Perception and Implementation of the Ohio Academic Content and Process Standards for Mathematics among Middle School Teachers

A dissertation presented to the faculty of The Gladys W. and David H. Patton College of Education of Ohio University

In partial fulfillment of the requirements for the degree Doctor of Philosophy

Suzanne D. Nichols August 2010 © 2010 Suzanne D. Nichols. All Rights Reserved.

2 This dissertation titled Perceptions and Implementation of the Ohio Academic Content Standards for Mathematics among Middle School Teachers

by SUZANNE D. NICHOLS

has been approved for the Department of Teacher Education and The Gladys W. and David H. Patton College of Education by

George A. Johanson Professor of Educational Studies

Renée A. Middleton Dean, The Gladys W. and David H. Patton College of Education by College of Education

3 ABSTRACT NICHOLS, SUZANNE D., Ph.D., August 2010, Curriculum and Instruction, Mathematics Education Perceptions and Implementation of the Ohio Academic Content Standards for Mathematics among Middle School Teachers (202 pp.) Director of Dissertation: George A. Johanson This dissertation describes findings of a qualitative study using a grounded theory methodology to explore teacher perceptions and implementation of the Ohio Academic Content Standards for Mathematics. Teachers who have knowledge of the Standards and have participated in professional development that builds on that knowledge do not always teach in a way that is indicative of standards-based instruction. This study examines the disconnect between teachers’ espoused beliefs about standards-based instruction and what students eventually experience in their classroom. Classroom practice of twelve teachers was explored through interviews, observations, and surveys of lesson plans and assessments. Not all teachers in this study had a thorough understanding of the Standards. For many, standards-based instruction meant teaching the Standards. The Standards involved mathematics content with little or no attention paid to the mathematical process standards. For many, Standards were a checklist of unitized grade-level indicators teachers were responsible for teaching, and the teachers’ effectiveness in teaching this checklist could be evaluated based on students’ test scores on standardized tests.

4 Teachers’ perception of their role and responsibilities could be categorized into three distinct groups- performance, compliance, and resistance- with each group having a differing perception made up of a compilation of ideas about the Standards, testing, teacher beliefs, and practice. Regardless of teacher perception, teachers’ decisions about classroom practice were purposeful. Teachers taught in a way they believed to be most likely to bring about desired results. Levels of teacher efficacy were associated with their success at achieving those desired results. Foundationally, teachers believed that their responsibility to students was based upon what was fundamental to their job. Some teachers approached teaching mathematics from a school mathematics perspective in which school mathematics was of the utmost importance. Their job was to prepare students to understand mathematics on a deeper, conceptual level in order to build a foundation for the mathematics students would encounter throughout life and, more immediately, throughout future mathematics course. Other teachers approached teaching mathematics from an assessed curriculum perspective in which teaching mathematics standards in order to prepare students to achieve at acceptable levels on high-stakes tests was their job. Approved: _____________________________________________________________ George A. Johanson Professor of Educational Studies

5

This one’s for you, Dad.

6 ACKNOWLEDGMENTS My deepest appreciation goes to my doctoral chair, Dr. George A. Johanson, who believed in me when I had trouble believing in myself. Your encouragement, persistent but subtle reminders to keep moving, and words of wisdom made this dissertation a finished product. As a show of lessons learned, I will tell you, sincerely, I am pleased with this work. Thank you, Dr. J. Thank you, also, to my committee members Dr. Tom Davis, Dr. Dianne Gut, and Dr. Craig Howley. Your wise counsel throughout the study design, my proposal, and the dissertation defense were invaluable. I have learned so much from you and for that I am grateful. I will always owe a debt of gratitude to the ACCLAIM leadership team, Dr. Vena Long, Dr. Bill Bush, Dr. Carl Lee, Dr. Jim Schultz, and Dr. Craig Howley for without them this doctoral degree would not have been possible. These amazing people had vision, imagination, drive, and the unique ability to make rural education at the mall a real experience. Kudos to you all. Thanks to my ACCLAIM cohort members. Rarely a day goes by that I don’t think of one of you and smile. You have changed my life in ways you will never know. Thank you especially to the “Git ‘R Done” cheerleaders who were encouraging me to the end. Most important, I would like to thank my family for supporting me through all these years of course work and then writing. Your understanding and unwavering belief in me encouraged me to continue working toward a dream. To my grandchildren who could not always have my complete attention but could not be put on hold because

7 childhood lasts but a season, Mamaw can go to the park every day now. And, you are only required to call me Dr. Mamaw for a few weeks longer, and of course, on special occasions. And last, but certainly not least, thank you, my dear husband, Roger, for being there through the tears and frustration, for understanding when you spent the evenings sitting on the porch alone, had cold cuts for dinner, and listened to my ranting and raving in moments of frustration. I love you more today than I ever have. And, you, my love, do not ever have to call me Dr. Sue.

8 TABLE OF CONTENTS Page ACKNOWLEDGMENTS .................................................................................................. 6   LIST OF TABLES............................................................................................................ 11   LIST OF FIGURES .......................................................................................................... 12   CHAPTER 1: INTRODUCTION ..................................................................................... 13   Background ................................................................................................................... 17   Statement of the Problem.............................................................................................. 19   Significance................................................................................................................... 20   Limitations and Delimitations of the Study .................................................................. 21   Definitions of Terms ..................................................................................................... 25   Summary ....................................................................................................................... 27   CHAPTER 2: LITERATURE REVIEW .......................................................................... 28   Student Achievement .................................................................................................... 30   Trends in International Mathematics and Science Studies (TIMSS).................................................... 31   National Assessment of Educational Progress (NAEP) ....................................................................... 32   Ohio Achievement Test......................................................................................................................... 33   Summarizing Achievement ................................................................................................................... 34   Standards....................................................................................................................... 34   NCTM Standards.................................................................................................................................. 35   Ohio Standards..................................................................................................................................... 37   Teaching........................................................................................................................ 39   Instructional Pedagogy ........................................................................................................................ 39   Current Classroom Practice ................................................................................................................ 43   Factors that Affect Classroom Practice ............................................................................................... 46   Teacher Beliefs ..................................................................................................................................... 51   Decisions about Practice ..................................................................................................................... 54   Teacher Efficacy................................................................................................................................... 56   Summary ....................................................................................................................... 60   CHAPTER 3: METHODOLOGY .................................................................................... 62   Prior Research Focus Groups as a Backdrop for the Study .......................................... 62   Focus Group Alpha .............................................................................................................................. 63   Focus Group Beta ................................................................................................................................ 64   Focus Group Gamma ........................................................................................................................... 67   Summary of Focus Groups: What Does It Mean? ............................................................................... 71   Qualitative Design Choice ............................................................................................ 73   Grounded Theory .......................................................................................................... 74   The Researcher..................................................................................................................................... 76   Participants .......................................................................................................................................... 77   Data Collection .................................................................................................................................... 80   Data Analysis ....................................................................................................................................... 86  

9 Verification of Interpretation ........................................................................................ 91   Summary ....................................................................................................................... 91   CHAPTER 4: DATA ANALYSIS AND RESULTS ....................................................... 93   Using Observations to Determine Practice ................................................................... 93   Physical Setting/Classroom Environment ............................................................................................ 94   Lesson Overview .................................................................................................................................. 94   Instructional Overview ......................................................................................................................... 95   Questioning .......................................................................................................................................... 96   Classroom Atmosphere ........................................................................................................................ 96   Analysis of Instruction Leading to the Development of Higher Order Skills....................................... 96   Overall Classroom Rating Profile........................................................................................................ 97   Mathematical Processes Benchmarks from Ohio Academic Content Standards: Mathematics.......... 97   Context: Who Are They? .............................................................................................. 98   Meet the Teachers ......................................................................................................... 99   Mr. Allen Anderson .............................................................................................................................. 99   Mr. Ben Brown ................................................................................................................................... 102   Ms. Carla Case................................................................................................................................... 103   Ms. Diane Davis ................................................................................................................................. 105   Ms. Ellen Early................................................................................................................................... 107   Ms. Faye Fout .................................................................................................................................... 109   Ms. Grace Gardner ............................................................................................................................ 110   Ms. Harriet Holmes............................................................................................................................ 113   Ms. Ingrid Ivy ..................................................................................................................................... 114   Ms. Jane Johnson ............................................................................................................................... 115   Ms. Kathy Kale ................................................................................................................................... 117   Ms. Laura Limley ............................................................................................................................... 118   Data Analysis .............................................................................................................. 120   Open Coding ...................................................................................................................................... 120   Axial Coding....................................................................................................................................... 122   A Procedural Aside ............................................................................................................................ 125   Themes........................................................................................................................ 128   Perception .......................................................................................................................................... 128   Purposeful Practice............................................................................................................................ 138   Grounded Theory ........................................................................................................ 140   School Mathematics Perspective........................................................................................................ 141   Assessed Curriculum Perspective ...................................................................................................... 143   Member Checking....................................................................................................... 146   Summary ..................................................................................................................... 146   CHAPTER 5: DISCUSSION AND IMPLICATIONS................................................... 149   Discussion of Research Questions in Light of Themes .............................................. 149   Research Question 1: What are teachers’ perceptions and understandings of the Ohio Academic Content Standards for Mathematics?................................................................................................. 149   Research Question 2: How do teachers translate Standards into classroom practice? .................... 151   Research Question 3: If teachers truly support the Standards and standards-based instruction, to what extent is this evident in their classroom practice? .................................................................... 155   Research Question 4: What conditions influence teachers enacting standards-based instruction in their mathematics classroom?............................................................................................................ 157  

Discussion of Findings................................................................................................ 159   Theme 1: Perception .......................................................................................................................... 159  

10 Theme 2: Purposeful Practice............................................................................................................ 161   Grounded Theory: Perspective .......................................................................................................... 162  

Interpretation of Findings ........................................................................................... 164   They Don’t Know That They Don’t Know.......................................................................................... 164   Teacher Efficacy................................................................................................................................. 165   Implications for Policy and Practice ........................................................................... 168   Recommendations for Future Practice........................................................................ 170   Contribution to Literature ........................................................................................... 173   Recommendations for Future Research ...................................................................... 174   Summary ..................................................................................................................... 175   REFERENCES ............................................................................................................... 178   Appendix A: TIMSS Test Scores ............................................................................... 190   Appendix B: Professional Development..................................................................... 191   Appendix C: Interview Prompts ................................................................................. 194   Appendix D: Mathematics Classroom Observation Instrument ................................ 195   Appendix E: Teacher Information .............................................................................. 198   Appendix F: MCOI Results ........................................................................................ 199   Appendix G: Data Collection Dates ........................................................................... 202  

11 LIST OF TABLES Page Table 1: Characteristics of Teacher Practice…………………………………………….41 Table 2: Key Characteristics of Efficacy Not Conducive to Reform Mathematics……...59 Table 3: Characterization of First Level of Axial Coding to Second…………………..124

12 LIST OF FIGURES Page Figure 1: Example of open coding to categories………...……………………………..126   Figure 2: Compliance descriptor codes……….………………………………………..131   Figure 3: Resistance descriptor codes………………………………………………….135   Figure 4: Performance descriptor codes………..………………………………………137 Figure 5: Relationship between themes and core concepts………….……………..…..148 Figure 6: Conceptual framework relating beliefs, attitudes, intentions, and behaviors………………………………………………………………….…..173

13 CHAPTER 1: INTRODUCTION As a result of the passage of No Child Left Behind legislation (No Child Left Behind Act, 2002), schools are, more than ever, being required to increase student achievement. The administrators and teachers of school districts failing to increase student achievement are being held accountable for lack of acceptable performance. Stakeholders are pressuring public educators to improve student performance (National Mathematics Advisory Panel, 2008). In an effort to fulfill the NCLB mandate requiring all students to achieve, mathematics teachers are being urged to use standards-based instruction in their teaching in order to increase student understanding and performance (Balfanz, MacIver, & Byrnes, 2006). The reputed need for standards-based instruction is not new in mathematics education and has been a suggested part of mathematics reform. While the push for major reform efforts in mathematics education has been ongoing for many years (Brownell, 1935; National Research Council, 1989), efforts have been renewed since 1989 when the National Council of Teachers of Mathematics (NCTM) released the original standards document, Curriculum and Evaluation Standards for School Mathematics. This document was intended to serve as a framework to guide mathematics education reform, and it called for a change in the way mathematics is taught (NCTM, 1989; National Research Council, 1989) by describing curriculum standards and standards concerning the processes of problem solving for mathematics education (Battista, 1994; NCTM, 1989, 2000). Educators in the United States were urged to begin changing their teaching in both content and pedagogy (NCTM, 1989). The foundation of change in content was a change from an emphasis on procedural

14 understanding to conceptual understanding and included “real-world” application of mathematics, using process skills such as reasoning, problem-solving, communication, and connections, and moving away from focusing on arithmetic to include other mathematical topics (Herrera & Owens, 2001). Changes in pedagogy called for studentcentered instruction, which included students “discovering and constructing mathematical relationships, rather than merely memorizing procedures and following them by rote” (Herrera & Owens, 2001, p. 89), using concrete materials to represent mathematics, and teaching in a way that requires students to actively engage in mathematics process skills (Herrera & Owens, 2001). State departments of education became catalysts for change. Ohio developed a state mathematics model in response to the NCTM Standards, and Ohio school districts were urged to rewrite local courses of study in mathematics to more closely resemble the state model. After 10 years of attempting to implement the original NCTM Standards, classroom instruction had changed very little (Balfanz et al., 2006; Hiebert & Stigler, 2004; Jacobs, Hiebert, Givvin, Hollingsworth, & Wearne, 2006; Schoenfeld, 2002). According to the Third International Mathematics and Science Study (TIMSS), overall, students in the United States were still being outperformed by their international counterparts. Fourth-grade students were now performing comparably in mathematics to other countries, but eighth-grade students were still performing below the international average. Students in Grade 12 were near the bottom of the international distribution list (Schmidt, Houang, & Cogan, 2002). Has mathematics reform in the United States been

15 effective? It remains an open question- one that can be answered, perhaps, only by learning about the perceptions of those directly responsible for delivering mathematics instruction. This type of insight can best be elicited using qualitative methods. The 1989 NCTM Standards were never intended as a final product (NCTM 2000). NCTM realized the standards put forth in 1989 served as a working document requiring continual change in order to remain pertinent. The NCTM Board of Directors appointed a multi-varied constituency committee to plan and launch a revision to the original NCTM documents (NCTM, 2000). As a result of the work of this committee, in 2000 NCTM released an updated standards document titled Principles and Standards for School Mathematics (PSSM). Despite the concerted efforts of many classroom teachers, administrators, teacherleaders, curriculum developers, teacher educators, mathematicians, and policymakers, the portrayal of mathematics teaching and learning in Principles and Standards for School Mathematics is not the reality in the vast majority of classrooms, schools, and districts. (NCTM, 2000, p. 5) NCTM’s newest standards document, PSSM, is currently the guiding force in mathematics education being less prescriptive than the previous document and generalizing expectations (or benchmarks) into grade bands rather than by grade-specific indicators. Concern and criticism persists, much of which relates to what has been called a “mile wide - inch deep” curriculum (Schmidt et al., 2002). NCTM, in developing the Principles and Standards for School Mathematics (PSSM), placed great emphasis on mathematical processes (2000). The importance of

16 process skills is evidenced by the fact that each of the process skills is designated a standard just as are the content standards-- Number and Operations, Algebra, Geometry, Measurement, and Data Analysis and Probability. The process standards address Problem Solving, Reasoning and Proof, Communication, Connections, and Representation (NCTM, 2000). Many of the differences in describing a traditional teaching approach as opposed to a standards-based instructional approach are closely linked to the emphasis placed on mathematical processes (Hamilton et al., 2003). In response to PSSM and No Child Left Behind, Ohio, like most other states, developed a new standards document for mathematics. Ohio’s Amended Substitute Senate Bill 1, which went into effect September 11, 2001, mandated that Ohio “develop and adopt clear academic content standards” (Ohio Department of Education, 2001, p. i). Using PSSM as a guideline, a group of mathematics educators from throughout the state met and developed the Ohio Academic Content Standards for Mathematics that Ohio teachers would use to prepare students to attain proficiency under a new accountability plan which included new standardized testing requirements. In December, 2001, the Ohio State Board of Education adopted the Ohio Academic Content Standards for Mathematics (Ohio Department of Education, 2001). Both PSSM and the Ohio Academic Content Standards for Mathematics announce a new paradigm in mathematics education. However, mathematics reform is a slow process. There has, in fact, been little change in classroom practice (Jacobs et al., 2006). While much of the research on theories of learning and mathematics calls for a studentcentered, constructivist approach to teaching mathematics (Bransford, Brown, &

17 Cocking, 2000; Cooney, 1994; Herrera & Owens, 2001), classroom teachers tend to maintain traditional approaches (Jacobs et al., 2006). Jerome Bruner, John Dewey, and Jean Piaget built their learning theories around research based in the field of cognitive psychology. They founded their theories premised on the belief that students learn by creating knowledge for themselves based on a network of ideas already developed and in place by life experience and prior learning (Brooks & Brooks, 1993; Heaton, 2000; Kennedy & Tipps, 1991). NCTM maintains that in order for students to learn, the knowledge they are building should be applicable to their lives or should, at the very least, be of interest to them (NCTM, 2000). On this view, teachers should attempt to supply such relevance by using hands-on or inquiry-based activities. This general theory, along with its classroom application, is commonly referred to as constructivism and seems to be the foundation on which one builds most of the tenets of standards-based instruction (Brooks & Brooks, 1993; Heaton, 2000; Kennedy & Tipps, 1991). Background As part of a research project in which I was involved, I had the opportunity to observe teachers’ practice in mathematics classes at the junior high and high school in a district where standardized mathematics test scores were substantially above the state average while socioeconomic status was very low and the transient rate quite high. Students from this school district were not, by several key indicators for success, expected to perform as well in mathematics as they had performed (Balfanz et al., 2006). In collecting data for this earlier research project, I observed mathematics instruction in

18 the classrooms of mathematics teacher from Grades 7-12. Because of my previous experience as a mathematics teacher and my views about effective instruction, I had preconceived expectations of what I would see in these mathematics classes. Going into classrooms, I expected to see standards-based instruction and was surprised instead to see a traditional method of instruction. As a proponent of standards-based instruction myself, I was astonished that in each classroom I visited students sat in rows and teachers used a classic, traditional, directed instructional teaching style. This did not seem to coincide with research indicators of good practice promoted by NCTM and the Ohio Department of Education. The research experience of observing in classrooms at this particular school for the purposes of the aforementioned study provoked a state of wonder and confusion in my mind. Having personally taught from a constructivist, hands-on, inquiry-based methodology when in the classroom, and believing this to be the best, most effective approach for students, I was philosophically unprepared to accept what I was seeing. According to the interviews I conducted, these teachers were knowledgeable about Standards and grade-level indicators and believed they were delivering a quality mathematics education to their students. Test scores seemed to validate this belief. In trying to make sense of these observations, I considered my personal experience. I began my teaching career using a traditional teacher-directed approach to instruction. As I worked toward a master’s degree and participated in a great deal of professional development, I changed my approach to teaching based on my new-found knowledge of the NCTM Standards. I gradually became a supporter of constructivist

19 mathematics reform as well as a proponent in my school district for standards-based instruction. What seemed to me to be a logical consequence of understanding mathematics reform did not seem to be manifesting itself in the practice of these classroom teachers. I wondered what was going on. Statement of the Problem Mathematics standards are not new. While it is true that there have been changes in the content of the standards throughout the past 20 years, the changes have been of a narrowing sort resulting in more definitive and precisely defined standards. Many teachers, both in-service and pre-service, have had a great deal of interaction with mathematics standards, many teachers profess to have a working knowledge of the standards, and many self-reportedly support the standards and standards-based instruction (Spillane & Zeuli, 1999). Research suggests that while teachers indicate they value standards-based mathematics instruction and use this approach regularly, there is little indication that many mathematics teachers are regularly using standards-based instruction, and, in fact, teach in a traditional manner. There appears to be a disconnect between what teachers say they believe and what teachers actually do in their practice. To understand this disconnect for Ohio teachers, the following questions need answering. 1. What are teachers’ perceptions and understandings of the Ohio Academic Content Standards for Mathematics (Ohio Department of Education, 2001)? 2. How do teachers translate those Standards into classroom practice?

20 3. If teachers truly support the Standards and standards-based instruction, to what extent is this evident in their classroom practice? 4. What conditions influence teachers enacting standards-based instruction in their mathematics classrooms? Significance Influential bodies of mathematics educators assert that mathematics reform is needed (NCTM 1989, 2000; National Research Council, 1989), and they contend that practicing classroom teachers are the ones who are in the best position to bring about that change (Heaton, 2000; NCTM, 1991, 2007; National Research Council, 1989). Thus, understanding mathematics reform from teachers’ perspectives is vital to developing strategies to promote the desired reform. The NCTM Research Advisory Committee (1995) responds to the question, “What is the point of doing mathematics education research?” (p. 302) with the following explanation: People often think the point of research is to answer questions definitively or to find solutions to problems. Rarely does either of these occur as a direct outcome of research. The point of doing research is more often to gain insight into problems, their sources, and their definition, or to open new ways of seeing what is currently taken as simple and obvious. (p. 302) This study is an attempt to understand or gain insight into what impedes mathematics reform on the level of classroom teaching from the perspective of middle school

21 classroom teachers-- an understanding that may be used in the future to foster mathematics reform. Limitations and Delimitations of the Study It is important for a researcher to divulge limitations of the study (Maxwell, 1996; Patton, 2002). A limitation of this study involved participants. While the study called for purposefully chosen participants in an attempt to collect information-rich data, their participation was voluntary, and selection was subject to willingness to participate. While many teachers with reportedly discrepant practice and espoused beliefs were identified, not all were willing to participate in the study. After many unsuccessful attempts, I was able to secure participation from teachers who, while perhaps not the most discrepant, readily took part in the study. Furthermore, accuracy of data collected was vital. It was possible that data obtained from observations did not represent practice that was typical for the participant. The “halo effect” suggests that teachers want to, and often do, try to present themselves in the best possible light or in a manner that they assume the researcher expects (Patton, 2002). In order to minimize this phenomenon, the researcher logged time in the classroom prior to the documented classroom observation and interviews. The amount of time in each class ranged from 9-11 days based on the rate at which students and teachers became accustomed to my presence. Some researchers contend that to a certain degree, all interview and observation data are subject to social desirability bias (Collins, Shattell, & Thomas, 2005). Social desirability is “a bias in self-report instruments created when participants have a tendency

22 to misrepresent their opinions in the direction of answers consistent with prevailing social norms” (Polit & Beck, 2004, p. 732). One strategy for controlling for the effect of social desirability bias was to present the study as an examination of effective practice and the participant as having been identified by professional development providers or curriculum directors as a provider of effective practice. This allowed participants to identify themselves as part of the prevalent culture. “When people become aware of the prevalence, they then believe that if they hold these attitudes or exhibit these behaviors they are part of the majority, rather than part of the minority. This legitimizes their own attitudes and behaviors, and they judge themselves less harshly. Once this happens, people are much more candid in their responses about these attitudes and behaviors.” (Thomas, Grawitch, & Scandell, 2007, p. 2752) With this approach, it appeared that an acceptable level of collegiality was achieved that lessened the effect of social desirability bias but the degree to which it was neutralized remains unknown. Another preventative measure to social desirability was that the researcher did not accept participants with whom there was a pre-existing relationship or knowledge. This lessened the likelihood of the participant attempting to act and respond in a way that they thought the researcher expected. Student behavior during the observations would have indicated a change in what the students considered normal routine and such behavior was documented in field notes. Additionally, lesson plan and assessment documentation, to the extent that it was descriptive, was an indicator of typical classroom procedure. As in any interview, “data

23 from and about humans inevitably represent some degree of perspective rather than absolute truth” (Patton, 2002, p. 569). Teachers’ self-reporting, while not usually intended to be misleading, is not always totally indicative of actual practice (Spillane & Zeuli, 1999). Therefore, teacher reporting was compared to other indicators such as observations, lesson plans, and student behavior. Another limitation of the study pertained to the selected method. Grounded theory is an inductive process and therefore operates from a postpositivist perspective (Patton, 2002). The accuracy of the results was dependent upon the analysis of the data gathered. In addition, it is difficult to separate the data gathered from the context in which it was collected (Maxwell, 1996). According to Denzin and Lincoln (1998): In so far as theory that is developed through this methodology is able to specify consequences and their related conditions, the theorist can claim predictability for it, in the limited sense that if elsewhere approximately similar conditions are obtained, then approximately similar consequences should occur. (p. 169) Transferability, therefore, is limited to the degree of similarity between the context of this study and the context of the study in question, and it is not the researcher’s task “to provide an index of transferability; it is his or her responsibility to provide the data base that makes transferability judgments possible on the part of potential appliers” (Lincoln & Guba, 1995, p. 316). I attempted to provide a thorough description of the context to allow for transferability. Finally, having been a mathematics middle school teacher, I have experiential knowledge of mathematics practice. I have formed an opinion of what I consider to be

24 effective mathematics teaching, and because of this prior experience, I took precautions to guard against bias in interpretation of data. I continued to strive for neutrality throughout the study. Patton (2002) cautions: The neutral investigator enters the research arena with no ax to grind, no theory to prove... and no predetermined results to support. Rather, the investigator’s commitment is to understand the world as it unfolds, be true to complexities and multiple perspectives as they emerge, and be balanced in reporting both confirmatory and disconfirming evidence with regard to any conclusions offered. (p. 51) While I do have a preconceived notion of good practice, I recognize my perspectives as unique in philosophical underpinnings and contextually grounded and do not have firm preconceived notions as to what other teachers’ reflect upon when they plan lessons or to how they determine what mathematics instruction in their classroom will be. I know how I planned and prepared to teach mathematics. I resisted allowing my experiences to influence my interpretations and analysis of the study. It is difficult to know what other teachers reflect upon when they plan lessons or how they determine what instructional methods to use in their classroom. To deal with this bias, interviews were transcribed verbatim (Maxell, 1996). I used an observational protocol instrument for observations, and I used triangulation, all of which are discussed in more detail in chapter 3.

