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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
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
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:
- 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?
- What are stakeholders’ experiences in mathematics classes, and how do those experiences relate to the mathematics teaching practices observed in their local elementary school?
- 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 stakeholders in 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.
Chapter 2 Literature Review
This literature review is structured in three parts: the “object” of study, the context in which it is being studied, and the lens through which it is viewed. The first part is a review of research regarding mathematics teaching practices, in order to present the ways in which education researchers have studied teaching practices in an effort to improve student learning when working with teachers. Research findings that relate to both “traditional” and “reform” oriented mathematics teaching practices are presented. The second part of the chapter is an examination of the theoretical perspective for this study, sociocultural theory, and how that theory has been developed and used within the field of mathematics education. This part will also include literature regarding educational stakeholders. The third part of the chapter is an examination of the concept of ritual, which provides a basis for the rationale and analytical framework for the study. The usefulness of the concept of ritual as a lens to view mathematics teaching practices is explored.
Mathematics teaching practice
At its core, this dissertation study explores mathematics teaching practices. The concept of teaching practice tends to be assumed as a collectively understood concept within the field of education; however, research indicates that this can be an elusive concept. Take, for instance, the dual phrasing of teacher practice and teaching practice, both of which generally refer to the same concept of what teachers do in their classrooms. As stated previously, this study refers to the concept as teaching practice so as place a focus on the teaching rather than the teacher; however, many use the phrase teacher practice, and therefore the phrase will be included in this dissertation as referring to the same concept as teaching practice. While most researchers seem to have an implicit understanding of the definition of teaching practice, there are nuances about the definition that tend to be distinct. For example, a majority of researchers tend to agree that teaching practice encompasses the actions of the teacher as they relate to the process of teaching. However, some researchers believe teaching practice includes and focuses on the relationships that evolve as a consequence of the actions of teachers (Lampert, 2001), whereas others believe teaching practice is more about the influences on the actions of teachers in the classroom (Gainsburg, 2012; Simon & Tzur, 1999; Speer, Smith III, & Horvath, 2010). This study utilizes the latter depiction and, as described in the previous chapter, defines teaching practice as the typically consistent and observable actions of the teacher within the classroom, and the multitude of influences on these actions.
Enduring mathematics teaching practices
The rationale for this study is based on well–documented indications that mathematics teaching practices have resisted reform efforts despite a need for improved student learning outcomes in the United States (e.g., Hiebert et al., 2005; Hoetker & Ahlbrand, 1969; Stigler & Hiebert, 1999). This section of the literature review explores research findings regarding teaching practices, with a focus on mathematics teaching practices, that have endured for many years in the U.S. education system.
In 1975, the Conference Board of the Mathematical Sciences [CBMS] found that
“Teachers are essentially teaching the same way they were taught in school” (CBMS, 1975, p.
77). This means that a 40–year–old 5th grade teacher in 1975 was teaching using methods from
1945, or a teacher of the same age in the present year is using teaching methods from 1975. The CBMS finding is consistent with other research findings that demonstrate that the average classroom continues to show little change in routines and practices, and in fact the same basic method of teaching mathematics has been used for at least a century (e.g., Dixon et al., 1998; Fey, 1979; Hoetker & Ahlbrand, 1969; Stake & Easley, 1978; Stigler & Hiebert, 1997; Stodolsky, 1988; Weiss, 1978). Take, for example, Welch’s (1978) description of an observed mathematics lesson:
First, answers were given for the previous day’s assignment. A brief explanation, sometimes none at all, was given of the new material, and problems were assigned for the next day. The remainder of the class was devoted to students working independently on the homework while the teacher moved about the room to answer questions. The most noticeable thing about math classes was the repetition of this routine. (p. 6)
This observation could be used to describe many of the mathematics classes taught this year in the United States, and is probably familiar to readers from at least one mathematics course experience in their past. When eighth grade mathematics teaching practices in the U.S. were compared to Germany and Japan in the Third International Mathematics and Science Study
(TIMSS), researchers found that 78% of the topics taught were stated rather than developed, and 96% of seatwork time was spent practicing procedures rather than applying concepts or inventing/thinking (Stigler & Hiebert, 1997). This implies that U.S. students are not heavily involved in thoughtful mathematical work during the school day, which is comparable to Welch’s (1978) description of a mathematics lesson that was observed at least twenty years prior to the findings described in TIMSS. More recent studies have continued to demonstrate a lack of rigorous intellectual development within K–12 schools (e.g.,Weiss & Pasley, 2004). As a result, numerous studies attempt to characterize mathematics teaching practices observed in public schools.
