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:

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Donald R. Livingston, Ed.D. Sharon M. Livingston, Ph.D.

Associate Professor and Project Advisor Assistant Professor and Project Advisor

Education Department Education Department

Spiral Assessments:

A STUDY IN AN ACCELERATED Mathematics Classroom

A project submitted

by

Lisa Morgan Skinner

to

LaGrangeCollege

in partial fulfillment of

the requirement for the

degree of

Specialist in Education

in

Curriculum and Instruction

LaGrange, Georgia

August 4, 2011

[WU1]

Spiral Assessments iii

Abstract

[WU2]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 aspositive changes to their learning environment based on the assessments. An extensive review of the literature led this study to seek practices in other countries where spiral curriculums and spiral assessments have been utilized and where those practices were deemed effective. Finally, this study used to surveys and interviews of math teachers and administrators to determine if a spiral assessment practice was feasible for broader implementation and if it would be supported. In addition[WU3], students were surveyed and interviewed about whether or not the practice led to improved attitudes about mathematics and if the practice raised confidence in their mathematics abilities.

Spiral Assessments iv

Table of Contents

Abstract...... iii

Table of Contents………………………………………………………...... iv

List of Tables and Figures…...…………………………………………………………….v

Chapter 1: Introduction…………………………………………………………………....

Statement of the Problem………………………………………………………….

Significance of the Problem……………………………………………………….

Theoretical and Conceptual Frameworks…………………………………………

Focus Questions…………………………………………………………......

Overview of Methodology………………………………………………………...

Human as Researcher……………………………..……………………………….

Chapter 2: Review of the Literature……………………………………………………….

Chapter 3: Methodology…………………………………………………………………

Research Design………………………………………………………………….

Setting…………………………………………………………………...……….

Sample / Subjects / Participants[WU4]…………………………………………………..

Procedures and Data Collection Methods………………………………………..

Validity and Reliability Measures.……………………………………………….

Analysis of Data……………………………………………….…………………

Chapter 4: Results………………………………………………………………………..

Chapter 5: Analysis and Discussion of Results……………………….…………………

Analysis………………………………………………………………………….

Discussion……………………………………………………..…………………

Implications……………………………………………………...……………….

Impact on Student Learning………………………………………………………

Recommendations for Future Research…………………………...……………..

References………………………………………………………………………………..

Appendixes………………………………………………………………………………

Spiral Assessmentsv

List of Tables and Figures

Tables

Table 3.1Data Shell …………………………………………#

Figures

Figure 4.1Independent t-test …………………………………#

Figure 4.2Independent t-test …………………………………#

Figure 4.3Independent t-test …………………………………#

Figure 4.4 Independent t-test …………………………………#

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 explains, “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) explains 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.Daniels, & Hyde, 2005, p. 106). According to the 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) explains 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 embeds the theory of social constructivism throughout. Bruner (1996) explains, “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 explains that students have a greater use for knowledge which has been acquired through discovering and making connections to prior experiences (p. xii). Fosnot (2005) further explains, “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 frameworks, ‘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, 2010).

This research is 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.

The National Board for Professional Teaching Standards [NBPTS] Core Proposition Three closely relates to this research. Proposition Three states 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 is 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 can 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?
  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 using 90 tenth-graders from five Accelerated Math 2 classes. The study took place throughout the 2010 school year and was compared to data collected in the 2009 school year using 75 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), the eighth grade CRCT scores were analyzed for significant difference within and between groups. First, an independent t-test on eighth grade CRCT scores was conducted to see if there was significant difference between the 2009 students and the 2010 students. Next, independent t-tests were conducted on the midterm and final 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.

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 5-point Likert Scale was used for the surveys and then 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?”, also used 5-point Likert Scale that was analyzed using the Chi Square method[WU5]. In addition, interviews were conducted with math teachers and administrators to ascertain attitudes about and feasibility of the spiral assessment method.

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 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 almost 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, make better sense of the content, and make them more successful on their Math 2 EOCT.

CHAPTER 2: REVIEW OF THE LITERATURE[WU6]

A review of current and past literature provides a justification for this action research study. The literature review presents evidence for each of the three focus questions in the study. The literature review includes 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”. [WU7]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 “aA 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) suggestsed 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) explains that iteration is the process of repeating itself over and over again and states, “nNothing is more central to this new beginning than the concept and practice of iteration” (p. 97). Doll (1993) also states that Bruner’s “spiral curriculum is worth looking at again and reframing in light of recursion theory” (p. 102). Doll (1993) further explains that “it is worth constructing a curriculum where students revisit with more insight and depth what they have done” (p. 102).

Harden (1999) supports the concept of a spiral curriculum, but adds that “Aa 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 cautions that a spiral curriculum often limits the depth of knowledge that students attain. Snider (2004) explains, “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 cautions 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 explains 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) explains that “classroom assessment has a higher calling – to improve the caliber of classroom instruction” (p. 117).

At the forefront of discussions about classroom assessments is the importance of using tests to gather information about students’ learning and to make decisions about future instruction. Dwyer (2008) states that “the central purpose of testing will be to inform and improve teaching and learning” (p. 5). Garcia, Spalding, and Powell (2001) define 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) believe 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) believe that using a variety of assessments to provide continuous feedback will nourish teaching and learning. They caution 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) explain, “A teacher who evaluates student learning and instructional practices solely on 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 (2000) states, “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.