SPIRAL ASSESSMENTS:
A STUDY IN AN ACCELERATED MATHEMATICS CLASSROOM
Except where reference is made to the work of others, the work described in this project
is my own or was done in collaboration with my Advisor. This project does not include
proprietary or classified information.
____________________________________________________________________
Lisa Morgan Skinner
Certificate of Approval:
______________________________
Donald R. Livingston, Ed.D.
Associate Professor and Project Advisor
Education Department
______________________________
Sharon M. Livingston, Ph.D.
Assistant Professor and Project Advisor
Education Department
Spiral Assessments ii
SPIRAL ASSESSMENTS:
A STUDY IN AN ACCELERATED MATHEMATICS CLASSROOM
A project submitted
by
Lisa Morgan Skinner
to
LaGrange College
in partial fulfillment of
the requirement for the
degree of
SPECIALIST IN EDUCATION
in
Curriculum and Instruction
LaGrange, Georgia
July 18, 2011
Spiral Assessments iii
Abstract
The purpose of this study was to research the effectiveness of spiral assessments
for mathematics students. The study compared one year of Accelerated Math 2 students
who received no spiral assessments throughout their course to the following year of
Accelerated Math 2 students who received numerous spiral assessments. In addition, the
spiral group was given appropriate feedback about their assessments as well as positive
changes to their learning environment based on the assessments. Students were surveyed
about whether or not spiral assessments led to improved attitudes about mathematics and
if the practice raised confidence in their mathematics abilities. Finally, this study used a
focus group of ten math teachers and interviews of two administrators to determine if a
spiral assessment practice was feasible for broader implementation and if it would be
supported by the faculty and administration at the school. Overall, this research study
produced positive results with regards to content retention and student confidence in
mathematics.
Spiral Assessments iv
Table of Contents
Abstract ……………………………………………………………………………….. iii
Table of Contents ……………………………………………………………………... iv
List of Tables …………………………………………………………………………... v
Chapter 1: Introduction ………………………………………………………………… 1
Statement of the Problem ……………………………………………………… 1
Significance of the Problem …………………………………………………… 1
Theoretical and Conceptual Frameworks …..…………………………………. 2
Focus Questions ………………………………………………………………... 4
Overview of Methodology …………………………………………………….. 5
Human as Researcher ………………………………………………………….. 6
Chapter 2: Review of the Literature …………………………………………………… 8
Spiral Curriculum ……………………………………………………………… 8
Formative Assessments ………………………………………………………... 9
Student and Teacher Attitudes ………………………………………………... 12
Organizational Change ……………………………………………………….. 14
Chapter 3: Methodology ……………………………………………………………… 18
Research Design ……………………………………………………………… 18
Setting ………………………………………………………………………… 19
Subjects and Participants ……………………………………………………... 19
Procedures and Data Collection Methods ……………………………………. 20
Validity and Reliability Measures ……………………………………………. 23
Analysis of Data ……………………………………………………………… 26
Chapter 4: Results ……………………………………………………………………. 30
Chapter 5: Analysis and Discussion of Results ………………………………………. 47
Analysis of Results …………………………………………………………… 47
Discussion ……………………………………………………………………. 55
Implications …………………………………………………………………... 57
Impact on School Improvement ……………………………………………… 59
Recommendations for Future Research ……………………………………… 60
References ……………………………………………………………………………. 62
Appendices …………………………………………………………………………… 66
Spiral Assessments v
List of Tables
Table 3.1
Data Shell ………………………………………… 21
Table 4.1
Independent t-test ………………………………… 31
Table 4.2
Independent t-test ………………………………… 32
Table 4.3
Independent t-test ………………………………… 33
Table 4.4
Independent t-test ………………………………… 34
Table 4.5
Independent t-test ………………………………… 35
Table 4.6
Independent t-test ………………………………… 35
Table 4.7
Dependent t-test ………………………………….. 36
Table 4.8
Chi-Square Statistic ………………………………. 38
Spiral Assessments 1
CHAPTER 1: INTRODUCTION
Statement of the Problem
In many classrooms throughout the nation, students are memorizing facts and
regurgitating those facts on their assessments. Little is being done by the student, or
perhaps the teacher, to ensure that content knowledge is being retained from year to year.
Johnson (2003) eloquently explained, “we all know that the way most secondary schools
work is that students spend about 179 days preparing for a three-hour Brain Dump in
some gymnasium in June. . . we are relatively sure that one year later they will have
forgotten just about everything from the year before” (p. 8).
In addition, with the new “Failure is Not an Option” [FNO] and retest policies of
some school systems, many students are waiting until the last minute to study for
assessments and many students often resort to just memorizing important content. As a
result, knowledge is not being constructed in a manner that guarantees “learning” has
actually occurred. Johnson (2003) explained that when students score well on a final
exam, teachers naturally conclude they ‘know’ the material and have ‘learned’ that
subject; when, in reality, this assumption may be further from the truth.
Significance of the Problem
According to Zemelman, Daniels, and Hyde (2005), “many [students] come to
believe that they are incapable of doing math. As they progress through the grades, fewer
and fewer students understand and enjoy math, leaving only a handful to continue with
capability and confidence” (p. 106). In addition, while in college, many students “take
only the minimum [math] courses required, despite the fact that many careers depend
upon mathematical knowledge” (Zemelman et al., 2005, p. 106). According to the
Spiral Assessments 2
National Council of Teachers of Mathematics [NCTM] (2000), “the need to understand
and be able to use mathematics in everyday life and in the workplace has never been
greater and will continue to increase” (p. 4). Usually, when students learn for the
moment and do not achieve a true understanding, negative results occur. With
mathematics being a subject that builds on mastery of prior topics, many students find
themselves “lost” and disliking the subject. NCTM (2000) explained that this lack of
mathematical understanding keeps the doors closed for many enhanced opportunities.
The purpose of this research was to find a way to encourage students to become
better learners. With such an emphasis being placed on assessments in schools, this
research aimed at making assessments become a more productive part of education. In
addition, with prior content being an integral aspect of the mathematics classrooms, it
seemed obvious to include this in the assessment process. Therefore, this research
attempted to answer the question, will spiral assessments have a positive effect on student
learning in the mathematics classroom?
Theoretical and Conceptual Frameworks
The LaGrange College Department of Education (2008) along with its teacher
candidates, strongly support a constructivist approach to learning. The philosophy of
constructivism is founded on the premise that individuals learn by reflecting on their own
experiences and by adjusting “mental models” to accommodate new experiences. In
particular, this research project embedded the theory of social constructivism throughout.
Bruner (1996) explained, “you cannot understand mental activity unless you take into
account the cultural setting and its resources, the very things that give mind its shape and
scope” (p. x). Bruner also explained that students have a greater use for knowledge
Spiral Assessments 3
which has been acquired through discovering and making connections to prior
experiences (p. xii). Fosnot (2005) further explained, “rather than behaviors or skills as
the goal of instruction, cognitive development and deep understanding are the foci; rather
than stages being the result of maturation, they are understood as constructions of active
learner reorganization. Rather than viewing learning as a linear process, it is understood
to be complex and fundamentally nonlinear in nature” (pp. 10-11).
The theoretical framework for this study was guided by LaGrange College
Education Department’s (2008) Conceptual Framework. Based on Tenet 2 from the
framework, ‘exemplary professional teaching practice’ students should be active
participants in the learning process, while teachers serve as mere facilitators. In addition,
teachers are expected to pull from a variety of resources in order to be effective in the
diverse classrooms of today. Active learning environments are required to help students
become active participants (LaGrange College Education Department, 2008).
This research was specifically connected to Competency Cluster 2.3 Assessment
Skills of the LaGrange College Education Department’s (2008) Conceptual Framework.
Teachers should understand and use formal and informal assessment strategies to
evaluate and ensure continuous intellectual, social, and physical development of students.
Secondly, teachers should involve students in self-assessment that helps them become
aware of their strengths and needs and should encourage students to set personal goals for
learning. Finally, teachers should monitor and adjust strategies in response to student
feedback. With the application of spiral assessments, teachers should be able to satisfy
all parts of cluster 2.3.
Spiral Assessments 4
The National Board for Professional Teaching Standards [NBPTS] Core
Proposition Three closely related to this research. Proposition Three stated that teachers
are responsible for managing and monitoring student learning. In addition, Element 1D
(Student Learning for Teacher Candidates) of the National Council of Accreditation of
Teacher Education [NCATE] Principles was directly related to this research. Both of
these standards adhere to the belief that teachers are responsible for assessing the learning
of students. This research used spiral assessments as a means by which teachers could
reflect on their students’ knowledge and as a result lead teachers to make positive
changes regarding their curriculum, learning environment, and future assessments.
Focus Questions
To determine if spiral assessments had a positive effect in the math classroom,
further questions were developed which included more specifics. The first focus question
addressed the math content that needed to be learned by the students. The second focus
question was an affective assessment of how students felt about incorporating spiral
assessments into their math class. Lastly, the third focus question addressed the area of
school improvement and whether spiral assessments could feasibly be adopted with the
school. As a result, the following three focus questions were developed to guide this
entire research process.
1. Will the use of spiral assessments, coupled with proper feedback, improve
students’ achievement on the Math 2 End of Course Test?
2. How will spiral assessments affect students’ feelings towards math, and will such
assessments improve student confidence?
Spiral Assessments 5
3. Will a system which employs the use of spiral mathematics assessments be
received positively by the math department and will the use of such a system be
supported by the administration?
Overview of Methodology
This action research occurred in a Georgia rural high school. The study took place
throughout the 2010 school year, initially using 83 tenth-graders from five Accelerated
Math 2 classes. The 2010 data were compared to data collected in the 2009 school year
with an initial 65 tenth-graders from four classes of Accelerated Math 2. Both
quantitative and qualitative data were used throughout the research.
In order to determine if spiral assessments would help students’ achievement on
the Math 2 EOCT (focus question one), several independent t-tests were run. First, an
independent t-test on eighth grade CRCT z-scores was conducted to see if there was
significant difference between the ability levels of the 2009 students and the 2010
students. When a significant difference was found in the two groups, the z-scores were
then screened and matched to equalize the ability levels of the two groups. The initial 83
students from 2010 were reduced to 49 and the 2009 initial 65 students were also reduced
to 49. Next, independent t-tests were conducted on two midterm exam scores for both
school years. The 2009 students were used as the control group, receiving no spiral
assessments throughout their course while the 2010 students received numerous spiral
assessments throughout the year, as well as adjustments to their curriculum. Finally, an
independent t-test was conducted on the Math 2 EOCT scores for both groups to ascertain
significant difference.