25 Definitions of Terms Standards-based mathematics instruction refers to practices that are consistent with content and process standards and guidelines published by the American Association for the Advancement of Science (1993) and the National Council of Teachers of Mathematics (2000). Common to all these documents is an emphasis on instruction that engages students as active participants in their own learning and enhances the development of complex cognitive skills and processes. Specific practices often associated with this approach include cooperative learning groups, inquiry-based activities, use of materials and manipulatives, and open-ended assessment techniques. All of these practices support active rather than passive learning, promote the application of critical thinking skills, and provide opportunities to apply mathematics and science learning to real-world contexts. In a comprehensive literature review synthesizing research conducted from 19932000 on reform in mathematics education, Ross, McDougall, and Hogaboam (2002) developed a list of 10 chief characteristics of classroom practice one would observe in a classroom where the teacher was attentive to mathematics education reform. The following list of characteristics has been paraphrased from page 125 of Ross, McDougall, and Hogaboam’s article: 1. a broader scope of mathematical content (as opposed to the over-emphasis on numbers and operations) 2. all students have access to mathematics (as opposed to teaching only basic mathematics to students demonstrating a lesser ability)

26 3. tasks are usually open-ended real life problems which sometimes have more than one answer and do not always contain all the information students need to solve the problem (as opposed to decontextualized tasks with one correct answer and all necessary information included in the problems) 4. instruction focused on building understanding of mathematical concepts through student discourse (as opposed to teaching content by the more traditional approach of presenting the material, practicing the skill, giving feedback, and remediating if necessary) 5. teacher taking the role of co-learner (as opposed to dispenser of knowledge) 6. mathematics is taught using manipulatives, calculators, and computers as teaching aids (as opposed to no teaching aids) 7. student discourse seen as a key learning mechanism (as opposed to being seen as a distraction) 8. authentic assessment (as opposed to formal end of chapter or unit tests) 9. mathematics as dynamic (as opposed to static) 10. increased importance of development of self-confidence (as opposed to correct mathematics These characteristics will be used to differentiate between standards-based and traditional instruction. Teacher efficacy is broadly defined by Wheatley (2005) as “teachers’ belief in their ability to influence valued student outcomes” (p. 748). Guskey and Pessaro (1994) went a bit further. “ In general, efficacy is perceived as teachers’ belief [sic] or conviction

27 [sic] that they can influence how well students learn, even those who may be considered difficult or unmotivated” (p. 628). Traditional instruction, according to Goldsmith and Mark (1999), is instruction that “focuses on memorization, rote learning, and the application of facts and [algorithmic] procedures” (p. 40); it is teacher-centered and uses drill and practice and direct instruction as the primary teaching formats. Summary Mathematics reform is ultimately a function of teachers’ practice (Battista, 1994; Swanson & Stevenson, 2002). Overall, there has been little evidence of widespread change in teacher practice even with a continuing emphasis on Standards and standardsbased instruction through pre-service preparation and in-service professional development. Understanding teachers’ perspectives and thought processes as they plan and think about what they will do in their classrooms offers promise for gaining insight into the process of reform in school mathematics. This study attempts to develop this understanding.

28 CHAPTER 2: LITERATURE REVIEW Education reform, and mathematics education reform in particular, has been a leading concern in education for decades. The past 25 years especially have shown an increased focus on school mathematics reform due in part to stakeholders’ lack of satisfaction with student achievement (National Research Council, 1989). To address persistent achievement concerns, those in the field of mathematics education developed and disseminated standards to guide mathematics educators in their decisions on what and how to teach mathematics. The mathematics principles and standards introduced and presented to the general public demonstrated a shift in the emphasis on the mathematics necessary to prepare students to be successful in an ever-changing world and were presented as an integral part of bringing about the desired changes in mathematics education. The majority of the suggested changes involve classroom practice. The demand for teachers to transform classroom instruction is argued as essential to the success of mathematics reform (Battista, 1994; Finley, 2000; Herrera & Owens, 2001). Teachers acknowledge the need to change their practice (Hiebert & Stigler, 2000). We are seeing changes in classrooms, but the changes are not always consistent with the vision set forth by NCTM in the 2000 Principles and Standards for School Mathematics, nor are the changes common practice. Many teachers still use a traditional approach to instruction in their classrooms (Hiebert & Stigler, 2000). Many teachers who profess to know about mathematics principles and standards, mathematics reform, and the shifts in emphasis in instruction, even while touting reform, do not significantly change their classroom practice (Hiebert & Stigler, 2000).

29 This literature review examines three bodies of literature that help to frame and illuminate the discrepancy between teachers’ avowal and implementation of standardsbased practice. The first section explores student achievement in mathematics-internationally, nationally, and on the state level-- as a catalyst for mathematics reform. The second section of the literature review gives a brief history and overview of the development and purpose of mathematics standards that teachers are asked to implement in classrooms. The last section, by far the largest due to the multi-faceted nature of instruction, examines teaching as a practice. Teaching as a practice is subdivided into six categories and draws on literature concerning instructional pedagogy, current classroom practice, factors affecting classroom practice, teacher beliefs, decisions about practice, and teacher efficacy. The first three of these categories provides a basis for understanding what one might see in the classroom. The differences between traditional and standards-based pedagogy are explored, and the current state of classroom practice is conceptualized while taking into account research indicating factors that affect classroom practice. Together, these three topics prepare the researcher for positioning this study in classroom practice. This literature describes what teachers actually do in their classrooms. The remaining three categories (teacher beliefs, decisions about practice, and teacher efficacy) deal with underlying, non-observable behaviors that are key to understanding any discrepancy between what teachers say they do and what actually occurs in their classrooms.

30 Without qualitative inquiry, it is almost impossible to learn about teacher beliefs, decisions about practice, and teacher efficacy (Patton, 1987). While there is a solid literature base dealing with teacher beliefs, most research relies upon teacher questionnaires and surveys, which limit one’s ability to contextualize data (Patton, 1987). Data about teacher beliefs and practice analyzed during Trends in International Mathematics and Science Studies (TIMSS) data collection cycles is a result of survey data. Surveys provide a way to collect a great deal of data from many different sources. The problem with surveys is that many use Likert scales, the most common measurement for teacher responses on surveys and questionnaires that do not allow responders to qualify their answers with additional information (Wheatley, 2005). Likert scales typically allow respondents to indicate their level of agreement with specific statements on a scale with varying incremental designations ranging from indicators of strongly disagree to strongly agree. There have been fewer studies and research on decisions about practice and teacher efficacy. Like teacher beliefs, effective research in these two areas lends itself to qualitative approaches. Student Achievement The interest in mathematics reform is motivated by a perceived need. The need is not new and has been an ongoing concern for many years. Students are not attaining what policymakers would consider to be an acceptable level of achievement in mathematics (NRC, 1989). The United States is, arguably, the most advanced country in the world, and policy-makers apparently believe student achievement in mathematics should reflect that status (NRC, 1989).

31 On October 4, 1957, the Soviet Union launched Sputnik 1, the first man-made satellite to orbit the earth. Americans, who had settled comfortably into their illusion of international intellectual superiority, found this turn of events unacceptable. Roger D. Launius (2006), a curator at the National Air and Space museum at the Smithsonian Institute, reports that on the evening of October 4, 1957, Lyndon B. Johnson, then the Senate Majority Leader, considered the events of the day and shared his “profound shock of realizing that it might be possible for another nation to achieve technological superiority over this great country of ours” (p. 6). The United States began planning to regain their status as world leader. The call for science and mathematics reform in the United States was reinvigorated, and a federally funded top-down approach to reform requiring major changes (mostly in curriculum) was suggested without input from teachers or school communities (Finley, 2000; Schoenfeld, 2004). This was the launch of the current round of emphasis on mathematics reform. Trends in International Mathematics and Science Studies (TIMSS) Since Sputnik, students in the United States have been compared internationally to other students and have not performed as well as their international counterparts (Findell, 1996). The results of the Third International Mathematics and Science Study were made available in a 1999 report from the National Center for Education Statistics (NCES). Students from the United States in 4th grade, with an average score of 545, scored above the international average of 529 while ranking 12th among participating countries (NCES, 1999). At the eighth-grade level, United States students scored an average of 500, which was below the international average score of 513 (NCES, 1999).

32 United States students completing their final year of secondary school, 12th grade, scored 461. This score was well below the international average of 500 (NCES, 1999). In 2003, the results of a more recent international comparison study, the Trends in International Mathematics and Science Study (TIMSS), were published. The study included only fourth- and eighth-grade students. Fourth-grade students from the United States again ranked 12th, but this time with an average score of 518, continuing to score above the international average of 495. Eighth-grade students showed improvement, scoring 504, which placed them above the international average of 466 but still ranking in 15th place internationally (NCES, 2004). These results, although showing improvement in scoring averages over previous studies, still indicate the United States trailing many of our international counterparts.1 National Assessment of Educational Progress (NAEP) Nationally, many states in cooperation with the federal government have, through their individual state departments of education, mandated accountability measures for local school districts for many years. State testing, however, is neither uniform nor does it allow for comparison to other states. Test scores can not be compared to determine how well students are achieving as these tests often do not measure the same content. One indicator of student achievement to this end is the National Assessment of Educational Progress (NAEP) scores. NAEP, or what is known as The Nation’s Report Card, is a project mandated by congressional legislation. NCES, which operates within the United States Department of Education, oversees the project. NAEP tests are administered to

1

See Appendix A for mean scores as well as standard deviation of mean scores for a sample of selected countries participating in TIMSS testing.

33 students in randomly selected schools throughout the United States. According to NCES (2005), over a period of 31 years beginning in 1973 and continuing through 2004, mathematics trend scores for students ages 9 and 13 have continued to increase over the 31 year period (219-241), whereas scores for 17-year-old students, although showing some fluctuation, have remained basically the same (304-307) as they were in 1973. Ohio Achievement Test The No Child Left Behind legislation required state level accountability assessments in order to maintain federal government funding (NCLB, 2001). Individual states, in an attempt to meet the requirements of NCLB, have developed statewide assessments addressing the accountability mandates. The Ohio Achievement Test, the OAT2, includes a mathematics assessment that is administered to students in grades 3 through 8. At the high school level, students are required to take and pass the Ohio Graduation Test (OGT), a high-stakes test that also includes a mathematics assessment. The extent to which Ohio teachers buy in to these and other objectives is uncertain. Statewide mathematics assessment results for Ohio indicate that students on the fourth-grade level have shown a slight decline in achievement since the inception of the current test in 2006. The average percentages of fourth-grade students scoring at or above proficient throughout the state for the three school years 2006-2008 are respectively 77%, 76%, and 74%, indicating a slight decline (ODE, 2008). In eighth grade, the mathematics portion of the OAT was first administered in 2005. For the years 2005-2008, the average percentage of eighth-grade students scoring 2

Effective in 2010, the Ohio Achievement Test (OAT) became known as Ohio Achievement Assessment. Only the name changed. Content remained the same. For purposes of this study, the test will be referred to as the OAT since the data was collected prior to the name change.

34 at or above proficient was respectively 60%, 68%, 71%, and 73%. These percentages indicate a significant jump followed by a steady increase in the average test scores (ODE, 2008). The high school graduation test, the OGT, was first administered in 2004. The average percentage of students scoring at or above proficient on this test, much like eighth-grade, showed a significant jump but then fluctuated over the next four years. For the years 2004 through 2008, the percentage of students passing the OGT has been 67%, 80%, 82 %, 81 %, and 79 % (ODE, 2008). Summarizing Achievement With the exception of secondary schools, where students’ long term test scores seem to have remained somewhat flat, these assessments seem to indicate that students are beginning to show improved achievement in meeting the new demands of mathematics. This improvement may be due in part to a focused mathematics curriculum aligning with the assessments being given. There was a time when every teacher, in every grade-level mathematics department, in every school district, in every state of the United States determined their own mathematics curriculum (National Research Council, 1989). This is no longer the case. Those in the mathematics field have determined a need for guidelines for mathematics education in the United States. Guidelines have been established in the form of standards. Standards Standards are “value judgments about what our students should know and be able to do” (Hiebert, 1999, p. 4). Prior to 1989, there were no mathematics standards in the

35 United States nor was there a mandated, uniform, official curriculum for school districts to follow. This does not mean that teachers had no guidance on what mathematics to teach. In reality, the United States had an unofficial national curriculum dictated by textbook companies and anonymous officials who selected which standardized tests to use in classrooms (National Research Council, 1989; Schoenfeld, 2004). The curriculum was said to be underachieving with arguably needless repetition year after year (National Research Council, 1989). This unofficial curriculum allegedly did not meet the changing needs of students in the United States (National Research Council, 1989). NCTM Standards In taking a leadership role in mathematics, the National Council of Teachers of Mathematics created the Curriculum and Evaluation Standards for School Mathematics in response to the call for reform in mathematics education (NCTM, 1989). The original NCTM (1989) mathematics standards were written to serve as a framework to guide reform in school mathematics. This renewed focus on reform called for fundamental change in two areas. The first area of recommended change was in content (NCTM 1991). The study of mathematics has traditionally emphasized computation as the most important component of mathematics, but now emphasis has shifted to problem solving and conceptual understanding. The NCTM Curriculum and Evaluation Standards (1989) urged a change in content priority and emphasis. The second area of suggested change is in pedagogy and how one views mathematics teaching and learning (NCTM, 1991). Because of the historical importance of computational skills, many teachers viewed teaching and learning from a behaviorist perspective, that of continuous practice. With a

36 new focus on conceptual understanding and problem solving, educators are urged to shift their approach to teaching and learning away from behaviorism to constructivism (Battista, 1994; Schoenfeld, 2004) and a more student-centered pedagogy emphasizing mathematical process skills. The 1989 Standards were organized into three grade bands-– K-4, 5-8, and 9-12. Each grade band had the first four standards in common. The common standards, focused on processing mathematics rather than with mathematical content, are mathematics as problem solving, mathematics as communication, mathematics as reasoning, and mathematics as connections (NCTM, 1989). The remaining standards on each grade level deal with the differing mathematics content deemed appropriate for that grade level (NCTM, 1989). The 1989 NCTM document was just the beginning of a process of change (NCTM, 2000). Curriculum and Evaluation Standards for School Mathematics (1989) along with Professional Standards for Teaching Mathematics (1991), and Assessment Standards for School Mathematics (1995) “represented an historically important first attempt by a professional organization to develop and articulate explicit and extensive goals for teachers and policymakers” (NCTM, 2000, p. ix). This trio of documents became the basis of the standards movement. In 1995, as a next step, NCTM organized and set into motion another project, Standards 2000, with the goal of building upon and strengthening their previous work (NCTM, 2000). The primary product of this project was Principles and Standards for School Mathematics (PSSM) which was released in 2000.

37 Principles and Standards is… a tool for better understanding the issues and challenges involved in improving mathematics education. It offers information and ideas that those with responsibility for mathematics education—whether the local, state or provincial, or national level—need in order to engage in constructive dialogues about mathematics teaching, curricula, and assessment. (NCTM, 2000, p. 380) Many mathematics educators currently recognize PSSM both as a guide and a standards document developed on a national level (even though the United States does not have an official national curriculum). As such, most states, including Ohio, have aligned their state standards documents with NCTM’s PSSM (Herrera & Owens, 2001; Ohio Department of Education, 2001). Ohio Standards In 1997, the state of Ohio, based on legislation from Amended Substitute Senate Bill 1 mandated a timeline for developing and implementing academic content standards. A joint council consisting of members from the State Board of Education and the Ohio Board of Regents established advisory groups to plan for development of standards. The mathematics content standards were prepared for adoption in the target year of 2001. Using the PSSM of NCTM as a framework, writing teams for the mathematics standards worked in grade-level groups creating the draft document. The Ohio Academic Content Standards for Mathematics document consists of six standards: Number, Number Sense and Operations Standard; Measurement Standard; Geometry and Spatial Sense

38 Standard; Patterns, Functions and Algebra Standard; Data Analysis and Probability Standard; and Mathematical Processes Standard. Benchmarks, or key checkpoints, were delineated by grade bands and clustered into K-2, 3-4, 5-7, 8-10, and 11-12 groups. The writing teams, 50% of which were classroom teachers (ODE, 2001), dissected the benchmarks into grade-level indicators across grade bands. Grade-level indicators are specific statements regarding what students should learn on each grade level (ODE, 2001). The process standards, however, are not grade-level specific. The mathematical processes include problem solving, reasoning, communication, representation, and connections. Writing teams attempted to incorporate mathematical processes into the writing of the content specific grade-level indicators in order to emphasize for teachers the importance of incorporating the mathematical processes into instruction on a daily basis as “good instruction consists of teaching mathematical content through mathematical processes” (ODE, 2001 p. 194). Once the draft document of the Ohio Academic Content Standards: K-12 Mathematics was completed, Ohio stakeholders were given an opportunity to review the work. Teachers, administrators, mathematicians, and the public reviewed the document. This generated thousands of responses that were considered, and changes were implemented. Ohio met the state’s legislative mandate for development of academic content standards in mathematics on December 11, 2001, with the adoption of the Ohio Academic Content Standards: K-12 Mathematics (Ohio Department of Education, 2001). This document defined the state’s official standards and was “intended to provide Ohio educators with a set of common expectations from which to base mathematics

39 curriculum” (Ohio Department of Education, 2001, p. 24). With standards in place, what the state says teachers in Ohio are supposed to teach is no longer in question. Teaching a standards-based curriculum, however, is not a trivial matter. Teaching The body of literature dealing with teaching is vast, and quite possibly, unending. In considering the research interest for this study, the researcher attempted to contain the magnitude of the literature by considering a logical progression of thought one might follow in contemplating the research questions. Instructional Pedagogy The practice of teaching is a compilation of numerous strategies and tactics for instruction. Characterizing a teacher’s specific teaching practice is usually difficult (even impossible) as very few teachers follow an immutable, clearly definitive procedure day after day (Cuban, 1983). Typically, even the staunchest practitioner will occasionally deviate from the familiar routine or pedagogy. It is more compelling, therefore, to describe characteristics of what one does in the classroom while teaching mathematics when categorizing instructional pedagogy. For the purposes of this study, I will categorize instruction as either standards-based or traditional, based on a preponderance of characteristics of practice. These two categories of instruction differ in that the role of most active participant in the classroom is placed with either the students or the teacher. In other words, the roles and responsibilities of teachers and students within the learning process change with a switch in instructional practice. Standards-based instruction, the goal of

40 mathematics reform, is claimed as student-centered while traditional instruction is claimed as teacher-centered. The applicable standards allegedly promote teachers moving from teacher-centered instruction to student-centered instruction. The move from teacher-centered instruction to student-centered instruction is described by certain constructs or what instruction looks like in the classroom. While there is no agreed upon, all-inclusive definitive description of what is standards-based instruction, certain characteristics are prominent features of this type of instruction. Likewise, traditional instruction is depicted by certain practices. In Table 1, Cuban (1983) described both types of instruction in observable measures.

41

Table 1. Characteristics of Teacher Practice Teacher-centered instruction

Student-centered instruction

Far more teacher talk than student talk

Student talk on learning tasks is at least equal

during instruction

to, if not more than, teacher talk

Most teacher questions call for reciting

Students ask questions as much as, if not more

factual information

than, the teacher

Most instruction occurs with whole

Most instruction occurs individually, in small

group rather than small groups or with

or moderately-sized groups, rather than whole

individuals

class

Teacher determines use of class time

Students help choose and organize the content

Teachers often rely upon textbooks

Varied instructional materials are available so

with lesser use of films, tapes, records,

students can use them independently or in

television, or other technology

small groups and use of materials can be determined by students or teacher

Tests usually concentrate on factual

Assessment is often open-ended or

recall of information

contextualized in real world applications

The classroom is usually arranged into

Classroom is usually arranged in a manner that

rows of seats facing a blackboard with

permits students to work together; no dominant

a teacher’s desk nearby

pattern exists

Note: The information contained in this table was excerpted from p. 160 of Cuban, L. (1983). How did teachers teach, 1890-1980? Theory into Practice, 22(3), 159-165. Side by Side Comparison of Teacher-centered and Student-centered Practices

42 Teacher-centered instruction began during a time when there was little printed material available and knowledge was considered to be purely objective. Nearly all schooling can be traced back to religious establishments, when teaching consisted of one who was knowledgeable with access to scarce printed materials bestowing their knowledge upon their passive students (Cohen, 1988). This view of education, teaching by telling, has not ceased. Many people within the general public, as well as some mathematics educators, believe the most effective approach to teaching mathematics is by teaching students how to follow rules and procedures (Battista, 1994; Roehrig & Kruse, 2005). The practice of emphasizing rules and procedures with little or no attention paid to developing conceptual understanding is the characteristic most associated with traditional teaching of mathematics (Hiebert, 1999). Research into the effectiveness of classroom practice is conflicting in many cases. Determining which teaching strategies and tactics best promote student learning is a controversial subject that mathematics educators have in the past, and will in the future, continue to debate (Franke, Kazemi, & Battery, 2007). However, studies dealing with student learning have identified deficiencies in traditional instructional practices associated with a behaviorist approach to teaching (Battista, 1994; Hiebert, 1999). Traditionally, the focus of mathematics has been on proficiency in computational procedures (Hiebert, 1999). With that outcome in mind, a behaviorist approach to teaching was appropriate as the focus was on observable behaviors rather than mathematical thinking (Battista, 1994). Battista (1994) expressed concern that at a time when learning mathematics is argued as essential to the nation, schooling persists in using

43 ineffective and obsolete teaching methods. Indeed, Astleitner (2005) argues that the preponderance of empirical research suggests that teachers must teach in a way that produces cognitive effects. Current Classroom Practice Most mathematics classroom teachers do not teach in a way that is considered conducive to mathematics reform (Hiebert, 1999). “Psychological and educational research on learning of complex subjects such as mathematics has solidly established the important role of conceptual understanding in the knowledge and activity of persons who are proficient” (NCTM, 2000, p. 20). According to this view, when students are learning mathematics in a traditional classroom, procedural proficiency often takes precedence over conceptual understanding. The TIMSS video study showed that in classrooms in the United States students spent 96% of the time they were engaged in seatwork doing procedural computation (Stigler & Hiebert, 1997). In Everybody Counts: A Report to the Nation on the Future of Mathematics Education, the National Research Council (1989) reported on what they considered to be the preferred mode of instruction: Research in learning shows that students actually construct their own understanding based on new experiences that enlarge the intellectual framework in which ideas can be created….Much of the failure in school mathematics is due to a tradition of teaching that is inappropriate to the way most students learn. (p. 5)

44 In spite of the Everybody Counts report, and ongoing calls for mathematics reform, current classroom practice has changed very little. Cuban (1983) attempted to determine how teachers taught from the early 1900s to 1980. He gathered data from a variety of sources including photographs of classrooms, textbooks, tests, recollections of students, reports by teachers and others visiting classrooms, student writing, research studies, and various other sources of information about classroom design. He discovered that, for the most part, teachers maintained teacher-centered instruction except for occasional periods of various school-adopted reforms (Cuban, 1983). Even though there was some deviation in elementary schools, this trend to maintain the status quo remains true, especially on the high school level (Cuban, 1983). While not all studies or researchers agree, many believe that student-centered instruction is the most appropriate pedagogical approach for mathematics teaching and learning. In addition, state mathematics standards call for mathematics instruction that focuses on mathematical process skills. Some observers assert that implementation of process skills into instructional practice, by nature, requires a student-centered focus. Research into the effectiveness of the instructional practices associated with the Standards gives only limited insight. It is impossible to assess the effectiveness of the Standards on student achievement without knowing what teachers are doing in the classroom (Hiebert, 1999). The effectiveness of Standards is susceptible to the nature of current classroom practice. Hiebert and Stigler (2000) reported on a TIMSS video study conducted during the 1994-1995 school year. The study consisted of videotaped lessons of 231 mathematics

45 lessons–each in a different classroom. Of the 231 lessons, 81were taught by teachers in the United States. Of these, 95% of the teachers reported being “somewhat or very aware of current ideas about teaching mathematics” (p. 5) which includes the ideals of mathematics reform. Additionally, 70% of the videotaped teachers from the United States reported that their recorded lesson was in keeping with current ideas in mathematics when, in reality, analysis suggested there was little evidence to support teachers’ assertions of reformed mathematics teaching (Hiebert & Stigler, 2000). In a study conducted in Michigan, researchers investigated classroom practice of self-reporting teachers using standards-based reform practices in Grades 3-8. Teachers completed the TIMSS teacher questionnaire about their instructional practices and questionnaires were scored based on the degree to which reported teacher practice matched the tenets of what NCTM standards, as well as state standards, identified as the goals and mission of mathematics instruction. The 25 teachers scoring in the top 10% were chosen for closer study. After observing in the classrooms of these teachers, the researchers determined that only 4 of the 25 teachers were really teaching in a way that promoted true integration of national or state standards (Spillane & Zeuli, 1999). A preponderance of research suggests that many teachers, even those who believe they are using standards-based instruction in their classrooms, adopt some aspects of reform such as cooperative grouping or use of manipulatives, but are less likely to make changes involving pedagogy such as discourse practices and requiring students to reason and justify their work (Spillane & Zeuli, 1999).

