.
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Winkler (2011) observed teachers who received professional development in
mathematic interactive lesson plans, and the results showed that the teachers participating
in this training led to higher student achievement on mathematics assessments.
Professional development was used to help teachers with a specific lesson plan format
that should help students to learn more easily. This training content is similar to the
content of training teachers on best practices in accordance with NCTM mathematics
NCTM middle school mathematics content standards. The teachers using the training
plan in this study should have students showing higher levels of mathematics
achievement such as shown in Winkler’s study (2011).
Parrish (2013) observed the effect of professional development on teacher
differentiation practices in Grades 3 to 5 mathematics and science achievement. The
results showed that for most professional development training, students who were part of
the study outperformed the district median level of achievement. Santau, MaertenRivera,
and Huggins (2011) observed ELL students whose teachers received professional
development with regard to science. These students also yielded assessment results that
were higher than students whose teachers were not part of the professional development.
Santau et al. demonstrated the need to provide effective professional development related
to the top mathematics instructional mathematics practices collected from the middle
school mathematics teacher panel in the modified Delphi research for this study. The result
of the training plan for this project study is to provide a project that generates higher levels
of student achievement in middle school mathematics when the teachers implement the
training.
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Caban-Vazquez (2010) conducted a study in which he determined that teacher
training positively impacted student mathematics achievement. An after school
mathematics program was the setting for this study. These findings support the idea of
using a training plan to educate middle school mathematics teachers regarding
recommended teacher instructional mathematics practices collected through the research.
The strategies shared in the Caban-Vasquez study are similar to the ones generated by the
panel.
Reading achievement. Reading instruction is a content area where research has
shown improved student achievement via professional development. Fisher, Frey, and
Nelson (2012) established that students received moderate gains in reading achievement
because of strategic professional development. Porche and Pallante (2012) also studied
the effects of professional development on fourth grade students. Porche and Pallante
(2012) concluded that most areas of reading statistically improved for the students. If
professional development helps students to succeed in reading instruction, mathematics
professional development may show similar results in the training plan for this project
study.
Research exists showing positive correlations between professional development
and state standardized test achievement scores (Jackson, 2014). In a middle school
population, the state achievement scores were analyzed to determine this correlation.
Content areas scores included both language arts and mathematics. A goal of this project
study plan is to help students achieve at a higher level, so professional development in the
form of a training plan might provide results similar to Jackson’s study.
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Effective Mathematics Professional Development
Beswick (2014) observed mathematics teachers and gathered their perceptions
about professional development. The findings indicated that mathematics teachers need to
communicate to make professional development meaningful. Liljedahl (2014) also
determined that teachers want input into their professional development and that single
session workshops are not a favored format. McConnell, Parker, and Eberhardt, (2013)
further established the need for pre-assessment of the teachers’ learning needs during
professional development in order to determine activities that are appropriate based on
teacher experiences. Kapanadze, Bolte, Schneider, and Slovinsky (2015) conducted
research on the professional development of teachers related to current teaching reforms
in science which led to higher student achievement. Cox (2015) noted the support behind
different choices that teachers have in professional development, and the training plan for
this project offers a choice of training that might be more flexible and in tune with
teachers. Suanrong and Herron (2014) emphasized that differentiated training helps to
amplify the event for teachers. The training plan allows for teachers to determine what
instructional mathematics practices are new and how it can apply to his or her needs,
similar to how the training plan from this study should be applied.
Polly, Neale, and Pugalee (2014) established that teachers went through
professional development felt better about mathematics instruction and demonstrated
improvement in teaching performance. The study included teacher observations, and the
results collected described the teachers as showing more knowledge and stronger
viewpoints regarding mathematical instruction. Jao and McDougall (2015) concluded that
mathematics teachers embrace professional development and appreciate the ability to
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implement what they had learned within a collaborative community. The training plan
will encourage teachers to pursue collaborative professional development, yet the
teachers will be able to use the checklist to personalize the training ot his or her personal
needs in relation to teaching the NCTM content standards.The training plan from this
study should impact mathematics teachers as instructional mathematics practices
generated by middle school mathematics teachers that can be used to improve pedagogy
and beliefs about how to best teach different NCTM middle school mathematics content
standards are provided. This plan can benefit the community of teachers who are all
sharing this instructional focus.
