A TENSION BETWEEN DIALOGIC AND DIRECT INSTRUCTION: ONE COMMUNITY’S MATHEMATICS TEACHING CULTURE

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A TENSION BETWEEN DIALOGIC AND DIRECT INSTRUCTION: ONE COMMUNITY’S MATHEMATICS TEACHING CULTURE

ABSTRACT

This study uses ethnographic methodology to describe and interpret one community’s mathematics teaching practices. Sociocultural theory is used as a theoretical framing to justify the study of the community’s educational stakeholders’ beliefs, values, and opinions regarding the mathematics teaching practices observed in their local elementary school. The study was conducted in two phases within a single school district. The first phase was focused on identifying and characterizing prominent mathematics teaching practices used in three fourth and fifth grade classrooms. The second phase was focused on eliciting and describing how stakeholders in the community (parents, teacher colleagues, principals, and professional development leaders) make sense of the identified mathematics teaching practices from the first phase of the study. Two prominent teaching practices were identified across the three observed fourth and fifth grade teachers’ mathematics lessons. One was similar to a direct instruction model of teaching, and the other was similar to a dialogic instruction model of teaching. Both teaching practices were represented in storyboards and used to elicit the beliefs, values, and opinions of stakeholders regarding mathematics teaching and learning.

Findings describe the two prominent mathematics teaching practices observed in the three teachers’ classrooms, and outline stakeholders’ beliefs, values, and opinions about mathematics teaching and learning. Discussion and implications explore how the findings illustrate the importance of examining values held by an educational community before attempting to improve aspects, such as teaching practices, within the community. Implications also explore the usefulness in using the concept of ritual as a tool for better understanding educational culture, rather than using educational rituals to rationalize unproductive teaching practices.

 

TABLE OF CONTENTS

LIST OF FIGURES……………………………………………………………………………………………….. vi

LIST OF TABLES………………………………………………………………………………………………… vii

ACKNOWLEDGEMENTS……………………………………………………………………………………. viii

Chapter 1 Introduction and Rationale………………………………………………………………………….. 1

Rationale for the study………………………………………………………………………………………. 5

Development of key terms…………………………………………………………………………… 8

Chapter 2 Literature Review……………………………………………………………………………………. 11

Mathematics teaching practice…………………………………………………………………………… 11

Enduring mathematics teaching practices………………………………………………………. 12

Mathematics teachers’ beliefs and their influence on practice…………………………….. 15

Best mathematics teaching practices…………………………………………………………….. 17

Theoretical framework: Sociocultural theory………………………………………………………… 24

What is sociocultural theory?……………………………………………………………………… 24

Use of sociocultural theory to study mathematics teaching practices……………………. 25

Analytical framework: Ritualized teaching practices………………………………………………. 27

Use of ritual in research in education and mathematics education……………………….. 29

Why ritual?…………………………………………………………………………………………….. 31

Chapter 3 Research Methods…………………………………………………………………………………… 33

Context of the study………………………………………………………………………………………… 36

Participating classrooms, teachers, and stakeholders………………………………………… 37

Data collection and analysis methods………………………………………………………………….. 43

Data collection and analysis: Phase I……………………………………………………………. 44

Data collection and analysis: Phase II…………………………………………………………… 57

Procedures to address trustworthiness and credibility……………………………………………… 62

Chapter 4 Findings from Classroom Observations: Mathematics Teaching Practices……………. 65

Connection to university–provided professional development…………………………………… 65

Professional development description…………………………………………………………… 66

Observed teachers’ mathematics teaching practices……………………………………………….. 69

Green teaching practice…………………………………………………………………………….. 70

Purple teaching practice…………………………………………………………………………….. 79

Observed teachers’ assertions……………………………………………………………………………. 89

Values and beliefs concerning mathematics teaching and learning………………………. 90

Values and beliefs about dialogic and direct instruction……………………………………. 92

Opinions regarding mathematics state testing…………………………………………………. 94

Chapter 5 Findings from Interviews: Stakeholders’ Values, Beliefs, and Sense Making………… 96

Four categories of stakeholders’ beliefs, values, and opinions…………………………………. 100

Stakeholder interpretations of each observed mathematics teaching practice……….. 100