46 Basically, teachers primarily teach in a traditional manner consisting of lectures, note-taking, and practice (Manouchehri, 2003; NRC, 1989). While this method may be considered effective for test-taking on standardized tests, it reportedly does little to promote understanding and long-term learning (NRC, 1989). Many observers claim that teachers usually teach as they were taught (Battista, 1994; Cuban, 1983; NRC, 1989; Thomas & Monroe, 2006) and, that even when motivated to reform classroom practice to standards-based instruction, may find it difficult to locate a colleague upon which to model instructional changes (Thomas & Monroe, 2006). School environments promote this traditional approach to teaching mathematics (Ross et al., 2002). Many parents, policymakers, school administrators, and testing program coordinators view traditional instruction as effective and prefer it to other less familiar pedagogy (Lubienski, 2002; Ross et al., 2002). Factors that Affect Classroom Practice Researchers have documented variability in classroom practice. They have also attempted to document various factors affecting and determining classroom practice. This task has proven to be quite difficult as there are many circumstances contributing to the end result of what one observes in a classroom. Research into such circumstances affecting classroom practice is often based upon survey instrumentation and observation, resulting in limited access to teacher interpretation and intention as a contribution to this body of research. In 2002, Ross, McDougall, and Hogaboam-Gray reported their review of studies conducted between 1993 and 2000, which offered empirical evidence of

47 mathematics reform. Ross et al. (2002) identified the following paraphrased commonly cited barriers to reform: 1. Teachers are being asked to use a pedagogy unfamiliar to them and much more difficult to learn; 2. Teachers lack the mathematical content knowledge necessary to effectively use rich mathematical problems to present content; 3. Textbooks can not provide support for unanticipated occurrences in classroom instruction using student-centered instruction; 4. Teachers feel less efficacious using strategies that put students in control of learning; 5. Teacher beliefs about mathematics often conflict with mathematics reform; 6. Parent expectations of what constitutes effective mathematics instruction often do not coincide with reform-style mathematics instruction; 7. Mandated assessments often place more emphasis on computational speed and accuracy than on conceptual understanding of mathematics; 8. Mathematics reform practices and content require a greater time commitment for teachers in their actual presentation of material as well as in their preparation for classroom instruction. The most common barriers included pedagogical concerns. Teachers are asked to teach mathematics in a way that is unfamiliar to them (Anderson & Piazza, 1996). Teachers’ beliefs about instruction provide the basis for many of their pedagogical

48 decisions. Those beliefs are based on their apprenticeship of observation (Lortie, 1975) which is the compilation of experiences teachers have had as students of mathematics in kindergarten through their teacher preparation programs (Lortie, 1975). Research has established that teachers teach much in the same way they were taught, and therefore, it is not surprising to note that one barrier to reform is teachers’ perception that they are being put in the position of agents of change for a teaching pedagogy they did not experience as students (Ross et al., 2002). Their apprenticeship of observation has not prepared them to teach reform mathematics (Anderson & Piazza, 1996; Laurenson, 1995). Teachers’ knowledge of how students learn seemingly affects their classroom practice. Traditional instruction stresses teaching students to systematically follow a set of rules or procedures. There may be no emphasis placed on the need to understand how students think or process information (Battista, 1994). However, the recommended pedagogy of mathematics reform teaching requires a deeper and more thorough understanding of, not only the mathematics content, but of how to design most effectively, engage students in, and maintain the rigor of worthwhile mathematical tasks (Henningsen & Stein, 1997). According to NCTM (1991), for instance, Teaching is a complex practice and hence not reducible to recipes or prescriptions…. Teaching mathematics draws on knowledge from several domains: knowledge of mathematics, of diverse learners, of how students learn mathematics, of the contexts of classroom, school and society. (p. 22) Circumstances that may affect classroom practice are those over which teachers have little or no control. One example would be the emphasis placed on mandated testing

49 programs (Laurenson, 1995; Wood, 2007). In many cases, the intent of state testing as an accountability measure is in direct conflict with mathematics reform practice (Ross et al., 2002). Teachers are reportedly concerned that effectively preparing students for highstakes testing precludes implementation of practices associated with mathematics reform, since student-centered approaches to teaching take more time in the classroom (Ross et al., 2002). Teachers are hesitant to submit to the time commitment that this type of instruction requires when there is so much emphasis placed on preparing students for high-stakes testing whether in relation to district testing regimes, state accountability mandates, federal Adequate Yearly Progress (AYP) requirements, or college entrance examinations. Additionally, administrative factors, such as time constraints, scheduling, and curriculum also affect classroom practice (Laurenson, 1995). Many teachers alter or avoid classroom practices that threaten their control over students (Laurenson, 1995; Wheatley, 2000). Teachers’ willingness to venture into unknown territory such as openended problem solving or the type of mathematics instruction called for with mathematics reform may be stifled by teachers’ lack of confidence in their knowledge of mathematics (Laurenson, 1995). All of these circumstances can reportedly influence classroom practice. In a study of 21 mathematics teachers, Manouchehri (2003) reported participants were knowledgeable and highly supportive of standards-based practice and rated themselves as confident in their ability to implement new strategies in their classrooms. Manouchehri looked for commonalities within this group that might have influenced their

50 disposition to embrace reform-style practice. This researcher noted four commonalities: (a) they were self-assured in their ability to control what and how well students learn and had a detailed view of what teaching and learning would occur in their classrooms; (b) their philosophical view of mathematics education was strongly supportive of mathematics as an apparatus for social change; (c) they envisioned themselves as change agents for education; and (d) they considered themselves to be life-long learners and their teaching as a work in progress ( Manouchehri, 2003). In another study, Raymond (1997) investigated the relationship between beginning elementary teachers’ beliefs and their classroom practice. She concurred with other researchers (Thompson, 1992) that practice is not always consistent with the teachers’ expressed beliefs. Further, she questioned whether the teacher was actually aware of the discrepancy. She wrote of Joanna, the participant highlighted in the study, “It was as if she thought that believing in good mathematics teaching practices was a way of practicing good mathematics teaching” (p. 569). Joanna reported inconsistencies in her classroom practice being influenced by many of the circumstances discussed previously in this literature review: time constraints on instruction, lack of resources, a need to control students’ behavior, and an emphasis placed on standardized testing. In a replication of Raymond’s study, Pittman (2002) researched experienced elementary teachers. She validated many of Raymond’s conclusions. However, one of the biggest differences was that while Raymond found that teachers who hold beliefs about teaching that are traditional and simultaneously have more non-traditional beliefs about mathematics pedagogy were still likely to teach in a way that was more influenced by

51 their traditional beliefs about mathematics than in their non-traditional beliefs about mathematics pedagogy. Their classroom practice, in short, was mostly traditional. By contrast, Pittman found the experienced elementary teachers’ practice to be more nontraditional than their beliefs. Teacher Beliefs Teacher beliefs play a critical role in mathematics education, and Battista (1994) asserts that beliefs directly influence what happens in the classroom. Beliefs affect the way teachers teach, the way they assess students, the way they relate to students, and their dispositions toward and interactions with students (Barkatsas & Malone, 2005). Simply put, it would seem that knowing about a teacher’s beliefs would allow one to predict the teacher’s practice. Teacher beliefs and their effect on practice are, however, anything but simple according to other observers. Green (1971) contended that “nobody holds a belief in total independence of all other beliefs. Beliefs always occur in sets or groups. They take their place always in belief systems” (p. 41). In addition, not only is what one believes important but also how one believes it. In attempting to qualify them, Green identified three dimensions of beliefs. The first dimension of beliefs is logical structure. Beliefs are held in a particular order. When examining one’s belief system, asking why one believes something in particular often leads to the expression of another belief. By continuing this process of asking why, eventually one arrives at a “why do you believe that?” question where the response is a simple reiteration of the belief. “I believe it just because I do.” There is no

52 further or underlying identifiable belief. That terminus of the litany of “why” questions is defined as a primary belief with all the others being held as derivatives of that primary belief. Green refers to this as a quasi-logical structure because it deals with the order of the beliefs in the belief system instead of with their logical relationship to one another (Green, 1971). Green’s second dimension of belief systems is psychological structure. Psychological structure refers to the strength with which one holds a particular belief. Green used the idea of concentric circles to explain psychological structure. The innermost circle contains the belief or beliefs held most strongly and are referred to as central beliefs. Central beliefs are accepted without question, held most dear, and are not easily debated. Each circle moving outward from the center represents beliefs held less strongly. These beliefs are peripheral and more susceptible to change (Green, 1971). Finally, a third dimension of belief systems as defined by Green is clustering. Clustering is grouping beliefs into groups that are disjunctive from other clusters of beliefs (Green, 1971). These beliefs can sometimes be conflicting, and clustering allows one to hold beliefs separate that are in direct contradiction with one another. There is typically a corresponding belief associated with this contradiction that allows one to continue to hold both beliefs even though they are contradictory (Green, 1971). Much of the literature on reform-based practice calls for change in teacher beliefs. The ability to change a teacher’s beliefs is dependent in part upon how the beliefs are held. Beliefs may be evidential, based upon evidence, or they can be non-evidential, based on other beliefs. “In other words, a person may hold a belief because it is supported

53 by the evidence, or he may accept the evidence because it happens to support a belief he already holds” (Green, 1971, p. 49). Non-evidentially held beliefs are more resistant to change than evidentially held beliefs because a belief based upon evidence can change when proof is presented that negates currently held evidence (Green, 1971). Teacher beliefs about mathematics can be divided into at least two distinct structures – beliefs about mathematics as a discipline and beliefs about mathematics teaching and learning. An important finding in a study by Barkatsas and Malone (2005) is that a teacher’s “prior school experiences and personal world-views, and ideologies were the main influence on the teacher’s beliefs about mathematics, but her own school experiences and her teaching experiences were the main influences on beliefs about teaching, learning and assessing mathematics” (p. 86). In other words, a teacher’s experiences in the classroom are the most influential factor in affecting classroom practice. This study is important in that it may indicate, like Green’s third dimension of belief systems, teachers may have clustered their beliefs about mathematics and their beliefs associated with teaching, learning and assessing mathematics in two separate and contradictory clusters. In order for teachers to work willingly toward changing instructional practice to a more standards-based approach, teacher beliefs, according to some observers, must be consistent with values of mathematics curriculum reform (Battista, 1994; Roehrig & Kruse, 2005). In a small study intended to assess how the use of reform-based curriculum materials altered the classroom practice of science teachers, Roehrig and Kruse (2005) found that teaching beliefs significantly impact classroom practice. The study revealed

54 the higher the levels of reform-based beliefs, the more the teachers exhibited reformbased classroom practice. Likewise, there was little change in the classroom practices of teachers holding traditional beliefs about instruction. Several studies on teacher beliefs look at the disconnect between espoused beliefs and classroom practice. One such study consisted mostly of survey questions using Likert-scale responses (Barkatsas & Malone, 2005). Analysis of the survey responses allowed the researchers to characterize secondary mathematics teachers based on their response data but it did not allow the researchers to query participants about their practice and the consistencies of practice and beliefs. In their ancillary case study of one teacher, however, Barkatsas and Malone probed for explanations behind evident discrepancies of beliefs and practice. According to the teacher’s interview, most discrepancies were related to classroom situations over which the teacher felt she had no control, her own prior experiences, and the social norms and expectations of her students and their families. A similar study by Raymond (1997) conducted at the elementary level resulted in explanations very similar to those uncovered by Barkatsas and Malone. Decisions about Practice Instructional planning is another reportedly important influence on classroom practice. What happens in the classroom is partly a result of how and what teachers plan (Little, 2003). The amount of time teachers spend planning also varies. Unlike novice teachers, teachers with more experience spend less time planning and tend to make many instructional decisions as the need arises during the course of the lesson. Teachers tend to

55 plan their lessons to the point at which they feel comfortably prepared to react on instinct and experience (Zimmerlin & Nelson, 2000). In a study comparing the planning process of fifth- and sixth-grade teachers in the United States and Japan, participants prepared and taught a lesson on finding the area of a triangle. Participating teachers were interviewed within both 24 hours before teaching the lesson and then after having taught it. Both groups of teachers reported similar issues while planning, with an emphasis on the mathematics. The biggest differences reported between the two groups was that teachers from Japan, unlike their American counterparts, spent a great deal more time considering how to promote student engagement and how to develop a more positive attitude toward mathematics learning (Fernandez & Cannon, 2005). Another study of 45 teachers in 16 schools in California, Michigan, North Carolina, and Vermont was undertaken in an attempt to understand why teachers fail to faithfully embrace the fundamental changes associated with mathematics reform. Kennedy (2004) looked at why teachers engaged in specific practices rather than trying to discern why they did not engage in other specific reform practices. Kennedy attempted to identify teachers’ concerns in specific classroom scenarios. Teachers, as well as researchers, watched a videotape of a previously recorded lesson of teachers teaching a lesson of their own choosing. During the viewing, teachers noted specific events during the lesson that they considered important or interesting. Those selections were later discussed in more detail. Two general patterns emerged. The first was that teachers’ thinking followed a similar line of thought whereby discussions started with teachers

56 telling either what they intended to do or giving their interpretation of what happened in the video. The second emergent pattern was that teachers usually voiced multiple intentions for a specific segment of their videotaped lesson. It seemed that perhaps teachers’ intentions were somewhat compatible with reform ideals while taking on even greater concerns such as long held principles about students learning. (Kennedy, 2004). In the final analysis of this study, Kennedy suggests that teachers attend to three reform ideals: (1) teachers are attentive to rigorous and important content; (2) teachers realize the importance of intellectual engagement; and (3) teachers try to ensure student participation (Kennedy, 2004). Kennedy summarizes: Reform ideals are indeed present in teachers’ thinking, though in somewhat different forms… Teachers interpret classroom situations in terms of six different areas of concern [content coverage and learning outcome, fostering student learning, maintaining momentum, student willingness to participate, classroom as a community, and personal needs], and rely on their own prior beliefs, values, and accumulated principles of practice to decide how to respond to situations as they arise. The problem reformers face may not be one of persuading teachers of their ideals, but instead one of persuading teachers to weigh different areas of concern differently as they make moment-by-moment trade-offs. (pp. 28-29) Teacher Efficacy Although there is extensive research on teacher efficacy, the literature base is limited in respect to the effect teacher efficacy has on classroom practice. While there appears to be a correlation between high levels of teacher efficacy and the likelihood of a

57 teacher using standards-based mathematics practices in the classroom, there also appear to be teachers with high levels of efficacy who choose to use a traditionalist approach in their teaching (Ross & Bruce, 2007). Teacher efficacy surveys, while quantifiable, have been insufficient in using data inductively to form a basis for a theory to explain this phenomenon. Efficacy is alleged to be associated with certain practices such as more studentcentered instruction, more hands-on learning opportunities, and greater emphasis on process skills- all consistent with reform-based classroom teaching. Teacher efficacy, according to Wheatley (2000), is developed as a combination of two theories--that of psychological constructivism and socio-cultural theory. As to the theory of constructivism, an individual creates knowledge within a framework of ideas held individually and is therefore influenced by the interpretation of content given by the individual. The socio-cultural perspective suggests that efficacy is developed within the context of the social structure and therefore any interpretation of efficacy has to be examined within the confines of the same social structure (Wheatley, 2000). Teacher efficacy, a construct for which exist many validated measurement instruments, is not inherently a measure that can predict a teacher’s endorsement of, or participation in, reform mathematics (Smith, 1996; Wheatley, 2000). Most efficacy measurement instruments are questionnaires with results determined on a Likert scale of high teacher efficacy (positive) or low teacher efficacy (negative). The usefulness of this type of instrument to measure teacher efficacy in mathematics is questionable as Likert

58 scales limit responses to rankings that are without context and use broad statements that are unrelated to any specific subject. Being efficacious, therefore, hardly implies being reform-oriented. Wheatley (2000) identified eight types of positive teacher efficacy that are not conducive to reform mathematics. These eight types include positive teacher efficacy grounded in the use of traditional teaching method or the achievement of traditional goals, competitive teacher efficacy, and a sense of independent and direct teacher control of student outcomes. They also include too-certain efficacy, the overly-optimistic efficacy of novice teachers, efficacy beliefs grounded in a hypothetical future, and pretend positive teacher efficacy beliefs. (p. 23) According to Wheatley (2000), these eight types of teacher efficacy are more likely to perpetuate traditional practice. For a description of characteristics associated with these eight types of teacher efficacy, see Table 2.

59 Table 2. Key Characteristics of Teacher Efficacy Not Conducive to Reform Mathematics Efficacy

Characteristics

Traditional teaching

Uncertain about level of student achievement with reform practices Prefer tried and true traditional methods

Traditional goals

Feel effective teaching “traditional mathematics” (computation)

Competitive teacher

Feel sense of superiority (effectiveness) over other teachers Unwilling to collaborate (share) teaching strategies

Teacher control

Fears lack of control over student learning with studentcentered instruction

Too-certain

Knows all about what’s important with teaching

Overly optimistic

Initial optimism about reform teaching gives way to security

novice

of traditional methods

Hypothetical future

Supports reform teaching if only other factors were different “Someday I will do this”

Pretend positive

Efficacy one feels forced to express due to high-stakes pressure (Praxis III requirements or social expectations)

Note: Excerpted and paraphrased from “Positive Teacher Efficacy as an Obstacle to Educational Reform,” by K. F. Wheatley, 2000, Journal of Research and Development in Education, pp. 18-20.

60 Summary The mathematics education of children in the United States is undergoing close scrutiny, not to say criticism. As a product of the educational system, student mathematical achievement levels based on standardized tests are seemingly unimpressive, and addressing the alleged problem is complicated by a tenuous view of what mathematics children might need to know and be able to do. Teachers are guided by national and, more specifically, state standards. Evidence strongly suggests that teachers’ practices have not yet responded to such guidance. Adding to the murkiness of what to teach, mathematics educators are immersed in a conflict of what constitutes effective instructional pedagogy. Current classroom practice, for many reasons, is still fairly traditional even though research on how students learn purportedly suggests the propriety of a more student-centered approach to teaching. Teaching, as a general rule, is steeped in tradition, and teachers often find themselves falling into a conventional mode of instruction even though they profess a non-traditional pedagogical intent. It seems clear, that for many teachers, their beliefs about mathematics and mathematics teaching are incongruous with their practice. Teacher belief systems are complicated, multi-dimensional, and can often be contradictory. Based on prior experiences as a student, an observer, and a recipient of advice from institutes of higher education, mathematics teachers’ true beliefs about mathematics may be much more traditional than their professed beliefs about mathematics teaching and learning. Standards-based mathematics instructional reform can hypothetically be compatible with beliefs about mathematics teaching and learning

61 and at odds with beliefs about mathematics as a discipline. While the classroom decisions teachers make derive in part from their beliefs, many other antecedents contribute to what one eventually sees in the classroom. According to Manouchehri (2003), “Research studies that reflect teachers’ own perspectives on issues concerning standards-based practice are rare,” (p. 78). The need to understand teachers’ perspectives is vital to understanding any dissonance phenomenon (if there truly is discord) between professed beliefs and practice. Significant to Manouchehri’s study as well as to this proposed study, Kennedy’s (2004) study regarding the rationale for attending to reform ideals and teacher intentions may not necessarily be in keeping with NCTM Standards, but the intentions and goals of the participants of the study are arguably lofty and show concern for student learning. It is in understanding those concerns and intentions, along with the many other considerations that are an everyday occurrence in classrooms that may allow us to realize a possibility of mathematics reform.

62 CHAPTER 3: METHODOLOGY In an era of mathematics reform, classroom practice is of the utmost interest to mathematics reformers, policymakers, parents, students, school administrators, community members, and most notably, teachers. Classroom teachers determine the extent to which practice changes or remains the same. This circumstance has been studied by researchers for quite some time. In fact, as the preceding chapter showed, what teachers espouse about mathematics and mathematics pedagogy has been studied. However, much of that knowledge has been obtained through methods that fail to confront why teachers make the decisions they make. This study attempts to discover how teachers understand the state standards, as well as how their understanding translates into classroom practice. Do teachers truly support the Standards and strive to provide standards-based instruction to their students, and if so, what is the evidence of this in their classrooms? (It is possible that this is not the case.) For those teachers who advocate for the Standards and standards-based instruction, what circumstances inhibit them from successfully implementing standardsbased instruction in the classroom? Prior Research Focus Groups as a Backdrop for the Study This study was based on the researcher’s personal prior teaching work with practicing teachers, and reading of current literature on classroom practice. What began as an inconsistency in my view of effective mathematics teaching and my observations of the instructional practice of mathematics teachers during a research study became a quest

63 for understanding. I decided to interview other teachers to see if I could make sense of this evident discrepancy. Initially, my sense making led me to conduct a focus group of mathematics teachers to find out how their knowledge of the mathematics standards affected what happened in their classrooms. I wanted to know how knowledgeable teachers were about the Standards, how they meshed the Standards with their instructional strategies, and how the Standards had (or had not) changed their teaching. Basically, I wanted to know how mathematics teachers defined what they do in the classroom. Focus Group Alpha The first focus group consisted of 11 teachers of Grades 2-7 from three different school districts within a single county. The majority of the teachers were teaching in selfcontained classrooms in Grades 3-5. The teachers were together on this occasion to attend the last session of an eight day sustained professional development series training arranged by their county educational service center. The eight professional development days were spread over a five-month period of time and were held during the teachers’ work day. I began the discussion with a request for them to share their thoughts with me. The question I asked the group was, “When you first start planning what you are going to teach and do in your mathematics classroom, what do you think about? Tell me about your planning.” Answers were short with little to no elaboration on the ideas put forth. Some of the considerations mentioned were the goals of the lesson and what they had to do to

64 achieve those goals, the indicators, what the state says they have to teach, time constraints put upon them by the need to prepare students for state achievement tests, lack of resources, and their student demographics. The teachers, for the most part, were hesitant to elaborate on their responses when I probed for additional comment. They were uncomfortable, fidgeted, and displayed signs of uncertainty and hesitation, so I ended the conversation. I surmise that my presence at a workshop arranged by their district leadership may have compromised the rapport needed for a more frank discussion. I did not feel I learned much about teacher practice by conducting this focus group. In addition to what I interpreted to be feelings of mistrust, I hypothesized that much of the problem had to do with the grade levels taught by the majority of these teachers. Mathematics is only a part of their teaching responsibilities, and many of them had expressed a dislike for mathematics. Perhaps the opinions of this group of teachers were atypical of educators specializing in mathematics. Focus Group Beta I chose to attempt another focus group with mathematics teachers who taught in departmentalized mathematics classrooms. I contacted a colleague who was preparing to facilitate a 2-week summer institute, primarily but not exclusively, for middle-school mathematics teachers. I arranged to attend the second day of the institute prior to most of the instruction that would take place since I did not want the content from the workshop to influence teachers’ conversations with me. I conducted two focus groups, which allowed me to meet with all teachers in attendance.

65 The teachers ranged in experience from a new teacher seeking employment for the upcoming school year to a teacher who had taught for 28 years. The incentives for participation in the workshop were such that the motivation for attendance was quite varied. Teachers, self-reportedly, participated for various reasons--free graduate credits for licensure renewal, stipends, free resources and materials, information, and networking. All were from rural school districts in southern Ohio. The workshop participants divided themselves into groups by choosing when to meet with me. This accident of timing prompted an interesting split between the participants as they self-selected their groups. The beta group was made up of 11 novice teachers--newly licensed teachers (some with and one without employment for the coming school year) and teachers who had some experience but generally no more than 5 years. The gamma group was made up of nine more experienced teachers. I met with the beta group over an extended lunch. I started the conversation with the same question from my earlier focus group. “When you first start planning what you are going to teach and do in your classroom, what do you think about? Tell me about your planning.” The first person who spoke said, “The [Ohio Academic Content] Standards. I think I can speak for all of us here. That is the first thing that comes to mind. The Standards.” Everyone around the table nodded their agreement. This group continued to talk about the Standards and the accountability system the state has adopted and how it has made teachers more accountable for teaching what is required. Many recounted using previous Ohio Achievement Tests to determine the testing emphasis placed on individual

66 indicators. Indicators that were most often addressed in testing questions were chosen as “power indicators.” When asked to explain further, they explained that power indicators helped them know exactly what indicators to hit the hardest. It helped them focus instruction. One teacher stated, I took the achievement tests over the last few years and mapped out what questions they asked. I matched them to the indicators. Those are the power standards. It helps me because I know what I can leave out and what I need to teach. All but two of the eleven teachers at the table had done this as part of their planning. This naturally led to a conversation about short-cycle assessments, which are a direct result of school districts trying to focus instruction on the Standards and gradelevel indicators. All teachers from the beta group spoke favorably of this practice. They were cognizant of the importance of alignment of practice and assessment. “I know what’s on the short-cycle assessment tests. I know what is on the achievement test.” Another teacher offered her perspective, “We all have our favorite things to teach. I like algebra. If I like teaching algebra, is that going to affect how I teach geometry? With what I like to teach…I can be more creative.” With no more prompting from the interviewer, the group eventually turned to a discussion of pedagogy. Most talked about different strategies for instruction, which included hands-on activities, worksheets, and making “real-world” applications. They agreed that what they actually do in the classroom is affected by time, cognitive ability levels of students, and students’ behavior. Teachers reported using a variety of strategies

67 based on those considerations, and they seemed to doubt the existence of “one right way” to teach. They admitted that strategies they employed were not even consistent throughout the day. “I might teach using a worksheet one period and then try something else the next.” Another added, referring to behavior, “What can you trust your kids to do?” Yet another added that it often happened that she found herself saying, “…This is second period and that didn’t work so now I will try something else.” The only male teacher in the group mentioned that he tried to find “real-world” applications for each topic he taught to which someone quickly added that it was important to make it meaningful to students. The beta group teachers spoke with animation and seemed confident in their classroom practice. Focus Group Gamma At the end of the day, I met with another group of teachers. This group consisted of more experienced teachers. Most of them had taught for many years. The exception was a middle-aged woman who was teaching a high school geometry course under alternative licensure while completing course work to obtain her license to teach. There were nine teachers in this group. Posing the same question, I expected a repeat of the beta group discussion. Things, however, were not exactly as I anticipated. Teachers began talking about the “big picture” and breaking down the year into “chunks” by grading period and then further by week. One teacher talked about selfevaluating what she had done the previous year and thinking about her weak areas and considering how to change or strengthen those. Several teachers talked about how

68 important it is in planning to know your students. “You have to pre-assess your students. I don’t need to spend time going over things they can already do.” Interestingly, they were 19 minutes into this conversation before anyone brought up the Standards. “They gave us a copy of the Ohio Academic Standards on a worksheet. And now, any time I talk about something on the sheet, I put a check by it so they [administration] know at the end of the year that I ‘covered’ everything.” Someone else reported, “Our new textbook has the Ohio Standards right in the front of the book. I made my kids sit down and read the indicators. They were bored and didn’t want to do it, but I made them anyway.” According to the gamma group, many of the school districts seemed to be taking measures to force students and teachers to focus on grade-level indicators. One teacher reported that the students in her school had to keep a check-sheet of the grade-level indicators in the front of their notebook, and whenever the class covered one of the indicators, the students and the teacher both had to initial that indicator on their checksheet. “That way they can never say they didn’t cover it when we get to the test.” Several teachers reported that although their schools were not currently following this procedure, they were planning to implement this for the next school year. Some reported they were required to write on poster paper and to highlight which indicators they were “covering” each day. Some reported that they had commercially made posters they were required to display in their rooms listing the standards and indicators. A few teachers discussed short-cycle assessments. One of the teachers said she did not use the short-cycle assessments because they were not aligned with the state tests.

69 (Incidentally, this person worked in the same school as one of the beta group interviewees who talked about the short-cycle assessments favorably citing them as well aligned to state indicators.) Another said she worked in a county where they used a pacing chart that dictated what each teacher on a grade level would be teaching on any given day along with prescribed and mandated short cycle assessments. She liked the pacing chart because many of the students in this county moved from school district to school district and claimed that the pacing chart helped to make sure “huge pieces of the curriculum” were not missed. In discussing what they did in the classroom, most teachers described classrooms where there was great disparity between the students’ ability levels. They talked about the difficulties they faced in meeting the needs of all their students, now and in the future. While most teachers talked about their struggles to meet the expectations of the state and their superiors, two admitted they were failing to meet the needs of both advanced and lower-level students because they felt they had no choice but to “teach to the middle.” They just hoped they could find time to help the others. The conversation then moved to how, in the next few years, all students will be required to take Algebra 2 as part of the core courses in high school. Several teachers expressed concern for their students who have individualized education plans (IEPs) and those who are struggling with more limited mathematics ability. Such students, they observed, had little recourse when it came to securing additional help. More than one teacher reported trying to make time outside of class to work with students who were struggling, since their parents were unable to help them with mathematics.

