1 Theorizing Teaching Practices in Mathematical Modeling Contexts

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Theorizing Teaching Practices in Mathematical Modeling Contexts Through the

Examination of Teacher Scaffolding

Dissertation

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy

in the Graduate School of The Ohio State University

Stephen T. Lewis

Graduate Program in the School of Education: Teaching and Learning

The Ohio State University

2018

Dissertation Committee

Azita Manouchehri, Advisor

Theodore Chao, Co-Advisor

James Fowler

Copyrighted by

Stephen T. Lewis

2018

Abstract

The purpose of this study was to examine the reflexive ways in which tensions between desired academic perspectives and life experiences, which are brought forth by learners when dealing with modeling contexts, are addressed by teachers so to advance learners’ modeling capacity. In particular I examined the activities of one 11th grade pre- calculus teacher as she attempted to establish a classroom culture of mathematical modeling. Of interest was unpacking how learners were enculturated into a mathematical modeling practice, and ways that the teacher’s enacted view of modeling advanced the development of both modeling capacity and curricular knowledge.

The multiple lenses of ethnomathematics (D’Ambrosio, 1985) and in particular ethnomodeling (Orey & Rosa, 2010) coupled with linguistic discourse analysis (Bloome et al, 2010) were utilized to more precisely identify the local views of mathematical modeling that the participating teacher held. The analysis of data revealed 23 specific components indicative of the teachers’ mathematical modeling practice.

Results of this study indicated that the teacher’s view of mathematical modeling was not as a vehicle for the learning of new mathematical concepts, nor was it that of a content in its own right, but rather a bridge between these two perspectives which served the opportunity to both advance modeling capacity and meet curricular demands.

Dedication

To my wife, Annie, and my children, Lindsey, Michael, and Jimmy, without whom this would not have been possible.

iii

Acknowledgments

Foremost, I would like to express my sincerest gratitude to my advisor, Dr. Azita

Manouchehri for all of her time in developing me as a mathematics education researcher.

I am grateful to have had the opportunity to learn from her experience and wisdom. Her sustained support helped better me as a scholar and a researcher. I am truly grateful to have worked with her.

I am grateful for my co-advisor Dr. Theodore Chao for his insights into equitable teaching practices in mathematics education. I am grateful for my committee member Dr.

Jim Fowler for providing the voice of academic mathematics. Our discussions have helped me recognize how student ideas connect to the boundary of human knowledge in mathematics.

I am grateful for Dr. David Bloome who provided me with substantial training in discourse analysis. Without him, I would not have been equipped to notice those small details in human interactions that signal important ideas.

I am grateful for Dr. Milton Rosa for his insights into ethnomathematics and ethnomodeling. Without he and Dr. Daniel Orey, the theoretical framework used in this study would be significantly lacking and this study incomplete.

I am grateful for my family, especially my wife, Annie, and my children, Lindsey,

Michael, and Jim. It was their love and support that allowed me to persevere through this

iv doctoral program. I am grateful for my mother, Linda, who taught me to never give up and to keep smiling no matter what.

I am grateful for my friends, especially Adam, Dug, Micky, and Nick, who helped remind me to take a break and play a game. It was in those moments of rest that insight into this work happened.

I am grateful for my academic brothers and sisters, in particular James, Azin,

Ding Ding, and Ayse. Without that community, their questions, and insights, my work would not be of the quality that it is today.

Finally I am grateful for my participating teacher Mae and her students for welcoming me into their classroom, allowing me to ask many questions, and share with me their insights into mathematics. Without this group, this study would not have been possible.

Vita

2005...... B.S. Mathematics

2008...... B.S. Psychology

2009...... M.Ed. Mathematics Education

Publications

Manouchehri. A., & Lewis, S. T. (2017). Reconciling intuitions and conventional knowledge: The challenge of teaching and learning mathematical modelling. In G. Wake, G. Stillman, W. Blum & M. North (Eds.) Crossing and Researching Boundaries in Mathematical Modelling (pp. 93-100). Springer: New York.

Noteman, N., & Lewis, S. T. (2017). Postscript: Support your side. Teaching Children Mathematics, 24(1), 64.

Fields of Study

Major Field: Education: Teaching and Learning Specialization: Mathematics Education

Table of Contents

Abstract ...... ii Dedication ...... iii Acknowledgments...... iv Vita ...... vi List of Tables ...... x List of Figures ...... xi Chapter 1: Introduction ...... 1 Chapter 2: Literature Review and Theoretical Framework ...... 5 2.1 Mathematical Models and Modeling ...... 5 2.2 Theoretical Perspectives on Mathematical Modeling in Educational Settings ...... 15 2.2.1 Modeling as a Vehicle ...... 15 2.2.2 Modeling as Content ...... 23 2.3 Ethnomathematics ...... 29 2.4 Empirical Summary ...... 37 2.5 Theoretical Framework ...... 41 2.5.1 Discourse Analysis in Educational Research ...... 41 2.5.2 An Ethnomodeling Cycle ...... 56 Chapter 3: Methods ...... 64 3.1 Research Design...... 64 3.2 Orientations Towards Data Generation ...... 66 3.2.1 Modeling Competence and Competencies...... 67 3.2.2 Sociomathematical Norms and Social Practices ...... 70 3.2.3 Ethnomathematics and Ethnomodeling ...... 73 3.2.4 Event Mapping ...... 73 3.2.5 Transcribing ...... 80 vii

3.3 Study Procedures ...... 81 3.4 Description of Research Site ...... 86 3.4.1 Background and Description of Cooperating Classroom Teacher ...... 87 3.4.2 Description of Classroom Contexts ...... 90 3.4.3 Researcher Positionality...... 91 3.5 Data Analysis Process ...... 92 3.6 Report on Pilot Study ...... 109 Chapter 4: Data Analysis and Results ...... 113 4.1 Mae’s Short-Term Goals, Long-Term Goals, and Orientation Towards Mathematics ...... 114 4.2 Outline of Modeling Sessions ...... 120 4.2.1 The Answer is Task ...... 121 4.2.2 Measure the Height Task ...... 127 4.2.3 Shortest Distance to the Cafeteria Task ...... 131 4.2.4 Trig Whips Task ...... 136 4.3 Analysis of Mae’s Actions During Modeling Sessions ...... 144 4.3.1 Analysis of The Answer is Task ...... 144 4.3.2 Analysis of Measure the Height ...... 148 4.3.3 Analysis of Shortest Distance to the Cafeteria ...... 152 4.3.4 Analysis of Trig Whips ...... 157 4.3.5 Analysis of Mae’s SMP Components Across Sessions ...... 161 4.4 Elaboration on the Components of Mae’s Sociomodeling Practice ...... 163 4.4.1 Sense-Making ...... 163 4.4.2 Conceptual Tool Use, Thinking about Mathematics, and Elaborating Students’ Mathematics ...... 167 4.4.3 Thinking About the Real World and Experiential Tool Use ...... 177 4.4.4 Teacher Humanizing ...... 182 4.4.5 Formalizing Curricular Mathematics and Asking Mathematical Questions.. 185 4.4.6 Enculturation into Mathematics and Activating Metacognition ...... 190 4.4.7 Physical Tool Use, Research, and Parameterization ...... 192 4.4.8 Assumption Making and Problem Generation ...... 198 4.4.9 Establishing Variables, Asking Contextual Questions, and Collaboration.... 205 4.4.10 Communicating Ideas and Sharing Solutions ...... 212 viii

4.4.11 Validating and Refining ...... 225 4.5 Mae’s View of Mathematical Modeling as Bridge ...... 232 4.6 Mae’s Scaffolding Interactions ...... 237 4.6.1 Provoking ...... 237 4.6.2 Acknowledging/Validating Learners’ Ideas ...... 238 4.6.3 Elaborating ...... 238 4.6.4 Emphasizing ...... 242 4.6.5 Clarifying ...... 242 4.6.6 Positioning/Stating ...... 243 Chapter 5: Conclusions and Implications ...... 245 5.1 Research Question 1 ...... 246 5.1.1 Modeling as Bridge ...... 247 5.1.2 Viewing Mae’s Practice Through an Ethnomodeling Lens ...... 250 5.2 Research Question 2 ...... 251 5.3 Teacher Decision Space and Pedagogical Vision ...... 254 5.4 Recommendations ...... 261 5.4.1 On Teaching ...... 261 5.4.2 on Teacher Education ...... 263 5.4.3 On Curriculum Development ...... 265 5.4.4 On Research ...... 266 5.5 Limitations and Barriers ...... 269 5.6 Concluding Remarks ...... 270 References ...... 272 Appendix A: Modeling Tasks ...... 284 Appendix B: IRB Approval Letter ...... 286

List of Tables

Table 1. Sample Event Map ...... 76 Table 2. Data Collection and Analysis Timeline ...... 82 Table 3. Overview of Observations in Data Collection ...... 93 Table 4. Analytical Components of SMPs ...... 98 Table 5. Analytical Coding Template ...... 104 Table 6. Sample Gestural Analysis ...... 107 Table 7. Distribution of Modeling Tasks ...... 116 Table 8. Evidence of Teacher Goal of Modeling ...... 118 Table 9. Event Map of Answer is Task...... 122 Table 10. Event Map of Answer is Task Debriefing Discussion ...... 124 Table 11. Event Map of Measure the Height Task ...... 128 Table 12. Event Map of Shortest Distance to Cafeteria Task ...... 133 Table 13. Event Map for Trig Whip Task...... 136 Table 14. The Answer Is Macro Analysis ...... 146 Table 15. Measure the Height Task Analysis ...... 150 Table 16. Shortest Distance to Cafeteria Analysis...... 154 Table 17. Trig Whips Analysis ...... 159 Table 18. Sense-Making ...... 165 Table 19. Use of Conceptual Tools...... 169 Table 20. Thinking about the Real World and Experiential Tool Use ...... 178 Table 21. Teacher Humanizing ...... 184 Table 22. Enculturation into Mathematics ...... 191 Table 23. Use of Technology, Research, and Parameterization ...... 193 Table 24. Reflective Interview of Answer Is Task ...... 197 Table 25. Problem Generation ...... 199 Table 26. Alyssa and Amber Make Assumptions ...... 204 Table 27. Establishing Variables and Constraints, Asking Contextual Questions, Collaboration...... 206 Table 28. Communicating Ideas and Sharing Solutions ...... 214 Table 29. Collaboration, Revision, Working Mathematically ...... 226 Table 30. Refinement of Model ...... 230 Table 31. Transition between Elaborating and Formalizing ...... 233 Table 32. Elaborating on Municipal Tax ...... 240

List of Figures

Figure 1. Modeling Cycle of Blum and Leiss (2007) ...... 7 Figure 2. Modeling Cycle of Pollak (1979) ...... 8 Figure 3. Fire Brigade Problem (Shukajlow, Kolter, & Blum, 2015) ...... 18 Figure 4. Solution Plan for Mathematical Modeling Tasks ...... 19 Figure 5. Cable Car Problem (Besser, Blum, & Kilmczak, 2015)...... 20 Figure 6. Personal Written Feedback (Besser, Blum, & Kilmczak, 2015) ...... 21 Figure 7. Ethnomodeling Cycle ...... 58 Figure 8. Data Analysis Protocol ...... 96 Figure 9. Sample Frequency Analysis of Component Occurrence ...... 105 Figure 10. Sample Ethnomodeling of Measuring ...... 109 Figure 11. Mae's Decision Space ...... 119 Figure 12. Aerial Map of Field Site ...... 132 Figure 13. Trig Whips Task Handout ...... 140 Figure 14. Back of Trig Whips Task Handout ...... 141 Figure 15. The Answer Is Component Distribution...... 147 Figure 16. Measure the Height Component Frequency ...... 151 Figure 17. Shortest Distance to Cafeteria Frequency Distribution ...... 156 Figure 18. Trig Whips Task Component Frequency ...... 160 Figure 19. Typology of Teacher Intervention by Component Frequency Across Modeling Sessions ...... 162 Figure 20. Formalizing Sine, Cosine, and Tangent ...... 187 Figure 21. Sine and Tangent Examples ...... 188 Figure 22. Formalizing Student Problem Contexts...... 189 Figure 23. Etic Ethnomodeling of Measuring the Umbrella...... 202 Figure 24. Measure the Height Referent Diagram ...... 211 Figure 25. Overview of Significant Findings ...... 246 Figure 26. Teacher Decision Space ...... 257

Chapter 1: Introduction

With the adoption of the National Governors Association (2010) Common Core

Standards for Mathematics (CCSM), there has been an increased interest in mathematical modeling in the United States as mathematical modeling has been listed as both high school content standard and a Standard for Mathematical Practice. While the CCSM describes the mathematical modeling process as a mechanism for the application of mathematical concepts to examining real-world issues, it offers little guidance in productive means for teaching mathematical modeling, and rather presents it as an extension of other standards. The absence of a research-based guide on how this standard might be effectively implemented is problematic (Cai et al., 2016), in particular when considering successful implementation of mathematical modeling as an open process depends on the critical and reflective analysis of the problems students face working on such tasks. There is agreement that more research is needed on the types of teacher interventions which are appropriate in modeling contexts (Borromeo-Ferri, 2018).

Evidence exists that facilitating modeling cognition among school learners involves intricate attention to cultural backgrounds and life experiences of children when considering the ways in which desired mathematical outcomes are to be treated. These outcomes should not be assumed as cherished prizes, but rather as humanizing capital which permits deeper reflection on these experiences by both teachers and school learners

(Manouchehri & Lewis, 2017; Rosa & Orey, 2017; Reilly, 2017; Rosa & Shirley, 2016;

Rosa & Orey, 2016). As such, the relationship among learners’ use of mathematics in making sense of real life events, teachers’ reflections on what they learn from children as they attempt to facilitate their modeling work, and the ways in which gulfs in knowledge are bridged demands meticulous scholarly attention within the international community.

It has additionally been pointed out that, even though there are agreed upon processes within mathematical modeling including assumption-making (Lesh, Hoover,

Hole, Kelly & Post, 2000), mathematizing (Freudenthal, 1983), parameterization (Ljung,

1998), selection and simplification of both contextual and systematic aspects (Niss, 2010;

Blum & Leiss, 2007), decision making (Gershenfeld, 1999), and validation (Niss, Blum

& Galbraith, 2011), various definitions of this process exist (Lesh & Fennewald, 2010).

Regarding the teaching of modeling, two distinct perspectives have aligned with its treatment: either as content motivated by a genuine, real-world context or as a vehicle with which to develop prescribed mathematics and service curricular needs (Julie &

Mudaly, 2007; Galbraith, Stillman, & Brown, 2010). These contrasting perspectives of the nature of mathematical modeling and its application within classrooms mandate a careful study of how teachers with an understanding of mathematical modeling facilitate modeling tasks in their settings. Such a study should take into consideration factors which teachers tend to have to reconcile, such as the presentation of deliberate mathematical concepts.

The overarching goal of my dissertation is to address these gaps in research by examining the reflexive ways in which tensions between desired academic perspectives

2 and life experiences which are brought forth by learners when dealing with modeling contexts are addressed by teachers so to advance learners’ modeling capacity. The proposed study examines, in particular, the activities of one teacher as she establishes a classroom culture of modeling practices. Of interest also is unpacking how learners are enculturated into this practice, and ways that the teacher’s enacted view of modeling advances development of modeling capacity and curricular knowledge. The research questions guiding this study are:

1. How are social practices in mathematical modeling contexts developed through

implementation of modeling tasks?

2. How does teacher scaffolding within these mathematical modeling contexts

promote the development of modeling capacity in learners?

Of interest within these research questions is developing an understanding of the particularities of teacher practice in mathematical modeling. Targeted scaffolds are examined as a way of theorizing teaching of mathematical modeling. The work is motivated through considering the authentic classroom experiences in which a teacher reconciles mathematical, professional, and personal obligations in presence of mathematical modeling based instruction. As will be detailed in the following sections, my view is motivated by the knowledge that school learners consider actively their cultural, historical, and family/life experiences when engaged in solving real-life problems. There is evidence that students draw on such non-academic intellectual capital when dealing with mathematical modeling tasks (Manouchehri & Lewis, 2017).

Adopting ethnomathematics (D’Ambrosio, 1985) and, specifically, ethnomodeling (Rosa

& Orey, 2010) as a framework for interpreting classroom interactions and, in particular, examining local practices in mathematical modeling, I intend to show how the teacher draws on learners’ extra-mathematical ideas to facilitate their modeling process. In doing so, I will analyze shared utterances using fine-grained textual discourse analysis (Bloome et al., 2010) to examine the means by which the teacher attempts to advance student learning while responding constructively to ideas shared in discussions, in which learning is characterized as “changing patterns of participation in specific social practices within communities of practice” (Gee & Green, 1998, p. 147).

Ethnomathematics affords consideration of the intersection of multiple cultures in the course of evolving group understandings. These multiple cultures include, but may not be limited to, the academic classroom culture negotiated and established by the teacher and school learners, the learners’ own espoused culture, as well as the culture associated with learning mathematics. When considering both mathematical and non- mathematical domains and their corresponding cultures, I present an argument that these multiple cultures form their own Funds of Knowledge (Moll, Amanti, Neff & Gonzalez,

1992), established through learners’ experiences and membership in those cultural communities, such as the classroom context. Each of these perspectives will be framed in the literature review and synthesized into a theoretical framework for analyzing classroom interactions in mathematical modeling contexts, tracing attempts at developing learners’ modeling capacity over time by examining participation patterns, and specific components of modeling process emphasized by the teacher and taken up by students as evident in their work.

Chapter 2: Literature Review and Theoretical Framework

In this chapter I offer an overview of the genres of research that ground my work and relevant literature associated with each genre, including mathematical modeling and ethnomathematics. I draw from this body of work to outline the theoretical perspective which I will use to examine teacher scaffolding in mathematical modeling.

2.1 Mathematical Models and Modeling

A mathematical model is described as a representation, in mathematical terms, of the behavior of objects (Dym, 2004) which is used to understand and interpret complex systems (Gershenfeld, 1999). Mathematical modeling reflects the action process of generating mathematical representations. This process involves understanding, validating, and detecting the limitations of tentative constructed models and refining them so to achieve greater precision (Erbas, Kerti̇l, Çeti̇nkaya, Çakiroğlu, Alacaci, & Baş,

2014; Dym, 2004; Lesh & Doerr, 2003). Mathematical modeling transitions from the examination of the physical characteristics of a system to translating them into a mathematical form for analysis; hence, knowledge of the external system, itself, must be acquired in order to engage in this process. In this process, assumptions must be made to simplify the situation being modeled, and variables which impact it must be identified

(Haines & Crouch, 2010; Verschaffel & De Corte, 1997; Erbas et al., 2014). Gershenfeld

(1999) claims that there is an element of decision-making associated with mathematical

5 modeling, in which correct and explicit or, more often, implicit assumptions concerning the system must be made in quest of an appropriate representational model.

Fundamentally, mathematical modeling involves the process of mathematizing, but contrasting perspectives of this process exists. Fueudenthal (1983) describes mathematization as detaching a context from reality through mathematics. Niss, Blum, and Galbraith (2011) characterize mathematizing as the process of removing a context from the confines of a problem so that mathematical work can be accomplished and the problem solving or modeling process can be advanced. Rosa and Orey (2010) contrast this view and argue that mathematization is the process in which members of distinct cultural groups come up with different mathematical tools which help them to organize, analyze, solve, and model specific problems located in the context of their own real-life contexts and situations. This allows groups to identify and describe specific mathematical ideas, procedures, or practices by schematizing, formulating, and visualizing problems in different ways. This yields the discovery of relations and regularities transferring real- world problems to academic mathematics through the process of mathematization. This cultural view of mathematization informs the current study, as cultural components play an integral part in local modeling processes and, in unpacking local conceptions of mathematical modeling. However, other views of mathematization, in particular, the decontextualization of the real world, are important to consider in concert with this ethnomathematical view.

Blum and Leiss (2007) describe mathematical modeling (Figure 1) as a process in which a problem situation, couched in real-world transitions into the mathematical world

6 through the process of mathematization, is analyzed and then, through interpretation of the results, moved back into the real world and validated. In order for this transition to occur, the problem first has to be interpreted by a problem solver and then, from that understanding, a situation model must be constructed. The situation model is then described by the modeler in the form of a real model. Blum and Leiss indicate that, in this phase of the modeling cycle, the modeler makes specific assumptions and controls for variables. Through the process of mathematization, a model is constructed. This mathematical result is then interpreted and compared against real-world conditions and constraints and validated. This cycle of constructing and verifying is then repeated until a sufficiently robust solution is reached (p. 226).

Figure 1. Modeling Cycle of Blum and Leiss (2007)

Pollak (1979) presents a compatible view of the process of mathematical modeling. His model identifies a relationship between Classical Applied Mathematics

(analysis, calculus, differential equations, integral equations, and theory of functions),

Applicable Mathematics (sets and logic, functions, inequalities, linear algebra, modern algebra, probability and statistics), and the Rest of the World (areas outside of these domains) (Figure 2). According to Pollak, doing applied mathematics involves transitions between these worlds. As a result, different mathematical concepts may be employed to construct a model for a given situation. In Pollaks’ view, the models which are produced are not necessarily cognitive, but rather explicit mathematical objects.

Kaiser (2005) indicates that, when doing applied mathematics, translating from a real-life situation into a mathematical problem is the heart of mathematical modeling (p. 101).

Figure 2. Modeling Cycle of Pollak (1979)

When examining the distinction between these different, but compatible, views of modeling, Blum and Leiss’ (2007) cycle appears to be more process-focused, whereas 8

Pollak‘s (1979) approach is more object-focused. This distinction becomes important, as they emphasize two different aspects of mathematical modeling: that there is an inherent cognitive process associated with the development of models and physical and mathematical objects which are constructed and acted on mathematically, respectively.

In examining mathematical modeling, these cycles emphasize that attention should be paid to both the mathematical objects as well as the processes involved in enacting this process.

Borromeo-Ferri (2006) discuss the cognitive nature of engaging in modeling cycles and claim that, in this process of modeling, students need to draw on their extra- mathematical knowledge of the situation under study. This extra-mathematical knowledge, according to the author, informs not only how they enter the modeling cycle, but also how they validate or reflect on the validity of their solutions (p. 149).

Transitioning between these various stages requires the modeler to connect their extra- mathematical knowledge (or knowledge of the context) to the mathematical ideas to which they have access. This can occur in a multitude of forms and across various levels of sophistication regarding the mathematical content. While the modeling cycles presented above address this particular notion, Borromeo-Ferri explicitly draws attention to the nature of the knowledge which learners draw upon from their extra-mathematical knowledge. Thus, careful attention needs to be paid to the ways in which this extra- mathematical knowledge manifests itself over the course of developing this disposition.

Borromeo-Ferri and Lesh (2015) indicate that when students engage in modeling, they do not always activate conceptual systems which are exclusively mathematical or

9 logical in their nature, but instead rely on their beliefs, dispositions, attitudes, and experiences. Learning to model then requires activation of these beyond-cognitive attributes (p. 63). Thus, when engaging students in mathematical modeling tasks, teachers need be prepared for instances in which students conceptualize the real-world context through an experiential lens which may be beyond cognitive attributes when thinking about problems. As such, in teaching mathematical modeling and facilitating a disciplinary view when examining these contexts, teachers may need to recognize how prior mathematical knowledge or how providing additional mathematical tools, concepts, or structures may help learners think about the situation under study. Further engagement in this process requires teachers to problemize learners’ considerations to support them in making mathematical connections or using mathematical structures.

The literature in mathematical modeling has also established differences in modeling performance among expert and novice modelers (Haines & Crouch, 2010;

Mousoulides, Christou, & Sriraman, 2008; Crouch & Haines, 2004). These authors characterize expert modelers as those who typically use mathematical modeling purposefully in related field work, and contrasts these modelers with novice modelers, i.e., those students learning the process of mathematical modeling. Findings in these studies indicate that novice modelers may experience greater difficulty in determining ways to account for the large number of variables and constraints in modeling contexts.

Additionally, these novice modelers rely heavily on their contextually-bound personal experiences or beliefs, and, as such, linking relevant mathematical theories to the context of the problem can be problematic (Haines & Crouch, 2010; Crouch & Haines, 2004).

For example, novice modelers tend to attempt to develop models which account for all identified variables or may have trouble in moving forward by not knowing what variables to control through making assumptions. This challenge has been linked to the modelers’ intuitive understandings of the systems under study. Mousoulides, Christou, and Sriraman (2008) found that a productive means for supporting advancement from novice to expert modeler is linked to exposure to modeling-focused problem-solving experiences coupled with deliberate teacher support (p. 129).

Doerr and English (2003) stress the importance of encouraging students to examine problem contexts several times in order for them to fully understand the intricacies the task demands. They argue that multiple entries allow learners to interpret the constraints, identify relevant variables, and perhaps delineate parameters of importance, as done in professional modeling. Through this process, modelers gain insights into the dynamics of particular contexts. For example, in endemic disease modeling, determining whether to examine the population, as a whole, as opposed to breaking up the population into subgroups and looking at the interactions between individuals based on a given rate of infection impacts directly how vaccines are distributed. Similarly, taking a continuous or discrete view of the system impacts the methods used to determine the relative infection and transmission of a given disease

(Earn, 2008; van Den Driessche, 2002; Hethcote, 2000; Hethcote & York, 1984; Keeling

& Eams, 2005). Thus, in structuring learning, it is important for teachers to find productive ways to capitalize on different views and highlight how those views inform the context.

Different types of mathematical modeling problems have also been discussed in the literature. For instance, English and Lesh (2003) propose ends-in-view problems as tasks that “present students with particular criteria for generating purposeful, complex, and multifaceted products that go beyond the given information” (p. 297). They further indicate that ends-in-view problems differ from typical in-class problems both in the level of complexity they require in constructions and in describing or justifying criteria to test the solution (p. 298). With regards to mathematical modeling tasks, this might involve a produced solution needing to be used for a specific purpose, i.e., not just to investigate the system and explore the components, but instead to satisfy the needs of an outside party.

Lesh, Cramer, Doerr, Post, and Zawojewski (2003) identify three types of modeling tasks useful in educational settings: Model Eliciting Activities, Model

Exploration Activities, and Model Adaptation Activities. According to Lesh et al., model eliciting activities typically require multiple days to complete in class, students are often encouraged to work in groups, and these activities are typically used before instruction on a given topic. The purpose of model eliciting activities is to make the students’ thinking apparent to the teacher and allow the teacher to identify conceptual strengths and weaknesses which might inform their instruction on the related unit (p. 46). The focus on model eliciting activities is unfolding the thinking processes of the students when engaged in mathematical modeling the representations they use to examine a problem.

Model exploration activities have the goal of helping students make sense of the target conceptual system through exploration using computer graphics or simulations. Again,

12 here, the mathematical representations are often prescribed, as the intended goal is for the teacher to gauge how the students interact with the conceptual systems and mathematical objects at their disposal (p. 46). Model adaptation activities involve taking the tools and conceptual systems developed during the model-eliciting activity and extending or applying them to a problem which would have initially been too difficult for the students to handle (p. 48). This hierarchy of tasks allows teaches to trace the development of a mathematical topic by considering how students come to interact with it in the given context. Importantly, the mathematical objects that are employed are often prescribed, as they have an aligned instructional intent. This is substantially different from open-ended problem situations and may require additional mathematical knowledge formalize (Doerr,

2007).

Niss, Blum, and Galbraith (2007) differentiate between Word Problems,

Application Problems, and Modeling tasks according to their cognitive demand. Niss et al. characterize word problems as dressed up mathematical problems in which words are borrowed from science or everyday life. These authors consider mathematizing a word problem to involve merely stripping away the context and using mathematics in a straightforward manner. Application problems are characterized by the fact that a known model for the given context is immediately at hand. Mathematization in application problems entails removing the context being modeled and establishing a purely mathematical question which can be solved without having to account for the context itself. In modeling problems, however, the question must be first specified and

13 operationalized, and the intended solution must be evaluated against both mathematical and contextual standards (Niss, Blum, & Galbraith, p. 11-12).

In considering the relationship between modeling contexts and application-based tasks, Niss et al. indicate that, in modeling, “we are standing outside mathematics looking in,” which involves looking at the processes involved in seeking mathematics to describe a context (p. 10). In contrast, applications involve “standing inside mathematics looking out” and taking on an object-focused view in which mathematics is linked to various contexts (p. 11).

In Niss et al.’s view, structuring tasks in certain ways affords different outcomes, either mathematical or contextual. When considering what constitutes mathematical modeling, few types of problems survive if this modeling process is used as criteria (p.

12). However, the perspective which is taken on mathematical modeling, in itself, determines the knowledge development process in mathematical modeling contexts. I distinguish between mathematical modeling as a vehicle for applying known mathematical concepts and mathematical modeling as a conceptual domain within mathematics itself. Looking beyond modeling as a vehicle to teach mathematics affords a view in which knowledge is linked to how mathematics describes a context and further opens up the notion of what competence in modeling looks like (Galbraith, Stillman, &

Brown, 2010).

Each of these types of tasks has been shown to foster the development of particular aspects of the mathematical modeling process, either in mathematization, problem formulation, working mathematically, or validating.

2.2 Theoretical Perspectives on Mathematical Modeling in Educational Settings

In examining the nature of mathematical modeling in educational settings, two distinct conceptual genres emerge: (1) teaching mathematics using modeling as a vehicle for instruction, and (2) developing mathematical modeling as a mathematical and conceptual domain of its own. Julie (2002) distinguishes these two conceptual genres, indicating that modeling as a content emphasizes the development of competences needed to model real-life situations. He further distinguishes modeling as a vehicle for using modeling to teach mathematical concepts (p. 3). In this section, I will provide an overview on each of these perspectives, compare them, and discuss their affordances and limitations when considering how the adopted view of modeling enhances the ability to examine learners’ engagement in the mathematical modeling process.

2.2.1 Modeling as a Vehicle

Using mathematical modeling as a vehicle for the development of mathematical concepts has been far more popular in the scholarship associated with mathematical modeling due to the fact that integrating mathematical modeling into the curriculum is easier. Galbraith, Stillman, and Brown (2010) argue that, within modeling as a vehicle, the focus is on developing prescribed mathematics. In this way, modeling is used as a means for learning new concepts, procedures, ways to conjecture or use context-driven justifications to solve problems. The authors argue that the intent of modeling as a vehicle is to service curricular needs.

An example of modeling as a vehicle can be seen in Model Eliciting Activities

(Lesh & Doerr, 2003). Lesh and Doerr (2003) discuss Model Eliciting Activities as thought-eliciting activities or tasks which emphasize the process that students undergo when solving problems. Lesh, Hoover, Hoyle, Kelly, and Post (2000) present six principles for the development of these particular types of thought-eliciting activities, one of which includes the model construction principle. According to Lesh et al. (2000), students should be able to construct a model through having access to appropriate mathematics to help them in describing the situation. As a result, teachers can see how students think and reason about both the context being modeled and its relationship to mathematical ideas. In designing modeling experiences for learners, this principle is accounted for by structuring situations based on idealized mathematical interpretations.

When planning and implementing modeling activities in the classroom, particular mathematical concepts can be highlighted to better understand how students think through and reason about interactions between the context and known mathematics. In adopting this particular view, difficulties in modeling can be attributed to an absence of important mathematical concepts and, as such, scaffolding practices take a particular form.

Scaffolding through the lens of modeling as a vehicle entails re-aligning student thinking to a particular mathematical form. For example, Schukajlow, Kolter, and Blum

(2015) propose a systematic scaffolding technique as a framework which is useful for helping learners’ progress through the modeling process. This framework is geared towards long-term development of learners’ self-regularity skills through providing

16 adaptive support and structures (Smit, van Eerde, & Bakker, 2013; Puntambekar &

Hubshcher, 2005), as cited in Schukajlow Kolter, & Blum, (2015). Schukajlow, Koler, and Blum offer a solution plan (Figure 4) as a means for supporting student thinking in mathematical modeling contexts. Here, the modeler is prompted with a series of techniques for progressing through the task, and these prompts align directly to phases in the modeling cycle (Blum & Leiss, 2007).

I illustrate how a solution plan scaffolds learners through a modeling as vehicle task by sharing its application on finding the maximum height that a fire-engine can rescue people in a burning building (Figure 3). The problem specifies that the truck must maintain a distance of at least 12 meters from the burning house, and the dimensions of the ladder and engine are provided. These questions correspond with using a particular mathematical means for representing the overarching context, specifically, the mathematical process of finding missing values using the Pythagorean theorem. The solution plan’s support aligns students’ view to an idealized mathematical process which determines the validity of solutions for this particular task. Through this particular means of scaffolding the responses, the modeler is led through this solution plan, coming to a specific classification of response while paying attention to particular mathematics used to represent the system.

Figure 3. Fire Brigade Problem (Shukajlow, Kolter, & Blum, 2015)

Schukajlow, Kolter, and Blum (2015) argue that following this alignment students can then be guided to further validate the solution by accounting for additional factors, for example, the height of a fireman climbing the ladder or possible jumping of the person being rescued (p. 1246).

Figure 4. Solution Plan for Mathematical Modeling Tasks

Besser, Blum, and Kilmczak (2015) also offer a feedback mechanism in mathematical modeling contexts. As illustrated in the solution plan presented above

(Figure 4), the authors examined the impact of providing both written and oral feedback in laboratory settings to students working on a mathematical modeling context. In this particular study, the students were given the horizontal and vertical distance of a cable car and asked to determine the cost of the rope suspending the car above the ground

(Figure 5).

Figure 5. Cable Car Problem (Besser, Blum, & Kilmczak, 2015)

Figure 6 highlights a sample response and scaffolding mechanism for the Cable

Car Problem. The feedback highlights the areas in which the instructor believed learners had interpreted the context appropriately and those in which the solvers had difficulties

(both mathematical and contextual). Lastly, suggestions for improvement in solving the problem are offered.

Figure 6. Personal Written Feedback (Besser, Blum, & Kilmczak, 2015)

In highlighting areas in which students approached the problem inaccurately, the researchers used mathematics as a framework for assessing whether or not the students were able to interpret the picture and extract relevant information from it. In this particular sample of feedback, the research team asked the solver to “think about the real situation concretely and have a closer look at the picture” as well as suggesting that “if extracting square root do so on both sides of the equation” (p. 473).

In reporting the value of these types of feedback mechanisms the researchers claimed there were no significant differences in performance as regarding either written or oral feedback when looking across experimental and control groups. Their analysis suggests that the typology of implementation of feedback with the intent to best improve performance on modeling tasks remains unknown (p. 476).

In considering the theoretical and philosophical orientations aligned with these mechanisms for scaffolding, I observed that feedback consisted primarily of looking at epistemological means for improving performance on assigned tasks using a deliberate model when exploring the situation. Fluency was measured by the use of the

Pythagorean theorem in analyzing these contexts leading to an idealized view of the problem context. As such, when structuring a task around the application of an idealized model, flexibility in that model’s use, particularly when performing calculations, is a fundamental part of interpreting the problem. However, when applying their feedback, the authors claimed that the students’ interpretations and reasoning within the context was problematic. There is little data reflected in the research report on how the students actually solved the problem or the mathematics they had associated with the context. The limited information on how students perceived the problem context may offer an explanation for why the results of this particular study did not reveal differences across feedback mechanisms. I emphasize that how problems are interpreted and solved from students’ perspectives may be beneficial in understanding how and why they mathematize contexts. I contrast this view of mathematical modeling with one in which,

22 through dialogue and social interactions, a better understanding of learners’ choices is obtained.

In conjunction with viewing modeling as a vehicle for teaching or exploring mathematical concepts, an alternate view is to consider mathematical modeling as a conceptual domain. In the next section, I illustrate how considering modeling as a concept affords an alternative view for how modeling skills may be supported in educational settings.

2.2.2 Modeling as Content

A second genre of mathematical modeling considers it to be a conceptual domain in its own right. Galbraith, Stillman, and Brown (2010) describe modeling as content in which mathematics is the outcome of the model development process. They further ague that this view of modeling entails the scrutiny, dissection, critique, extension, and adaption of existing models with the purpose of assessment and evaluation of constructed models. They also argue that this view of modeling is motivated by genuine, real-world contexts rather than advancing mathematical knowledge. Modeling as content affords mathematics to be treated as chosen by the learner, not imposed by an authority, in which the goal is improvement in modeling cognition and learners may need to be convinced that alternate views are viable or appropriate.

