Many studies have been done on the impact of online mathematics courses. Most studies concluded that there is no significant difference in student success between online and face-to-face courses. However, most studies compared "traditional" online and face-to-face courses. Mathematics educators are advocating a shift from traditional courses to student-centered courses where students argue and defend the mathematics under the guidance of the teacher. Now, the differences in online and face-to-face student-centered mathematical courses merit a more in-depth investigation. This study characterized student mathematical discourse in online and face-to-face Calculus lab sections based off of a framework derived from an NCTM standard for the students' role in discourse. Results showed that the discourse in both the face-to-face and online environments can be rich and productive. Thus, both environments can be viable arenas for effective mathematical discourse. However, this effectiveness is contingent on whether or not the teacher as the facilitator can help the students avoid the ways in which online discourse can be impeded. The characteristics of discourse, how they compare, and the resulting recommendations for teachers are discussed.

This action research study examined my attempts during a six-lesson unit of instruction to implement five practices developed by Stein, Engle, Smith, and Hughes (2008) to assist novice teachers in orchestrating meaningful mathematical discussions, a component of inquiry-based teaching and learning. These practices are anticipating student responses to a mathematical task, monitoring student responses while they engage with the task, planning which of those responses will be shared, planning the sequence of that sharing, and helping students make connections among student responses. Although my initial anticipations of student responses were broad and resulted in unclear expectations during lesson planning, I observed an improvement in my ability to anticipate student responses during the unit. Additionally, I observed a high-level of interaction between my students and me while monitoring their responses but these interactions were generally characterized by low-levels of mathematical thinking. The actual sharing of student responses that I orchestrated during discussions, and the sequencing of that sharing, generally matched my plans although unanticipated responses were also shared. There was a significant amount of student interaction during the discussions characterized by high-levels of thinking, including making connections among student responses. I hypothesize that task quality was a key factor in my ability to implement the five practices and therefore recommend implementing the five practices be accompanied by training in task selection and creation.

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Thesis (Ph.D.)--University of the Witwatersrand, Faculty of Humanities, School of Education, 2011 / This study is a located in a foundational undergraduate mathematics course designed to facilitate the transition from school mathematics to advanced mathematics. The focus of the study is on two innovations in the course; the use of practical problems that make links to non-mathematical practices and a learner-centred pedagogy. While these innovations are part of the discourse of the mathematics education community in terms of access to school mathematics, this study investigates the relationship between these innovations and access to advanced mathematics. The texts of three practical problems from the course and texts representing the verbal and non-verbal action of 17 students as they worked collaboratively in small groups on these problems were analyzed. The analysis of these texts is used to describe and explain, firstly, how the practical problems in the foundational course represent the practice of foundational undergraduate mathematics and its relationship to other practices in the educational space (for example, school mathematics, calculus reform, advanced mathematics, and non-mathematical practices). Secondly, the students‟ enabling and constraining mathematical action on the practical problems is described and explained. Answering the empirical questions in this study has required theoretical work to develop a socio-political perspective of mathematical practice. This theoretical perspective is based on Fairclough‟s social practice perspective from critical linguistics, but has been supplemented with recontextualized theoretical constructs used by Morgan, Moschkovich and Sfard in mathematics education. These constructs are used to conceptualize the notion of mathematical discourse and action on mathematical objects in this discourse. The methodological work of this study has involved supplementing Fairclough‟s method of critical discourse analysis with Sfard‟s method of focal analysis to analyze mathematical, discursive, social and political action in a socio-political mathematical practice. The central finding of this thesis is that foundational mathematical practice represents both continuities and disruptions in its relationship to other practices in the space. As a result, participation in the foundational practice is complex, requiring control over the how and when of boundary crossings across practices, social events and texts. On the basis of this complexity, innovative foundational practice is positioned paradoxically in the higher education space. On the one hand, it represents an alternative to the dominant representation of mathematical practice and positioning of the foundational student in higher education. On the other hand, the complexity of foundational practice makes access to advanced mathematics problematic and foundational practice thus reproduces the dominant ordering.