70 Teachers discussed the need to make mathematics meaningful for their students. One teacher said, “I try to use ‘real-world’ applications whenever I can. You know…. I don’t care how many golf ball problems there are. My kids don’t golf. Put it back on the farm. Take it back to everyday.” Another expressed concern, “What if they don’t want to go back and farm? They may go cut timber or tobacco.” The group erupted with indignation that what students want to do with their lives is not honored within the educational system. The conversation was not moving forward in a way that I could determine what was actually happening in the classroom and why they chose certain instructional practices. Intervening, I asked the gamma group, “When you think about your instruction, what kinds of things are you doing in your classroom?” Teachers talked about paperfolding, acting out problems, trying to use “real-world” applications, using tangrams to prove the Pythagorean Theorem, and using dry erase boards for group computational activities. While these were strategies reported as a direct result of a specific question, throughout earlier discussion, most talked about doing problems on the board for students to copy and starting assigned homework problems by working problems together as a group before giving students time to work on the problems prior to leaving the classroom, indicative of traditional methodology. After much venting of their frustrations over testing, one teacher suggested that value-added evaluation would be beneficial to their students as the value-added system judges performance as individual growth (over time) rather than in reference to a norming group. One teacher expressed her thoughts on the matter.

71 It will also help these kids. Even though they don’t make the score, they can see that they have come up from here to here (using hand motions). They compare them to themselves. It is important for that kid to realize the growth they have made. I mean why compare these kids to kids in California? I have kids who have never been to a mall. I had a kid who didn’t know what a pineapple was. You tell me, is it fair to compare that kid to some kid in Sandusky? Those kids don’t know what a farm is and could never milk a cow but my kids…. Well….they can take care of a cow! Another teacher told about a student who had no interest in school but sure knew how to hotwire a car. This group seemed to me to have a whole-child view of the purpose of education. Summary of Focus Groups: What Does It Mean? When I thought about the differences in these three focus groups, what kept coming to mind were their attitudes and feelings about the Standards. It was not even something they all identified as such. The first group seemed to view the Standards as an administrative mandate to which they were being held accountable. Although they indicated some appropriate supports were being offered, they were concerned that the expectations set forth by administrators were unrealistic and that teachers would be viewed unfavorably if test scores were not acceptable. The beta group interpreted their responsibility to students in terms of the Standards: This is what we are supposed to be doing and these are all the ways in which we are accomplishing this goal. The gamma group, by contrast, seemed to see the Standards as almost a hindrance to preparing their

72 students for life. As one teacher explained, “When I started, I felt like I was preparing the kids for their life. Now I feel like I am preparing the kids for a test.” These teachers seemed not to doubt that this change was a disservice to their students but they also seemed at a loss as to what they might do about it. For this group of teachers, the Standards seemed to be an obstacle to overcome. Their stories were about the challenges they faced due to changes they seemed to consider a detriment to education. These challenges made their jobs much harder. With this group, I never really could decide if they truly understood what was meant by the term “standards” as they seemed to use it loosely and sometimes erroneously. The discussions with practicing teachers did little to alleviate my confusion about what teachers did in their classrooms and why. It did confirm that teachers seem to have differing degrees of understanding and that they place differing degrees of value on mathematics standards. This is not surprising as, according to research, one would expect debate and difference. The discussions did not give me great insight in to what part mathematics standards play in determining classroom practice because, although all the teachers in the focus groups reported teaching according to the [Ohio] Standards, their descriptions of classroom practice varied considerably and their use of the term “standards” was sometimes, as it seemed to me, inconsistent with their meaning. In addition, some of the teachers’ reports of standards-based classroom practice were arguably more characteristic of traditional instruction. The testimony of the focus groups seemed to confirm a disconnect between what teachers profess to believe and do and what may be actually happening in their classrooms. Their statements of practical

73 intentions may not be their resultant practice. These discussions reinforced my desire to understand how teachers determine what they teach, what teachers do in their classrooms, and what happens between planning and implementation that changes what an objective observer might see in the classrooms. Qualitative Design Choice The purpose of a research study dictates the methodology of the study. Some types of research questions are well suited to analyzing great quantities of information in order to make it possible to use standardization and aggregation to generalize predetermined quantifiable facts or information from a comparison of various subjects (Patton, 2002). Other types of research questions require methodologies that allow a researcher to investigate a topic in-depth and within a social constraint (Patton, 1987). Ragin (1994) characterizes quantitative studies as data condensers (using deductive processes to take quantities of information and bringing it down to a rather simple quantifiable answer to a question) but portrays qualitative studies as data enhancers (using inductive processes to take information gleaned and pulling it all together to make meaning and show relationships). Using qualitative research, one can focus on learning about key information known only to the research participants to construct a welldetailed, in-depth knowledge of social data. The goal of this research study was to understand what teachers actually take into consideration when planning instruction because there seems to be a disconnect between espoused beliefs and intentions and actual practice of middle school mathematics teachers in relationship to standards-based mathematics reform practices in the

74 classroom. While this phenomenon is regularly documented, the situation is contextual and depends on teachers’ individual interpretation. What circumstances affect the interpretation? Given similar circumstances, can the same response be expected from every teacher? Without context, I believe the underlying reasons have superficial meaning and offer little in the way of insight about thoughtful reform. Given the goal of the study, I used qualitative methods, in particular, grounded theory. Grounded Theory According to Strauss and Corbin (1998), grounded theory is appropriate when “all of the concepts pertaining to a given phenomenon have not yet been identified, at least not in this population and place. Or, if so, the relationships between the concepts are poorly understood or conceptually undeveloped” (p. 40). I chose grounded theory because my goal was to understand the nature of what happens and the interplay of relationships involved in the planning, preparing, and implementing classroom practice. Unlike theory that is developed conceptually and then tested, grounded theory is derived from data that is collected within the context of the phenomena of interest (Maxwell, 1996). Grounded theory is based on the use of inductive strategies and constant comparison of data for theory development (Patton, 2002). It develops theory rather than tests existing theory (Patton, 2002). “Grounded theory depends on methods that take the researcher into the real world so that the results and findings are grounded in the empirical world” (Patton, 2002, p. 125). According to Strauss and Corbin,

75 Theory denotes a set of well-developed categories (e.g., themes, concepts) that are systematically interrelated through statements of relationship to form a theoretical framework…the statements of relationship explain who, what, when, where, why, how, and with what consequences an event occurs. Once concepts are related through statements of relationship into an explanatory theoretical framework, the research findings move beyond conceptual ordering to theory… A theory usually is more than a set of findings: it offers an explanation about phenomena. (1998, p. 22) Understanding the phenomena is the intended outcome and that, in this view, can only occur in context. Phenomena related to teacher beliefs, teacher efficacy, and classroom practice have typically been studied with surveys. The usefulness of that approach is questionable. Likert scales limit responses to rankings that are without context, use broad statements that rarely engage mathematics per se, and seldom allow teachers to qualify their responses. The resulting research can not engage teachers’ thought processes or understandings. It can, in short, be misleading when such thought processes and understandings are relevant (as in this study). Grounded theory is emergent. The mission is to find the theory embedded in the research data as it currently exists. Data are collected and then compared to previously collected data. Constant comparison helps theory emerge, and data are compared to emerging theory. Ideally, the final result is a theory that, at least for this set of data, is sound. It fits, and it works, and it helps people to better manage the situation under study.

76 Grounded theory, not unlike other qualitative methodologies, is only as valid and reliable as the research design. A major part of the research design for grounded theory involves the researcher. The researcher must be knowledgeable, qualified, and implement the study faithfully. The Researcher I began teaching in the public school system in 1989 and continued teaching in the classroom until May of 2004. Thirteen years of my career were spent teaching mathematics in Grades 6-8. Four years of that time, I also served as the mathematics specialist for my school district. I left teaching in the public schools to serve for one year as a professional development team coordinator with a program associated with a National Science Foundation project, ACCLAIM (Appalachian Collaborative Center for Learning, Assessment, and Instruction in Mathematics), working with teams in Kentucky, North Carolina, Ohio, Tennessee, and Virginia. In 2005, I began teaching mathematics methods courses in the College of Education at Ohio University in Athens, Ohio. My teaching certification is as a generalist in elementary Grades 1-8, but my concentration area is in mathematics and all my post-baccalaureate coursework has been in mathematics and mathematics education. In addition, I have aggressively pursued a deeper understanding of mathematics teaching from the perspective of a practitioner, both receiving and delivering mathematics teaching professional development. I secured National Board Certification in Early Adolescence Mathematics. I have had extensive experiences in mathematics education. Serving as a mathematics specialist in my school district afforded me an opportunity to participate in

77 sustained mathematics professional development opportunities as well as to observe, model, and team-teach in many classrooms. I served the Ohio Department of Education as a member of the writing team for the Ohio Academic Content Standards for Mathematics. In addition, I trained for and delivered professional development modules for the Ohio Mathematics Academy Project (OMAP). I served as a consultant and writer for three projects for educational technology centers in Ohio. I have delivered many hours of professional development throughout schools in southern Ohio and served as an educational consultant to a junior high school mathematics department in a school district where teachers were struggling to improve mathematics test scores on the Ohio Achievement Tests. During this time, I worked with, observed, and talked with many mathematics teachers in rural Appalachian schools. My interactions with teachers have been varied, and my work usually required me to establish rapport with teachers. I believe I have been especially successful in getting teachers to open up and speak about their feelings and concerns. Participants I chose to focus on middle-school mathematics teachers in part because of my greater familiarity with that content and pedagogy. More substantively, research about middle-level teachers’ planning and thinking is token with respect to contemporary expectations for reform-style practice. Finally, I harbored the intuition that middle-level mathematics teachers may prove to be the teachers most receptive to the “constructivist”

78 message. If this intuition was warranted, results of this study could prove particularly useful to the improvement of middle-level mathematics teaching. With grounded theory, the selection of participants is vital, and, therefore, the researcher used purposeful sampling as the selection strategy. The participants chosen were selected based on criteria that offered the study an opportunity to explore participants’ insights and reasoning about classroom practice. Participants were: 1. teaching only mathematics at a middle-school level, 2. identified by colleagues in teacher education programs and local school district curriculum directors who provide professional development to inservice teachers as having been exposed to extensive standards-based instruction and pedagogy specific to Ohio Academic Content Standards (see Appendix A for examples of extensive professional development in standards-based instruction and pedagogy), 3. willing to be observed teaching in their classroom and to be interviewed individually and as part of a focus group, and 4. willing to share lesson plans and samples of assessments with the researcher. This research involved an in-depth study of 12 teachers. Of the 12 teachers who participated in the observations and interviews, I chose six teachers for the final focus group discussion. To capitalize on early indications drawn from the constant comparative nature of grounded theory, the teachers invited to participate in the focus group were the six teachers whose practice and intentions were most discrepant. Six participants allowed

79 the focus group to be small enough to involve each participant in discussions and large enough to provide rich interaction (Brotherson, 1994). After repeated attempts to coordinate a date, place, and time for the focus group meeting, a meeting was scheduled but only five of the six invited teachers attended the focus group. The two teachers whose practice and intentions were most discrepant did not wish to participate in a focus group. A total of five teachers attended. In selecting potential participants, the researcher solicited from colleagues who were professional development providers and curriculum directors the names of teachers who had previously participated in standards-based professional development. Given the names of prospective teachers, I contacted the teachers and discussed the proposed study in order to obtain their consent to participate in the research study. Following the teachers’ agreement to participate, I contacted building administrators to seek approval for the teachers’ participation in the study, as the study required observation of classroom instruction. While 12 participants were secured initially, adding new participants was an option should it have become necessary by a need identified for theoretical sampling as theory began to emerge. However, this did not occur. Confidentiality is usually required in order to obtain full cooperation and forthrightness from participants in order to protect the individuals from harm or other punitive damages (Patton, 2002). In maintaining confidentiality, teachers were assured that the names and associated school districts would be omitted from the final study reports so as to protect participants and to help ensure reliability of data.

80 While the researcher did not intentionally solicit similarity in participating school districts, the majority of the school districts were of like typology. The 12 participants represented 11 different schools districts in southern Ohio. Note: The two teachers from the same district taught on different grade levels in different buildings and reported minimal contact. Of the 11 districts, 10 were categorized by the Ohio Department of Education as rural/agricultural. Nine of the 10 were high poverty and low income districts located in Appalachia. The other rural/agricultural district was low poverty and low to median income located outside Appalachia. The final district was categorized as rural/small town with moderate to high median income.3 During the data collection phase, each participating teacher was asked to share their local building report card designation for school effectiveness. All 12 teachers were able to obtain a copy of the report. Three schools were rated excellent. Six schools were rated effective. And, three were rated continuous improvement.4 Data Collection Data collection began in September soon after the proposal was approved, the institutional review process was completed, and the first participant was identified and had agreed to participate. Participant selection continued until 12 participants had been interviewed and observed.

3

The Ohio Department of Education developed a typology for school districts in an effort to identify similar districts. Due to confidentiality, specific information linked to participating schools is not included but the typology can be found at http://education.ohio.gov/GD/Templates/Pages/ODE/ODEDetail.aspx?page=3&TopicRelationID=390&Co ntentID=12833&Content=78585 4 Local report cards are part of Ohio’s accountability system. Due to confidentiality specific information linked to participating schools is not included but school district report cards can be obtained at http://ilrc.ode.state.oh.us/

81 The aim of data gathering at [the time of initial sampling] is to keep the collection process open to all possibilities…To ensure openness, it is advantageous not to structure data gathering too tightly in terms of either timing or types of persons or places, although one might have some theoretical conceptions in mind, because these might mislead the analyst or foreclose on discovery. (Strauss & Corbin, 1998, p. 206) Data collection followed naturalistic inquiry. Initially, the researcher began by briefly interviewing each teacher to establish demographics and to answer any questions the participants had. This meeting also included discussion and signatures for informed consent. Following the initial interview, the researcher logged time in the classroom to get a feel for the typical classroom setting and behaviors of students and teacher. This time allowed the teacher and students to become comfortable with the researcher’s presence in the classroom. Once a level of comfort had been established, the researcher observed each participant teaching in their classroom two times prior to conducting an indepth interview. The observations provided context for the interviews that allowed for a deeper understanding than just interviews alone (Hoepfl, 1997). A Mathematics Classroom Observation Instrument (MCOI), created from several classroom observation protocols, was used as the protocol instrument for the structured classroom observations.5 See Appendix D. This protocol had not been rated for reliability or validity and the results were not reported statistically. It was used merely as an organizer for recording 5

The major format and content for the MCOI is based on the Mathematics Classroom Assessment Instrument from Leadership by Design. Content from Unit Standards from the Student Teaching Final Evaluation forms as well as the Professional Internship Progress Record from Ohio University have been incorporated and are formatted to look similar to the other items of the MCOI. The items included on the observation tool are reflective of reform-based mathematics practice and effective teaching strategies.

82 data observed in the classroom. During the second observation visit, the researcher attempted to obtain copies of teachers’ lesson plans and assessment to see to what extent they aligned with standards-based instructional strategies. Lesson plans, when available, gave some indication of typical classroom practice, and the types of questions and tasks on classroom assessments offered insight into the teacher’s understanding and adoption of the principles associated with the Standards. Field notes were made by the researcher during the interviews and observations. Field notes are written descriptive accounts of any phenomenon the researcher feels is important to the study. They included time, dates, location, and descriptions of what was observed. Field notes should be rich in fact as well as researcher reaction. They should be insightful and analytical (Patton, 2002). Field notes were recorded during and immediately following any observations or interviews. As soon as possible following the second observation, the participant and the researcher arranged a mutually convenient time for an in-depth interview. “Interviewing is necessary when we cannot observe behavior, feelings, or how people interpret the world around them,” (Merriam, 2001, p. 72). Uncovering teachers’ perceptions requires the participant to explain or, at least, talk about their unobservable thoughts and the reasoning behind their various actions (which are observable). The in-depth interview with the first participant was a semi-structured interview using a general interview guide approach that “involves outlining a set of issues that are to be explored with each respondent before interviewing begins. The guide will serve as a basic checklist during the interview to make sure that all relevant topics are covered.” (Patton, 2002, p. 342)

83 The researcher had a list of topics to address if the topic did not occur naturally in conversation. Researcher notes reflected which topics occurred naturally. A list of interview topics or questions for the initial interview is included as Appendix C. The interviews were recorded using a digital voice recorder and the researcher took notes during the interview to record unspoken data, formulate new questions, note insights, and make a record of any other important data not captured by the recording. Participants were asked to talk about their current teaching assignment and any constraints placed upon them that may be outside his/her control. The second question was the same as the question posed to focus groups, “When you begin planning for instruction, what do you think about? Tell me about your planning.” The placement of this question so early in the conversation allowed respondents to answer without having ideas about standards, reform, and classroom practice subconsciously interjected into their thought process. This conversation was followed up, as necessary, with probes into the teacher’s knowledge and familiarity with the Standards, mathematics reform, and what the teacher perceived to be their role in mathematics reform. Participants were asked to describe a typical day in their classroom and to expound upon factors they felt influenced their teaching in either a positive or a negative way. Teachers were asked to compare and contrast their experience at the same grade level as a student to the experience their students were currently having. Finally, the previous observations, lesson plans, and assessments were discussed and the researcher asked for explanations of any inconsistencies brought to light during the observations and compared and contrasted what teachers said with what was observed to look at inconsistencies between the teachers’ professed beliefs and their

84 practice (Thompson, 1992). These were the basic topics of conversation planned for and needed for the interview in terms of the research study, but the interview conversation often moved in many other directions. The interviews were conducted as much as possible as an open-ended discussion allowing the teacher to talk freely about their teaching, their beliefs about teaching, their background, their education, their current position, and anything else they perceived as relevant to the discussion. “An open-ended interview… permits the respondent to describe what is meaningful and salient without being pigeon holed into standardized categories” (Patton, 2002, p. 56). Following each interview, the recordings of the interviews were transcribed verbatim. As soon as feasibly possible following the interview, the interview notes were reviewed and anything omitted was added, notes that did not make sense were clarified, and details about the setting, the conditions of the interview, and the observations of the researcher were added. As noted by Patton (2002), “the period after an interview or observation is critical to the rigor and validity of qualitative inquiry. This is a time for guaranteeing the quality of the data.” (p. 383) Once the interviews and observations were concluded, the data may have indicated a need for clarification of previously collected data or additional new information (Patton, 2002). If this had been the case, follow-up interviews may have been necessary based on what the researcher learned from the two observations, lesson plans, assessments, and interviews. On only two occasions was a follow-up question necessary. In both cases, the follow-up questions were interview questions from the original topics

85 list which had been inadvertently omitted during the interview. As each new participant was interviewed and observed, the data was compared to previously collected data and interview questions and points of discussion were adjusted accordingly. A final step in the data collection process occurred toward the end of the study. After most of the data analysis had been compiled, five of the participants were brought together as a focus group for a dinner meeting to discuss themes that emerged from the study. Lincoln and Guba (1985) refer to this focus group meeting as member checks. “The purpose of this comprehensive check is not only to test for factual and interpretative accuracy but also to provide evidence of credibility” (Lincoln & Guba, 1985, pp. 373374). While discussing emergent themes to check for accuracy, it is possible that new information or insight could have been gathered. This process allows focus group participants an opportunity to hear the opinions and thoughts of other study participants and to respond to their comments as well as to the reported comments they themselves made (Patton, 2002). Lincoln and Guba indicate “it is quite likely that the member-check process will highlight the need for further revisions and extensions. New information needed to complete earlier unfinished sections should be added. Factual errors should be corrected.” (p. 378). This focus group interview, like all other interviews, was recorded and transcribed verbatim. The focus group interview data was used by the researcher as a source of data and as a part of the data analysis process prior to officially completing the data analysis phase of this research project.

86 Data Analysis The goal of this study was to understand what teachers think about and take into consideration in teacher preparation for mathematics instruction and how this affects practice in the classroom. With this purpose in mind, the researcher could not define emerging categories or themes prior to completion of data collection and analysis. The approach was inductive. The researcher continually analyzed the data in order to draw out understanding (Patton, 2002). Grounded theory requires constant comparison of new data collected to previously collected data, and this process began as soon as data collection started. New data was compared to previously conducted interviews and observations and theory began to emerge. The process continued and as theory began to emerge, data was compared to theory (Denizen & Lincoln, 1998; Patton, 2002; Ragin, 1994). The question of how to know when one has collected enough data must be addressed. Ideally, one collects data to the point of saturation. Saturation occurs when the researcher is not learning anything new as data is being collected (Ragin, 1994). Lincoln and Guba (1985) suggest that data collection stop at the point of “saturation of categories” which they describe as the point at which “continuing data collection produces tiny increments of new information in comparison to the effort expended to get them” (p. 164). If after collecting all relevant data from 12 participants, saturation of categories had not occurred within data analysis, additional participants would have been added to the study. However, this was unnecessary.

87 The researcher used Weft QDA qualitative software for data analysis, starting with a line-by-line analysis beginning with the completion of the first case. Patton (2002) describes this coding process as vital to the analysis process. He states: This descriptive phase of analysis builds a foundation for the interpretative phase when meanings are extracted from the data, comparisons are made, creative frameworks for interpretation are constructed, conclusions are drawn, significance is determined, and in some cases, theory is generated. (p. 465) The qualitative analysis of grounded theory is based on a series of iterations of synthesizing and interpreting data to pull out existing themes (Strauss & Corbin, 1998). There are six levels of analysis (Harry, Sturges, & Klingner, 2005; Strauss & Corbin, 1998). Analysis begins with open-coding (Bernard, 2000; Patton, 2002; Straus, 1987; Strauss & Corbin, 1998) by looking for themes, patterns, or categories during an initial evaluation of the data collected during the first participant’s interview (Harry, Sturges, & Klingner, 2005; Strauss & Corbin, 1998). Following open coding, the researcher began to further analyze data using axial coding. Axial coding is the process of relating categories to subcategories (Harry, Sturges, & Klingner, 2005; Patton, 2002; Strauss, 1987; Strauss & Corbin, 1998). Developing themes or selective coding allowed the researcher to look for themes within and between categories and subcategories (Harry, Sturges, & Klingner, 2005). Themes were tested, explanations were interrelated and finally, in looking at the themes that emerged from the data, the researcher began to develop theory that explains or makes sense of the data (Harry, Sturges, & Klingner, 2005; Strauss & Corbin, 1998).

88 Data analysis began with open coding. The first set of interview notes, transcripts, observation and field notes, lesson plans, and observation protocol reports were assessed with attention being given to merely answering questions about what was happening, what the teacher was thinking, what the teacher planned, what the teacher did, and why things happened as they did. It was important to become familiar with the data and identify categories suggested by the data. The coding was grounded in the specific data. The researcher, during the open coding phase, gave names or codes to events, activities, and occurrences. Each time a code was repeated, that item was compared to all other items with the same code. “The basic, defining rule for the constant comparative method is that, while coding an incident, the researcher should compare it with all previous incidents so coded, a process that soon starts to generate theoretical properties of the category” (Glaser & Strauss, 1967, p. 106). When new pieces of information are compared to existing similarly identified categories, the researcher must determine if the items belong together and if the code is an accurate categorization of the data (Harry, Sturges, & Klinger, 2005). One piece of data may be associated with more than one code and codes may be revised or omitted throughout the coding process (Bogdan & Biklen, 1991). A codebook was created and used throughout the study to keep accurate and running records of codes. In addition, throughout the process of data analysis, the researcher was memoing while analyzing data. Memoing during data analysis is much like field notes during data collection. It serves as a written record of analysis of text including the researcher’s thought process as themes and relationships between data become clear (Bernard, 2000).

89 When the second participant’s transcripts, observation notes, lesson plans, assessments, and field notes were collected, they were analyzed just as was the first participant’s data. Then, that data was compared data to data, keeping in mind the previously studied case. Once this had been accomplished, axial coding began. Axial codes are the conceptual categories that relate the codes developed during the open coding phase of data analysis. The codes are analyzed for commonalities and clustered together around points of intersection (Strauss & Corbin, 1998). When the researcher is engaged in axial coding, the decisions made take into account the researcher’s interpretation of data. At this point, the researcher has developed familiarity with the data and has begun to abstract meaning from this data as well as previously collected data (Harry, Sturges, & Klingner, 2005). This procedure was iterative and continued throughout the process of data analysis and collection. The development of themes began with examining the conceptual categories. Some categories are more predominant than others. Themes are developed when one summarizes the content of conceptual categories that seem to be related. The relevance of the developed themes to incoming data from subsequent data collection is examined as part of the iterative nature of the constant comparison associated with this third level of analysis (Harry, Sturges, & Klingner, 2005). The researcher has the final decision in determining which categories are important enough to develop into a theme. Predominant categories naturally develop into themes. However, based on the intent of the study, the researcher may decide to develop

90 a theme from less predominant categories in order to maintain the focus of the study (Harry, Sturges, & Klingner, 2005; Strauss & Corbin, 1998). Once themes are developed, the researcher tests the themes. To test the themes the researcher will “apply them to all interview and observational data” (Harry, Sturges, & Klingner, 2005, p. 9) to determine whether the themes account for all the data. If not, additional codes may be needed. This process, as with other levels, is iterative with constant comparison of data. At this point, themes have been developed from the data. It is grounded in the data and tested against the data. The themes are now referred to as explanations (Harry, Sturges, & Klingner, 2005. The researcher then looks for interrelationships within the explanations. This level of data analysis consists “of trying to come to conclusions about contradictions within an explanation… and then comparing across the explanations to see how they related to each other. In the model we call these ‘interrelated explanations.’” (Harry, Sturges, & Klingner, 2005, p. 10) Sometimes a central finding, one that can stand alone, explains all the phenomena associated with a study. Often, however, this is not the case. “For research concerned with nuances of human behavior,” (p. 10) it is more likely that one will not have a central finding because measuring social processes is difficult (Harry, Sturges, & Klingner, 2005). Delineation of theory follows. There are two categories of theories. Theories can be substantive or formal. A theory is substantive if it works within a specific context. For

91 a particular setting and at a particular time, the theory holds true. Theory is formal if the theory holds true within many different contexts and it has been tested and can be applied to many similar topics (Harry, Sturges, & Klingner, 2005). Theory developed from this study is substantive because it was developed in a specific context and setting. In order for substantive theory to become formal, the substantive theory would have to be tested in various situations and have the theory hold true before it could be considered formal theory (Harry, Sturges, & Klingner, 2005). Verification of Interpretation According to Patton (2002), it is a misconception to think that the purpose of triangulation is to verify that different sources of data will yield the same results. Many times different data sources will offer a deeper albeit different relationship between the data source and research (p. 556). In this study, the researcher used interviews, observations, assessments, and lesson plans to explore the participants’ intended and actual practice in the classroom. The researcher anticipated inconsistencies across the data, and it is that relationship that one hopes to illuminate. Additionally, the researcher worked with a committee throughout the data collection and data analysis process to ensure that rigor was established and maintained for grounded theory qualitative studies. Summary The idea for this research study was formulated as a result of a prior research project in which the researcher participated. Because of the nature of the research questions, the only logical methodology was qualitative research and, more specifically, grounded theory. The complexity of understanding the factors that influence classroom

92 practice became more evident through focus study interviews conducted during the summer of 2007. Preliminary planning and study design was completed with the knowledge that qualitative research, and grounded theory in particular, may require minor changes to the initial design as theory begins to emerge from the data collected. A review of unanticipated literature may be likely as the study unfolds. Moreover, working in the chosen method, specific data analysis decisions with the grounded theory approach cannot be determined prior to data collection. Particular features in the design of this study were purposely left open or unspecified by necessity. Those features for which one could plan are described in this chapter.

