Annotated Bibliography: Mathematics Teacher’s Beliefs and Change

Pamela R. H. Bailey

George Mason University

Research Questions

What teacher practices or beliefs will encourage mathematical growth in secondary mathematics students? How can a mathematics leader encourage secondary mathematics teachers to adopt reform practices?

Becker, J. R., Pence, B. J., & Pors, D. (1995). Building bridges to mathematics for all: A small scale evaluation study. Presented at the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Ohio: PME.

Teachers are exposed to professional development but concern is expressed as to why or why they do not decide to implement the practices learned and prepared. The researchers want to look at facilitating change in beliefs of teachers involved in a three year professional development program. Students in California are expected to take Algebra 1 by their ninth grade year in high school which has led to a concern for equity and accessibility to quality instruction.

Five participants were selected from the professional development program according to specific criteria set forth by the researchers so that a diverse group would be probed. Part of the criteria was based on the results from the Instructional Practices Scale instrument that all program participants took along with observations made during coaching experiences. The main source of data was through interviews with questions centered on curriculum, pedagogy, and equity. Triangulation was conducted between the interviews, the instrument responses, and the qualitative data from the coaching. The findings showed that the longer a participant was involved in the professional development led to an increase in the beliefs of teachers toward a more student-centered instructional approach with their classrooms exhibiting the change. All of those interviewed stated that they understood and agreed with the goals set for in the sessions. It was a unanimous response that the networking with other teachers meant the most to them.

Due to space, the researchers were unable to elaborate on the instrument or the methodology; instead they opted to focus on the results. The construct is a self reported analysis which leads to participant bias. Lastly, it is a three year program so we should know that those who remain as participants will acknowledge that their instruction has been impacted.

Breyfogle, M. L. (2005). Reflective states associated with creating inquiry-based mathematical discourse. Teachers and Teaching: Theory and Practice, 11(2), 151-167.

Breyfogle focuses on what a teacher experiences as he tries to change the discourse in his classroom in an effort to become more inquiry-based and student-centered. What the teacher actually does in the classroom and what is written when reflecting on his practices may not correlate. Breyfogle wants to see if videos of the teacher and students in action will help the teacher perceive accurately his actions and to adjust them to be more student-centered.

The teacher has taught six plus years and is considered to be innovative and a leader in reform mathematics. He was videotaped weekly with the researcher making notes about times, the discourse, and any additional comments he thought might help. Breyfogle then selected a segment of the video that was assessed by the teacher the next day using an instrument he created, Discourse Reflection Tool (DRT). Additionally the teacher was interviewed three times to clarify and expound on reflections to that time. The results showed a graduated state of change in beliefs and reactions by the teacher. He started out being defensive and explaining his actions that changed to questioning his actions and being frustrated with himself. This led to a question and explore state where he still questioned his actions but also sought how to change his practices. The final state was exploring and indicated that the teacher was more thoughtful about his actions like an outsider looking in. Breyfogle stages of change correlate with Guskey’s (2002) model of teacher change in a classroom practice that leads to student improvement and engagement will ultimately result in a change in teacher beliefs.

A weakness is definitely that this was the view of only one teacher who was already considered to be innovative. I wonder if the teacher was reluctant to change if the practice would also be effective.

Cooney, T. J., Shealy, B. E., & Arvold, B. (1998). Conceptualizing belief structures of preservice secondary mathematics teachers. Journal for Research in Mathematics Education, 29(3), 306-334.

Cooney, Shealy, and Arvold address the problem of teacher beliefs and changing beliefs through the type of experiences encountered in their teacher education programs by developing stages similar to research by Belenky, Clinchy, Goldberger, and Tarule (1986) and Perry (1970). The purpose of the paper is to determine that once teachers are exposed to reform practices why or why do they not continue those practices in the classroom.