Spiral Assessments 6
To address the second focus question, “How will spiral assessments affect
students’ feelings towards math, and will such assessments improve student
confidence?”, surveys and reflections were administered throughout the year. A sevenpoint Likert Scale was used on the surveys and each question was analyzed using the Chi
Square method. Student reflections focused on changes in attitudes and confidence levels
toward mathematics and provided qualitative data for the research project.
The third focus question, “Will a system which employs the use of spiral
mathematics assessments be received positively by the math department and will the use
of such a system be supported by the administration?”, was addressed using a focus
group and two interviews. At the end of the study, math teachers participated in a focus
group discussion about spiral assessments and qualitative data were collected.
Additionally, two administrators were interviewed to ascertain attitudes about spiral
assessments and the feasibility of implementing a spiral assessment program within the
school.
Human as Researcher
Over the course of 17 years of teaching mathematics, I have learned that my
students tend to “memorize for the test” and then quickly lose the knowledge that was
assessed. As a student, I too, exhibited this behavior. Although math was my favorite
subject, it wasn’t until I taught the subject that I truly grasped what I was doing and why I
was doing it.
Last year I was faced with teaching the new Georgia Accelerated Math 2 course
where my students were required to take the regular Math 2 end-of-course test. Over half
of the content assessed on the Math 2 EOCT was taught the previous year in Accelerated
Spiral Assessments 7
Math 1. Although the students in Accelerated Math 2 are typically stronger in their math
ability, I quickly realized there was a ton of content knowledge they lost from the
previous year. As a result, I spent a month in April reviewing content they were expected
to already know. With this research, I hoped to find that a spiral assessment approach,
when implemented throughout the year, would help my students retain content longer and
make them more successful on their Math 2 EOCT.
Spiral Assessments 8
CHAPTER 2: REVIEW OF THE LITERATURE
A review of current and past literature provided a justification for this action
research study. The literature review presented evidence for each of the three focus
questions in the study. The literature review included information regarding spiral
curriculums, formative assessments, student and teacher attitudes, and organizational
change.
Spiral Curriculum
Most people have heard the following phrases on more than one occasion:
“practice makes perfect” and “use it or lose it.” Whether it’s practicing the piano or
perfecting a golf swing, people recognize the importance of practice. Perhaps the most
applicable place for practice is inside a mathematics classroom. Even math teachers who
go a few years without teaching a certain concept have to spend time reviewing the
material before they can explain it successfully to their students.
Bruner (1960) introduced the idea of a spiral curriculum and the importance of
revisiting concepts throughout one’s education. He explained that “a curriculum as it
develops should revisit these basic ideas repeatedly, building upon them until the student
has grasped the full formal apparatus that goes with them” (p. 13). Bruner (1960)
suggested that students initially learn a general idea which can then be used as the “basis
for recognizing subsequent problems as special cases of the idea originally mastered” (p.
17). Doll (1993) explained that iteration is the process of repeating itself over and over
again and stated, “nothing is more central to this new beginning than the concept and
practice of iteration” (p. 97). Doll also stated that Bruner’s “spiral curriculum is worth
looking at again and reframing in light of recursion theory” (p. 102). Doll further
Spiral Assessments 9
explained that “it is worth constructing a curriculum where students revisit with more
insight and depth what they have done” (p. 102).
Harden and Stamper (1999) supported the concept of a spiral curriculum, but
added that “a spiral curriculum is not simply the repetition of a topic taught. It requires
also the deepening of it, with each successive encounter building on the previous one” (p.
141). Snider (2004) also cautioned that a spiral curriculum often limits the depth of
knowledge that students attain. Snider (2004) explained, “in a spiral curriculum, many
topics are covered but only briefly…The result of teaching for exposure is that many
students fail to master important math concepts” (p. 31).
Jensen (1990) also cautioned that spiral curriculums are hindering our efforts to
improve the educational system in our country because they “rob both students and
teachers of the excitement and motivation that is inherent in anticipating learning
something new” (p. 4). According to Jensen (1990), this lack of curiosity prevents
students from learning any topic in depth and keeps students from reaching a level of
meaningful understanding. Instead, students scratch the surface of numerous topics and
revisit those topics without any further depth. Jensen (1990) also explained that countries
with high mathematics achievement have curricula without a high degree of repetition.
Formative Assessments
Now, more than ever before, schools, administrators, and educators are being held
to higher accountability standards for student achievement. Whether one supports or
opposes today’s high-stakes tests, there is no denying the importance of them. With the
emergence of these tests, educators are being forced to closely examine their curriculum,
their teaching practices, and their own classroom assessments. Popham (2001) explained
Spiral Assessments 10
that “classroom assessment has a higher calling – to improve the caliber of classroom
instruction” (p. 117).
At the forefront of discussions about classroom assessments was the importance
of using tests to gather information about students’ learning and to make decisions about
future instruction. Dwyer (2008) stated that “the central purpose of testing will be to
inform and improve teaching and learning” (p. 5). Garcia, Spalding, and Powell (2001)
defined formative assessments as those assessments used to gather information “while
work is still in progress in order to influence the final outcome” (p. 303). Today, there is
a greater push for the use of formative assessments within the classroom, as opposed to
the traditional summative assessments that occur at the end of a chapter or unit.
Andrade and Cizek (2010) believed that formative assessments “offer great
promise as the next best hope for stimulating gains in student achievement” (p. 3). While
formative assessments can take many different forms (observations, oral questioning,
class discussions, projects, portfolios, homework, group work with peer feedback, student
self-assessment), the primary goal of such assessments is to provide information for the
purpose of making adjustments within the classroom. These formative assessments have
the potential to “provide missing linkages between classroom practice and large-scale
assessments” (Andrade & Cizek, 2010, p. 4).
Jones, Carr, and Ataya (2007) believed that using a variety of assessments to
provide continuous feedback will nourish teaching and learning. They cautioned against
using one test score to provide a picture of what a student does or does not know and that
a test score is simply a snapshot in time and subject to error. Jones et al. (2007)
explained, “a teacher who evaluates student learning and instructional practices solely on
Spiral Assessments 11
the basis of test scores is missing valuable information . . . The more information a
teacher collects, the more valid the inferences based on that information” (p. 74). The
National Council of Teachers of Mathematics [NCTM] (2000) stated, “assembling
evidence from a variety of sources is more likely to yield an accurate picture” (p. 24).
Teachers who implement a variety of testing techniques have a better picture of their
students and their classroom and can make more effective decisions regarding both.
In a mathematics classroom, successful completion of problems depends greatly
on successful completion of prior problems. Jones et al. (2007) explained the
importance of formative assessments “when correct procedure is crucial to later success”
(p. 77). Waiting for a summative assessment to discover that a student missed important
information early in the learning process is extremely frustrating to the teacher as well as
the student. Formative assessments must be ongoing and must provide “feedback and
direction to students as they proceed toward a goal” (Jones et al., 2007, p. 77).
The National Council of Teachers of Mathematics [NCTM] (2000) strongly
believed that assessment should “support the learning of important mathematics and
furnish useful information to both teachers and students” (p. 22). An assessment should
no longer just come at the end of instruction, but should be something administered
throughout instruction. NCTM (2000) stated, “assessment should not merely be done to
students; rather, it should also be done for students” (p. 22). In addition, NCTM (2000)
explained that “assessment and instruction must be integrated so that assessment becomes
a routine part of the ongoing classroom activity rather than an interruption” (p. 23).
Zemelman et al. (2005) explained that teachers in ‘Best Practice’ classrooms use
assessments for more than just providing a grade on a report card. To gain a deeper
Spiral Assessments 12
understanding of student learning, progressive teachers “monitor students’ growth in
richer and more sophisticated ways” (Zemelman et al., 2005, p. 252). According to
Zemelman et al. (2005), teachers should use gathered information to guide instruction, to
make critical daily decisions about helping students grow, and most importantly, to help
students set goals, monitor their own work, and evaluate their efforts.
Student and Teacher Attitudes
To quote Winston Churchill (n.d.), “Attitude is a little thing that makes a big
difference.” When taken to heart, what a difference this statement could make in one’s
life. For students and teachers, nowhere is this statement more applicable than inside the
classroom.
Teachers’ attitudes affect the way they interact with their students, the way
they teach, and the way they assess. Likewise, students’ attitudes affect the way they
interact with their teachers, the way they behave, the way they learn, and how well they
perform. When looking at improving student achievement, student and teacher attitude is
an area that must be examined.
There was plenty support in the literature connecting positive attitudes in
mathematics to positive student achievement (Farooq & Shah, 2008; Marzano, 2003;
McMillan, Cohen, Abrams, Cauley, Pannozzo, & Hearn, 2010; Wigfield & Meece,
1988). According to Farooq and Shah (2008), “students’ success in mathematics depends
upon attitude towards mathematics. It also influences the participation rate of learners”
(p. 75). Marzano (2003) also explained the importance of motivation to classroom
success. He stated, “if students are motivated to learn the content in a given subject, their
achievement in that subject will most likely be good” (p. 144).
Spiral Assessments 13
Regarding attitudes towards a spiral curriculum, there were mixed reviews in the
literature. According to Caldwell (2008), when a tenth grade student was asked about
repetition in her math class and why teachers revisited the same topics from year to year,
the girl responded, “so that it’s, like, branded in our brains, so that we know it forever”
(p. 3). Caldwell further explained that discussions with other students revealed feelings
of boredom and frustration towards repetitious work and that students did not appreciate
being reminded that they had covered the work before.
Student and teacher attitudes surrounding formative assessments also impacted
learning. There were several references which supported the use of formative
assessments in the classroom and most revealed a positive correlation between their use
and resulting positive attitudes, both in teachers and in students. McMillan et al. (2010)
explained, “the statistically significant positive relationships between overall formative
practices and class averages of student motivation, suggest an association between at
least some formative practices and student motivation” (p. 10). They also reported that
when teachers employed several types of formative assessments, that students reported
higher levels of motivation (McMillan et al., 2010).
Although there appeared to be a positive connection between the use of formative
assessments and an increase in student motivation, McMillan et al. (2010) discovered that
many secondary teachers “do not use formative assessment practices nearly as much as
what is suggested in the literature, and report that several critical components of
formative assessment, especially instructional correctives, are not widely used” (p. 11).
A possible reason cited for the discrepancy in teacher use of formative assessments was
that teachers may “perceive formative assessment practices as impractical or time-
Spiral Assessments 14
consuming” (McMillan et al., 2010, p. 11). Additionally, in a study conducted by
Volante, Beckett, Reid, and Drake (2010), while teachers acknowledged the importance
of feedback, many teachers found that their feedback was useless and a waste of time if
the students did not even bother to look at it. Finally, although teachers noted difficulties
in utilizing self- and peer assessment within their classrooms, they did concur that
“involving students in the assessment process is vital to student learning” (Volante et al.,
2010, p. 13).