93 CHAPTER 4: DATA ANALYSIS AND RESULTS This study involved exploring the thoughts and practice of 12 teachers by spending time in their classrooms to become familiar with the teacher, the students in the classroom, the teacher’s practice, and the teachers’ thoughts and opinions about their teaching. After an initial interview, the researcher spent time in each teacher’s classroom before conducting two structured observations followed by an in depth interview with the participating teacher. Data consisted of field notes, interview transcripts, structured observation notes, lesson plans, and sample assessments. The practice of logging time in the classroom was used in order to more accurately determine classroom practice. Logging time prior to conducting two structured observations provided a more valid assessment of typical classroom practice. While the practice of logging time with classroom visits was maintained throughout the study, there was very little, if any, difference in classroom behavior on the part of the teacher or the students from the first day to the last day of the researcher’s time in the classroom. Using Observations to Determine Practice The Mathematics Classroom Observation Instrument (MCOI) was used as an organizational tool for recording information helpful in categorizing teacher practice as traditional or standards-based. While the instrument was too general to offer definitive information, it did serve to organize characteristics of classroom practice in a way that helped the researcher to look for commonalities between the teachers’ actual practice and how research defined traditional and standards-based practice. The instrument was divided into eight sections, each dealing with a different aspect of what was observed in

94 the classroom during the structured observation. Each of the criteria from the eight sections, and the subsequent categorization is discussed. For more information on the MCOI form, see Appendix D. Physical Setting/Classroom Environment This section of the MCOI solicited two pieces of information. The first considered overall student seating and how it facilitated classroom learning, and the second concerned the materials readily available to students. Student seating was disregarded as an indicator of practice because it became apparent early on in the study that teachers, for the most part, did not have control of student seating. In most cases, student seating was determined by furnishing the new school buildings most schools had recently acquired. The teachers were not necessarily consulted as to their choice of furniture or seating arrangements. The second criterion, classroom environment, was indicative of practice as it included the availability and accessibility of mathematics tools such as manipulatives, calculators, computers, and informational displays. Lesson Overview The lesson overview section of the MCOI recorded information on (a) major instructional resources, (b) content delivery, (c) the lesson placement within the instructional sequence, (d) seating arrangement for this particular lesson, and (e) content focus. Two of the criteria, lesson placement and content focus, were mostly insignificant in terms of this study for determining classroom practice. Another, content delivery, was useful in assessing accuracy in teacher knowledge as far as content taught and teacher

95 knowledge of grade-level indicators of Standards. Content delivery, however, contributed little to distinguishing between traditional and standards-based instruction. The criterion, major instructional resources used in the lesson, was suggestive of classroom practice especially when coupled with field note observations of typical classroom activities during classroom observations. For example, if the textbook was the major instructional resource most of the time, it was suggestive of traditional instruction. In the case of the 12 teachers observed, sole and consistent textbook use was highly indicative of traditional instruction. Seating arrangement for the lesson was suggestive of classroom practice. The concept of seating had more to do with what students were expected to do than with the physical placement of seats. Small group arrangements, whether students are doing the same or different tasks, are indicative of student-centered instruction. Instructional Overview The instructional overview section required looking at two different criteria. The first, primary instructional strategy, focused on what the teacher was doing throughout the lesson. Teacher led activities such as lecture, discussion, and demonstration were suggestive of traditional instruction. The second criterion, student activity, focused on what students were doing throughout the lesson. While it is rare that students will be doing only one type of activity throughout a class period, the predominant activity was recorded and secondary activities were noted to provide a more complete assessment of what was happening in the classroom. Listening to or observing the teacher, completing a

96 practice worksheet, reading the text, or working on an assignment from the textbook characterized traditional practice. Questioning The two criteria regarding questioning, quality of questions and questioning techniques, were very important indicators of classroom practice. Traditional instruction was more likely to include either no questions or questions that were mostly narrow and focused on factual recall or required one word responses. However, in standards-based instruction questioning is key. According to Ross, McDouggal, and Hogaboam (2002), student discourse is a key indicator of standards-based instruction. Classroom Atmosphere Student involvement and classroom culture/learner attitudes are two criteria associated with Classroom Atmosphere. Student engagement was an indicator of standards-based instruction to the extent that students were either sitting listening or they were actually doing something that involved them in the lesson. As for atmosphere, while cooperation with the teacher and/or other students is expected in any classroom, the attitudes on the MCOI protocol beyond cooperation are indicative of standards-based instruction. Analysis of Instruction Leading to the Development of Higher Order Skills This section of the MCOI requires an examination of the type of instructional activity focusing on three criteria. They are (a) the amount of student investigation or research, (b) the level of student engagement in the activity, and (c) the mathematics skills being developed. The progression of student investigation begins with none,

97 increases to an activity which focuses on lower level mathematical process skills, and finally to an activity that emphasizes higher level mathematical process skills. In order to categorize instruction, traditional instruction is rated as no investigation or research and a primary focus on only one or more of the first seven mathematical skills listed. The mathematics skills list is a hierarchy of mathematics skills ranging from basic to higher order. There are 17 skills with the first seven suggestive of traditional instruction and the remaining skills being more indicative of mathematics instruction that is standards-based. Overall Classroom Rating Profile The seventh category on the MCOI is a statement about the overall effectiveness of classroom instruction. There are five statements about the quality or effectiveness of the lesson with consideration given to student engagement and alignment with standards. The defining characteristics of these statements involve the extent to which the lesson is aligned with standards and requires students to use higher level thinking skills. The first two statements are associated with higher level thinking skills indicative of standardsbased instruction. The last two suggest no or minimal alignment with standards-based instruction. The middle statement is indicative of instruction that does not clearly line up with any of the other four statements. Mathematical Processes Benchmarks from Ohio Academic Content Standards: Mathematics This section provides a list of the process benchmarks from the Ohio Academic Content Standards for Mathematics in middle school. There are 11 benchmarks and the

98 evidence of three or fewer benchmarks in a lesson, for the purposes of this study, was considered traditional instruction. Context: Who Are They? This study represents teachers throughout a four county area in southern Ohio. While teachers participating in the study may have been casually acquainted through various professional development experiences, none of them were familiar with any of the other participating teachers. The number of years of teaching experience for this group of teachers ranged from seven years to thirty-seven years. Classroom settings were varied. Of the 12, one teacher’s classroom was housed in an elementary building separate from the high school, one was housed in a separate middle school building, and the rest of the classrooms were housed in a campus setting with a middle school or junior high occupying a distinct section of the high school wing of a campus or, in one case, a sixth grade occupying a distinct section of an elementary wing of a campus. All teachers worked with middle-school students with the majority of them teaching a single grade-level. Teacher certification was primarily elementary with eight teachers certified to teach grades one through eight. Additionally, other certifications included one teacher certified in each of the following: grades 4-9 mathematics, high school mathematics, grades 4-9 combined with high school mathematics, and high school mathematics combined with elementary. See Appendix E for more specific information on teacher certification, grade-level, and longevity. For the purpose of reporting information associated with this study, teachers were each given a pseudonym. Teachers’ pseudonyms reflect the gender as well as the order of

99 initial commencement of data collection indicated by the alphabetical ordering of the first letter of the pseudonym. For example, as signified by the fourth letter of the alphabet, Ms. Diane Davis, a female, was the fourth teacher interviewed prior to beginning classroom observations. Meet the Teachers In order to provide background for the study, a brief overview of each teacher is presented. The overview provides basic information about the classroom and practice of each teacher while identifying the instructional practice as standards-based or traditional as defined in chapter. A summary table of teacher practice derived from structured observations is included as Appendix F. Mr. Allen Anderson Mr. Anderson, a teacher of 11years, teaches mathematics to 6th through 8th grade students. He seems to be a teacher whose primary mode of instruction is traditional. During visits to his classroom, the only instructional strategy observed was lecture with mathematics computation demonstrations on the board. His communication with students required one word responses dealing with math facts or procedures. His lesson plans were printed copies of lesson plans from the textbook publisher with the dates hand-written. Mr. Anderson considers himself to be a good teacher who teaches using standards- based instruction. He readily admits, however, that he doesn’t attend to the process standards or use manipulatives in his teaching and considers that it might be detrimental to his students’ understanding of mathematics.

100 I don’t believe I get that [mathematical processing] from each individual student because a lot of that [mathematical process skills] I believe is above some of the students’ heads. Because of not doing some of the group work, not doing some of the hands-on stuff, I think I lose a lot of it [conceptual understanding]. (Anderson) Mr. Anderson has an abundant supply of mathematics manipulatives obtained from various workshops and professional development opportunities in which he has participated. All of the materials were stored in a locked closet in the back of the room and are not used. He cites the homogenous ability groupings of his classes as a reason for not using the manipulatives. Mr. Anderson feels that use of manipulatives with high groups slows them down. He says that low groups are unable to perform using manipulatives and, while some of the middle group might benefit from using manipulatives, he does not think it is fair to students if the teacher treats one group differently than another so he does not use manipulatives with any of his students. Neither does Mr. Anderson have students work in groups or do any type of project that requires application of mathematics although he frequently mentioned in his lectures possible applications of mathematics. Data is important to Mr. Anderson, and he frequently mentioned using the state achievement data reports generated by the district central office to determine what he needs to focus on when planning for future instruction. Mr. Anderson reports his students’ regularly achieve high test scores, and he takes pride in those testing results. When asked if he was doing standards-based instruction, he referred to the textbook, which according to the supplementary material from the textbook publishers, is aligned

101 with Ohio Standards. He then referred back to test results which break down students’ scores into individual areas of difficulty for each student. Mr. Anderson exhibited a comfortable working knowledge of state testing data generated from his students’ test scores. Mr. Anderson views the Standards as an exact guideline of the mathematics content he is required to teach each year regardless of students’ prior knowledge. To clarify this assertion, Mr. Anderson said while waiting for his 7th grade class to begin, “If one of these students goes down and tells Mr. Jackson [the principal] that I am teaching 6th grade standards in here, I get in trouble. I am not supposed to teach 6th grade standards in here.” (Anderson) Mr. Anderson’s knowledge of the Standards was drawn from the textbook lesson planning software tool that allowed one to click on a lesson in the text and the software will display the lesson plan that lists the state indicators for that lesson. Mr. Anderson was unfamiliar with the actual Ohio Academic Content Standards for Mathematics book although he reported owning a copy which he said was somewhere in his classroom. To summarize his feelings about the Standards, Mr. Anderson said, “Teachers, we know what kids need if we’re doing what our job is. If they’re truly a teacher, truly here for the kids, they’re going to do their job. We don’t need standards to go off of. However it’s nice to have those Standards to see where the kids need to be at when we’re done.” (Anderson)

102 Mr. Ben Brown Ben Brown teaches eighth grade mathematics. The school district requires all eighth grade students to take Algebra 1 for high school credit. Having taught seven years as a mathematics teacher, Mr. Brown uses direct instruction as his primary teaching method and asserts that consistency is the driving force behind his teaching practice. The driving force for me is probably consistency…. I found that if I tried to get fancy with this stuff and show them some of the stuff I had learned in college that they just got confused. And, I am probably one of the most boring teachers here because I am so methodical about everything. But I want to make sure they understand and if I’m boring, I’m boring. But, if I go through material in the same way and I’m consistent with it, it gives them a steady platform to work from. If I use the same concepts and same terminology every time we do something then they don’t feel like I’m throwing new material at them. They just think they’re getting the same thing with something a little extra on the end. I give them something that they can work from basically. (Anderson) This sense of consistency prevailed in Mr. Brown’s classroom throughout every visit and observation. Instruction, as well as the classroom routine, remained the same. Instruction from one class period to the next was, as a general rule, the same. Mr. Brown used the same mathematical examples, and he presented them in the same order. In addition, his wording, timing, and extemporaneous conversation was almost identical from one period to the next.

103 Mr. Brown professed knowledge of the Standards. With this in mind, he said that in the month leading up to the state achievement test he has to move away from his Algebra 1 curriculum in order to review the indicators that are required for eighth grade but are not covered in the Algebra I coursework. Although he complies, Mr. Anderson does not seem to view this practice as his primary responsibility. I hate the standardized tests. I always tell my kids, I’m here to get you ready for high school. That’s my number one job. If you do well on the standardized tests, great. But my job is to get you ready for algebra II and that’s the way I push myself during the year. (Anderson) Ms. Carla Case Ms. Case has been teaching seventh grade mathematics for all but the first year of her nine-year teaching career. She is very student-oriented and reportedly participates in all the school-sponsored motivational events such as hat day and pajama day, both of which were activities to encourage students to attend school during “Count Week.” (Count Week is the one week selected in October of each year on which school district funding is determined based on average daily attendance for that week.) Based on conversations and interactions between Ms. Case and students entering and visiting her classroom, she seemed to be a popular teacher with the students. She is animated and declared a passion for teaching mathematics and for teaching mathematics in a nontraditional manner. I feel this is my passion for why I teach math. I feel a lot of students struggle with math, and it’s their hardest subject. Most of the time when they come in at the

104 beginning of the year, they hate it. And, so, I feel that my job is to get them to enjoy it and I just want them to know for one year that math doesn’t have to be hard. It can be fun. There are projects and things we can do that make math exciting. It doesn’t have to be notes, bookwork, notes, bookwork. And that’s why I teach this, and that’s why I teach that way. (Case) According to the MCOI analysis and field notes, Ms. Case is most characteristic of a non-traditional teacher in practice. (See Appendix F.) Students have mathematics textbooks they use but Ms. Case does not rely on the textbook nor does she work through the textbook systematically. “There’s some things I’ve got to leave out because I’ve got to get to this. I’ve got to do this, and that’s what the state tells me. You know, so some things I do leave out.” (Case) As a general rule, Ms. Case reports that she uses the text book as a resource to build activities she uses to present the material in the classroom. As a result of the many professional development activities in which she has participated, Ms. Case reports having a large repertoire of activities and teaching ideas for mathematics. She makes songs, cues, and repetitive mantras a part of her instruction. She explains that using these tricks triggers associations for students. “That’s why I do those little things. As soon as I say ‘happy machine’ [absolute value], they all know it’s a positive. And, uh, those are just little tricks so you don’t have to teach your whole lesson. You can just say one thing and they remember it.” (Case) One of the determining factors in the sequence of instruction for Ms. Case is a county-wide initiative for short-cycle assessment. School districts within the county joined together to develop a pacing chart that would be in place at each grade level for all

105 schools in the county. This pacing chart, according to Ms. Case, was an attempt to ensure transient students, which are common in the area, would not be missing out on curriculum as they moved from school district to school district. Ms. Case reported that the pacing chart changed her teaching dramatically. I used to go in the order of the book the first couple years we taught without the [county short-cycle assessment plan]. Cause you know, they wrote the book. Kind of makes sense. Now I wouldn’t always use the book. You know what I mean. Or I’d use it for projects in there. That’s probably what I did. I used it for projects in there. Then when the [county short-cycle assessment plan] came along, you’re just like picking from this chapter, and this chapter, and this chapter, trying to get all the Standards. I mean, I was hitting all the Standards before but they were just in a different order and it was like, you know, we do all that number sense now where before I’d probably do bits and pieces throughout the entire year. (Case) Ms. Diane Davis Diane Davis, a sixth grade mathematics teacher, has been teaching for seven years. She is on her school building leadership team and is a part of the decision-making body for her school district. One of the main goals of this team is to focus on ways to improve student test scores. Ms. Davis has participated in the adoption of short-cycle assessment as a means of improving student achievement and is knowledgeable about ongoing intervention programs being implemented for at-risk students in her district. When questioned about the Standards, Ms. Davis reported that she knew the Standards and was very familiar with them but was required to follow the lead of her

106 district, and they had a curriculum map and pacing charts in place that she was required to use. I have the big book with the standards and I’ve been through classes with the Standards but when I’m doing my planning, I use my curriculum map that is designed by our district which is kind of based on the Standards. (Davis) Upon inspection of those documents, it was easily discernible that the district’s curriculum map was an exact reproduction of the grade-level indicators for sixth grade from the Ohio Academic Content Standards for Mathematics (OACSM). It appears Ms. Davis does not recognize this. Students in Ms. Davis’ classroom were seated in groups of four students. The primary collaborative activity of the groups seemed to be trading and grading homework papers. On one occasion, students were asked to discuss the problems they missed on their homework assignment after having traded and graded another student’s work. Other than this one incident, none of the visits to the classroom offered evidence of student collaboration or group interaction in this group seating arrangement. Evidence from the structured observations indicates that Ms. Davis uses traditional methods of instruction. She is focused on presenting students with mathematics content that she feels they will need to pass the state achievement test. She does so using direct instruction and views repetitive practice as an important part of ensuring that students retain the information. When asked about mathematics reform, Ms. Davis does not see reform as necessary for her students.

107 I’m sure I, and you know this probably changes depending on where you teach at, I don’t think out here in good old Townsend County there is, you know, they don’t push. We don’t hear a lot about math reform. But I’m sure there’s probably all kinds of wonderful things happening in larger populated areas. Cause I mean, here, I don’t really see a need for reform per se. (Davis) Ms. Davis also spoke at times about the need for students to learn more practical applications of mathematics such as learning how to use checkbooks and the computation necessary to keep a checkbook. Ms. Ellen Early Ms. Early teaches eighth grade mathematics in a campus style school building where all grades are housed under one roof. She teaches all eighth grade students in the district. A 17 year veteran, Ms. Early is a highly energetic teacher who expresses excitement about her job. She plays music in between class changes and is often seen dancing as her students enter the classroom. Her rapport with students appeared to be excellent, and she engages in conversations with her class about topics that indicate she is aware of their activities and lives outside the classroom. Ms. Early’s teaching style is student-centered. In addition to students engaging in mathematics content through projects, Ms. Early requires students to write about their findings and mathematical understanding, justify their reasoning, and explore various solution paths. As students work together, Ms. Early is often found in the midst of a group questioning students and trying to extend their thinking by asking questions that begin with “what if.” A cursory examination of her lesson plans suggested that problem

108 solving and projects were a regular part of her instructional planning. Often three days per week were spent in these types of activities. When given a chance to talk about the Standards, Ms. Early described them as an outline, a goal, and a set of rules to go by. She did not seem to view this as detrimental to what she was attempting to do in the classroom, however. “I mean, really, since they’ve come down on all that stuff, we’ve become better teachers because it makes us more accountable.” (Early) She readily accepted the accountability factor that goes hand in hand with the responsibility of being a teacher. You just have to take ownership. I just think that we as teachers, if this is the job that we are going to do, we’ve got to take ownership. We can’t just say it’s where they come from. You can blame it on mom and dad. You can blame it on whoever but, in reality, I have them more than their parents do so it’s my job. (Early) While accepting accountability for teaching what students need to know, Ms. Early also holds her students accountable for knowing what they are supposed to be learning and making every effort to learn the material. She requires students to keep an academic notebook that contains their homework, learning logs, daily problems, projects, and a list of the eighth grade standards. If they know what they need to know, I think that gives them a goal. You know, this is what I have to know by the end of the nine weeks or by the end of this chapter. It really makes them accountable. (Early) Students are required to correct any errors and use their notebooks as a source for review during an intervention class at the end of the day.

109 Ms. Faye Fout Faye Fout, a teacher of 37 years, plans to retire from teaching at the end of the current school year. Seemingly full of energy, she meets her students at the door and converses with them about extra-curricular activities, summer plans, and the latest movies on DVD. There is an easy familiarity filled with banter and teasing between the students and Ms. Fout. With the ringing of the tardy bell, the classroom atmosphere immediately takes on a serious tone, all talking ceases, and students begin working on the problem of the day as Ms. Fout takes care of administrative tasks. Ms. Fout and her students focus on mathematics for the next 42 minutes. Although her certification is a permanent elementary certificate, her knowledge of mathematics was evident in her teaching. She reports having taken some college-level mathematics classes over the years to increase her knowledge and to serve as a refresher for her personal growth. She tutors high school students who struggle with mathematics. When asked about the Standards, Ms. Fout expressed support and a level of satisfaction with the Standards. Well, the Standards have done a lot to kind of equalize curriculum over the state, and I think that’s a good thing in most cases. I understand that those are minimal requirements, and I should just not be going that far but be extending beyond that when I can. So, I see those as a minimum set of standards that each child needs to master. Effectiveness? I think if you just look at their purpose as being uniformity and a minimum level, I think they’re effective. (Fout)

110 Ms. Fout reports having mathematics labs once or twice a week, a fact which is substantiated by a survey of her very detailed lesson plans. While the labs are a regularly scheduled part of her teaching, Ms. Fout also uses simple activities and tasks throughout her lessons that allow students to explore and practice what they are learning. An example of this was on one occasion when students were reviewing translations, reflections, and rotations, she asked students to stand up beside their desks and did a version of Simon Says with all actions requiring students to demonstrate an understanding of the transformations terminology. Using information from the MCOI and classroom visits, it appears that Ms. Fout is using standards-based, student-centered instruction in her classroom. She consistently incorporated more than half the mathematical processes benchmarks from the OACSM in her lessons. Ms. Fout always required her students to explain or justify their thinking, both orally and in writing. She encouraged students to explore alternate ways to do the mathematics on which they were working. Her students, whether at her urging or on their own, demonstrated persistence. Ms. Grace Gardner Ms. Grace Gardner teaches 8th grade in a school district that has never met the state standard for mathematics on the state-issued local report card in the middle school. Ms. Gardner reportedly attended several professional development activities with Ms. Early, another participant in this study who taught in a neighboring school district. Ms. Gardner made reference to Ms. Early on several occasions as she talked about students, teaching, and community.

111 Based on criteria considered in the structured observations as well as classroom observations leading up to the structured observations, Ms. Gardner uses a traditional approach to teaching. On one occasion she did attempt to do a group project activity. Students appeared to be unfamiliar with any procedures or routines for working in groups. Following the lesson, Ms. Gardner talked about the lesson. Should I stand up there and teach? Should I stand up there and just give them formulas and stuff? No, I probably shouldn’t do that but to begin with, a lot of this stuff in eighth grade they’ve never seen before. Part of it, like that group work, that group work that we did today, puts me on edge because I’m like who’s doing what? Or, I can’t get to every group. Or, are they just copying? I do think different things should be done but sometimes it’s hard to mix it up. I mean, it’s really cause I’m from the stand up there, question, answer, discuss, whatever at the board school, and sometimes I have a hard time letting go and getting them in groups. (Gardner) Ms. Gardner, in her eighth year of teaching, also attributed her directed instruction strategy to her lack of experience and the fear of losing control in the classroom. Ms. Gardner expressed concern that the difficulties for the school district as far as meeting goals for achievement testing results were overwhelming. Students were apathetic. They don’t care. They don’t care one bit. You know, you can preach to them all about it. All about it. You know. Let’s get our achievement scores up. Let’s just

112 get down and get to it. There are some kids that are like, ‘If I don’t pass the eighth grade test, I pass whether I pass it or not so who cares.’ (Gardner) In addition to the apathy on the part of students, Ms. Gardner attributes another part of their lack of success to parents who are not involved in their child’s education and are unable to help them with the mathematics students are learning in eighth grade. Ms. Gardner views the Standards as “what the kids should have learned by the end of their 8th grade year.” (Gardner) Having served on the committee that developed the pacing chart for the county school board, Ms. Gardner admits that she did not know what the Standards were at the time and struggled with participating in the development of a pacing chart that was coherent and cohesive. Ms. Gardner found it helpful that this committee pulled together a series of questions for each grade level indicator from the Standards. She still uses those questions along with the released test items from the OAT on a daily basis to prepare students for the achievement test. Ms. Gardner, aware of state department initiatives, talked about the proposed revision of the Standards. I think there’s too many for each grade level and they said they’re going to change those to where we don’t have to teach as many but I think they’re just going to shorten it so it doesn’t look like there’s as many…A lot of that stuff I think is stuff they probably won’t need to know until they’re freshmen or sophomores. I think it’s stuff that I should not be teaching. (Gardner)

113 Ms. Harriet Holmes Harriet Holmes has been teaching for 25 years. Currently teaching mathematics to seventh and eighth grade students, Ms. Holmes seems very frustrated with her job. She expressed this succinctly. “And there are days I’ve gone home and cried cause I haven’t accomplished anything and these kids are not going to pass this test. I don’t know what to do.” (Holmes) Ms. Holmes has attended many mathematics professional development programs including a two-year collaboration between two institutions of higher education that promoted networking with middle school mathematics teachers in her region. Ms. Holmes reports faithful attendance at these collaborative meetings and expressed satisfaction with the materials and ideas she obtained from these sessions. There was no evidence during the observations of use of any materials other than test preparation resource worksheets. Ms. Holmes is a traditional teacher who relies solely on a direct instruction model for teaching. She taught from a seated position at her desk and at no time during the observations did she ever get up from her desk other than to walk to the hallway during a class change. When students in the class had questions, they went to her desk and waited in line for help. Copies of grade level indicators for both seventh and eighth grade were attached to the front of Ms. Holmes’ lesson plan book. When asked about the purpose of the Standards, Ms. Holmes replied,

114 Basically to guide us through from one grade to the next. But some of them are so complex it’s hard to understand what they want. If they were a little bit more, I guess, dummied down, but they’re so broad on some of them that anything can fit in there. I don’t like that. (Holmes) Ms. Holmes went on to say that students today are asked to learn mathematics content that she, as a student, did not encounter until much later in her school career. Ms. Ingrid Ivy Ingrid Ivy is a fifth grade teacher who has been teaching for 10 years in a small rural school. Her classes each day are blocked into three sections lasting 90 minutes. The students are ability-grouped according to their reading scores on the OAT which, according to Ms. Ivy, may or may not be indicative of their mathematics ability. Ms. Ivy uses a traditional approach to teaching. Her students are seated in rows and spend the 90 minutes of class time following a routine procedure. Individual students are called on to put homework problems on the board and each student checks his/her own work. Homework is collected and the teacher, using the smart board which allows her to project material from her computer to a screen in the front of the room, goes over several OAT test item samples from previous OAT tests based on the mathematics content covered in the homework assignment. Students work the problems at their desk and then the teacher goes over the correct answer. Finally, the lesson of the day is presented with Ms. Ivy putting examples on the board. She then makes a homework assignment, and students begin their homework. Any student who finishes the assignment before class time has ended is to read silently until dismissed.