Nadelson et al. (2013) studied how teachers perceived professional development
with regard to their own value teaching STEM (science, technology, engineering, and
mathematics) content. The end result maintained that teachers were confident in their
teaching and student learning after professional development. Similar results were
obtained in other research (Lane et al., 2015) with regard to teacher feelings of efficacy
after professional development in assessment intervention strategies. Abilock, Harada,
and Fontichiarof (2013) suggested that professional development opportunities based on
specifi teacher needs increase teacher effectiveness in the classroom. Renninger, Cai,
Lewis, Adams, and Ernst (2011) advocated the view that teachers preferred training that
is learner directed based on each teacher’s needs. Teachers will progress through the
training plan in this study and will be educated about top instructional mathematics
practices collected from the outcomes in the modified Delphi methodology. As a result,
teachers may move towards a more positive viewpoint of their own abilities to improve
student learning basedon student learning needs
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Implementation
The implementation plan in this section was created to direct the training planhe
plan also ensures materials, resources, supports and a reasonable timetable are in place. It
also defines the roles and responsibilities of the stakeholders who will take part in the
plan.
Potential Resources and Existing Supports
The plan includes several resources and existing supports. Resources include the
internet and email which allows the training plan to be shared within the school district
mentioned in the problem or any other one in the country. Other resources include the
data or chosen effective instructional mathematics practices from the panel in the
modified Delphi group and research available related to the instructional mathematics
practices that can be shared with teachers. Supports include personnel who are already in
place to help teachers with mathematics instruction coaching in the school district. These
people would be able to communicate with teachers and share the plan and materials.
Potential Barriers
The training plan includes potential barriers as well. One barrier would be time
required to complete the training plan. Teachers are busy professionals, and the plan
needs to be manageable so that they can fit it in to their schedules and apply the new
information along with their regular teaching duties. The training plan is asynchronous,
so teachers can fit it in during his or her free hours rather than attending training at a set
date, time, and place. Another barrier would include the buy-in from administration
within the school district in order to implement it. To help with this issue, the training
plan is shared with administration in a debriefing meeting by the researcher as a tool to
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improve middle school mathematics achievement, which is the overarching problem of
the study and throughout the nation. The purpose of the study was to gather researchbased
teaching and learning instructional mathematics practices that follow best practice and
align with the concepts taught and CO’s mathematics standards, which, ultimately, may
positively impact middle school students’ mathematics achievement. This purpose is
helpful to explain to administration why the project is worth the district’s time and effort
for implementation as the project since school district’s might be interested in
implementing these instructional practices that could lead to better student achievement
in mathematics.
Proposal for Implementation and Timetable
The project implementation and timetable are discussed in this section. First, the
purpose and background along with the training plan outline, project explanation and
details, pre-assessment checklist, NCTM content standard modules, and project
evaluation rubric evaluation regarding the project goals is discussed. The timetable would
include 1 day to preview the materials, 1 day to read the project explanation and
background, scan the information in the NCTM content standard modules, and to
complete the Current Instructional Mathematics Practices Being Used Pre-assessment
Checklist that is included in the plan, and 1-3 days to review the new instructional
mathematics practices and plan implementation included in the NCTM content standard
modules in the project. Additionally, the participant would designate a term (i.e. 6 or 9
weeks term) to use the teaching activities generated from the instructional mathematics
practices, and also set aside time on 1 day to evaluate the integration based on the project
goals.
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Roles and Responsibilities of Student and Others
Roles and responsibilities are shared for the stakeholders—a group that includes
the researcher, administrator, teachers completing the training plan, and the students. The
researcher is responsible for sharing the project study with local school administration
and sharing how to implement the timeline. The administration shares the training plan
with local middle school mathematics teachers. The teachers complete the components of
the training plan and share results with the researcher. The researcher collects any data
generated from the participant evaluations. The students actively participate in lessons
that teachers have generated using the new instructional mathematics practices.