Organizational Change
It is impossible to live in today’s society and not recognize the rate at which
things are changing. Cameron and Green (2009) noted, “the rate of change and discovery
outpaces our individual ability to keep up with it” (p. 1). Lieberman (2005) explained
that the changes educators have to deal with occur one on top of another at “an
increasingly frenetic speed” (p. x). If schools expect to stay current in the lives of their
students, they must be willing to change with the world in which they operate. The
question then becomes, how do schools most effectively bring about change?
In terms of implementing change in any organization, Cameron and Green (2009)
recognized the importance of the people on the receiving end of the change. They
explained, “Whatever the level or degree of organizational change, the people on the
receiving end are individual human beings. It is they who will ultimately cause the
change to be a success or a failure. Without looking at the implications of change on
individuals we can never really hope to manage large-scale change effectively” (p. 3). In
a school system, individual school, or even a classroom, change will not be successful
unless the teachers and students are considered an important piece of the puzzle.
Spiral Assessments 15
One important organizational change strategy supported in the literature was
increased teacher involvement in leadership roles. Gabriel (2005) explained that with No
Child Left Behind and the increased importance of high-stakes tests, many principals are
turning to teachers to provide more effective organizational behavior. Gabriel (2005)
argued, “the only leadership that will make a difference is that of teachers. They alone
are positioned where all the fulcrums are for change. They alone know what the day-today problems are and what it takes to solve them” (p. 1). Gabriel (2005) also stated,
“inviting teachers to participate in the decision-making process by elevating them to
leadership roles should be viewed as a means to accomplish significant change in the
field of education” (p. 156). He also recommended that schools encourage leadership in
teachers and equip them with the skills to become productive leaders in today’s changing
society.
Marzano, Pickering, and Pollock (2001) stressed the importance of teacher desire
and commitment to bring about change within a school. They explained that “a small
group of educators within a school who are enthusiastic about a particular innovation can
‘infect’ an entire staff with that enthusiasm” (pp. 157-158). Cameron and Green (2009)
also pointed out, “individual change is at the heart of everything that is achieved in
organizations. Once individuals have the motivation to do something different, the whole
world can begin to change” (p. 9). Based on the literature, teacher leaders are excellent
change agents within a school; however, teacher leaders must have the desire and will to
see the change occur for there to be any real hope for the change to be a success.
In addition to utilizing teachers as leaders within a school, professional
development was mentioned as a means of bringing about change. Unfortunately, the
Spiral Assessments 16
professional development literature raised some concerns about successful
implementation. Volante et al. (2010) pointed out that “teachers often begrudge topdown, mandated professional development and do not hold value in its execution” (p.
16). Marzano (2003) also claimed that the regularly scheduled staff development in most
schools “is not necessarily meaningful or useful in terms of impacting student
achievement” (p. 65). Fullan (2005) recognized the importance of professional
development in any change strategy but cautioned that the impact of such development
has been minimal through the years.
Since professional development was recognized as an important part of
organizational change, it made sense to find ways to make it more effective. Marzano
(2003) noted that professional development “must be constructed in ways that deepen the
discussion, open up the debates, and enrich the array of possibilities for actions” (p. 66).
According to Marzano, an extensive study on staff development conducted by Michael
Garet and his colleagues revealed three important features of staff development: (1)
focus on content knowledge, (2) opportunities for active learning, and (3) overall
coherence of staff development activities. Marzano summarized Garet’s findings and
explained that staff development should be specific and relevant to a teacher’s subject
area, teachers should be able to apply and test strategies within their classrooms, and staff
development should be coherent and build on one another. Additionally, Volante et al.
(2010) explained that staff development that involved teacher input or staff-development
that was self-directed, tended to lead to more sustained changes in classroom practice.
Based on the literature review conducted for this study, there appeared to be a
natural connection between a spiral curriculum and the use of formative assessments in
Spiral Assessments 17
the classroom. While no current literature exists on ‘spiral assessments’, the study tied
the two concepts together in hopes of improving student achievement. The assessments
in the study revisited concepts previously covered in the curriculum and were considered
‘spiral’ in nature. Spiral assessments as well as daily warm-up exercises in the study
were ‘formative’ in nature and were used to inform students and teachers of progress
being made in the classroom. The literature review also revealed the importance of
teacher leaders and proper staff development within schools if sustained change was
expected to occur.
Spiral Assessments 18
CHAPTER 3: METHODOLOGY
Research Design
This study was conducted using both action and evaluation research and consisted
of both quantitative and qualitative data. Tomal (2003) explained that action research is
“one of the most practical and efficient methods of conducting research by educators” (p.
vii). He further explained that research is not finding a known solution to a problem,
rather it entails “a careful undertaking to discover new facts and relationships concerning
a solution that is unknown” (Tomal, 2003, p. 1). Educators are constantly looking for
solutions to existing problems in their classroom. Action research provides educators a
“systematic process of solving educational problems and making improvements” within
their classrooms (Tomal, 2003, p. 5). In addition, since the goal of action research is to
solve a problem and make improvements, researchers rely less on scientific inquiry and
inductive reasoning, and more on the “practicality and feasibility” of solving a given
issue (Tomal, 2003, p. 9). McNiff and Whitehead (2006) explained that, “action research
can be a powerful and liberating form of professional enquiry because it means that
practitioners themselves investigate their own practice as they find ways of living more
fully in the direction of their educational values. They are not told what to do. They
decide for themselves what to do, in negotiation with others” (p. 8). McNiff and
Whitehead (2006) also supported using action research because anyone and everyone can
do it: “all you need is curiosity, creativity, and a willingness to engage” (p. 13).
Supplementing the action research methodology, this study was conducted using
evaluation research. Charles and Mertler (2002) explained that evaluation research “is
done to determine the relative merits of various products and approaches used in
Spiral Assessments 19
education” (p. 311). The purpose of evaluation research is to assess the product or
program developed in action research (Charles & Mertler, 2002). While action research
was used to seek a solution to a problem in the classroom, evaluation research was used
to determine the effectiveness of the proposed solution. Was student achievement
increased and was the program accepted by the teachers and students?
Setting
This study took place in a rural high school in west Georgia. The school’s student
population was approximately 1300, of which 39% were minority, 54% White, and 7%
other. Of the student population, 47% were economically disadvantaged and 6% were
classified as students with disabilities. This setting was chosen for the study because, as
a mathematics teacher working at the high school, I had easy access to the subjects’ test
records. In order to conduct research in the classroom, permission was obtained from the
high school principal, the county school system, and the institutional review board at
LaGrange College.
Subjects and Participants
There were 148 subjects included in this study. All of the subjects were tenth
grade students enrolled in an Accelerated Math 2 course during the 2009-2010 and 20102011 school years. All of the students were 15-16 years old. Of the 65 students in
Accelerated Math 2 during 2009-2010, 35 were males and 30 were females. Of the 83
students in Accelerated Math 2 during 2010-2011, 44 were male and 39 were female.
In addition to the 148 subjects participating in the study, there were ten math
teachers who participated in a focus group and two administrators who were interviewed.
Five of the ten math teachers participating in the study had less than five years teaching
Spiral Assessments 20
experience, two teachers had more than ten years experience, and three teachers had more
than twenty-five years experience. Both administrators interviewed in the study had
more than twenty-five years experience in education. One administrator had been the
principal at the high school for fifteen years and was an elementary principal for five
years prior to that. The second administrator had been the registrar and math department
supervisor at the high school for the last five years and was a middle school assistant
principal for 5 years prior to that.
Procedures and Data Collection Methods
In order to answer the study’s focus questions, procedures were followed, data
were collected, and statistical tests were used for analysis. Table 3.1 provides an
explanation of the procedures and statistical tests used throughout the study.
Spiral Assessments 21
Table 3.1. Data Shell
Focus Question
Literature
Sources
Type: Method,
Data, Validity
How data are
analyzed
Rationale
Will the use of
spiral
assessments,
coupled with
proper feedback,
improve students’
achievement on
the Math 2
EOCT?
Bruner
(1960)
Doll (1993)
Dwyer
(2008)
NCTM
(2000)
Method:
Midterm
assessments;
End-of-course
assessment
Quantitative:
Independent
t-test
Quantitative:
Determine if
there are
significant
differences
Quantitative:
Dependent
t-test
Chi Square
Cronbach’s
Alpha
Quantitative:
Determine if
there are
significant
differences
How will spiral
assessments affect
students’ feelings
towards math, and
will such
assessments
improve student
confidence?
Caldwell
(2008)
Jensen
(1990)
McMillan
et al.
(2010)
Reeves
(2007)
Data:
Interval
Validity:
Content
Method:
Math Anxiety
Questionnaire,
Student
Reflections
Data:
Nominal;
Qualitative
Validity:
Construct
Will a system
which employs
the use of spiral
mathematics
assessments be
received
positively by the
math department
and will the use of
such a system be
supported by the
administration?
Fullan
(2005)
Gabriel
(2005)
Marzano
(2003)
Method:
Focus Group
Interview
Data:
Qualitative
Validity:
Construct
Qualitative:
Coded for
themes:
Recurring
Dominant
Emerging
Qualitative:
Coded for
themes:
Recurring
Dominant
Emerging
Qualitative:
Look for
categorical and
repeating data
that form
patterns of
behaviors
Qualitative:
Look for
categorical and
repeating data
that form
patterns of
behaviors
Spiral Assessments 22
To determine whether or not spiral assessments could improve student
achievement on the Math 2 End-of-Course Test [EOCT], two groups of students’ test
scores were compared after the implementation of such assessments. The 65 students
from the 2009-2010 Accelerated Math 2 classes were used as the control group and
received no spiral assessments throughout their course. The 83 students from the 20102011 Accelerated Math 2 classes received spiral assessments for each mid-unit and final
unit test throughout their course. There were a total of ten spiral assessments
administered throughout the 2010-2011 school year. In addition, the 2010-2011
experimental group received daily spiral reviews at the beginning of class and received
immediate feedback on the reviews.
When looking at whether spiral assessments could improve students’ attitudes
towards, and confidence in, mathematics, the Math Anxiety Questionnaire [MAQ] (see
Appendix A) was administered as a pre- and post- implementation. The questionnaire
was given to the experimental group prior to spiral assessments and again at the
conclusion of the implementation. The scores were analyzed to determine if students’
math anxiety changed following the use of spiral assessments in the classroom.
To determine if spiral assessments would be received positively by the math
department, as well as the administration, a focus group and two interviews were
conducted at the conclusion of the study. In May, 2011, ten math teachers participated in
a focus group to discuss the general purpose of assessments and to discuss whether or not
spiral assessments could be feasibly administered in their classrooms (see Appendix B).
