One of the important discussions in the philosophy of mathematics, is that centered on Benacerraf’s Dilemma. Benacerraf’s dilemma challenges theorists to provide an epistemology and semantics for mathematics, based on their favourite ontology. This challenge is the point on which all philosophies of mathematics are judged, and clarifying how we might acquire mathematical knowledge is one of the main occupations of philosophers of mathematics. In this thesis I argue that this discussion has overlooked an important part of mathematics, namely mathematics as it is exercised by ordinary people (almost everyone without a mathematics degree). I do so by looking at the different theories that have been put forward in the recent debate, and showing for each of these that they are unable to account for the mathematical practices of ordinary people. In order to show that these practices do need to be accounted for, I also argue that ordinary people are (sometimes) doing mathematics, i.e. that they engage in properly mathematical practices. Because these practices are properly mathematical, they should be accounted for by any philosophy of mathematics. The conclusion of my thesis, then, is that current theories fail to do something that they should do, while remaining neutral on how well they perform when it comes to accounting for the practices of professional mathematicians.

Students’ mathematics identity has become a more prominent concept in the research literature (Jackson & Wilson, 2012). The experiences of African Americans are still underreported, with African American elementary students receiving the least attention. This dissertation uses a case study method to explore two learners’ experiences. The purpose of this qualitative study was to explore African American third grade students’ classroom interactions with mathematics in order to better understand factors that promote positive mathematics identities. This research study explored the mathematics classroom influences on three third grade African American learners’ mathematics identities in a K-8 school in a north central Midwestern city in the United States. The school was classified as 100% free and reduced lunch and served approximately 900 students, with the vast majority of students classified as African American. The three student participants and their teacher were all African American. The student participants wore glasses that video recorded their perspectives. A stationary camera was also used to capture the wider classroom environment. Each student participant completed three interviews (Seidman, 2013). The teacher participant completed one interview. Additionally, the student participants completed a mathematics interest questionnaire. Findings showed the importance of an explicit focus on the Standards for Mathematical Practice, a growth mindset, and positioning for promoting positive mathematics identities. In one case study, Janae’s experiences in lessons about fractions highlight the relevance of the Standards for Mathematical Practice, specifically attending to precision and making sense of and persevering in solving problems. In both the classroom and in interviews, she shows the importance of making sense of problems and persevering in solving them and of attending to precision. In the second manuscript, I explore Jaane and Kayla’s different experiences. Janae was positioned more positively and faces limited resistance in maintaining a positive mathematics identity. Kayla, on the other hand, regularly rejected and renegotiated the positions offered to her as she aimed for success and a positive mathematics identity. Kayla’s growth mindset and negotiation of positions offered to her in the classroom were critical factors in how she maintained a positive mathematics identity.

Administrators who observe mathematics teachers need to have knowledge and an understanding of mathematics teaching and learning to effectively evaluate teachers and how their instructional practices relate to student thinking. This research study was conducted to illustrate the importance of understanding the thought process of administrators as they make decisions about teacher effectiveness based on what they notice during observations of mathematics classrooms. The purpose of this study was to examine what administrators attend to in the instructional environment and how what they notice influences their ability to identify the Common Core State Standards, Standards for Mathematical Practice. A purposive sample of six administrators engaged in cognitive interviews, known as think alouds, while observing two mathematics classroom videos. This study was designed to explore how administrators* instructional leadership knowledge or skills influence what they notice during mathematics instruction. There was evidence that administrators did notice aspects of the instructional environment pertaining to teachers, students, and, content. However, in this study it was found that administrators with an understanding of mathematics teaching and learning attended more to student*s mathematical thinking during instruction. It was also found that there was an increase of the administrators* mathematical language and attention to student interactions with mathematics content when the administrators were presented with a tool describing the elements of a classroom engaged in the Standards for Mathematical Practice.