Project Evaluation
The project evaluation goal-based design is discussed in this section to include a
description, the justification, the overall project goals and evaluation goals, and key
stakeholders. The justification for the training plan is based on the goals of the overall
project. The purpose of the study was based on including successful teaching and learning
instructional mathematics practices infused into existing curricula which could lead to
major gains could decrease the achievement gap and encourage student proficiency in
mathematics. This purpose connects to the he overall goal for the project which would be
to raise student achievement in middle school mathematics based on sharing instructional
mathematics practices with teachers to use from the modified Delphi study. One key way
to do this is through a training plan where teachers can learn and apply new instructional
mathematics practices based on each of their training needs. The project goals are based
on a rubric provided to teachers to complete at the end. The rubric has 4 levels with a
rating of 4 being the highest. Teachers rate their gain in overall knowledge regarding
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effective mathematics teaching instructional mathematics practices and rate their
perception of student gains in classroom assessment after infusing the new instructional
mathematics practices. The key stakeholders are the teachers and the students.
Implications Including Social Change
Local Community
The training plan addresses the needs related to student learning in my local
community. Positive implications are stimulated by the plan via the stimulation of higher
student achievement. Higher student achievement can demonstrate that students are
learning as a result of better teaching and learning practices. The overarching problem of
the study is based on low achievement scores for middle school students in mathematics.
The modified Delphi study allowed a middle school mathematics teacher panel to share
instructional mathematics practices that work well for each of the NCTM middle school
mathematics content standards. The effect related to teachers and students engaging in the
instructional practices shared could be that students score higher on assessments. When
students demonstrate higher test scores, it can be connected to the conclusion that more
mathematics learning is occurring as a result from the use of more effective instructional
mathematics practices integrated into learning activities. All levels of the educational
community might benefit from the implementation of recommended mathematics
instructional practices.
The instructional mathematics practices shared and ranked at the top by the
modified Delphi study middle school mathematics teacher panel include high energy
activities involving small group collaboration and inquiry learning. These suggested
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instructional mathematics practices energize students to learn. These instructional
mathematics practices lead to positive social change in regard to enthusiasm to learning.
The students in the local community develop non-cognitive skills and connections
to teachers because of the training plan. Non-cognitive skills include general knowledge,
inquisitiveness, art and culture awareness, leadership, interpersonal skills, and public
responsibility (Sommerfeld, 2011). These skills developed include ones that are often
predictive of future success in academics especially in college. Instructional mathematics
practices suggested by the panel and highlighted in the project include ones like small
group collaboration, real life application, and inquiry learning that lead to improvement in
some of these soft skills.
As teachers self-assess and look for instructional mathematics practices from the
modified Delphi study that are not being used, the students may benefit. The students
benefit because the teachers build a repertoire of teaching practices that can guide
students to mastery of challenging mathematics concepts. Expanding the methods or
practices to help students learn mathematics could be a result from teachers stepping out
of the traditional mathematics teaching role.
Instructional mathematics practices suggested and used in the project help to build
learning more than academic testing and other measurement focused learning
requirements. These outside of the box types of skills may lead to success in the
workforce (Levin, 2015) The project instructional mathematics practices guide students to
develop the non-cognitive skills that are a necessity to be productive in society.
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Far-Reaching
In the larger context, the project study provides better instructional practices for
students across the country facing a similar problem with low mathematics achievement.
The results collected from the modified Delphi study which were used to design the
training plan help just about any middle school mathematics classroom across the
country. The instructional mathematics practices rated by the middle school mathematics
teacher panel help students no matter their location. Teachers across the country enhance
instructional practices by applying the data collected and using the training plan.
Conclusion
The training plan genre and project was the focus of Section 3. It included the
description and goals of the project, the rationale and a literature review related to the
project genre chosen. This section also contains information regarding implementation of
the project and its implications.
Next, in Section 4, the project limitations, strength and scholarship are discussed.
This section also allows for reflection on analysis and the project’s study’s impact on
social change. Implications and future research based on the findings are also shared.