Lastly, following a comparison of the Math 2 EOCT scores for the experimental and
control groups, two administrators from the participating high school were interviewed
Spiral Assessments 23
about the use of spiral assessments, the feasibility of such assessments, and the
organizational change process for the school (see Appendix C).
Validity, Reliability, Dependability, Bias, and Equity
To determine if the use of spiral assessments could improve students’
achievement on the Math 2 EOCT, teacher-made midterm exams and a state required
end-of-course test were used to gather interval data about students’ content knowledge of
Math 2. Each assessment was examined and found to be valid, reliable, and free from
bias. According to Popham (2009), “assessment reliability refers to the consistency with
which a test measures whatever it’s measuring” (p. 25). Additionally, Salkind (2010)
explained, “a valid test is a test that measures what it is supposed to” (p. 151). The
midterm exams, as well as the Math 2 EOCT, exhibited content validity because they
were all aligned with the Georgia Performance Standards and covered content that was
taught in the classroom. Questions from each assessment were also checked and found to
be free of unfair or offensive bias and were also free of disparate impact.
To determine if spiral assessments would affect students’ feelings towards math
and improve student confidence, both nominal and qualitative data were gathered through
student surveys and student reflections before and after the implementation of spiral
assessments. Measures were taken with both instruments to assure validity, reliability,
and dependability. The Math Anxiety Questionnaire exhibited construct validity and was
considered reliable because it was constructed and tested by a reputable group of
statisticians, including Meece (1981) from the University of North Carolina. According
to Popham (2009), “…internal consistency reliability requires only a single testadministration to a single group of students. This is because it represents the degree to
Spiral Assessments 24
which a test’s items appear to be doing the same consistent measurement job” (p. 25). To
determine internal consistency reliability, Cronbach’s Alpha was run on both surveys.
In order to ensure dependability of the surveys and the student reflections, several
steps were taken. The surveys and reflections were administered at the beginning of each
class period on the same school day, and subjects were consistently given the same
directions throughout the day. There were an adequate number of subjects (eighty-three)
completing the surveys and reflections, and all subjects were given the option of
anonymity. The amount of time allowed between the first and second survey was
approximately seven months, following the use of approximately ten spiral assessments.
Finally, the student reflections were absent of bias because the subjects were simply
asked to reflect on their feelings towards math and towards spiral assessments. The
language used to question the subjects was fair and inoffensive and showed no sign of
disparate impact.
To determine whether or not spiral assessments would be received positively by
the math department and supported by the administration, a teacher focus group and two
administrator interviews were used to collect qualitative data. The data obtained from
these two sources were found to be valid, reliable, and dependable. Focus group
questions, as well as interview questions, exhibited construct validity because they
accurately measured what the focus question asked. All questions were submitted to, and
approved by, a LaGrange College advisor and were found to be reliable and free from
bias. The questions contained no unfair or offensive language and were found to be free
of disparate impact. There were an adequate number of participants in the teacher focus
group, and the participants represented a diverse sample of the high school’s math
Spiral Assessments 25
department. The focus group, as well as the administrator interviews, occurred at the end
of the spiral assessment implementation and all participants were given accurate results
from the student data.
In addition to checking for validity, reliability, dependability, and bias when
answering the study’s focus questions, an equity audit was performed by the researcher in
the school where the study was being conducted. Skrla, McKenzie, and Scheurich (2009)
explained, “equity audits are a systematic way for school leaders . . . to assess the degree
of equity or inequity present in three key areas of their schools or districts: programs,
teacher quality, and achievement” (p. 3). The purpose of the equity audit was to ensure
that all students, regardless of age, gender, race/ethnicity, national origin, or disability,
were receiving equal educational opportunities.
At the beginning of the study, the equity audit revealed that several efforts were
currently being taken to increase equity for students. Specifically, in the math
department, high quality teachers with varying levels of teacher education and experience
were distributed to all levels of classes. Additionally, great emphasis was placed on
collaborative planning for teachers, which provided students with equitable lessons and
common assessments. Lastly, while the equity audit revealed that achievement equity is
still an area in need of improvement; it also revealed that the school’s primary focus was
to close this achievement gap. The school was providing the students with several
opportunities to increase their achievement in classes and on standardized tests, and as a
result hoped to decrease the dropout rate.
Spiral Assessments 26
Analysis of Data
The data collected in this action research were centered on answering the three
focus questions presented in the study. When looking at whether or not spiral
assessments could improve Math 2 EOCT test scores, several independent t-tests were
run. To determine if there was a significant difference in the starting ability levels of the
experimental and control groups, an independent t-test was conducted on the eighth grade
CRCT z-scores. When a significant difference was found in the two groups, the z-scores
were then screened and matched to equalize the ability levels of the two groups. In
October, 2010, the experimental group took a mid-term exam and their scores were
compared to last year’s October midterm scores of the control group. Another
independent t-test was conducted to determine if a significant difference existed. In
March, 2011, the experimental group took a second midterm exam and their scores were
again compared to last year’s March midterm scores of the control group. Another
independent t-test was used to determine significance. Lastly, in May, 2011, the
experimental group’s Math 2 EOCT scores were compared to the control group’s Math 2
EOCT test scores and one last independent t-test was run. For all of the independent ttests, the decision to reject the null hypothesis was set at p<.05. Additionally, the effect
size was measured using Cohen’s d. A small effect size ranged from 0.0 to .20; a
medium effect size ranged from .20 to .50; and a large effect size was set for any value
above .50.
To determine whether spiral assessments affected students’ feelings towards
math, and whether such assessments improved students’ math confidence, the Math
Anxiety Questionnaire [MAQ] was administered to the experimental group two times
Spiral Assessments 27
during the study. The questionnaire was given at the beginning of the study and again at
the conclusion of the spiral assessment implementation. To determine if there was a
significant difference between the pre and post questionnaire scores, a dependent t-test
was run on the data and the decision to reject the null hypothesis was set at p < .05. The
questionnaire used a seven-point Likert scale and was analyzed using Cronbach’s Alpha
to determine internal consistency reliability. Additionally, each question was analyzed
using the Chi Square method to determine significance. The significance level was
reported at the p<.05, p<.01, and the p<.001 levels.
In addition to using the Math Anxiety Questionnaire to analyze students’ attitudes
about math, student reflections were collected towards the end of the study. Students
were asked to reflect on their feelings about math and also their feelings about the use of
spiral reviews and assessments throughout the year. The reflections were then examined
and coded for themes. Highlighters were used to color code recurring, dominant, and
emerging themes that surfaced in the reflections. Coded themes were used to look for
categorical and repeating data that formed patterns of behaviors.
To determine whether or not spiral assessments would be positively received by
the math department and supported by the administration, a focus group of math teachers
and interviews of administrators were conducted at the end of the study. The focus group
discussion and the interviews were all recorded and transcribed. The transcriptions were
then examined and coded for themes. As in the student reflections, highlighters were
used to color code recurring, dominant, and emerging themes found in the data.
Spiral Assessments 28
While data were collected and analyzed to answer three focus questions, the study
was also analyzed holistically to ensure it had validation, credibility, transferability, and
to ensure that it was transformational.
Validation
This study exhibited ‘consensual validation’ because it was approved, reviewed,
and supported by the LaGrange College faculty. At the conclusion of the study, findings
were presented and the study was defended to the LaGrange College faculty. The study
also contained ‘epistemological validation’ because data results were compared to the
review of the literature to determine whether or not the literature supported the findings.
Credibility
Credibility of the study was ensured through the use of multiple data sources. To
answer the three focus questions, data were collected and analyzed from teacher-made
assessments, state assessments, student questionnaires, student reflections, a teacher
focus group, and administrator interviews. Eisner (1991) calls this process ‘structural
corroboration,’ where a confluence of evidence comes together to form a compelling
whole. Within this concept of structural corroboration are embedded the concepts of
‘fairness’ and ‘rightness of fit’. Fairness was achieved in the review of the literature by
presenting opposing points of view regarding the concept of a spiral curriculum. In order
to achieve precision and rightness of fit, extensive research of the literature was
conducted to present a tight argument and a coherent case for the use of spiral
assessments. Additionally, great care was taken when analyzing the data, both
quantitatively and qualitatively, in order to provide strong evidence for judgments made.
Spiral Assessments 29
Transferability
This study was transferable because it can be used by others and applied in
other classroom settings. Eisner (1991) calls this process ‘referential adequacy’ and
explained how perception and understanding by others will increase because of the
research conducted during this study. This study can be easily replicated and can be used
by others for future research to help increase student retention of any content.
Transformational
This study was transformational and contained ‘catalytic validity’ (Lather as cited
by Kinchloe & McLaren, 1998) because of the positive change it brought about in the
researcher, the students, and other math teachers. Students, as well as teachers,
discovered the positive effects that spiral assessments had on content retention. With this
discovery, teachers hoped to find additional ways to include spiraling within the
classroom.
Spiral Assessments 30
CHAPTER 4: RESULTS
The following quantitative and qualitative data were collected during this action
research study. The results of the data were organized around the three focus questions.
To answer focus question one and determine whether or not spiral assessments
would improve student test scores on the Math 2 EOCT, several independent t-tests were
run throughout the study. The purpose of the t-tests was to compare the means of two
different groups (control and experimental), and to determine if there was any
significance between the differences in the means (Salkind, 2010). Additionally, to
determine the magnitude of the difference, the effect size was measured for each t-test
using Cohen’s d. A small effect size ranged from 0.0 to .20; a medium effect size ranged
from .20 to .50; and a large effect size was set for any value above .50.
First, to determine if there was a significant difference in the ability levels of the
experimental and control groups, an independent t-test was conducted on the eighth grade
CRCT z-scores (see Table 4.1). The null hypothesis was there will be no significant
difference between the ability levels of the 2009 control students and the 2010
experimental students.
Spiral Assessments 31
Table 4.1. Independent t-test: Eighth Grade CRCT z-scores
t-Test: Two-Sample
Assuming Unequal
Variances
Mean
Variance
Observations
Hypothesized Mean
Difference
df
t Stat
P(T<=t) one-tail
t Critical one-tail
P(T<=t) two-tail
t Critical two-tail
t(130) = 2.464, p < .05
z-score 2009
5.444
7.916
65
z-score 2010
6.708
9.467
67
0
130
-2.464
0.008
1.657
0.015
1.978
The data from the eighth grade CRCT z-scores showed the mean z-score for the 2009
students was 5.44 and the mean z-score for the 2010 students was higher at 6.71. Since
the obtained value, 2.464, was greater than the critical value, 1.657, the null hypothesis
was rejected. The results of the eighth grade CRCT z-scores showed that there was a
significant difference between the ability levels of the two groups. A Cohen’s d value of
0.431 indicated a medium effect size in the difference.