Stakeholder beliefs and values concerning mathematics teaching and learning…….. 107

Prevalent stakeholder experiences from grade school mathematics classes………….. 119

Stakeholder opinions regarding state mathematics testing……………………………….. 123

Categories of stakeholder assertions compared and contrasted………………………………… 125

Stakeholder beliefs, values, and classroom interpretations compared to direct or dialogic instruction teaching practices…………………………………………………………………………………… 126

Stakeholders’ beliefs and values about mathematics teaching and learning compared to their interpretations of the green and purple classrooms…………………………………………………………….. 129

Stakeholders’ beliefs and values about mathematics teaching and learning compared to their opinions regarding mathematics state testing……………………………………………………………. 130

Comparison between stakeholder assertions and Ana, Beth, and Fred’s assertions………. 130

Stakeholder beliefs and values compared to those of the Ana, Beth, and Fred………. 131

Stakeholder opinions regarding mathematics state testing compared to those of the Ana, Beth, and Fred………………………………………………………………………………………………………….. 131

Chapter 6 Discussion, Implications, and Conclusion…………………………………………………… 133

Answering the research question……………………………………………………………………… 134

Answering the research sub–question A regarding mathematics teaching and learning 135         

Answering the research sub–question B regarding stakeholders’ experiences………. 139

Answering the research sub–question C regarding state mathematics assessments… 140

Discussion………………………………………………………………………………………………….. 141

Impact of the professional development on this study…………………………………….. 142

Tension between two types of teaching practices…………………………………………… 144

Difference between experiences and values………………………………………………….. 148

Implications………………………………………………………………………………………………… 150

Implications for teacher professional development………………………………………… 150

Implications for research methodology……………………………………………………….. 153

Implications for existing research………………………………………………………………. 156

Future research…………………………………………………………………………………………….. 158

Concluding remarks………………………………………………………………………………………. 160

References………………………………………………………………………………………………………… 162

Appendix A: Teacher Interview Questions……………………………………………………………….. 179

Appendix B: Storyboards for Green and Purple Classrooms…………………………………………. 181

Appendix C: Stakeholder Interview Protocol…………………………………………………………….. 193

Appendix D: Mock Assessments for Stakeholder Interview………………………………………….. 198

Chapter 1 Introduction and Rationale

Research has demonstrated that many mathematics teaching practices persist within the American educational system, despite various attempts at change. Studies have revealed that, as a collective group, U.S. teachers continue to use the same mathematical teaching practices that they have for many years (e.g., Gainsburg, 2012; Jacobs et al., 2006). Furthermore, findings from additional research studies indicate that mathematics teaching practices in the United States tend to focus on procedural competency rather than on understanding concepts and mathematical relationships (e.g., Malzahn, 2002; National Council of Teachers of Mathematics [NCTM] , 2014; Oakes, 1985; Stigler & Hiebert, 2004; Whittington, 2002). This finding is in contrast to standards documents that call for a focus on conceptual learning, problem solving, and development of mathematical reasoning (The Conference Board of the Mathematical Sciences [CBMS] , 2010;

Council of Chief State School Officers (CCSSO), 2010; NCTM, 2007, 2014; National Research

Council [NRC] , 2012)

Mathematics education researchers have made great strides in trying to address the discrepancy between observed mathematics teaching practices and desired mathematics teaching practices within the United States. There are multiple studies regarding what types of mathematics teaching practices that researchers consider to be best teaching practices for the type of student learning described above. Identified teaching practices include raising the level of the cognitive demand of mathematical tasks (Stein, Smith, Henningsen, & Silver, 2000), encouraging students to justify their reasoning and make connections with other students’ reasoning (McClain, 2002), and leading discussions that are student oriented as well as directed towards specific mathematical learning objectives (Stein, Engle, Smith, & Hughes, 2008). In addition, there are teaching standards documents that outline best practices according to these and similar research studies (NCTM, 2007, 2014). For example, NCTM (2014) suggests eight productive mathematics teaching practices: establish mathematics goals to focus learning, implement tasks that promote reasoning and problem solving, use and connect mathematical representations, facilitate meaningful mathematical discourse, pose purposeful questions, build procedural fluency from conceptual understanding, support productive struggle in learning mathematics, elicit and use evidence of student thinking.