Section 4: Reflections and Conclusions
Introduction
This section includes my comments on my strengths, limitations,
recommendations, and reflections on the project, which included creating and evaluating
the training plan. I also comment on my learning process, the study’s implications,
applications of the study, and directions for future research.
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Project Strengths
In the project study, several project strengths were evident. The first strength is
that the genre of the project and the content were directly connected to the problem of the
study and the data collected in the modified Delphi study. The problem of low middle
school achievement in mathematics was the focus of the study, and the project provides
instructional mathematics practices teachers can use to find better ways to instruct
students in each of the NCTM mathematics NCTM middle school mathematics content
standards. The instructional mathematics practices shared in the project were generated
by the middle school mathematics teacher panel using the modified Delphi research
method.
A few other project strengths stand out. One helpful characteristic of the training
plan genre is that is allows teachers to integrate it into his or her own time schedule. The
plan is something to be used on one’s own without attending pre-scheduled professional
development sessions. Another strength is that the project is geared to each teacher’s
unique learning needs. The teacher completes a checklist at the beginning of the training
that helps to narrow down what instructional mathematics practices are new to his or her
teaching experiences. Then, the teacher is able to focus on instructional mathematics
practices that he or she has not tried in mathematics instruction based on the information
from the training plan.
Recommendations for Remediation of Limitations
Some limitations are evident when examining the project training plan. The panel
in a modified Delphi study generally contains a small number of middle school
mathematics teachers. This small group helps to build a consensus in a more efficient
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manner, but a different type of study that samples a large number of middle school
mathematics teachers could also be useful in collecting a larger variation in opinions
regarding effective mathematics instructional mathematics practices.
Another limitation is the ability to control whether teachers actually participate in
the training. Since the training is one that can be completed on a teacher’s own schedule
in any location, the district loses some control over whether mathematics teachers have
actually participated. I recommend districts provide a suggested timeline for completing
the training plan and follow-up to see if teachers have completed it. The district could
also provide incentives for the teachers to complete it early that might include comp time,
extra planning time instead of training on planning days, or something similar.
Some districts may have mathematics teachers strictly following one mathematics
curriculum, and this type of plan could limit or interfere with the ability for teachers to
apply teaching instructional mathematics practices outside of specific curriculum lesson
plans. To avoid this problem, district administrators need to make it clear that teachers
may work outside of specified curriculum so they can infuse specific instructional
mathematics practices recommended within the training plan.
One limitation to any training plan is getting teachers motivated to utilize the
shared instructional mathematics practices. One way to do this would be for teachers to
have some say in the training that is part of his or her professional development plan. The
district could provide teachers with this training plan and other ideas throughout the year.
Teachers are often looking for ideas that can help them to implement effective
instructional mathematics practices into instruction, so many may gravitate towards this
project training plan.
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Scholarship
Scholarship is an area where I have grown throughout the process. The teacher
leadership courses helped to prepare me for this journey of growth, and the research
process leading to the completed project study has guided me towards the proper forms of
research and writing at the doctoral level. I have gained experience in evaluating the
quality of my research and that of others along with the ability to understand and use a
whole new level of academic vocabulary. I have reached the highest levels of academic
scholarship because of the long process from creation to approval of each section of the
study.
In the scholarship process, I have learned that there is limited recent research
related to mathematics achievement at the middle school level. This knowledge
empowered me to move forward and create a study that not only helps teachers, but does
so quickly through the completion of the training plan. The knowledge from the study
may spur others to use scholarship to study some of the instructional mathematics
practices suggested by the panel in the future. I also gained new ideas for personal use in
teaching mathematics from my middle school mathematics teacher panel who
participated in the modified Delphi study. I will not view mathematics instruction in the
same way as a result of my scholarship experiences.
Project Development and Evaluation
The idea to create a project study related to helping students with low
mathematics achievement in middle school came from frustration within my own
experiences as a mathematics teacher at this level. In my role, I encountered ideas that
seemed to help, but much of the curriculum was prescribed; and there was little guidance
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that helped teachers or room to supplement and add to prescribed instructional
mathematics practices. I wanted to study and create a project that helped this problem.