The researchers begin the study with a survey of 15 preservice secondary math teachers. The focus was on beliefs about teaching and exploring reactions to student responses to problem solving situations was the focus. Four of the teachers were interviewed, for a total of 5 each, and took a second survey to determine changes in the teacher’s beliefs and growth

The survey’s and interviews revealed that a teacher’s inner voice effects teacher change and aids in reflecting critically on experiences. Suggested the following levels of growth: naïve, idealist, isolationist, and connectionist; a change in the levels presented by Belenky, et al. (1986) and Perry (1970) and recommended that education programs concentrate on students becoming reflective connectionists. In addition, they proposed creating activities that challenge teachers causing them to doubt evidence they currently hold and to reflect and adapt current practices.

I am not sure why or how they narrowed down their participants from 15 to 4 as they continued on with the interviews. The relationship to the Belenky, et al. (1986) and Perry (1970) studies did not seem to play a huge part as the researchers developed their own levels so I am not sure why they focused on them. No constructs were developed or used.

Cross, D. I. (2009). Alignment, cohesion, and change: Examining mathematics teachers’ belief

structures and their influence on instructional practices. Journal of Mathematics Teacher

Education, 12, 325-346.

The study addressed five teachers’ mathematical beliefs by analyzing how they correlated with their instructional practices and was a help or heed when incorporating reform practices. The purpose of the paper is to understand how beliefs support or deter implementing reform-oriented practices by teachers by looking at three different models of belief structures. Cross believed that teacher beliefs must correlate to the expectations. The goal of the study is to assess the degree of beliefs aligned with instruction, how beliefs help or hinder teachers incorporating reform based practices and resources.

Two formal observations were conducted with follow-up interviews. The interviews focused on “…teachers’ views of mathematics as a discipline (Cross, 2009, p.330)”, content pedagogy, and student learning. Participants reviewed and approved the interview transcripts. Weekly observations continued, two per week for 10 weeks, followed by informal discussions. Lesson plans and student work was collected. Results showed three themes, the nature of mathematics (computation versus a way of thinking), mathematics teaching (demonstration versus guidance), and student learning (practice versus understanding). Educating teachers and changing their practices may be accomplished by understanding how teachers believe mathematics should be taught with regard to these areas.

I would feel that the study had substance if there had been some type of quantitative analysis conducted along with the qualitative. We know the types of beliefs so now what and could there also be other factors? Future suggestions for research were made. There were no constructs developed or used.

Cwikla, J. (2002). An interview analysis of teachers' reactions to mathematics reform professional development. Presented at the Annual Meeting of the American Educational Research Association, New Orleans, LA.

The study addresses what characteristics influence teachers’ reactions and interactions with students and what activities affect teacher’s classroom practices. Four hypotheses considered are 1) when teachers’ thinking and learning is centered around how students’ think and learn then instructional practices improve; 2) collaborative environments lead to improved instructional practices; 3) change in instructional improvements occur in small increments; and 4) experimentation and inquiry lead to instructional improvement. Overall the goal is for teachers to increase their understanding of student thinking so that there will be a change in instructional practices that are based on the National Council of Teachers of Mathematics (NCTM) Process Standards.

Middle school teachers (110) involved in a statewide professional development on how to use new curriculum materials. It is a three year program with all teachers completing a survey midway in the first year. The survey consisted of teacher views on how students learn and on constructivist aspects, reactions to activities that encourage students’ mathematical thinking, and demographic information. Follow-up surveys and interviews were conducted with thirteen teachers to delve deeper into the hypothesis.

Quantitative and qualitative results supported each other. Key findings were that years of experience, advanced degrees, and mathematical content knowledge affected teacher insight into student thinking. Each hypothesis was discussed separately. We do not know the validity or reliability of the survey. Factor analysis, Promax rotation, Structure Matrix, and t-tests were used in the analysis.

Derry, S., Wilsman, M., & Hackbarth, A. (2007). Using contrasting case activities to deepen teacher understanding of algebraic thinking and teaching. Mathematical Thinking and Learning, 9(3), 305-329.

The problem addressed how teacher’s beliefs change when exposed to problem solving tasks using nontraditional approaches and how reflecting and/or evaluating case studies affect teachers understanding of student ability to reason algebraically. Goals were to have teachers compare and interpret tasks containing multiple representations and solutions and to explain their thinking and those of the students. Creating an approach to instruction for teachers will aid them in acknowledging their own ability to understand algebraic thinking and representational fluency.