In developing learners’ analytical skills, a teacher or researcher may need to engage in reflexive (but also critical) inquiry (Qualley, 1997) in which they attempt to discover, examine, and critique claims and assumptions in response to an encounter with

23 an idea, text, person, or culture (p. 3) and help learners become critically aware of their own ideas (p. 4). Reflexivity in teaching modeling also means that teachers may have to abandon their perceived notions of appropriate or viable mathematics in a given context, listen carefully to students’ views and interpretations, and help them evaluate and expand those views. This situates the teacher as a co-learner with the students and addresses the democratic structures of classroom interactions in opposition to traditional authoritarian views of teaching. Given the nature of curriculum, this view of teaching can be difficult to implement, as it requires flexibility and willingness to balance these features with academic expectations.

Niss (2012) claims that “applications, modeling and modeling for the learning of mathematics deal with the activation of mathematics in extra-mathematical contexts for extra-mathematical purposes can foster motivation with some students for the study of mathematics and help support and consolidate their concept formation, sense-making and experience of meaning in and of mathematics” (p. 50). Hence, using modeling as a context with which to explore new mathematical ideas can assist students in developing an understanding of some underlying ideas associated with the mathematical concept being taught. Similarly, Gravemeijer (2007) argues that focusing on learning mathematics within a particular context and relating that context to the formal mathematical structure could be used as a point of inquiry for exploring the concept in detail (p. 140). This notion supports the idea that modeling could be used as a point of reference for developing new mathematical topics. Niss, Blum, and Galbraith (2007) further claim that a modeling perspective begins with the conceptualization of some

24 problem situation, then, through simplifying and structuring, makes the situation more precise (p. 9). Niss, Blum, and Galbraith further elaborate that this precision occurs according to the problem solvers’ knowledge, goals, and interests, leading to a particular specification relative to the language and concepts of the situation (p. 9). I focus here on the aspects of simplifying and structuring and, in particular, the role that language plays in the scaffolding process.

Modeling as content demands that learners come to learn how modeling can be a useful tool in describing behaviors or practices within groups or systems. In this way modeling is used to support real-life problem solving. This view is consistent with scholars affiliated with the ethnomathematics field. Rosa and Orey (2010) define ethnomodeling as the process of elaborating problems or questions which arise from contexts outside of the formal domain of mathematics and are drawn from realities and experiences of the individuals, which, in turn, create an idealized version of the knowledge needed for a particular group to thrive and survive (p. 61). Accordingly, the ethnomodeling perspective focuses on how knowledge is produced, the social conventions in which the knowledge is institutionalized, and the means by which it is transmitted across the cultural group for educational purposes. Thus, it is a study of the system in its entirety, in which an understanding of the system in itself is acquired as well as the interrelationships across systems (p. 61). Rosa and Orey argue that this evaluation comes from examining the social contexts in which this production and dissemination occur and should begin with the interests/motivations of the students rather than a focus on idealized curricular objectives or processes (p. 61). In these regards, ethnomodeling

25 relates teaching to the development of new knowledge through dialogical inquiry (Freire,

1973).

Regarding teaching practices associated with ethnomodeling, Rosa and Orey

(2010) argue that the integration of ethnomodeling into pedagogical practices allows us to view perceived deficits in mathematical knowledge as “conceptual mathematical richness” (p. 62) but warn that studying the knowledge systems of cultural groups in an idealized and western-mathematical view can be problematic. Rosa and Orey (2010) argue further that when drawing on an ethnomodeling perspective the teacher should be willing to adjust their own mathematical ways of thinking and practices so to accept notions that differ from conventional mathematics.

Further, Rosa and Orey (2010) argue that research grounded in ethnomodeling should involve examining ways in which individuals or groups draw on traditional or curricular mathematical ideas in the course of their problem solving experiences, not to idealize these as correct or appropriate ways of thinking, but rather to highlight the relationship between cultural groups and the deeply embedded mathematics in their daily activities (p. 64). This construct of ethnomodeling becomes powerful since it does not disregard prior exposure to western mathematical conventions, but, instead, accounts for their enactment in problem solving contexts. It looks at not only how individuals employ those ideas, but also how they navigate through the problems while drawing on various types of knowledge. Further, Orey and Rosa state that one important (if not the most) characteristic of ethnomathematics and ethnomodeling is that they help stretch the notion of what constitutes conventional mathematical knowledge and linking it to what is known

26 in academic western mathematics (p. 64). Barbosa (2006) offers a means for examining the nature of discourse which emerges in modeling discussions and uses these as a basis for a socio-critical approach to facilitating modeling discussions.

Barbosa (2006) describes critical epistemology as inclusion of critique in the knowledge-construction process. Barbosa argues that, when engaging in modeling discussions, three types of talk typically emerge: mathematical, technological, and reflexive discussions (p. 298). In Barbosa’s view, mathematical discussions relate to ideas belonging to the pure mathematics field, technological discussions refer to the techniques associated with the development of models, and reflexive discussions involve the nature of the model, the criteria used, and associated consequences (p. 297).

According to Barbosa, the constitution of a socio-critical approach to modeling is associated with the reflexive discussions which serve as an entry point into considering the role of mathematics in society (p. 298). An important implication regarding modeling as a conceptual genre emerges from Barbosa’s work: there are differences in the types of talk which emerge within a discussion of modeling contexts, and these different types of talk correspond to different aspects of the development of models. Barbosa argues that, depending on the teachers’ perspective, each type may be weighed differently (p. 298).

This suggests that when examining the entirety of the modeling process, the formation and validation of models plays a key role, and aspect is a fundamental part of modeling.

While the two perspectives of modeling as content and modeling as vehicle are different, it is not to say that they are not compatible. Julie and Mudaly (2007) indicate that the range between modeling as content and vehicle is a spectrum and the two

27 perspectives are not mutually exclusive; however, in most studies dealing with mathematical modeling, one particular perspective is adopted and findings support the advancement of either modeling as content or vehicle. Each function of teaching mathematical modeling aligns with planned and in-the-moment instructional goals established by the classroom instructor or educational researcher. One fundamental difference between modeling tasks in light of content and vehicle is that their formation, facilitation, and extension might be linked to different types of probing questions that are used to further development. Should an instructor wish to pursue a deeper understanding of the nature of the so called “real” context, itself, questions may revolve around the utility of the model in terms of predicting or describing behaviors in the situation. If the instructional outcomes are intended to be more mathematical in nature, in terms of understanding and developing mathematical objects, the discussion should be framed according to an inquiry into the nature of the model, itself. Similarly, paying heed to mathematical ideas which the students already have access to may help teachers in predicting how students may perceive situations from a mathematical perspective and structuring those ideas into new mathematical concepts to explore.

It is critical to note that the instructor sets the nature of these conversations. Niss

(2012) brings to light two important points reflected in the research on mathematical modeling. First, that knowledge of pure mathematics, alone, is insufficient to transfer these skills to contexts outside of mathematics, as the ability to transfer knowledge requires learners to be equipped with knowledge of extra-mathematical contexts and the ability to make assumptions about the contexts in order to control the vast number of

28 variables which are part of a real-world situation. Second, that modeling and development of mathematical models can, in-fact, be taught, but that explicit curricular decisions need to be addressed by the instructor in order to establish a learning environment which is conducive to the instructional goal set (p. 51).

These differences in perspectives of modeling (vehicle or conceptual genre) ultimately inform the ways in which knowledge is considered viable in modeling contexts. Aligning one’s view with modeling as a mathematical conceptual genre allows one to situate understanding of modeling in the social and interactional contexts in which learning occurs. It also allows for a treatment of mathematics as a humanizing capital promoting deep reflection by teachers and school learners as they engage in considering these contexts. Aligning one’s view to modeling as vehicle enables one to consider how mathematical and curricular knowledge is developed across instruction and curricular goals are satisfied. Reconciling these views of modeling as concept and vehicle and theorizing how using them in concert can support the teaching and learning of mathematical modeling requires further contemplation.

Drawing on the work of ethnomathematics, one can conceptualize the different ways in which people interact based on their membership in various cultural domains and how they develop, communicate, and use scientific and mathematical knowledge to solve emergent problems within their culture.

2.3 Ethnomathematics

The history and ontology of ethnomathematics draws it roots from aspects of critical education (Freire, 1973). Freire advocates for an active, dialogical, critical, and

29 criticism-stimulating method of educational reform which changes the programmatic content of education, itself, and draws on various dialogical techniques to foster the production of new knowledge in people (p. 40). D’Ambrosio (1985) defines ethnomathematics as the “mathematics which is practiced among identifiable cultural groups” (p. 45), such as tribal natives or various professional classes. He distinguishes this form of mathematics from what he characterizes as academic mathematics or the mathematics taught in school.

D’Ambrosio (1985) describes culture as strategies for societal action, including aspects such as jargon, codes, myths, symbols, utopias, and ways of reasoning and interpreting (p. 46). Further, D’Ambrosio claims that the mathematical structural relationships within the domain of ethnomathematics may never reach a traditional formalized and symbolic form; rather, ethnomathematical concepts take on a form of their own relative to the culture and manifest structural changes based on societal changes over the course of time. D’Ambrosio refers here to the ad hoc practices associated with ethnomathematics, as opposed to a structured body of knowledge (p. 47).

Augmenting D’Ambrosio’s view, Powell and Frankenstein (1997) look more closely at the development of these culturally bounded conceptual forms and the structures that emerge as they are developed.

Powell and Frankenstein (1997) indicate that mathematics, being a cultural product, is created by human beings and thus is interconnected to culture (p. 119).

Powell and Frankenstein express that mathematical structures, themselves, are developed through the cultural activity of learning to speak a language (p. 120). The ability of a

30 learner to engage in a particular mathematical means of communication is linked to being able to access the linguistic norms associated with communication in that particular domain. This, however, does not limit mathematical structure to one particular register and thus warrants the close examination of how learners engage in particular mathematical events while drawing on the linguistic features to which they have access.

From this perspective learning is viewed as the range of interactions which are mediated through engagement in mathematical situations. Lave (1988) deepens this view by considering the form of mathematical practices within various cultural settings.

Lave (1988) argues that context in itself shapes mathematical activity (p. 154) and that culture cannot be divorced from cognition since a context is stretched across various actors that shape and influence the activities within the relevant setting. The discursive moves that actors take in the activity setting as well as the nature of their context experiences interact with the nature of the mathematical tasks under study. Lave further describes “dissolving problems” as problems in which the mathematical task or activity disappears into the activity rather than being formally solved, for example, finding a package of cheese and looking for a similar package to compare prices rather than finding the price per unit of the cheese. She indicates that such transformations of the solution challenge the scholastic assumptions concerning the bounded nature of curricular problems (p. 120). In Lave’s view, one can then expect ways of thinking as well as solutions to these culturally-bounded problems to take on forms relative to the context of these problems. Martin (1997) expands on ways that culture shapes mathematical activity and argues that these ways of reasoning are indeed mathematical.

Martin (1997) advocates the view of mathematics as “a product of society and it can both reflect and serve the interests of particular groups” (p. 155). He equates mathematical objectivity to social conventions whereby mathematicians institutionalize a set of behaviors for how to use semiotic systems to reason and argue (p. 155). From this perspective mathematics learning demands enculturation into these particular social conventions. Martin further argues that removing western ideas and conventions as a mandatory means for engagement in mathematics warrants examination of how members of cultural groups interact in social contexts and their discursive practices, i.e., what gets established, what gets taken up through the language in use, and what social consequences arise from these interactions. Eglash (1997) supports this view and argues that, from the ethnomathematical perspective, there is the distinct possibility that there is indigenous intentionality in the mathematical patterns represented in different cultures (p.

85).

Eglash (1997) proposed an epistemological difference in ethnomathematics from non-western mathematics in that the structural forms are open to interpretations and not limited to the accepted conventions, using fractal geometry in African tribes as an example. He characterizes fractals as the reputation of patterns on a diminishing scale (p.

5) and traces such design in the layout of tribal villages. In this example, Eglash brings to light that one cannot assume the African tribes employ fractural designs to understand the underlying structure associated with fractal geometry, nor should one dismiss this possibility, but rather take this as an indication of the intentionality of the design which emerges in talking with the designers (p. 5). Extending Eglash’s ideas to classroom

32 learning grounded in an ethnomathematical perspective, this concept implies that developing an understanding of culturally-grounded mathematics demands the recognition of mathematical concepts as they emerge and then, through dialogue, determining the genesis of those mathematical ideas. To satisfy this integration of ethnomathematics and academic learning, Borba (1997) offers a means for the incorporation of ethnomathematics into the school curriculum.

Borba (1997) offers a plan for how ethnomathematical ideas might be incorporated into the school curriculum. He advocates for the use of thematization and project organization, in which students and teachers co-construct the curricular activities in the classroom, students investigate actively thematic problems established by themselves and their teacher, and the teacher’s role is to assist learners in developing a critical view of the world (Freire, 1973). In accomplishing this task, Borba (1997) calls for the ethnoknowledge developed by students in the context of working on tasks, compares this sort of knowledge to academic disciplines, and treats this knowledge as viable and culturally bounded in the sense that teachers and students can engage in dialogue to discuss the nature of different interpretations or solutions and viability of each method (p. 269).

Rosa and Orey (2010) argue that ethnoknowledge is acquired by students in the pedagogical action process of learning mathematics in a culturally relevant educational system. This process enables the development of discussions between teachers and students about the efficiency and relevance of mathematics in different contexts. This approach should permeate instructional activities in which the ethnoknowledge which

33 students develop should be compared to their academic mathematical knowledge. In this process, the role of teachers is to help students to develop a critical and reflective view of the world by using mathematics through modeling.

Orey and Rosa (2016) similarly claim that mathematical thinking is influenced by a variety of factors acting on the environment in which cultural groups exist, including language, religion, economics, as well as social and political activities. The authors advocate the use of ethnomodeling, a methodological tool designed to help in the conceptualization of mathematical activity. They characterize ethnomodeling as a means for people to engage in humanistic mathematics (p. 11) or introducing members of a particular cultural group to the mathematical ideas of their culture and, as such, calling for an examination of problems relevant to their particular community (p. 12). As an example, they discuss the 2014 FIFA World Cup, which was held in Brazil, and the rise in the cost of transportation in Brazil and its impact on Brazilian citizens as a result of the games (p. 14). The subjects in this ethnomodeling investigation were 110 Brazilian students enrolled in an online seminar on Mathematical Modeling. In examining the issues associated with transportation, ethnomodeling was able to support these students in generating data-based arguments to protest the rising costs of Brazilian transportation.

In this sense, the mathematical structures which were examined by these students helped them to generate a mature argument (in the form of a YouTube video), in which these local problems were shared with the global community (p. 15).

In looking at some of the literature associated with ethnomathematics and its practical application of ethnomodeling, trends emerge relative to the importance of

34 structuring mathematical arguments to advance societal actions by its members

(D’Ambrosio, 1985; Rosa & Orey, 2016; Eglash, 1997; Orey & Rosa, 2010). Moreover, symbolic representations are not the sole forms of communication for mathematical ideas and mathematical activity is shaped by the culture in which it resides (Martin, 1997).

There are different ways to describe mathematical relationships in the world, and an ethnomathematical perspective considers these as formalized mathematics merely shaped by a different cultural context (Lave, 1988). In further synthesizing the aforementioned work, one can extrapolate that much of the extraction of understanding related to mathematics and its use by students comes from the perspective of the researcher or observer in this context. As such, it becomes an important role of the teacher or observer to locate mathematical meaning while drawing on their own perspective and expanding their own views based on how their students consider the problems at hand. Then the teacher or observer, through discussion, should create a shared understanding reconciling their view with that of the students and create a shared way of understanding the context through the establishment of a new cultural view (Orey & Rosa, 2016).

From an ethnomathematical perspective, rather than judging students’ work against conventionally endorsed mathematics, one should consider what the student/child/learner is doing and how they engage in mathematical tasks in light of what they have done in the past; the audience that they are producing these representations for; who they are working with; their past experiences in doing these types of problems; when in time these problems are being done with respect to current political, cultural, religious, and economic views; as well as where in space this learning is being done. This is

35 consistent with Bloome, Carter, Christian, Otto and Shuart-Faris’s (2010) view that in examining classroom-based practices, ways that people act and react to each other should be considered when viewing mathematical learning.

There are seemingly a number of affordances for considering mathematical modeling from an ethnomathematical perspective (Rosa & Orey, 2010, 2015, 2016,

2017). First, this perspective recognizes that students enter classrooms with different types of knowledge, which are contextual and mathematical in their nature.

Ethnomathematics affords researchers the chance to consider the interplay between these types of knowledge and situates the learners’ views as valid ways of thinking, as both context and mathematics plays a crucial role in the advancement of the modeling process.

When using mathematics in contexts that have a social flavor, culture plays an integral part in the ways in which these contexts are considered, which may serve as a barrier to furthering the modeling process or the mathematical tools used to describe and further the context. Additionally, rather than emphasizing one type of mathematics as being appropriate to describe the context, it offers different views, which has been shown to provide additional insights into the modeling process. For example, in endemic disease modeling, the recognition of different subgroups of people and their activities within a particular infected community help to create vaccine plans to aid in the abolition or decrease of particular diseases (Hethcote & Yorke, 1984). Considering additional components associated with contextual knowledge affords the development of mathematical models to remedy these issues. This is seen frequently across formal mathematical modeling in professional areas.

In the next section, I will present an argument which considers a classroom as a cultural unit in which various individual views are integrated. This argument, in turn, affords the use of ethnomathematics as a means for examining interactions within this culture in consideration of both mathematics and the production of knowledge across cultural groups.

2.4 Empirical Summary

In looking across the multiple perspectives of mathematical modeling and ethnomathematics, I make the following case for the integration of these methods as a platform for conducting research. I present the argument that the classroom, itself, acts as a cultural unit in its own right, where the members of that group inform ideologies of that cultural unit and knowledge is validated based on those ideological views. Being that the classroom is a part of a social institution with its own set of agendas; there is an aspect of conventional learning associated with that particular space. The members of this particular culture, however, go beyond the walls of the classroom, as I recognize that the class is situated within a particular school, governed by administrative policies and different conceptions of learning, as well as within the space of a particular community with a diverse group of individuals. Each of the members of this particular community comes from a variety of cultural and institutional backgrounds warranting a diverse membership. These institutions (classroom, school, community, members) are not, and should not be, treated as isolated entities, but rather as a socially constructed culture.

Orey and Rosa (2016) outline three different ways in which cultures can come together and link these ways to equitable interactions. Orey and Rosa (2016) indicate that, when cultures meet, there are three possible outcomes: (1) one culture eliminates the other completely, (2) one culture is absorbed by the other, or (3) the two cultures come together to produce a third distinct culture (p. 18). The fist perspective suggests one

(dominant) culture wiping out another other culture entirely. Standardized assessment exemplifies this perspective since these measures are not concerned with student thinking on particular problems, but rather investigating a student’s ability to competently respond to a particular form of representation, and situate knowing with being able to produce that particular form.

In cultural absorption (the second view), one culture exists within another as an isolated member. Views do not necessarily cross, and each group maintains their own distinct perspectives. This could be consistent with students’ development of secondary intuitions about mathematics, in which students actively distinguish mathematical ways of behaving or thinking from real-life experiences (Fischbein, 1979). In this way, students develop views of school mathematics that may contrast with problem solving in real-world contexts and vice versa.

The third view offered by Orey and Rosa indicates a shared approach to coming to understand a concept, one in which views, experiences, and knowledge are integrated from these multiple views. Mathematics education theorized from this view suggests that the role of the teachers is not to impose a particular type of thinking on learners, but, instead, to engage in their community by embracing students’ views and validating their

38 life experiences and background (p. 19). When establishing such an inclusive classroom environment, the teacher needs to construct a classroom culture which is conducive to students communicating their ideas as they engage in problem solving. The teacher should also consider actively multiple approaches to student responses to be interpreted through their own mathematical lens. Mediating the formal mathematization of a context while using particular mathematical language or forms should occur as a collaborative effort with the goal of communicating ideas as a shared activity. This approach is different from the traditional practice of standardizing thoughts and actions, which Martin

(1997) characterizes as an industrial technique of production.

Ethnomethodology, or the use of ethnographic methods to produce accounts of ways people negotiate everyday situations (Denzin, 1994), and research in funds of knowledge propose that considering learning as constructing new knowledge is far more productive than viewing learning as how knowledge is transmitted across mediums. An ethnomathematical view affords using both an anthropological and mathematical lens to examine the process by which knowledge is generated and how scaffolding practices are sensitive to the views which the students bring to the learning environment. In distinguishing from a western academic and privileged view of what mathematical knowledge looks like, ethnomathematics assumes that the basis of mathematical knowledge as observed is, in fact, sophisticated in its own right. Thus, one can examine the learning aspects within this cultural group by considering how the individual members of the community come to construct understanding socially and solve problems

39 relevant to their community, which I constitute as funds of knowledge related to this classroom culture.

My view also recognizes that the perspective of each member of the cultural community is informed by their various funds of knowledge and how these funds of knowledge construct individuals’ identities. Drawing on the research outlined above, I consider these funds of knowledge as viable means for supporting the learning process in which students rely on their experiences to frame their interests and views in ways that deem knowledge to be important and worth learning in greater detail. I argue that, if a teacher places value on these aspects of knowledge and uses these as a means for furthering investigation from a mathematical perspective related to thinking and reasoning, students will not only be motivated to pursue lines of thinking consistent with mathematical interpretations of that context and also that these lines of reasoning can connect and be expanded to deep mathematical principles relating to those systems of behaviors.

In considering how mathematical modeling is developed with student learners, attention must be paid to how related funds of knowledge emerge through interactions.

Additionally, consideration of how these funds of knowledge inform the selection of variables, assumptions linked to those variables, and mathematical ways contexts are perceived becomes important in helping to foster validation of those models. I argue that reflexive and dialogic teaching serves as a means for enhancing knowledge and establishing normative behaviors associated with mathematical modeling. To warrant

40 and test this preliminary theory, I rely on discourse analysis to evaluate the conversations that emerge during the course of mathematical modeling instruction.

2.5 Theoretical Framework

In this section, a theoretical framework will be established by using a discourse analytical perspective for conducting educational research (Bloome et al., 2010) which integrates the principles of mathematical modeling and ethnomathematics to structure unpacking the nature of teacher/student interactions and how they elicit deep and meaningful instances of learning.

2.5.1 Discourse Analysis in Educational Research

Voigt (1994) characterizes mathematical sense-making as a negotiated practice among its participants, highlighting that “in the course of negotiation, the teacher and the students (or the students among themselves) accomplish a network of mathematical meanings taken-to-be-shared” (p. 283). From his perspective, mathematical meaning is not transmitted across participants in class, but rather co-constructed by participants through social interactions. During this process of negotiation, the single meaning of an idea and the underlying context inform each other (p. 289). Further, this negotiation is a necessary and fundamental condition for learning, especially if students’ background knowledge differs from that of the teachers (p. 291).

Voigt (1994) distinguishes two types of negotiation in classroom learning environments, implicit and explicit. Implicit negotiation relates to accepting ideas which

41 are presented without challenging those ideas outwardly. Explicit negotiation concerns direct challenges to ideas through discussion or questioning (p. 281). The absence of a socially visible challenge does not imply it does not exist. It is important to note that implicit challenges are more difficult to perceive, as they are not overt. I perceive these implicit negotiations through evidence presented in social contexts through uptake in social interactions.

Drawing on Voigt’s (1994) construct, one can conceive mathematics as a social construction and negotiated amongst the members of its community. For example, in the notion of proof by contradiction, the basic idea is to assume that a statement is false and then exhibit that this assumption leads to false claims, which is considered by the community to be a weaker form of proof (Beth, 1970). Beth (1970) indicates that

“according to widespread opinion, a proof by contradiction, except in those cases where it is the intention to justify a negative conclusion, is not as valuable as direct proof” (p.

15). While a proof by contradiction follows a sound, logical sequence of reasoning corresponding to truth-values, formal deductive proofs are often considered to be, in some sense, more robust, a conception that Beth challenges in his work by offering case examples where proof by contradiction is the most viable option for constructing a proof.

Similarly, there has been, and continues to be, much debate within the mathematics community as to the acceptance of a proof using a computer to run a large number of simulations, which would have been impossible prior to the development of the modern computer (Dreyfus, Nardi, & Leikin, 2012). The fundamental argument here is that there are different degrees of rigor associated with types of proofs, as established socially

42 by the community of academic mathematicians. This point provides evidence supporting this claim; that mathematics, itself, is not immune to aspects of negotiation and perceptions of truth are subject to cultural and historical biases.

In this research, I argue that consideration of mathematics from a social and negotiated perspective offers many affordances. First, it situates mathematical knowledge within a particular cultural group and is reflected through participatory actions by the members of the community drawing on an ethnomathematical perspective.

Recognizing that mathematics has a variety of appropriate and acceptable forms which emerge through aspects of language offers a means to unpack ways in which members of a particular group interact in mathematical ways. However, understanding the ethnomathematics of a particular group requires acknowledgment or valorization by the researcher that distinct cultural perspectives exist which inform mathematical actions.

Second, a social and negotiation perspective of mathematics highlights the dynamic nature of mathematics as it adapts over the course of time. The endorsement of new mathematical ideas or acceptable forms of mathematical arguments drives from negotiations among the members of that particular cultural unit. In order to gain insight into these cultural aspects of knowledge negotiation, scholars advocate for the development of an insider’s view (Bloome et al., 2010; Rosa & Orey, 2016; Hymes,

1996; Silverstein, 1993). Establishing this analytical and theoretical framework for looking at data through principles associated with discourse analysis.

Discourse analysis is not just a tool for examining dialogic interactions, but an emerging perspective for data generation and analysis (Bloome et al., 2010; Scollon &

Scollon, 2003; Agar, 2013; Gee & Green, 1998). Wortham and Reyes (2015) describe discourse analysis as a method of research which provides evidence about social processes through the examination of speech, action, and other semiotic signs and characterizes learning as systematic changes in behaviors across events, in which empirical evidence is situated in the examination of pathways across linked events (p. 1).

Gee and Green (1998) offer that discourse analysis studies ways in which opportunities for learning are constructed across time, groups, and events; how knowledge is constructed in educational settings; how that knowledge shapes and is shaped by the discursive and social practices of its members; how these practices pertain to structuring aspects of the space, such as curriculum; and how learning is influenced by actions of others through interactions within and beyond the setting (in places such as textbooks, curriculum, legislators, the community, and so on) (p. 119).

Gee and Green (1998) further advocate for a logic of inquiry (Birdwhistell, 1977), in which the use of theory informs and interacts with the methodology as well as the analytical plans and is not in isolation or exclusive to analysis. In this view, theoretical knowledge is critical in informing methodology, as each decision in theory informs the method of the study. In addition, the theory through which the method was adopted also informs the nature of the claims which are warranted through analysis (p. 121). Gee and

Green (1998) indicate that, in conducting a discourse analytic study, there are a number of important aspects which help to bring to light these changes and instances of learning in context, including contextualization cues, such as pitch, stress, intonation, pause, juncture, proximity between speakers, eye gazes, and gestures, in addition to grammatical

44 structures (Gee & Green, p. 122); how culture informs interactions and knowing as situated knowledge as a set of principles for acting and interpreting experiences (p. 124); or how cultural models are operated on and refined across time (Geertz, 1973). Gee and

Green indicate that this situated knowledge is tied closely to the ways in which the researcher frames, identifies, and reports on instances during classroom events and, in particular, influences and informs the research questions and interprets the actions and reactions of members of the group (p. 125).

In considering these important implications for conducting discourse analysis research in this dissertation study, careful attention to the social aspects and their uptake across participants becomes paramount, demanding that the holistic picture and the multiple components in interactions be considered. Here, the discourse analytic perspective advocates for the use of an ethnographic perspective in which dialogue is central to the work in terms of providing a conceptual approach for data analysis from an emic perspective (Gee & Green, 1997) and, as such, examining meaning in practice rather than from a static or idealistic view (p. 126). In considering how this applies to mathematical modeling or mathematics education, in general, this perspective offers a means for examining the ways meanings are presented in contexts through actions and situating this as knowing. For example, in discussing aspects of graphs or functions pertaining to a context, students or teachers may use informal terminology (for example, peaks and troughs when talking about oscillatory behaviors) which, when used in context, pertains to understanding a particular repetitive pattern, in which aspects of period and amplitude are implicit in the discussion, but explicit in context. In other words, I use this

45 perspective to establish that deep mathematical thinking may not necessarily be connected to use of mathematical terminology.

Lave (1996) indicates that examining activities within the learning environment using historical, dialogical, and social practices allows the conceptualization of learning as the process of shaping participation over time, or, explicitly, “the social ontological, historically situated, perspective on learning…reflects ways of becoming a participant, ways of participating, and ways that participants and practices change” (p. 157). This particular view accounts for the role that history plays in the conceptualization of learning. When referencing historical ontology, I not only talk about the institutional structures of school, but also how learning looks over the course of time (year to year, moment to moment), along with ways that it is informed by the institutional power structures in play.

Discourse analysis also accounts for intertextuality, or the relationship between various texts; intercontextuality, or the relationship between events and other events; and interdiscoursivity, or the relationship between discourses (Bloome et al., 2010, p. 44).

Bloome et al. (2010 claim that, for connections to be ratified by others, participants must recognize and exhibit awareness of these connections. Bloome and colleagues define intertextuality as the juxtaposition of texts, in which this juxtaposition is not connected implicitly to the text itself, but rather must be proposed, acknowledged, recognized, and have some social consequence or uptake (p. 41). For example, in the context of validating mathematical models, one aspect of study could be identifying how the concept of real- world business is juxtaposed with and in school mathematics application tasks when

46 modeling is the focus for an academic year. Discourse analysis offers a means for critical examination of this juxtaposition and its uptake.

Bloome et al. (2010) describe intercontextuality as the social construction of relationships across contexts, past or future (p. 144). They argue that intercontextual links may be related to events in which the members are not directly involved, for example, a school board adopting a particular curriculum that informs instruction (p. 45).

Similarly, drawing back to the work of ethnomathematics, Gattegno (1980) recognizes that students come to classes with prior academic experiences which are a part of the ways in which they conceptualize and work with mathematics. In this sense, experiences which impact the events are influenced by both local and global factors, some immediately visible, some hidden, and paying attention to each of these is important to analysis.

Interdiscursivity, or the relationship among institutional discourses (Bloome et al.,

2010, p. 144), is important to research on learning mathematical modeling since the contexts used concern real-world situations. In such contexts, multiple discourses are at play, the discourse associated with conventional mathematical funds along with the various contextual funds of knowledge. In mathematical modeling, problem situations explicitly attempt to draw interdiscursive relationships, bringing in mathematical ways of knowing in connection to conventional ways of knowing. Interdiscursivity acts as the bridge between these funds of knowledge. How learning occurs (based on the discursive perspective) requires careful examination of how these interdiscursive relationships are proposed and accepted and the social consequences that emerge as a result (Bloome et al.,

2010, p. 144). Mathematical modeling requires interdiscursive connections to be made between mathematics and real-world contexts, a point which is exhibited in mathematical modeling competence (Kaiser, 2007; Maass, 2006; Blum & Borromeo-Ferri, 2009), in particular, in how these extra-mathematical contexts are mathematized. How these discourses interplay goes beyond simply using mathematics to structure the problem, but rather examines how these connections are taken up socially through interaction.

Discourse analysis affords the means with which to consider how learners respond to context-based tasks and establishes the validity of those responses with respect to both their experiences in real life and in mathematics.

Bloome et al. (2010) identifies two crucial aspects that define and distinguish discourse analysis: implied personhood and the foregrounding of events (p. 2). Citing the work of Gergen and Davis (1985), they claim that personhood is not pre-determined, but rather socially constructed through symbolic action. They further claim that, in people acting and reacting to each other, they are constantly negotiating the concept of personhood amongst themselves by establishing a shared reference for how they define each other as well as the characteristics which they assign based on treating each other as persons. Bloome et al. establish that this notion of personhood emerges through language in its use and elaborate that these construct negotiations of personhood always occur and cannot be divorced from the events in which they occur (p. 5). Bloome et al. argue that the examination of the use of language within the event in which it occurs can offer a means for determining meaning within that interaction, an argument consistent with what

Silverstein (1985) denotes as the “Total Linguist Fact” of an event.

Silverstein (1985) defines the Total Linguistic Fact as “the datum for a science of language, irreducibly dialectic in nature” in which interactions of sign forms are contextualized to situations of interested human use and mediated by cultural ideology (p.

220). Silverstein concludes that there is an ideological focus on particularities in structuring indexical usage as well as that any linguistic form has multiple indexical values which may not be readily apparent to its users. Similarly, Silverstein indicates that any linguistic form consists of action, which causes the user to be conscious of how the language system is being used and who it is being used with, particularly how the language refers to what has been said, and what these indices refer to (p. 256).

In extending these ideas to mathematical modeling, I have argued above that the predominant view of scaffolding and teaching practices associated with this domain falls within the vein of remedying student thinking and normalizing their responses to a perceived ideal solution. Particular attention is paid to the formal registers which are employed to talk about and solve these problems. However, when students or teachers draw on mathematical ideas in casual ways in unpacking or understanding the context of a problem, the terms used are often interrelated or used incorrectly in formal mathematical registers, such as students casually discussing aspects of an oscillatory function, as presented above, without the use of conventional mathematical terms.

However, I argue that when considering the Total Linguistic Fact of this event though careful examination of how these statements are being made, what these statements mean in the context of their use, as well as how they are taken up, offers a stronger indication of understanding of mathematical concepts. This is in contrast to placing a perceived

49 lack of understanding on the misuse or lack of conventional terms. As such, I make the case that it is essential to look at students doing mathematics from a holistic perspective, regardless of the formal registers which are being used in the dialogue, and look beyond to how they are being used. Halliday (1978) supports this notion by elaborating that one needs to look beyond the sentence, alone, and rather pay attention to the meaning of a text (written, spoken, or otherwise) in its entirety (p. 5).

To reinforce the notion that one cannot examine work alone, but that processes are indicative of student understanding of a particular context, Goffman (1981) indicates that “when individuals in the presence of others respond to events, their glances, looks and postural shifts carry all kinds of implication and meaning” (p. 1). In this sense, to understand how students come to know mathematical ideas, one must, in effect, see them in action, not just in terms of approaching a problem, what they write, or only the mathematical ideas which are reflected, but also in the absence of these particularities and their relationship to what the student is doing regarding their posture, glances, and ways they interact with others. These important semiotic relationships can offer further clues for the unpacking of the nature of learners’ responses in mathematical modeling contexts to an individual who is deemed an authority. For example, if, given a particular context, the class’ attention is drawn to a particular member, this reaction might exhibit that the individual has contextual knowledge which is deemed viable in this situation, say, for example, the teacher. If that teacher then offers their perspective, this validates the initial claim and positions them as an authority. As time progresses, if the authority transitions

50 to members of the class, one can document this transition in authority and connect it to learning in that domain.

These semiotic relationships (Halliday, 1978) are couched in mathematics education and mathematical modeling, as well. Morgan (2006) indicates that “studying language (with respect to mathematics) and its use must thus take into account both the immediate situation in which meanings are being exchanged (the context of situation) and the broader culture within which the participants are embedded (the context of culture)”

(p. 221). This relationship is consistent with the view that in order to gain an understanding of how learners come to develop mathematically, one needs a broad view of the content with respect to the culture in which this learning is couched and that learning may look different across these cultural contexts.

Morgan (2006) further indicates that socially-situated meaning-making arises through interactive participation by individuals in that social context and argues that individual cognition and the social constructions are compatible (p. 221). The author deduces two important conclusions in her consideration of the use of social semiotics in mathematical contexts: (1) social semiotics serve as a basis with which to investigate how various texts which might present different views of mathematics are taken up by students and what happens when students produce mathematical interpretations of a context different from the teacher’s interpretations, and (2) social semiotics afford the ability to look at the aspects of mathematical texts produced and where power relationships lie, along with the various forms which they take (p. 238). Further, Morgan argues that interactions occurring between two or more participants provide a much

51 richer corpus of data for examining the means with which participants in the dialogue are forming identities and social positions (p. 238) consistent with the research presented previously. In this sense, I extend these claims to make the case that an optimal way to examine the means by which students develop a mathematical modeling disposition is by observing them actually working on generating mathematical questions and considering mathematical means for describing their interpretations of the problem context as well as how the use of mathematical ideas comes to inform their understanding of the context of the problem posed. Morgan concludes by arguing that, in adopting this social semiotic view of mathematical learning, the focus on how this learning comes about must occur in the context of a communicative exchange (p. 239).