One of the more prevalent types of mathematics teaching practices described through such research is the IRE pattern of discourse. As reported in the TIMSS video study, “most U.S. mathematics classrooms maintain an Initiation–Response–Evaluation (IRE) interactions pattern, where the evaluation move on the part of the teacher focuses on students’ answers rather than the strategies they use to arrive at them” (Franke, Kazemi, & Battey, 2007, p. 229). The IRE pattern of discourse describes a three–step conversational pattern between teacher and students: a teacher presents an academic question to students, one or more student(s) responds to the question, and the teacher replies to the answer with some type of valuation. The prevalence of the IRE discourse pattern within mathematics classrooms is well documented (e.g., Cazden, 2001; Mehan, 1985; Pierson, 2008; Silver, Smith, & Nelson, 1995; Spillane & Zeuli, 1999), and students often expect this type of discourse pattern during a mathematics lesson.
In addition to studies on patterns of discourse in mathematics education across all grade levels, there has been extensive research on the mathematics teaching practices used by elementary and middle school teachers. Findings consistently indicate that elementary teaching practices tend to emphasize procedures, memorization, and correct answers to exercises (e.g.,
Epstein & MacIver, 1989; Gill & Boote, 2012; Goodlad, 1984; Oakes, 1985; Stodolsky, 1988). Stodolsky (1988), specifically, found that learning in fifth grade classes was almost entirely devoted to facts and skills, with 97% of lessons having low–level cognitive objectives, and an average of three–fourths of the mathematics class periods in the study were spent engaging students in recitation of mathematical content. Both elementary and middle school teachers tend to engage students in discourse that result in students providing correct answers to exercises that required procedures, memorization, and basic arithmetic (Spillane & Zeuli, 1999). Likewise, Boaler (2000) writes, “dominant school practices in the mathematics classroom are memorization, reproduction of procedures, and individualized work, all of which play a limited role in situations outside the mathematics classroom” (p. 391). Even after the Principles and Standards of School Mathematics (NCTM, 2000), which calls for mathematics teachers to focus more on conceptual learning, an emphasis on procedural learning has still been found to be prevalent in school classrooms (Hiebert et al., 2003; Lampert, Beasley, Ghousseini, Kazemi, &
Franke, 2010). Hiebert et al. (2003) recorded that the U.S. was the only country in their study in which teachers utilized each of following practices: low–level mathematical challenge (which included a prevalence of routine exercises, practicing familiar procedures, and an absence of mathematical reasoning), an emphasis on procedures and review, and lessons that were both mathematically and pedagogically fragmented. By fragmented, they mean lessons that addressed multiple mathematical topics within a single lesson and a higher level of non–mathematical discussion during lessons.
As a result of such research findings, Cobb and Jackson (2011) call for large change in practice through creating a support system for teachers. Findings presented in their article indicate that changes in teaching practices need to be supported by multiple types of educational leaders within the school system. This indication supports the rationale for the current study, as it acknowledges that more than just the teacher has an influence on teaching practices. Furthermore, prevailing cultural beliefs tend to be an impediment to changing teaching practices. Teachers’ beliefs (e.g., Handal, 2003; NCTM, 2014; Phillip, 2007) as well as parents’ beliefs (e.g., Herbel– Eisenmann, Lubienski, & Id–Deen, 2006; NCTM, 2014; Sam & Ernest, 2000) about education continue to influence the ability for educators to create lasting change in teaching practices.
Mathematics teachers’ beliefs and their influence on practice
As described in the development of key terms, the second dimension of the definition of teaching practice involves teachers’ beliefs about the teaching and learning of mathematics. This section describes research that has demonstrated a connection between mathematics teachers’ beliefs and their teaching practices in order to reinforce the notion that the two are often interrelated.