Following this independent t-test, the CRCT z-scores from the 2010 group were
matched with the closest CRCT z-scores from the 2009 group. The matching of CRCT zscores left 98 students in the study with equal ability levels, 49 students from each group.
To verify no significant difference in the ability levels existed, another independent t-test
was conducted on the eighth grade CRCT z-scores (see Table 4.2). The null hypothesis
was there will be no significant difference between the ability levels of the 2009 control
students and the 2010 experimental students.
Spiral Assessments 32
Table 4.2. Independent t-test: Equalized Eighth Grade CRCT z-scores
t-Test: Two-Sample
Assuming Equal
Variances
Mean
Variance
Observations
Pooled Variance
Hypothesized Mean
Difference
df
t Stat
P(T<=t) one-tail
t Critical one-tail
P(T<=t) two-tail
t Critical two-tail
t(96) = 0.016, p > .05
z-scores 2010
6.274
8.641
49
7.745
z-scores 2009
6.283
6.850
49
0
96
-0.016
0.494
1.661
0.987
1.985
The equalized data from the eighth grade CRCT z-scores showed the mean zscore for the 2009 students was 6.28 and the mean z-score for the 2010 students was
slightly lower at 6.27. Since the obtained value, 0.016, was less than the critical value,
1.661, the null hypothesis was accepted. The results of the equalized eighth grade CRCT
z-scores verified there was no significant difference between the ability levels of the two
groups. A Cohen’s d value of 0.004 indicated a minimal effect size in the difference.
In October, 2010, the experimental group took a mid-term exam and their scores
were compared to last year’s October midterm scores of the control group. Another
independent t-test was conducted to determine if a significant difference existed between
the two groups (see Table 4.3). The null hypothesis was there will be no significant
difference between the October midterm scores of the 2009 control students and the 2010
experimental students.
Spiral Assessments 33
Table 4.3. Independent t-test: October Midterm Scores
t-Test: Two-Sample
Assuming Equal
Variances
Mean
Variance
Observations
Pooled Variance
Hypothesized Mean
Difference
df
t Stat
P(T<=t) one-tail
t Critical one-tail
P(T<=t) two-tail
t Critical two-tail
t(96) = 1.8295, p < .05
Oct Midterm 2010
82.367
98.446
49
166.929
Oct Midterm 2009
77.592
235.413
49
0
96
1.830
0.035
1.661
0.070
1.985
The mean score of the 2009 students was 77.59, which was lower than the mean
score of the 2010 students at 82.37. Since the obtained value, 1.830, was greater than the
critical value, 1.661, the null hypothesis was rejected. The results of the October
midterm scores revealed there was a significant difference between the midterm scores of
the 2010 experimental group and the 2009 control group. A Cohen’s d value of 0.370
indicated a medium effect size in the difference.
In March, 2011, the experimental group took a second midterm exam and their
scores were again compared to last year’s March midterm scores of the control group.
Another independent t-test was used to determine significance (see Table 4.4). The null
hypothesis was there will be no significant difference between the March midterm scores
of the 2009 control students and the 2010 experimental students.
Spiral Assessments 34
Table 4.4. Independent t-test: March Midterm Scores
t-Test: Two-Sample
Assuming Equal
Variances
Mean
Variance
Observations
Pooled Variance
Hypothesized Mean
Difference
df
t Stat
P(T<=t) one-tail
t Critical one-tail
P(T<=t) two-tail
t Critical two-tail
t(96) = 5.329, p < .05
March Midterm
2010
89.265
109.074
49
146.236
March Midterm
2009
76.245
183.397
49
0
96
5.329
3.25574E-07
1.661
6.51149E-07
1.985
Again, the mean score of the 2009 students, 76.25, was lower than the mean score
of the 2010 students at 89.27. Since the obtained value, 5.329, was greater than the
critical value, 1.661, the null hypothesis was rejected. The results of the March midterm
scores revealed there was a significant difference between the test scores of the 2010
experimental group and the 2009 control group. A Cohen’s d value of 1.078 indicated a
large effect size in the difference.
Lastly, in May, 2011, the 2010 experimental group’s Math 2 EOCT scores were
compared to the 2009 control group’s Math 2 EOCT test scores and one last independent
t-test was run (see Table 4.5 and Table 4.6). The null hypothesis was there will be no
significant difference between the Math 2 EOCT scores of the 2009 control students and
the Math 2 EOCT scores of the 2010 experimental students.
Spiral Assessments 35
Table 4.5. Independent t-test: Math 2 EOCT Raw Scores
t-Test: Two-Sample
Assuming Equal
Variances
Mean
Variance
Observations
Pooled Variance
Hypothesized Mean
Difference
df
t Stat
P(T<=t) one-tail
t Critical one-tail
P(T<=t) two-tail
t Critical two-tail
t(96) = 0.513, p > .05
eoct(math2) 2010
448.510
669.880
49
549.761
eoct(math2) 2009
450.939
429.642
49
0
96
-0.513
0.305
1.661
0.609
1.985
Table 4.6. Independent t-test: Math 2 EOCT Percentile Scores
t-Test: Two-Sample
Assuming Equal
Variances
Mean
Variance
Observations
Pooled Variance
Hypothesized Mean
Difference
df
t Stat
P(T<=t) one-tail
t Critical one-tail
P(T<=t) two-tail
t Critical two-tail
t(96) = 1.035, p > .05
eoct(math2) 2010
86.429
28.792
49
25.773
eoct(math2) 2009
87.490
22.755
49
0
96
-1.035
0.152
1.661
0.303
1.985
Although the mean percentile score of the 2010 experimental group, 86.43, was
lower than the mean percentile score of the 2009 control group, 87.49, there was not a
Spiral Assessments 36
large enough difference to make it significant. Since the obtained value, 1.035, was less
than the critical value, 1.661, the null hypothesis was accepted. The results of the Math 2
EOCT scores revealed there was no significant difference between the test scores of the
2010 experimental group and the 2009 control group. A Cohen’s d value of 0.209
indicated a medium effect size in the difference.
To answer focus question two and determine if spiral assessments would affect
students’ feelings towards math, and improve student confidence, a Math Anxiety
Questionnaire was administered to the 2010 experimental students at the beginning and at
the end of the study. To determine if there was a significant difference between the preand post-questionnaire scores, a dependent t-test was run on the data (see Table 4.7). The
null hypothesis was there will be no significant difference between the scores on the
Math Anxiety Questionnaire administered at the beginning and the end of the study.
Table 4.7. Dependent t-test: Pre- and Post-Math Anxiety Questionnaire
t-Test: Paired Two Sample
for Means
Mean
Variance
Observations
Pearson Correlation
Hypothesized Mean
Difference
Df
t Stat
P(T<=t) one-tail
t Critical one-tail
P(T<=t) two-tail
t Critical two-tail
t(82) = 18.493, p<.05
PreTotal
39.422
158.320
83
0.993
0
82
18.493
2.71382E-31
1.664
5.42764E-31
1.989
PostTotal
36.157
174.353
83
Spiral Assessments 37
The result of the dependent t-test revealed that the mean score for the prequestionnaire was 39.42 and the mean score for the post-questionnaire was 36.16,
showing a decrease in the average survey score from pre to post. Since the obtained
value, 18.493, was greater than the critical value, 1.664, the null hypothesis was rejected.
The results of the dependent t-test showed there was a significant difference between the
totals obtained from the pre-Math Anxiety Questionnaire and the post-Math Anxiety
Questionnaire. An effect-size r value of 0.125 indicated a small effect size in the
difference.
With each questionnaire, the chi-square test statistic was calculated to compare
what was observed on the questionnaire to what would be expected by chance (Salkind,
2010). Table 4.8 revealed the results of the chi-square tests for both the pre- and postquestionnaires. Since the questionnaire used a seven-point Likert scale, the degrees of
freedom for the chi square test was 6 and, at a .05 level of significance, the critical value
was 12.59. The obtained value for each question are listed in the table and the
significance level was reported at the p<.05, p<.01, and the p<.001 levels.
Spiral Assessments 38
Table 4.8. Chi-Square Statistic for Math Anxiety Pre- and Post-Questionnaire
Survey
Items
n=11
Item 1
Survey Question
PreQuestionnaire
χ2
When the teacher says he/she is going to ask 30.6667 ***
you some questions to find out how much
you know about math, how much do you
worry that you will do poorly?
When the teacher is showing the class how
to do a problem, how much do you worry
that other students might understand the
problem better than you?
When I am in math class, I usually feel at
ease and relaxed.
When I am taking math tests, I usually feel
nervous and uneasy.
Taking math tests scares me.
Item 2
Item 3
Item 4
Item 5
Item 6
Item 7
I dread having to do math.
Item 8
Item 9
Item 10
Item 11
It scares me to think that I will be taking
advanced high school math.
In general, how much do you worry about
how well you are doing in school?
If you are absent from school and you miss
a math assignment, how much do you worry
that you will be behind the other students
when you come back to school?
In general, how much do you worry about
how well you are doing in math?
Compare to other subjects, how much do
you worry about how well you are doing in
math?
PostQuestionnaire
χ2
43.5833 ***
50.3333 ***
72.1667 ***
65.8333 ***
55.3333 ***
7.5
22.5 ***
35.5 ***
45.3333 ***
99.8333 ***
48.5 ***
46.1667 ***
122.8333 ***
17.6667 **
5.5
8.6667
9.3333
5.3333
7.6667
14.1967 *
11.1667
* p<.05, **p<.01, ***p<.001
The results of the chi-square statistic for the Math Anxiety pre-Questionnaire
highlighted several significant items. Questionnaire items 1, 2, 3, 5, 6, 7, 8, and 11 were
all found to be highly significant when p < .05, .01, and .001, meaning that there were a
high percentage of students that answered a certain way on these questions. However,
items 4, 9, and 10 were not significant at all, which means that there was no significant
difference on these questions between what was observed in the answers and what would
have been expected to occur by chance.
Spiral Assessments 39
The results of the chi-square statistic for the Math Anxiety post-Questionnaire
also highlighted several significant items. Questionnaire items 1, 2, 3, 4, 5, 6, and 7 were
all found to be highly significant when p < .05, .01, and .001, meaning that there were a
high percentage of students that answered a certain way on these questions. However,
items 8, 9, 10, and 11 were not significant at all.
To determine the internal consistency reliability of the items on the Math Anxiety
pre- and post-Questionnaires, the Cronbach’s Alpha test was conducted using the
questionnaire responses. According to Salkind (2010), “[internal consistency reliability]
is used when you want to know whether the items on a test are consistent with one
another in that they represent one, and only one, dimension, construct, or area of interest”
(p. 147). For the Math Anxiety pre-Questionnaire, the computations gave a Cronbach’s
Alpha of 0.98 and for the Math Anxiety post-Questionnaire, the Cronbach’s Alpha was
0.99. Therefore, both of these surveys showed a high level of internal consistency
reliability using the results of the Cronbach’s Alpha test.