The purpose of this study was to examine traditionally aged developmental mathematics college students’ experiences of mathematical practices, in a 4-week summer learning community, using a qualitative explanatory single case study approach (Yin, 2009). This study used the methodological framework of Local Communities of Mathematical Practices (Winbourne & Watson, 1998), the conceptual theory of situated cognition (Brown & Duguid, 1988), and the theories of communities of practice (Lave & Wenger, 1991), and learning communities (Tinto, 1997). The objectives were to highlight contextual factors that allowed participants to be academically successful as evidenced by their mathematical practices (Ball, 2003). The research question was: How does participating in a 4-week summer learning community shape developmental mathematics college students’ experiences of mathematical practices? The participants of this case study were one group of four women. Data were collected in the form of video and audio tape of classroom interactions, observations and reflections, diagnostic pretest, and participant interviews. Findings revealed that participants’ mathematical practices were shaped in part by: a) the way students identified with mathematics reflected their ‘success’ or ‘failure’ in the mathematics course; b) the students level of participation within the community; c) the students collaboration with purpose, discussion, and reflection; d) the students shared repertoire confirmed the consensus of knowledge; e) the students mutual engagement played a large part in their motivation, and f) the students joint enterprise within the learning community led to a self supporting system verifying that learning is the intersection of activity, concept, and the classroom.

This thesis confronts Penelope Maddy's Second Philosophical study of set theory with a philosophical analysis of a part of contemporary set-theoretic practice in order to argue for three features we should demand of our philosophical programmes to study mathematics. In chapter 1, I argue that the identification of such features is a pressing philosophical issue. Chapter 2 presents those parts of the discursive reality the set theorists are currently in which are relevant to my philosophical investigation of set-theoretic practice. In chapter 3, I present Maddy's Second Philosophical programme and her analysis of set-theoretic practice. In chapters 4 and 5, I philosophically investigate contemporary set-theoretic practice. I show that some set theorists are having a debate about the metaphysical status of their discipline{ the pluralism/non-pluralism debate{ and argue that the metaphysical views of some set theorists stand in a reciprocal relationship with the way they practice set theory. As I will show in chapter 6, these two stories are disharmonious with Maddy's Second Philosophical account of set theory. I will use this disharmony to argue for three features that our philosophical programmes to study mathematics should have: they should provide an anthropology of mathematical goals; they should account for the fact that mathematical practices can be metaphysically laden; they should provide us with the means to study contemporary mathematical practices.

The researcher in this case study sought to determine the ways in which teachers support their students to create viable arguments and critique the reasoning of others (SMP3). In order to achieve this goal, the self-conceived classroom roles of two teachers, one experienced and one novice, were elicited and then compared to their actualized roles observed in the classroom. Both teachers were provided with professional development focused on supporting student engagement in SMP3. This professional development was informed by the guidelines that describe the behaviors students should exhibit as they are engaged in the standards for mathematical practice contained in the Common Core State Standards for Mathematics. The teachers were observed, video recorded, and interviewed during and immediately after the professional development. A final observation was performed four weeks after the PD. The marked differences in the teachers' characteristics depicted in each case added to the robustness of the results of the study. A cross-case analysis was performed in order to gauge how the novice and experienced teachers' roles compared and contrasted with each other. The comparison of the teachers' self-perception and their actual roles in the classroom indicated that they were not supporting their students as they thought they were. The analysis yielded specific ways in which novice and experienced teachers might support their students. Furthermore, the cross-case analysis established the support that teachers are able to provide to students depends on (a) teaching experience, (b) teacher content and pedagogical knowledge, (c) questioning, (d) awareness of communication, (e) teacher expectations, and (f) classroom management. Study results provide implications regarding the kinds of support teachers might need given their teaching experience and mathematics content knowledge as they attempt to motivate their students to engage in SMP3.