There are various research studies that examine mathematics teachers’ beliefs about teaching and learning as well as their beliefs about the content they teach. Teachers’ beliefs have been found to have an impact on student learning as well as teaching practices (e.g., Handal, 2003; McCombs & Whisler, 1997; Wilkins, 2008). In one instance, Carpenter and Fennema (1992) report a positive correlation between first grade teachers’ knowledge of their students’ mathematical abilities and those students’ achievement. They also report a positive correlation between a belief that teaching is building upon students’ existing knowledge in order to construct new knowledge and students’ abilities to engage in problem solving. Similarly, Thompson (1984) and Raymond (1993) reported findings that teachers’ beliefs about mathematics teaching and learning influence their mathematics teaching practices. Specifically, Thompson (1984) reported on a case study of three junior high school teachers that demonstrated how their beliefs on mathematics teaching and learning had a significant impact on their teaching practices. And, Raymond (1993) found that among six case studies, teachers’ beliefs about mathematics influenced their teaching practices. Raymond (1993) also demonstrated a connection between teachers’ past experiences and their beliefs about mathematics teaching and learning.
In a more recent study, Polly and colleagues (2013) found that teachers who believed that teaching requires the teacher to transfer a set amount of knowledge to students, engaged heavily in teacher–oriented practices. Furthermore, multiple researchers report a connection between student–centered teaching practices and student motivation, both in classwork and on assessments (Kelley, Heneman, & Milanowski, 2000; McCombs & Whisler, 1997; Ryan, Ryan, Arbuthnot, & Samuels, 2007). For example, findings indicate that teacher–oriented practices correlate with smaller gains on curriculum–based assessments than student–oriented practices, and that teachers who believe that mathematics teaching should promote the use of discovery learning and encourage students to make connections within the content, self–report a higher use of student– centered practices (Polly et al., 2013).
Teachers’ beliefs are examined in this study as a potential influence on the enduring quality of mathematics teaching practices. While most of the studies referenced in this section discuss how teachers’ beliefs influence student–learning outcomes, this dissertation is one step removed from those outcomes because it focuses on teaching practices, rather than on student outcomes.
Best mathematics teaching practices
Often in the opposite direction of enduring teaching practices, research findings describe what researchers deem to be best teaching practices. The phrase best teaching practices can be misleading because it does not immediately address the desired learning outcome that is associated with those teaching practices. Teaching standards documents tend to lead teachers and researchers into certain directions about best teaching practices for mathematics. For example, NCTM (2014) presents eight research–informed teaching practices that “represent a core set of high–leverage practices and essential teaching skills necessary to promote deep learning of mathematics” (p. 9). Recent teaching standards documents tend to promote the belief that best teaching practices are those that support engaging students in problem solving and developing mathematical reasoning. Research on best teaching practices will be used to guide the data collection for this study.
Some researchers provide lists of findings from studies of teachers who are considered effective in increasing student achievement in their relative school locations. For example, according to student achievement levels on the Iowa Test of Basic Skills, Good and Grouws (1977) found that effective teachers in Iowa request more work and higher achievement from students, provide immediate, nonevaluative feedback that was relevant to the mathematical task, and have an ability to make clear presentations of content. In another study, Schoen, Cebulla, Finn and Fi (2003) report that student achievement on the Iowa Test of Educational Development seem to correspond to the following teaching practices: an increased use of pair and group work among students along with a decreased use of teacher presentations and whole group discussions, limited use of class time for nonacademic activities, a use of multiple assessment strategies including student interviews, high expectations on homework, and grading that focused more on academic factors than on attitude or effort. In a related study, focused on teacher quality rather than on student achievement, Arbaugh, Lannin, Jones, and Park–Rogers (2006) report that teachers with high lesson quality tend to use high cognitive level tasks, encourage students to solve problems using their own methods, ask students to discuss their problem solving methods with each other, and encourage students to critique mathematical tools.
The lists of practices discussed here are considered best teaching practices because research has shown that they support student achievement and align with teaching standards documents (e.g., NCTM, 2014). The remainder of this section will focus on three of the more prominent categories of best practices: classroom discourse, the use of mathematical tasks, and establishing classroom norms. These three are the focus because of their importance and prominence in the literature base, and they are reasonable as categories of practice within which teachers can be expected to engage their students.