In addition to using the Math Anxiety Questionnaire to analyze students’ attitudes
about math, student reflections were collected towards the end of the study and
qualitative data were collected. Students were asked to reflect on their feelings about
math and also their feelings about the use of spiraled warm-up exercises and “blast from
the past” questions on their assessments. Out of 82 reflections, 72 students expressed
positive feelings towards math and 68 students expressed positive feelings towards
warm-ups and spiral assessments. A struggling student explained, “I like the warm-up
reviews and the ‘blast from the past.’ I forget things very quickly and need to be
reminded of formulas and how to work problems. Sadly, I think my grade might be even
Spiral Assessments 40
worse if we didn’t do these.” Another student stated, “I understand math easily enough, I
just don’t always remember it. This [blast from the past] will help me on the EOCT not
to forget how to work the problems.” An additional student commented, “Coming into
class and having warm-ups helps me remember how to do previous things I’ve learned.
I’ve always picked up on math quickly, but sometimes I don’t always hold onto it.”
Another student who is a big fan of the reviews stated, “We NEED warm-ups and ‘blast
from the past.’ Do NOT get rid of them! I have a bad habit of memorizing a formula or
a way to solve a problem for the test instead of learning and understanding the concept.
Those review problems help refresh my memory and show me what I need to focus on
for the EOCT.”
Although the majority of students agreed that revisiting content would help them
retain information and help their scores on the EOCT, some students suggested that the
“blast from the past” questions should count as bonus points on their tests. One student
explained, “Warm-ups and reviewing helps me out a lot and I think the more I see it, the
more I can remember it. Same goes for ‘blast from the past’; I just hate how it brings my
grade down sometimes.” This student’s sentiment was echoed on several reflections.
Another student commented, “I like the ‘blast from the past’ but it should be bonus.
Sometimes you don’t know what the blast from the past is and sometimes it’s not fresh
on your memory. The last test we took, ‘blast from the past’ is what killed me.”
Offering bonus points for “blast from the past” questions appeared on 23 different
reflections.
To answer focus question three and determine whether or not spiral assessments
would be received positively by the math department and supported by the
Spiral Assessments 41
administration, a teacher focus group and administrator interviews were used to collect
qualitative data. During the teacher focus group, six questions were asked regarding the
purpose and types of assessment, as well as the prospect of implementing spiral
assessments. Throughout the focus group discussion, the following three themes
emerged: (1) assessments are important sources of information in the beginning, middle,
and end of a unit, (2) students retain information better with more exposure, and (3) time
is a major issue.
When asked about the purpose of classroom assessments, teachers listed several
uses. Assessments are used to check for student understanding, to determine if the
students are ready to “move on”, to allow students to self-evaluate, and to provide
information for grades. Secondly, teachers explained that the best time to administer
assessments was in the beginning, middle, and end of a unit. Although Teacher 1 stated
the importance of pre-testing a unit to show student gain, other teachers agreed that there
was not enough time to add yet another assessment to the many others currently being
administered in their classrooms. Teacher 3 said the time for testing was “continuously.”
Next, teachers were asked what different forms of evaluation were currently being
utilized in their classrooms and if there were any forms they liked better than others. The
types of assessments being used fell into two categories, traditional and formative. The
type, and amount, of technology teachers had in their classrooms also affected what they
were able to do with assessments. Teacher 1 explained how she used graphing
calculators to conduct “quick polls” throughout the unit to provide immediate feedback to
her students. Teachers without this technology have never had the opportunity to
administer quick polls, but did express an interest in doing so. The majority of teachers
Spiral Assessments 42
listed traditional forms of assessment such as homework, bellwork, quizzes, and tests. A
couple of the teachers also listed formative types of assessment such as, “tickets-out-thedoor”, individual student white boards, group work, and classroom questioning.
Finally, following an explanation of spiral assessments, teachers were asked
whether or not spiral assessments were valid forms of assessments, if they were feasible
in their classrooms and what reservations they had with such assessments. Teacher 3 said
formative assessments were “absolutely” valid, “absolutely” feasible, and that he had no
reservations with such assessments. Teacher 3 also explained that he had been using
spiral assessments in his classroom for the last thirty years and planned to continue doing
so until he no longer taught. Teacher 4 stated, “Math is all about spiraling. Students
have to know previous material in order to master future material.” Teacher 5 added,
“Just like us, when students don’t use it, they lose it.”
Despite these positive comments and the consensus that spiral assessments were
valid and worthwhile, the overall opinion of the group was that spiral assessments were
not very feasible and would be difficult to implement in their classrooms. The majority
of teachers in the focus group said the biggest deterrent to using spiral assessments was
time. Teacher 6 expressed her concern that students would complain, “you didn’t tell us
this was going to be on the test.” She explained that students always want to know
exactly what is on the test with “no surprises”, and because of that, students would expect
her to keep reviewing material and “[she] would never have time to get done what [she]
needed to get done.” Other teachers expressed a concern that preparing spiral
assessments would take extra time and Teacher 7 added, “It will take more time to assess
students because you have to add more questions to every test.”
Spiral Assessments 43
Following the arrival of the Math 2 EOCT scores, interviews were conducted
with the principal and an assistant principal of the high school to share the results of the
study and to determine if the administration would support the use of spiral assessments.
The first six questions of the interview mimicked those of the teacher focus group and the
answers provided by the administrators were very similar to the answers provided by the
teachers. Two additional administrator questions provided suggestions for implementing
change in the classroom and in the school as a whole. During the administrator
interviews, the following three themes emerged: (1) formative and summative
assessments provide valuable information to students, parents, and teachers, (2) practice
makes perfect, and (3) change is a gradual process.
When asked about the purpose of classroom assessments, both administrators
listed reasons similar to those listed by the teachers. Administrators expected teachers to
use assessments throughout a unit to ascertain what their students have mastered and to
make decisions regarding their instruction. They also valued assessments as a source of
documentation and as a device for reporting progress to both parents and students.
Administrator 1 explained, “Teachers and students should use assessments to gather
information about content knowledge. Assessments not only show what the student has
learned, but also how the teacher has taught.” Administrator 2 explained, “Formative
assessments are just as important, if not more important, than the summative ones.
Teachers can’t change what they’re doing if they don’t know what their students are
learning.”
Due to the important role assessments play in teachers’ classrooms, both
administrators agreed that a variety of assessments should be utilized everyday in every
Spiral Assessments 44
classroom. Administrator 1 stated, “Whether formally or informally, students should be
assessed everyday throughout the entire school year.” Administrator 2 commented,
“Assessments should be an on-going, continuous part of every class.” When asked about
the types of assessments observed most in classrooms, Administrator 1 distinguished
between formal and informal assessments. In terms of formal assessments, all teachers
used quizzes, unit tests, benchmarks, midterms, and final exams as part of their grading
practices. Regarding informal assessments, the use of student questioning was observed
the most. Some teachers incorporated “tickets-out-the-door” and other quick feedback
activities; however, the majority of teachers simply asked questions to gather information
about their students. Administrator 1 explained, “Some teachers are more skilled
questioners than others. Some teachers have a natural ability to gather information from
their students, simply by the questions they ask.”
Following general assessment questions, administrators were asked about the
validity and feasibility of spiral assessments in the classroom. Similar to focus group
responses, administrators believed spiral assessments were valid and extremely useful in
the classroom. Administrator 2 stated, “Students wouldn’t be allowed to forget what they
had learned. They would know they were responsible for the content over and over and
over again.” Administrator 1 recounted a statistics class taken in college where spiral
assessments were administered throughout the entire course. She claimed to have learned
more in that course than in any other and said that at the end of the course, “it all fit
together.” Administrator 1 credited this statistics class, and the assessments used in the
class, for her own implementation of cumulative tests when she was a classroom teacher.
Spiral Assessments 45
Although she administered a separate unit test and then a cumulative review and then a
cumulative test, the spiral concept was there.
When asked about concerns or potential problems with spiral assessments,
Administrator 1 was quick to say, “Some teachers just won’t do it…not because they
don’t believe in it, but because they’re too lazy to put forth the effort.” Changing to
spiral assessments would mean teachers would have to re-create tests they’ve been using
for years. Some teachers would be willing to make the change, others would not.
Administrator 2 also noted the problem with teachers not taking corrective action once
spiral assessments were administered. He explained, “If a teacher doesn’t do anything to
correct a problem a student is having, then that student is going to get penalized over and
over again for the same gap in knowledge.”
At the conclusion of the interviews, both administrators were asked whether or
not they thought teachers would be open to administering spiral assessments in their
classrooms and what was the best way to bring about change within a school. Again,
Administrator 1 explained that some teachers would resist the change, “no matter how
good the change was.” Administrator 2 commented, “Some teachers come on board
immediately, some wait until they see others do it, and others never come on board.”
Both administrators suggested a gradual change policy when implementing something
new in the school. Similar to changes that take place now, research and data should be
collected to ensure spiral assessments are worth doing in the first place. The faculty
should then be presented the research and data, and given time to study and understand
the change. When the time comes to move to spiral assessments, Administrator 1
suggested each teacher choose one class to start the process with instead of trying to do
Spiral Assessments 46
all of their classes at one time. The next year each teacher could pick up another class
and keep going from there. Administrator 2 also recommended working with other
teachers in the same subject to collaboratively create the spiral assessments. He added,
“Working with someone else to lessen the burden always makes change easier.”
Spiral Assessments 47
CHAPTER 5: ANALYSIS AND DISCUSSION OF RESULTS
Analysis
The purpose of this study was to determine the effectiveness of spiral assessments
on content retention and student attitudes in a mathematics classroom. To determine the
effectiveness of spiral assessments, both quantitative and qualitative data were collected
throughout the study and used to answer three focus questions.
To answer focus question one, “will the use of spiral assessments, coupled with
proper feedback, improve students’ achievement on the Math 2 End of Course Test?”,
data were collected on two teacher-made midterm exams and a state-made Math 2 EOCT,
for a 2009 control group and a 2010 experimental group. The control group received no
spiral assessments, while the 2010 received numerous spiral assessments. The teachermade midterm exams were identical for the 2009 and 2010 groups, while the state-made
Math 2 EOCT was created by the state with no information on its consistency from one
year to the next. Additionally, 8th grade CRCT data were used in the study to screen zscores in an attempt to equalize the control and experimental groups.
Following an equalization of ability levels in the control and experimental groups,
data were collected on the midterm and EOCT test scores, and three independent t-tests
were run to determine whether or not a significant difference existed. The results of the
October midterm revealed a mean score of 77.89 for the 2009 control group and a higher
mean score of 82.37 for the experimental group. The independent t-test revealed that a
significant difference existed in the groups and a Cohen’s d value determined a medium
effect size. From the results of the October midterm, it appeared that spiral assessments
were having a positive effect on content retention for the 2010 experimental group.