115 Ms. Ivy seems to view the Standards as directive. Well, I assume they’re to tell us what to do because pretty much everything we do revolves around that. I mean that’s the basis of every day. That means this is what I want you to do and this is what you’re going to do basically. (Ivy) Ms. Ivy seems distant from her students and views her responsibility to them, as well as to parents and district administrators, to be covering the Standards and preparing students to take the achievement test. She expresses her concern that students are not being prepared for life. “I feel there’s a lot of things kids need to know that we don’t get the focus on. They’re probably not going to go to college.” (Ivy) Ms. Jane Johnson Jane Johnson has been teaching mathematics for 21 years. She reports more than 90 credit hours of coursework from professional development prior to beginning her postgraduate degree. Ms. Johnson credits the large amount of professional development and her years of teaching experience with giving her the flexibility to individualize instruction for students who struggle with a concept. I’ve always been very flexible. If they didn’t understand the concept we spend more time on it. And like I said, that’s one good thing about, I guess, 22 years now, hard to believe. But, you know, I can come up with 10 problems out of anything, out of any concept we’re doing quickly that I can add to their (pause), give them more assignments, or more practice on that concept. And I have so many resources that I’ve accumulated over the years. I can grab something quickly. (Johnson)

116 Ms. Johnson also credits her experience working with the high school curriculum as well as middle school curriculum as another factor in helping her see the big picture as it relates to the Standards, thus, giving her an advantage over some of her colleagues who do not have such an extensive mathematics background. Ms. Johnson views the Standards as an improvement over previous practice when everyone developed their own course of study. She is cautious, however, admitting that some of the Standards are a minimal learning goal and teachers are free to interpret them as they see fit. “They wanted to close the gap. Make sure that everyone was learning at least a minimal amount, not that all the standards are minimal. You can write them as extensive as you would like.” (Johnson) In her classroom, during her participation in this study, it was evident that Ms. Johnson frequently had students working in groups, discussing mathematics, and finding applications of the mathematics. On many occasions, Ms. Johnson would begin to ask a question and students in unison would complete the questions. “So what? And, who cares?” Ms. Johnson explained that students need to know why the mathematics is important, what the real world applications are, and who would use the mathematics. The structured observations in Ms. Johnson’s room would seem to indicate that she is using standards-based instruction in her teaching. Her lesson plans validate that distinction. Students more often than not are doing group projects most of which they present to the class at the conclusion of a unit of study. When teaching, Ms. Johnson is attentive to the mathematical process standards and her students spend more time with

117 their own explanations and justifications than with teacher lecture. An answer to a question is always followed by a request for reasoning or justification. Ms. Kathy Kale Ms. Kale teaches seventh, eighth, and ninth grade mathematics at a small rural high school in southern Ohio. The school had gone through a restructuring process prior to the beginning of school, and Ms. Kale was assigned to teach four different grade levels of mathematics. In addition, students with special needs were included in regular classrooms for the first time in many years. Ms. Kale had no experience working with students with special needs and expressed that this was a great source of stress for her during the current school year. Ms. Kale speaks of the need for setting higher standards for students. She says, “Even with the Standards we are just meeting the basic. That’s still the bottom of the bucket in a way. You’re not excelling over that.” (Kale) She seemed to relate the idea of minimal Standards to what she sees happening in schools as far as expectations for students. She expressed frustration for situations when students move into the school district and report loving the school because it is easy. She contends that educators should not allow socioeconomic factors to justify having low expectations for students. Working in a school district that has a large population of transient students, Ms. Kale reportedly likes having a county-wide pacing chart for the Standards to ensure that students’ mathematics learning is not disrupted more than necessary as families move from school district to school district. She feels that teachers need to be held accountable for teaching mathematics and being responsible for the Standards accomplishes that.

118 I can remember… my first days at [the elementary building] coming into my room and looking for the course of study and finding it all dusty on a shelf cause the truth is, back then, people did not follow any kind of guidelines. So the Standards, having Standards for the whole state, I like. (Kale) Ms. Laura Limley Ms. Limley teaches seventh grade mathematics. She teaches general mathematics primarily but has two classes of students taking seventh grade pre-algebra. Ms. Limley seats students in groups of four with desks facing each other and frequently instructs students to work together or discuss the mathematics she is presenting. Even given this directive, students worked individually and there was seldom any discussion about mathematics during classroom observations. Ms. Limley has taught mathematics for 12 years and describes her teaching style as focused on problem-solving. When asked how often she did activities, projects, or handson lessons she described a changing practice. Not as much as what I used to because before, when I taught sixth and seventh, when I first started, it was almost like everyday. Now it’s limited. I can’t do as much it seems like because of the time factor. But certain areas lend themselves more easily cause it’s like you got to get to something and then get on with it. Some of the topics that I did more hands-on stuff with when I first started isn’t considered as important now as other stuff so you know, like with volume I still do the building stuff but only for one day just to kind of get that concept why that formula is what it is. You know, putting the numbers up so they can see those. But, it just seems like because of the

119 test, I feel like my instructions changed. And, you know, a lot of people would say that shouldn’t be because if you’re still teaching, your kids are going to pass the test anyway. But it has changed it. A lot I think. (Limley) Last year, Ms. Limley’s students performed well enough on the state achievement test to earn the school a point on their local report card. She prepared her students by using test questions from previous tests as her problems of the day which serves as the opener for class each day. She considers it part of her job to be knowledgeable about the test and to use that knowledge to help her students. “I feel like I have a responsibility, since 2005, they’ve been taking this test. I’ve, you know, I look over the old test. I look and see what’s going to be on it. That’s called teacher responsibility.” (Limley) Part of that responsibility, in her view is to know which grade level indicators from the Standards appear most often on the test and to make sure students are familiar with those concepts. “We focus on power standards and those are the ones we focus on and we try to incorporate the others in those power standards but just maybe not stress the importance as much as those power standards.” (Limley) The analysis of classroom practice from the observations and a cursory inspection of lesson plans indicate that Ms. Limley teaches using traditional methods of instruction. The textbook is her main resource and there was nothing in her lesson plan book through December that would indicate any teaching strategy other than directed instruction. Student notebooks were filled with practice worksheets from the textbook publisher and student written released test items from the Ohio Department of Education.

120 Data Analysis Data analysis for this study was completed in part using Weft QDA, a software program created by Alex Fenton for the analysis of textual data (Fenton, 2006). The software program was downloaded from the internet and is a basic code-and-retrieve program that, according to the author, is easy for first time users to use (Fenton, 2006). The first step was to import all documents from the study into the program. This included interview and observation transcripts, field notes, and structured observation records. Prior to beginning the coding process, two categories were formed for the purpose of organization and easy retrieval of data throughout the analysis process. The first category created was Participants. The researcher then formed subcategories for each participating teacher in order to organize all documentation associated with a particular teacher participant into a subcategory. This allowed the researcher to recall and view individually all documents associated with each teacher as a whole. The second category created was Interview Questions which allowed the assimilation of data by all participating teachers’ responses to each interview question. A subcategory for each interview question was created. With this basic structure in place, the data analysis process began with open coding after the first participant was interviewed. Open Coding As each document was added to the software program, it was coded with a one word descriptor or phrase that summarized ideas presented in the data. Transcripts and field notes were considered line by line and general codes were assigned. All subsequent documents were added and reviewed in the same manner. As the data was coded, it was

121 compared to previously coded data. Large numbers of codes were created throughout the process of open coding of data. New codes were developed as each participant’s document files were added to the program. While the number of new codes being added lessened as new participant information was added, it wasn’t until the last two teachers’ final interviews were added that no additional codes were necessary. Often times, based on content and context, a piece of data was assigned multiple codes. An example of open coding where one statement might be coded using multiple codes follows. This statement was made in response to the question ‘what is standards based instruction?’ To me that is a lesson that is driven from what the kids need to master. So to me, if I, the more standards that I can combine in one lesson to just keep instilling it is better for me and better for them. (Early) This passage (“a lesson that is driven from what kids need to master”) was assigned the code what is standards based instruction because it was how the teacher defined standards based instruction. It was also assigned a code, focus of instruction, because with part of this statement, “what kids need to master,” the teacher seems to be saying that she focuses instruction on the Standards. The Standards tell us what students need to master. The code, repetition (strategy), was added for the phrase “just keep instilling” because it suggests the teacher believes students need multiple encounters with the content. The phrase “the more standards that I can combine in one lesson” was suggestive of the code bang for buck which categorizes teachers’ attempts to cover more than one standard per lesson. In more general terms, this statement was coded teacher efficacy

122 because Ms. Early seemed to be inferring that her success, being better for her as a teacher, was based on how well students learn and understand the mathematics taught in her classroom. The order in which data collection for each participant case was started was successive—first, second, third, and so forth as indicated by the alphabetic ordering of their pseudonyms. Completion of data collection, however, did not necessarily occur in the same sequential order. Additional participants of the study were added prior to completion of a participant. Most participant data collection was being conducted simultaneously, often scheduling three or four teachers at a time due to varied class schedules and the need to log time in several classrooms each day. Consideration of this fact does not alter conclusions drawn from the study but may have been a consideration in the constant comparison of data. (For beginning and completion dates, see Appendix G.) Axial Coding Throughout the process of open coding, as new data were coded, the researcher continued to go back and explore previous data for new connections based on insight from each successive participant. The ideas for categories began to form during the process of axial coding and a review of all previous codes allowed the researcher to begin looking at commonalities that existed in the data. Open codes were clustered together into broad categories forming the first level of axial coding. These broad categories were Standards as a mandate, value of Standards, knowledge of Standards, applicability of the Standards, standards-based instruction, teaching strategies, real world applications,

123 planning for instruction, state testing, teaching to a test, teachers’ view of mathematics, teacher beliefs, teacher efficacy, accountability relationships, responsibility, and barriers. As these categories were considered further, which represented a second level of axial coding, some of the categories seemed related as did the data from which the categories were selected. Open coding from the data in the teaching strategies, barriers, and planning for instruction categories often overlapped, fitting into more than one category. Many pieces of data were connected to all three categories and all three categories seemed to inform the researcher how teacher participants thought about their practice. The same was true for teachers’ view of mathematics, teacher beliefs, efficacy, and responsibility. All were constructs created in the teachers’ mind and could be examined on a level of teacher beliefs. It seemed logical to think about the categories on a more conceptual level. Therefore, the second round of axial coding revealed four core concepts which accounted for most of the data. The core concepts were Standards, testing, beliefs, and practice. Table 3 shows the categories from which core concepts were developed.

124 Table 3. Categorization of First Level Axial Coding to Second Level First level categories

Second level categories

Standards as a mandate value of Standards Standards knowledge of Standards applicability of the Standards state testing accountability

Testing

teaching to a test teacher beliefs teachers’ view of mathematics Beliefs teacher efficacy responsibility planning for instruction real world applications teaching strategies Practice standards-based instruction barriers relationships

125 A Procedural Aside The number of codes developed during open coding seemed overwhelming to me, a novice researcher. The use of Weft QDA software allowed data to be coded and then subsequently retrieved based on coding assignations. This format, while simplifying the coding process and streamlining the retrieval of similarly coded material, presented a problem for me. Relationships between codes and comparison of coded data were difficult to see due to opening program windows, switching between program windows of data, and going back and forth between files in general. I needed data that did not minimize to the window tray every time I tried to compare it to another piece of data or look for a similar statement. I needed visuals and a new plan. To resolve this dilemma, I printed the files from the original open coding of data that allowed me to look at every piece of data that had been coded with the same keyword or phrase. I then wrote all those open coded keywords and phrases on separate post-it notes so that I could physically manipulate the codes. I used the keywords and phrases post-it notes to search for relationships between open codes. As I moved the postit notes into similar or related piles, categories became more defined resulting in names describing relationships. For example, when the post-it notes labeled hands-on activities, memory devices, practice in class, warm-up/review, homework, and cooperative learning were compared and contrasted, strategies emerged as a category. This, a result of first level of axial coding, is reflected in figure 1.

126 practice in class

warm-up/ review

hands-on activities

memory devices

homework

Teaching Strategies

cooperative learning

Figure 1: Example of open coding to categories. Categories were developed at this first level of axial coding. Resulting categories can be found in Table 2. In thinking about the categories and the connections that existed between them in relation to the research questions, the evidence suggested there were four core concepts at play. Most of the data could be addressed by one of four core concepts-- Standards, testing, beliefs, and practice. The next logical step in analyzing the data seemed to be taking each of the core concepts in turn and examining the interrelationships between the pieces of coded data within the core concept. Laying aside the data associated with the core concepts of testing, beliefs, and practice, initially I began working with the pages of coded data that were synthesized into the core concept of Standards because I intuitively surmised that all the data would eventually be related to the Standards. I cut into strips each individual piece of data from the pages of data which were associated with the Standards core concept. The strips were then sorted using the open codes from earlier in the data analysis

127 process which were written on post-it notes for categorizing the data. I placed the individual strips of data that I had cut apart on the post-it note category papers from the open coding phase of data analysis. Still struggling to make sense of the data, I noticed my rainbow centimeter cubes sitting on the shelf. Visuals are very important organizational tools for me so I assigned each of my participants a color and placed a centimeter cube on each piece of data associated with each participant. For example, I chose to make Mr. Anderson dark green. I went through all the data laying on the table in categories and placed a dark green centimeter cube on every piece of data attributed to Mr. Anderson that was represented in the Standards core concept. I repeated this process assigning the color yellow to Mr. Brown. I continued building this visual until each piece of data had a colored centimeter cube on it representing that the data was from specific teachers. Standing back and looking at the table filled with rainbow cubes, I noticed very distinct patterns. Although there were some discrepancies, certain colors seemed to cluster together. For instance, the code standards as minimal requirements was surrounded by red, black, white, and brown cubes. These were the same colors that were clustered to the bang for buck and cohesive progression codes. Similarly, yellow, pink, light green, and orange clustered together on other Standards codes lacks knowledge, ambiguous, too many, and too rigorous. When assimilating the relationship between similarly colored codes, I began to differentiate characteristics of various perceptions of the Standards. I repeated this process for the other three core concepts, testing, beliefs, and practice. As I considered these groupings of codes, three themes emerged based on

128 teacher perceptions. Discussion of the emergent theme and its three descriptors-performance, compliance, and resistance-- will follow in depth later in the chapter. This procedure was basically the same process the Weft QDA software utilized. Reviewing the qualitative analysis program results provided confirmation of those I was able to “see” with the colored cubes. The need to compare one category to another required frequent switching between computer program windows. Often this resulted in so many minimized windows at the bottom of my screen that some of them were no longer visible in the tray. With the computer program, if I reconsidered a code, it was a bit more complicated than physically moving a slip of paper from one pile to another. Sorting the rainbow cubes provided a color visual which was very beneficial in analyzing the data. Themes Data collection continued and new data were added to previously collected data and reviewed in light of the axial coding. Conceptual categories were developed which led to core concepts. After looking at the core concepts and comparing the relationships among them, themes began to emerge from the data. The themes that developed were, initially, perception (which was broken down into three subcategories, or descriptors, of perception) and, later, purposeful practice. Perception Perception as a theme describes how teachers process and assimilate the factors affecting their role and responsibilities as a mathematics teacher. Previously categorized data could be sorted or disregarded based on how it related to the teachers’ perceptions of

129 the Standards. Looking at the data with a focus on the Standards, new relationships began to develop. As categories and then data within each category were considered repeatedly and compared to other pieces of data, new relationships began to form. Eventually, the data could be sorted into three descriptors of perception—compliance, resistance, and performance. The first theme descriptor is compliance. Teachers’ statements which fall into this category indicate a view of the Standards as guidelines or mandates set forth by the state and accepted by local school districts. Teachers, reportedly, are instructed to teach the Standards and prepare students to pass the state achievement test. According to Ms. Ivy, “My responsibility is to teach them everything that’s covered in the Standards and prepare them for that achievement test.” Many of the teachers referred to the Standards as their bible. “The Standards are truthfully like the classroom bible. It’s what we have to go by.” (Davis) Teacher who value compliance interact with the Standards as individual gradelevel indicators rather than as a cohesive document. Ms. Gardner recounts her experience in putting together a yearly plan. “(I) wasn’t really with the understanding of what I was doing so I fit all this stuff in and it’s just, it’s just so much to do in one nine weeks. I don’t know how to change it.” Ms. Johnson offered similar experience. We (the county schools) have pacing charts that I really don’t care for too much. Overall, they’re not too bad but they (the teachers who developed the pacing chart) had to get all the indicators in somewhere and they threw some in that don’t

130 belong. I was wondering who on earth did these, but we do the quarterly assessments so definitely I hit those indicators within that nine week period. (Johnson) Because teachers for whom compliance is important adopt a view of themselves as teachers based on how well their students do on the achievement test, they place great stock in test scores and look at them as “checkpoints” in self-evaluation. I think you have to be able to back yourself up with the data and I think that’s what the OAT does, is you can back your teaching up with the data, and I think that’s comforting towards those administrators. I think they know that person is doing everything they possibly can. I honestly can’t think of a teacher who does a horrible job that gets a good score on the OAT. You know, you always know those people who don’t pull their weight in a building but those people also don’t do well on the OAT. (Limley) Ms. Case corroborates Ms. Limley’s thinking. “(Take) the achievement test. It’s nice to know if you’re being an effective teacher.” Much of Ms. Case’s interview conversation dealt with testing and test scores. Planning and teaching is focused on preparing for the state achievement test. Teachers for whom compliance statements were common set teaching goals in relation to standardized testing. “My only goal is for my kids to be on track with eighth grade standards and to pass the achievement test.” (Gardner) Teachers who exhibited compliance traits were most likely to accept claims made by others in reference to the Standards. Mr. Anderson, who takes pride in covering the

131 whole textbook before the end of the year, asserted that his textbook aligns with Ohio standards because the textbook company representative told them the textbook covered Ohio standards, and the textbook publishers included the Standards in the front of the textbook. Several school districts subscribed to test preparation programs for students which teachers supported and used on a regular basis based on information supplied by the software program representatives. Ms. Limley, for a warm up as a regular part of her class procedure, downloaded questions from the test preparation program based on which indicator she was working. Students worked on the questions during the warm up review. During the observations, this warm up segment regularly consumed the first 20 minutes of her 44 minute class. On a very basic level, teachers who are high on compliance are doing their job. Figure 2 shows early open codes that were used to develop the compliance descriptor.

checkpoints/ accomplishments accepting others interpretation

testing

guidelines

Compliance

Figure 2: Compliance descriptor open codes.

doing my job

132 The second theme descriptor is resistance. Resistance is categorized by an almost romanticized view of education. Teachers’ statements that align with this descriptor lament about how education has changed. They contend that teachers no longer have the time to do the things that are important, like reading to students or really getting to know them, because they have Standards to teach. Used to be the teacher would read, you know? We’d have a big chapter book or something like the Ramona books, and the teacher everyday would take so much time and they would read those to us and it’s, I know in math, I don’t teach the reading but you know, unless you’ve got a major activity worked in with it, you can’t take 20 minutes everyday to do that. You have Standards to teach. You know it’s like lots of little things that I don’t feel like I have time to do. You know. (Ivy) The Standards, according to many views, do not represent the mathematics that students really need to know. Statements that were consistent with resistance indicated that students need to know the mathematics they will use in everyday life such as balancing checkbooks (which was the only specific example teachers offered of mathematics students need to know for everyday life.) A lot of students don’t know the basic math to survive in life. We don’t teach checkbooks. That’s something every human needs and we don’t teach it unless you do it personally. Like, it’s decimals as your standard and unless you do that as a project, you’re not going to. They could go all the way through school and not have checkbooks. (Case)

133 Teachers who made more statements which could be described as resistance types of statements also seemed to be more tentative in their knowledge of the Standards and often used the terms pacing charts, Standards, grade-level indicators, and curriculum maps interchangeably to mean the same thing. Likewise, pacing charts or curriculum maps which were made up of grade-level indicators assigned for coverage during specific grading periods were viewed as different content than the Standards. Some teachers seemed to see the Standards and subsequent testing as detrimental to the education of students. According to teachers who exhibited traits of resistance, the number of standards is a problem. “I think there’s just too many for each grade level. (Gardner) Resistance statements also encompass time. There isn’t enough time to cover the mathematics well enough for students to get a firm understanding of the material because you have to keep moving on to the next topic. Mr. Brown described how time changed his instructional practice when he went from block scheduling to a single class period of 40 minutes. I felt the first two years I taught I had block classes first of all which changed how much time I had, but I was able to put so much more into algebra where I felt the kids going into high school knew it so much better. I was able to slow down and cover things at a slower pace, cover all possibilities, really make sure they felt comfortable with it. And now, it’s get through it, get through it, get through it cause I not only have to do algebra, I have to cover geometry and measurement, all the others that aren’t in the book. So I have to get through my book in order to

134 get out of the book and pull in other material. So, yeah, it definitely has changed the way I teach at least. I’ve seen it with everyone around me too. (Brown) Additionally, teachers who made resistance statements indicate the Standards require that students learn much harder mathematics at a lower grade level than ever before. Ms. Gardner reports, “Oh no, we’re doing high school math. I learned this stuff in high school. Not in sixth grade.” Ms. Holmes not only finds the mathematics challenging for students but also talks about not understanding the Standards herself. “Some of them are so complex it’s hard to understand what they want.” (Holmes) These confining factors contribute to students who do not have a solid foundation in mathematics and, in many cases, are identified as students who just don’t care and can’t be motivated. They don’t care. They don’t care one bit. You know, you can preach to them all about it. All about it. You know, let’s get our achievement scores up. And there are some classes that are very excited about it like this last period class. If we told them so and so is beating you, just that group of kids is like that. Let’s just get down and get to it. Then there are some kids that are like if I don’t pass the eighth grade, I pass whether I pass it or not so who cares. (Gardner) Figure 3 shows early open codes that make up the resistance descriptor.

135

kids don’t get it wrong math focus

changing education

motivation

too many standards Resistance

Incomplete understanding of standards

too rigorous time

Figure 3: Resistance descriptor open codes. The third category of teacher perception is performance. Teachers who made performance statements view the Standards as minimal competencies. They see them as broad statements that allow teachers to move beyond the literal generalities of the indicator to help students develop to a more sophisticated level of understanding. The Standards, according to performance-minded teachers, form a coherent and cohesive logical progression of the study of school mathematics. From a performance stance, teachers attempt to teach toward a bigger goal. They view mathematics as applicable to real life and seek out real-world applications of mathematics as a basis for instruction. Mine has always been to prepare them for life, more so than anything else. That they have the necessary math skills that they need to function in everyday life.

136 Some of the things I teach possibly aren’t going to be used on an everyday basis but could be. If they know how to use it, they will. In doing so, I hope that they do well on the standardized test. I don’t feel that I am one of the teachers who teaches to the test. I want to see connections. (Johnson) This philosophy of teaching mathematics as connections, allows teachers who value performance to address multiple grade-level indicators in every lesson. They are looking for more “bang for the buck”. (Johnson) Covering grade-level indicators is secondary to preparing students to be mathematically literate. As mentioned by Johnson above, performance teachers do not consider teaching to a test as appropriate practice. Quite honestly, I don’t worry about that state test too much. I just really don’t. I mean, I know it’s important and I’m sure I cover everything that could be on there but as far as preparing them for that test, I don’t. That’s just the way I view it. (Fout) Ms. Kale became quite animated when discussing teaching to a test. I refuse to teach that test. I don’t use that Buckle Down6 stuff ever. I refuse to teach a test. I will never ever do that. I hate that. Oh. Because, because if you’re teaching the thinking and you’re having those discussions all the way through the year, I think you’re going to be just fine. And, how freaking boring is teaching a test to these kids? If I had to sit and do those workbooks or any of that stuff-- how boring is that? That is really pretty boring and I like math so, you know?

6

Buckle Down is a state-specific test preparation series published by Buckle Down Publications to prepare students for the state achievement tests.

137 “That test,” while a major focus for compliance-minded teachers and a major distraction for resistance-minded teachers, also has significance for teachers who value performance traits. These teachers view the Standards as having made other teachers accountable for doing their jobs. I think a lot of the reform that has taken place has been very good because, especially from the middle school perspective, that we’ve, there have been teachers that say I don’t teach that cause I’m not comfortable with the material or I don’t like that and the reform and Standards have forced them. You have to teach it anyway. (Johnson) Figure 4 shows early open codes associated with the performance descriptor. Standards are minimum

Real world applications

bigger goal

Refuse to teach test

coherent, cohesive progression

Performance

Figure 4. Performance descriptor open codes.

accountability

138 Purposeful Practice The second theme developed from axial coding, purposeful practice, was developed intentionally to answer the research questions. Purposeful practice as a theme describes a teacher’s implementation of instructional strategies that best compliment the relationship between perception and teaching practices which allow teachers to fulfill their roles and responsibilities. Classroom practice is a function of utility. Teachers plan to do what they deem necessary to accomplish a goal. In the case of many of the participants, the goal is to teach in a way that allows students to successfully pass the OAT. This also allows teachers to be successful in performing their job. They determine their responsibility to various stakeholders of the educational process, set goals for meeting those responsibilities, and then decide upon the most efficient and effective manner for meeting the goal. For most of the teachers in this study, the desired outcome of their teaching, the thing that verified their status as a “good teacher,” was for their students to pass the state achievement test. Whether by making decisions based on the teacher’s own learning experiences (which all teachers reported to be very traditional) or by making decisions based on expediency (as time was mentioned as a determining factor of practice by every participating teacher), teachers’ practice was pretty standard. Decisions about practice usually did not vary from teacher to teacher as to what was considered appropriate in the classroom as far as their teaching methods and strategies were concerned. Classroom observations were extremely similar from day to day and from classroom to classroom, and most teachers, when asked, provided basically the

139 same general outline as provided by other teachers of what one would see on a daily basis as far as classroom structure was concerned. Generally I take attendance then we settle down and go over the homework. Then I start working on whatever we’re doing. And of course, ‘do you understand it’, yada yada. I ask a question and they just sit there and look at you. Then it’s time to start the work and that’s when you find out who doesn’t understand it. And that’s generally what I do in every class. We have homework. They have time to get started on that. (Ivy) Only four of the teachers regularly used strategies that were not in keeping with traditional instruction. For those four teachers, there was a mix of traditional and nontraditional instruction. When asked if she had a typical organizational structure for her class, Ms. Kale whose practice could be described as standards-based replied, Not always. Maybe some times I mix it up too much but that’s kind of my personality too. I don’t have a set procedure and it goes day to day depending on how they (the students) are doing with the content. You’ve got to be real flexible and adjust things as you go. (Kale) For the most part, teachers are not particularly concerned with mathematics reform movements. Several teachers said they had never heard of mathematics reform. Ms. Davis, on the other hand, is aware of mathematics reform movements but does not find the possibility beneficial to her students. I don’t think out here in good old Townsend County… we don’t hear a lot about math reform. But I’m sure there’s probably all kinds of wonderful things

140 happening in larger populated areas…Here, I don’t really see a need for reform per se. (Davis) For others, like Mr. Brown, it is a conscious decision not to consider mathematics reform. “I would say I completely ignore it and just teach what works.” Not all teachers, however, shared this apathy toward mathematics reform. I think a lot of the reform that has taken place has been very good because, especially I see from the middle school perspective, that there have been teachers that I don’t teach that cause I’m not comfortable with the material or I don’t like it and the reform and Standards have forced them. You have to teach it anyway. (Johnson) Ms. Fout views mathematics reform as a necessary part of the changing mathematical needs of the world. She said, “When I look at math reform, I look at how we’re changing math and math curriculum to keep up with the needs of our world today.” Reportedly, mathematics reform played a minimal role in determining teacher practice for the majority of participants. The concern in planning for instruction solidly linked what participants deemed to be the goal of their teaching and the most effective course of action to achieve that goal. The goal was intimately tied to how the participants viewed or made sense of their role as a mathematics teacher. Grounded Theory The themes, Perception and Purposeful Practice, provide insight into characteristics, beliefs and practice of participants. While most participants aligned closely with one perception, seldom did perception account for all the data associated

141 with a participant. Teachers who were primarily compliant might have performance tendencies. Compliant teachers might make resistance statements. It was only when perspective was considered that participant statements, beliefs, and actions became somewhat predictable. Perspective is a point of view through which one sees or makes sense of something. Based on the data from this study, it appears that the way teachers make sense of their role in mathematics classroom instruction is grounded in their perspective on mathematics teaching. From one perspective, which will be called School Mathematics Perspective, teachers believe that, above and beyond anything else, it is the mathematics and helping students to develop a deep conceptual understanding of the mathematics they will use throughout their school career and life that is important. From the other perspective, which will be called Assessed Curriculum Perspective, teachers believe that, above and beyond anything else, it is preparing students to pass the state achievement test by teaching the mathematics most likely to be tested that is important. Student success is measured in part by their ability to achieve proficiency on the OAT. These perspectives directly influence how teachers address classroom practice. School Mathematics Perspective From one perspective, teachers perceive school mathematics as the most important element in fulfilling their responsibilities as a teacher. School mathematics is the guiding force. It is through the perspective of school mathematics that teachers interpret and give meaning to mathematics teaching. The school mathematics is important for future mathematics courses, beyond school, and without regard to a standardized test.