Classroom discourse is one of the more researched mathematics teaching practices, and includes several characterizations, which are included among the list of best teaching practices. When utilizing discourse with and among students as a deliberate teaching practice, mathematics teachers need to pay attention to the students if they are going to help them develop a productive disposition (NRC, 2001) towards the mathematical content. For example, teachers have to decide when they need to speed up or slow down a conversation (Rittenhouse, 1998), and they need to focus on what the students are saying so as to rephrase or ask further questions about the discourse (Lampert, 1990; Lampert, Rittenhouse, & Crumbaugh, 1996; Rittenhouse, 1998). Furthermore, Lubienski (2002) found that classroom discourse that included students made those students who came from families with a higher socio–economic status feel more confident and empowered, but made students who came from families with a low socio–economic status feel frustrated in those same conversations. This indicates that discourse may need to be sensitive to individual students. Evertson, Anderson, Anderson, and Brophy (1980) found that among junior high and high school mathematics teachers, more effective math teachers were the ones who asked more questions of their students. These teachers tended to focus on whole class instruction with some individual mathematics work, and were also “active, well organized, and strongly academically oriented” (p. 58).
Franke, Kazemi, and Battey (2007) note three teaching practices that stem from discourse in the classroom that will be further developed here: revoicing, interrogated meaning, and the use of mathematical tasks. Revoicing is defined as “the reuttering of another person’s speech through repetition, expansion, rephrasing, and reporting” (Forman, McCormick, & Donato, 1998; O’Connor & Michaels, 1993, 1996). Revoicing students’ language within classroom discourse can allow a teacher to incorporate precise mathematical language into the conversation, and maintain students’ understandings and focus on a particular mathematical goal within the lesson (Franke, et al., 2007; Herbel–Eisenmann, Drake, & Cirillo, 2009). Researchers have reported that the teaching practice of revoicing has been found to support students’ mathematical ideas as well as their identities as math learners (Forman, Larreamendy–Joerns, Stein, & Brown, 1998; O’Connor& Michaels, 1993, 1996; Strom, Kemeny, Lehrer, & Forman, 2001). Interrogated meaning is a phrase introduced by Rosebery, Warren, and their colleagues (Rosebery, Warren, Ballenger, & Ogonowski, 2005; Rosebery, Warren, & Conant, 1992) to engage students in questioning the content being discussed. Through interrogated meaning, students are encouraged to make meaning out of intellectual activity that takes place in the classroom, and clarify any misunderstandings they may have about the content.
Choosing high level mathematical tasks is a teaching practice that involves understanding one’s students well enough to know which tasks will provide high–level cognitive problem solving and engage students in using multiple strategies, connecting ideas across content areas, and provide tasks that are interesting to students (e.g., Franke et al., 2007; Stein, Grover, Henningsen, 1996). Parlady & Rumberger (2008) found that first grade teachers who spend more time engaging students in explaining how a mathematical problem is solved before allowing students to solve the problem saw an increase in test scores. Whole–class discussions around a mathematical problem tend to be at a higher cognitive level if the teacher introduces mathematical tasks in a manner that maintains high cognitive demand (Jackson, Garrison, Wilson, Gibbons, and Shahan, 2013). The same study found that the quality of whole–class discussion around a mathematical task was positively correlated to attention given to the mathematical relationships within a task during introduction of that task, even when the task was not a problem–solving task (Jackson et al., 2013).
In their research with the Mathematical Tasks Framework, Stein and Lane (1996) found that mathematics teaching practices that engaged students in doing mathematics tasks or procedures with connections tasks corresponded to higher achievement levels on assessments that tested students abilities to engage in high–level mathematical thinking and reasoning. These researchers also found that if teachers engaged students in mathematical tasks that “were both set up and implemented to encourage the use of multiple solutions strategies, multiple representations, and explanations,” their students tended to perform better than those who were engaged in tasks that did not do those things (Stein & Lane, 1996, p. 50). Similarly, the researchers on the Quantitative Understanding: Amplifying Student Achievement and Reasoning (QUASAR) project found that engaging students in higher–level mathematical tasks resulted in higher student achievement (Silver & Stein, 1996).
Establishing classroom norms is an important aspect of teaching practice because the practice allows for productive mathematical discourse between students (Franke et al., 2007). Norms are not created solely by the teacher, but can be fostered by teachers through certain activities and language that require students to work together. Teachers’ use of mathematical notation has been shown to help establish productive classroom norms (McClain & Cobb, 2001). According to some researchers (Franke et al., 2007; Yackel, Cobb, & Wood, 1991; Yackel, Cobb, Wood, Wheatley, & Merckel, 1990), there are four types of norms that are built in classrooms: how students and teachers work together, ways for students to think for themselves, what types of mathematical reasons are acceptable, and what explanations are acceptable. Additionally, Kazemi and Stipek (2001) recorded four social norms within fourth and fifth grade classrooms that they found to support conceptual learning among students: “(a) an explanation consists of a mathematical argument, not simply a procedural description or summary; (b) mathematical thinking involves understanding relations among multiple strategies; (c) errors provide opportunities to reconceptualize a problem, explore contradictions in solutions, or pursue alternative strategies; and (d) collaborative work involves individual accountability and reaching consensus through mathematical argumentation” (p. 64).