Twenty teachers from three middle schools completed five assignments based on readings, problem solutions for Sample Algebraic Modules (SAM), daily journal reflections, and were videotaped with researchers taking field notes. The professional development was in two parts, a summer workshop and a school year sessions. Teachers were assessed on content knowledge, pedagogical content knowledge, and their ability to analyze student thinking by watching two videos of student’s actions during a problem solving activity.

Teachers showed an increase in comparing multiple representations and interpreting the solutions. They had improved their ability to identify and explain student’s algebraic thinking, mathematical vocabulary, classroom discourse but they had not increased their ability to reflect on the go about what they were doing and students responses.

Excerpts from teachers helped to validate statements. Content was shown as a weakness but nothing was done to help them improve. The researchers used the design research methodology by Hall (2001) and Turning to the Evidence (Driscoll, Goldsmith, Seago, and Mumme (2004).

Harel, G., & Lim, K. H. (2004). Mathematics teachers’ knowledge base: Preliminary results. Proceedings of the 28^th Conference of the International Group for the Psychology of Mathematics Education, Vol. 3, 25-32. Norway: PME.

Does a teacher’s practice depend on the teacher’s knowledge level or the teacher’s beliefs about how mathematics should be taught and learned was the problem addressed by Harel and Lim (2004). The purpose of the paper was to look closely at the types of knowledge and how it affects instructional practices. Harel and Lim state that “Student learning depends on the teacher’s actions, which are, in turn, dependent on the teacher’s knowledge base…(p.25)”.

The study involved three teachers in a public high school with low income student population, on a block schedule, and offering a rigorous college preparatory education. Classroom observations were conducted once or twice a week for a total of 14 observations for each teacher. Meetings followed each observation to discuss goals of the lesson and to reflect. Some observations were videotaped and the conversations audio taped.

Harel and Lim (2004) looked at knowledge as a way of understanding (WoU) and a way of thinking (WoT). The teacher in the case study was traditional, self-centered, and focused on the lesson from a mathematical standpoint. Part of the revealed transcript showed that professional development needs to focus on all aspects of knowledge in order to maximize effectiveness. A change in knowledge will affect teacher beliefs. Teachers need to wrestle with the mathematics and then reflect on their own learning which will lead to a better appreciation and understanding of “…epistemological and pedagogical issues (Harel & Lim, p.32).”

The theoretical framework contained definition of terms. No clear statement of the research question(s). They observed three teachers but only went into depth discussing one. There were no constructs developed or used

Handal, B., & Bobis, J. (2004). Teaching mathematics thematically: Teachers’ perspectives. Mathematics Education Research Journal, 16(1), 3-18.

Teaching using themes in mathematics has been praised as a way to provide students with relevant learning experiences instead of focusing on mathematical concepts. Empirical research has not been conclusive on the impact thematic approaches have on student learn and attitudes. Problems with the studies include the design of the studies, misunderstanding of the materials used, and methodologies used. The purpose of the study was to determine beliefs and practices of ten secondary mathematics teachers when implementing thematic instruction. Instruction, curriculum, and organization are hypothesized as the reason why the thematic approach is not implemented.

Participants first completed the questionnaire and then participated in a semi-structured interview based on questions centered on instruction, curriculum, and organization. The common themes were the result of the questionnaire. Thematic instruction is reform mathematics that is highly structured. Teachers found that if the theme did not interest the student then they did not pay attention. Beliefs and practices was the aim of the researchers and they revealed a great deal of conflict with the three factors. Appreciation was shown for the approach but there are problems with preparing and obtaining materials, being pressured to use the theme approach, and student concerns with behavior and repetition.

Weakness was a lack of observations to correlate statements with actions. Also they cannot be assured that the teachers actually implemented the units as intended. I have a concern that the approach is too thematic and not centered on a set of concepts. No information was given about the questionnaire.

Lepik, M., & Kaljas, T. (2009). Facilitating change in teachers’ views of teaching mathematics.