Le platonisme mathématique, la thèse selon laquelle les mathématiques portent sur des objets abstraits existant de manière indépendante à notre esprit et notre langage, est un des sujets les plus débattues dans la philosophie des mathématiques. L’image des mathématiques qui s’en dégage est souvent perçue comme se heurtant à des problèmes épistémologiques considérables : si il est vrai que les mathématiques sont une science qui porte sur des objets en dehors de l’espace et du temps, comment nous, des êtres situés spatio-temporellement, pouvons avoir une quelconque connaissance mathématique ? En conséquence, la défense du platonisme et le débat sur l’ontologie des mathématiques se sont largement concentrées sur cette dimension épistémologique. Dans ce travail de thèse, nous essaierons de réitérer le rôle de la métaphysique et de la pratique des mathématiques dans le débat sur l’ontologie des objets mathématiques. Notre objectif principal est plus particulièrement le développement et l’application d’un programme métaphysique général, capable de rendre compte des aspects ontologiques des mathématiques qui sont propres à une interprétation platoniste des mathématiques. Pour ce faire, notre stratégie consiste à insister tout d’abord sur le besoin de clarification des thèses platonistes concernant la nature abstraite des objets mathématiques et l’indépendance de ces objets et à essayer d’étendre la portée du platonisme au-delà des concepts et théories mathématiques habituelles. / Mathematical platonism is the idea according to which mathematics is about a domain of abstract objects, existing independently of our though and language. It is one of the central subjects in philosophy of mathematics, and is often considered to face important epistemological problems. If, as the platonist thinks, mathematics really are a science of objects outside of space and time, then how is mathematical knowledge even possible? As a consequence of the epistemological problem, the debate has focused mainly around the epistemological dimension of platonism. In this study however, we will try to move away from epistemology and restate the role of metaphysics and mathematical practice in the ontological debate on mathematical objects. Our main objective will be to develop and apply a general metaphysical program in order to explain the ontological aspects of a platonist interpretation of mathematics. In order to do this, it will be necessary to clarify the abstract nature of mathematical objects and the ontological independence of these entities, and to extend the scope of platonism beyond the usual concepts and mathematical theories.

This thesis is about the nature of proofs in mathematics as it is practiced, contrasting the informal proofs found in practice with formal proofs in formal systems. In the first chapter I present a new argument against the Formalist-Reductionist view that informal proofs are justified as rigorous and correct by corresponding to formal counterparts. The second chapter builds on this to reject arguments from Gödel's paradox and incompleteness theorems to the claim that mathematics is inherently inconsistent, basing my objections on the complexities of the process of formalisation. Chapter 3 looks into the relationship between proofs and the development of the mathematical concepts that feature in them. I deploy Waismann's notion of open texture in the case of mathematical concepts, and discuss both Lakatos and Kneebone's dialectical philosophies of mathematics. I then argue that we can apply work from conceptual engineering to the relationship between formal and informal mathematics. The fourth chapter argues for the importance of mathematical knowledge-how and emphasises the primary role of the activity of proving in securing mathematical knowledge. In the final chapter I develop an account of mathematical knowledge based on virtue epistemology, which I argue provides a better view of proofs and mathematical rigour.

In this confirmatory factor analysis, factors previously identified to explain the variability in Middle School Mathematics Teachers’ perception of the Common Core State Standards of Mathematics were considered as factors hypothesized to effect high school math teachers’ perceptions of challenging math tasks (CMTs). The factor of student characterization (i.e., disposition, academic preparation, and student behavior) was additionally considered as a factor hypothesized to explain teachers’ perceptions of CMTs, as well as site-based variables (i.e., curriculum, assessment and evaluation, professional development, and collaboration). In addition, teachers’ understanding of the importance of the mathematical practice standards and teacher familiarity with enacting CMTs were factors considered in the model. The original septenary factor structure was modified and good model fit was achieved. In addition to the confirmatory factor analysis model which provides a structure for considering teachers perceptions of CMTs, descriptive statistics are presented from the survey developed that captured teachers’ perceptions of CMTs relative to their sites.

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