The purpose of this study was to examine secondary mathematics teachers' questioning, responses, and perceived influences upon their instructional decisions regarding questioning and response to students' ideas. This study also compared the questioning practices, responses, and influences of beginning teachers to more experienced teachers. Previous studies on teacher quality in mathematics education have focused on general characteristics of mathematics teachers' instructional practice including a broad range of instructional strategies. Little is known about mathematics teachers' questioning practices and responses to students' ideas that research has repeatedly reported are critical to student mathematics learning in secondary classrooms. Furthermore, it is not clear how different novice teachers are in questioning and responding to students from experienced teachers. This understanding can provide significant insights into teacher education programs for mathematics teachers. With those issues in mind, this study was designed to answer the following questions: (1) What similarities and differences exist in questioning patterns between novice and experienced teachers when guiding a classroom mathematical discussion? (2) What similarities and differences exist in responses to students during pivotal teaching moments between novice and experienced teachers when guiding a classroom mathematical discussion? (3) What perceived factors impact the responses teachers give to students' ideas, and how are these factors of influence different among novice and experienced teachers? This study employed a multiple case study research design to compare the questioning practices and responses of three beginning teachers and three experienced teachers. Multiple sources of data were collected, including two interviews (i.e., initial interview and follow-up interview) for each teacher, five days of classroom video footage for each teacher, and field notes by the researcher for each interview and observation. The researcher conducted initial interviews with each teacher to gain a general sense of the teacher's philosophy and use of questions in guiding classroom discussion. Five instructional days of observation followed the initial interview, and then the researcher conducted a follow-up interview by use of video-stimulated response. All interviews were transcribed verbatim for analysis. The data was analyzed mainly using the constant comparative method to identify regularities and patterns emerging from the data. Results showed differences between beginning and experienced teachers in the frequency and variety of questions asked. Although all teachers showed the largest number of questions in the Socratic questioning category, differences were prominent in the semantic tapestry and framing categories. Results regarding teacher responses to pivotal teaching moments showed that four teachers favored a procedural emphasis in their responses to students, and two teachers used responses to direct students to make clear connections within or outside of mathematics. Perceived influences identified include: (1) reflection on experience and mathematical knowledge for teaching, (2) time, and (3) relationship with students, teachers, and parents, and knowledge of student background. Practicing teachers can expand the types of questions they use in the classroom, making particular efforts to include those areas that this study showed to be most lacking: semantic tapestry questions that help students build a coherent mental framework related to a mathematical concept, and framing questions that help frame a problem and structure the discussion that follows. The comparison between beginning and experienced teachers also shed light on important practices for teacher education. The beginning teacher participants from this study had no trouble noticing pivotal teaching moments in their lessons but were less developed in their responses to them. Recommendations for mathematics teacher education programs are to provide opportunities to develop content, pedagogical knowledge including specific instruction on questioning strategies, and also to provide parallel field experiences where pre-service teachers can apply the knowledge and skill they are learning.

The purpose of this study was to determine whether a professional development (PD) for teachers focused on improving teacher pedagogical content knowledge (PCK) related to operations with integers would improve teacher PCK and if there was a relationship between their level of PCK and the change in the understanding of their students as measured by pre- and posttest of teacher and student knowledge. The study was conducted summer 2010 in a large urban school district on two campuses providing a district funded annual summer intervention, called Jumpstart. This program was for grade 8 students who did not pass the state assessment (Texas Assessment of Knowledge and Skills), but would be promoted to high school in the Fall 2010 due to a decision made by the Grade Placement Committee. The Jumpstart program involved 22 teachers and 341 students. For purposes of this study, changes were made to the PD and typical curriculum for a unit on integer operations to promote teacher and student conceptual understanding through a process of mathematical discussion called argumentation. The teachers and students explored a comprehensive representation for integer operations called a vector number line model using the Texas Instruments TI-73 calculator Numln application. During PD, teachers engaged in argumentation to make claims about strategies to use to understand integer operations and to explain their understanding of how different representations are connected. The results showed statistically significant growth in teacher PCK following the professional development and statistically significant growth in student understanding from pre- to posttest compared to the students who participated in the program the previous year. The findings also showed that there was a statistically significant association between teacher posttest PCK and student improvement in understanding even when controlling for years of teaching experience, teacher pretest knowledge, and student pretest score. This adds to the research base additional evidence that professional development focused on teacher pedagogical content knowledge can have a positive effect on student achievement, even with just a short period of PD (6 hours in this case). / text

Thesis (M.Ed. (Mathematics Education)) -- University of Limpopo, 2022 / A classroom in which learners are afforded opportunities to engage in meaningful mathematical discourse (Sfard, 2008) is desirable for the effective teaching and learning of mathematics. However, engagement in mathematical discourse requires learners to use appropriate mathematical vocabulary to think, learn, communicate and master mathematics (Monroe & Orme, 2002). Hence, I have undertaken this study to explore how mathematical vocabulary is used during mathematical classroom discourse using the lens of the commognitive framework. I chose a qualitative approach as an umbrella for the methodology with ethnography as the research design whereby participant observation, structured interviews and documents were used to collect data. One Grade 9 mathematics classroom with 25 learners and their mathematics teacher were purposefully selected as participants in the study. During data analysis, I looked at Sfard’s (2008) constructs of the commognitive theory to analyse the data and identify the mathematics vocabulary used in the discourse. This was followed by the use of realisation trees that I constructed for the teacher and learners’ discourse, which I used to identify learners thinking as either being explorative or ritualistic. Results indicate that both the teacher and learners use mathematical vocabulary objectively with positive whole numbers to produce endorsed narrative regulated by explorative routines. However, with algebraic terms both positive and negative, the teacher and learners’ discourse is mostly disobjectified, and produces narratives that are not endorsed and are regulated by ritualistic routines. It also became evident that the mathematical vocabulary that the teacher and learners use in the classroom discourse includes words that are mathematical in nature and colloquial words that learners use for mathematical meaning. v Furthermore, learners’ responses to the given mathematics questions which they are solving are mostly correct, hence it can be argued that the learners’ narratives were endorsed. However, their realisation trees indicates that learners were not working with mathematical objects in their own right (Sfard, 2008) and hence their narratives were not endorsed. I have recommended in this study, that teachers need to be cautious when operating with entities and not separate operations from their mathematical terms. Furthermore, the department of basic education, during workshops should encourage educators to always request reasons from learners substantiating their answers to questions in order to enhance their explorative thinking.