The idea to research the problem I was experiencing with ineffective mathematics
curricula and teaching strategies was cultivated as I went through my teacher leadership
courses and considered possible research focus for my student and as I developed my
prospectus. I focused on a project study because with the goal of creating something that
could be applied immediately to help address the problem. I focused on the modified
Delphi method as two of my colleagues used this research method to address similar
problems related to language arts instruction and professional development. The modified
Delphi method was recommended by a colleague’s chair at Walden University.
As I conducted this research, I had instructional mathematics practices generated
from my middle school mathematics teacher panel for each of the NCTM mathematics
middle school NCTM middle school mathematics content standards and needed a project
that could be used to share and apply the results to mathematics instruction. The project
genre that seemed like the most efficient way to do this was professional development.
After further discussion, my chair and URR guided me toward a training plan project.
The consensus was that this would be the most useful tool for middle school mathematics
teachers not just in my location but anywhere throughout the United States.
The key parts to the training plan that were most helpful and a focus of my
development were the goals and evaluation pieces. The goals were directly tied to the
problem of low mathematics achievement and to the instructional mathematics practices
collected from the modified Delphi research. The checklist at the beginning of the
training plan helps teachers to customize the plan to their unique instructional practice
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experiences and the top recommended instructional mathematics practices from the panel.
The rubric at the end helps teachers to evaluate the effectiveness of the plan based on the
learning goals.
Leadership and Change
I learned more about myself and mathematics instruction especially in the areas of
leadership and change. Leadership and change are natural paths that follow when trying
to remedy a problem such as low achievement in mathematics. Change needs to occur
because the status quo is not working or showing the desired or required results.
Leadership is needed to generate the change.
In my study, I initially was very focused on change since the current mathematics
achievement results were so low. After researching the NCTM Principles and examining
current research, it was clear that there were instructional mathematics practices the
mathematics teachers could be using that would help students to achieve higher results.
The modified Delphi research method was used to gather the data from a middle school
mathematics teacher panel to help find instructional mathematics practices that could be
helpful in making changes to current instructional practices.
I gained in leadership in several ways throughout the study. One place I developed
in leadership skills was in the creation of the research surveys and locating and directing
participants on my panel in the modified Delphi research. The other place I gained
leadership skills was in the creation of the project training plan. I had to think as a leader
when looking at the most effective way to bring the instructional mathematics practices to
teachers. Looking at teachers as individuals with unique training needs was one
leadership principle that I used. I believe the ability to customize the training plan to the
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individual needs of the teacher will lead to higher teacher motivation to actually apply the
knowledge and skills learned from completion of the training plan.
Analysis of Self as Scholar
My analysis of self in the scholarship process included examining the information
I gained throughout the doctoral process. My scholarship process included the extensive
amount of research I conducted in all phases through the research process and ending
with the creation of the project training plan. Throughout all of my work, I have gained in
all areas of scholarship.
I learned how to write a problem statement that related to circumstances regarding
low mathematics achievement in my school district. Everything I researched and learned
was directly based on this problem including searching for current research and
researching the NCTM principles. I learned about gaps in mathematics achievement
research and some instructional mathematics practices and techniques that appeared to
help. My investigation saturated the literature. I felt confident that this study was needed
to address the instructional gap in practice.
In the modified Delphi research process, I learned how to go through the process
of soliciting and securing participants, how to create and manage surveys, and how to
analyze the data I collected. I had little experience in these areas, but now I feel
competent in my understanding of other research and in my ability to conduct more
research of my own in the future. I think I can only become better at the process if I do
move forward with further research in the future. I also plan to focus more on the data
collected from this study and find more ways to apply the research to help with the
underlying problem of low mathematics achievement in middle school.
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In the project creation phase, my scholarship levels again improved, and I learned
even more about scholarship and research. I learned quickly that the project needed to
connect directly to the problem and research data collected. I learned that there are
several models to consider, and I established from the literature that the training plan
option works well to connect the results of the study to a useful product for teachers. The
components, as well as the stakeholders, were carefully considered in the planning
process. Evaluation was something I had not given much consideration before the process
began, but I grew to see how important evaluation is in scholarship to determine if the
project is effective and how to improve the project.