Drawing on the findings about best practices, researchers have made attempts to support and improve mathematics teaching practices across the country (e.g., Ball, Hill, Rowan, & Schilling, 2002; Stein, Smith, Henningsen, & Silver, 2000). There have been multiple large–scale studies regarding professional development attempts to improve mathematics teaching practices (e.g., Bell, Wilson, Higgins, & McCoach, 2010; Borko, 2004; Carpenter, Fennema, & Franke, 1996; Stein, Grover, & Henningsen, 1996). Despite attempts at improving mathematics teaching practices, studies on school policy highlight a scarcity of professional development programs working with a large amount of teachers and school districts proving successful at creating permanent changes in the teaching practices of participating teachers (Cobb & Jackson, 2011; Elmore, 2004; Gamoran et al., 2003; McLaughlin, 2006).

Current research tends to focus on how to improve mathematics teaching practices by working directly with the teacher; for example, trying to improve his or her knowledge of mathematical content or mathematical teaching pedagogy. Cobb and Jackson (2011), as well as others outside of mathematics education (e.g., Bryk, Sebring, Allensworth, Luppesco, & Easton, 2010; Elmore, 2004; Sebring, Allensworth, Bryk, Easton, & Luppesco, 2006), call for improving teaching practices through “supporting schools’ and broader educational jurisdictions’ development of the capacity to scaffold teachers’ (and others’) ongoing learning” (Cobb & Jackson, 2011, p. 7). Following in the spirit of this call for research, the intent of this study is to focus attention on potential influences on teacher practice outside of the classroom teacher. This study explores the culture surrounding mathematics teaching practices that have endured within elementary mathematics classrooms to the point that they have become ritualized.

The research presented in this study is founded on the perspective that mathematics teaching both influences and is influenced by the culture of the educational community of which the teacher is a member. A definition of culture is adopted from Bohannan (1995) and Nuthall (2005) as “the customary ways of acting, thinking, and feeling that are common to the members of a society and sustain their relationships (Nuthall, 2005, p. 896). The perspective that an educational culture is entwined with mathematics teaching practices provides a basis for this study, which explores one educational community’s beliefs, interpretations, and values regarding mathematics teaching practices. I choose to focus on ritualized mathematics teaching practices because of the notion that “within a culture, the ritualized routines that structure social interactions are unlikely to survive without a web of supporting beliefs or myths that explain and justify the way these routines are played out” (Nuthall, 2005, p. 920). Therefore, this research reveals insights into an educational culture regarding the use of a ritualized mathematics teaching practice. Specifically, the research question guiding this study is: How do educational stakeholders make sense of mathematics teaching practices observed in their local elementary school? The phrase “make sense of” is further explored through the following research sub– questions:

  1. What are stakeholders’ beliefs, values, and opinions regarding mathematics teaching and learning, and how do those beliefs, values, and opinions relate to the mathematics teaching practices observed in their local elementary school?
  2. What are stakeholders’ experiences in mathematics classes, and how do those experiences relate to the mathematics teaching practices observed in their local elementary school?
  3. How do stakeholders’ beliefs, values, and opinions about mathematics teaching practices relate to the state mathematics assessment administered to students in their local elementary school?

The main research question is intentionally broad so as to allow for conversations and data collection to be extended by the participating members of the educational community. For the practical purposes of the research study, this question is examined through the three sub– questions listed above. While the main research question, and the ethnographic methodology, stood as a reminder to remain open to whatever topic of conversation the participants introduced during interviews, the sub–questions helped to direct interviews with stakeholders towards specific conversational topics that may provide insight into how they make sense of mathematics teaching and learning that was observed in their local elementary school.