Pedagogy Studies, 94, 73-76.

Estonian mathematics is experiencing problems with instruction being drill and practice and students losing interest in the subject. They are in the midst of reform so are seeking avenues for teachers to improve instruction. Reflection that is meaningful and taken seriously by those participating is seen as a way to improvement. A community of practice made up of university teachers and high school mathematics teachers was formed for this purpose.

The teachers met regularly with the researchers to reflect on their practices and to participate in seminars and lectures. Reflection sharing involved videotapes, lesson plans, along with discussions on problems and experiences. Only 23 of the 34 original participants completed the year and a half study. Teachers took a questionnaire at the beginning and end of the project that contained open-ended questions about their views on mathematics and reflected on their teaching practices. The change in questionnaire responses showed that teacher‘s initial beliefs regarding rules and routines, understanding concepts, and problem solving was varied. By the end of the study beliefs were more homogeneous, focusing on discovery and meaningful learning. Teacher metacognition developed with participation in the community of practice.

A major weakness is that the researchers did not validate the teacher’s statements with observations. They did include some qualitative statements that were collected during the meetings which did correlate with and show growth. Only information given about the questionnaire the participants took was that they were open-ended and about teacher beliefs. I wonder about its validity.

Orrill, C. H. (2006). What learner-centered professional development looks like: The pilot studies of the InterMath Professional Development Project. The Mathematics Educator, 16(1), 4-13.

Orrill questions how teachers can implement reform methodology without ever experiencing it. The initial goals are to improve content knowledge and skills using open-ended explorations, to use technology to support explorations and thinking skills, and to collaborate with and have support from peers.

Two groups of certified math middle school teachers met once a week for a semester at the University of Georgia to explore algebra, geometry, number sense, and statistics/probability as they would be presented in mathematical reform classes. Three weeks of the field notes were analyzed. Twelve randomly chosen participants and both instructors participated in tape recorded interviews. Pre- and post-surveys were taken by all participants to rate their technology usage, comfort level using technology, and their usage of open-ended explorations with students.

Categories from data centered on support, interaction, barriers, presentation, and adoption. Concerns flagged by the researcher were an over-reliance on the instructors, wanting correct procedures, not wanting to expand content knowledge, viewing course as a focus on technology, and lack of reflection. The goals of the participants did not match those of the professional development. Teachers wanted make and take sessions, activities ready to use in the middle school classroom. The teachers did not know how to work in groups, how to apply their knowledge and skills, or use technology therefore proving their hypothesis.

Need to determine and address content weaknesses. I also felt that the literature review was very weak. Research questions were difficult to determine. Orrill did not say much about the survey and its construction.

Sztajn, P. (2003). Adapting reform ideas in different mathematics classrooms: Beliefs beyond mathematics. Journal of Mathematics Teacher Education, 6, 53-75.

The problem addressed in the study is if suggestions for changing mathematics instruction is affecting all children being able to learn mathematics. “The goals of this study are to understand elementary school teachers’ beliefs and practices and to unveil factors that influence the way teachers adapt mathematics reform rhetoric when trying to adopt it (Sztajn, p.53, 2003).” Understanding teacher beliefs and instructional practices as well as what factors influence how teachers interpret and apply reform language will be researched.

A case study of two experienced elementary school teachers from different schools provided data collected for four weeks of observations. Five interviews were conducted with principals, some teachers, and some parents. Final themes were students’ needs, beliefs about children and how they learn, society, and education. Handouts and lesson plans were gathered.

Comparing and contrasting the teachers was a major portion of the analysis. Sztajn (2003) also developed a motto for each showing that the teachers not similar. The findings included that teachers are not receiving support and assistance to meet the expectations of reform in mathematics. Their beliefs drive what they deem the best for their students and influence what and how they interpret reform in mathematics. Current reform movement did not alter the teachers’ ideological vision, it was the opposite. The teacher’s ideological vision influences the understanding and execution of mathematics reform.

The process she uses in the research was clear. She has a section for each of the teachers detailing her analysis of the observations along with excerpts from the teacher followed by themes and contrasting views on practices. Weakness would be that there are only two participants. No constructs were developed or used.