Due to the increasing diversity of mathematics classrooms today, teachers need guidance on how to support English Learners (ELs) in mathematics classes in a way that situates language learning within mathematical activity. Unfortunately, neither mathematics education research nor EL education research is sure how to navigate the complexity of teaching ELs mathematics while supporting both their language development and their mathematical development through their participation in mathematical activity. This study examined ELs' participation in mathematical Discourse, investigating both the mathematical purposes ELs accomplished by using multiple symbol systems, and the way ELs used non-English language (NEL) symbol systems to support their spoken English. The participants were college-aged ELs beginning their studies at the English Learning Center at an American university. The students all had fluency with basic conversational English, and had many different levels of mathematical experience. I identified five categories of purposes in which ELs engaged during mathematical Discourse. I also developed the Replace Augment Learn (RAL) framework that describes how ELs used NEL symbol systems to make up for their decreased English literacy and facilitate their participation in mathematical Discourse. Analysis of the data suggests ELs' use of NEL symbol systems (1) played a significant role in achieving many of the purposes associated with mathematical Discourse, and (2) opened up a space for effective language acquisition. These findings indicate that authentic mathematical activity can be a productive site for language development, and that ELs with basic conversational English and literacy with a variety of symbol systems can participate meaningfully in mathematical Discourse.

The purpose of this study is to describe how university honors calculus students negotiate meaning and language for conceptually important ideas through mathematical discourse. Mathematical discourse has been recognized as an important topic by mathematics education researchers of various theoretical perspectives. This study is written from a perspective that merges symbolic interactionism (Blumer, 1969) with personal agency (Walter & Gerson, 2007) to assert that human choice reflects, but is not determined by, meanings that are primarily developed through social interaction. The process of negotiation of meaning is identified, described, and analyzed in the discourse of four students and their professor as they draw conclusions about the volume of water in a reservoir based on graphs of inflow and outflow. Video data, participant work, and transcript were analyzed using grounded theory and other qualitative techniques to develop three narrative accounts. The first narrative highlights the participants' use of personal pronouns and personal experience to negotiate meaning for the conventional mathematical terms "inflection" and "concavity." The second narrative describes how the participants' choices in discourse reflect an effort to represent both their mathematical and experiential understandings correctly as they negotiate language to describe critical "zero points." The third narrative describes the participants' process of mapping analogical language and meaning from the context of motion to the context of water in a reservoir. Analysis of these three narratives from the perspective of conventional and ordinary mathematical language suggests that the contextualized study of mathematics may provide students access to mathematical discourse if the relevant mappings between mathematical language and language from other appropriate contexts are made explicit. Analysis from the perspective of social speech (Piaget 1997/1896) suggests that specific uses of personal pronouns, personal experience, and revoicing (O'Connor & Michaels, 1996) may serve to invite students to become participants in mathematical discourse. An agency-based definition of mathematical discourse is suggested for application in research and practice.

This study examines teacher questioning with the purpose of identifying what types of mathematical questions are being modeled by the teacher. Teacher questioning is important because it is the major source of mathematical questioning discourse from which students can learn and copy. Teacher mathematical questioning discourse in a conceptually-oriented classroom is important to study because it is helpful to promote student understanding and may be useful for students to adopt in their own mathematical questioning discourse. This study focuses on the types of questions that comprise the mathematical questioning discourse of a university teacher in a conceptually-oriented mathematics classroom for preservice elementary teachers. I present a categorization of the types of questions, an explanation of the different categories and subcategories of questions, and an analysis and count of the teacher's use of the questions. This list of question types can be used (1) by conceptually-oriented teachers to explicitly teach the important mathematical questions students should be asking during mathematical activity, (2) by teachers who wish to change their instruction to be more conceptually-oriented, and (3) by researchers to understand and improve teachers' and students' mathematical questioning.