Analysis of Self as Practitioner
Throughout the process, I saw myself as a practitioner. I was looking for ways to
improve in the practice of the art of teaching mathematics so that students could more
successful. This part of my background led me to the problem statement, research
method, and the idea to share what I had learned with other teachers in the training plan. I
recently moved to a new role as a mathematics coach, and the instructional mathematics
practices collected from the middle school mathematics teacher panel along with the
project will provide resources for mathematics teachers I coach. Through this role, I am
in connection with other mathematics coaches who might also utilize the project in their
roles supporting teachers. The training plan is a practical tool that I could share with other
practitioners depending upon his or her experience and needs.
Analysis of Self as Project Developer
I learned that I had room to grow as a project developer through this process. I
learned that to set effective goals, there had to be an underlying problem and research
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behind the problem. Once those things were established, I had to choose a project that
connected the research to the desired outcome that was also feasible. After selecting the
project, I was responsible for looking at the components and setting up an evaluation plan
to help determine whether the goals were met. Having completed this process, I am now
confident that schools and businesses should consider following a similar process when
implementing projects.
The Project’s Potential Impact on Social Change
The project has a great potential impact for positive social change in my local
community as well as the national community. As shared in Section 1, the problem of low
mathematics achievement is one that is a problem starting at a small community level but
continues to be a trend when looking at the overall mathematics achievement in the
United States. The project provides teachers with middle school mathematics
teachersuggested instructional mathematics practices to use when teaching middle school
mathematics for each of the NCTM middle school mathematics content standard. The
results could include better understanding and higher mathematics achievement at all
levels. The instructional mathematics practices are ones that should work anywhere.
Implications, Applications, and Directions for Future Research
The findings from this project study provides implications, applications, and
future direction for research in similar topics of study. The problem of low mathematics
achievement is one that is common across the United States, and some ideas can be taken
from the data and project to use for related research. Any of the suggestions in this
section are ones that would take the exploratory nature of the modified Delphi research
and use the results to explore further.
114
I have several suggestions for future research as a result of the work from this
study. The instructional mathematics practices collected from the modified Delphi
research could be explored further. Researchers could focus on one NCTM middle school
mathematics content standard and try adding a few of the effective instructional
mathematics practices into a quantitative study to see if adding them did result in higher
mathematics achievement. A case study could be another potential study. Researchers can
locate teachers who implement one or more of these instructional mathematics practices
for each of the NCTM middle school mathematics content standards in their own research
study. Future research could include a mixed methods study where the mathematics
achievement and motivation of student learners could be study base on one or more
instructional mathematics practices suggested from the modified Delphi research. The
project itself could be the basis of the research as teachers go through the training and
actually apply it to teaching. The possibilities are vast and more research related to the
problem of this study would add to the limited amount of current research available on
the problem of low middle school mathematics achievement.
Conclusion
In this modified Delphi mixed methods project study, I examined the problem
related to middle school mathematics low achievement. My modified Delphi
methodology allowed for me to have a middle school mathematics teacher panel in the
field reach consensus on mathematics instructional mathematics practices that were
recommended to help students. The results of this study included a list of effective
instructional mathematics practices that the panel recommended for instruction in each of
the NCTM middle school mathematics content standards. Based on the problem and my
115
results, I designed a training plan where teachers learn to self-evaluate their skills and
experience in the recommended instructional mathematics practices for each NCTM
middle school mathematics content standards. Then, the teachers are provided with
application ideas for the NCTM middle school mathematics content standards that they
can use in future middle school mathematics instruction.
When designing the project, I considered my findings from the modified Delphi
methodology, the problem of low mathematics achievement, and the literature review
with regard to using the genre of professional development/training plan to help teachers
learn and apply the information from the research. When the teachers apply the plan, the
results may lead to an increase in student achievement in middle school mathematics.
116
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