142 It is a foundation. “It’s the building block of science. So you can’t do anything else without it.” (Brown) Teachers, from this perspective of school mathematics as the big picture, view classroom practice and their responsibility as a teacher by looking, first and foremost, at the importance of school mathematics. [My goal] has always been to prepare them for life, more so than anything elsethat they have the necessary math skills that they need to function in everyday life. Some of the things I teach possibly aren’t going to be used on an everyday basis but could be. If they know how to use it, they will. In doing so, I hope that they do well on the standardized test. I don’t feel that I am one of the teachers who teaches to the test. I want to see connections. I really drive that home. Sometimes too much they tell me, but I think it’s really important….I always try to tie it back to who would use this, and my million dollar questions are “so what” and “who cares.” Why would somebody need to know how to do this? And I try to show them jobs and possibilities. With the Algebra I, it’s really not an accelerated class as the kids perceive it. It’s a foundational class for all the rest of the high school math. And, if they’re lacking in one of those foundation areas, it’s going to collapse. So I really want them to have thorough knowledge of Algebra I skills so that they can succeed in future math classes or in their job area. (Johnson) Through this school mathematics perspective, teachers view the Standards as statements that describe minimally what students should know and be able to do. They view their responsibility for teaching mathematics content as delving deeper than what is

143 minimally required by the Standards. Mathematical connections are important and presenting lessons that build on mathematical connections promotes teaching in a manner that integrates more than one and often several grade-level indicators in each lesson. The focus of mathematics instruction is on ‘the bigger picture” and how mathematics is vital to real world applications. We have our Standards and indicators and benchmarks and stuff but they want the learning to be more authentic, you know. Experiences in class that are going to, and projects and things that are going to, not so much, I teach this indicator, teach this indicator, but you know, making the math meaningful in class. (Kale) Teachers for a school mathematics perspective are not oblivious to the Standards. The value placed upon the Standards from the school mathematics perspective, however, has more to do with the value of the Standards as a mechanism for ensuring teacher accountability in teaching mathematics to their students. Ms. Fout believes that the Standards along with the OAT have increased teacher accountability. “At least now, with Standards, everyone is teaching the same math. And, although I don’t really like that OAT test, teachers aren’t showing movies on Friday afternoon anymore. They have a test to prepare for.” (Fout) Assessed Curriculum Perspective From this perspective, teachers perceive the Standards as the most important element in fulfilling their responsibilities as a teacher. The Standards are the guiding force. It is through the perspective of assessed curriculum that teachers interpret and give

144 meaning to mathematics teaching. From an assessed curriculum perspective, the value of mathematics is defined by the emphasis put upon the mathematics through the Standards. From this perspective, teachers view the Standards as “a set of rules we’re supposed to go by to have the kids master” (Early), and the purpose of Standards is “basically to guide us through from one grade to the next” (Holmes). Teachers who seemed to be operating from an assessed curriculum perspective had lesson plans which listed one grade-level indicator per lesson and had, usually in the front of lesson plan books, a copy of the grade-level indicators which had check marks to indicate which indicators they had already covered. In many cases, students were provided with the same list of indicators and were required to mark off or check indicators as they were taught. During classroom observations, the researcher noted there was usually little evidence of teachers connecting current content to previous or future content. It appeared that each indicator was taught in isolation. Teachers indicate that Standards provide a sense of direction for what they are to teach. “I like the math Standards. I like saying this is what you need to teach each year.” (Anderson) The use of pacing charts, however, might seem to imply that teachers had little control over what they taught and when. This is less a factor than one would expect. Pacing charts were used in eight of the schools and in seven of the eight cases, the teacher who participated in the study had a major influence on designing the pacing charts. In five of the eight schools, the teacher was the sole designer of the pacing chart for their grade level, and in two schools, the teacher served as one of three teachers who worked

145 together to create the pacing chart. (The eighth teacher was not employed by her current school district when the pacing charts were designed.) Teachers with an assessed curriculum perspective view their responsibility for teaching mathematics content in terms of teaching the Standards. In particular, they teach grade-level indicators. Rather than focusing on the importance of each mathematical topic, the teacher with an assessed curriculum perspective determines the amount of time spent on individual mathematical topics based on to what degree a topic is representative of content tested on the state achievement test. The historical occurrence of each individual grade-level indicator on previous achievement tests is frequently used to determine which indicators are power indicators. We have some power indicators that we have, you know, looking at what’s been on past OATs and what has been really hit hard in the past. You know, we kind of call those our power indicators and we make sure if they don’t know anything else, we make sure that they know these. (Davis) Whether these power indicators are referred to as such, the mathematics taught in the classrooms of a teacher who holds an assessed curriculum perspective is based to a certain degree on how frequently the content occurs on the state achievement test. When asked about how she decided what to teach, Ms. Limley replied, “The Standards. Like I said, that’s what I do. I pick the ones that are strongest for me to cover that I’ve seen the most on the test.” The importance of mathematical content seems to be directly related to how much emphasis is placed upon a topic in testing materials.

146 Member Checking In December, following data collection and much of the data analysis, five of the participants met with the researcher to discuss preliminary findings of the study. The researcher presented the core concepts, Standards, practice, beliefs, and testing, as elements teachers consider when planning for classroom instruction. The theme of purposeful practice was also discussed. Participant discussion, while corroborating both the core concepts and purposeful practice as valid considerations in planning, did not provide new insight. However, it was valuable to the researcher in that it confirmed saturation of data. Summary The analysis of data collected throughout this study began with open coding. Through axial coding the researcher synthesized open codes into categories. A second level of axial coding brought four core concepts to light. These four core concepts, Standards, testing, beliefs, and practice, were used to organize all the data. Once all the data were organized by core concepts, comparing and contrasting of the data within each core concept allowed a theme to develop. The theme, perception, addressed the insight into the interaction between the Standards, testing, beliefs, and practice. While perception was the theme relating the core concepts, each of the core concepts could be divided into three distinct subcategories of characteristics which seemed to relate specific types of teacher behaviors and beliefs. The descriptors, compliance, performance, and resistance, were assigned to each of these distinctive subcategories of the theme perception.

147 Another theme was developed selectively in order to inform the study as to teacher practice. The theme, purposeful practice, explores the relationship between teachers’ goals or beliefs and the way they choose to teach mathematics. Figure 5 illustrates the relationship of themes and core concepts.

148 Theme 1 Performance Perception

Core Concepts

Theme 2

Standards Testing Beliefs Practice

Compliance Perception

Standards Testing Beliefs Practice

Resistance Perception

Standards Testing Beliefs Practice

Purposeful Practice

Figure 5: Relationship between Themes and Core Concepts. Finally, after looking at the interrelationships of the themes, a grounded theory emerged. Teachers look at mathematics teaching from one of two perspectives. They either view mathematics teaching through a school mathematics perspective in which mathematics is the focus and the Standards play a minor role, or they view mathematics teaching through an assessed curriculum perspective in which the Standards are the focus and school mathematics plays a minor role.

149 CHAPTER 5: DISCUSSION AND IMPLICATIONS Looking back at the focus groups held prior to beginning this study, it is obvious that the results of the study were fore-shadowed. Teacher statements made then are very similar to interview data from the study. The same issues and concerns were voiced. Teacher statements fell solidly in place with the data analysis of the study. The only missing component was the observation of classroom practice. Based on the results of the study, the focus group teachers would have only strengthened the evidence gathered. While research questions are designed to guide the direction of a study, there aren’t always definitive answers to those questions. In this case, the research questions can be answered in somewhat general terms but the definitiveness of the answer is questionable as the interpretation of some of the questions comes into play. This chapter begins with a brief statement in relation to each of the research questions. Discussion of Research Questions in Light of Themes Research Question 1: What are teachers’ perceptions and understandings of the Ohio Academic Content Standards for Mathematics? Not all teachers participating in the study have a thorough understanding of the Standards. For many, their perception of the Standards lacks knowledge, understanding, or both of the original intent which was to guide educators to teach mathematics in a way that would help students meet the changing needs of the world today. (NCTM, 1989) Computation, which in the past was the primary focus of mathematics, has become less emphasized as a need for problem-solving and application of mathematics has become more emphasized. For this reason, the focus of the Standards has increased the need for

150 students to develop mathematical process skills such as problem solving, reasoning, communication, connections, and representations. The NCTM standards documents treat the process skills as standards just as they do content. On the state level, that distinction is less evident as the process standards are embedded in grade-level indicators rather than being addressed as the content standards are. The intent of the ODE writing team was for teachers to provide effective instruction to students. The ideal for accomplishing this was by “teaching mathematical content through mathematical processes” (ODE, 2001 p. 194) rather than dealing with the process standards independent of the content standards. As a member of this original writing team, I recall much discussion about whether to have process standards as grade level indicators or to include them as verbs, or the action of teaching, within the content standards. The decision was made to write the process skills into the content standards so as to convey the message that mathematics content should be taught using the mathematical processes as an everyday part of instruction. While a lofty pursuit, this apparently is not the message received by all teachers. This may be due in part to the unfamiliarity with the original Standards document brought about perhaps by the use of pacing charts which list only the grade-level indicators for each grade level. Few of the teachers interviewed were familiar with the original Ohio Academic Content Standards for Mathematics book in which authors of the supplemental material found in the back of the Standards book described the philosophical approach to the embedding of process standards within the content standards.

151 Many teachers interviewed for this study perceived the Standards as a “grocery list” of what they are to teach on each grade level. Not only did they not attempt to promote mathematical connections for their students, the teachers, themselves, did not view the mathematics content as connected. The Standards, as a set of “things to teach,” can be checked off as they are completed. One teacher, Ms. Case, shared her checklist during a classroom visit and explained that the checklist helped keep her organized. She also listed grade-level indicators on the windows of the classroom with washable markers as she finished teaching each one. Additionally, she provided students with a list of grade-level indicators. This list was kept in a mathematics notebook, and students marked each indicator with the date it was covered in class. Nine of the participating teachers followed a similar procedure with student mathematics notebooks. Some teachers, however, perceived the Standards as minimal guidelines for what teachers should be teaching. These teachers viewed mathematics as the bigger picture that allows a teacher a great degree of professional discretion as to the depth and extent to which they take the mathematics. To these teachers, the Standards are representative of the very least that students should know. Additionally, these teachers think about the mathematics in terms of real world applications and as a necessary foundation for learning higher levels of mathematics. Research Question 2: How do teachers translate Standards into classroom practice? For many of the participating teachers, in particular those with compliance and resistance stances, the content and process standards were discrete. It would appear that while teachers have adopted the content standards, the process standards are being

152 neglected in favor of mathematics content being taught as procedure. The following excerpt of an interview, part of which was discussed earlier, expresses this distinction well. Researcher: Process standards. Process skills. How important are they in mathematics, and how do you work that into instruction? Mr. Anderson: What do you mean by processing standards? Researcher: The NCTM process standards are problem solving, communications, connections, representation, and reasoning and proof. Is this something you consider in instruction? Mr. Anderson: Well, a lot of times, well, do I get it from each individual student? No, I don’t believe I get that from each individual student because a lot of that I believe is above some of the students’ heads. A lot of because of not doing some of the group work, not doing some of the hands-on stuff. I think I lose a lot of it. However, when I do something on the board, when I do something in the class and the kids are explaining, giving an answer in class, normally I make them explain it. You know what I’m saying? For example, the magic line. I think you’ve seen that there. The kids jump all over it. Such an easy process. So much easier than the process that’s shown in the book. I wish other people would show it but a lot of people have never met, a lot of people have never seen it, you know. You go back to last year whenever the aide was in the classroom, I’d say probably once a week she was copying something off the board she had never seen in her life how it was introduced. You know. And once you see that easier process, it’s a

153 lot easier for kids to explain that process back to you. To show their work on paper or what have you…. I find that when you take and find an easier process, it’s going to be an easy process to explain. You have a long drawn out process, especially for students that struggle, it’s going to be a hard process to put that back into words. So, I don’t usually fool with that. I try to teach something that works every time. Every time. (Anderson) In some cases, as with Mr. Anderson, it appeared that process standards may have been a topic with which he was vaguely familiar but it was not something he reflected upon on a regular basis. Perhaps as a result of this focus on procedure, a behaviorist approach to teaching, that of procedural and repetitive practice, is the dominant teaching strategy being used by many teachers as they try to teach a Standards-driven curriculum to their students. With respect to the practice of the 12 teachers participating in the study, it is important to remember the presence of the researcher in the classroom was limited to a total of approximately 13 days in most cases. This consisted of two structured observations as well as approximately 11 days during which the researcher was present and writing field notes. Additionally, teachers’ lesson plans provided historical information on teacher practice. Documentation suggests 4 of the 12 teachers used standards-based instruction on a daily basis. (See Chapter 2 for a description of standards-based instruction.) It should be noted, however, that while some of the class sessions for these four teachers appeared to be aligned with characteristics of traditional instruction, the use of process skills was highly evident and tended to move the

154 instructional episode into a more standards-based category. It appeared that 5 of the 12 teachers were strictly traditional with minimal student and teacher interaction. The interaction consisted of students giving one word responses to step-by-step procedural demonstrations on the board as the teacher worked through problems and students took notes. It appeared that 2 of the 12 teachers made attempts to use projects or hands-on activities occasionally although the effectiveness of using the process skills during the class sessions was indeterminate. The final teacher had no interaction with students other than to give directions for assignments and on no occasion during the study did she get out of her chair which was placed at her desk. (She reported no physical reason requiring her to remain seated.) Lesson plans were examined to provide a sense of teachers’ regular practice. Most teachers used some type of documentation for lesson plans but details were sometimes absent or minimal. This absence of detailed lesson plans, especially for veteran teachers, was not unexpected as a previous study found that teachers’ planning progressed to a point where teachers felt comfortably prepared to teach spontaneously (Zimmerlin & Nelson, 2000). Traditional instruction, because of the structure associated with it, seemed to require minimal planning. A survey of lesson plans written by teachers who were traditional in their instructional approach showed minimal detail. A typical lesson plan gave the page number of the assignment and the numbers of the problems assigned. Another common component of these teachers’ lesson plans was the prominence of lists of grade-level indicators or pacing charts with planning materials.

155 Teachers indicated that these documents were used on a regular basis for planning instruction. Research Question 3: If teachers truly support the Standards and standards-based instruction, to what extent is this evident in their classroom practice? Prior to beginning this study, the researcher anticipated that this question would be the most straight-forward and easy to answer. This did not turn out to be the case. It appears that teacher support of the Standards is an ambiguous construct. There are varying degrees of support that may or may not be separable from purpose and value. It would appear from this study that most teachers support the Standards. The basis for that support may not be consistent with the goals of the Standards document, however. Some teachers support the Standards as defining the expected outcomes of their job which is to teach certain mathematical concepts. Their perception of the Standards does not move them beyond that level of support. Other teachers have a good understanding of not just the content standards but the interrelationship between the content standards and the process standards as well as the goals that brought about a need for standards. The same could be said for the teachers’ support of standards-based instruction. While it is evident that some of the teachers in this study have a good grasp of what is involved with standards-based instruction as it pertains to the call for mathematics reform, not all teachers hold that understanding of standards-based instruction. Many of the teachers in this study view and define standards-based instruction as teaching the Standards.

156 With this in mind, the extent to which support of the Standards and standardsbased instruction is evident in classroom practice depends upon the interpretation of this research question. If teachers view standards-based instruction as teaching the Standards and their understanding is that the Standards are mathematics content, then support of the Standards and standards-based instruction is highly evident in classroom practice. Teachers are doing what they interpret to be standards-based instruction. They are doing standards-based instruction in the way they feel is the most effective method of teaching which is basically drill and practice. Their practice is purposeful. If teachers truly interpret standards-based instruction in this way, it would seem that there is no disconnect between what teachers say they do and what teachers do. Teachers for whom compliance is important all fall solidly into this category. For teachers who have a performance perception, the answer to this question is a bit more problematic. Performance-minded teachers support the Standards, and they express support of standards-based instruction. Their practice, however, typically falls somewhere along a continuum between traditional and student-centered. This scenario was the situation for which exploration was intended originally with this research study. Initially, walking into the classroom of some teachers who exhibit a performance perception, one might assume a traditional approach to teaching, but after a bit of observation it becomes obvious that the teacher is attending to the mathematical process standards in her/his interactions with students. Students are explaining, describing, and justifying their thinking to other students as well as to the teacher. Connections are being made within the mathematics classroom as well as to real world applications of the

157 mathematics. Teachers aligning with performance perceptions seem to be able to balance some characteristics of traditional instruction with standards-based instruction through the use of mathematical process skills. Teachers who make statements aligning with a resistance perception, on the other hand, appear to support neither the Standards nor standards-based instruction. To say these teachers do not accept the Standards is inaccurate. They do not see much of the mathematics content as relevant to their students, but they teach the mathematics mandated by their school district leadership. While they teach in ways that are very similar to those with compliance perceptions, they do not have a clearly delineated understanding of the Standards or of how to be effective as a mathematics teacher. Classroom practice, based on observations, is traditional. The difference, albeit slight, between teachers who voice compliance perceptions and teachers who voice resistance perceptions is that resistance-minded teachers tended to spend more time teaching mathematics by using released test items as the mathematics content and by addressing specific strategies for passing the state achievement test. Research Question 4: What conditions influence teachers enacting standards-based instruction in their mathematics classroom? From the participating teachers’ points of view, one has to report that most teachers expressed the belief that they were doing standards-based instruction. When asked to describe the factors most likely to impact their ability to do standards-based instruction, teachers cited a few conditions that negatively influence enacting standardsbased instruction in the mathematics classroom. The most commonly mentioned barriers

158 teachers reported facing were lack of time followed very closely by students’ deficient levels of prior knowledge or mathematical foundation. Teachers, nearly across the board, cited “the Standards” as the most common factor taken into consideration when planning for instruction. Most teachers, when asked to describe their planning process named the Standards as their initial consideration. The OAT was the second most common determining factor. Other factors were, in the order of prevalence, pacing charts, students’ ability level or prior knowledge, pacing charts, students’ understanding of the current material, and time constraints. Prior research substantiates these findings (Laurenson, 1995; Ross et al., 2002). From the point of view of the researcher, the factors that influence teachers enacting standards-based instruction in their mathematics classroom begin with an incomplete understanding of the Standards and standards-based instruction. Teachers appeared to know the mathematics content standards, but most were unfamiliar or uncomfortable with the process standards. Teacher use of instructional methods promoting process skills was evident in only four classrooms. The majority of the teachers participating in this study interpreted standards-based instruction to mean simply teaching the Standards. Given this circumstance, the conditions that influence teachers ability to enact standards-based instruction in their mathematics classroom is a lack of knowledge as to the scope of the Standards and characteristics (or even the definition) of standards-based instruction.

159 Discussion of Findings Themes emerge from the conceptual categories during constant comparison of data. Some conceptual categories, predominant categories, are more evident than others and develop into themes as a natural consequence of data analysis. This is the case with the first theme which is perception. Oftentimes, the researcher will choose to develop a theme that is less prominent if the theme lends itself to enlightening the study. (Harry, Sturges, & Klinger, 2005; Strauss & Corbin, 1998) This is the case with the second theme, purposeful practice. Theme 1: Perception Perception emerged from the data as a theme. Perception is attaining awareness or understanding. (Agnes, 2003) The second level of axial coding produced four core concepts. The core concepts, Standards, testing, beliefs, and practice were dependent upon how one understood the Standards. Considering perception and within those four core concepts one could clearly delineate three differing perceptions of the Standards. Descriptors of the three perceptions, performance, compliance, and resistance, were chosen to represent those differing perceptions. The following summaries give a brief, general description of characteristics of each perception descriptor mined from the data. A performance perception allows one to see the Standards as minimal requirements which form a cohesive, coherent progression for learning mathematics. Mathematics is necessary for our futures, academically and in reality, and should be taught using real world applications. This approach to teaching promotes instruction where multiple indicators are addressed by authentic learning opportunities.

160 Another perception of the Standards is compliance. Compliance perception represents an understanding of the Standards as guidelines mandated by those in administrative positions. The Standards are the focus of instruction and the state achievement test is the measure of how effective the teacher has been. Student test scores relate directly to the effectiveness of the teacher. Teachers focus on preparing students to pass a standardized mathematics achievement test, and instructional decisions are made based on the importance of this test. From a resistance perception, teachers lack understanding of the Standards. The Standards do not represent the mathematics students, especially students raised locally, need to know. They are too rigorous and too many numerous. The educational process is less student-oriented, and students no longer care about learning. Many have a poor foundation in mathematics. And teachers use released test items frequently in their instruction. In short, the collection of teacher beliefs causes them to question the purpose, utility, and appropriateness of the Standards. Perception did not seem to be tied to teacher certification. Of the three teachers who most closely related to a performance perception, two of the teachers held elementary certification and one held secondary certification. Of the five teachers who most closely related to compliance, three held elementary certificates, one held secondary certification, and one held a middle childhood mathematics certification. All four teachers who closely related to resistance held elementary certification. An extended background in mathematics as part of undergraduate teacher preparation did not

161 necessarily predispose a teacher to a performance perception. There was nothing suggested by the data that explained this tendency toward performance. Theme 2: Purposeful Practice The second theme, purposeful practice, was developed because implementation is a key factor in addressing the research questions. It became apparent through this study that teachers ultimately teach in a manner that is purposeful to their perception of the Standards and perspectives on mathematics teaching. Standards-based instruction as it pertains to mathematics reform was not high on the priority list for most of these teachers. Hiebert, in a study from 1999, contends that most teachers do not teach in a manner that lends itself to mathematics reform. This observation proved to be true in this study as well. For teachers with compliance and resistance stances, however, their expressed goals were not to teach mathematics in a reformed manner. The main goal expressed by a majority of the teachers in this study was to teach so that students passed the mathematics achievement test. Teachers with a compliance stance saw this as the focus of their job. Because there is an emphasis on computational speed and accuracy on most mandated standardized tests (Ross et al., 2002), choices for instructional practice tend to be traditional as traditional instruction has been considered to be effective for testtaking preparation. (NRC, 1989) Teachers teach in a manner that they feel is most effective in accomplishing the goals they have set for their students and traditional instruction satisfies that need. From a performance stance, teachers who view mathematics as a cohesive, coherent discipline teach with the big picture in mind. Their focus is not on the Standards

162 as they pertain to testing. Their focus is on school mathematics as a discipline. These teachers view their job as helping students to understand mathematics on a deeper conceptual level that will allow them to use the mathematics in future endeavors. Grounded Theory: Perspective Perspective, according to the Webster New World Dictionary (Agnes, 2003), is the capacity to view things in their true relations or relative importance. Teachers of mathematics obviously consider mathematics to be important. Teaching mathematics then is an important undertaking and one that the teachers in this study take seriously. When considering the teaching of mathematics, it became clear that teachers thought about mathematics teaching from different perspectives. The relative importance of what teachers do in the classroom is based on the relationship between school mathematics as a discipline and the assessed curriculum. Some teachers, mostly those whose perception aligns with compliance, view mathematics through a perspective called Assessed Curriculum Perspective. The Standards are held in a position of importance. The importance of the mathematics is then viewed in relationship to the Standards. Because of this perspective of assessed curriculum, teachers unitize the Standards into grade-level indicators or single objectives. They view the Standards as a list of things to teach and standards-based instruction as taking indicators from that list and teaching them. Nearly, all teachers for whom compliance was evidenced when asked what is standards-based instruction gave a reply that was equivalent to “teaching what you’re supposed to be teaching” (Case). Teachers who make resistance statements also view mathematics through an Assessed Curriculum Perspective.

