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 test- taking 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,
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.
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
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
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 standards- based, 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
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
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.
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
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 standards- based 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
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)
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 standards- based 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.
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