The perspectives on dialogic learning are consistent with the points addressed by

Bloome et al. (2010) concerning heeding both the interactions and the event in which those interactions occur in framing an analytical view. This analytical view aims to account for not only what is happening during the event, but must also take into account what happened prior to the event and what will happen in the future. These interactions are grounded in a particular cultural context, and, as a result, the mathematics that emerges may not translate automatically across cultural boundaries (such as across school and jobs, or in and across classroom contexts). Often, we take for granted that there are difficulties in translating mathematical concepts presented in the context of a mathematics class over to scientifically embedded concepts (algebra to physical science, geometry to biology, calculus to physics). Those aspects of the culture associated with a particular classroom, subject, location, participants, and so on, provide information on

52 how the actors in that particular culture engage in thinking about the discipline and how learning manifests itself.

I have, at length, presented arguments highlighting the importance of culture in how people talk and think about particular contexts in which they are engaged.

However, there are varying definitions of what constitutes culture. For example, culture has been described as the complex whole, including knowledge, beliefs, art, morality, and any habits acquired by humans as a member of society (Taylor, 1871); what one must know in order to behave in a manner acceptable to society (Goodenough, 1981); or various standards for perceiving, evaluating, believing, or doing, not so much in determining how people act, but rather in standardizing behavior (Culture and Language

Society, 1981). Geertz (1973) describes culture as both the actions which people take and the import of the meaning of those actions. Street (1993) considers culture not as a noun, or what it is, but rather as a verb or what culture does for individuals. This definition affords the ability to look at culture from a more local perspective and then find meaning in the social system, rather having it be due to individual differences.

Bloome (2015) argues that awareness of these different aspects of culture grants us the ability to create a much richer perspective and allows flexible thinking about interactions which occur within the culture of where the learning is taking place.

In the current study I rely on an open definition of culture to support flexible thinking about the interactions which occur. Recognizing that these varying definitions of culture are not mutually exclusive, but, instead, interactive, can support the framing of a particular view of a context in different ways. As such, when considering aspects

53 associated with the learning of mathematical concepts, ones view of what culture is and does informs the interpretation of the interactions which are observed. One can consider aspects about what mathematical ideas the students draw on and the representative forms which those ideas take as a function of not only how the students view these ideas, but also where these views manifest themselves. They can be seen as a product of prior instruction or possibly in terms of how the individuals perceive the associated contexts and draw on past life experiences to gain a better understanding of the dynamics of the overarching system.

Bloome and Green (2015) discuss aspects of language as culture and societal action as well as the form versus the functional aspects of language (Bakhtin, 1986;

Volosinov, 1929; Bordieu, 1977). Bloome and Green point out that there are views concerning how language is treated as an asset or cultural capital which can be exchanged and bartered with and for (Bordeieu, 1977; Bloome and Green, 2015). The authors characterize “forms” of language as things, such as a question, response, statement, exclamation, quote, repeating, lecture, report, or narrative. These aspects indicate the particular form that the language takes. Whereas functional aspects are those such as holding the floor, bidding for the floor, making a request, opening or closing an interaction, validation, elaborating, and so on. In this sense, then, the functional aspects of language dictate how individuals interact in a given social context, i.e., what they are doing, while the form aspects are related to the particular shape that language takes. This commodification of language applies to mathematics, whereby having access to certain mathematical registers or being able to use mathematics in a certain way and

54 communicate mathematical results using a particular form affords it to be viewed as an asset different than other forms based on the conventions of school. In establishing a framework for analyzing interactions which occur during mathematical modeling, experiences which remove conventional ways of representing mathematics allow for a more open view of the mathematical ideas and their intersection with the various contexts in which they reside. Thus, in considering this disposition for mathematical modeling, both formative and functional aspects of language become important in analysis.

In summary, discourse analysis of a microethnographic study offers multiple affordances for developing a theoretical framework for examining how a modeling disposition is developed. First, this framework accounts for the particular cultural unit of the classroom and its members (teacher, students, and researcher) and situates it as such.

Second, an ethnomathematical view affords situating knowing within the context of that cultural unit and in the event in which it occurs. Third, it recognizes the individual background knowledge and experiences which each member of the classroom draws upon when engaging in solving problems situated in contexts relevant to the group, in addition to the funds of knowledge of the particular classroom cultural unit. Finally, in drawing on principles associated with discourse analysis, one can carefully examine multiple facets of the interactions which occur in the classroom and identify the significance of those interactions based on their uptake. Situating this in the context of a classroom focused on developing mathematical modeling practice in learners allows for the establishment of a framework suitable for determining how components of this practice are developed over time. In considering the ways in which texts are produced

55 and taken up in a particular classroom aligned to developing mathematical modeling with its students, careful consideration of the intertextual, intercontextual, and interdiscursive links which are developed frames an analytical view for determining how this practice is socially constructed through unpacking the social and linguistic tools which teachers and learners use in constructing and developing new knowledge.

2.5.2 An Ethnomodeling Cycle

In my view the added lens of ethnomodeling coupled with discourse analysis offers additional insight into the construction of a modeling practice. In developing a systematic process for conducting ethnomodeling research and as a supporting theoretical framework, I propose the use of an ethnomodeling cycle (Figure 7) to frame the research perspective. Permeating this cycle is the transition between local, global, and dialogical ways of understanding the ethnomathematics of a group. Orey and Rosa (2015) indicate that, when engaging in studying the mathematics of other cultural units, in order for a better understanding of how that culture engages in ethnomathematics, the observer’s own culture of mathematics should interfere minimally with interpreting the culture under study. This is not to say that a researcher’s mathematical views should be disregarded, but rather that one’s view should not assume similar interpretations. These mutual understandings of mathematical events should emerge through dialogic processes

(p. 365). As such, Orey and Rosa (2015) differentiate between etic (outsiders/global), emic (insiders/local) ,and glocal (dialogical) approaches to ethnomodeling. The ontogenesis of the terms emic and etic (Pike, 1954) arises through an analogy of the

56 linguistic terms phonemic, or the sounds used in a particular language; and phonetic, or the general aspects of vocal sounds (cited in Orey and Rosa, 2015). In ethnomathematics, an emic view corresponds an insider’s view, or the view held by members of the group under study, which is informed by the shared history and experiences of the members of the group. An outside, or etic view, corresponds to the perspective which the observer brings into the research process and, similarly, is informed by their historical, life, and background experiences. General practices in ethno-methodological research advocate for the development of an emic view, which is established by a researcher or observer getting to intimately know and participate in the cultural group being studied (Hymes, 1962).

A glocal view is mediated through a dialogical approach in which etic knowledge is not privileged over emic (Rosa & Orey, 2016, p. 377) and, through the process of translation, creates a spectrum spanning between etic to emic views. In this sense, there may be times during the dialogical process in which one type of knowledge may come to the forefront of the other for the sake of coming to a better understanding of the mathematical practices of the group.

Emic practices in mathematical contexts Local

Valorization of Cultural differences

Discovering Glocal Theory of Mathematical Translation Emic Systems Practices (ethnomodeling)

Elaboration (ethnomodeling)

Etic Mathematical Models Global

Figure 7. Ethnomodeling Cycle

Beginning at the right side of this cycle, I identify this spectrum as ranging from local to global. In the design of this cycle, local knowledge is deliberately placed above global, as this placement emphasizes the importance of individuals interacting within their local communities. I warrant this view based on the necessity of capturing diverse forms of mathematical ideas and procedures (Orey & Rosa, 2016). This is contrary to the views typically expressed whereby western-academic mathematical knowledge is the highest form of mathematical knowing and trickles down to inform educational processes

(e.g., The Didactic Transposition of Brousseau, 1970). Rather, I take this local to global transition as a means for fostering views of mathematics built on respect and admiration

58 for those cultural differences. I denote this spectrum by using a double-sided arrow ranging between local and global on the cycle in Figure 7.

At the highest geographical point of the cycle, I situate the emic practices which occur in real-life contexts. These are, at their core, the interactions between members of the cultural unit as they engage in solving relevant problems. The right side of this cycle conveys the process of translation, or the interaction between the emic and the etic views, which is mediated through ethnomodeling. Again, Orey and Rosa (2016) describe translation as the process of modeling local cultural systems which may have western- academic mathematical representations (p. 9). Their view of translation advocates that this process is bi-directional: first, in helping an observer to better understand the emic ways of solving these problems, and, second, in how the etic mathematical interpretations can be shared and systematized. For example, indigenous designs can be interpreted from the perspective of the observer, and, through dialogue, intentionality can be confirmed. In this process of translation, a reflexive view (Qualley, 1997) ought be taken up by the researcher so as to foster this process in which both researcher and members of the group under study situate themselves as learners from each other, based fundamentally on respect for each other. At the same time, when an outsider observes patterns, it becomes their responsibility not to infer that these local perceptions apply to their own etic mathematical knowledge or processes, but rather they should validate their findings through discussion to better understand those emic perspectives for problem solving.

The left-hand side of the cycle captures the process of discovering emic systems.

In this phase of ethnomodeling, a researcher considers the observed interactions in a cultural unit through their own mathematical lens. This is not for the purpose of imposing their own mathematical ideas onto the culture, but rather to identify patterns which emerge in this local problem-solving and use those discoveries to foster discussion to better understand their genesis. This discovery process leads to a dialogue in which observations are weighed against local perceptions of behaviors and emic ethnomathematical knowledge emerges. Orey and Rosa (2015) indicate that:

Researchers and investigators, if not blinded by their own cultural backgrounds, are profoundly influenced by the paradigm in which they are immersed, which includes all prior theory and ideology that they have absorbed. If they are aware of this, they should come out with an informed sense of distinction that makes a difference from the point of view of the mathematical knowledge of the work being modeled. In so doing, in the end, they will be able to tell outsiders what matters to insiders. (p. 372)

This is to say that one need not un-know all that they have come to accept in their own mathematical conventions, but rather use those recognizable and personal forms of mathematics to help convey to the rest of the world how various cultural groups come to understand and operate within their own ethnomathematical practices.

Eglash (1999) presents a spectrum for considering the ethnomathematics of a particular cultural unit ranging from non-mathematical unintentional and unconscious structures to pure mathematics. Eglash characterizes unconscious structures as naturally exhibited patterns with no intentionality in their construction and indicates that these unconscious structures do not count as mathematical knowledge, even though they may be mathematically describable by an outsider (e.g., fractal patterns found in termite

60 mounds) (p. 5). Next, Eglash characterizes conscious and implicit mathematical knowledge as conscious patterns with no explicit knowledge attached. Eglash referrers to applied (ethno) mathematics as the ability to make explicit the components of the pattern and develop a set of rules for how these patterns can be employed. Finally, Eglash characterizes pure mathematics as the ability to have words for comparing properties in identifying areas where those patterns work, why they work, and how to find new patterns (p. 5-6). According to this view of ethnomathematics, this distinction in classification affords a means with which to consider not all behavior as mathematical knowledge as well as to classify various degrees of mathematics when observing a cultural group. It is important to note, however, that we need to listen carefully to what designers and users of these structures have to say as what may appear accidental or unconscious regarding patterns may, in fact, have an intentional mathematical component

(Eglash, 1999, p. 5).

It is important to note that this distinction is not meant to be restrictive and rule out the idea that a culture has developed mathematical ways of understanding, but rather offers a means with which to help the researcher or observer in understanding where and how these various mathematical ways emerge. It assumes that they exist and seeks to find and bring to light these differences. It classifies them as being mathematical and offers visible evidence for their existence.

These ongoing processes of discovering and translating lead to the construction of a glocal theory of mathematical practices through two routes: (1) valorization of cultural differences by bringing to light the sophisticated nature of these practices, and (2)

61 elaboration of emic ethnomodels translated into etic forms in order to make these practices more familiar to those not immersed in the cultural practices of the group.

Rosa and Orey (2016) indicate that the process of valorization affords the recognition of how learners assess and translate contextual problems and elaborate these models across contexts (p. 10). This valorization creates awareness of differences in perspective as well as empowers those views creating an equitable framework for recognizing mathematical differences. Greer, Mukhopadhay, and Roth (2012) discuss how the process of valorization is related to overarching issues of power. In particular, they indicate that, typically, the mathematics which is valorized is related directly to the mathematics of groups with power (p. 5). By valorizing the mathematics of different groups, it shifts the dynamics of power and strengthens those voices typically unheard or ignored and, as such, creates a platform for justice for oppressed groups (Blanco-Alvarez

& Oliveras, 2016). Blanco-Alvarez and Oliveras further indicate that this valorization process ultimately needs to impact the school setting by creating an environment sensitive to both historic evolution of these processes as well how these practices support the political interests of the group. In this sense, the ethnomodeling perspective affords a means with which to fulfill this critical need when students are exposed to problems relevant to their communities and the worlds in which they reside.

The last key component in developing a glocal theory for the mathematical practices of a cultural group is the elaboration of etic mathematical models. This process serves the function of both deepening the understanding of the development of those processes from a systematized and idealized perspective as well as aiding in the reporting

62 and valorizing these emic processes to an outside community. Orey and Rosa (2015) claim:

The application of ethnomathematical techniques and the tools of modelling allow us to see a different reality and give us insight into mathematics done in a holistic manner. The pedagogical approach that connects the cultural aspects of mathematics with its academic aspects is named ethnomodeling, a process of translation and elaboration of problems and questions taken from systems that are part of any given cultural group. (p. 378)

Further, the elaboration of these cultural mathematical practices can support their development as well as offer pedagogically a connection between the emic practices and the etic mathematical knowledge which a school tries to instill in student learners. As such, ethnomodeling can be a seemingly powerful pedagogical tool in the presence of relevant cultural mathematical activities.

From the perspective of research, elaboration serves as a means with which to foster further dialogue between the observer/investigator and the cultural unit. In this way, these cultural perceptions merge into a shared glocal view of the mathematical practices which occur.

In the next chapter, I will outline a methodology for employing this theoretical framework and describe the research site at which this framework will be employed.

Then, I will offer a method for generating and warranting data from the microethnographic study conducted and use this method to present, warrant, and analyze data from this study to examine the ways in which mathematical modeling capacity is developed in learners.

Chapter 3: Methods

This chapter describes the protocol for the collection, generation, and analysis of data for this study. Section 3.1 outlines the research design, establishing the overarching goals and the theoretical and methodological orientation towards data generation. Next,

Section 3.2 reports on the procedures used and timeline for this study. After that, Section

3.3 offers a description of the research site and participants. Further, Section 3.4 provides an overview of the data analysis process. Finally, Section 3.5 offers a commentary on pilot study findings to exemplify the research process.

The research questions which served as a guide for this study were:

1. How are social practices in mathematical modeling contexts developed through

implementation of mathematical modeling tasks?

2. How does teacher scaffolding within these mathematical modeling contexts

promote the development of modeling capacity in learners?

3.1 Research Design

This study generated data from a broad micro-ethnographic study which took place across an academic semester in Fall of 2017 and examined the interactions between a teacher who endorsed mathematical modeling and her students in an 11th grade pre- calculus classroom in the Midwestern United States. The intent of this study was to 64 examine what social practices are developed when a teacher engages her students in mathematical modeling (RQ1) and how this process is developed through teacher scaffolding (RQ2). Acknowledging that there are many different definitions of mathematical modeling (Lesh et al., 2010), it becomes important to unpack the participant teacher’s view of mathematical modeling through careful examination of her practice and those components which establish her view and then consider how learners are enculturated into that practice. A discourse analytic perspective (Bloome et al., 2010) was adopted for implementation of the research design in concert with ethnomodeling

(Orey and Rosa, 2010) to support the collection, generation, and analysis of data.

Bloome et al. (2010) indicate that the adoption of such a perspective entails the claiming of a history of traditions which must be argued not based necessarily on its prior establishment, but based on what is happening in the present and how it looks to the future. Further, Gee and Green (1998) indicate that an ethnographic perspective places dialogue (spoken or action) at the heart of the work in which discourse analysis forms a basis for identifying what members of the class should know, produce, predict, interpret, and evaluate in the given social setting and acquiring cultural knowledge of the group.

Gee and Green claim that discourse analysis affords examination of both what has the potential to be learned as well as what is actually learned through social interactions (p.

126). They characterize learning as “changing patterns of participation in specific social practices within communities of practice” (Gee & Green, 1997). Ethnomathematics assumes mathematical practices to be a cultural product (Powell & Frankenstein, 1997) and, as such, manifests in forms indicative of that culture (Martin, 1997). Further, in

65 ethnomathematics, mathematical modeling, itself, can be a useful tool in documenting mathematical activity within a cultural group (Orey and Rosa, 2010) and further necessitates that a researcher not be blinded by their own mathematical background, but rather rely on it as a way of interpreting and describing behaviors within the culture

(Martin, 1997; Orey and Rosa, 2015). In this way, discourse analysis calls for an ethnographic approach to data generation and analysis (Gee and Green, 1998), and ethnomathematics offers tools suitable for capturing mathematical activity within the cultural unit.

In this study, it was my intent to capture how mathematical modeling cognition was nurtured in learners while it was simultaneously used as a vehicle to fulfill curricular objectives. In order to understand the development of mathematical modeling in learners, it is necessary to have a method for the generation of data based on both a theoretical frame regarding mathematical modeling as well as a sound mechanism for data generation and processing prior to analysis. As such, I outline below my theoretical orientation towards data generation as a way of framing my view of the study.

3.2 Orientations Towards Data Generation

The development of the theoretical and methodological orientation towards data generation stems from work in mathematical modeling competence (Blomhoj & Jensen,

2003; Maass, 2006), sociomathematical norms (Yackel & Cobb, 1996), discourse analysis (Bloome et al., 2010), ethnomathematics (D’Ambrosio, 1985; Martin, 1997;

Powell & Frankenstein, 1997; Eglash, 1999), ethnomodeling (Orey & Rosa, 2010, 2015,

2016), funds of knowledge (Moll et al., 1992), the mathematical modeling cycle (Blum &

Leiss, 2007; Borromeo-Ferri, 2006), and theoretical perspectives on the teaching of mathematical modeling (Julie & Mudaly, 2007; Galbraith, Stillman, & Brown, 2010).

These combined perspectives allow for a detailed examination of the ways in which a modeling capacity is developed across teacher/student interactions in a local classroom context drawing on the teachers’ view of the mathematical modeling process.

3.2.1 Modeling Competence and Competencies

Blomhoj and Jensen (2003) describe modeling competence as the ability to carry through a whole modeling process with the destination being mathematical action.

Blomhoj and Jensen present a case for integrating a holistic approach to modeling with an approach designed to examine the various sub processes associated with the modeling process. The sub processes which Blomhoj and Jensen identify consider ways in which modeling tasks are formulated, how relevant objects are selected for mathematization, and how those objects and relations are translated into mathematical forms. Additionally how mathematical methods are used to generate mathematical results, how those results are interpreted, what conclusions are generated, and how conclusions are evaluated against the context are included as sub processes (p. 125). Blomhoj and Jensen find that there are benefits to examining particular phases associated with modeling through pre- established problems, in addition to examining problems that transcend the entire modeling process. They argue that neither of these approaches are adequate on their own, but offer that careful consideration should be given to the inadequacy of the

67 atomistic approach, even though focusing on those sub processes can be tempting (p.

137).

When discussing the generation of data which considers the development of the social practice of mathematical modeling, Blomhoj and Jensen (2003) indicate that this process is not straightforward, but, instead, when modeling is at the heart of a class, the development of competencies comes from a multi-focal view of this process. Hence, it mandates that different types of pedagogical interactions take place between teachers and students which are designed to help scaffold the students into becoming competent mathematical modelers and learn the game of mathematical modeling. Blomhoj and

Jensen (2003) further call attention to the myriad questions which students consider when engaged in the modeling process, including the nature of modeling, its purpose and intent, what is to be learned, what constitutes success or completion of a task, what assumptions need to be made, what should be disregarded, what model is optimal, and what aspects should be critiqued (p. 128).

In developing a platform for data generation, Blomhoj and Jensen’s (2003) points are important to consider as they encompass ways in which students enact the practice of mathematical modeling. These questions served as a means for focusing the data collection and analysis process, in particular, as a way to identify how the participating teacher responded to these conceptual issues as they arose and as she enculturated students into mathematical modeling and how she defined success in this domain.

Developing modeling capacity also involves the development of practices which support the use of mathematics in solving everyday problems and enable learners to carry

68 out mathematical modeling processes autonomously (Kaiser, 2007; Maass, 2006; Blum,

1996). These authors have established a wide variety of competencies associated with successful modeling processes. Kaiser (2007), in particular, presents a broad overview of the various competencies associated with mathematical modeling, including being able to solve real-world problems using mathematics, reflecting on the modeling process through activation of modeling knowledge, having insights into connecting mathematic and reality, having insights into the processes of engaging in mathematics, understanding the subjective nature of modeling, judging solutions, making arguments, as well as mastering the social factors associated with group work and collaboration (p. 111-112). Kaiser finds that there is a high degree of variation in how these processes develop over time and indicates ,in lieu of these results, that competencies could be developed through modeling-focused courses. However, the instability in the progress which students exhibit calls for long-term and over-time development to serve as a foundation for improvement

(p. 118).

In light of these findings, there is a need to closely examine the development of modeling practices over time as it offers a view of both the holistic and atomistic

(Blomhoj and Jensen, 2003) modeling approaches. This assumes that, with the teacher’s long-term goal of developing modeling capacity, modeling will transcend individual instances and be integrated across the entirety of instruction. As I will outline below, my participating teacher couches her instruction in mathematical modeling and, as such, offers a platform for examining data in this capacity. In seeking data, this allows

69 examining not only particular modeling experiences, but also what is learned and how learning transpires over time.

Another key area which has been understudied is the notion of how students go about connecting mathematics to their everyday lives (Kaiser, 2007; Maass, 2006). This notion links complementary ideas including, intuitions, experiences, and backgrounds from which students draw when encountering problems relevant to their lives

(Manouchehri & Lewis, 2017) and couple with known or learned mathematics.

Manouchehri and Lewis (2017) find that learners rely on their own personal experiences and intuitions when faced with problems with which they are familiar. These authors argue that developing modeling capacity necessitates the exploration of this intuitive and background knowledge in mathematical ways. Within this study, data generation and analysis sought how the teacher engaged in this scaffolding practice, in particular, her receptivity to student solutions and how she expanded on those solutions using both curricular and student mathematical ideas.

3.2.2 Sociomathematical Norms and Social Practices

Another important feature of developing a practice in mathematical modeling is the recognition of the social and communicative factors associated with discussions in mathematical modeling. To examine the longitudinal factors associated with the development of mathematical modeling practice, I looked towards sociomathematical norms (Yackel & Cobb, 1996) as a way of unpacking the local practices which emerge as learners engage in mathematical modeling tasks.

Yackel and Cobb (1996) discuss sociomathematical norms as a means of interpreting classroom interactions which capture how students develop mathematical beliefs and values and, as such, how they become autonomous mathematical thinkers and doers of mathematics (p. 458). They distinguish these from general social norms associated within school or cultural settings. Further, they argue that sociomathematical norms are established through participation in a particular cultural setting and that reasoning and sense-making by individuals resides in the interactive aspects in which meaning and knowing are taken to be shared by the individuals in the group (p. 460).

Yackel and Cobb define sociomathematical norms as those normative aspects of mathematics discussion which are specific to the students’ exhibited mathematical activity and examine the particularities of mathematical differences, mathematical efficiency, and mathematical elegance, in addition to the means which students use to make mathematical explanations and justifications (p. 461). They argue that sociomathematical norms are established in all mathematics classrooms, regardless of method of instruction, and, as such, are particular to the classroom context (p. 462) and seen as a product of the teacher responding to and interacting with their students.

According to these authors, sociomathematical norms offer a way for analysis and discussion of mathematical aspects of teacher/student activity within the mathematics classroom. Further, these norms are not predetermined criteria to be imposed, but rather they are continuously generated and modified by students and teachers through their interactions (p. 474). Stephan (2014) elaborates on the role of sociomathematical norms stating that these normative behaviors involve negotiating the criteria for what constitutes

71 an acceptable mathematical explanation, a different solution, or an efficient/sophisticated solution and links these criteria to the goals or philosophies of instruction in the classroom (p. 563). In this sense, these sociomathematical norms are linked directly to the ideological views of mathematics, as established in the classroom.

In my view, the concept of sociomathematical norms becomes important when considering the context of mathematical modeling, in particular, when looking at how behaviors are sustained over time. One problematic aspect, however, is the linking of these social factors exhibited in mathematics classes and the statistical process of normalization. While some behaviors might normalize across observations, more important are those particular aspects of practice which are emphasized, elaborated on, and validated within certain contexts. It is in this light that I consider the term practice to be better suited for characterizing these behaviors because practices are shared (by different people and groups), social, and part of the cultural model for interacting in particular contexts (Gee & Green, 1998). It is also in this light that I consider sociomodeling practices (SMPs) to be those social practices which establish activity within mathematical modeling from an emic and dialogic perspective. Because SMPs are cultural models used for enacting mathematical modeling, they are, as such, informed by emic views of the practice, and, in this way, the particular components which make up the SMPs may vary from context to context. In generating data for this study, examining what specific components of SMPs emerge, are emphasized, and are negotiated as a teacher engages in mathematical modeling with her students becomes important to distinguish.

3.2.3 Ethnomathematics and Ethnomodeling

Using ethnomathematics to support the data generation process involves reconciling the mathematical perspectives of the observer with that of the local cultural practices which emerge in solving problems. This requires an observer to not cast aside their own mathematical experiences, but rather use them to better explain what matters to the insiders of a particular group to other outsiders (Rosa & Orey, 2017). In this way, I draw on my own mathematical experiences and insights into the mathematical modeling process to examine activity within the group, discover emic practices indicative of the classroom view of mathematical modeling, engage in acts of dialogic translation to better understand those emic practices, develop an insider’s perspective, and then elaborate mathematically on those practices. Rosa and Orey (2017) further argue that mathematical modeling, in itself, can be a useful tool for describing local and cultural activity and, as such, elaborate on those cultural practices to make them recognizable to outsiders. In this way, my own mathematical background acts as a lens for data generation, but I do not assume that my own view is consistent with the views of the members of the group. It is through the facilitation of discussions within these modeling contexts that emic perspectives emerge, so I adopt this ethnomodeling perspective as a tool to support the dissemination of what happens within selected events.

3.2.4 Event Mapping

Methodologically, Hennessey et al. (2016) draw on ethnographic tools to document structures of interactions and look across communicative acts (i.e., those

73 speech acts defined by their interactional functions which are given status through social context as well as grammatical form and intonation), communicative events (i.e., the series of turns which occur within an interaction with a common purpose, orientation, or topic), and communicative situations (i.e., the general context in which communication occurs) (p. 18). Analysis of the dynamic interactions across these communicative acts, events, and situations offer a means for the development of interpretative narratives of the lessons with respect to how they unfold over the course of time. Similarly, this analysis mechanism affords examination of questions regarding the nature of participation in a lesson in which activities change over time (lecture/open discussion/other forms), in addition to how responses are taken up and their respective stability (p. 19). Hennessey et al.’s model provides a useful framework for organizing data in the current study, in particular, when developing a breakdown between communicative events within a communicative situation. Hennessey et al. suggest event mapping as a means for facilitating this event breakdown.

Hennessey et al. (2016) describe event mapping as a way to develop conceptualizations of dialogic interactions by looking at shifts within and across macro- level conversational situations, communicative events at the meso-level, and conversation acts at the micro-level of dialogue (p. 18). One crucial aspect of their event mapping process is the documenting or describing of the conversational situation, the general context of the lesson, or the sequences which the lesson follows (p. 19). Drawing on this process within this work, I use event mapping in order to document major shifts which occur over the course of a lesson and gain a sense of the general context of instruction.

Table 1 offers a sample event map of an event narrative in which a class discussion is facilitated regarding problems linked to the profit, costs, and revenue of a hypothetical business (elaborated on in further detail later). Over the course of this discussion, the nature of the conversation shifts to aspects associated with mathematical modeling and its relationship to the discourse of business. In order to capture these dialogic shifts, I used the event map to highlight these different sections in order to frame my analytical view.

In the event map in Table 1, the column to the far left indicates the time-stamp from the video collected during the observation. This was done in order to offer ease of access to key moments in the videos for deeper review. The second column offers an overarching description of the initiating action which framed a particular period of time in the event. Column 3 offers response actions to the initiating event documenting the dialogic nature of the discussion. The intent was to highlight the social nature of the conversations and not focus explicitly on the teacher as the sole individual responsible for the flow of the lesson. The fourth column offers the particular message unit from the transcript in the voice of the individual who initiated this piece of the event in order to reference the transcript for further analysis of the key event and the shifts which occur.

Finally, the last column offers preliminary notes/referencing of mathematical tasks or initial insights into the specific conversational events which occur over the course of the lesson to serve as another means for further inquiry.

Table 1. Sample Event Map

Time Initiating Action Response Action Message Unit Notes 00:23:52.18 Students selects Teacher writes the T: So, for 64, a general Mathematical task: problem to discuss problem on the board formula for the way Find when P>0 if from their homework and opens a discussion businesses work, and 푃 = 푅 − 퐶 assignment the of the problem. they say this right here. 푅(푥) = 0.0125푥2 + 412푥 previous night. Economists for Smith 퐶(푥) = 12225 + 0.00135푥3 Brothers Incorporated, some company, find that the company profit P is equal to the revenue minus the cost. 00:26:09.06 Teacher articulates the Students offer T: How many During this instance, students

specific requirements suggestions for finding customers must the offer suggestions for solving for the problem a solution to the Smith Brothers have to problem and problem based on their be profitable? mathematical procedures for attempts on the finding when the business is homework. profitable: - Graphing - Algebraically - Solve Function on calculator 00:30:18.16 Students determine Teacher prompts Mi: It has to be 30 or Students found exact solution at one solution occurs at students to consider greater 29.7 people and discussed 30 people alternate solutions aspects of rounding (up or down) based on degrees of polynomials.

Table 1. continued Time Initiating Action Response Action Message Unit Notes 00:31:04.24 Teacher suggests that Students indicate that T: Based on our Teacher probes students to there may be more there should be at discussion yesterday, determine the other points of solutions than the one most 3 solutions where we're talking intersection of the two equations identified about the number of and consider which is “better” or zeros the function has more appropriate. and we were talking about the degree of the polynomial, you guys what was the degree, the highest degree, of those two

77 polynomials….how

many solutions do you think it has? 00:32:29.22 Students graph Teacher draws graph Ag: There's three Teacher refers to the student- equations on problem on the board T: Aww...look how generated graph of functions on calculator and notice for reference pretty it looks her calculator. that there are 3 Okay, we're going to do solutions. that...so, when you Using graphs, the class zoomed out determines that there is a range what is your scale for where profit is positive between that by the way? 30 and 542 people. 00:36:34.16 Teacher calls for Students deliberate on T: So, guys, we had two Students deliberate about which students to unpack the the range of viable answers, 30 and graph is cost and which graph is meaning of these two profitability and try to 542 people… How revenue solutions determine where could you decide which revenue exceeds cost. one is going to Class determines that between 30 00:38:46.24 to be a better solution; and 542 the business would be 30 or 542? profitable

Table 1. continued Time Initiating Action Response Action Message Unit Notes 00:39:05.19 An recognizes that the Teacher invites “real- An: so the cost is too Students bring up real-world cost would exceed world” into the high um so implications for the scalability of revenue beyond 542 discussion to evaluate T: So the cost is too the business as the size customers models high here, like literally approaches 542 people in real world terms, not math terms Students consider a number of The cost is too high for perspectives, including: this business model to - discrete views of profit work with respect to number of people - raising price of the

78 product to increase profit

- viability of particular model when scaling up business

00:40:11.13 Teacher re-connects Cl picks up on T: Hmm…what does The teacher’s proposal for discussion to the connection and offers that probably say about connecting the previous application problem at to align the solutions their current business discussion on profitability to the hand (30 and 542) to their Cl: It’s somewhere current business at hand taken up current range between 30 and 542 by Cl when he affirms through his response

When framing dialogic shifts which merit analysis of the development of SMPs, careful consideration is paid to the uptake of these shifts and how they index aspects of mathematical modeling practice. In this particular example, while working on what might be conceived of as a conventional application problem, through the course of discussion, this problem allowed for development of SMPs by creating intercontextual links between the problem and legitimacy of models in actual business. This case highlights how event maps can structure an analytical view of an event as intercontextual links emerge through segmenting the lesson in this fashion. On an interactional level during whole- group discussions, noticing the frequency by which the teacher elicits the revisiting of particular solutions to the contexts and then whether and how the students begin to engage in this autonomously affords a means for considering how these practices are developed over time. Whole-group discussions afford the teacher the ability to mediate the learning process and synthesize points made across student participants to meet specific goals. In the event map referenced above, one can draw particular attention to important points in the discussion, for example, 00:39:05.19, in which the teacher invites “real-world business” into the discussion, and offer evidence on how this transition meets the objectives which the teacher established prior to the lesson.

Additionally, one can look for evidence over the course of a whole-group interaction.

When considering interactions during explicit modeling tasks, a focus on a small group of students interacting with the task becomes an alternative means for studying the research questions since these interactions reveal genuine evidence of students’ own initiations which may not become visible during public discourse. In examining these

small-group interactions, event maps can be adjusted to focus on those explicit instances, capture similar transitions, and frame further the analytical view. Similar principles of conversation analysis will be employed, and message units will be parsed and coded to examine how, in these cases, students are picking up on the normative behaviors co- constructed by the class.

3.2.5 Transcribing

Aside from event mapping, the methodological process of constructing data involves transcribing key events to be used in the analysis. Geertz (1973) indicates that, with respect to ethnographic description, there are three characteristics: “it is interpretive; what it is interpretive of is the flow of social discourse; and the interpreting involved consists in trying to rescue the ‘said’ of such discourse from its perishing occasion and fix it in perusable terms” (p. 20). Geertz further argues that this process is microscopic

(p. 21). In this process of creating a thick description of cultural activity, Geertz (1973) involves the process of transcribing selected events to fix them for this interpretive process. Bloome et al. (2010) argue that the process of transcribing establishes boundaries within an event and supports individuals in the process of understanding what is happening and how to construct meaning within an event (p. 14). In their process of transcribing, they advocate for identifying message units by identifying contextualization cues consisting of pauses, stresses, intonation, stylistic changes, or gestures. This they further argue is established post hoc from the event in which it occurred (p. 19).

I have adopted this perspective to support my data generation. Following the construction of event maps, I engaged in transcribing sections of key events which merited further analysis. In this process, I reviewed my audio and video data, listening for intonational shifts, gestures signaling mathematical thinking or other mathematical activity, and other contextualization cues reflecting mathematical modeling. This transcription process involved using the transcription software InqScribe. My raw transcriptions were then imported into Microsoft Word and generated as tables in which speaker, message unit, and line number could be distinguished.

I employed these theoretical and methodological contexts to inform my orientation for data generation across my study. Below, I offer a detailed description of the research process from conception of the pilot study to final analysis of data used in the current report.

3.3 Study Procedures

The purpose of this study was to examine how learners are enculturated into a mathematical modeling practice through teacher scaffolding and, through these observations, consider a means for theorizing about teaching practices in mathematical modeling. This comes through examining, in detail, how a teacher positions mathematical modeling, the particular components which establish her view of mathematical modeling (RQ1), and how she scaffoldings the learning of that practice in learners to advance her personal and curricular goals (RQ2). In order to address these research questions, I conducted a micro-ethnographic study of her teaching practices in

an 11th grade pre-calculus classroom. This study consisted of a pilot study during the

2016/2017 academic year, formal data collection during the fall semester of the

2017/2018 academic year, and analysis during spring and summer of 2018. Table 2 reflects the data collection and analysis timeline. A brief description of the procedure is offered following this presentation.