The first sub–question allows for an examination of the beliefs, values, and opinions of the participating stakeholders, specifically how stakeholders’ beliefs, values, and opinions regarding mathematics teaching and learning relate to observed mathematics teaching practices in their local elementary school. The definitions of belief and value are adopted from Phillip (2007) who uses a combination of published literature and dictionary definitions to succinctly capture the meaning of each term. He defines belief as “psychologically held understandings, premises, or propositions about the world that are thought to be true,” and value as “the worth of something” (p. 259). Furthermore, Phillip (2007) clarifies the distinction between the two meanings as he writes, “Whereas beliefs are associated with a true/false dichotomy, values are associated with a desirable/undesirable dichotomy. Values are less context-specific than beliefs” (p. 259). The definition of opinion is adopted from The New Oxford American Dictionary as “A view or judgment formed about something, not necessarily based on fact or knowledge” (Opinion, 2015). Discussion with stakeholders about their beliefs, values, and opinions is one way to provide insight into the beliefs and values of the educational community. These discussions can help to illustrate the aspects of education that the community values as well as the statements about education in which the community believes.

The second sub–question allows examination of how the experiences of stakeholders relate to their interpretations of the observed teaching practices. This question is motivated by research findings that demonstrate teachers’ experiences with mathematics teaching and learning may influence their teaching practices (e.g., Raymond, 1993). As teachers are among the group of stakeholders for this research study, it makes sense to examine their past experiences, as well as extend the notion that past experiences may influence any stakeholder’s interpretations of mathematics teaching practices. Conversations with stakeholders about their experiences in mathematics classes may provide insight into why they interpret mathematics teaching practices in a certain way.

The third sub–question allows for insights into how the stakeholders in the local community relates to the larger, national community. Many of the stakeholders are directly influenced by the state standards, which are adopted from the Common Core State Standards for Mathematics (CCSSO, 2010), because they either work directly with students in the state, or they work with teachers in the state. The implementation of standards and corresponding assessments across multiple states in the country may have an influence on how the community interprets the mathematics teaching practices in their local community.

Rationale for the study

The theoretical perspective for this study is a sociocultural theory of teaching and learning. Sociocultural theory originated with Vygotsky (1978) as a way of understanding how people learn. Distinguishing this theory from previously defined theories of learning is the interpretation that learning is a process that involves more than just a child and a teacher. As Cobb (2007) states, “sociocultural theory characterizes the individual as a participant in established, historically evolving cultural practices” (p. 25). Explored in more detail in the subsequent chapter, sociocultural theory describes multiple ways in which culture impacts the teacher and the learner. For the purpose of this study, learning (whether learning to do teaching or learning to do mathematics) is regarded as an induction of a person into an educational community, rather than solely a generation of new individual knowledge.

It follows from sociocultural theory that learning is influenced by cultural practices and norms. As Wenger (1998) writes, “learning is an issue of engaging in and contributing to the practices of their communities” (p. 7). For mathematics education, this means students are engaging in established practices of a few different, yet related communities, as seen in Figure 1-1. These communities are: the community within the particular classroom (peers and teacher within a student’s current mathematics class), the community of the school site and school district (other mathematics teachers, other subject teachers, the entire grade level of students, possibly students in other grade levels, and administrators), the community of family (parents, siblings, other relatives), and the community of the general society surrounding the learner (members of the town, county, state, and nation).

Figure 1-1. Communities in which a learner is engaging in established practices.

Sociocultural theory is used in two ways to illuminate potential influences of the opinions of the relevant educational cultures on the mathematics teaching practices used by teachers within that community. First, as teachers engage students in the learning of mathematics, their mathematics teaching practices are influenced by mathematical learning goals that are deemed important by the relevant educational community. Second, as teachers assume the role of learner throughout various aspects of their careers, their teaching practices also have the potential to be influenced by the relevant educational community’s opinions regarding advantageous mathematics teaching practices. When teachers are engaged in learning about mathematics teaching practices, either before the start of their career (through 12 years observing the act of teaching from a student perspective, or engagement in university courses as college students), at the start of their career (through interactions with teachers and administration at their local school district), or during the extent of their career (through formal or informal professional development or conversations with other educators), they are potentially influenced by the relevant educational community. In summary, the relevant educational community can potentially influence both a teacher’s decisions regarding what mathematics content to include in lessons as well as the teacher’s mathematics teaching practices used during lessons.