Spiral Assessments 48
The March midterm revealed more promising data on spiral assessments. While
the 2009 control group only averaged 76.24 on the test, the 2010 experimental group
boasted an 89.27 average. Again, an independent t-test revealed a significant difference
in the data with a large Cohen’s d effect size. From the results of the March midterm, it
appeared that, given more time, spiral assessments had an even greater positive effect on
content retention for the 2010 experimental group.
Unfortunately, the Math 2 EOCT results did not echo the same promising data of
the midterm data. The 2009 control group averaged a raw score of 450.94, percentile
score of 87.49, whereas the 2010 experimental group averaged lower scores of 448.51
and 86.43 respectively. If the results of the Math 2 EOCT are viewed in isolation, it
would appear that spiral assessments had a negative effect on content retention for the
2010 experimental group. However, since the EOCT data were in direct contrast to data
collected from both midterms, one has to question whether or not the state altered their
versions of the Math 2 EOCT from one year to the next.
Although the results in the study contradicted each other from teacher-made to
state-made tests, a possible reason for the contradiction was supported in the review of
literature. Jones et al. (2007) cautioned against using one test score to provide the whole
picture. They explained, “A test is a snapshot in time, affected by numerous sources of
error . . . The more information a teacher collects, the more valid the inferences based on
that information” (Jones et al., 2007, p. 74). Therefore, when interpreting results
produced in the study, it is important to not just look at the end result, but also to look at
what happened along the way.
Spiral Assessments 49
In the review of the literature, Bruner (1960) and Doll (1993) emphasized the
importance of revisiting content in order to increase student mastery. Schulman (1998)
also supported the spiral concept and added, “by revisiting the same questions throughout
the school year, children are encouraged to amplify their mathematical thinking” (p. 72).
Schulman (1998) further explained how teachers could focus their instruction based on
gathered information from their student “revisits” (p. 72). Based on the literature and the
results from the teacher-made midterms, revisiting problems throughout the year helped
the 2010 students with content retention.
Also supported in the literature were that formative assessments contributed to the
positive results of the teacher-made midterms. Throughout the 2010 school year, greater
attention was placed on using assessments as an informational tool rather than just a
means for providing a student grade. Reeves (2007) warned, “as long as we use
[assessments] only as a means to rank schools and students, we will miss out on their
most powerful benefits” (p. 27). He further stated, “when teachers’ classroom
assessments become an integral part of the instructional process and a central ingredient
in efforts to help students learn, the benefits of assessment for both teachers and students
will be boundless” (p. 28). Throughout the study, daily warm-ups, quizzes, white board
activities, and summative assessments, all spiral in nature, were used to guide
improvements in instruction. The increased use of formative assessments was heavily
supported in the literature and appeared to have a positive effect on content retention.
To address the second focus question, “how will spiral assessments affect
students’ feelings towards math, and will such assessments improve student
confidence?”, surveys and reflections were administered throughout the 2010 school year
Spiral Assessments 50
to the experimental group. At the beginning and end of the study, quantitative data were
collected on the Math Anxiety Questionnaire [MAQ] using a seven-point Likert Scale. A
student answering with extreme anxiety on each of the questions would have scored a 71.
Both the pre- and post-questionnaire mean scores were well below this extreme case.
The pre-questionnaire mean score was only a 39.42 with the post-questionnaire mean
score not far away with 36.16. Although the Accelerated Math students that participated
in the study exhibited low levels of anxiety even at the beginning, they did show a
decrease in their levels at the end. The dependent t-test run on the pre- and postquestionnaire data revealed that there was a significant difference in the two scores. The
drop in pre and post scores revealed students had lower anxiety following the use of
spiral assessments. Internal consistency reliability was proven with a Cronbach’s Alpha
score of a .98 and .99 for pre- and post-questionnaire respectively.
Following the dependent t-test, each question was analyzed using the Chi Square
method to determine whether or not students answered significantly in one direction.
Item 1, “When the teachers says he/she is going to ask you some questions to find out
how much you know about math, how much do you worry that you will do poorly?”, and
Item 2, “When the teacher is showing the class how to do a problem, how much do you
worry that other students might understand the problem better than you?”, were both
significant at the p<.001 level, with students predominantly answering on the low end of
the scale, with a 1 “not at all”. On the pre-questionnaire, 63% answered between 1 and 3
on Item 1 and 75% on Item 2. These percentages grew to 76% on Item 1 and 80% on
Item 2 on the post-questionnaire which indicated more students were less worried at the
end of the study.
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Another item of interest was Item 3, “When I am in math class, I usually feel at
ease and relaxed”. On the pre-questionnaire, only 18% responded with a 7 “very much at
ease”; however, 31% responded with a 7 on the post-questionnaire. This increase from
18% to 31% indicated that more students felt comfortable in math class at the end of the
study. Item 4, “When I am taking math tests, I usually feel nervous and uneasy”, did not
show any significance on the pre-questionnaire because students’ responses were evenly
spread along the scale. However, following spiral assessments, the question became
significant at the p<.001 level, with students predominantly choosing on the low end of
the scale with a 1 “not at all nervous”.
Item 5, “Taking math tests scares me”, and Item 6, “I dread having to do math”,
produced similar results to Items 1 and 2. Both questions were significant at the p<.001
levels because students predominantly chose numbers on the lower end of the scale with
a 1 “never”. Again, pre and post totals revealed an increase in percentages of students
choosing lower numbers on the post-questionnaire than the pre-questionnaire.
Item 7, “It scares me to think that I will be taking advanced high school math”,
was the most significant question in terms of chi square numbers, but the least significant
question in terms of the study. Accelerated Math students have already chosen the
advanced math path and therefore chose mostly 1’s and 2’s for this question. However,
even with this extreme case, the percentage of students choosing a 1, “not at all”, grew
from 42% on the pre-questionnaire to a 54% on the post-questionnaire.
Items 8 through 11 did not offer much evidence for this study. Item 8, “In
general, how much do you worry about how well you are doing in school?”, did show
significance at the p<.01 level with students choosing on the higher end of the scale.
Spiral Assessments 52
However, this question addressed anxiety towards school and not specifically towards
math. Also, students appeared to be more worried about school at the beginning of the
year than they did at the end of the year. Item 9, “If you are absent from school and you
miss a math assignment, how much do you worry that you will be behind the other
students when you come back to school?”, was probably not significant because
Accelerated Math students, in general, have good attendance rates. Item 10, “In general,
how much do you worry about how well you are doing in math?”, was also not
significant. If students thought this question was referring to their grade in math class,
then this lack of significance could be attributed to the fact that most Accelerated Math
students have high math grades. Lastly, Item 11, “Compared to other subjects, how much
do you worry about how well you are doing in math?”, exhibited a small significance in
the pre-questionnaire and no significance in the post-questionnaire. Again, students may
have been a little more worried at the beginning of the year, but at the end they knew
their class average and knew how their math grade compared to other subjects.
In addition to the pre- and post-Math Anxiety Questionnaire, student reflections
were administered at the end of the study. Student reflections provided qualitative data
for the second focus question and focused on attitudes about math and the use of spiraled
reviews and assessments. Because the study included subjects that were enrolled in
Accelerated Math, it was not surprising that the majority of students expressed positive
feelings towards math in their reflections. However, I was a little surprised that students
reflected so positively on the spiraled warm-ups and assessments. Most students
recognized the importance of practicing content in order to master it. Additionally, most
students were in favor of the “blast from the past” test questions and believed the
Spiral Assessments 53
questions would help them perform better on the Math 2 EOCT. The student’s comment,
“Do NOT get rid of them”, demonstrated how important he thought the spiraled content
was.
Although the students in this study exhibited less anxiety and more positive
feelings towards math from the beginning, their decrease in anxiety demonstrated in the
MAQ results and positive comments in the student reflections were supported in the
literature. Reeves (2007) explained, “when students are involved in the classroom
assessment process, they are more engaged and motivated, and they learn more” (p. 31).
McMillan et al. (2010) explained, “…overall formative practices showed a positive
relationship with student motivation” (p. 11). Throughout the 2010 school year, students
contributed to the creation of the spiraled warm-ups and were provided immediate, daily
feedback. Reeves (2007) added, “students are highly motivated when they have more
choice during the learning process and receive more quality feedback” (p. 36).
To answer the third focus question, “will a system which employs the use of
spiral mathematics assessments be received positively by the math department and will
the use of such a system be supported by the administration?”, a focus group of ten math
teachers and interviews of two administrators were conducted at the conclusion of the
study. Qualitative data were collected from the focus group discussion and the interviews
and then coded for themes.
As discussed in Chapter 4, the math teacher focus group revealed the following
three themes: (1) assessments are important sources of information in the beginning,
middle, and end of a unit, (2) students retain information better with more exposure, and
(3) time is a major issue. Basically, teachers agreed that student retention is an ongoing
Spiral Assessments 54
problem and a problem that they have not been able to remedy over the years. They also
agreed that the more a student practices a concept, the more they understand it.
Additionally, teachers viewed assessments as a major component of their classroom and
agreed that assessments provided them a wealth of information.
The discouraging data that emerged from the focus group was that, although
teachers supported the spiraling concept, they were leery to make a change because of the
time and effort it would take on their part. I believe this lack of teacher willingness
resulted from enduring years of constant change. Each year, teachers are forced to
participate in school improvement efforts which they may or may not believe in. Further
frustration arises when the improvement effort is short lived and another change is
implemented before their prior efforts have time to work.
Similar to the teacher focus group, the administrator interviews revealed the
following three themes: (1) formative and summative assessments provide valuable
information to students, parents, and teachers, (2) practice makes perfect, and (3) change
is a gradual process. In general, teachers and administrators viewed assessments in the
same light and stressed the important role assessments play in classroom instruction.
Additionally, both administrators expressed an expectation for teachers to use
assessments frequently and to use a variety of assessments to make informed decisions in
their classrooms. Administrators also recognized the importance spiraling content
because it allowed students to practice content over and over. Lastly, administrators
recommended a gradual time frame for implementing a change with teachers.
Although both administrators were in full support of using spiral assessments in
the classroom, neither was ready to jump on board with a school-wide implementation. I
Spiral Assessments 55
believe this lack of willingness to bring about school-wide change also stemmed from
tolerating a multitude of changes over the last few years. I believe administrators are as
frustrated as teachers when it comes to school improvement. Some changes are forced
upon them by the state or county, and other changes result from their own efforts.
Regardless of where the change originates, I believe administrators recognized teacher
frustration and feared what another change may do to the morale of the school.