Table 2. Data Collection and Analysis Timeline

Research Phases Time Period Protocol Pilot Study Data Collection Aug.-Mar. 2016/2017 Audio/Video Recordings and Interviews Pilot Study Analysis and Apr.-May 2017 Event Mapping & Indexical Reporting Discourse Analysis Pre-Meeting Interviews June-Aug. 2017 Audio and Video Recording with Participating Teacher, and interviews Creation and preliminary selection of modeling tasks and curriculum planning Formal Data Collection Aug. – Nov. 2017 Audio and Video Recording/ Teacher Interviews/ Case Student Interviews/Field Notes Event Mapping for Dec. 2017 – Jan. 2018 Hennessey et al. (2010) Collected Data Selection, Transcription, Feb. 2018 – Mar. 2018 Transcriptions (Bloome et and Description of Key al., 2010); Thick Description Events (Geertz, 1973) Analysis and Reporting of Mar.-May 2018 Discourse Analysis (Bloome Key Events et al., 2010); Ethnomodeling (Rosa & Orey, 2015) Final Document June 2018 Preparation Final Document Reporting July 2018

The pilot study began during the 2016-2017 academic year. Observations were conducted on a weekly basis and audio/video data gathered during these observations, accompanied by field notes. Data generation during the pilot study involved seeking instances in which modeling was discussed, creating event maps of those instances based on major instructional shifts during each session, and transcribing those events into message units. Pilot study data was then analyzed using indexical discourse analysis

(Scollon & Scollon, 2003) to examine those ways in which modeling emerged within discussions. It was during this analytical process that SMP was conceptualized and the decision was made to focus on this establishment of practice as the primary source of data for this study.

Formal data collection for the dissertation research commenced during the summer of 2017 with preliminary planning and information gathering meetings between the cooperating teacher and myself. It was during these meetings that the scope of the instruction and preliminary modeling encounters was planned. I also collected information from the participating teacher about her short-term and long-term instructional goals pertaining to mathematical modeling. Additionally, the participating teacher outlined four preliminary ideas for modeling tasks which would likely occur during the observational period. Observations commenced during late August of 2017 and ran through October 2017. The participating teacher’s first block class was selected as the focus for these observations. The 10-week period observation period running from

August – October covered three instructional units with four distinct modeling sessions.

Data collection occurred daily during instruction and audio data, video data, and field

notes were gathered. The audio recording process consisted of using a Swivl camera with primary microphone worn by the teacher, one microphone worn by the researcher for quiet commentary, and four microphones used for capturing discussion during student work. In video recordings, two cameras were primarily used, with one camera focused on the teacher and synced with her microphone to capture specific interactions between the teacher and her class, and the other camera focused on whole-group discussions and individual students during small-group work. Field notes focused on documenting researcher reflections, broad descriptions to be used for event mapping, and outlining particular observations to support the analysis process. Daily debriefing interviews were also conducted, giving the teacher a place for reflection and examination of key instances that occurred during class sessions, and were audio recorded, video recorded, and documented through field notes, as well.

Daily event maps (Hennessey et al., 2016) were created for each observation documenting major shifts in instruction to outline the progression of each lesson. From these event maps, key episodes were selected for further analysis, in particular, those events in which mathematical modeling tasks were facilitated. These events were transcribed into message units and analyzed to determine how the participating teacher established a sociomodeling practice (RQ1) and how that practice was refined and taken up by learners (RQ2). This analytical process began during November of 2017 and concluded in late May of 2018. Following this analysis period, the final reporting occurred during June 2018 with final presentations scheduled for July of the same year.

In progressing through this iterative design methodology, a thick description of classroom events (Geertz, 1973) was established which was geared toward documenting the social components which make up the cooperating teacher’s view of mathematical modeling and how those components were refined and taken up across the 10-week observational period by learners. With the teacher’s goals of fostering growth in modeling capacity, while simultaneously imparting curricular knowledge, analysis focused on ways in which modeling capacity was nurtured in a fashion sensitive to both personal ideas and intuitions of learners, along with curricular mathematics. In order to generate this description of practice, I looked across four distinct modeling experiences and examined how the teacher facilitated those tasks and couched them in her instruction to accomplish her content-focused goals. Yackel and Cobb (1996) indicate that enactment of sociomathematical norms is fundamental in reporting on teacher and student activities in a mathematics classroom (p. 474), a construct easily extendable to mathematical modeling contexts. Further, Yackel and Cobb argue that sociomathematical norms can cut across areas of mathematical content through looking at the validity of solutions, differences and similarities in solutions, as well as sophistication and efficiency of solutions and, thereby, help judge the quality of mathematics explanations (p. 474).

Additionally, these authors argue that sociomathematical norms provide an opportunity for a teacher to clarify their role as a representative of the mathematical community and develop practices consistent with the current agendas of that community (p. 474). Yackel and Cobb relate this to drawing on reform- and inquiry-based practices within mathematics education.

In order to report on the cultural behaviors which emerge within the data collection process and understand those local conceptions of mathematical modeling, mathematical learning, and those social practices which result, it is paramount to develop a clear picture of the context in which the research was conducted and gain insight into the teacher, her goals, and her experiences. To do so, in the next section, I offer a description of the research site and background/objectives identified by the cooperating teacher in this study.

3.4 Description of Research Site

This dissertation study took place at a private academy in the Midwestern United

States within an 11th grade pre-calculus classroom during the fall of 2017. This school indicates on its website that it is “an independent, coed, PreK-12, college preparatory day school committed to developing the entire individual – mind, body and character” with a curriculum designed to promote critical thinking and creativity grounded in a tradition of academic challenge and personal support. The school, itself, is divided into three distinct sections: a lower school, middle school, and upper school.

My research occurred in the upper school, which is dedicated to developing students who are college-bound. The upper school curriculum offers a series of required core courses and selective specialty electives which meet a broad range of student interests and career potentials. The school advocates for relatively small class sizes in order to meet individual student’s instructional needs, resulting in typically fewer than 20 students per class. In establishing common cultural practices, each student is required to

give junior speeches over the course of the year in which they take a position on a topic which is important in their lives and articulate its impact on their personal growth. The mission statement of the upper school is: “The Academy strives to develop and sustain a community of thoughtful, responsible, capable and confident citizens eager to engage in a pluralistic and ever-changing world.” The upper school runs on a rotating block schedule in which students attend their regular classes for 70-minute instructional sessions 4 days a week. Observations for data generation and analysis were conducted during these 70- minute class periods. The school’s mathematics department supports problem solving and encourages teachers to implement instructional strategies outside of direct instruction. Teachers at this school also have the opportunity for continued professional development, and teachers are encouraged to present at national and international conferences. Teachers are also encouraged to align their curriculum across courses and use common assessments across these classes. Additionally, their departments meet frequently to discuss current goings-on in their courses to better align their instruction.

3.4.1 Background and Description of Cooperating Classroom Teacher

The participating teacher Mae, a pseudonym, is a twelve-year veteran of teaching mathematics at the secondary level. The 2017-2018 academic year, in which this study took place, was her second year teaching at the research site. Prior to her appointment at the academy she had taught in a rural district in the Midwestern United States (9 years).

She had spent one year early in her career at a private charter school. During the data collection phase for this work, Mae was working on and successfully completed her

Master’s Degree in Mathematics Education from the overseeing university. Mae and I took common classes, which was where we met. Additionally, we shared a major faculty advisor and were encouraged to collaborate as a result. Mae has additionally engaged in presenting on mathematical modeling in numerous contexts, ranging from local communities to teachers to international research conferences. She considers herself to be an expert on mathematical modeling teaching and uses it frequently in her instruction.

The rational for the selection of Mae for this study was based on this development and our shared interest in mathematical modeling in classrooms, in particular, in supporting learners to use mathematics in solving problems relevant to their daily lives.

The cooperating teacher identified that her long-term instructional goals for the academic year included developing mathematical modeling skills among her students.

As an additional goal, the teacher intended to use technology as a means for mathematical inquiry and exploration. She has expertise in the use of technology in instruction and holds certificates from Texas Instruments as a T^3 instructor. She has led numerous teacher-focused seminars and workshops on mathematical modeling and collaborated on prior research projects with myself.

During an information-gathering interview with Mae, she expressed that she viewed mathematical modeling as a response to the question “when am I ever going to use mathematics in real-life”. She indicated that, as a student, she always questioned the utility of the mathematical concepts she studied in class and was rarely satisfied with her own teachers’ responses. As such, she felt that modeling contexts should be personally meaningful to her students. In her early years of teaching, she often tried to use textbook

application problems as contexts for modeling but felt unsuccessful, as her students had minimal exposure to the contexts. Mae also expressed that, in her current view of modeling, aligned contexts should have some social or experiential weight with her students. Further, she believed that mathematical modeling could be helpful in making these real-world contexts meaningful to learners, allowing her to help in making mathematical connections as contexts are explored. To her, engagement with mathematics yields meaningful learning, and, because of this notion, raising student interest in tasks is a primary concern.

In orchestrating discussions around mathematical modeling, Mae indicated that, when preparing for explicit modeling experiences, she tries to consider current events or real-world scenarios which her students have mentioned previously or have recently encountered so to assure their familiarity with situations. She saw her role as an aid in structuring these situations so that students could grasp that realized mathematics is what they were exploring. She advocated for the use of technology due to the support it could provide learners during their problem solving. According to her, in modeling contexts, technology allows students to make conjectures; test their responses and refine them as they engage in the modeling process, removing the constraint of getting bogged down with tedious calculations; and allows deeper thinking about the contexts (personal correspondence regarding the nature of modeling, 12/1/2016).

Mae felt that modeling should be treated as both content and vehicle as she recognized that there were particular aspects of modeling which learners should explore and understand relative to that practice. Additionally she expressed that she viewed

modeling as a vehicle with which to impart curricular knowledge. In selecting tasks, Mae looked for opportunities in which learner’s ideas could be explored and then aligned to curricular topics. In this way, she felt that she could operate flexibly with modeling as both a content and vehicle (personal correspondence, Fall 2017).

3.4.2 Description of Classroom Contexts

In studying those ways in which sociomodeling practices are established through teacher-student interactions, it becomes important to consider the classroom context, itself, regarding the make-up of the students as well as initial observations concerning and insights into their backgrounds. The pre-calculus class under study met together four times per week for 70 minutes from 8:05-9:15AM. A typical lesson was covered across one classroom session, and a variety of instructional strategies were used to cover curricular topics, ranging from inquiry-based learning, problem solving, mathematical modeling, direct instruction, guided notes, and hands-on tasks.

The class which was studied consisted of 18 students from various ethnic and cultural backgrounds. During discussions between students, it became apparent that the vast majority of students came from affluent families in business, law, medicine, or other relatively high-paying professions. As a result, many students expressed interest and familiarity in these areas as potential venues for their future careers. As students explored mathematical modeling tasks in which they were presented, this information served useful in unpacking their responses to contextual issues and questions and informed additionally those scaffolds which Mae initiated.

In facilitating modeling tasks, Mae’s view allows students to be curious about ways in which mathematics is applied to real-world events without imposing her questions on those experiences, but rather supporting students in generating and responding to contextual questions themselves. Secondly, Mae advocated strongly that she wished for her students to both feel successful and enjoy mathematics, as she believed that a number of her students did not have the opportunity to consider mathematics in this way in the past (personal correspondence, Spring 2017).

Considering these classroom, mathematical, and pedagogical goals, my initial analysis of modeling capacity was based on two factors: (1) students’ inquisitiveness about practical applications of mathematics through modeling, and (2) students’ dispositions towards mathematics. These two factors were used as indicators of how these features developed over the course of the observation period. The teacher, during an initial interview, indicated that she believed that she could accomplish these goals through modeling. As such, in considering the context of the classroom, I examined how these factors emerged across interactions and an SMP was developed over time.

3.4.3 Researcher Positionality

Identity markers that I draw on are that of a white, heterosexual, cisgender male who has spent most of his adult life in the Midwestern United States. Prior to conducting this doctoral research, I worked as a middle and high school classroom teacher for four years teaching 7th through 12th grade mathematics. In teaching in a more difficult context where many students were disengaged from mathematics, I found successes in

incorporating real-world projects into my instruction. It was through these projects that I was able to make mathematics relevant to my students. It was through incorporating these projects that I was invited to take a year-long sabbatical from classroom teaching to develop an integrated mathematics unit for a local educational nonprofit. It was through this curriculum development that lead me to pursue my doctoral studies in mathematics education as I realized that there was still much to learn about developing robust tasks.

In my doctoral studies I began to research mathematical modeling, and found it to be tightly connected to those mathematical projects that I had previously facilitated as a teacher of which culminated in this dissertation study.

3.5 Data Analysis Process

In looking at the development of SMPs over an observation period, I focused my analysis on those initial days during which mathematical modeling tasks were explicitly implemented. I did this in order to capture how Mae presented the practice of mathematical modeling to her students and how she used modeling as a bridge towards concept formalization. In contemplating the evolution of practice, I sought intercontextual links between events and identified common components indicative of mathematical modeling and contemplated how teacher interventions promoted their development over time. Table 3 offers an overview of the conducted observations across my data collection.

Table 3. Overview of Observations in Data Collection

Day # Date Topic Notes 0 8/11/17 Pre-Planning Meeting with Teacher Overview year’s objectives, scope and sequence of instruction, and facilitation of modeling tasks 1 8/22/17 The Answer Is Task Modeling Task Facilitated 2 8/23/17 Answer is Discussion; Syllabus Overview; Modeling Task Measure the Height Task Discussion, Modeling Task Facilitation 3 8/24/17 Debrief Discussion for Measure the Height; re-Assessment Trigonometry Skills 4 8/28/17 Review answers to Pre-Assessment; Modeling Task SOHCAHTOA introduction, Presentation Discussion of Solutions to Measure the Height Task 5 8/29/17 Trigonometry Practice Problems: Sides of Sine, Cosine and Missing Triangles Tangent Formalized 6 8/31/17 Trigonometry Squares Puzzle Activity Matching trigonometric expression with result 7 9/1/17 Trig Squares Puzzle Wrap-Up, Trig and Demonstrated Unit Circle Problems Lecture, Trig and Solving; Student Unit Circle Problems Student Practice Practice – ill- defined problems leads to student assumption making 8 9/6/17 Trig Test 1 Review Packet Students work individually or in groups practicing solving problems 9 9/7/17 Triangle Trigonometry Assessment 10 9/12/17 Determining the Shortest Distance to Modeling task Cafeteria Task (Law of Sines) facilitated and discussed 11 9/14/17 Formalization and Practice of Law of Sines (problems from book)

Table 3. continued Day # Date Topic Notes 12 9/15/17 Law of Sines TI Calculator Discovery Technology- Activity; Ambiguous Law of Sines Focused Lesson 13 9/19/17 More Ambiguous Law of Sines Practice 14 9/20/17 Law of Cosines and Heron’s Formula Application/Practice 15 9/22/17 Review Packet for Test (Individual Work) 16 9/25/17 Law of Sines and Law of Cosines Assessment 17 9/27/17 Trig Identities Day 1: Sine, Cosine and Teacher positions Tangent Identities unit as a “lack of application”; lecture and practice in groups; students exhibit confusion about teacher preferred method of solving 18 9/28/17 Trig Identities Day 2: Clarification of Teacher positions Process and Sum and Difference Identities these problems as puzzles, reviews one-sided manipulating 19 10/3/17 Trig Identities Day 3: Solving for Missing Guided discussion Variables Using Trig Identities with student contributions for problem solving 20 10/4/17 Radians/Degrees Conversion PME-NA Conference 21 10/6/17 Review Packet (Individual Work) PME-NA Conference 22 10/9/17 Review for Identity and Radians Assessment 23 10/12/17 Trig Identities and Radian/Degree Conversion Assessment 24 10/16/17 Radians and Arc Length 25 10/17/17 Radians Assessment 26 10/19/17 Radians Additional Practice 1 (Packet) Teacher Attended Conference 27 10/20/17 Radians Additional Practice 2 (Packet) Teacher Attended Conference

Table 3. continued Day # Date Topic Notes 28 10/24/17 Discussion of Radians Packet; Discussion Modeling Task of Radians and Area of Circles; Discussion Facilitated of Angular Speed and Velocity; Trig Whips Activity 29 10/25/17 Analysis and Unpacking of Trig Whips Modeling Activity; How Fast To Run Problems. Discussion Facilitated 30 10/27/17 Formalizing Angular Velocity; Sharing Modeling Task Solutions for Student Generated Problems Bridged to Angular Velocity Note: Events highlighted in yellow are those events explicitly analyzed

While numerous instances of mathematical modeling occurred across the observation phase, my rational for selecting these initial encounters was to provide an in- depth documentation and analysis of the intricacies and particularities of component construction and to see how my participating teacher developed mathematical modeling capacity and used it as a bridge to academic mathematics. This necessitated a robust unpacking of these preliminary interactions from presentation of modeling task through concept formalization.

In my data analysis protocol (Figure 8), I identified specific components which were indicative of Mae’s view of the mathematical modeling process and then sought to find intercontextual links between this framing of mathematical modeling and its enactment over time.

Figure 8. Data Analysis Protocol

I began my analytical process with the creation of daily event maps using the audio data, video data, and field notes collected during my observations. These events were then transcribed by partitioning verbal utterances and gestures into message units.

A fine-grained analysis was conducted across each event and examined specifically how to unpack deliberate teacher intervention which supported the development and refinement of SMPs, which I characterized as scaffolds.

This analysis took place across four distinct phases, each of which will be summarized in turn. Phase one of my analysis involved looking across initial modeling encounters, beginning with the first interaction between the teacher and her students. In this initial observation, I identified the specific components of the SMP which informed

Mae’s short- and long-term goals in developing mathematical modeling. The second phase of my analysis involved looking at how those components were refined and negotiated across subsequent discussions, building a more robust view of my teacher’s

SMP. Of particular interest was determining which components of the SMP were 96

emphasized and how those components aligned with her known goals of promoting mathematical modeling. Phase three of my analysis involved looking at which particular components were taken up by students as they engaged in a modeling task to provide evidence that learning had taken place. Phase four consisted of ethnomodeling events in order to connect emic practices to etic mathematics. In this way, I was able to trace how mathematical modeling capacity was nurtured across discussions.

Phase One: Component Identification

Phase one involved identifying specific components indicative of Mae’s view of mathematical modeling in the first event. In doing so I examined the video, audio, and transcript data looking for common themes which either were explicitly emphasized or referenced frequently across the event. These components were then labeled and categorized according to their intent. Table 4 offers a complete list of the analytical components of SMPs which were identified across these interactions.

Table 4. Analytical Components of SMPs

Analytical Codes Description Problem Generation The act of problem posing or development of specific contextual questions framed by an overarching problem Communicating Ideas The linguistic act of making public those aspects of cognition, including thoughts, ideas, insights, or intuitions Parameterization Considering the impact of particular parameters within a situation context, including estimation of a value believed close to the actual amount Refining Taking a generated solution or model and adjusting it based on contextual information or re-exposure to the context Assumption-Making Accounting (implicitly or explicitly) for particular variables which impact the system through assuming values or explicitly controlling for those variables/components assuming them away Thinking about the Real World Considering explicitly contextual characteristics which inform how a system behaves and how those considerations mediate interactions within the problem Validating Considering the utility of a constructed model or problem in light of the context in which it is housed Collaboration Working with other individuals to generate models, develop solutions, or share ideas about contextual or mathematical activities Physical Tool Use Using physical tools to advance the problem-solving process or develop contextual information used in the process of solving Researching Seeking out information to support the process of sense-making, parameterization, or assumption-making Establishing Variables Addressing explicitly particular variables in the context of a problem

Table 4. continued Sense-Making Thinking and reflecting about the mathematical and contextual components within a problem, and using those considerations to advance the problem- solving process Asking Mathematical Questions Consideration of the mathematical components of a problem context and asking/generating mathematically answerable questions Asking Contextual Questions Consideration of the contextual components of a problem context and generating questions or ideas about those mediating contexts to enhance that contextual view Thinking about Mathematics Thinking explicitly about mathematical ideas, processes, or procedures relative to a problem context Sharing Solutions The act of broadcasting solutions to worked-on problems as a way of public assessment or discussion Experiential Tool Use Considering individual experiences, ideas, intuitions, or cultural knowledge in light of a particular context or problem Teacher Humanizing Sharing personal experiences or information as a means of broadcasting aspects of identity or lived experiences Enculturation into Mathematics Connecting or inviting ideas and considerations into a particular mathematical register or form Activating Metacognition Activation of thinking about thinking or reflection on particular aspects of the sense-making process Formalizing Curricular Mathematics Formalizing curricular mathematical topics though deliberate problem selection, presentation of particular mathematical procedures, using specific mathematical notation Conceptual Tool Use Using conceptual tools such as mathematics in solving problems or facilitating discussion

In my analysis and results section, I will elaborate on how these components were identified across events, but for the purpose of making clear my methodological process,

I offer the example of the components Technology Use and Research. Within the initial event, the cooperating teacher observed students working on a related modeling task and noticed that her students were using their cell phones to research contextual information necessary for the problem. In observing this, she stopped the class in their work and explained it was permissible for students to use their cellphones as a tool to research unknown information. In this way, she established these components of technology use and research as being a part of her SMPs and, as such, they were labeled to correspond to this type of behavior.

Phase Two: Component Refinement

To look at how components identified in phase one were refined across discussions, I conducted a line-by-line analysis of transcriptions, bringing to light the particularities of discussion in terms of how these components were addressed and refined by the participating teacher. Each message unit was mapped to identified components of SMPs. I considered further the nature of conversational functions (Green

& Wallat, 1981; Bloome, 1989) which emerged across these discussions. This provided further insight into the underlying meaning of each message unit with respect to advancing students in mathematical modeling.

Bloome et al. (2010) indicates that conversational functions aren’t a set of discrete and mutually exclusive categories, but, instead, highlight the ways in which a teacher and learners respond to each other (p. 68). I identified the prevalent conversational functions

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in my data as provoking, acknowledging/validating, clarifying, elaborating, emphasizing, and positioning (which includes establishing/stating and sharing). I considered provoking actions to be those message units which incite an elicited response.

Acknowledging and validating is considered to be interactional, in which a response is affirmed and positioned as viable. A clarifying action is one in which an individual re- states or expands on one particular issue or point which was addressed previously. I distinguish this from elaboration, which I define as adding additional details to a previous utterance. Finally, taking a particular stance or making a particular claim exemplifies the acts of establishing/positioning/stating or sharing.

Bloome et al. (2010) argue that in conducting this degree of analysis in classroom events we have the ability to recognize that those cultural practices which occur in classroom contexts are indicative of both the broad cultural models of that context and shared local views simultaneously. In particular, they state that “any instance of a classroom literacy event may sufficiently match the shared cultural model of classroom literacy practices at one level while at other levels it may diverge from the shared cultural model” (p. 68). I interpret this to mean that while these interactions are occurring in a mathematics class along with the fact that sharing solutions to mathematical problems is common-place within this context, the ways in which teachers employ these conversational functions to advance the discussion and what aspects of learning transpire as a result may not always match the expected or anticipated responses for that context.

In other words, while, at times, responses may seemingly omit deliberate connections to mathematical processes and procedures, these interactions can indeed inform future and

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subsequent mathematical interactions, as the context of discussion is different, but still connected. The identification of these conversational functions affords the ability recognize how Mae values student responses and how she positions them to explain, enhance, and mathematize these considerations in light of problem-solving within a modeling context.

This coding process of identifying components of SMPs and their manifestation through conversational functions supports what Geertz (1973) calls a thick description; an interpretation of the flow of social discourse, and he says that the interpreting process involves fixing social discourse in careful and detailed terms (p.17). Geertz indicates that generation of a thick description affords the process of theorizing, where the aim lies in drawing large conclusions from “small, but very densely textured facts” (p. 27). In the presentation and unpacking of the selected events within my study, I seek further to establish a typology of teacher responses to student utterances as a mechanism for theorizing about how the establishment of the classroom culture leads students to considering their problems and solutions in ways that support the mathematical modeling process.

Table 5 offers a template of the analytical process of unpacking key events. In organizing data across the top row, the components of SMPs are listed. Each following row contains a referent line number, a transcribed message unit, the speaker of that message unit, and a place to check (denoted with an x) which disposition that message unit corresponds to. In order to emphasize and organize the interactions, rows in which the teacher is speaking are denoted in a light grey. Further, each linking disposition

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which the teacher establishes in the course of the interaction is color-coded to match the outlined conversational functions listed previously. In this organization, I have attributed the following colors to each conversational function: provoking – red; acknowledging/validating – orange; clarifying – yellow; elaborating – green; emphasizing – blue; and establishing/positioning/stating/sharing – purple. The intent of using this color scheme is to offer quickly a visualization of the dynamics of the teacher scaffolding to better assist in creating a typology of teacher interventions.

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Table 5. Analytical Coding Template

Mathematics

Making

ofModel

and Experiences Ideas

Making

Individual

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Color and Symbols Key: Provoking; Acknowledging; Clarifying; Elaborating; Emphasizing; Establishing a position

Following the coding of this data set, I looked across these events and tracked which conversational functions and components most frequently emerged. I then tabulated and graphed this data in order to provide a visualization of the nature of the discussion and easily identify which components were most predominant. A sample frequency distribution of components is offered in Figure 9.

Figure 9. Sample Frequency Analysis of Component Occurrence

Phase Three: Component Uptake

After the initial events were coded and tabulated, I turned to observing students collaborating on modeling tasks to see which specific components of SMPs they enacted.

This analytical process consisted of analysis of spoken words and gestures and mapping those to the components of SMPs. I used this as a way of providing evidence of learning

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in that those most frequently referenced components were those which were taken up by students during their work. Table 6 offers a sample of this analytical process. In this table, I include the identity of the specific speaker, the words said to each member of the group, a brief description of their activity, the related components of SMPs referenced by the students, and, finally, an accompanying image. Considering the uptake of these components constitutes the change in practice which Gee and Green (1998) claim provides evidence of learning.

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Table 6. Sample Gestural Analysis

Line Speaker Message Unit Description of Activity Related Component Image Al and Am play with inclinometer trying to

determine how to use it 810 Am It's like 15 Am and Al find the Working degrees angle of elevation mathematically – and draw a diagram measuring; of the situation collaboration

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811 Am Here, let me hold Am takes tape Collaborating; this measure end working mathematically (measurement) 812 Al Move it over Al prompts Am to Collaborating; stretch the tape working measure mathematically - attending to precision 813 Yeah, over to Am stretches tape Collaborating; there measure to umbrella; working minor adjustments mathematically observed to level tape measure with respect to the ground

Phase Four: Ethnomodeling

Phase four of analysis entailed the ethnomodeling of student strategies to help uncover aspects of mathematical work which the students and teacher engaged in which may not be immediately visible. Rosa and Orey (2017) argue that ethnomodeling offers a means for translation and elaboration of mathematical activity which occurs within a cultural group. They claim that ethnomodeling affords those who are engaged in the research process to study the system of mathematics “regarding their own contextual reality in which there is an equal effort to create an understanding of all components of these systems as well as the interrelationship among them” (Rosa & Orey, 2017, p. 155).

According to these authors, by using mathematical modeling, a researcher can translate emic (cultural) practices into etic mathematical forms to establish relations between local contextual designs and academic mathematics. In this study, I used ethnomodeling

(Figure 10) as a means to valorize the mathematical activity of the students and to further showcase the particular components of the SMPs as well as their connection to the mathematical modeling process as outlined in the literature.

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h=x+z

Figure 10. Sample Ethnomodeling of Measuring

Using these four phases, I was able to trace the development of mathematical modeling practice from initial conceptualization to formalization. In the next chapter, I offer an in-depth account of this process while examining how the various components of

SMPs are established, refined, enacted, and connected to the short- and long-term goals of the teacher.

3.6 Report on Pilot Study

Prior to formal data collection, a pilot study was conducted during the 2016-2017 academic year in Mae’s classroom across two pre-calculus classes. The purpose of this initial work was to gain an initial overview of Mae’s practice and see how she facilitated mathematical modeling tasks. During this study, I examined those ways in which

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modeling emerged across her teaching and looked for distinct instances with which to determine the consistency of her facilitation of tasks and establish a baseline for her practice. I observed Mae on a weekly basis and video recorded whole-class discussions and small-group work with case students. My data generation process within the pilot study involved creating event maps for observed sessions, transcribing select sessions into message units, and using a conversation mapping worksheet (Bloome et al., 2010) in order to examine the particular ways in which Mae facilitated modeling discussions with her students. Data was then analyzed using indexical discourse analysis (Lewis &

Manouchehri, in press; Silverstein, 1993; Scollon & Scollon, 2003) to determine how particular ideologies associated with real contexts emerged within modeling discussions and application problems.

Pilot study analysis revealed that Mae’s practice evoked mathematical modeling both within and outside of formalized tasks. For example, in one particular instance during the solving of an application problem involving business profit potential, Mae facilitated a brief and seemingly unimportant discussion on real-world business and engaged students in a brief iteration of the modeling process (Lewis & Manouchehri, in press). It was determined, based on these findings, that methodologically consecutive observations needed to be conducted across multiple units of instruction to better understand how mathematical modeling discussions were sustained across events and how they impacted her formalization of mathematical concepts.

While the conversation mapping tools used proved to be useful in unpacking the structure of discussions within events, it was determined that a more robust tool was

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needed to capture the particularities of Mae’s practice which were indicative of mathematical modeling. As such, an emergent coding scheme (Glaser & Strauss, 2017) was adopted in which codes were generated based off of particular components of Mae’s practice observed during discussions and sustained across events. Additionally, while indexical discourse analysis was determined to be a useful tool in terms of understanding how modeling emerged from application tasks and how real-life was indexed within a discussion, it was not able to support the development a typology of teacher practice.

Hence, additional holistic coding and analysis methods were deemed necessary in order to trace the progression of practice across events. In a way, the component analysis used within the larger dissertation study indexed particular aspects of her practice, but it was those components that were traced across the events.

This pilot study supported the refinement of the research questions, as well, specifically, by focusing on the construction of modeling practice and how teacher scaffolding enculturated learners into her view of modeling. Further, with modeling transcending the formalization of mathematical concepts, data from the pilot study offered initial insights into the development of modeling as a bridge between concept and vehicle, as it revealed the non-trivial view of mathematical modeling held by Mae.

Further, with this enculturation into the modeling process, specific focus on teacher scaffolding became necessary in the larger study.

From a methodological standpoint, I also opted to add additional cameras and audio markers, in particular, the Swivl, in order to focus specifically on teacher interactions and pick up on scaffolding discussions between the teacher and her students.

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In this way, pilot data generation and analysis supported the overarching study by streamlining the process of data collection, generation, and analysis. In the next chapter, I provide an overview of the data and significant findings of this study and offer a discussion on these findings.

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Chapter 4: Data Analysis and Results

The goal of this study was to examine, in the naturalistic stetting of a classroom, how one teacher, Mae, went about building mathematical modeling capacity among learners. This was investigated through careful examination of specific practices which seemed to be common in her interactions with students, as a way of documenting how she establishes the culture of the classroom. Moreover, this goal is not to view the classroom as an outsider, but rather develop an emic view of how the teacher’s epistemology interacts with the type of scaffolds on which she relies. I examined these issues by planting myself in her classroom across twenty-one class sessions. Across these observations, I examined how Mae developed an SMP, what components of that practice appeared to be visible in her interactions with students, and then traced across data to see how this practice was carried over and if additional components emerged.

The overall results of this study revealed that Mae vacillated constantly between building mathematical modeling capacities, or the skills needed for effective modeling practice, and linking these capacities to the targeted mathematical concepts of her curriculum. I propose that the teacher’s paradigm, or view of mathematical modeling, was neither modeling as content nor modeling as vehicle, but rather modeling as a bridge between these two epistemological perspectives. In the remaining sections, I will offer evidence through my data of how Mae enculturated students into her view of mathematical modeling and developed the practice of modeling as bridge. 113

In this chapter I begin by outlining Mae’s orientation towards mathematics, and how her orientation manifests across her short- and long-term instructional goals of anchoring students’ mathematical thinking (4.1). Next, I offer a brief description and accompanying event maps for the five modeling sessions which occurred across the observations (4.2). Then I offer an analysis of the specific teacher actions which occurred during these modeling sessions and identify what components of Mae’s practice were prevalent across each session (4.3). In section 4.4, I unpack each identified component of

Mae’s SMP and analyze its impact on developing mathematical modeling capacity. I then consider Mae’s view of modeling as bridge and exemplify how she uses mathematical modeling to advance modeling as content and, simultaneously, as a tool for advancing curricular knowledge (4.5). Finally, I discuss those scaffolding interactions observed by

Mae and contemplate how they promote the development of components of Mae’s SMP

(4.6).

4.1 Mae’s Short-Term Goals, Long-Term Goals, and Orientation Towards Mathematics

During the summer of 2017, Mae and I met frequently to discuss aspects of this study, in particular, her short- and long-term instructional goals pertaining to her teaching. One primary topic of discussion involved the frequency and selection of mathematical modeling tasks, as Mae had indicated that one of her main long-terms goals was to develop mathematical modeling capacity in her students.

When considering the frequency with which mathematical modeling emerged, roughly one-third of the observed sessions (10 out of 31) involved the direct facilitation,

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discussion, or extension of mathematical modeling tasks. Further, the facilitation of these modeling tasks did not occur in one instance, but was distributed across the entirety of the observations (Table 7; reflected in yellow). In comparing the facilitation of modeling tasks to those other instances of instruction, notice further that, across these events, 8 of the 31 were devoted towards summative assessment in some capacity, and the remaining

13 sessions were given over to formalizing mathematical concepts through direct instruction or problem-solving.

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Table 7. Distribution of Modeling Tasks

Date Topic Modeling 8/22/17 Problem Posing Yes 8/23/17 Problem Posing/Measurement/Triangle Trigonometry Yes 8/24/17 Measurement/Triangle Trigonometry Yes 8/28/17 Measurement/Triangle Trigonometry Yes 8/29/17 Triangle Trigonometry/Measurement Problems Yes 8/31/17 Trigonometry Puzzle Activity No 9/1/17 Unit Circle No 9/6/17 Test 1 Review No 9/7/17 Test 1 (Triangle Trigonometry) No 9/12/17 Law of Sines/Optimizing Distance Yes 9/14/17 Law of Sines No 9/15/17 Ambiguous Law of Sines/Use of Technology No 9/19/17 Ambiguous Law of Sines No 9/20/17 Heron’s Formula/Law of Cosines No 9/22/17 Test 2 Review No 9/25/17 Test 2 (Law of Sines/ Law of Cosines) No 9/27/17 Trigonometric Identities (Sine, Cosine and Tangent) No* 9/28/17 Trigonometric Identities (Sum and Difference) No* 10/3/17 Trigonometric Identities (Solving) No* 10/4/17 Radian and Degree Conversion No 10/6/17 Test 3 Review No 10/9/17 Test 3 Review No 10/12/17 Test 3 (Trigonometry Identities/Radian and Degree No Conversion) 10/16/17 Radians and Arc Length No 10/17/17 Radian Quiz No 10/19/17 Radians and Arc Length Practice No 10/20/17 Radians and Arc Length Practice No 10/24/17 Angular Speed/ Trig Whips Yes 10/25/17 Angular Speed/ Trig Whips/ Problem Generation Yes 10/27/17 Angular Speed/Sharing Individual Problems Yes Yellow: Reflects Deliberate Connection to Mathematical Modeling Blue: Reflects Absence of Real-World Application (according to teacher)

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In conversing with Mae about her willingness to implement mathematical modeling tasks, she identified two factors she considered while designing tasks, time and capability to bridge. Mae believed that, in order to facilitate modeling tasks in an authentic and meaningful way, students should be given time to explore their own unique solutions motivated by contextual variables which they deemed important. Further, she required that facilitating this task would still allow her to remain on pace in her content instruction. Mae also felt that these selected tasks should allow the opportunity to bridge learners’ solutions to those desired mathematical concepts. This capability to bridge further required Mae to have a certain level of comfort in both anticipating solutions, then having the mathematical insight into how to support their formalization. She further indicated that she needed to be able to identify at least one solution pathway which was directly linked to her curricular goals. It was only when these aspects of her practice were met that she would be comfortable in facilitating modeling tasks.

In order to support this assertion, I looked at those instances in which modeling did not occur and saw no evidence of bridging. Additionally, Mae would often explain to her students why modeling was not being facilitated. One such example where this occurred was during the unit on Trigonometric Identities. In this unit, Mae specifically voiced that she would not be using mathematical modeling because she didn’t recognize any connections between trigonometric identities and real-world problems. This is not to say that these connections didn’t exist, but rather that she lacked familiarity with their application (Table 8).