Development of key terms

The focus of this study is on the members of each community depicted in Figure 1-1, who are considered educational stakeholdersin the learning of the students within that community. A characterization of stakeholder for this study originates from Krainer (2014), who writes that educational stakeholders “have effects on students’ knowledge and at the same time [stakeholders] are affected by [students’] knowledge or lack of knowledge” (p. 54). Whereas Krainer (2014) refers to the effects between stakeholders and students, the term stakeholders will be utilized in this study to describe all people within a community who potentially influence or are influenced by mathematics teaching practices. If the mathematical knowledge of students is of consequence to educational stakeholders, then the ways in which students are engaged in learning is also of consequence, which implies that mathematics teaching practices matter to stakeholders as well. In addition, because teachers can be a specialized type of learner as they engage in learning about the practice of teaching, educational stakeholders have the potential to effect teachers’ mathematical teaching knowledge and have the potential to be affected by teachers’ mathematical teaching knowledge or lack of that knowledge in a similar way to how Krainer describes the relationship between students and stakeholders.

Krainer (2014) lists possible stakeholders in education as “parents, principals, superintendents, mathematicians, teacher educators, educational publishers, test developers, companies, (education) policy–makers, and even the whole society can be regarded as ‘stakeholders’” (p. 54). This list serves as the source for potential categories of educational stakeholders included in this study. Figure 1-1 includes four categories of educational stakeholders who may potentially influence the learning of a group of students within the community. The outermost layer of Figure 1-1 lists the more general society as educational stakeholders, including local, state, and national communities. This study concentrates on local educational stakeholders, within a single school district, with some connection to the larger society through the statewide mathematics standards used within the district. As can be seen in Figure 1-2, there are five main groups of educational stakeholders included in this study. This list is specific to the participating school district, and each stakeholder group will be discussed more thoroughly in the methods section of this dissertation.

Figure 1-2. Groups of stakeholders who may have an influence on teaching practices.

One way to study the culture surrounding mathematics teaching practices is to examine a ritualized mathematics teaching practice within a single community. A ritualized teaching practice is an action, or a collection of actions, in which a teacher engages within a classroom setting that is directed towards student learning, and “that aspect of action that is formalized, traditionalized, symbolic performance” (McCloskey, 2013, p. 25). A ritualized teaching practice has endured in a particular culture for at least multiple years and therefore should be familiar to educational stakeholders within the local community. As such, it has potential to provide information about stakeholders’ beliefs, interpretations, and values about not only the particular ritualized mathematics teaching practice, but also about mathematics teaching and learning in a more general sense. The goal of this study is to understand the culture surrounding a ritualized mathematics teaching practice by examining how educational stakeholders make sense of that practice.

The phrase teaching practice is used frequently in this dissertation, rather than the phrase teacher practice so as to focus on the actions of teachers, rather than the teachers themselves; however, the two phrases are often used interchangeably and therefore any mention of teacher practice in relevant research literature will be assumed to have a similar understanding. Both terms are often assumed to be in the common vernacular of readers of educational research literature, but it is an elusive phrase that does not necessarily hold the same meaning for all. For example, some studies focus more on knowledge and skills (e.g., National Research Council,

2001) and others focus more on beliefs and relationships (e.g., Lampert, 2001; Simon & Tzur, 1999). For the purposes of this study, teaching practice is defined using the works of a combination of education researchers (Gainsburg, 2012; Grossman et al., 2009; Simon, 2000; Simon & Tzur, 1999) to mean the typically consistent actions of a teacher within a classroom, and the multitude of influences on these actions. This definition has two dimensions; the first dimension is the observable actions of the teacher within the classroom. These actions must be fairly consistently observed within the classroom to be considered a practice. In other words, the action is not something the teacher does only once or twice, but rather something that is done repeatedly throughout the school year. The second dimension, as described by Simon and Tzur (1999), is “everything teachers think about, know, and believe about what they do. In addition, teachers’ intuition, skills, values, and feelings about what they do are part of their practice” (p. 254). Both dimensions of teaching practice will be explored through this study. The first dimension is useful in identifying a ritualized mathematics teaching practice, and the second dimension is useful in accessing the culture in which the ritualized mathematics teaching practice resides.

A TENSION BETWEEN DIALOGIC AND DIRECT INSTRUCTION: ONE COMMUNITY’S MATHEMATICS TEACHING CULTURE

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