The ability of teachers and administrators to greatly impact change reform in
school was heavily supported in the literature (Fullan, 2009; Gabriel, 2005; Marzano et
al., 2001). The concerns and reluctance of teachers and administrators to implement
spiral assessments were also evident in the literature. Marzano et al. (2001) commented,
“busy teachers who have been doing things the same way for a fair amount of time will
have many valid reasons for not trying a new strategy. What is clearly required to alter
the status quo is a sincere desire to change and a firm commitment to weather the
inevitable storms as change occurs” (pp. 157-158). They further commented,
“Administrators and classroom teachers are often overwhelmed by the sheer amount of
change attempted and the work involved” (Marzano et al., 2001, p. 159). Lieberman
(2005) agreed with the amount of changes schools attempt and referred to those change
numbers as “staggering” (p. vii). It was clear in the literature that teachers and
administrators across the nation feel as frustrated as the teachers and administrators in the
study when it comes to change implementation.
Discussion
Overall, this research study produced positive results with regards to content
retention and student confidence. Quantitative data showed spiraled students with an
Spiral Assessments 56
advantage over non-spiraled students on teacher-made midterms. Quantitative data on
the Math Anxiety Questionnaire also revealed spiraled students with less math anxiety
following the implementation of spiral assessments.
Although quantitative data were inconsistent between teacher-made and statemade tests, qualitative data were consistent among students, teachers, and administrators.
Qualitative data revealed a general consensus that students lose content knowledge when
they do not use the knowledge over and over. Students, teachers, and administrators also
recognized the importance of assessments in the classroom, and in particular, the positive
effect spiral assessments could have over student retention and student confidence.
Students’ not only made positive comments about ‘blast from the past’ questions, but also
about feeling prepared for their EOCT.
This study was successful in highlighting the importance of using meaningful
assessments in the classroom to make informed decisions about students’ content
knowledge and to provide students with effective feedback. This study also focused on
the importance of revisiting concepts throughout a course in order to ensure student
mastery and produced results that quantitatively and qualitatively supported the spiral
concept.
This study achieved structural corroboration and credibility through the use of
multiple data sources (Eisner, 1991). Quantitative data were collected through teachermade midterms, a state-made EOCT, and a Math Anxiety Questionnaire. Additionally,
qualitative data were collected through student reflections, a teacher focus group, and
administrator interviews. Fairness was achieved by the presentation of opposing views in
the review of literature, as well as the presentation in the discrepancy of student scores in
Spiral Assessments 57
the teacher versus state-made tests. Finally, with an extensive research of the literature
and a careful analysis of the data, this study attained a tight argument, a coherent case,
and rightness of fit.
Implications
Since statistical significance was not found using the Math 2 EOCT scores, the
results of this study cannot be generalized to a larger population. However, the results of
the teacher-made midterms did show statistical significance and were not affected by the
possibility of varying difficulty levels in the state-made test. Being able to control the
consistency of state tests would greatly benefit future research and would hopefully yield
results that could be generalized to a larger population.
While the quantitative data did not produce results that could be generalized to a
larger population, the qualitative data provided great insight into the benefits of spiral
assessments. The qualitative data from students revealed an overwhelming support for
the use of spiral assessments. Students recognized their inability to retain information if
they were not provided with opportunities to practice the content. Student reflection
comments supported the use of daily spiral reviews and unit spiral assessments. In
addition to student comments, teacher comments reflected a similar support of spiral
assessments. Teachers also recognized their students’ difficulty retaining information
and supported the concept of spiraling. The qualitative results of the study revealed an
overwhelming support of the spiral concept and could be generalized to a larger
population.
As mentioned above, this study was transferable and had ‘referential adequacy’
(Eisner, 1991). Content retention is a problem experienced by teachers throughout the
Spiral Assessments 58
county, state, and nation. Teachers from any discipline can apply this study to their
classroom in an attempt to increase content retention. Teachers can easily replicate and
expand this study to include spiral assessments, as well as other activities that are spiraled
in nature.
This study was transformational and contained ‘catalytic validity’ (Lather as cited
by Kinchloe & McLaren, 1998) because of the positive change it brought about in the
students, other math teachers, and me. As a whole, students revealed in their reflections
that spiral assessments helped them remember information they normally would have
forgotten. Students recognized the importance of practicing content over a period of time
and believed this practice would help them perform better on the Math 2 EOCT.
Hopefully, this study encouraged students to apply the saying “practice makes perfect” to
all of their classes.
In the math teacher focus group discussion, teachers expressed an overwhelming
support for spiraling content. Every teacher expressed a concern that their students
memorize information to do well on a test and then quickly lose that knowledge.
Although teachers were concerned about the time involved implementing spiral
assessments, they also appeared very open to finding ways to change. Hopefully, this
study made teachers aware of the potential spiral assessments have to bring about
positive outcomes in their classrooms.
While I believe this study had a positive effect on students and teachers, I believe
it had the greatest impact on me. This study not only led me to recognize the importance
of spiraling my assessments, but also the need to spiral content on a daily basis.
Additionally, I recognized that spiraling content is not only important for success on state
Spiral Assessments 59
tests, but more importantly for preparing students for future math classes. This study
brought about a determination to use spiral assessments throughout the remainder of my
teaching career. This study also brought about a desire to research vertically what
content future math teachers need my students to be experts in. Not only do I plan to use
spiral assessments to prepare my students for state tests, but with this vertical knowledge,
I plan to better prepare students for future success.
Impact on School Improvement
According to Tomal (2003), action research provides a means for educators to
solve problems and make improvements within their classrooms. Hopefully, with
classroom improvements come school-wide improvements. Marzano et al. (2001)
insisted that it only takes a few enthusiastic educators to “infect an entire staff with
enthusiasm” (pp. 157-158).
Following this research study, I personally witnessed two teachers within my
department discussing ways to change their upcoming tests to include questions from
previous units. I also watched as they divided up the tests to start changing over the
summer. With “time” emerging as a major deterrent for implementing spiral assessments
during our focus group discussion, I was pleased to see these teachers finding a way to
overcome this problem.
If nothing else, this study has lead to many discussions regarding classroom
assessments and how assessments can be better utilized by teachers. Not only has the use
of spiral assessments been greatly discussed, but also the importance of formative
assessments and teacher feedback to students. Not only have such discussions occurred
within my department, but also between members of other departments. It is my hope
Spiral Assessments 60
that teachers become increasingly aware of how they are using assessments within their
classrooms and that teachers start looking for ways to embrace the spiral concept.
Recommendations for Future Research
The greatest obstacle this research study contended with was the frequent
changing of the math requirements from the State Department of Education. The 2009
control group was considered the “guinea pig” group because, each year they moved up,
starting from the 6th grade, the new GPS curriculum rolled in and their teachers were
novices to the new standards. Additionally, in the middle of the study, the state
department decided to drop the GHSGT requirement and move to an EOCT requirement.
Therefore, it is uncertain whether the state department changed the 2010 Math 2 EOCT
from the 2009 Math 2 EOCT and whether a change affected test scores from one year to
the next.
As a result of the numerous math changes over the last few years from the state
department, I don’t feel confident in the state test scores that resulted during the study. I
strongly recommend the study be conducted in the future, after things are more settled
and consistent within the state department. Otherwise, the study needs to be conducted
without the use of state tests.
One way to ensure that a change in the state EOCT does not affect test scores
from one year to the next, is to use a control and experimental group from the same
school year. A couple of classes could receive spiral assessments throughout the year,
while a couple of other classes would not. Therefore, all four classes would take the
exact same EOCT and there would be no doubt which students performed better. The
Spiral Assessments 61
problem with this method is, if spiral assessments improve student achievement, then the
control group would be getting the raw end of the deal.
Additionally, I would like to see this action research study conducted on nonaccelerated math students. It became apparent in the MAQ results and the student
reflections that most of my accelerated math students were already confident in their
math ability, and most of my students already liked math. This makes sense because
students taking accelerated math are doing so because they are better at math than their
peers. Therefore, when trying to ascertain whether or not spiral assessments improve
student attitudes and confidence, many of these students did not have much need for this
improvement. I believe surveys and reflections from non-accelerated students would
reveal more information towards focus question two.
Spiral Assessments 62
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Spiral Assessments 66
Appendix A
Math Anxiety Questionnaire (MAQ)
1) When the teachers says he/she is going to ask you some questions to find out how
much you know about math, how much do you worry that you will do poorly?
1
Not at all
2
3
4
5
6
7
Very much
2) When the teacher is showing the class how to do a problem, how much do you worry
that other students might understand the problem better than you?
1
Not at all
2
3
4
5
6
7
Very much
3) When I am in math class, I usually feel at ease and relaxed.
1
2
Not at all at ease
3
4
5
6
7
Very much at ease
4) When I am taking math tests, I usually feel nervous and uneasy.
1
2
Not at all nervous
3
4
5
6
7
Very much nervous
5) Taking math tests scares me.
1
Never
2
3
4
5
6
7
Most of the time
4
5
6
7
Most of the time
6) I dread having to do math.
1
Never
2
3
7) It scares me to think that I will be taking advanced high school math.
1
Not at all
2
3
4
5
6
7
Very much
8) In general, how much do you worry about how well you are doing in school?
1
Not at all
2
3
4
5
6
7
Very much
Spiral Assessments 67
9) If you are absent from school and you miss a math assignment, how much do you
worry that you will be behind the other students when you come back to school?
1
Not at all
2
3
4
5
6
7
Very much
10) In general, how much do you worry about how will you are doing in math?
1
Not at all
2
3
4
5
6
7
Very much
11) Compared to other subjects, how much do you worry about how well you are doing
in math?
1
2
Much less than other subjects
3
4
5
6
7
Much more than other subjects
Spiral Assessments 68
Appendix B
Teacher Focus Group Questions
1. How can assessments be utilized within the classroom? What is the purpose of
classroom assessments?
2. When is the best time for assessing knowledge in the classroom?
3. What other forms of evaluation do you use in your classroom? Do you think some
forms are better than others?
4. Do you think spiral testing is a valid assessment of skills?
5. Are spiral assessments a feasible form of assessment?
6. What problems or reservations do you have with spiral assessments?
Spiral Assessments 69
Appendix C
Administrator Interview Questions
1. How should assessments be utilized within the classroom? What is the purpose of
classroom assessments?
2. When is the best time for assessing knowledge in the classroom?
3. What forms of evaluation do most teachers use in their classrooms? Do you think
some forms are better than others?
4. Do you think spiral testing is a valid assessment of skills?
5. Are spiral assessments a feasible form of assessment for our school?
6. What problems or reservations do you have with spiral assessments?
7. Do you think teachers at our school would be open to changing the type of
assessments they administer in their classrooms?
8. What is the best way to educate our teachers about spiral assessments and how is the
best way to bring about change in our school?