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Table 8. Evidence of Teacher Goal of Modeling

1 Teacher Okay, so, girls and guys, we’re going to start with something new today, and 2 normally we start with an activity, but I’ll be honest with you. I don’t have 3 an activity this chapter. This chapter is not like a real-life application of trig. 4 There’s not a lot of real-life stuff to do with it. But it’s still one of my favorite 5 chapters, which completely surprises me, because I really love everything 6 to be real-world and hands-on and things like that. And this is not that, 7 but I really like it because it’s like puzzles.

Reflected in the table above, Mae was explicit in noting that she has done mathematical modeling and real-world applications in the past (line 2), that students should not expect there to be real-world applications within this unit (line 4), and that she commonly advocates for real-world applications in her teaching (line 6).

Figure 11 offers a conceptual model for Mae’s decision space (Manouchehri, in progress) as she engaged in planning and facilitating lessons over the course of the year.

Based on observational data and discussions with Mae, the long-term goals expressed were that students develop modeling capacity as well as realize the value of mathematics in making sense of everyday life. Evidence that this reflects her long-term goals comes from her explicit mention of this during the course of planning and the fact that she started her school year by setting the tone with mathematical modeling. In supporting this advancement, coupled with the necessity to impart curricular mathematics, the question then became how she navigated her instruction to meet both the short-term needs mandated by the curriculum while still staying true to her long- term goals. 118

Depicted in this teacher decision space, short- and long-term goals are distinguished within each column. Within the category of short-term goals are included mastery of specific mathematical concepts and, in particular, those that were observed over the course of the data collection phase, such as trigonometry. In accomplishing these short-term goals, I have also listed those specific components indicative of modeling as a vehicle, as that framework aligns with developing curricular mathematics, such as quantifying, formalizing, and computation using specific mathematical procedures.

Short-Term Goals Long-Term Goals

Mastery of Specific Development of Mathematics Mathematizing through • Trigonometry • Modeling Capacity • Pythagorean Theorem • Asking questions • Special Right Triangles • Problem Posing • Radian/Degree • Use of Experiments Conversions • Collecting and Analyzing • Angular Velocity Data • Etc. • Convincing • Argumentation Modeling As Vehicle • Sense-Making • Formalization • Quantification Modeling As Content • Computation • Relevance of • Contextual Meta-Level mathematics in daily Components lives • Asking and answering Short-Term Modeling contextual questions Discussions • Making Sense of Real World

Sustained Modeling Discussions

Figure 11. Mae's Decision Space

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In looking at the long-term goals of the teacher, I identified the development of mathematizing and sense-making in mathematical ways. This came through the teacher’s continued and sustained use of modeling questions, provoking students to ask contextual and mathematical questions, provoking reflection and insight into the dynamics of real- world contexts and how mathematics might afford solutions to problems that arise, problem posing, collecting and analyzing data, convincing through argumentation, and general aspects of sense-making and logical reasoning. As short-term goals are primarily linked to the acquisition of particular mathematical skills, Mae’s long-term goals appear as integral components of her instruction regardless of a specific task or problem and appear frequently across the development of particular concepts. It is also important to note that some of these long-term goals were not exclusive to mathematical modeling encounters but also targeted because of their placement in a mathematics classroom.

Other long-term goals were exclusive to mathematical modeling tasks, such as developing the mathematization practices of students. In this way, the broad mathematical practices and the intent to engage in mathematical modeling are tightly linked, and thus cannot be divorced, as they are motivated by targeting long-term development to help students understand the applicability of mathematics in making sense of and solving real-life problems.

4.2 Outline of Modeling Sessions

In this section, I offer a brief overview of the five modeling sessions which occurred across ten days of instruction. For each session, I provide an event map with an

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accompanying description of that session highlighting the major instructional outcomes.

These event maps are organized into three columns which outline different aspects of the activity. The first column outlines the initiating action which progresses that section of the event. The second column lists the response action and, in particular, identifies who responded. Finally, the last column identifies key results or outcomes of that event.

4.2.1 The Answer is Task

The Answer is Task occurred during the first two days of the academic year. In this encounter Mae facilitated a problem-posing task in which students were asked to write a mathematical problem for which the answer was $73.13. This session is organized into two events, each of which is outlined in an event map with accompanying description. The first event within this session (Table 9) involves the teacher launching the Answer is Task and supporting students as they work through the problem. Event 2 reflects the sharing of student-generated problems and the orchestration of a debriefing discussion (Table 10).

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Table 9. Event Map of Answer is Task

Initiating Action Response Action Notes Teacher asks class for permission to not Students respond that this is Teacher establishes her go over syllabus and rather engage in okay practice outside of what she problem posing tasks that will better felt the students would reflect the nature of her course. expect.

Teacher initiated discussion Teacher indicates that her course is about Students get up and help to Teacher establishes a stance asking questions and asks students to get pass out paper on solving problems and paper for the task thinking about the real world in mathematics Teacher introduces the problem “the Students give a dollar value Teacher constructs the answer is” by asking students for dollar of 73 and cent value of 13 problem on the board “The and cent values, which she writes on the answer is $73.13” board. Teacher initiates a discussion about what Students generate contextual There is a LONG pause the students are thinking about the variables impacting the from the time the teacher Answer is Task answer being $73.13 initiates the question and the first student response Teacher synthesizes student contextual Teacher informs students to Criteria Outlined: factors and establishes explicit criteria for work on their own, then - start with a simple the task collaborate with other problem students to refine and - develop a more enhance the problem complex problem by adding details - work with others to refine problems and agree on a final problem Students begin to work and collaborate Teacher interjects with Teacher draws explicit on the problem acknowledgment of student attention to behaviors of strategies and vocalizes pausing, reflection, revision, these areas to the class. and asking questions Students continue to work on task Teacher calls attention to Teacher draws explicit development and refinement additional behaviors and attention to use of clarifies why she likes them technology to research contextual factors. Teacher openly asks group to consider Student responds that tax is Teacher praises this the variable tax in their problem irrelevant if bought on a tax- behavior, indicating that the free weekend student outsmarted her; uses as a means for advocating thinking about the real world and when it trumps math. Teacher closes the Answer is Task by Students compile documents Teacher advocates that she asking students to reflect on their process and staple together their wants her students to be and having students turn in problems work to turn it in reflective of the process and discussion that happened among the groups as they worked. 122

In this event, Mae initiated instruction by asking students for permission to operate outside of common expectations and not cover the content of the course syllabus in detail, but rather engage in an activity that better reflects her long-term goals. Mae then wrote on the board “The answer is…” and asked students to generate a dollar value between 60 and 100 and a cent value between 13 and 83. The students selected the values of 73 and 13 respectively. These values were added to the board, generating the problem

“The Answer is $73.14.” Mae then asked the students to consider this problem and, together, they generated a list of contextual factors which might influence this solution.

Mae then outlined a list of criteria for students as they engaged in working on this problem in that they first must start with a simple problem, develop a more complex problem by adding additional details, and then work with other students to refine their problems and select one to share during the discussion. After these criteria were outlined, the students began constructing their problems, and during this time Mae walked around the classroom monitoring their work, asking clarifying questions about their solutions, and encouraging them to add additional contextual details. During this monitoring, Mae highlighted exemplary behaviors which were observed to the class as a whole. Following a period of approximately 10 minutes, Mae asked students to wrap up their discussions and to reflect on and document the discussion process which occurred during their collaborative efforts.

In the next phase of this session, Mae facilitated a debriefing discussion and the sharing of student-constructed problems to the Answer is Task. A broad overview of the event is reflected across the event map in Table 10. 123

Table 10. Event Map of Answer is Task Debriefing Discussion

Initiating Action Response Action Notes Teacher asks students Students begin to Students were asked to share one to review their work review and discuss their problem and their construction from previous day. solutions, then move process; considerations Teacher informs into a social discussion students they will be sharing their solutions Teacher asks for Amazon Shipping “A Hershel Supply Dawson volunteer group to Problem shared Backpack (Nylon/tan) is $51.99 share first and a pencil-pouch is $9.29. If the tax is $3.90 and shipping is $7.95, what is the total cost of your items?” Teacher acknowledges Students explain how Teacher emphasizes research and shipping component they researched the tax thinking about the real world and asks how tax was and shipping values for determined those items Teacher asks for Tax-Free weekend and “On Monday, August 21, if you another group to share Percent of Value bought a cold steel pole-axe and Problems shared only had $41.26, how much money do you need? (The axe is $114.39)” “What is 5% of a $1452.60 item?” Teacher asks students Students respond that Teacher acknowledges differences to consider distinctions percent of value is in both shared problems between two problems more mathematically (mathematical and contextual) and oriented, and tax-free establishes marrying of real and weekend is more mathematical worlds as contextually oriented mathematical modeling Teacher asks for third Notebook and Gum “If you have $100 & you buy 8 group to share their Problem shared notebooks @ $3 each w/a tax of problem $138 and you buy a pack of gum for $149, how much do you have left? – The answer is $73.13” Teacher acknowledges Teacher claims that Contextual Factors Valorized: how students are sense-making is a - Cost of items thinking carefully useful tool in - Adjusting what was bought about contextual mathematics to yield solution factors and engaging in - Sense-making about the sense-making context of purchasing

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Table 10 continued Initiating Action Response Action Notes Teacher asks for Grocery Shopping “The man went to the store & another group to share Problem shared bought $100 worth of groceries. He spent $40 on meat, $20 on bread and $40 on fruit. He returned $35 worth of things and then rebought an item for $8.13. How much money does he have left?” Teacher valorizes the Teacher references how Teacher humanizes herself in choice to have a “man” she appreciates it when presence of her students as she rather than a woman her husband does the validates their solution going grocery shopping shopping Teacher calls on final Students indicate they “$67.94 before tax, assuming group to share hadn’t finished writing bought in Ohio w/average their problem, but can municipal tax for Ohio and $73.13 share what they have after taxes.” thus far Teacher validates Teacher clarifies the In context of discussion of choice of municipal tax difference between difference of municipal tax student as a factor in their state tax and municipal asks if this difference is a lot a lot solution and asks class tax and, in particular, if they know what how municipal tax rate municipal tax is impacts purchasing price Researcher asks student Student clarifies his Researcher asks “what do you to elaborate on what he view with a lot is 7 mean by a lot a lot?” and “why is means by A Lot A Lot cents, but a lot a lot is that important to you?” 10 cents Teacher validates his Teacher elaborates on Synthesis of a lot a lot discussion formalization, situating this term by re- using it in other contexts to further it as a technical emphasizing the humanize herself in presence of definition difference in municipal students taxes in other contexts Teacher wraps up Teacher transitions into Purpose of activity, according to discussion by the Measure the Height teacher, was to help students summarizing the intent task understand how to ask of the activity mathematical questions and the contextual details needed to go into that process

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In debriefing discussions of the Answer Is task, the teacher asked students to review their problems from the previous day and consider those contextual factors which they opted to include in their problem context and what motivated those decisions. Following a period of three minutes, Mae reconvened the class and asked each group to share their constructed problems, and, as each group shared, Mae deliberately valorized one particular consideration, either contextual or mathematical, and avoided any criticism or judgment.

When considering the first solution, Mae focused on the students’ use of technology to determine precise costs of items and cost of shipping. The second group shared two problems, one contextually focused on purchasing items on a tax free weekend, and the other a straightforward and decontextualized percent of value problem.

In valorizing these responses, Mae first recognized the importance in considering real- world factors of not needlessly overcomplicating the mathematics and, in the second problem, valorized its importance in the field of academic mathematics. She used these two contrasting views to establish a definition of mathematical modeling as the marrying of real-life and mathematics. In the third solution, Mae referenced the students’ process of considering what items might be necessary to yield a value close to the desired solution and further advocated for the sense- making that this requires. For the fourth group, Mae humanized herself by sharing her personal experiences with her husband going shopping as a way to authenticate the students’ choice of details within their problem, then used the context of municipal tax to discuss how she uses mathematics to make decisions about purchasing items out of state during her travels.

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In referencing the difference between the cost of each item depending on the municipal tax rate, one student questioned if this difference was significant by asking if the quantity was “a lot” or “a lot a lot.” In response to this question, the researcher provoked the student to elaborate on its meaning. In light of his response, Mae adopted this phrase and contextualized it in light of her own personal experiences as a way of further humanizing herself and validated how mathematics can support sense-making.

4.2.2 Measure the Height Task

The next mathematical modeling session occurred immediately following the

Answer is Task on the second day of school. Mae intended for this event to act as a pre- assessment for triangle trigonometry. In this task, students were asked to determine the height of three different objects in three different ways. The sequence of this event is reflected below (Table 11).

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Table 11. Event Map of Measure the Height Task

Initiating Action Response Action Notes Teacher introduces Teacher demonstrates how Teacher draws a diagram measure the height task to use an inclinometer to on the board of a triangle measure an angle to a point with referent angle of elevation and person inscribed in the triangle Teacher leads students Students commence Al and Am opt for outside working on measurement measuring the height of an task umbrella, the height of a tree, and the height of a light post Al and Am measure the Al and Am abandon use of Students consider the slant angle of elevation, distance angle for direct of the ground; how the to the umbrella from the measurement strategy for angle might be used to point of measurement, and the umbrella determine the height height of Al up to her eyes Al and Am transition to Al and Am discuss the Am uses Al’s height and estimating the height of the height of the tree and what height of umbrella for tree point to measure to and determining estimate verify estimate Al and Am notice that their Al and Am discuss what Measurement error estimate is off by almost could have caused their attributed to slant of the one foot measurement to be off umbrella surface and the uneven plane at the base of the tree Interviewer conducts brief Al and Am explain their Al and Am explain that interview with Al and Am process for determining they couldn’t remember height using direct how to use the angle of measurement and elevation to determine estimation height Teacher walks over Al and Am re-explain their towards the end of the process and indicate they interview and asks students can’t remember how to use what they are working on the angle to determine the height Teacher asks students to Am indicates they could consider the missing sides measure the hypotenuse by of triangles and how they laying on the ground using might determine those eyesight and Pythagorean sides theorem

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Table 11. continued Initiating Action Response Action Notes Al and Am develop a Teacher expresses approval Teacher listens to students’ strategy for determining of this strategy and tells mathematical ideas and the height using girls to do it expands on their strategy Pythagorean theorem Teacher takes students Teacher asks students to back into classroom write their measurements on the board Teacher facilitates a brief Class period ends and Teacher indicates they will wrap-up discussion on students pack up and leave continue this debriefing measuring the height as bell rings discussion the next day of class

This event commenced with Mae indicating that they were to begin with a unit on trigonometry and that this task serves as an introduction into these concepts. Mae then demonstrated the use of an inclinometer to find an angle of elevation and passed out inclinometers and tape measures to each group of students. While Mae referenced the inclinometer in the framing of the task, she in no way mandated that this tool or trigonometry needed to be used as a strategy, but rather offered it as an option. After this demonstration, Mae formalized the constraints for this task by writing the task on the board. She then took the students outside and asked that they work in groups to measure the height of three different objects in three different ways.

During this period of student work, Mae walked around monitoring each group and engaging with them to support their measurement strategies. Within these discussions, I observed Mae reflexively listening and supporting students on expanding on the strategies they were considering. I opted to focus on a case group of students,

Alyssa and Amber (both pseudonyms) to see how they enacted this modeling task. After

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a brief period of playing with the inclinometer and trying to replicate the teacher’s use,

Alyssa and Amber chose to determine the height of a table umbrella using direct measurement, the height of a tree using estimation and then verifying their estimate, and the height of a light post using the Pythagorean Theorem. In between measuring the heights of the tree and light post, Mae approached this group and asked them to re-voice their strategies and inform her of what they were currently thinking about. These students informed her that they weren’t aware of how the angle measurement could support their determination the height, but that they were also thinking about how to use their knowledge of triangles and the Pythagorean Theorem for the light post. Mae supported the students’ ideas and helped them to organize their diagram to see where the triangles existed which would allow them to use this concept, which the students then enacted.

Following this phase, Mae brought the students back into the classroom and facilitated a debriefing discussion on the student’s selected strategies by asking each group to share one of the methods they used for determining a height. Similar to the

Answer is Task, Mae then validated one particular aspect of each group’s mathematical or contextual considerations. During the following two days of class, Mae asked students to put their measurements and strategies on the whiteboard and gave each group a chance to share their entire measurement processes. She then used these solutions to bridge into a discussion of triangle trigonometry, in which this concept was presented as another method for determining the height of an object. She then asked students to apply these new mathematical ideas to their determined measurements as a way of verifying the

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heights and to determine if there were any missing pieces on their diagrams. She also modeled this process by using one group’s diagram and collected measurements for a basketball hoop. In this way, Mae created a bridge between this mathematical modeling task and her curricular goals.

4.2.3 Shortest Distance to the Cafeteria Task

The Shortest Distance to the Lunch Room Task (Table 12) occurred midway through the instructional unit on trigonometry on the tenth day of school. The intent of

푠푖푛(퐴) sin⁡(퐵) sin⁡(퐶) this task was to introduce students to the Law of Sines, = = , with A, 푎 푏 푐

B, and C the angle measurements of a triangle, and a, b and c being the lengths of the respective sides. In this task, Mae opened the discussion by asking whether the upper, middle, or lower school on their campus had the shortest distance to the cafeteria. To support the problem-solving process, Mae handed out an aerial map of the campus

(Figure 12) with guiding questions and had students identify routes from each building to the doors of the cafeteria. Mae then passed out tape measures to each group and indicated that they were to go outside and determine which building was indeed closest to the cafeteria.

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Figure 12. Aerial Map of Field Site

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Table 12. Event Map of Shortest Distance to Cafeteria Task

Initiating Action Response Action Notes Mae tells students that they Mae asks students which Context is grounded in will be starting a new unit school they think (upper, learners’ experiences with on trigonometric laws and middle, or lower) has the campus and daily lunch doing another outside shortest distance to the routes activity lunch room Mae passes out activity Mae indicates that they will Task: sheet with aerial map of be exploring triangles and Use aerial map to find field site on one side and measuring distances distance from corner of task instructions on the between each school and upper, middle, and lower other the lunch room to verify schools to cafeteria. assumptions Students work in groups of Case students observed As learners adjusted the three to measure distances opted to measure from the parameters of the task, outlined on the instruction door to the lunchroom Mae responded positively sheet rather than the corner as it as she monitored their is more accurate to use the work and allowed them to path traveled make these adjustments to the task Mae monitors students as Students respond to Mae Active discussion they collect measurements with their ideas and regarding what routes supporting their tool use measurements, and adjust should be determined, and identifying distances practices based on Mae’s specifically whether on map. Mae also asks feedback. For example, measurements should be students to explain their placement of tape measure along the sidewalks or process during this or measuring across plants. through the grass monitoring Mae brings students back Students provide distance Mae averages the into class and asks them to values and Mae documents responses based on student share their measurements measurements on the suggestion to obtain a best for each route between Smartboard, where she has guess for the distance. As buildings a large picture of the aerial these discussions happen, map up on the board Mae asks about what contextual factors caused this distance. Mae passes out protractors Students measure angles Students maintain working to students and asks them between routes and check in groups of three. Mae to identify angle measurements with their also documents measurements between group members measurements on the measured distances on Smartboard as groups their papers determine them

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Table 12. continued Initiating Action Response Action Notes Mae introduces Law of Bell rings dismissing Mae had intended to Sines as a way to students, so Mae asks facilitate this portion of the determine measurements students to generate their lesson in class, but bell on the aerial maps which own contextual questions disrupted the lesson and were not originally and use Law of Sines to she opted to assign it for collected support their solutions as a homework homework assignment Mae summarized the Mae indicated that they Day 2 of Shortest Distance activity of the previous would be continuing the to Lunch Room day activity through the second day and would begin with sharing out their questions Mae asks students to share Student responses their contextual questions considered distance which they generated the between upper and middle night previously school as both share a common sidewalk leading towards lunch room Through the share-out, it Mae takes students back was revealed that student outside to measure this measurements differed for distance to check which distance between buildings was more accurate Students measured the Mae encouraged students distance between middle to be precise in their and upper school with measurements and record yardsticks their findings Mae takes students back Students attributed error to Mae attributed into class and facilitates a different measurements applicability of Law of discussion on comparing between buildings given a Sines and trigonometry as actual measurement to route a way to validate calculated measurement measurements where and asking for what could distances are known and to have caused error find new measurements given pre-existing information on the diagram Mae transitions to Students spend the rest of Students were assigned calculator exploration the day completing this Law of Sines practice activity on Discovering activity problems for homework Law of Sines

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While students were outside measuring distances, Mae supported them in both their measurement techniques and through considering different authentic routes which students would take. During my observation, I followed a group of case students who opted to measure the distance from the upper school to the cafeteria. During their measurement process, they realized that in order to make their measurements more authentic, they needed to measure from door to door rather than from the corner of the building, which they had elected to measure. Following this re-measuring, Mae took the class back inside and asked each group for their determined measurements and constructed a diagram on the board of the distances between buildings using triangles.

Finally, Mae referred students back to their aerial maps, handed out protractors, and asked students to determine the angles between each building. Mae facilitated a brief lecture on the Law of Sines as a way of generating new contextual information given the distances and angle measurements from the activity. Following this lecture, Mae asked the students to generate their own contextual questions about distances or angle measurements within that diagram and to find those distances using the Law of Sines or in other mathematical ways.

At the start of the next class period, Mae summarized the activity, asked students to share their determined measurements, and then took them back outside to validate their solutions by examining the errors between calculated and actual measurements. In this way, she positioned mathematics as a useful tool in not only validating measurements, but also in determining missing pieces of a system given limited contextual information.

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Mae then transitioned into a technology-based investigation of the Law of Sines and referenced this throughout the discussion of this concept.

4.2.4 Trig Whips Task

The Trig Whip Modeling Task (Table 13) occurred following a unit of instruction on degree-radian conversions and arc length and was designed to introduce students to the concept of linear and angular speed. In this session, Mae began by introducing the concept of angular speed as a joint trigonometry and physics idea. She then asked students what they could tell her about this concept and what real-world contexts involved angular speed. Students responded by arguing that anything round that moves, such as a gear, has angular speed. Mae built on this notion by telling students that they would be engaging in a “Trig Whip” task, in which they would be determining the speed of each person in a people chain made by linking arms and moving around in a circle, then passed out the corresponding activity sheet (Figures 13 & 14).

Table 13. Event Map for Trig Whip Task

Initiating Action Response Action Notes Teacher introduces Teacher asks students Teacher starts by asking concept of angular speed what they can tell her students what they know as a "trig and physics idea" about angular speed and (ideas and experiences) what they know that has angular speed Students respond that Teacher asks students Aligned to variable gears and anything that what angular speed looks identification in modeling moves have angular speed at, in particular, and they process identify distance and time as factors

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Table 13. continued Initiating Action Response Action Notes Teacher tells students they Teacher asks students Teacher demonstrates the will be doing a “trig whip” what will happen if more task with another student activity and explains students are added to the parameters of task as chain as they move around linking arms with one person turning in a circle and those on the outside moving around as the center person turns Teacher passes out Teacher takes class into Students were encouraged handout and puts students the hall and tells students to use their cell phones as in groups of 5 (one center to start working on the tools for keeping time and 4 runners) task together Students collect data on Teacher supports Students observed adjusting distance from each student measurement process by strategies by having the to center student and time helping determine inner person move slower so to complete revolutions measurement distances that outer students can keep (indicated by handout) and logistics of revolutions up and stay in line

Students document measurements on their handouts Teacher transitions Period concludes with the Data collection went almost students back into bell ringing to the bell, so the teacher classroom after they have didn’t engage in follow-up finished their discussion on first day measurements Teacher initiates Teacher re-establishes the Variables identified: debriefing discussion on prevalent variables that Number of people, distance trig whips data collection impact the system around circle, distance and says intent of task was between students, rate of to “see how the math center person’s turning works to help better understand the context” Teacher writes “Angular Teacher indicates they are Connecting these topics to Speed” on board following a trajectory speed, distance, and time moving from within activity circumference to arc length to angular speed

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Table 13. continued Initiating Action Response Action Notes Teacher discusses how to Teacher transitions Teacher monitors students compute angular speed students to back page of as they work through task. I using collected data and handout and asks them to opted to focus on a case suggests that student compute angular speed for group of students, and the convert inches to feet to all group members and following event map is better conceptualize then respond to the final indicative of their work distance traveled and question concerning 20th speed person’s speed Case group of students Case student Krista Case students generate new determine angular speed of postulates that she modeling context based on each member of their believes Olympian Husain determined data group and determine the Bolt could sustain that speed of the 20th person to speed be 377ft/sec based on their angular speed Case student Beau doubts Beau works through Beau exhibits the working that he could sustain that conversion and determines mathematically phase of speed and suggests speed to be 257 miles/hour modeling cycle; Krista converting that speed to exhibits research. miles per hour Krista researches Husain Bolt’s top speed to be 27.8 miles/hour Beau argues that they Krista argues that, if the Students engage in would need a race car for inner person turned validation and refinement of the 20th person because of slower, then Husain Bolt solution. calculated speed could act as the 20th Students generate multiple person, but also states that contextual questions based she doesn’t want to think on their findings about doing the *Interesting to note that calculations Krista and Beau were willing to generate contextual questions, but, without provoking, didn’t re-engage in the cycle Teacher brings class back Students share their During discussion teacher together and asks students solutions to determine the brings up differences in to discuss their findings speed of group members. speeds attributed to the Krista and Beau were variables distance and speed encouraged to share their of rotation findings about the 20th person’s top speed

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Table 13. continued Initiating Action Response Action Notes Teacher asks students to Teacher puts formulas for Derivation further highlights consider the relationship arc length, linear speed, relationship between between angular and linear and angular speed on the variables, and equations speed board and asks class to with the contextual factor of derive angular velocity speed using those formulas Mae assigns computation Bell rings and students problems for homework exit the class using these formulas

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Figure 13. Trig Whips Task Handout

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Figure 14. Back of Trig Whips Task Handout

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Mae facilitated this task by first asking one student to stand in the center of a circle and linked arms with that student. As this student pivoted and she walked, she asked the class what would happen if they added more people. The students responded that those who were on the outer edge would be moving faster than those in the middle and would have to run to keep up. Mae then indicated that they would be investigating this event further, in particular, they would be determining how fast each person in a chain would be moving and predict the outer speed given a much larger chain of people.

The teacher put students into groups of five with one student designated as the center and the remaining four students assigned to run and passed out a handout to accompany their work which included a place for determined measurements and speeds.

Next, Mae took all students into the common area and asked them to complete this activity and determine the amount of time per revolution and the distance of each concentric circle. Students attempted a series of trial runs and then collected data on the time elapsed to complete three revolutions. During this time, Mae supported the students in their data collection process by helping them identify what needed to be measured and from what point to determine these measurements from. Students were encouraged to use their cell phones in this task for timing and documenting the process of their work.

Once the data collection was completed, students were asked to determine the angular speed of the outer and inner-most person and then respond to a series of extension questions on the back of the handout, in particular, predicting how fast a person on the outer edge would have to move if the chain were 20 people long. These predictions were then shared with the whole group, and the teacher then transitioned to

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deriving the formula for angular velocity and demonstrated how they could use this formula to verify their predictions.

During this task, students were asked to generate and respond to their own contextual problems. I observed a group of case students who determined that the 20th person in a chain would have to move at a pace of roughly 337ft/sec. In contemplating this speed, the case students considered whether or not an Olympian, Husain Bolt, could act as the 20th person in their chain. Their process of determining the answer to this question involved converting feet per second to miles per hour and determined that this individual would have to run approximately 257mph, realizing that, even at his top speed of 29 miles per hour, he would be nowhere near fast enough to keep up and that it would take a race car to do so. Realizing this, another case student in this group postulated that, if the innermost person turned at a slower rate, then this athlete could keep up and asked at what speed must the innermost person turn to satisfy this condition. Following this problem generation and response, the teacher facilitated another discussion in which students shared their solutions and constructed problems as a way of highlighting the applicability of these formulas in responding to their ideas, thus further engaging in the modeling process. With this task, learners were not only enculturated into the process of solving modeling tasks, but also into refining models and generating and responding to their own questions.

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4.3 Analysis of Mae’s Actions During Modeling Sessions

In this section, I will offer an analysis of each session with respect to Mae’s scaffolding interactions. I identify those components of Mae’s SMP that appeared to be deliberate across each session. I will then present a detailed description of each of these components and highlight how they impact the development of mathematical modeling capacity.

4.3.1 Analysis of The Answer is Task

The holistic coding (Table 14) and frequency distribution (Figure 15) of the

Answer Is task revealed that those components which were most prevalent across teacher behaviors were: Experiential Tool Use (n=14), Thinking about the Real World (n=11),

Teacher Humanizing (n=10), Sense-Making and Reflection (n=10), and Establishing

Variables or Constraints (n=9). Mid-range components were: Asking Contextual

Questions (n=6), Sharing Solutions (n=5), Researching (n=5), Conceptual or

Mathematical Tool Use (n=5), Asking Mathematical Questions (n=4), Physical Tool Use

(n=3), Elaborating Student Mathematics (n=3), Enculturation into Mathematics (n=3),

Communicating Ideas (n=3), Thinking about Mathematics (n=2), Collaboration (n=2),

Activating Metacognition (n=1), and Establishing Parameter Values (n=1). The components absent from this session were: Formalizing Curricular Mathematics,

Validating and Refining Models, and Assumption-Making.

This distribution indicates that the majority of teacher interactions involved inviting learners to consider how to incorporate their individual experiences and ideas

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into the context of developing mathematical questions. The teacher spent a considerable amount of time across this session in valorizing those lived experiences, both her own and her students’, in the context of purchasing. Across this event, Mae humanized herself and discussed intimate aspects of her life, such as her family and family travels, as a way of inviting students to bring in their own contextual experiences into their problems they explored. Rather than focusing on developing curricular mathematics within this task, Mae elaborated on the mathematics that her students considered. The discussion that culminated were concerned primarily with how students make sense of how and why they opted to choose certain contextual factors within their problems.

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Table 14. The Answer Is Macro Analysis

Variables/Constraints

Making

Physical)

Making/Reflection

ratingStudent Mathematics

Tool Use (Conceptual/Mathematics) Use Tool (Experiential) Use Tool CommunicatingIdeas Parameterization Refinementof Model Assumption ThinkingtheWorld Real about Validationof Model Collaborating ProblemGeneration Researching Establishing Sense Mathematical Asking Questions Contextual Asking Questions Thinkingabout Mathematics SharingSolutions Humanizing EnculturatingMathematics into ActivatingMetacognition FormalizingMathematics Curricular Elabo Event ( Use Too Framing of Expectations for the Course X x x X

146 Establishing the Value for “The Answer x x x x x is Task”

Establishing Contextual Factors for “The x x x x x x x Answer is” Task Establishing Operational Constraints for x x x x x x x Answer is Task Launch of Task and Student Work x x x x x x x x x x x x

Teacher Valorization of Student x x x x x x x x x x Considerations and Strategies Backpack Problem Discussion x x x x x x X x x

Tax-Free Weekend and Percent of Value x x x x x X x x Problems Discussion Notebook and Gum Problem Discussion x x x x x X x

Grocery Shopping Problem Discussion x x X x

Municipal Tax Problem Discussion x X x x x x X x

A lot A lot Discussion-asked by x X x x x x researcher Teacher Synthesis of A Lot a Lot x x x x x x

Final Wrap-up Discussion x x x x x x x x x

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Figure 15. The Answer Is Component Distribution

4.3.2 Analysis of Measure the Height

In analyzing the Measure the Height Task and how Mae formalized triangle trigonometry across subsequent lessons, the components which appeared to be most prevalent are reflected across Table 15 and its accompanying frequency distribution in

Figure 16. In this event, the components of Sense-Making (n=7), Thinking about

Mathematics (n=7), Conceptual Tool Use (n=6), Thinking about the Real World (n=5),

Asking Mathematical Questions (n=5), and Elaborating Student Mathematics (n=5) occurred most frequently. Next, this analysis revealed that Establishing Variables (n=4),

Enculturation into Mathematics (n=4), Establishing Parameter Values (n=4), Physical

Tool Use (n=3), Communicating Ideas (n=3), Collaborating (n=3), and Formalizing

Curricular Mathematics (n=3) appeared at the mid-range level. The least frequently occurring components were Experiential Tool Use (n=2), Teacher Humanizing (n=2),

Asking Contextual Questions (n=2), Sharing Solutions (n=2), Validating Constructed

Models (n=2), and Assumption-Making (n=1). The components absent from this event were Activating Metacognition, Refining Constructed Models, Problem Generation, and

Researching.

This breakdown reveals that, across this session, Mae encouraged contextual sense-making and connecting student-known mathematics to these ideas to a higher degree. This process involved not just establishing which variables impact the system, but also linking them to mathematical ideas. This is in contrast to the previous modeling session in which the goal was to generate contextual problems and make sense of the details needed. During this session, I also observed Mae elaborating on student

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mathematics by considering their exhibited strategies and then linking these mathematical ideas to her own curricular goals. This did not occur during the initial phases of the modeling task and was not imposed on students by devaluing their own strategies, but rather after individual solution strategies were valorized as viable options for determining the height. Further Mae deliberately did not impose the curricular process as being THE way to determine the height, but rather another method equally as viable as those exhibited by her students. This bridging between the facilitation of a modeling as content task and linking it to the curricular objectives came about by drawing on student solutions and marrying them with additional strategies, consistent with how elaboration in the ethnomodeling process is described.

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Table 15. Measure the Height Task Analysis

tics

Curricular Mathematics Curricular

Making

ofModel

Making/Reflection

Tool Use (Conceptual/Mathematics) Use Tool (Experiential) Use Tool CommunicatingIdeas Parameterization Refinement Assumption ThinkingtheWorld Real about Validating Collaborating ProblemGeneration Researching Variables/Constraints Establishing Sense Mathematical Asking Questions Contextual Asking Questions Thinkingabout Mathematics SharingSolutions Humanizing EnculturatingMathematics into ActivatingMetacognition Formalizing ElaboratingStudent Mathema Event (Physical) Use Too Introduction of Measure the x x x x x x

150 Height Task

Teacher Monitoring of Student x x x x x X x x x x x Strategies Teacher Scaffolding of Alyssa x x x x x x x x x x x x x and Amber Measure the Height Wrap-Up x x x x x x x X x x x Discussion Student Sharing of Each Strategy x x x x x x x x X x x x Formalizing Sine, Cosine, and x x x x x x Tangent Using Sine, Cosine, and Tangent x x x x x x x to Find the Missing Sides of a Basketball Hoop (drawn from student data)

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Figure 16. Measure the Height Component Frequency

4.3.3 Analysis of Shortest Distance to the Cafeteria

Within the Shortest Distance to the Cafeteria Task, those components of Mae’s practice which were most prevalent were: Conceptual Tool Use (n=12), Sense-Making

(n=10), Thinking about Mathematics (n=10), Elaborating Student Mathematics (n=10),

Physical Tool Use (n=10), and Enculturation into Mathematics (n=9). The mid-range components exhibited by Mae were: Formalizing Curricular Mathematics (n=7),

Thinking about the Real World (n=6), Asking Mathematical Questions (n=6),

Establishing Variables and Constraints (n=6), Teacher Humanizing (n=5),

Communicating Ideas (n=4), Experiential Tool Use (n=4), Sharing Solutions (n=4),

Validating Models (n=4), and Refining Models (n=4). The components which appeared least frequently within this session were: Asking Contextual Questions (n=3),

Assumption- Making (n=3), Collaborating (n=2), Problem Generation (n=2), and

Activating Metacognition (n=1). The components of Mae’s practice which did not appear within this event were: Establishing Parameter Values and Researching.

Across this task, I observed that formalizing curricular mathematics was far more prevalent than in previous encounters. This formalizing was still motivated by elaborating on student mathematics in initial phases of the lesson, but contrary to the distinct phases of elaboration and formalization, here, it was instead threaded throughout the entire modeling task. In fact, the teacher initiated this task by stating that they would be using the Law of Sines to help them determine which school has the shortest distance to the cafeteria. Students were still encouraged to explore their own solution strategies,

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but rather than each solution being deemed correct, they instead were compared against each other to derive a “best guess” for each distance.

Considering student measurements, this analysis revealed far more attention to both the use of conceptual tools, such as mathematics coupled with physical tools such as a tape measure or yardstick. Experiential tools came into the discussion but primarily during the measurement portion of the task, in particular, when determining the specific route each school takes to the cafeteria. This task also afforded the first instances of refinement of models, while, in previous tasks, each group’s solution was taken as valid.