163 From another perspective, some teachers view mathematics teaching through a perspective called School Mathematics. School mathematics is of utmost importance and the Standards are viewed as a tool which organizes the various mathematics topics. Teachers for whom performance is most important tend to have this perspective. They translate Standards into classroom practice by looking at the big picture in mathematics. “I look at how everything fits together with what I’ve already taught. The progression, I guess, and how it flows. I kind of want a fluid transition from one area to another.” (Fout) Teachers’ perspectives provide the foundation on which each individual teacher forms the undergirding of their teaching practice. One’s belief as to the relative importance of the value of school mathematics as opposed to the importance of the value of the assessed curriculum to teaching practice positions their perspective as a central belief in the psychological structure of one’s belief system (Green, 1971). Every other belief and decision is based upon that perspective. Teacher perception and purposeful practice are predictable based on teacher perspective. Teachers who have a school mathematics perspective will generally teach in a manner that takes on the perception of performance. In order to teach school mathematics and help students do mathematics, it is imperative they develop a conceptual understanding of mathematics (Hiebert, 2000). Without process skills, this is unlikely to happen. Practice will be standards-based because the process skills are necessary. Teachers who have an Assessed Curriculum perspective will generally teach in a manner that takes on the perception of compliance or resistance. The focus is on teaching a checklist. The checklist prepares students for the state test. Teachers will teach in the

164 manner that has been proven in the past to allow teacher to efficiently and expeditiously cover a prescribed curriculum. Generally, this will be traditional instruction. The theory generated from this study is substantive. Repeated studies are needed to test this theory to determine generalizability. Interpretation of Findings They Don’t Know That They Don’t Know A push for mathematics reform has been at the forefront of mathematics education for many years (Battista, 1994; Finley, 2000; Herrera & Owens, 2001). Mathematics standards have been developed to form a framework for the mathematics curriculum (ODE, 2001). Standards books were distributed to teachers throughout the state with the assumption that teachers would read the books in order to learn about the Standards. An instructional commentary in the back of the book describes the process standards in detail. Based on the interviews with participating teachers, the Ohio Academic Content Standards book more often than not has been lost or misplaced. Only three teachers from the study could locate their copy of the book when asked. Teachers have copies of their own grade-level indicators tucked in lesson plan books or desk drawers. In many cases, the grade-level indicators have been transferred to pacing charts where Standards and related grade-level indicators have been separated to accommodate 9-week grading periods. As there are no grade-level indicators for the mathematical process standards, they are not included in pacing charts and seem to have become a non-issue. Of the 12 teachers in the study, five of them did not know what the mathematical processes were

165 when asked. Through logging time, observations, and field notes, it was apparent that incorporating the process skills in their instruction was not a priority for most of the teachers. Incorporation of the process skills is an indicator of standards-based instruction. The absence of this in most of the classrooms was not uncommon. Previous research by Spillane and Zeuli (1999) corroborate the findings in this study. Teachers sometimes profess to be doing standards-based instruction while adopting some indicators of student-centered instruction such as cooperative groups or seating arrangements but maintaining a teacher-centered focus. Unique to this study, it became apparent through interviews that in many cases, teachers reported their classroom practice as standardsbased, and when probed, described their definition of standards-based instruction as teaching the Standards. Even though all of the teachers who participated in the study had been identified as teachers who had participated in professional development and should have developed a vocabulary that included a definition of standards-based instruction, this did not seem to be the case. Statements proclaiming standards-based practice appeared to be made earnestly. Teacher Efficacy Common to most participants of this study was the emphasis teachers felt was placed on standardized test scores. There was a tremendous amount of pressure to do well on the OAT. Throughout the study, during time spent interviewing teachers, teachers frequently mentioned the OAT. They described their responsibility as a teacher to be

166 preparing students for the OAT. For a majority of the teachers, during class time, rarely did a class period go by without some mention to students of the OAT. Because of the emphasis placed on this test on so many levels, when teachers begin to seek validation of their effectiveness as a teacher, they look at their students’ achievement test scores as an indicator of their own effectiveness. Teachers based much of their feelings of efficacy on how well their students scored on a standardized test. They viewed passing test scores as proof of their quality as a teacher, as well as, validation that their practice was effective. This, as the sole basis for measuring effectiveness is troublesome as there are too many variables with standardized testing for which one can not control (Ernest, 1991). As student test scores have more and more become the standard against which teacher effectiveness is measured, teacher efficacy has becomes intimately tied to student achievement. Test results are splashed across front pages of newspapers. News programs seem to preface local reports of financial woes and proposed fund raising and school levies with overall student test data. Accountability measures to ensure that schools are performing well have become an everyday part of life. With such an emphasis on student test scores, there is pressure on the local level of school government to raise student test scores. Superintendents and principals talk to teachers about students’ test scores. The OAT, as well as its high school equivalent the OGT, is an everyday fact of life for students and teachers. Because of this scrutiny of student test scores, teacher efficacy, “a teacher’s belief in their ability to influence valued (emphasis added) student outcomes” (Wheatley, 2005, p. 748), is deeply seated in student test scores. There doesn’t seem to be

167 any doubt in the minds of teacher participants that passing student test scores are valued by principals and school administrators. This yields a convention of teacher efficacy that is contingent upon student test scores. While research suggests a distinct connection between high teacher efficacy and the use of standards-based instruction, one can reach the same level of efficacy while using traditional teaching methods (Ross & Bruce, 2007) especially when high student test scores appears to be a valued result of teaching mathematics. While the basis may not be the same for both groups, a high sense of efficacy is apparent in the case of teachers who are compliant and those who operate from a performance stance. Traditional teaching efficacy (Wheatley, 2000) is evident in compliance statements because efficacy is based on student achievement. From a performance stance, teachers assert that they can make a difference in student achievement. The difference is a deeper understanding that will serve their students well at the current time as well as in the future. Both groups have a high sense of efficacy. Not all teachers have a high sense of efficacy however. Smith (1996) discusses the detrimental effect low teacher efficacy has on students as well as on teachers. He contends teachers with a low sense of efficacy, which seems to be the case for teachers who make resistance statements, were less inclined to attempt any innovative types of practice in their classrooms. Teachers with a low sense of efficacy identify students’ lack of ability, lack of motivation, and poor family support as factors that contribute to the teacher’s ineffectiveness and lower test scores (Smith, 1996). The teacher can not control for these factors.

168 Efficacy does play a part in classroom practice and teacher focus. Ironically, it would appear accountability measures intended to improve mathematics teaching have removed the focus from school mathematics and shifted it to preparing students to pass a test. This shift in itself does not account for the assessed curriculum perspective, but the number of times 12 teachers brought up preparing students for state testing during interviews in different contexts, 127, indicates that teachers are focused on preparing students for a test. High test scores are the valued outcome of a year of teaching mathematics and therefore, the importance of the test scores situates teacher efficacy in that realm. The need to prepare students to take this test could be one of the factors prompting teachers to use a method of instruction that is comfortable and familiar and lends itself to preparing students to take a standardized test. It has been noted by researchers that oftentimes standardized testing, the vehicle intended to improve teacher quality, runs contrary to its purpose (Ross et al., 2002). Evidence from this study substantiates this. Implications for Policy and Practice Because of the emphasis placed on computation prior to 1989, a time which served as an apprenticeship of observation (Lortie, 1975) for teachers who are teaching today, many teachers still judge how successful students are based on their ability to learn basic facts and computation. And, arguably, most standardized tests are written with an emphasis on computation and memorization. Writers of the Ohio Graduation Test purport to use a majority of moderate and high complexity levels of questions which employ higher order thinking skills when in reality nearly half the available points on the

169 Spring, 2009 test were attached to low complexity test items relying on recall, fact recognition, definitions, and procedural computation (Regan, 2010). Regan performed an analysis of the mathematics section of the Spring, 2009, test administration and determined that the cut score for proficiency allowed a passing score without having answered a single question of moderate or high complexity (2010). The behaviorist approach to teaching serves that purpose well (Battista, 1994). Many teachers, in general, and the majority of the teachers participating in the study, still teach mathematics from that frame of mind. The behaviorist approach, however, does not lend itself readily to teaching mathematics as problem solving or reasoning which are the goals of standardsbased instruction. School mathematics is perceived to be a procedural activity where we have practice, repetition, and consistency. These actions are procedural and can be completed step by step. Understanding is not procedural. If testing is to be used as a determinant of student achievement and the goal for students is a deep conceptual understanding of mathematics, then testing will have to be changed to reflect this. With the adoption of the Common Core State Standards for Mathematics (CCSSM), a change in state-wide testing will be necessary if achievement tests are to reflect curriculum. This would provide an ideal time to consider changing the format of future tests to reflect the goals educational leaders say they are striving to obtain. A viable option might be to model new tests after TIMSS tests which are more performance-based. Teachers, in being efficacious, will prepare students to pass that test. The question then becomes whether a test can be written that actually assesses deep

170 conceptual understanding. And, if so, can this be taught from a behaviorist’s perspective? The current state of reality for mathematics teachers brings this issue to the forefront. Recommendations for Future Practice The state is preparing to launch a new campaign to introduce newly adopted CCSSM to teachers and other stakeholders. The structure of the new standards is different in that process standards coupled with standards for mathematical proficiency are combined into what is termed Standards for Mathematical Practice. Mathematical practice involves the work of doing mathematics. The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy). (Common Core State Standards Initiative, 2010)

171 In addition to the Standards for Mathematical Practice, this new document has Standards for Mathematical Content. As with the OACSM, the writers of the new standards document have tried to incorporate the Standards for Mathematical Practice into the wording of the content standards by using terminology from the Mathematical Practice Standards such as apply and extend, represent and analyze, develop understanding, summarize and describe to name a few. This attempt to solicit standardsbased instruction in classrooms is very similar to the previous endeavor by the writers of the OACSM document which has proven less than successful in changing the nature of mathematics instruction. A concerted effort must be made to change teacher perceptions concerning the nature of school mathematics as well as the nature of student learning if there is to be an increase in the occurrence of standards-based instruction in classrooms. The first step in this quest should be to ensure all teachers are receiving appropriate professional development that provides them with an introduction to the new standards, purpose and need for new standards, and time to collaborate with other teachers to explore what adoption of the standards should look like in their classroom. This can not occur in a day. Teachers need time to interact with the new standards. Effective professional development should be sustained. It should enhance content (the new standards) and pedagogy while allowing time for collaboration and collegiality (Guskey, 2003). Without effective professional development, teachers may deal with the new standards as they dealt with OACSM. Effective professional development, as described above, would essentially prevent the current situation discussed earlier in the section entitled They Don’t Know that They Don’t Know.

172 Additionally, it is important to include school administrators in this focused professional development. Teachers’ sense of efficacy is based on their belief that they can affect valued student outcomes (Wheatley, 2005). The spoken as well as unspoken message teachers are receiving from school administrators and other stakeholders is that positive student test scores are valued over teaching reform mathematics. “Even when supervisors promote reform efforts and give teachers in-service education on what they perceive to be constructivist methods, they still evaluate teachers using the old paradigm” (Anderson & Piazza, 1996, p. 53). This practice serves to validate the importance of achieving high test scores for teachers who view their responsibilities as a mathematics teacher from an assessed curriculum perspective. It, also, has the potential to suppress mathematics reform even further as legislators are hinting at merit pay based on performance. If performance is assessed using a paradigm of traditional instruction with student test scores being valued as an indicator of success, teachers will be unwilling to consider adopting a standards-based approach to teaching. It is imperative that educational leadership be educated in standards-based instruction. It is important to realize teachers’ basic perceptions as discussed in this study will not change quickly. Teachers beliefs and perceptions of school mathematics affects and is affected by the teachers’ attitudes. Attitude affects what the teachers’ intentions are as to classroom practice. Intention determines behavior which in turn affects beliefs. A conceptual framework created by Fishbein and Ajzen (1975) describes the relationship between beliefs, attitudes, intentions, and behaviors. This relationship is represented in Figure 6.

173

Belief about object X

Attitude toward object X

Behavior with respect to object X

Intentions with respect to object X

Figure 6: Conceptual Framework Relating Beliefs, Attitudes, Intentions, and Behaviors This framework conceptualized by Fishbein and Ajzen is based on the same premise as work by Roehig and Kruse (2005). Beliefs, or perceptions, will significantly affect classroom practice. There will be little change in practice until teacher beliefs can be changed. Teacher perceptions must become more aligned with mathematics reform and standards-based instruction before there will be significant change in classroom practice. Contribution to Literature This study provides a juncture for the combination of literature on current classroom practice, factors affecting classroom practice, and teacher efficacy. The three literature bases are all abundant in their own right, but this research pulls the three together in a meaningful way. With this study, teachers are given voice in the context of

174 their classrooms to talk about teaching, the decisions they make, and reasons behind those decisions. Each construct is intimately related and dependent upon the others and studies looking at each independent of the others are without context and speculative. The literature on teacher efficacy is validated by the actions and assertions of participants of this study. Literature on classroom practice is accurate and tells a story of a predominance of traditional methods over standards-based instruction. Factors affecting classroom practice were identified by the participants of this study. The contribution to literature from this study involves the interplay between these forces. Without qualitative methods, the researcher may not have been able to ascertain that some teachers have illconceived notions of what is standards-based instruction. It is now necessary to add a misunderstanding of what standards-based instruction actually is to the list of factors affecting classroom instruction. Literature on teacher efficacy should include discussion of efficacy that is based on the expectations of significant stakeholders in the educational community that can be spoken or unspoken. And, current classroom practice should address the affect goals and purpose has in teachers’ decisions about teaching practice. Recommendations for Future Research This study was designed in an attempt to allow the researcher to understand the disconnect between teachers’ espoused beliefs about the Standards and standards-based mathematics teaching and what actually happens in their classroom. An unexpected finding from the study indicated that some teachers interpret standards-based instruction as teaching the Standards. It was determined that perception was a significant factor in teacher practice, specifically as it relates to holding an understanding of standards-based

175 instruction. To explore the original research questions, future study is needed that builds upon these results but looks specifically at the practice of teachers who hold a performance perception. It was obvious through this study that teacher beliefs play a key role in classroom practice. While teachers’ perspectives are central to their belief systems, it would be interesting to explore how other beliefs are related to or a result of teacher perspective. This information would be key to inducing teacher change. Given that teachers from this study participated in vast amounts of professional development focused on mathematics education but still had an incomplete understanding of the Standards, further research designed to provide insight into this conundrum would be beneficial. Is it possible that professional development has morphed into a vehicle to perpetuate and enforce compliance? If so, to what degree is this happening and why? If not, what prevents teachers from taking away a more performance-based perception of their role in teaching? Many other issues came to the forefront during this study that call for investigation. Where do performance teachers “get their stuff?” On what are their perceptions based? How do the high levels of efficacy for performance and compliance differ? How would this same study pan out from an administrator’s point of view? Summary We are setting a standard for teachers to meet that they don’t recognize. Standards-based instruction is a construct that some teachers define differently than the definition mathematics educators attach to the concept. Teachers believe they are doing

176 standards-based instruction if they are conscientiously covering the Standards written by the Department of Education. This, on some level, sheds light on the absence of mathematics reform. Much can and should be done to change teacher practice. Valuing school mathematics is a highly desirable trait to nurture in mathematics educators and could bring about significant change in the way we teach mathematics. Teachers need a thorough understanding of the Standards rather than the limited view of the Standards as mathematical content that is predominant in education today. With the introduction of the CCSSM, educational leadership will have a chance to remedy mistakes of the past mistakes that include the use of standardized tests emphasizing computation and rote memorization. Teachers are efficacious by nature and do what is necessary to feel successful. As long as success is tied to student test scores, teachers will teach in a manner that prepares students to pass the test. True assessment measures what is valued and sought after. It is important, then, to determine what is valued and sought after. Teacher practice will follow. This is especially relevant in light of the adoption of the new standards which became a topic of conversation during the final focus group discussion. Yeah, math is important. And these new standards ain’t going to be any different. I saw a draft copy, you know, and they didn’t take any out. All they really did was move stuff around and, kind of, they added more stuff because now everything is connected to something else. Why do I need to know that? To teach my kids? My

177 kids ain’t going to get that. Why do they need to know that stuff? Are they going to change the test? That’s what I want to know. (Anderson) Yes, it is important to determine what is valued and sought after. Undoubtedly, teacher practice will follow.

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190

Appendix A: TIMSS Test Scores TIMSS Average Scores and Standard Deviation for Selected7 Countries 1995 2003 Grade 4

Grade 8

Grade 12

Grade 4

Grade 8

Australia

546 (92)

530 (98)

522 (97)

499 (81)

478 (84)

Cyprus

502 (86)

474 (88)

446 (73)

510 (85)

459 (81)

England

513 (91)

506 (93)

531 (87)

498 (77)

Hong Kong

587 (79)

588 (101)

575 (63)

586 (72)

Hungary

548 (88)

537 (93)

529 (77)

529 (80)

Japan

597 (81)

605 (102)

565 (74)

570 (80)

Netherlands

577 (71)

541 (89)

540 (55)

536 (69)

Singapore

625 (104)

643 (88)

594 (84)

605 (80)

Slovenia

552 (82)

541 (88)

512 (87)

479 (78)

493 (71)

United States

545 (85)

500 (91)

461 (91)

518 (76)

504 (80)

483 (92)

560 (90)

Note: 546 (92) indicates average mean score of 546 with a standard deviation of 92. This table was compiled from information available on the International Association for the Evaluation of Educational Achievement website (Boston College, 1996, 1997, 1998, 2004).

7

Countries were selected based on availability of data for all reported test cycles with the exception of England, Hong Kong, Japan, and Singapore. The latter three were chosen because of their performance and England was chosen because of the relationship between them and the United States.

191 Appendix B: Professional Development Extensive Professional Development Opportunities Attended by Participants Name

Description

Math Summer Camp Grades 4-6 and Grades 5-7

A program that focuses entirely on improving teaching & leaning in grade 4-6 and 5-7 mathematics. Teachers learn from practitioners who share classroom tested lessons that address Ohio’s Academic Content Standards in Mathematics. Focus sessions concentrate on hardto-teach concepts and student assessment. Teachers will receive a variety of instructional resources for the classroom.

Grade 4-6 Math Network

Participants are provided with instructional support, best practice instruction modeled at each session, and resources. The goal is to establish a partnership of colleagues who teach grade 4-6 math. Participants will be invited to share ideas that have proven to be successful for their students.

Grade 7-8 Math Network

Participants are provided with instructional support, best practice instruction modeled at each session, and resources. The goal is to establish a partnership of colleagues who teach grade 7-8 math. Participants will be invited to share ideas that have proven to be successful for their students.

OMAP(Ohio Mathematics Academy Project): Algebra, Number, and Probability – Grades 3-6

Participants learned how to apply key concepts and benchmarks for the Numbers, Probability, and Algebra content standards, create collaborative problem-based activities in mathematics for students, study tools and strategies to evaluate and monitor the progress of students, model specific teaching strategies that can be used in the classroom, build knowledge in mathematics content and pedagogy and join a community of teachers who want to learn.

192 OMAP: Geometry- Grades 3-6 and Grades 7-10

Participants learned how to apply the new geometry and spatial sense and measurement mathematics academic content standards, create collaborative problem-based activities in mathematics for students, study tools and strategies to evaluate and monitor the progress of students, model specific teaching strategies that can be used in the classroom, build knowledge in mathematics content and pedagogy and join a community of teachers who want to learn.

OMAP: Mathematical Processes- Grades 3-6

Participants learned how to apply key concepts and benchmarks from the Academic Content Standards, create collaborative problem-based activities in mathematics for students, study tools and strategies to evaluate and monitor the progress of students, model specific teaching strategies that can be used in the classroom, build knowledge in mathematics content and pedagogy and join a community of teachers who want to learn.

ENABL (Enabling and Nurturing Activity Based Learning) Grades 4-8 Mathematics

This professional development program will help you integrate the content standards into your instruction; help you develop activity-based teaching strategies; enhance your ability to teach reading comprehension as it relates to mathematics; and infuse differentiated instruction into lessons for your students with different learning abilities and styles.

Discovery Institutes

Participants will strengthen their understanding of concepts in mathematics through inquiry. Cooperative grouping techniques provide support for all learners.

iDiscovery: Sustaining Professional development through Web-based Learning Communities

Provides web-based learning communities that support professionals as they strive to implement strategies and techniques learned during Discovery Institutes and workshops.

193 eSMILES (Enhancing Science and Mathematics Instruction and Learning with Electronic Support

Participants analyze and student performance, discover new ways to teach difficult standards, and further content knowledge in mathematics and science

CMP (Connected Math Project)

Teachers will examine issues related to mathematics reform, enhance their understanding of mathematics content, develop intervention strategies, and practice using CMP teacher support materials. The course will assist educators in creating classrooms consistent with the vision promoted by the National Council of Teachers of Mathematics in Principles and Standards for School Mathematics and the Ohio Academic Content Standards for mathematics.

194 Appendix C: Interview Prompts 1) Tell me about your current teaching position. 2) When you begin planning for instruction, what do you think about and do? 3) Tell me what you understand about Ohio’s mathematics standards. 4) Tell me what parts of the Ohio Academic Content Standards book you have looked at. 5) Tell me about standards-based mathematics instruction. 6) “Mathematics reform” is a frequently discussed topic in mathematics education. Tell me about mathematics reform. 7) What is your role in mathematics reform? 8) Describe an average day in your classroom. 9) What are the factors that influence instruction in your classroom? 10) Compare and contrast the experiences of students in your classroom on an average day to the experiences you had on an average day when you were in ____ (insert grade level of teacher being interviewed) grade. 11) For you, describe what would be an ideal teaching situation. 12) What should I have asked to talk about that I did not?

195 Appendix D: Mathematics Classroom Observation Instrument

196

197

198 Appendix E: Teacher Information Certification, Position, and Longevity Teacher

Certification

Position

Longevity

Grades 6, 7, 8

11

Grade 8 (Algebra I)

8

AA

5 Year Professional – Elementary (1–8)

BB

5 Year Professional – Middle School (4-9) Mathematics

CC

5 Year Professional – Elementary (1–8)

Grade 7

9

DD

5 Year Professional – Elementary (K–8)

Grade 6

7

EE

5 Year Professional – Elementary (1–8)

Grade 8

17

FF

Permanent – Elementary (K–8)

Grade 7

37

GG

5 Year Professional – Elementary (1–8)

Grade 8

8

HH

5 Year Professional – Elementary (1–8)

Grade 7,8

25

II

5 Year Professional – Elementary (1–8)

Grade 5

10

JJ

5 Year Professional – Elementary (1–8) 5 Year Professional – High School (7–12) Mathematics

Grade 7, 8, 9, 10

21

KK

5 Year Professional – Middle School (4–9) Mathematics & Soc St 5 Year Professional – High School (7–12) Mathematics and Data Systems

Grade 7, 8, 9

20

LL

5 Year Professional – Elementary (1–8)

Grade 7

12

199 Appendix F: MCOI Results Name

AA

10

I. Physical Setting

Seating

II. Lesson Overview

Environment Major Resources

III. Instructional Overview

Content Delivery Place in instructional sequence12 Seating Arrangement Content Focus13 Primary instructional strategy

IV. Questioning

Student Activity Quality

V. Classroom Atmosphere VI. Analysis of Instruction Leading to the Development of Higher Order Skills

VII. Overall Classroom Rating Profile VIII. Mathematical Processes Benchmarks

8

Techniques Student Involvement Culture/Attitudes Amount of student investigation Level of Student Engagement in problemsolving Mathematics skills being developed Scale of 1-5 with 1 being lowest and 5 being most effective Number of mathematical processes benchmarks addressed (average of two observations)

BB

CC

DD

T8

N9

T

N

T

N

T

N

* X11 X X

*

* X X X

*

*

* X X X

* X X X

*

*

*

*

*

*

*

*

*

X *

*

X *

*

*

X *

X *

*

-14

-

X X -

X

X

X X X X X

X X X

X X X X X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

1

2

4

2

1

1.5

7

3

Indicative of traditional instruction Indicative of standards-based instruction 10 This information was not suggestive of teacher practice because teachers often were not in control of the types or placement of student desks. 11 X indicates teacher practice falls into the category listed at the top of the column 12 This information was not suggestive of teacher practice. The focus of the lesson would change based on where in the instructional sequence the observed lessons fell. 13 Content of the lesson was for information purposes only and did not suggest teacher practice. 14 A dash indicates that lessons that were the focus of observations were mixed (one traditional, one nontraditional) 9

200 Appendix F: MCOI Results Name I. Physical Setting II. Lesson Overview

III. Instructional Overview IV. Questioning V. Classroom Atmosphere VI. Analysis of Instruction Leading to the Development of Higher Order Skills VII. Overall Classroom Rating Profile VIII. Mathematical Processes Benchmarks

EE Seating Environment Major Resources Content Delivery Place in instructional sequence Seating Arrangement Content Focus Primary instructional strategy

T *

FF N * X X X

T * X X

*

*

*

X *

-

-

GG

HH

N * X

T * X X X

N *

T * X X

N * -

*

*

*

*

*

*

X *

*

X *

*

X *

*

X

X

X

X X X

X X X

-

-

Student Activity Quality Techniques Student Involvement

X X X

Culture/Attitudes Amount of student investigation Level of Student Engagement in problem-solving Mathematics skills being developed Scale of 1-5 with 1 being poor and 5 being effective for all

X

X

X

X

X

X

X

X

Number of mathematical processes benchmarks addressed (average of two observations)

X X X

X

X

-

X

X

X

X

X

X

X

X

-

4.5

3.5

2.5

1

6.5

7

1.5

1

201 Appendix F: MCOI Results II I. Physical Setting II. Lesson Overview

III. Instructional Overview

IV. Questioning V. Classroom Atmosphere VI. Analysis of Instruction Leading to the Development of Higher Order Skills VII. Overall Classroom Rating Profile VIII. Mathematical Processes Benchmarks

Seating Environment Major Resources Content Delivery Place in instructional sequence Seating Arrangement Content Focus Primary instructional strategy

T * X X

N *

*

*

*

*

LL

T * X X X

N *

*

*

*

*

*

*

*

*

X *

*

X *

*

-

-

X

-

-

X X X

X X -

X

-

-

X

X

-

-

X

X

X

X

-

X

X

X

X

X

X

X

X

X

X

-

X X X X

Culture/Attitudes Amount of student investigation Level of Student Engagement in problem-solving Mathematics skills being developed Scale of 1-5 with 1 being poor and 5 being effective for all

X

T *

KK N * X X X

Student Activity Quality Techniques Student Involvement

Number of mathematical processes benchmarks addressed (average of two observations)

JJ

-

-

X

-

T *

N * X X

-

2

4.5

2

2.5

1

7.5

1

1.5

202 Appendix G: Data Collection Dates Data Collection Beginning and Ending Dates Participant

Beginning Date

Ending Date

Alan Anderson

09/10/2009

09/24/2009

Ben Brown

09/29/2009

10/26/2009

Carla Case

09/30/2009

10/23/2009

Diane Davis

10/14/2009

11/24/2009

Ellen Early

10/15/2009

11/18/2009

Faye Fout

10/20/2009

11/16/2009

Grace Gardner

10/26/2009

11/30/2009

Harriet Holmes

10/29/2009

12/17/2009

Ingrid Ivy

11/06/2009

12/17/2009

Jane Johnson

11/10/2009

12/14/2009

Kathy Kale

11/10/2009

12/18/2009

Laura Limley

11/16/2009

12/14/2009