Kazemi and Stipek (2001) found that the teaching practice of establishing a norm specifically designed to increase opportunities for conceptual thinking led students to a deeper understanding of the mathematical content. As a part of the previously mentioned QUASAR project, Silver et al. (1995) reported that middle school teachers needed to create a classroom norm that included trust and mutual respect if they were going to develop communities that would engage in productive academic discourse. They also found that the teacher should model logical arguments and justifications in order to increase students’ abilities to engage in logical arguments and to successfully utilize mathematical justification as a strategy for communicating mathematical ideas to other students. Research on Cognitively Guided Instruction (CGI) demonstrated that teachers from first grade through third grade cultivated higher–achieving students if they engaged in the teaching practices of listening to their students’ mathematical reasoning and thinking, used that thinking to influence their lessons, engaged students in using multiple strategies for problem solving, and engaged students in more problem solving and word problems than exercises with number facts (Fennema et al., 1996; Franke, Carpenter, Levi, & Fennema, 2000).
Another aspect of establishing classroom norms is the relationships that are formed between teacher and students. Teaching practices should include building relationships among students so that the students feel comfortable having open and honest conversations among each other and can fully explore their own mathematical ideas within the context of the classroom (Franke et al., 2007). In addition, teachers should build relationships with students so they can attend to students’ individual mathematical needs within their lessons. This includes knowing the ways students tend to think about mathematical topics and being informed about potential mathematical trajectories (Simon, 1995; Simon, Tzur, Heinz, & Kinzel, 2004). When building relationships, it is important for teachers to attend to students’ cultural identities and practices (Franke et al., 2007).
As discussed, research studies have characterized multiple best practices in mathematics education, but three commonly researched practices were used as guides for this study: productive mathematical discourse in the classroom, the use of mathematical tasks, and establishing classroom norms. These three categories of teaching practices are used to guide classroom observations by providing a focus for the observations conducted for this dissertation study. While not all ritualized teaching practices fall into the category of best practices (in fact, many are considered not best practice), the focus on best practices allows for a positive relationship between the observed teachers and me.
Direct instruction and dialogic instruction
While the previous three categories of best teaching practices were used to guide the observation focus during data collection, the each of the two prominent identified teaching practices were similar to either direct or dialogic instruction. One group of mathematics education researchers came together throughout the 2011/2012 academic year to characterize and discuss two opposing types of mathematics teaching practices, traditional and reform, which they characterize as direct and dialogic, respectively. The group of nationally recognized experts in mathematics, mathematics education, and education in general met for a series of discussions regarding mathematics teaching practices. The organizers of this collection of discussions, Munter, Stein and Smith (in press), make it clear that both practices are seen as valid forms of teaching among different groups of researchers, and the goal of their orchestrated discussion series was to clarify how each type of practice is characterized by the discussion participants. According to their discussion group, direct instruction is characterized as: a collection of activities that includes: presenting a lesson goal that is explained and connected to previous lessons, describing the necessary concepts and procedures through examples of problems, and allowing students time to practice on similar problem types, during which the teacher provides support as needed. Contrary to direct instruction, dialogic instruction is characterized as: a collection of lessons during which teachers provide students with opportunities to grapple with mathematical content, justify their own mathematical claims, analyze claims stated by their peers, and practice mathematical problems that were thoughtfully created by a teacher. Notably, the authors point out that while teachers use direct instruction, “lessons should be made engaging” (p. 17), but do not include the same caveat to the description of dialogic instruction. Rather, the dialogic instruction definition is supplemented by a description of two types of tasks in which teachers should engage their students: “tasks that initiate students to new ideas and deepen their understanding of concepts, and tasks that help them become more competent with what they already know” (p. 17). The undertones of the two definitions are that there is a clear mathematical goal for both types of lessons, but direct teaching practices include more straight–forward presentation and practice of mathematical content, whereas dialogic teaching practices include more student–centered, exploration of mathematical content.
A TENSION BETWEEN DIALOGIC AND DIRECT INSTRUCTION: ONE COMMUNITY’S MATHEMATICS TEACHING CULTURE