Here, the teacher was visibly active in supporting student refinement of measurements and even took the class back outside after the conclusion of data collection to verify and re-measure distances which they had not acquired properly previously.

Up to this point modeling encounters had previously occurred at the beginning of the unit and prior to any formalization of curricular topics. This particular task different as it occurred in the middle of the unit. This sheds light as to why there was a higher degree of formalization, conceptual tool use, thinking about mathematics, and enculturation into academic mathematical processes (such as the Law of Sines). Other aspects of modeling, such as assumption-making, establishing parameter values, and establishing variables, were less frequent across the session. They did occur but did so mainly in the early phases of the task.

153

Table 16. Shortest Distance to Cafeteria Analysis

Making

ofModel

Making/Reflection -

Determining the Shortest Distance to

Tool Use (Conceptual/Mathematics) Use Tool (Experiential) Use Tool CommunicatingIdeas Parameterization Refinement Assumption ThinkingtheWorld Real about Validationof Model Collaborating ProblemGeneration Researching Variables/Constraints Establishing Sense Mathematical Asking Questions Contextual Asking Questions Thinkingabout Mathematic SharingSolutions Humanizing EnculturatingMathematics into ActivatingMetacognition FormalizingMathematics Curricular ElaboratingStudent Mathematics the Cafeteria (Physical) Use Too Shortest Distance to Lunch x x x x x x x Room Pre-Discussion

154 Teacher Monitoring of Shortest x x x x x x x x x x Distance

Shortest Distance To Lunch x x x x x x x x x Room Debrief Discussion Connecting Measure the Shortest x x x x x x x Distance to Triangle Concepts Students Determine Angle x x x x x x x x Measurements on Aerial Map (Protractor) Comparing Measurement x x x x x x x x x x x Discussion Generating Questions from Map x x x x x x x x x x x x x Summary of Shortest Distance x x x x Task Sharing of Strategies for x x x x x x x x Determining the Missing Sides of Triangles

Table 16. continued

tics

Making

ofModel

(Conceptual/Mathematics)

Making/Reflection - Determining the Shortest Distance to

the Cafeteria

Tool Use Use Tool (Experiential) Use Tool CommunicatingIdeas Parameterization Refinement Assumption ThinkingtheWorld Real about Validationof Model Collaborating ProblemGeneration Researching Variables/Constraints Establishing Sense Mathematical Asking Questions Contextual Asking Questions Thinkingabout Mathematics SharingSolutions Humanizing EnculturatingMathematics into ActivatingMetacognition FormalizingMathematics Curricular ElaboratingStudent Mathema (Physical) Use Too Validation of Measurements x x x x x x x x x x x (Outside) Comparing Collected & x x x x x x x x 155 Calculated Values x x x x x x X x x X

“Best Guess” Discussion Using x Averages Teacher Discusses Law of Sines x x X x x Calculator Activity Students Work Through x x x x x X x x Calculator Lesson on Exploring the Law of Sines

156

Figure 17. Shortest Distance to Cafeteria Frequency Distribution

4.3.4 Analysis of Trig Whips

Across the Trig Whip session, the components of Mae’s practice which most frequently emerged were: Conceptual Tool Use (n=8), Thinking About Mathematics

(n=8), Sense-Making and Reflection (n=7), Enculturation into Mathematics (n=7),

Elaborating Student Mathematics (n=6), Formalizing Curricular Mathematics (n=6),

Thinking about the Real World (n=6), and Asking Mathematical Questions (n=6). The components which appeared at a mid-range were: Experiential Tool Use (n=5), Asking

Contextual Questions (n=5), Establishing Variables (n=4), Teacher Humanizing (n=4),

Communicating Ideas (n=4), Collaboration (n=4), and Establishing Parameter Values

(n=4). The components which occurred least frequently were: Refining Models (n=3),

Physical Tool Use (n=2), Sharing Solutions (n=2), Validation of Models (n=2),

Assumption-Making (n=2), Researching (n=2), and Problem Generation (n=1). The sole non-emergent component in this task was Activating Metacognition.

This analysis reveals that, regarding the facilitation of the Trig Whip task, a higher degree of focus on formalizing curricular mathematics was observed throughout the event. Again, this is not surprising as across the first two months of school, the students had covered a wider array of mathematical concepts and had more of those experiences to draw from in facilitating the solving of this task. Additionally, a large portion of this event was devoted to formalizing a model for angular speed from known parameter values. There was a higher degree of structure imposed on this task regarding the collection and analysis of real-life data; however, Mae allowed space for students to generate and respond to their own contextual problems within this system. There was

157

also a higher degree of model refinement as students began considering the linear speed of a sole object and formalized a new model which could determine angular speed.

Evidence supporting the claim that Mae still actively invited learner experiences and contextual interpretations was observed in the relatively high frequency of humanizing, experiential tool use, and thinking about the real world.

158

Table 17. Trig Whips Analysis

Model

Making

ofModel

Metacognition

Making/Reflection

ConceptualUse Tool l Use Too Experiential CommunicatingIdeas Parameterization Refinement Assumption ThinkingtheWorld Real about Validationof Collaborating ProblemGeneration Researching Variables Establishing Sense Mathematical Asking Questions Contextual Asking Questions Thinkingabout Mathematics SharingSolutions Humanizing EnculturatingMathematics into Activating FormalizingMathematics ElaboratingStudent Mathematics Event Use PhysicalTool Teacher Introduces Concept of x x x x x x x Angular Speed and Velocity Introduction of Trig Whip x x x x x x x x x x

159 Activity

Teacher Monitoring/Scaffolding x x x x x x x x of Student Work on Trig Whips Trig Whip Debrief Discussion x x x x x x x x Introduction of Angular Speed x x x x x x Formulas Applying Angular speed to Trig x x x x x x x x Whip Discussion Derivation of Angular Speed x x x x x x x x from Linear Speed and Arc Length Teacher Monitoring of x x x x x x x Computing angular speed of Trig Whips Teacher Monitoring of Trig X X x x x x x x x x x x x x x x x x Whip Extension Angular Speed and Extension x x x x x x x X x x x Final Discussion Practicing Angular Speed x x x x x x x Problems

160

Figure 18. Trig Whips Task Component Frequency

4.3.5 Analysis of Mae’s SMP Components Across Sessions

Figure 19 outlines a typology of teacher interactions across all modeling sessions based on observed component frequency providing insight into Mae’s view of SMPs.

The predominant component across all sessions involved the provoking of sense-making and reflection within these modeling tasks in both contextual and mathematical ways.

Evidence to support this claim comes from the large number of occasions of encouraging student to think about the real world and mathematics. Reconciling lived contextual experiences with that of mathematics involves the use of conceptual tools, such as mathematical processes or concepts, which explains the high frequency of use of conceptual tools. This reconciliation in Mae’s practice comes from establishing contextual variables through inviting learners’ experiences and teacher humanization of contextual experiences, elaborating on those experiences mathematically, and connecting students’ mathematical ideas to curricular objectives. While appearing less frequently, the components of activation of metacognition, problem generation, establishing parameter values, assumption-making, researching, collaborating, sharing solutions, and communicating ideas inform this process throughout by offering a mechanism with which to engage in reconciling real-life with mathematics. Looking across this data,

Mae’s view of mathematical modeling lies in this reconciling of real-life ideas with mathematics, exhibited by the components in the upper half of the typology listed below.

In the next section, I offer a detailed description and accompanying microanalysis of the emergence of each component of Mae’s SMP and describe how Mae uses these components to inform her instruction of mathematical modeling as a bridge.

161

162

Figure 19. Typology of Teacher Intervention by Component Frequency Across Modeling Sessions

4.4 Elaboration on the Components of Mae’s Sociomodeling Practice

Across observations, I identified 23 key components of Mae’s practice which appeared to be deliberate. These components were: (1) provoking sense-making and reflection; (2) conceptual tool use, such as mathematics; (3) thinking about the real world; (4) thinking about mathematics; (5) experiential tool use; (6) elaborating student mathematics; (7) enculturating students into academic mathematics; (8) establishing variables or constraints; (9) asking mathematical questions; (10) teacher humanizing as a co-participant; (11) physical tool use; (12) formalizing curricular mathematics; (13) asking contextual questions; (14) communicating ideas; (15) sharing solutions; (16) collaboration; (17) validating models; (18) refining models; (19) research; (20) assumption-making; (21) establishing parameter values; (22) problem generation; and

(23) activating metacognition. These components were stressed across each modeling session. A constant comparative effort was undertaken to trace back how these components were manifested along with their impact on the development of modeling capacity in learners. With Mae having started the year with the goal of developing mathematical modeling and claiming those beliefs outlined above, I examine each component of her practice, in turn, and interpret her actions in terms of these articulated beliefs and goals.

4.4.1 Sense-Making

The component of provoking sense-making and reflection was observed to be most frequent across all modeling encounters. Data analysis revealed that Mae’s view of

163

promoting personal sense- making and reflection was linked to considering actively contextual and mathematical factors within a given problem context, making sense of those factors, and incorporate them into the problem-solving process. One example in which this view of sense- making emerged was in Mae’s first encounter with her students during the Answer is Task. Table 18 offers a microanalysis of the opening discussion of this event in which Mae is actively provoking students to engage in sense- making in order to establish the contextual constraints for this problem.

164

Table 18. Sense-Making

Making

ofModel

and Experiences Ideas

Making/Reflection

Message Unit Individual

Speaker ProblemGeneration CommunicatingIdeas Parameterization Refinement WorkingMathematically Assumption ThinkingtheWorld Real about Validationof Model Collaborating Use Technology Researching Variables Establishing Sense Mathematical Asking Questions Contextual Asking Questions Thinkingabout Mathematics SharingSolutions Humanizing EnculturatingMathematics into ActivatingMetacognition Formalizing/Quantifying Line 52 T so the answer is seventy- x x three dollars and thirteen cents. x 165 53 What do you think 54 Jo || What do we think x about the answer being $73.13 55 T yeah what do you think x 56 Bi Are you buying x X something 57 Jo That's a good answer x x X 58 T (smiles) so first question x x x that came up is 59 are we buying something x x x 60 what else is in your x brains right now 61 Ay If we did buy something x x x what did we buy 62 T What else is in your x brains 63 Am How many did you buy x x x 64 Bi Did we get change? x x x 65 T Did we get change x x x 66 How many did we buy x x x

Table 18. Continued

Making

ofModel

and Experiences Ideas

Metacognition

Making/Reflection

Message Unit Individual

166 69 T What am I making you do x x

on the first day that other teachers don't? 70 Ca Making us think. x 71 Bi Ask questions. x x 72 T Yeah, x 73 you're thinking, x X 74 you're asking questions, x x X 75 and you guys don't even x X realize it, 76 but when I say what's in x x your brains, 77 you have to make sure x x you understand what's in your brains 78 before you ask your x x question, 79 and it’s like a big part of x x x math.

Mae initiated this discussion by asking students the question what do you think

(line 53), and, as students offered their own personal insight into the problem, she further provoked them to deepen their considerations by asking what else is in your brains (lines

60, 62, and 67). In lines 76-77, Mae clarified that when I say what’s in your brains, you have to make sure you understand what’s in your brain before you ask your question, she advocates for the personal sense-making and reflection which students undergo in the modeling process as a means to support their communication of ideas in this modeling context and further emphasized its importance by relating the importance of that personal sense-making to the domain of mathematics itself, in particular, by claiming it to be a big part of math (line 79). The fact that Mae couched this component as relevant to both the task at hand along with mathematics, in general, offers further explanation as to its prevalence across events, as she views it as fundamental to her long term goal of developing the mathematization skills in her students, and, as such, it is not surprising that it appears so frequently.

4.4.2 Conceptual Tool Use, Thinking about Mathematics, and Elaborating Students’ Mathematics

The use of conceptual tools, such as mathematics, appeared as an integral component of Mae’s SMP. This component was exemplified as Mae facilitated discussions amongst her students, in particular, during her scaffolding encounters.

Within these encounters, Mae elicited students’ mathematical ideas and elaborates on those in order to advance their problem-solving processes. Table 19 reflects a brief exchange between Mae and two of her students, Alyssa and Amber, as they worked 167

through the measure the height task. During this interview, Mae provoked the students to summarize their mathematical work to that point and then elaborated on the considered conceptual tools to support their solving process.

168

Table 19. Use of Conceptual Tools

Questions

Making

ofModel

nicating Ideas nicating

Making/Reflection

Speaker ProblemGeneration Commu Parameterization Refinement ConceptualUse Tool Assumption ThinkingtheWorld Real about Validationof Model Collaborating Physicaltool Use Researching Variables Establishing Sense Mathematical Asking Contextual Asking Questions Thinkingabout Mathematics SharingSolutions Use Tool Experiential Humanizing EnculturatingMathematics into ActivatingMetacognition Elaborationof Student Mathematics FormalizingMathematics Curricular Line Message Unit 1065 T So what’s going x

169 on with you guys 1066 What’d you x x guys decide to do 1067 Am We did x x x x measuring for one 1068 and estimation x x x x for another 1069 and then we’re x x x trying to figure out something

1070 cause there’s got x x x x to be something to do with the angles 1070 for like || x x x x x figuring out like a ratio

Table 19. continued

Curricular Mathematics Curricular

Making

ofModel

Making

Message Unit Use Tool Experiential

Speaker ProblemGeneration CommunicatingIdeas Parameterization Refinement ConceptualUse Tool Assumption ThinkingtheWorld Real about Validationof Model Collaborating Use PhysicalTool Researching Variables Establishing Sense Mathematical Asking Questions Contextual Asking Questions Thinkingabout Mathematics SharingSolutions Humanizing EnculturatingMathematics into ActivatingMetacognition ElaboratingStudent Mathematics Formalizing Line 1072 T Why do you x x x think that 1073 Am Because of x x x

170 geometry

1074 it’s like x x x geometry 1075 Am I don’t like x x x x x algebra but geometry all makes sense 1076 Like, it’s like x x x x puzzle pieces that connect 1077 and I feel like I x x x remember something about the angles 1078 but I haven’t x x x taken geometry in like 2 years 1079 so I can’t x x x remember what it is

Table 19. continued

World

Mathematics into g

Making

ofModel

Making

Message Unit Use Tool Experiential

Speaker ProblemGeneration CommunicatingIdeas Parameterization Refinement ConceptualUse Tool Assumptio Thinkingthe Real about Validationof Model Collaborating Use PhysicalTool Researching Variables Establishing Sense Mathematical Asking Questions Contextual Asking Questions Thinkingabout Mathematics SharingSolutions Humanizing Enculturatin ActivatingMetacognition ElaboratingStudent Mathematics FormalizingMathematics Curricular Line 1080 Am [Laughs] x x 1081 T No I hear you x 1082 Am But it’s got to be x x 171 something x x 1083 T So let’s think about it 1084 what things x x x x x x could we measure in a triangle 1085 T Cause you said x x x x 1086 well we could x x x x do something with the angles 1087 Right x x 1088 Am Length of the x x x sides 1089 T Length of the x x x x sides 1090 okay x x x x

Table 19. continued

Making

ofModel

MathematicalQuestions

Making

Message Unit Use Tool Experiential

Speaker ProblemGeneration CommunicatingIdeas Parameterization Refinement ConceptualUse Tool Assumption ThinkingtheWorld Real about Validationof Model Collaborating Use PhysicalTool Researching Variables Establishing Sense Asking Contextual Asking Questions Thinkingabout Mathematics SharingSolutions Humanizing EnculturatingMathematics into ActivatingMetacognition ElaboratingStudent Mathematics FormalizingMathematics Curricular Line 1091 So what would x x x x x that look like 1092 Am I mean | if it’s x x x 172 like all the angles are the

same then it’s like 푎2 + 푏2 = 푐2 1093 Or but I can’t x x x 1094 I mean I’m x x x trying to 1095 but the shape of x x x this is so weird 1096 like it x x 1097 I’m trying to x x think of how to do that 1098 T So you’re saying x x x x we could use this if we had triangles 1099 but this is a x x x x weird shape

Table 19. continued

Tool Use Tool

Making

ofModel

Making

Message Unit Experiential

Speaker ProblemGeneration CommunicatingIdeas Parameterization Refinement ConceptualUse Tool Assumption ThinkingtheWorld Real about Validationof Model Collaborating Use PhysicalTool Researching Variables Establishing Sense Mathematical Asking Questions Contextual Asking Questions Thinkingabout Mathematics SharingSolutions Humanizing EnculturatingMathematics into ActivatingMetacognition ElaboratingStudent Mathematics FormalizingMathematics Curricular Line 1100 Am Yeah x x 1101 T Are we allowed x x x x x x to draw in extra

173 lines to make

shapes look a little bit better 1102 Am Yeah | but then x x x X x we have to re- measure 1103 I don’t know X X 1104 It’s | cause if we x x X X x did from the top down to here 1105 Al Down to my feet x x X X | we could do that 1106 Am Yeah x X X 1107 Al So from the top x X x of the umbrella 1108 Am But then how do x x x x x x we do the angle

Table 19. continued

tics

Making

ofModel

Making

Message Unit Use Tool Experiential

Speaker ProblemGeneration CommunicatingIdeas Parameterization Refinement ConceptualUse Tool Assumption ThinkingtheWorld Real about Validationof Model Collaborating Use PhysicalTool Researching Variables Establishing Sense Mathematical Asking Questions Contextual Asking Questions Thinkingabout Mathematics SharingSolutions Humanizing EnculturatingMathematics into ActivatingMetacognition ElaboratingStudent Mathema FormalizingMathematics Curricular Line 1109 Am Do we just have x x x X x you lay on the

174 ground 1110 Al Yeah x x x x 1111 T You could do x x x x that 1112 Am Really 1113 T Yeah x x x x 1114 Am Okay 1115 T So you’re not x x x x x x x going to look crazy 1116 Am For something else 1117 T you’re just x x x going to look like the rest of your class 1118 Am Okay 1119 Al [Laughing]

Table 19. continued

Making

ofModel

Making

eriential Tool Use erientialTool

Message Unit Exp

Speaker ProblemGeneration CommunicatingIdeas Parameterization Refinement ConceptualUse Tool Assumption ThinkingtheWorld Real about Validationof Model Collaborating Use PhysicalTool Researching Variables Establishing Sense Mathematical Asking Questions Contextual Asking Questions Thinkingabout Mathematics SharingSolutions Humanizing EnculturatingMathematics into ActivatingMetacognition ElaboratingStudent Mathematics FormalizingMathematics Curricular Line 1120 Am So for the third thing | for the

175 streetlight | you want to do that 1121 Al Okay yeah 1122 Am Okay let’s try that 1123 T I’m excited x x x

In the analysis above, Alyssa and Amber encountered a number of mathematical barriers, in particular, how to use the angle of elevation (line 1070) and how to construct right triangles to be able to use the Pythagorean Theorem (line 1092). In listening and responding to these barriers, Mae validated the use of those mathematical strategies which these students were referencing and then elaborated on those mathematical ideas by helping them identify triangles by drawing axillary lines (line 1101).

Mae’s scaffolding process, in this regard, was acknowledging conceptual tools and supporting their elaboration in a mathematical way. With respect to Mae’s long-term goals of promoting modeling capacity, this offers further evidence that in facilitating modeling tasks Mae’s practice incorporates coming to understand the contextual aspects of a learner’s solution, listening to those mathematical tools which they are considering, and finally elaborating on those ideas to bridge the gap between formal mathematics and the ideas.

Across this encounter, Mae further encouraged students to think actively about mathematics as a way of anchoring their problem-solving strategies. Reflected in line

1101, Mae scaffolds students to consider the particular problem-solving heuristic of drawing auxiliary lines (Polya, 1959) as a way of advancing their solution process. This scaffolding was not done to promote her idealized solution strategy of use of trigonometry, but rather was informed deeply by Alyssa and Amber’s conceptualization of the task and a reference to their mathematical ideas of using the Pythagorean Theorem.

Evoking this scaffold and asking them to think mathematically about where extra lines might be drawn so that the Pythagorean Theorem applied offers evidence of Mae’s view

176

of facilitating modeling tasks in which learners’ ideas are explored and elaborated upon using conceptual tools which they have access to.

4.4.3 Thinking About the Real World and Experiential Tool Use

The components of thinking about the real world and promoting the use of learners’ experiential tools appeared frequently across modeling sessions. Mae invited real-world considerations and experiences to promote their mathematization and further elaboration. The utility of these components further supported learners to better understand the complexities of those contexts being modeled. Mae further stresses that these contextual and experiential factors motivate the decisions which we make in our daily lives. In Table 20, Mae facilitates a discussion as her students work through constructing their problems in the Answer is Task and highlights these components of thinking about the real world and experiential tool use within her interactions with students.

177

Table 20. Thinking about the Real World and Experiential Tool Use

Mathematics

Making

ofModel

Making

Experiential Tool Use Tool Experiential

Speaker ProblemGeneration CommunicatingIdeas Parameterization Refinement ConceptualUse Tool Assumption ThinkingtheWorld Real about Validationof Model Collaborating Use PhysicalTool Researching Variables Establishing Sense Mathematical Asking Questions Contextual Asking Questions Thinkingabout Mathematics SharingSolutions Humanizing Enculturatinginto ActivatingMetacognition ElaboratingStudent Mathematics Formalizing Curricular Mathematics Line Message Unit 173 T Is there tax on that x x x backpack 178 174 H There is the state x x

a sales tax and sometimes there is a local sales tax on that too 175 Jo Unless you buy it x on a tax-free weekend 176 T Oh buddy you just x x x out smarted me 177 guess what x x x 178 Well you just x x x brought up a good point

Table 20. continued

ion

Making

ofModel

Making

Message Unit Use Tool Experiential

Speaker ProblemGeneration CommunicatingIdeas Parameterization Refinement ConceptualUse Tool Assumption ThinkingtheWorld Real about Validationof Model Collaborating Use PhysicalTool Researching Variables Establishing Sense Mathematical Asking Questions Contextual Asking Questions Thinkingabout Mathematics SharingSolutions Humanizing EnculturatingMathematics into ActivatingMetacognit ElaboratingStudent Mathematics FormalizingMathematics Curricular Line 179 that's a real life x x event that just got

179 you out of doing more complicated math 180 you write that into x x x x your problem and your gold.

181 T It's John right x 182 John the one who x outsmarted me 183 yeah see that's the x x x cool thing about real life 184 sometimes it works x x x in our favor right 185 and you've always x x x x got to keep real life in mind when you’re doing math

Table 20. continued

ematics

Making

ofModel

Making

Message Unit Use Tool Experiential

Speaker ProblemGeneration CommunicatingIdeas Parameterization Refinement ConceptualUse Tool Assumption ThinkingtheWorld Real about Validationof Model Collaborating Use PhysicalTool Researching Variables Establishing Sense Mathematical Asking Questions Contextual Asking Questions Thinkingabout Math SharingSolutions Humanizing EnculturatingMathematics into ActivatingMetacognition ElaboratingStudent Mathematics FormalizingMathematics Curricular Line 186 cause sometimes it x x x doesn't need to be that complicated

180 187 Jo My mom talks x x

about tax weekend all the time so that's like on my brain 188 T Include it include it x x x 198 T put the dates on x x there 190 put the dates on x x there 191 The idea is we kind x x of stick with one problem and keep with it 192 and just think a x x little bit more thoughtfully about it

This analysis revealed that, as the teacher provoked Harry and his group to consider the variable tax (line 173), and when John intervened with the tax-free weekend consideration (line 175), she immediately validated this thinking and initiated an important discussion about how contextual considerations can outweigh the need for complex mathematics (line 176). While Mae had anticipated students would consider tax as an important contextual factor, she had prepared to support their mathematization in that regard. However, when John brought up tax- free weekend, Mae’s demonstrated flexibility by moving beyond this anticipated variable when John problemized it with the context of a tax-free weekend (line 75).

The State of Ohio Department of Taxation had indicated that on Friday August 4th through Sunday, August 6th beginning at 12:00AM on Friday and ending at 11:59PM on

Sunday, a tax holiday was in effect during which clothing and school supplies or instructional materials would be exempt from taxes. Rather than insisting that the students must consider the tax burden on the items regardless of those dates, i.e., insisting on her own perspective, she instead commended John’s intelligence and uses this point to elaborate on her SMP. In particular, Mae stated that, in this case, the real life event got them out of doing more complicated math (line 179) as long as this information was reflected in the problem. She further clarified to the class, at large, that sometimes mathematics doesn’t need to overcomplicate the real life context, and, if a simpler solution exists which meets the criteria for the problem, then it is sufficient.

Another important aspect of this interaction is related to the use of experiential tools and cultural knowledge when considering a problem context. In line 187, John

181

states that this tax- free weekend is constantly on his mind, as his mother referenced it frequently at home. This valorization of his use of experiences and the encouragement he received to incorporate them into the problem establishes a democratic space in which this knowledge is not only appreciated, but also encouraged. This is reflected through the teachers’ comments about thinking more thoughtfully (line 192) about the problem. It is important to note that, being a pre-calculus class, merely considering the base-price per unit of an item and how these items combine to yield a particular value, while not accounting for tax, would be a seemingly low-level or limited response in the view of etic curricular mathematics. However, this interaction exhibited the teacher’s view that sometimes the simplest solution which adequately describes the event is sufficient, based on her claim that mathematics need not necessarily be that complicated as long as it adequately describes realistic factors.

4.4.4 Teacher Humanizing

As reflected across the macro analysis of modeling sessions, teacher humanizing appeared with great regularity. Data analysis revealed that Mae humanized herself in the presence of her students in order to promote learners’ thinking about the real world and invite learners’ experiences and intuitions into the problem-solving process. As depicted in Table 20 in the previous section, when John brought up the context of a tax-free weekend, Mae positioned herself as being outsmarted (line 176), challenging the notion of the teacher as the keeper of knowledge and having the right answer. When John shared his problem during the debriefing discussion (Table 27), Mae re-references the

182

tax-free weekend and further elaborates that, in light of this variable, she will go shopping (Line 476), and that she learned something (line 478), indicating that she has changed her practice in light of this student information (Table 21).

183

Table 21. Teacher Humanizing

Making

ofModel

Tool Use Tool

Solutions

Making - Message

Unit Use Tool Experiential

Line Speaker ProblemGeneration CommunicatingIdeas Parameterization Refinement Conceptual Assumption ThinkingtheWorld Real about Validationof Model Collaborating Use PhysicalTool Researching Variables Establishing Sense Mathematical Asking Questions Contextual Asking Questions Thinkingabout Mathematics Sharing Humanizing EnculturatingMathematics into ActivatingMetacognition ElaboratingStudent Mathematics FormalizingMathematics Curricular 474 T so you did x x it on a tax- 184 free weekend 475 Jo yeah it's x tax- free this week till like Friday so 476 T I gotta go x X shopping 477 so that's x x beautiful 478 see I X learned something today

The significance of this interaction is that, through this humanizing process, Mae established a cultural practice which encouraged other students to share their experiences and offer insight into other problems. Evidence that this component was further taken up occurred across all modeling events, in particular, when students began generating their own problems given a context. One such example occurred during the trig whips activity, in which two students determined the velocity of the 20th person in the chain to be 337 feet per second and actively discussed whether their favorite athlete could maintain that speed, generating a new modeling context as a result. Further, once they determined that this was impossible, they adjusted the question to consider at what speed the inner-most person could turn to allow for this athlete to maintain this position as the

20th person.

4.4.5 Formalizing Curricular Mathematics and Asking Mathematical Questions

While Mae’s long-term goals involved the development of modeling capacity in her students, she was also concerned with formalizing the curricular concepts mandated by the course. There were some instances in which Mae enculturated students into these mathematical concepts through direct instruction or problem-solving as opposed to modeling. When she recognized the connection between contextual problems and her overarching content goals, she was more likely to include modeling in her instruction.

During these instances, I traced how Mae engaged in formalizing academic concepts, drawing on the previously facilitated modeling sessions. Analysis of Mae’s practice revealed that, in typical instances, Mae would start with mathematical modeling and

185

solving contextual problems, allow students to explore their ideas and intuitions, and then bridge those solutions to the curricular goals which she wished to impart. This typically manifested through establishing intercontextual links between student solutions and these concepts. It was through asking contextually-bound mathematical questions during discussions and giving students time to explore those concepts in those familiarized contexts that this transition between the elaboration of student mathematics and formalizing curricular mathematics ensued.

To exemplify this formalization of curricular mathematics through asking mathematical questions, I briefly overview a discussion which Mae facilitated after the measure the height task formalizing sine, cosine, and tangent. Mae began this discussion by defining these trigonometric relationships as corresponding to the sides of a right triangle (Figure 20).

186

Figure 20. Formalizing Sine, Cosine, and Tangent

Mae listed the following definitions with the support of her students:

표푝푝표푠푖푡푒 푠푖푛푒 = ℎ푦푝표푡푒푛푢푠푒⁡

푎푑푗푎푐푒푛푡 푐표푠푖푛푒 = ℎ푦푝표푡푒푛푢푠푒

표푝푝표푠푖푡푒 푡푎푛𝑔푒푛푡 = 푎푑푗푎푐푒푛푡

After defining these terms, Mae drew a corresponding diagram identifying the side relationship based on a referent angle theta. She then walked students through two decontextualized examples for determining the missing side of a right triangle (Figure

21).

187

Figure 21. Sine and Tangent Examples

After solving the two examples as a class, Mae established a link between trigonometry and the measure the height task. In particular, she used one group’s context of determining the height of a basketball hoop and highlighted how these trigonometric relationships could be used to determine the value of the missing side given their collected data (Figure 22).

In this example, the Mae began by re-drawing the student diagram on the board and labeling the corresponding measurements. Mae then informed the rest of the class that one of these groups’ deliberations concerned whether or not their determined angle measurement using the inclinometer was accurate. Mae then stated that trigonometry affords the ability to determine the accuracy of a tool by validating whether or not the measured angle corresponds to the calculated value. Mae then demonstrated how to use inverse trigonometric functions and calculated the angle of elevation using the inverse tangent as 24.23 degrees. She then asked the students to compare this calculated value to

188

their measured value of 20 degrees. This caused the students to discuss whether or not this measurement error of 4 degrees was significant and, as a class, determined that it was good enough.

Figure 22. Formalizing Student Problem Contexts

Across this encounter, Mae formalized curricular mathematics through the facilitation of discussion in which she asked mathematical questions and demonstrated mathematical procedures. This notion of error-checking supported the reconciling of the contextual experience of determining the height of real-world objects and formal concepts in trigonometry.

189

4.4.6 Enculturation into Mathematics and Activating Metacognition

In calling specific attention to the underlying mathematics, Mae helped enculturate students into her view of what mathematicians do when engaged in solving real-world problems through the activation of metacognition in sense-making. Almost daily across all observations (modeling and non-modeling), Mae explicitly referenced this notion of what mathematicians do.

Recall that, in Table 24 in section 4.4.1, Mae openly asked her students to consider what she was doing that other teachers typically may not (line 69), and I argue that, by explicitly referencing that she was making students not only think, but think carefully, about what was in their brains, what they are doing, and how to ask their questions, she was activating metacognitive awareness of mathematics. She further established that this process is a big part of math (line 79), and that, further, they don’t even realize they are doing it (line 75).

This frequent discussion of what constitutes mathematics or what mathematicians do served as a means with which to enculturate students into the discipline of mathematics beyond those skills which are advocated within the curriculum. One example of this was during her monitoring of student work. As students were deliberating contextual factors as they generated problems, Mae praised their asking of questions and informed them that mathematicians are good question-askers (line 133), further encouraging this behavior and enculturating them into mathematics (Table 22).

190

Table 22. Enculturation into Mathematics

gnition

Making

ofModel

and Experiences Ideas

Making/Reflection

Message Unit Individual

191 do all year long 132 I want you guys x x x

to be great question- askers 133 because that’s x x x what mathematicians are

4.4.7 Physical Tool Use, Research, and Parameterization

In looking at how Mae incorporated the use of physical tools, such as technology, research, and parameterization, into her SMP, evidence exists that this occurs as she engaged in monitoring her students’ progress. Within the Answer is Task problem development, Mae noticed that students had considered values of items relatively close to

$70. She used this to refine and expand the components which she has previously established by adding in a marker for technology use as a tool for research (Table 23).

Ljung (1998) indicates that the system identification process in mathematical modeling supports the transition between observed data and a mathematical model (p.

163). According to Ljung, navigating the modeling process is related to establishing a procedure which reconciles observed data, a set of candidate models, and a criterion of fit and validates this relationship (p. 163). In considering this process through the aforementioned event with the answer being a fixed $73.13 corresponds to the observed data component of parameterization. This Ljung describes as the information required to build and validate models (p. 164). Ljung further indicates that, in using this data, one can consider candidate models with which to find a suitable model structure which can account for the behaviors of the overarching systems. In this approach, Ljung discusses aspects of this parameterization process, in particular, that of selecting candidate models which cover as many parameters as possible and then consider the physical properties of the systems through creating a state space where parameters are then estimated (p. 167).

While Ljung discusses aspects of modeling related to abstract and advanced systems, I see a connection in this process to my observed data.

192

Table 23. Use of Technology, Research, and Parameterization

Making

ofModel

and Ideas

Making/Reflection

Individual

something that’s about 70 dollars 158 and I said get out x x your phone and look it up 159 as I’m going to x x x trust you as adults 160 to use your x x phone when it's appropriate and put it away when it’s not 161 so if you're using x x x phones for research because you need to know how

Table 23. continued

Making

ofModel

and Experiences Ideas

Making/Reflection

ctivatingMetacognition

Message Unit Individual

194 look up what a

sales tax rate is get out your phones 164 use them x x responsibly 165 now I’ll call you x x out if you're not 166 but use your x x phones responsibly 167 use your iPads x x responsibly 168 or I’ll make you x x put them away

The analysis of this exchange further reveals elaboration of Mae’s SMP through establishing a mechanism for the use of technology as a tool for research, coupled with how use of technology in this manner can support the establishment and refinement of parameter values. Mae used this exchange to emphasize other components which were addressed previously, such as establishing variables (lines 161-163).

When Mae brought up Harry’s question of what is approximately $70 (line 157), she is calling attention to the fact that Harry is estimating a parameter value for the overarching system, which Ljung (1998) argues is crucial in the modeling process. In this regard, Harry selected a value in line with the total cost reflected in the problem, close to that which will yield the desired value. In refining his problem to generate a value close to the exact amount, he added additional components to the initially selected item. In light of Harry’s parameterization process, the teacher used this observation to highlight the merits of technology as a useful tool in authenticating parameter values, thus making them less arbitrary.

In a further refinement of the components of her SMP, Mae advocated, in this capacity, for her students to “research” either what items might fit the estimated parameter values or find the actual sales tax rate in order to further authenticate the problems’ contexts and support the development of their resulting models, in particular, their refinement. In a broad sense, regarding modeling, Mae advocated that they were allowed to use technology when appropriate (line 160) and framed it as a useful tool in research (line 168).

195

In considering further the use of physical tools, Mae explicitly stated this as one of her major objectives within a debriefing interview which occurred immediately following this task (Table 24).

196

Table 24. Reflective Interview of Answer Is Task

300 R Any reflective thoughts 301 T My reflection has to be very quickly cause convocation is starting and that's 302 something that you don't miss at all 303 R Uh huh 304 yeah absolutely 305 T so I liked it and its interesting cause I like 306 Chris and I talked about the different things we wanted to do today 307 and I pushed to do this activity just the way I executed it 308 cause I wanted to talk like informally kind of about like habits of mind 309 that we were doing in class 310 R yep 311 T and I wanted to like bring up this idea of using your cell phone to research 312 and thinking about the real world when it trumps math 313 R yeah 314 T and stuff like that so I loved it 315 it was like totally my agenda 316 R yes I noticed that too and I made I documented and noted those exact same things 317 T yes 318 R you established that we research in this class 319 we think about the real world 320 T yeah 321 R we um yeah we think about when it doesn't match the mathematics 322 and even you're advocating for technology 323 get your phone out 324 use it 325 T and use it responsibly 326 R I trust you 327 T yeah 328 because I think a lot of the like 329 almost all of the teachers are thinking about like the basket now 330 like collecting them all on the way in 331 and I'm like I just can't commit to that like 332 I think I feel like it’s better to make sure that people are using them responsibly 333 so 334 This is something I did new though 335 I've never had that like had them brainstorm together first 336 I've always just had them start writing their questions and like awkward silence 337 R will you do it again 338 T yes I think I will 339 I think it was just enough of a tone setting thing

197

In this Mae referenced the components of thinking about the real world (line 312) and using technology for research (line 311). Mae clarified her privileged use by contrasting it with inappropriate technology use (line 332) and stated that she wasn’t prepared to adopt other teachers’ policies to avoid these devices in their classes. In this way, she warranted this tool use as being an integral component to her SMP.