Wilson, M. R. & Lloyd, G. (1995). High school teachers’ experiences in a student-centered

mathematics curriculum. Presented at the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Ohio: PME.

Researchers were concerned with teacher beliefs about student-centered instruction. A focus was on how teachers transitioned from teacher-centered to student-centered practices and specifically about pedagogical authority. The authority is about determining if student actions are correct and that understanding the concepts lies with the teacher, the student, or the textbook. Ascertaining if the teacher or the student handles and accepts the transition will be uncovered as many teachers blame students for why they do not change.

Over a six week period three traditional teachers and target student groups, as determined by the teacher, were observed and videotaped daily. Teachers participated in four or five interviews: students in the target groups were interviewed periodically. All the teachers taught a unit out of the Core-Plus Mathematics Project (CPMP) text. The researchers found that the students adapted to student-centered learning faster and easier than the teachers. Teachers were afraid that their students could not do the mathematics and make the connections, transitioning from whole to small groups were problematic, and felt that they had to interject in order for the students to understand the concepts fully. The study stated that the teachers were committed to change and had just begun to use the CPMP books. Perceptions of preconceived student concerns were stronger than what actually existed. If there was resistance it would decrease as time goes on. We need to understand how our students think and learn. The teachers did not receive any assistance with implementing new curriculum. No constructs were developed or used.

References

Becker, J. R., Pence, B. J., & Pors, D. (1995). Building bridges to mathematics for all: A small scale evaluation study. Paper presented at the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Ohio: PME.

Belenky, M. F., Clinchy, B. M., Goldberger, N. R., & Tarule, J. M. (1986). Women’s ways of knowing: The development of self, voice, and mind. New York: Basic Books.

Breyfogle, M. L. (2005). Reflective states associated with creating inquiry-based mathematical discourse. Teachers and Teaching: Theory and Practice, 11(2), 151-167.

Cooney, T. J., Shealy, B. E., & Arvold, B. (1998). Conceptualizing belief structures of preservice secondary mathematics teachers. Journal for Research in Mathematics Education, 29(3), 306-334.

Cross, D. I. (2009). Alignment, cohesion, and change: Examining mathematics teachers’ belief

structures and their influence on instructional practices. Journal of Mathematics Teacher

Education, 12, 325-346.

Cwikla, J. (2002). An interview analysis of teachers' reactions to mathematics reform professional development. Paper presented at the Annual Meeting of the American Educational Research Association, New Orleans, LA.

Driscoll, M., Goldsmith, L., Seago, N., & Mumme, J. (2004). Turning to the evidence: Professional development using classroom artifacts. Paper presented at the National Council of Supervisors of Mathematics 36^th Annual Meeting, Philadelphia, PA.

Guskey, T. R. (2002). Professional development and teacher change. Teachers and Teaching: Theory and Practice, 8(3-4), 381-391.

Hall, R. (2001). Schedules of practical work for the analysis of case studies of learning and development. The Journal of the Learning Sciences (Special Issues on Design Research), 10, 203-222.

Handal, B., & Bobis, J. (2004). Teaching mathematics thematically: Teachers’ perspectives. Mathematics Education Research Journal, 16(1), 3-18.

Lepik, M., & Kaljas, T. (2009). Facilitating change in teachers’ views of teaching mathematics.

Pedagogy Studies, 94, 73-76.

Orrill, C. H. (2006). What learner-centered professional development looks like: The pilot studies of the InterMath Professional Development Project. The Mathematics Educator, 16(1), 4-13.

Perry, W. G. (1970). Forms of intellectual and ethical development in the college years. New York: Holt, Rinehart, & Winston.

Sztajn, P. (2003). Adapting reform ideas in different mathematics classrooms: Beliefs beyond mathematics. Journal of Mathematics Teacher Education, 6, 53-75.

Wilson, M. R. & Lloyd, G. (1995). High school teachers’ experiences in a student-centered

mathematics curriculum. Paper presented at the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Ohio: PME.