4.4.8 Assumption Making and Problem Generation

While many of the components of Mae’s SMP emerged during class discussion, others were manifested in tasks she selected and how she chose to facilitate the tasks.

Establishing variables, assumption-making, and problem generation were three of those components.

Regarding the component of problem generation, developing new problems given a context came about first during the planning of the answer is task. In her planning, and confirmed through debriefing interviews, Mae indicated that her goal for that particular problem was to use a heavily contextualized problem-posing task to introduce students to developing mathematical questions and focusing on real-world aspects which inform their development. Table 25 offers an analysis of the conclusion to the Answer is debriefing discussion. Within this discussion, Mae states explicitly to her students that the intent of the activity was to get them to ask questions (line 650) and find those details which needed to go into the problem generation process (line 651).

198

Table 25. Problem Generation

Making

/Quantifyin

Ideas and Experiences Ideas

MathematicalQuestions

Making

Message Unit Individual

Line Speaker ProblemGeneration CommunicatingIdeas Parameterization Refinementof Model WorkingMathematically Assumption ThinkingtheWorld Real about Validationof Model Collaborating Use Technology Researching Variables Establishing Sense Asking Contextual Asking Questions Thinkingabout Mathematics SharingSolutions Humanizing EnculturatingMathematics into ActivatingMetacognition Formalizing 629 T it's kind of nice in Ohio x 630 T I'm not gonna lie x 631 so um I've heard the same thing x about the West Coast

199 632 but I haven't had the pleasure of x being over on the West Coast

very much 633 anybody got some experience x x with that 634 Be It's a lot out there x x 635 T It's a lot a lot or a lot x 636 Be It's a lot a lot out there x 637 T a lot a lot x x 638 Be and it's not even just the um x sales tax 639 it's just everything’s a lot more x x expensive out there 640 T got it x 641 I mean I’ve heard that x 642 but like I said I haven't had a x personal experience 643 So um I try to defer to people x who have personal experience 644 by the way that's our first x defined word in PC 645 a lot a lot x 646 now we all know what it means x 647 I like it x 648 [teacher smiles] 649 Okay so really nice job with this x

Table 25. continued

Making

/Quantifyin

Ideas and Experiences Ideas

Making

Message Unit Individual

Line Speaker ProblemGeneration CommunicatingIdeas Parameterization Refinementof Model WorkingMathematically Assumption ThinkingtheWorld Real about Validationof Model Collaborating Use Technology Researching Variables Establishing Sense Mathematical Asking Questions Contextual Asking Questions Thinkingabout Mathematics SharingSolutions Humanizing EnculturatingMathematics into ActivatingMetacognition Formalizing 650 the whole goal of this activity x x x was to get you guys thinking about how we ask mathematical questions 651 and the details needed to go into x x x x x x x x x 200 those

Regarding the component of assumption making and its relevance to choice of task, I observed that students engaged in this process as they worked on measuring height for the measuring the height task. While not explicitly referenced within this session, the task of measuring mandated that students consider contextual factors which impact the height of the objects and account for those factors. For example, while monitoring

Alyssa and Amber during this process, they opted to measure the height of a table umbrella and estimate the height of a tree and then verify that measurement with a direct- measuring strategy. In their discussion, they openly assumed contextual factors, such as the slant of the ground and the height of the tree to be to the top of the highest branch rather than trunk or leaves. In this assuming, they were able to advance within their measurement process. Their process of measurement is exhibited within the ethnomodel below (Figure 23). Further, during the shortest distance to the cafeteria task students openly deliberated which path to measure between buildings and assumed certain factors to be true, i.e., that students don’t typically walk through plants, but do walk through the grass. It was the selection of this particular task which motivated the process of assumption-making and furthered a part of Mae’s SMP.

Ethnomodeling affords the ability to elaborate on local cultural practices and valorize those practices as viable mathematics. Figure 23 offers an ethnomodel of Alyssa and Amber’s process of measuring the height of the umbrella. Reflected in the ethnomodel below, Alyssa and Amber assumed the distance y to be parallel to the ground and, further, that the height of the umbrella, x+z , was perpendicular to the ground. In their measurement process, they determined the angle of elevation (theta), Alyssa’s

201

height to eye (x), and the height from eye-level to the tip of the umbrella, (z). The resulting model for height was ℎ = 푥 + 푦, which was chosen due to access to tools, previously established variables, and access to mathematical concepts with which they were familiar. This model was weighed against alternate models involving angle measurements and formalized in Alyssa and Amber’s work.

h=x+z

Figure 23. Etic Ethnomodeling of Measuring the Umbrella

Measuring the horizontal distance elevated off of the ground alleviated the need to account for slant from the point of measurement. Through the process of ethnomodeling,

I was able to consider the enacted task in light of the methods which these students used

202

for determining the height and notice the viability of their method, how they assumed and considered contextual factors, and how they accounted for these mathematically.

The researcher, in a brief exchange with Alyssa and Amber, validated these considerations by asking them to re-voice their process of measurement (Table 26).

Here, Alyssa and Amber discuss openly the assumptions they made and how they were thinking about the real world, thus taking up these components of practice.

203

Table 26. Alyssa and Amber Make Assumptions

Line # Speaker Message Unit Description of Activity Components of SMP 857 Am mmhm Am turns towards researcher 858 R What strategy are you using for Provoking reflection; figuring out the height formalizing 859 Am We were trying to figure out Al walks back toward researcher Working mathematically; what the calculated angle and Am formalizing 860 cause if you stand right here Am walks back to the spot where Establishing variables; working you get the angle from like the they collected the measurement; mathematically; thinking about top of the umbrella to our eyes points to the top of the umbrella real world 861 and then it's 90 degrees to the Waves hand in a horizontal arc Assumption-making; thinking ground assuming its completely palm facing toward the ground at about the real world; working

204 flat flat mathematically

862 so we calculated the other angle Working mathematically, to be um 75 degrees thinking about the real world, thinking about mathematics 863 and we're trying to figure out Am looks at her paper; gestures Thinking about the real world, we can use these angles to find towards angles 1 and 2. thinking about mathematics the height 864 R Why is it important to think about the ground being flat 865 Am um to get it to be 90 degrees Points and motions up and down Assumption-making; thinking from the straight up height of towards the umbrella about the real world the umbrella to the ground 866 that's 90 degrees cause Brings hand down and motions Assumption-making; thinking otherwise we'd have to do a horizontally from body towards about the real world; thinking bunch of crazy math to account umbrella about mathematics for the ground

4.4.9 Establishing Variables, Asking Contextual Questions, and Collaboration

The components of establishing variables, asking contextual questions, and collaboration emerged typically during the initial phases of presenting the modeling tasks and then across subsequent discussions between Mae and her students. Table 27 outlines a brief discussion in which Mae operationalizes the measure the height task by engaging in these components of her practice. Within this discussion, Mae had just demonstrated how to use an inclinometer to determine the measure of an angle from sight and transitioned into asking students what objects should be measured and what constraints should be imposed on the task.

205

Table 27. Establishing Variables and Constraints, Asking Contextual Questions, Collaboration

Making

ofModel

and Experiences Ideas

Generation/Establishing

Making

Message Unit Individual

Line Speaker Problem CommunicatingIdeas Parameterization Refinement WorkingMathematically Assumption ThinkingtheWorld Real about Validationof Model Collaborating Use PhysicalTool Researching Variables Establishing Sense Mathematical Asking Questions Contextual Asking Questions Thinkingabout Mathematics SharingSolutions Humanizing EnculturatingMathematics into ActivatingMetacognition Formalizing/Quantifying 745 T what should we probably measure x 746 what should be like some some x x x x x x x

206 like constraints on this 747 Am nature x x

748 T nature okay x x x 749 measure something out in nature x x x x 750 um should we be measuring like if x x x x x x we find a Cicada shell on the ground 751 Sx [shakes head "no"] x 752 T no ideally like what do you think x x x x these are for measuring 753 short things or tall things x x x 754 Am tall x x 755 T so let’s make a rule um x x x 756 you have to measure three things x x x that are taller than yourself 757 okay x x x 758 I’m going to put it in writing x x x x because I'm a visual person so I like this 759 um and if you're a visual person it x x x might be good to write this down 760 I'm not going to be giving out x worksheets 761 you know you're just going to x X kind of work this out

Table 27. continued

tifying

Making

ofModel

and Experiences Ideas

Making

Message Unit Individual

Line Speaker ProblemGeneration CommunicatingIdeas Parameterization Refinement WorkingMathematically Assumption ThinkingtheWorld Real about Validationof Model Collaborating Use PhysicalTool Researching Variables Establishing Sense Mathematical Asking Questions Contextual Asking Questions Thinkingabout Mathematics SharingSolutions Humanizing EnculturatingMathematics into ActivatingMetacognition Formalizing/Quan 762 but you're going to measure three x x x things taller than yourself [writes on the board at same time] 763 measure three things taller than x x yourself 207 764 but there's an added challenge x x

765 like I think right now you're kind x x x x of like thinking in your brains 766 I feel like x x 767 You all remember doing x x x something like this in geometry 768 Sx (shake heads "no") x

769 T ooh [teacher clasps hands together x X and smiles excitedly] 770 ooh [smiles] x X 771 okay that's good x x 772 I like this x X 773 I like that it's new x X 774 T So measure three things that are x x x taller than yourself and here's the stipulation 775 measure it three different ways x x X 776 measure or calculate it. x x x

Table 27. continued

Making

ofModel

and Experiences Ideas

Making

Message Unit Individual

Line Speaker ProblemGeneration CommunicatingIdeas Parameterization Refinement WorkingMathematically Assumption ThinkingtheWorld Real about Validationof Model Collaborating Use PhysicalTool Researching Variables Establishing Sense Mathematical Asking Questions Contextual Asking Questions Thinkingabout Mathematics SharingSolutions Humanizing EnculturatingMathematics into ActivatingMetacognition Formalizing/Quantifying 777 so I want you to dig deep in your x x x x x brains and I want you to think really carefully about different strategies that you have in um

208 geometry 778 like how you can figure out x x x

779 like a lot of you guys have kind x x x x of seen pictures like this [Figure 30] 780 you may not have gotten to go x x outside 781 but a lot of you guys probably x x x x saw pictures that look like this 782 any of those pictures coming back x x to mind 783 [Students affirm by nodding] okay x x x 784 so I'm gonna kind of leave it at that spot 785 of course I'm walking around and x x x x x of course I’ll answer any questions that you guys have when you're out there 786 but I just want to give you guys a x x x x chance to think 787 like a lot of times we don't get that x time

Table 27. continued

World

Making

ofModel

and Experiences Ideas

Making

Message Unit Individual

Line Speaker ProblemGeneration CommunicatingIdeas Parameterization Refinement WorkingMathematically Assumption ThinkingtheReal about Validationof Model Collaborating Use PhysicalTool Researching Variables Establishing Sense Mathematical Asking Questions Contextual Asking Questions Thinkingabout Mathematics SharingSolutions Humanizing EnculturatingMathematics into ActivatingMetacognition Formalizing/Quantifying 788 so what I want you guys to do is I x want you to be in groups of either two or three 789 three would probably be pretty x x x ideal cause then you have one 209 person to hold on to your paper and do the writing for the group 790 you can have another person who x x x x would cite things 791 and then the third person could X x kind of be the leveler 792 kind of you know just kind of an x idea of this 793 but what I want you guys to do is I X x want you to go ahead and grab your supplies 794 all of these do the same thing X 795 like I said you would just line up X the top like that and have someone read it for you 796 so you have someone like reading X the thing 797 someone lining it up X 798 and guys we have x 799 we have until 9:10 so let’s do this x 800 we're going to go outside for x x twenty minutes measure three things taller than yourself 801 but you have to measure in three x x different ways

In this episode, Mae initiated the discussion by asking students to generate constraints for the accompanying task (line 746) and gave them agency in their construction, i.e., nature (line 747). Across this discussion, Mae asked the students contextual questions in order to help facilitate the establishment of variables. For example, Mae drew attention to the utility of the inclinometer for measuring the height of something larger than oneself, which becomes a constraint on the problem context.

In order to facilitate this task in a way that allowed students the ability to draw on their own mathematical ideas while simultaneously acting as a bridge to her intended goals, Mae imposed an additional challenge to the task. She asked students to measure the height of each object in three different ways (line 775). I argue that it was through establishing these constraints that this task was transformed from a typical measurement task frequently seen in geometry classes into a modeling context in which students were presented with the opportunity to make assumptions, validate, and refine, each of which are key components in the modeling process.

In establishing the constraints for this task, Mae drew a diagram on the board

(Figure 24) and asked students if they recognized it from previous classes (line 779). In generating this diagram, she focused students on both the contextual aspect of measuring and mathematical representations of the system. In this way, through asking this contextualized mathematical question, she readied the students to link their experiences with the mathematical ideas embedded in the task. In drawing this picture and asking a pseudo-contextualized question concerning familiarity, she further established constraints on this system and linked it to the underlying mathematics of the task.

210

Figure 24. Measure the Height Referent Diagram

It was further in this discussion that Mae advocated for collaboration. It is noteworthy that, across each modeling session, Mae asked students to work collaboratively and outlined a process for students to support each other during these tasks. Across the depicted event in Table 20, Mae referenced numerous times how she expected students to support each other during their work. Across task enactment, this aspect of collaboration was picked up by students and is depicted further in the scaffolding interaction between Mae, Alyssa, and Amber (Table 25, lines 1102-1109), in which these students collaborate to develop a third strategy for measuring the height of a light post. It is also noteworthy that the image which Mae drew on the board for

211

supporting her students in determining the height was similar to that which was enacted by Alyssa and Amber, revealed through ethnomodeling.

4.4.10 Communicating Ideas and Sharing Solutions

Across each modeling session, learners were encouraged to both communicate their ideas and share their solutions for tasks. In communicating ideas and sharing solutions, Mae was able to better elaborate on learners’ mathematics so to bridge the gap between those ideas and formalized curricular concepts.

In the measure the height debriefing discussion (Table 28) Mae encouraged students to not only share their solutions of the measurements they determined during their work, but also to give insights into their process and discuss their ideas while they were working. Mae was democratic in this sharing process as she gave each group the opportunity to talk about these solutions. She acknowledged, elaborated on, and clarified certain points. As there were a wide variety of strategies, the solution alone was insufficient for provoking reflection. However learners’ solutions coupled with the orchestration of discussion provided a picture of each group’s strategies for solving the task.

As an example, in line 1200, Mae asked one group to highlight their solutions to the task, and when the students responded that they used the Law of Sines to determine the height, the teacher asked whether they knew it intuitively or had to look it up (line

1206). When the students responded that they used Google to determine what it was

(Line 1207), Mae not only validated this method, but used it to promote the use of

212

physical tools, as well. This provides a further example that these components are tightly linked and inform each other in developing Mae’s SMP.

213

Table 28. Communicating Ideas and Sharing Solutions

Model

Making

ofModel

and Experiences Ideas

Making/Reflection

Individual

Speaker ProblemGeneration CommunicatingIdeas Parameterization Refinement WorkingMathematically Assumption ThinkingtheWorld Real about Validationof Collaborating Use Technology Researching Variables Establishing Sense Mathematical Asking Questions Contextual Asking Questions Thinkingabout Mathematics SharingSolutions Humanizing EnculturatingMathematics into ActivatingMetacognition Formalizing/Quantifying Line Message Unit 1200 T Okay so what was one x x x x strategy that was used

214 outside that you really liked 1201 Ad Law of Sines x x x 1202 T you used the Law of x x x x Sines 1203 Ad yeah x 1204 T okay what is the Law of x x x x x Sines 1205 Ad it's um sine of A over a x x x equals sine of B over b equals sine of C over c 1206 T and did you remember x x x x x that from geometry or did you Google it 1207 Ad we Googled it x x 1208 T okay nice x x 1209 Google’s a good thing x x right 1210 Google’s a great thing x x 1211 so you had to have sine x x x x x of A over a

Table 28. continued

ons

Making

ofModel

and Experiences Ideas

Making/Reflection

Message Unit Individual

1212 wait what's the X X x x x 1213 are those the same A x x x x x X

215 1214 Ad no sine of big A over x x x x little a

1215 and big A would be the x x x x angle and little a would be the side 1216 T Interesting x x x x x 1217 so even if we didn't have x x x x x x a strategy for that third strategy you had enough measurements that you could pull from Google to get some ideas 1218 nice x x x x 1219 and you have x 1220 I mean think about it, x

Table 28. continued

gnition

Making

ofModel

and Experiences Ideas

Making/Reflection

Message Unit Individual

216 understanding it would have literally been Greek 1222 but you had that sense- x x making 1223 that was cool 1224 T what was one of your x x x strategies that you really liked for figuring out one of your objects 1225 Ki We used the Pythagorean x x x Theorem 1226 T How did you use that x x x x x x 1227 Ki um we got all the angle x x x x x measurements and then we had like 1228 we were able to measure x x x x x from like the object to where we took the angle measurement

Table 28. continued

Making

ofModel

and Experiences Ideas

Making/Reflection

Message Unit Individual

1232 we assumed that we were x x x x x working at about half way though because it made it easier to figure out 1233 so we just used half of x x x the objects height um squared plus the um distance between where we stood to take the angle measurement to the object squared that and then added them together and then um figured out what the c squared was 1234 T cool x x x 1235 so you found that that x x x distance like along the diagonal right

Table 28. continued

tacognition

Making

ofModel

Use

and Experiences Ideas

Making/Reflection

Message Unit Individual

218 objects 1238 Ki well it would have but x we just found 1239 we just did it ourselves x x 1240 T okay okay x x 1241 T so one of your methods x x x x x x x was just measuring it right another was uh using Pythagorean Theorem and assuming it was half of the side 1242 what was your third x x method 1243 Ki um it wasn't really a x x x x different one we just used the length of the measuring it ourselves from where we stood to the object

Table 28. continued

Making

ofModel

and Experiences Ideas

Making/Reflection

Message Unit Individual

219 1246 and I mean that not in X X X like a 1247 you know how X X X X sometimes you're trying to figure out what your teacher means by the word interesting 1248 like I really am interested X x X X 1249 like my brain is working X X X a lot right now 1250 so when I pause and I go X X X interesting 1251 it's like a very positive X X X thing 1252 what you've done is X X X you've gotten my brain thinking in a new way 1253 so I like that x X X

Table 28. continued

Making

ofModel

and Experiences Ideas

Generation

MathematicalQuestions

Making/Reflection

Message Unit Individual

Speaker Problem CommunicatingIdeas Parameterization Refinement WorkingMathematically Assumption ThinkingtheWorld Real about Validationof Model Collaborating Use Technology Researching Variables Establishing Sense Asking Contextual Asking Questions Thinkingabout Mathematics SharingSolutions Humanizing EnculturatingMathematics into ActivatingMetacognition Formalizing/Quantifying Line 1254 T interesting in this word X X X X* in this class means a very good thing X x X X* 220 1255 so I’m pausing cause I'm just thinking 1256 okay X* 1257 we have about one more X* minute 1258 oh darn I don't have time X* to assign homework cause I've got to go over these problems 1259 are you guys okay with X* that 1260 Sx it's going to be hard X 1261 Sx but if it's what you've got X to do 1262 T okay so uh really quickly x x x Ravi’s group give me one of your strategies that you guys did

Table 28. continued

Making

ofModel

and Experiences Ideas

Making/Reflection

Message Unit Individual

221 1265 | measured the angle of the tree um 1266 and we would measure x x the distance between Colin and the tree, 1267 and then the eye his eyes x x to the tree 1268 T where did this idea come x x x from 1269 Ra um just naturally X 1270 T it just felt natural x X 1271 Ra yeah x 1272 T nice nice x X 1273 T and what about you guys x x x 1274 what was one of your x x x most interesting strategies

Table 28. continued

Questions

Making

ofModel

and Experiences Ideas

Making/Reflection

Message Unit Individual

222 car was about 6'4"

1276 T nice so you kind of you x x x x eyeballed it based on your knowledge of Henry. 1277 T what about you guys x x x what was one of your most interesting thoughts or methods 1278 Am we did the same thing x x x x x where we tried estimation using Alyssa's height and failed miserably 1279 T how do you know you x x x x failed miserably 1280 Am we measured it x x x afterwards to see how close we were

Table 28. continued

Making

ofModel

and Experiences Ideas

Mathematically

Making/Reflection

Message Unit Individual

Speaker ProblemGeneration CommunicatingIdeas Parameterization Refinement Working Assumption ThinkingtheWorld Real about Validationof Model Collaborating Use Technology Researching Variables Establishing Sense Mathematical Asking Questions Contextual Asking Questions Thinkingabout Mathematics SharingSolutions Humanizing EnculturatingMathematics into ActivatingMetacognition Formalizing/Quantifying Line 1281 T so there was that idea x x x x that since you had three ways of doing this 223 1282 T you could actually x x X x

measure it 1283 and kind of see how x x x x x your math was doing 1284 Am yeah x x x x 1285 T nice x 1280 you guys I love this x x x x X x like I'm geeking out X 1281 about this what I want to do is X x when you bring this tomorrow or the next 1282 day we meet I'm going to ponder X X X some of the things that you guys said and and kind of think about 1283 where we go from there

Table 28. continued

Making

ofModel

and Experiences Ideas

Making/Reflection

Message Unit Individual

Um I'm going to think about it overnight where I want to go with this 1285 cause I love it I love the ingenuity and x x X X the creativity and the 1286 Googling okay

4.4.11 Validating and Refining

The components of validation and refinement initially emerged within the measure the height task as Mae outlined the process for problem construction (Table 29).

In this interaction, Mae encouraged her students to develop a problem, think about how to make the problem more contextually precise, and refine that problem based on those considerations (lines 93 and 94). Additionally, she asked her students to validate that their solution yielded the target $73.13 (line 92). Within her students’ working processes, she explicitly observed her students refining and validating their models (Table 30). Mae stated, in this case, that she observed students refining their problems (line 126) by adding additional contextual details (line 127).

Holistically, these components were also manifested at a macro-analytical level.

Mae’s process of elaborating on her students’ mathematics and then linked those mathematical ideas to her desired curricular goals. She enculturated learners into new methods for solving problems. During this process, students’ mathematical ideas were contextually validated, then these new mathematical tools were offered to support their refinement. In this way, validation and refinement were integral components to Mae’s view of modeling as a bridge.

225

Table 29. Collaboration, Revision, Working Mathematically

Questions

Making

ofModel

and Experiences Ideas

Making/Reflection

Individual

226 here's what I want you guys to do 81 I want you to x write the problem whose answer is 73 dollars and 13 cents 82 and I want you to x x start off for the first two minutes write your own 83 problem on your x x own beautiful color of paper 84 Okay X 85 and you guys X 86 you know what X 87 people are like X

Table 29. continued

Making

ofModel

and Experiences Ideas

lizing/Quantifying

Making/Reflection

Message Unit Individual

Speaker ProblemGeneration CommunicatingIdeas Parameterization Refinement WorkingMathematically Assumption ThinkingtheWorld Real about Validationof Model Collaborating Use Technology Researching Variables Establishing Sense Mathematical Asking Questions Contextual Asking Questions Thinkingabout Mathematics SharingSolutions Humanizing EnculturatingMathematics into ActivatingMetacognition Forma Line 88 I don't want to do x x this because it's too hard x x x 227 89 so start with a simple problem 90 start with a very x x x simple problem 91 and then after x x x you write a simple problem and you feel good about your 92 simple problem x x x x and you know that the answer is 73 dollars and 13 cents 93 I want you to x x x draw a line under it and then write a more complex problem below it 94 add details add to x x x it

Table 29. continued

ons

Making

ofModel

and Experiences Ideas

Making/Reflection

Message Unit Individual

228 with a simple problem 96 written at the top x x of your paper 97 your beautiful x paper because I love beautiful paper 98 two minutes of x x just working on your own 99 and then you’re x x x going to start to work with each other and start to add details 100 Okay x x x x

Table 29. continued

hematics

Making

ofModel

and Experiences Ideas

Making/Reflection

Message Unit Individual

Speaker ProblemGeneration CommunicatingIdeas Parameterization Refinement WorkingMathematically Assumption ThinkingtheWorld Real about Validationof Model Collaborating Use Technology Researching Variables Establishing Sense Mathematical Asking Questions Contextual Asking Questions Thinkingabout Mat SharingSolutions Humanizing EnculturatingMathematics into ActivatingMetacognition Formalizing/Quantifying Line 101 but sometimes x we have to understand our own thoughts 229 first x 102 I’m putting on a timer because I get really distracted 103 so a timer is x never to make you feel stressed or hurry 104 a timer is to x make sure that I keep track of time 105 so two minutes x x x for your own scenario 106 and then you'll x x x start to work together to fill in more details

Table 30. Refinement of Model

Making

ofModel

and Experiences Ideas

Making/Reflection

Message Unit Individual

123 so first of all I x saw you guys taking your time 124 second of all I x saw you guys 125 like some of you x have already moved into trying different questions and 126 some of you x were revising the question you already had 127 and adding more x details to it

Table 30. continued

Making

ofModel

Use

and Experiences Ideas

Metacognition

Making/Reflection

Message Unit Individual

In the next section, I expand on how the components of Mae’s SMP promoted her view of mathematical modeling as a bridge and outline the explicit process which Mae used to accommodate this view.

4.5 Mae’s View of Mathematical Modeling as Bridge

In the course of discussion during which she facilitated learners’ modeling work components of Mae’s practice emerged. While the literature outlines and distinguishes between the epistemological view of mathematical modeling as content and vehicle (Julie

& Mudaly, 2007; Galbraith, Stillman, & Brown, 2010), Mae’s view of mathematical modeling was that of a bridge between these two views. On the one hand, Mae adopted a holistic view of modeling as the interaction between mathematics and real life (Kaiser,

2007). On the other hand, she also supported the organization of social practices to establish arguments and support their decisions, adopting a socio-epistemological perspective (Cantoral, Moreno-Durazo, & Caballero-Perez, 2018; Arrieta Vera & Diaz,

2015). In this way, the curricular objectives, alone, did not drive her teaching process of mathematical modeling, but are coupled with the stance that learners should be equipped to consider real-world problems through the lens of their own experiences and draw on conceptual tools to support developing a well-conceived solution. Mae’s view of modeling as bridge emerged as she vacillated between elaborating student mathematics and formalizing curricular mathematics (Table 31). In this way she was able to reconcile both the goals of advancing curricular knowledge and supporting learners’ real-world problem-solving and decision-making.

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Table 31. Transition between Elaborating and Formalizing

Elaborating Formalizing Student Curricular Event Phases Mathematics Mathematics The Answer is Pre-Discussion (Course Objectives) Introduction of Answer Is Task x Discussion of Learner Interpretations of Answer Is Task x

Establishing Explicit Criteria for Answer is Task x Student Generate Solutions for Answer Is Task x Teacher Monitoring of Student Solution Generation and x Scaffolding Responses Sharing Solutions to Answer is Task x Elaborating A Lot a Lot x Teacher summary of Answer is Task and Transition to x

The Answer Is The Task Measure the Height Introduction to Measure the Height Task and x x Demonstration of Inclinometer Establishing criteria for Measure the Height Task x Students work to determine height of three objects three x

different ways

Teacher supports learners in strategy selection and x implementation Debrief discussion of Measure the Height x Sharing solutions for Measure the Height Task x Teacher introduction of Sine, Cosine and Tangent as x x one way to determine height Students practice Sine, Cosine and Tangent Problems x Validation of Measurements for Measure the Height x X

Measure the Height Task the Measure using Sine, Cosine, and Tangent

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Table 31. Continued Elaborating Formalizing Student Curricular Event Phases Mathematics Mathematics Shortest Distance to Lunch Room Pre-Discussion Teacher Monitoring of Shortest Distance x Shortest Distance To Lunch Room Debriefing x Discussion Connecting Measure the Shortest Distance to Triangle x x Concepts

Students Determine Angle Measurements on Aerial x x

Map (Protractor) Comparing Measurement Discussion x x Generating Questions from Map x Summary of Shortest Distance Task

Lunch Room Lunch Sharing of Strategies for Determining the Missing Sides x to to of Triangles Validation of Measurements (Outside) x Comparing Collected and Calculated Values x x “Best Guess” Discussion Using Averages x x Teacher Discusses Law of Sines Calculator Activity x Students Work Through Calculator Lesson on Exploring x

Shortest Distance Distance Shortest Law of Sines Teacher Introduces Concept of Angular Speed and x Velocity Introduction of Trig Whip Activity Teacher Monitoring/Scaffolding of Student Work on x Trig Whips Trig Whip Debriefing Discussion x Introduction of Angular Speed Formulas x x Applying Angular Speed to Trig Whip Discussion x x

Derivation of Angular Speed from Linear Speed and x Arc Length Teacher Monitoring of Computing Angular Speed of x x Trig Whips Teacher Monitoring of Trig Whip Extension x x Angular Speed and Extension Final Discussion x

Trig Whips Task Whips Trig Practicing Angular Speed Problems x

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Table 31 depicts the transition across events between elaborating student mathematics and formalizing curricular mathematics. In every event, Mae began the discussion by elaborating on student mathematics and transitioning into formalization. In this way, she began by always valorizing learners’ ideas and then supporting the mathematization of those ideas. In addition, these elaborations were sustained across the formalizing discussions through intercontextual links to those solutions. As depicted previously in the event maps, Mae typically used student solutions to anchor the development of her intended curricular goals and, more importantly, portrayed them as not the means for solving the problem, but as an alternative means. In this way, she maintained the validity of learners’ solutions and ideas in the modeling process, while, at the same time, advanced her own agenda.

In order to validate these findings, I discussed with Mae her rational for implementing tasks and factors that motivated her decision to incorporate modeling into her instruction. In this interview, she informed me that her number one factor for implementation of mathematical modeling was that she felt comfortable in anticipating student contextual considerations and advancing them mathematically. This confirmed why occasions of elaborating student mathematics were so frequent. Mae also indicated that she, herself, needed to have a clear path linking her desired curricular objectives to the context. That she needed to be able to construct a mathematical model using the mathematics she wished to convey in order to support this transition. In the facilitation process, Mae argued that she also needed the time to allow students to explore their own solutions to completion and still have time to teach the necessary concepts. As such, Mae

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opted for using tasks where she could either hit on remedial skills for the forthcoming lessons or hit on multiple standards within those modeling tasks.

Following the implementation of each modeling task, I conducted another brief reflection interview with Mae asking her about whether she would adjust any of her pedagogical or task choices. It was in these interviews that Mae reported feeling positive about their overall implementation because of what she was able to learn from her students about their mathematics. As an example, in the measure the height task, Mae voiced her reflection to the class after they had just submitted their solutions. In this reflection, Mae indicated that this modeling task acted as a pre-assessment for a unit of instruction and that it served that function adequately. Mae facilitated modeling tasks supporting the development of those components most closely associated with modeling as content, such as thinking about the real world; establishing variables; refining; validating; researching; asking contextual questions; and physical, conceptual, and experiential tool use, then, through discussion, transitioning to developing curricular knowledge while still maintaining the integrity of student solutions for the task. It was in this way that modeling as bridge was established.

In the next section, I offer an analysis of those particular scaffolding techniques which Mae used within her interactions with students and highlight the sequence that afforded her SMP to manifest itself.

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4.6 Mae’s Scaffolding Interactions

In order to address my second research question on how modeling capacity was nurtured among learners, it became necessary to look at the particularities of discussion in terms of how Mae scaffolded students into her view of mathematical modeling. To examine teacher scaffolding, I looked to conversational functions (Green & Wallat, 1981;

Bloome, 1989) which emerged through modeling discussions in different events.

Bloome et al. (2010) indicates that conversational functions aren’t a set of discrete and mutually exclusive categories, but, instead, highlight the ways in which teachers and learners respond to each other (p. 68). I identified those prevalent conversational functions among the data as provoking, acknowledging/validating, clarifying, elaborating, emphasizing, and positioning (which includes establishing/stating and sharing), each of which will be discussed, in turn.

4.6.1 Provoking

Classifying a message unit as a provoking action involves observing inciting an elicited response. As reflected in previous discussions, these provoking actions

(identified in red within the analysis) were observed through Mae provoking sense- making and reflection or provoking learners to use their experiential tools or in thinking about the real world. Analysis revealed that provoking students to consider these contextual factors afforded Mae the opportunity to support the elaboration of student mathematics by foregrounding those ideas. This process is depicted in Table 28 when

Mae encourages her students to share their strategies.

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4.6.2 Acknowledging/Validating Learners’ Ideas

Acknowledging and validating learners’ ideas (depicted in orange in Table 20) is characterized by those ways in which Mae affirms student ideas and positions them as viable. This component is seen typically following her provoking actions and is foundational to elaborating on learners’ mathematical ideas. Across these modeling encounters, Mae engages in the process of reflexive listening, validates some aspect of the students’ ideas, and then expands on their validity either by affirming their contextual thinking or through the elaboration of their mathematical ideas. These components are manifested when Mae responds initially to learners’ ideas. This is exemplified in Table

20 on line 126 in how she responds to John’s claim of tax being irrelevant if purchasing during a tax-free weekend.

4.6.3 Elaborating

Mae engaged in elaborating by adding additional details to learners’ statements during the orchestration of discussions. There were two distinct ways in which Mae elaborated on learners’ ideas, mathematically and contextually. Mathematical elaboration was described earlier in this document as a distinct component of her SMP. Contextual elaboration occurred when Mae interjected her own experiences to support learners’ contextual ideas. For example, during the sharing of the municipal tax problem, Mae asked her students if they understood what municipal tax was (Table 32, Line 559) and

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then elaborated on what that tax rate is and how it impacts the price of purchase (lines

561-565), offering additional information on the impact of that variable.

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Table 32. Elaborating on Municipal Tax

Making

ofModel

/Quantifying

and Experiences Ideas

Metacognition

Making

ormalizing

Message Unit Individual

Line Speaker ProblemGeneration CommunicatingIdeas/Process Parameterization Refinement WorkingMathematically Assumption ThinkingtheWorld Real about Validationof Model Collaborating Use Technology Researching Variables Establishing Sense Mathematical Asking Questions Contextual Asking Questions Thinkingabout Mathematics SharingSolutions Humanizing EnculturatingMathematics into Activating F 549 T So when I cut you off yesterday X you were in the middle of what part of your research 240 550 Sx We were finishing up tax x 551 Kr Like adding up the tax in Ohio x X

552 Be Conor was trying to figure out x what it would if if the whole tax, including tax 553 T so he was trying to find what it x x would be before tax 554 So working backwards X 555 like like here's this price X 556 I gotta figure out what tax works x x x x x for this area or 557 Co Well I just took the average for x X x x the state of Ohio for municipal then took the state tax 558 T Interesting x x x x x 559 do you guys know what he means x x by municipal tax 560 Sx Local X x 561 T Like the city tax x x 562 for instance if you buy something x x at Polaris which is Franklin County the taxes are going to be different than if you go over to Newark and buy something

Table 32. continued

estions

Making

ofModel

/Quantifying

and Experiences Ideas

Making

ormalizing

Message Unit Individual

Line Speaker ProblemGeneration CommunicatingIdeas/Process Parameterization Refinement WorkingMathematically Assumption ThinkingtheWorld Real about Validationof Model Collaborating Use Technology Researching Variables Establishing Sense Mathematical Asking Qu Contextual Asking Questions Thinkingabout Mathematics SharingSolutions Humanizing EnculturatingMathematics into ActivatingMetacognition F 563 they have a different sales tax x x amount 564 where if you go down to x 565 I think the most expensive one is x X 241 down by Cincinnati

4.6.4 Emphasizing

I define emphasizing to be those ways in which Mae explicitly referenced the use of particular components of her SMP or specific contextual factors which she wanted to focus students’ attention on. This scaffolding technique is depicted in Table 23, lines

161-163, where Mae discussed how use of physical tools, such as technology, could support their establishment of variables by helping them research those contextual factors. In this way, Mae is elaborating on appropriate tool use by describing contexts in which it is appropriate and, in this process, emphasizing the component of establishing variables and constraints. In this way, these scaffolds are tightly linked together with the components of Mae’s SMP and, without careful and descriptive discourse analysis, would not be able to be identified.

4.6.5 Clarifying

Mae engaged in clarifying by restating or expanding on a particular issue or point which was addressed by her or her students in order to promote the elaboration of student mathematics through their real-world considerations or within the process of formalizing curricular mathematics by clarifying how the specific mathematical content offers utility in solving those contextual issues. As depicted in Table 32, Mae engaged in process- based clarifications as she elaborated on the students’ process of computing municipal tax

(lines 555-556). In this example, Beau clarifies that he was attempting to determine the municipal tax rate based on comparing the state tax with their municipality (line 552).

Mae clarifies this process by stating that these students were working backwards by

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starting with the purchase price and trying to figure out which tax worked for that price

(line 555-556). This clarifying action gave Beau the opportunity to elaborate on his process and also allowed the rest of the class to better understand his thinking. Further clarifying afforded Mae the opportunity to determine if her observations were indeed valid by checking with the students in her class and offering her insights into their processes, as well.

4.6.6 Positioning/Stating

Finally, taking a particular stance or making a particular directed claim exemplified the act of positioning. Mae used positioning to adopt a specific contextual or mathematical viewpoint and expand on its relevance contextually or mathematically.

From a mathematical standpoint, this scaffolding mechanism supported how Mae formalized curricular concepts. During the trig whip task, Mae formalized the concept of angular speed as 푣 = 푟휔 and established each variable in this equation as a realistic factor within the problem. She then encouraged students to conceptualize where those variables occurred in the context of the trig whip activity as a way to further signify their relevance. This is depicted within the analysis of the trig whip problem (Table 17) in the vacillation between elaborating on student mathematics and formalizing curricular mathematics. Formalizing the relationship between angular speed and focusing students on their contextual implications supported the development of their modeling processes.

It is interesting to consider that these scaffolding interactions parallel the ethnomodeling process. Provoking students to share their own ideas and insights into the

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problem allowed Mae to discover the emic ideas which her students held. By acknowledging and validating those ideas, Mae valorized the deep and complex ways in which her students considered the problems. In elaborating and clarifying, Mae engaged in the process of translation, where, through dialogic inquiry, a shared view of learners’ ideas and solution methods were established. In emphasizing and positioning, Mae further elaborated on the connection between the emic ideas and etic mathematics in her curriculum. It was in this way that Mae herself engaged in the process of ethnomodeling to inform her teaching practice, and it was through the orchestration of discussion that this process ensued.

It was through the interaction of these scaffolding interactions that Mae was able to enculturate her students into her sociomodeling practice and accomplish her curricular goals. Based on this analysis, in the next section, I offer my broad conclusions on the development of SMP through modeling as bridge and offer implications for teaching, learning, and future directions for research.

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Chapter 5: Conclusions and Implications

The purpose of this study was to develop a theory for the teaching of mathematical modeling by examining how a teacher facilitated mathematical modeling tasks with her students, along with what students seemingly came to learn through teacher scaffolding interactions. In this chapter, I will address my findings relative to the guiding research questions:

1. How are social practices in mathematical modeling contexts developed through

implementation of modeling tasks?

2. How does teacher scaffolding within these mathematical modeling contexts

promote the development of modeling capacity in learners?

Data analysis revealed that in establishing a sociomodeling practice the teacher relied on her professional vision (Goodwin, 1994) and pedagogical resources when planning for and facilitating tasks. Components of sociomodeling practice emerged as a result of the teacher’s enactment of these tasks. In facilitating task enactment Mae treated modeling neither as a content, nor as a vehicle but as a bridge between these two methodological approaches to implementation. This view of modeling emerged as she equivocated between elaborating student mathematics and formalizing curricular mathematics. These results are summarized below (Figure 25) and elaborated on in this chapter.

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Figure 25. Overview of Significant Findings

5.1 Research Question 1

How are social practices in mathematical modeling contexts developed through implementation of modeling tasks?

Lesh, Hoover, Hole, Kelly, and Post (2000) argue for the importance of task design in fostering modeling capacity in learners. While task design is indeed important to advancing this goal, data and analysis from this study revealed the importance of targeted discussions in which components of mathematical modeling are deliberately addressed so as to enculturate students into a sociomodeling practice. It is through the interaction of tasks with careful orchestration of modeling discussions that modeling capacity was promoted. Data from this study suggests that, when adopting an ethnomodeling perspective, local cultural practices indicative of mathematical modeling within a classroom culture become apparent. Through the processes of discovery and

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translation one can better identify the cultural components which inform this practice.

Analysis revealed 23 distinct components which informed the participating teacher’s view of mathematical modeling, and it is through enactment of these components that mathematical modeling practices was supported in the classroom.

5.1.1 Modeling as Bridge

In looking at the development of modeling capacity in learners, this study identified 23 explicit components referenced by the teacher when interacting with students as they worked on modeling tasks. These components included (1) provoking sense-making and reflection; (2) conceptual tool use, such as mathematics; (3) thinking about the real world; (4) thinking about mathematics; (5) experiential tool use; (6) elaborating student mathematics; (7) enculturating students into academic mathematics;

(8) establishing variables or constraints; (9) asking mathematical questions; (10) teacher humanizing as a co-participant; (11) physical tool use; (12) formalizing curricular mathematics; (13) asking contextual questions; (14) communicating ideas; (15) sharing solutions; (16) collaboration; (17) validating models; (18) refining models; (19) research;

(20) assumption-making; (21) establishing parameter values; (22) problem generation; and (23) activating metacognition, and were detailed in the previous chapter. Each of these components informed the mathematical modeling process and was emphasized differently across different phases of task enactment. It was observed that students took up those components most frequently emphasized or elaborated on as they engaged in

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modeling practices. Further, Whole group discussions served as the primary vehicle for the teacher to target these components of the modeling practice.

Keiran (2003) identifies problem solving, critical thinking, reasoning, and modeling as global meta-level skills involved in the learning of mathematics. These skills are central to mathematical thinking as they support learning of abstract concepts.

Data analysis revealed that the teacher’s emphasis on real-world aspects of a problem context targeted these metal level skills as she held students accountable for identifying contextual factors important to analyzing modeling tasks. In facilitating modeling, it was also observed that the teacher attempted to balance multiple goals and objectives as she navigated her daily instruction, ranging from curricular, mathematical, pedagogical, personal, and ideological goals. It was through reconciliation of these goals that a particular focus across a discussion emerged. This implies that discussions that focus on unpacking contextual factors can support learners in considering variables or factors central to that system. While past research has indicated that novice modelers find it difficult to strip away irrelevant or unnecessary contextual factors (Haines & Crouch,

2010), this challenge can be met through discussions in which students are encouraged to explore and contemplate the relevance of various factors on the derived solution. The analysis also revealed that an explicit focus on certain components, in particular, thinking about the real world, sense-making, elaborating students’ mathematics, and formalizing of curricular topics allowed the teacher to fulfill both short- and long-term instructional goals. For example, in opting to implement the Measure the Height problem as a mathematical modeling task rather than a basic measurement task, the teacher reconciled

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teaching particular aspects of modeling as content, while using the task itself as a vehicle for teaching desired content. Rather than demanding that students use specific mathematical techniques to solve this problem, the teacher allowed students to consider the problem in their own ways. She then supported their mathematization process drawing on learners’ own considerations. This approach validated students’ ideas both mathematically and personally. The teacher accomplished this by stressing the importance of thinking about the real world, and sense-making. When she transitioned to formalizing mathematical ideas, different components were stressed to meet specific content objectives. Combining contextual and mathematical ideas allowed the teacher to advance mathematical modeling and curricular mathematics in tandem.

Certain barriers did impact the frequency of enactment of mathematical modeling tasks. Mae was willing to implement modeling tasks when she felt confident in anticipating contextual and mathematical elements that students may reference when examining particular problems. She considered also whether or not the tasks could be linked to mathematical objectives of her curriculum. Of important to her was also her assessment of whether or not she could carry out the activity in a timely manner. With respect to the ability to anticipate learners’ contextual and mathematical considerations

Mae believed it essential to take into account learners’ assumptions and interpretations when making mathematical modeling a real and meaningful process to them. This desire greatly impacted her choices. That is, if she had limited knowledge or experience with a particular context, she felt it necessary to either research for additional information she

249

felt she needed to productively guide students or abandon the use of modeling in favor of other instructional methods.

Mae also believed it important to be able to recognize connections between student-anticipated considerations and her own curricular objectives. In places where she felt secure in this domain specific knowledge she explored students’ problem solving strategies and connected their ideas to her mathematical goals. In contrast with traditional teaching practices where the teacher acknowledges one solution as optimal

Mae valorized different solutions and highlighted their merits. In transitioning to academic mathematics, she often stressed that her approach to a problem was only one, and not the, way to solve the problem. In this way, modeling was treated not necessarily as a vehicle towards reinforcing a particular mathematical concept or approach, but rather as a bridge between student strategies and academic mathematics.

It was Mae’s vision (Goodwin, 1994) which allowed her to enact modeling tasks sensitive to learners’ ideas while keeping an eye on her curricular objectives. If she was unable to recognize an explicit connection to mathematics, she was less likely to facilitate modeling skills within that topic. Her follow-up questions on tasks she implemented aimed to first elaborate on ways that learners had interpreted the context, and she then introduced mathematics to support students’ ideas.

5.1.2 Viewing Mae’s Practice Through an Ethnomodeling Lens

Mae’s enactment of modeling tasks was consistent with the ethnomodeling cycle.

Frequently, she relied on her etic mathematical lens to better understand how learners were interpreting and solving tasks. She engaged in translating ideas as she monitored

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her students’ work on tasks and supported their mathematization process. This came through reflexive listening to how different students considered a problem and acknowledged those ways as viable solutions.

In considering how Mae supported model validation and refinement, observed interactions revolved primarily around the development and expansion of student considerations and linking those ideas to mathematical concepts with which they were familiar. This practice is consistent with the elaboration phase outlined in the ethnomodeling cycle and consistent with practices which connect cultural ways of problem-solving with the etic mathematics of the observer (Orey and Rosa, 2017, 2016,

2015, 2010). Analysis revealed further that the process of validating and refining models was also exhibited as Mae formalized curricular mathematics. She acknowledged students’ models as viable solutions, and then she introduced mathematics as a way of validating their solutions.

Previous research reports identify major scaffolding techniques in mathematical contexts as the teacher shifts learners’ considerations to idealized models or mathematics

(Skakjlow, Kolter, & Blum, 2015). Contrasting this notion of scaffolding to modeling as a bridge, teacher sensitivity to learners’ ideas, experiences, intuitions, and their genesis informed Mae’s interventions in the modeling process, in that those ideas were given contextual merit and teaching involved supporting students in elaborating on those ideas.

5.2 Research Question 2

How does teacher scaffolding within these mathematical modeling contexts promote the development of modeling capacity in learners? 251

In considering teacher scaffolding and its impact on the development of modeling capacity in learners, analysis revealed the importance of coordinated discussions throughout the modeling process. Further, repeated and sustained reliance on SMP components motivated students to uptake those components during task enactment. As an example, when two students (Alyssa and Amber) were measuring the height of three objects, they considered the contextual aspects of those objects and how they impacted their measurement techniques and the assumptions they had made about the impact of the slope of the ground. As a result of this contextualization process, Alyssa and Amber established a shared definition for what constituted the height of an object and then validated their measurements. They made sense of their calculations and revisited their initial estimates. The components of the SMP that the teacher referenced acted as a scaffold supporting their work.

With regards to scaffolding practices during task enactment, analysis also revealed that when learners encountered barriers in the modeling process, teacher’s reflexive listening was foundational to her ability to advance learners’ mathematizing process, allowing the students to re-enter the task with greater accuracy and success.

Instances were observed where the teacher was unsure of how to progress students based on what they had initially considered. In these episodes, she disclosed her vulnerability, but continued to research learners’ ideas herself in order to offer pedagogical guide that could advance their work. This approach demonstrated to learners the need to acquire additional contextual or mathematical information to better develop their models. In this way, Mae further modeled productive modeling practices. 252

During the whole group discussion of tasks Mae tried to make connections across multiple solutions (mathematical and contextual) by carefully selecting and structuring students’ responses. She used these post-task discussions as occasions to bridge the gap between learners’ solution strategies and the mathematical ideas which she wished to convey. These discussions served as a bridge connecting students’ ideas to conventional mathematical representations.

Finally, it was common for Mae to interject and introduce a modeling stance across seemingly non-modeling experiences. When a contextual question was raised or referenced, she deviated from her intended plan to highlight the connection between what was proposed to modeling process or previous modeling sessions. In this way, intercontextual links were established across tasks, and previous and current experiences were referenced to reinforce modeling practices in various mathematical events. Mae’s desire to create and reinforce intercontextual links nurtured learners’ modeling capacity across tasks and over time.

In summary, teacher scaffolding mechanisms occurred through the orchestration of modeling-related discussions in which learners’ ideas and experiences were actively explored and developed in mathematical ways, starting from their views and transitioning to idealized mathematical ways of considering those contexts. In this way, Mae was able to simultaneously develop processes of mathematical modeling as a content along with mathematical modeling as a vehicle.

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5.3 Teacher Decision Space and Pedagogical Vision

Goodwin (1994) characterized professional vision as those specific and specialized ways which members of a group use to examine events of interest to them.

With respect to the practice of teaching, this professional vision allows teachers the ability to notice and interpret actions within a classroom context (M. Sherin, Russ, B.

Sherin, & Colestock, 2008).

In establishing insight into Mae’s decision-making as she facilitated mathematical modeling tasks, I consider Professional Vision in concert with Resources (Schoenfeld,

2009) to unpack her practices. Schoenfeld (2009) coined the notion of professional vision and argued it as the driving force for what transpires in the course of interactions around a professional agenda. According to Schoenfeld (2009) understanding these visions is central to research around practice in a natural setting. He urged researchers to remain sensitive to the cultural context of the site and encouraged collaboration between curricular designers and researchers with a proposed agenda, which includes: (1) the means by which experienced designers make principled decisions as they undertake revisions, (2) the orientations and goals which shape their decision-making, and (3) the overarching knowledge base involved, i.e., the design principles and techniques, heuristics, and professional vision drawn on in this decision-making process (p. 18).

In contemplating how professional vision can foster growth of modeling capacity and how Mae relied on her vision and resources to advance learners’ modeling practices while accomplishing her curricular goals, analysis of the data revealed that the careful and deliberate structuring of lessons and task treatment were key players. Across this

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study, I conducted numerous reflective interviews with Mae in which she openly discussed her goals, both short- and-long term; reflections on classroom activities she used; insights into her practice; and future directions. During these discussions, she frequently addressed ways in which she could engage students in the mathematical modeling process, while, at the same time, not lose instructional time needed to cover those overarching mathematical concepts which the curriculum demanded. Mae’s practice was consistent with this vision. Her instruction was framed with a reflexive stance where she valorized learners’ intuitions, background knowledge, experiences, and the like, while, at the same time, enculturated students into curricular mathematics, allowed Mae to balance her short and long-term instructional goals.

Figure 33 offers a conceptual model of Mae’s decision making space

(Manouchehri, in progress) as she planned and facilitated lessons over the course of one instructional unit. Mae’s long-term goals, which she expressed during the interviews, were for students to not only develop modeling capacity, but also to realize the value of mathematics in making sense of everyday life. She was specific about this particular goal and stressed it throughout the data collection phase.

I distinguish between these two types of goals that mediated Mae’s interactions with her students to further frame her decision-making during the course of instruction.

Notice that, under the category of short-term goals, I identify the mastery of specific mathematical concepts and, in particular, those that were observed over the course of data collection, especially concepts associated with trigonometry. In accomplishing these

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short-term goals, I have also listed those specific components indicative of modeling as a vehicle, as the framework aligns with developing curricular mathematics.

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Short-Term Goals Long-Term Goals

Mastery of Specific Development of Mathematics Mathematizing through • Trigonometry • Modeling Capacity • Pythagorean Theorem • Asking questions • Special Right Triangles • Problem Posing • Radian/Degree • Use of Experiments Conversions • Collecting and Analyzing • Angular Velocity Data • Etc. • Convincing • Argumentation Modeling As Vehicle • Sense-Making • Formalization 257 • Quantification Modeling As Content • Computation • Relevance of

• Contextual Meta-Level mathematics in daily Components lives • Asking and answering Short-Term Modeling contextual questions Discussions • Making Sense of Real World

Sustained Modeling Discussions

Figure 26. Teacher Decision Space

In looking at the long-term goals of the teacher, I acknowledge the development of mathematizing or sense-making in mathematical ways. This came through the teacher’s sustained use of modeling questions, provoking students to ask contextual and mathematical questions, provoking reflection and insight into the dynamics of the real- world context and how mathematics might afford solutions to problems that arise. She stressed problem posing, collecting and analyzing data, convincing through argumentation, sense-making and logical reasoning. Similarly, these long-term goals relate to those components she most regularly expressed in class. Although Mae’s short- term goals were primarily linked to enacting specific tasks or initiating specific content- focused discussions to target growth in those areas, the long- term goals were integral components of her instruction and appeared frequently across the development of particular concepts. It is also important to note that while some of Mae’s long-term goals were targeted because of their relationship to specific curricular mathematics, other long- term goals surrounded supporting student mathematization and developing modeling capacity. Due to this, building mathematical dispositions and engagement in mathematical modeling are tightly linked. They are motivated by targeting the long-term development of students in terms of understanding the applicability of mathematics in making sense of and solving real-life problems.

In considering how a teacher accomplishes their short- and long-term goals related to their professional vision, Schoenfeld (1985) discusses the importance of the resources available to individuals to support the advancement of knowledge in a particular domain. Resources, according to Schoenfeld, are an inventory of what an

258

individual has at their disposal (p. 46) and “the informal and intuitive knowledge about the domain; facts, definitions, and the like; algorithmic procedures; routine procedures; relevant competencies; and knowledge about rules of discourse in the domain” (p. 55).

While primarily linked to aspects of mathematical problem solving, it is important to consider how teachers draw on their resources when making pedagogical decisions in solving problems pertaining to the teaching of mathematical modeling and reconciling views of modeling as content and modeling as vehicle.

One key resource in Mae’s willingness to implement mathematical modeling tasks was her ability to relate an overarching context to the mathematical concepts which she wished to impart. This notion of mathematical access is consistent with that of modeling as vehicle. If Mae had difficulty in recognizing how this context could be mathematized in a way that aligned to desired topics, it was less likely that modeling attempts would be pursued. Modeling discussions did emerge across lessons, and modeling practices were emphasized when they arose in the moment, but the teacher’s ability to mathematize played a direct role in task facilitation.

Another resource that seemingly impacted implementation of modeling task was that of time. Mae recognized that enacting modeling tasks with fidelity took time to explore and discuss. It was only when Mae felt comfortable that through modeling she could accomplish her short-term content specific goals in a timely manner that she was willing to implement such tasks.

The teacher’s mathematical and contextual knowledge also served as resources that shaped her practice. Her ability to recognize explicit mathematical ideas that

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students considered when working on tasks was observed to support her scaffolding approaches. In the measure the height task, Mae engaged in reflexive listening, in which she attempted to make sense of her students’ ideas. She adjusted her interactions based on those interpretations. She was open during the debriefing interviews as she expressed that her task selection was motivated by her ability to recognize the underlying mathematics and formalize those concepts.

Mae’s long-term goal of developing mathematical modeling skills was aligned to a deep and intimate understanding of the modeling process, coupled with the belief that developing this capacity was fundamental to the learning of mathematics. This knowledge and view of the modeling process acted as a resource for advancing modeling capacity, as it helped inform when modeling discussions were initiated and pursued across lessons and how those modeling experiences were re-addressed across the unit of instruction. For example, the measuring the height task wasn’t merely facilitated and then abandoned, but, instead, considerable time was devoted to unpacking the work, connecting learners’ ideas to etic mathematical concepts, and creating intercontextual links to those experiences through additional discussions, future modeling tasks, homework assignments, and summative assessments. It is important to note that these resources aren’t mutually exclusive and that they do not occur in isolation; instead, they are deeply integrated into the teachers’ belief structures.

A teacher’s vision and resources impact how modeling is facilitated or nurtured in classrooms. It is, therefore, important to support the development of a professional vision which allows teachers to accomplish their content-focused goals, which are typically

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mandated by a curriculum, while using modeling as a vehicle for learning this content.

Further, teachers need to be equipped with an arsenal of resources that can support their attempts at implementing modeling tasks. Among many include: deep mathematical content knowledge to support mathematizing of learners’ considerations as they engage in tasks; familiarity with contextual factors embedded in tasks, and vision to see how contextual interpretations can bridge into curricular topics.

5.4 Recommendations

The findings of this study give light to recommendations for the development of teaching, teacher education, and research in mathematical modeling. In this section, I outline the major implications of this study, discuss limitations of this research, and offer concluding remarks and directions for future research.

5.4.1 On Teaching

Regarding the teaching of and teacher development for mathematical modeling,

Borromeo-Ferri (2018) indicated a need for more research on the type of interventions which are appropriate for model-based instruction, in particular, those that are adaptive to learners’ difficulties in modeling (p. 2). In this work I offered a detailed analysis of how a classroom teacher implemented mathematical modeling tasks in a way that was sensitive to learners’ ideas, intuitions, and background knowledge as she facilitated their work on modeling tasks . In adopting contexts with visible solution paths aligned to curricular mathematics, the participating teacher in this study was able to reconcile her

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short- and long-term goals. In orchestrating discussions in modeling contexts, it was also revealed that alignment between the phases of ethnomodeling (Rosa & Orey, 2015) promoted equitable interactions in which student ideas were valorized or challenged and then mathematized as viable solutions. Further, through the careful selection of tasks

(Lesh et al., 2010), modeling was advanced as a bridge.

To support the development of teachers’ use of mathematical modeling in their classrooms, it is productive to: (1) expose pre-service and in-service teachers to the ethnomodeling cycle as a teaching framework with support for orchestrating modeling discussions through that cycle, (2) allow teachers to engage in mathematical modeling as learners to better understand the process, and (3) brainstorm ways in which these tasks could be linked to curricular topics. Cursory and episodic exposure to modeling tasks is not sufficient in providing teachers with a vision of how modeling tasks might be taken on in a way that encourages finding multiple solutions and leads to formalization of mathematical concepts. In this light teachers need to explore tasks both mathematically and pedagogically where they gain experience with how to link student solutions to curricular ideas. Further, the practices used for the orchestration of modeling discussions should be distinguished from those for other mathematical topics, with explicit emphasis given to allowing agency for learners’ ideas and transitions throughout the modeling phases.

Manouchehri (2017) called for additional research on generating theoretical descriptions which support teachers in developing tools for navigating instruction and assessing mathematical modeling. Further, she argued that teachers should be supported

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in overcoming the mathematical, epistemological, and pedagogical barriers to teaching mathematical modeling. Data highlighted that it was only when Mae recognized explicit connections between modeling context and curriculum that she embarked on modeling.

Further, time constraints in terms of facilitating tasks which were sensitive to learners’ ideas impacted her willingness to engage in modeling tasks. Supporting teacher development in light of these findings mandates that teachers recognize explicit ways in which modeling tasks connect to their curriculum, coupled with adequate time to enact these tasks with fidelity.

Contemplating ways to further support teachers in connecting modeling experiences to curricular mathematics requires time to examine pathways between task and desired mathematical concepts and, further, how to connect thoughtfully anticipated strategies without imposing a particular method on this work. As a final step, the teacher looks across these ideas and develops an implementation plan to use in the classroom.

5.4.2 on Teacher Education

Mae offered an example of a dedicated teacher committed to supporting mathematical modeling practices in her classroom. She was knowledgeable about the mathematical modeling but, at the same time, reluctant to initiate tasks if she was unsure of how tasks could be linked to her curriculum. This implies that illustrative examples of mathematical modeling experiences which are content-based need to be discussed, in great detail, in teacher preparation programs. While experiences with mathematical modeling processes as learners is necessary for teachers to develop an understanding of

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the process , these experiences are not sufficient in advancing the pedagogical knowledge of teachers pertaining to implantation of mathematical modeling. Explicit discussion of how modeling as a bridge might be enacted can assist teachers in nurturing modeling skills whilst remaining responsive to students’ ideas and desired mathematical outcomes.

The results of the study revealed that teacher beliefs, professional vision, and resources were instrumental in how and whether modeling tasks were facilitated in classroom. While past research has considered the impact of teacher knowledge and beliefs on practice (Rowland & Ruthven, 2011; Rowland, Huckstep, & Thwaites, 2003), in this study, the teacher needed to see the mathematical destination of the tasks in order to use it. Rowland Huckstep, and Thwaites (2003) coined the term contingencies as a teachers’ response to events which may not have been planned in advance of instruction including misconceptions that students bring forth that deviate from a planned lesson (p.

98). While such deviations were exhibited by Mae, it was her vision and ability to anticipate learners’ responses to tasks that informed her interactions across tasks that motivated her shifts. In other words, it was not that Mae was hoping that she would be able to deal with such unexpected responses, but that she needed to be able to identify a link between tasks, learners’ ideas, and desired outcomes. She was willing to improvise when she felt secure in her ability to see the ultimate mathematical outcome. This was informed by her global view of modeling as a fundamental component of her instruction.

This suggests that teacher education programs may need to make front and center the improvisional nature of teaching and the important role that teacher curiosity and willingness to deal with ambiguity plays in effective teaching. This disposition is

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certainly most important in places where open ended tasks, such as modeling situations, are used in instruction.

5.4.3 On Curriculum Development

Mae’s attempts at relying on modeling as bridge merits further attention on how modeling as content and modeling as vehicle might be linked, specifically in areas that teachers may not have recognized ways of how mathematical concepts may apply to real- world issues. In recent years ample resources and research-based modeling tasks have been produced on modeling tasks appropriate for educational use. However, evidence exists that these resources are currently underused by classroom teachers. This, I suspect, is due to an absence of detailed instructional resources for teachers regarding their implementation intricacies. Modeling as a bridge, as theorized in this study, requires delicate treatment of mathematics, mathematical modeling, and those pathways that effectively bridge tasks to curricular objectives. These knowledge bases are sophisticated and need time to develop. As echoed by research of the 1990’s and 2000’s on reform- based curriculum implementation instructional guides for teachers play an integral part on whether materials are used by teachers as designers had intended (Van de Walle,

1990; Ball & Lubienski & Mewborn, 2001; Handal & Herrington, 2003; Ball 1994).

Consistent with the findings of previous research, results of the current study further highlight the need for development of teaching resources, such as facilitation guides, highlighting ways to address multiple solution paths and how each path could target specific curricular outcomes of modeling based tasks. Simultaneously, there exists a

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need for implementation plans and mechanisms for orchestrating classroom discussions in which learners’ ideas are explored and analyzed against a common task.

Julie and Mudaly (2007) state that modeling as content and modeling as vehicle are not dichotomous; however, there are limited studies which adopt this spectrum approach, and most studies typically discuss modeling from one aspect, content, or vehicle. In looking at Mae’s practice, it could be argued that her global treatment was that of modeling as a vehicle. The difference, however, resided in her worldview about mathematical modeling as not only as a motivator for mathematics learning but an essential skill that needs to be developed. While Mae was motivated by a desire to link the modeling situations she used to curricular mathematics, it became evident that her approach vacillated between two goals, which at times were competing. In light of this result, I offer implications for future research by considering the means with which I was able to notice this subtle view.

5.4.4 On Research

This study offered one teacher’s conception of mathematical modeling for instruction by identifying components she emphasized in her interactions with students. It is necessary to examine alternate classroom contexts and study how other teachers link modeling to curricular mathematics. Further contemplation is also needed on ways in which learners’ ideas are used as a resource for mathematical modeling instruction and whether similar methods to those observed in Mae’s work are employed by others. More

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consistent and systemic research is certainly needed to operationalize such practices and examine its value to mathematical modeling instruction.

A number of findings of work reported here warrant future research and elaboration of theory. First, the components of SMP require further unpacking. Given

Mae’s advocacy stance towards the use of technology in modeling it is important to examine how digital technology and simulations support mathematical modeling. When considering how to motivate students to think about the real world problems and pursue solutions by elaborating on student mathematics, Mae referenced mathematical ideas with which learners were previously familiar. This aligned with Lesh et al.’s (2000) idea on the import of coordinating mathematical modeling task demands with concepts that students know. However, one aspect of modeling is that tasks may require the development or learning of new mathematics (Dym, 2004). More research is warranted to better understand how to support the development of mathematical modeling when learners’ ideas are hidden. When learners’ solution strategies differ across groups and each group requires different mathematical guidance to their solutions, it remains unclear how a teacher might best structure instruction to optimize learners’ work whilst remaining sensitive to time constraints.

Rosa and Orey (2013) argued for ethnomodeling as a useful methodological tool for conducting ethnomathematics research. Additionally, D’Ambrosio & Rosa (2008) framed ethnomathematics as a useful tool for examining cultural practices of mathematics outside of indigenous groups. This study extends the view of ethnomodeling and advocates its merit as a theoretical framework for conducting

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classroom-based research. Analysis of teacher interventions in this study aligned to the phases of ethnomodeling, but explicit use within teaching contexts is still under- theorized. Further studies that examine classroom ecology through the ethnomodeling lens are needed.

I was recently criticized that this research doesn’t fit within the paradigm of ethnomathematics, as I opted to study classroom culture in an affluent setting with an economically homogeneous population in the Midwestern United States. The ethnomodeling process however, isn’t exclusive of context as it affords a researcher the ability to account for, but not be blinded by, their own perspective. It allows the researcher a lens to understand practice from an insider’s point of view. The components of Mae’s sociomodeling practice were a result of her worldview, along with the context in which she taught, and coupled with the backgrounds and funds of knowledge of her students. Ethnomodeling certainly granted me the ability to “see” and “experience” her practice as the theory claims it to do. This mandates an expansion of the theory of SMP by looking at the construction of modeling practice in other settings, as those identified components are indicative of that local view of modeling. It is through the analysis of other classrooms that SMP theory could be generalized.

Pre-Calculus is arguably one of the few places in the high school curriculum in which modeling can occur easily. Regardless of the context, examining teacher knowledge and how that knowledge motivates or inhibits incorporation of modeling is important. Supplementary to this issue is how the orchestration of mathematical

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modeling discussions can motivate development of modeling capacity and, more still, what knowledge of mathematical modeling teaching might look like.

Finally, this study’s adoption of ethnomodeling, in concert with discourse analysis, offered great value in unpacking teacher practices across discussions. This leads to two major implications for advancing research in this area: first, other discursive tools should be contemplated in light of unpacking teacher practice, and, second and much bigger still, is how the adopting of multiple theoretical lenses to examine the same phenomenon can reveal insight into the analytical process. Ways in which other theoretical perspectives might complement each other in the analysis of classroom based data is integral to future research.

5.5 Limitations and Barriers

While this study yielded insights into the development of modeling practice, there are both methodological and theoretical limitations that could inform future work in this area. First, developing a trusting relationship with a teacher is time and labor intensive.

Hence, studies of the type reported in this work would be difficult to conduct across multiple classrooms. Prior to data collection for this study, a total of two years was spent getting acquainted with the cooperating teacher and her classroom contexts. This investment was a necessary mark of research informed by ethnomathematics.

Additionally, the analytical process of fine-grained textual analysis takes a considerable amount of time and energy, in particular, the development of a coding system. Having an established set of components to examine prior to making observations could be useful,

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in particular, general components to mathematical modeling, such as assumption-making, validating, refinement, and the component of thinking about the real world. Additionally, as much information as possible concerning the teacher’s short- and long-term goals prior to observation could be productive.

The intent of this study was to examine the relationship between practice, teacher scaffolding, and student learning. Due to time constraints. the first two aspects of this research were addressed, that is, how practice was established and how a teacher scaffolded students into this practice. Additional research is still needed to examine student learning within these modeling contexts. In this study, Mae made assumptions regarding students’ interests and background experiences and sought to validate her assumptions as she facilitated tasks. Developing a deeper understanding of learners’ background knowledge and cultural heritage could inform future analysis of how learners approach modeling contexts. Current research is underway which targets this aspect of practice, but additional research in this area is needed.

5.6 Concluding Remarks

Results of the study further punctuate the need for careful examination of the particular ways in which teacher interventions promote the development of modeling capacity in learners. I highlighted specific scaffolding methods which one teacher employed and ways in which she enculturated learners to desired modeling skills. It was revealed that those specific components of practice advocated by the teacher were most likely to be adopted by learners. It was also revealed that, when a teacher remains

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sensitive to curricular objectives and learners’ ideas, intuitions, and background knowledge, this sensitivity affords the development of modeling as a bridge between content and vehicle.

Recently, Cai et al. (2016) identified some key questions they believed worthy of further investigation to advance the research in the teaching and learning of mathematical modeling. From an instructional perspective, the report highlights the need to characterize essential features and essential habits of mind and competencies which need to be developed in order for students to become competent in modeling (p. 2). Results of this study respond to this call as I identified the SMP components of Mae’s practice. Cai et al. (2016) indicated also that research is needed to identify those factors which have an impact on learners’ generation of researchable questions (p. 3). Again this study offered development in this area by examining how Mae helped her students engage in problem- posing tasks as they inquired about and examined real life contexts. Cai et al. advocated for additional research into documenting and unpacking instruction when modeling is used in classrooms and, further, factors that motivate the selection of tasks and their use

(p. 4). This, they argue, demands examining the nature of classroom discourse which supports learners’ modeling practices. This study documented those factors and identified how teacher belief motivated the selection and scaffolding of modeling tasks.

Going beyond one classroom and one teacher provides addition needed insights on these important issues.

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Appendix A: Modeling Tasks

The Answer Is Task: The Answer is $73.13. What is the question?

Measure the Height Task: Measure the height of three different objects in three different ways.

Shortest Distance to the Cafeteria Task: Of the Upper, Middle or Lower School, which has the shortest distance to the cafeteria?

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Trig Whips Task: In groups, students stand shoulder to shoulder. One student shoulder acts as the center, and the group works together as a radius, walking in a circle. Each group records: • Individual student distances from center • Time required to complete 3 revolutions • Angular displacement Perform three trials, switching positions to experience the effect of being the outside point. Each team will need an official timer and the rest are whippers!

Measurement Activity: What is each team member’s distance from the center?

Outdoor Activity: Whip it good! How long did it take you to complete 3 revolutions, to the nearest tenth of a second?

Classroom Activity: Perform the following calulations. For the 3 revolutions, how far did each team member travel (in feet), what was their velocity (in feet per second), and what was their angular speed (in radians per second)?

Extension Activity: If you had 20 people in your group, how fast would the outside person have to “walk” to keep up?

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Appendix B: IRB Approval Letter

09/27/2016

Study Number: 2016B0308 Study Title: FOSTERING STUDENT GROWTH IN MATHEMATICAL MODELING CONTEXTS

Type of Review: Initial Submission

Review Method: Expedited

Date of IRB Approval: 09/26/2016 Date of IRB Approval Expiration: 09/26/2017

Expedited category: #5, #6, #7

Dear Azita Manouchehri,

The Ohio State Behavioral IRB APPROVED the above referenced research.

In addition, the following were also approved for this study:

• Children (permission of one parent sufficient)

As Principal Investigator, you are responsible for ensuring that all individuals assisting in the conduct of the study are informed of their obligations for following the IRB-approved protocol and applicable regulations, laws, and policies, including the obligation to report any problems or potential noncompliance with the requirements or determinations of the IRB. Changes to the research (e.g., recruitment procedures, advertisements, enrollment numbers, etc.) or informed consent process must be approved by the IRB before implemented, except where necessary to eliminate apparent immediate hazards to subjects. 286

This approval is issued under The Ohio State University's OHRP Federalwide Assurance #00006378 and is valid until the expiration date listed above. Without further review, IRB approval will no longer be in effect on the expiration date. To continue the study, a continuing review application must be approved before the expiration date to avoid a lapse in IRB approval and the need to stop all research activities. A final study report must be provided to the IRB once all research activities involving human subjects have ended.

Records relating to the research (including signed consent forms) must be retained and available for audit for at least 5 years after the study is closed. For more information, see university policies, Institutional Data and Research Data.

Human research protection program policies, procedures, and guidance can be found on the ORRP website.

Michael Edwards, PhD, Chair Ohio State Behavioral IRB

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