This research study investigated how professional mathematicians understand and operate with highly-abstract, advanced mathematical concepts in their own work. In particular, this study examined how professional mathematicians operated with mathematical concepts at different levels of understanding. Moreover, this study aimed to capture what factors influence professional mathematicians' level of understanding for particular mathematical concepts. To frame these research goals, three theoretical levels of understanding were proposed, process-level, pseudo-object-level, object-level, leveraging two ways that Piaget (1964) described what it meant to know or understand a mathematical concept. Specifically, he described understanding an object as being able to "act on it," and also as being able to "understand the process of this transformation" (p. 176). Process-level understanding corresponds to only understanding the underlying processes of the concept. Pseudo-object-level understanding corresponds to only being able to act on the concept as a form of object. Object-level understanding corresponds to when an individual has both of these types of understanding. This study is most especially concerned with how professional mathematicians operate with a pseudo-object-level understanding, which is called pseudo-objectification. For this study, six professional mathematicians with research specializing in a subfield of algebra were each interviewed three times. During the first interview, the participants were given two mathematical tasks, utilizing concepts in category theory which were unfamiliar to the participants, to investigate how they operate with mathematical concepts. The second interview utilized specific journal publications from each participant to generate discussion about influences on their level of understanding for the concepts in that journal article. The third interview utilized stimulated recall to triangulate and support the findings from the first two interviews. The findings and analysis revealed that professional mathematicians do engage in pseudo-objectification with mathematical concepts. This demonstrates that pseudo-objectification can be productively leveraged by professional mathematicians. Moreover, depending on their level of understanding for a given concept, they may operate differently with the concept. For example, when participants utilized pseudo-objects, they tended to rely on figurative material, such as commutative diagrams, to operate on the concepts. Regarding influences on understanding, various factors were shown to influence professional mathematicians' level of understanding for the concepts they use in their own work. These included factors pertaining to the mathematical concept itself, as well as other sociocultural or personal factors. / Doctor of Philosophy / In this research study, I investigated how professional mathematicians utilize advanced mathematical concepts in their own work. Specifically, I examined how professional mathematicians utilize mathematical concepts that they do not fully understand. I also examined what factors might influence a professional mathematician to fully understand or choose not to fully understand a mathematical concept they are using. To address these goals, six research-active mathematicians were each interviewed three times. In these interviews, the mathematicians engaged with mathematical concepts that were unfamiliar to them, as well as concepts from one of their own personal research journal publications. The findings demonstrated that professional mathematicians sometimes utilize mathematical concepts in different ways depending on how well they understand the concepts. Moreover, even if mathematicians do not have a full understanding of the concepts they are using, they can still sometimes productively leverage this amount of understanding to successfully reach their goals. I also demonstrate that various factors can and do influence how well a professional mathematician understands a given mathematical concept. Such influences could include the purpose of use for the concept, or what a mathematician's research community values.

Thesis advisor: Lillie R. Albert / To explore student mathematical self-efficacy and understanding of graphical data, this dissertation examines students solving real-world problems in their neighborhood, mediated by professional urban planning technologies. As states and schools are working on the alignment of the Common Core State Standards for Mathematics (CCSSM), traditional approaches to mathematics education that involves learning specific skills devoid of context will be challenged. For a student to be considered mathematically proficient according to the CCSSM, they must be able to understand mathematical models of real-world data, be proficient problem solvers and use appropriate technologies (tools) to be successful. This has proven to be difficult for all students--specifically for underrepresented students who have fallen behind in many of the Science, Technology, Engineering and Mathematics (STEM) fields. This mixed-method design involved survey and case-study research to collect and examine data over a two-year period. During the first year of this study, pre- and post-surveys using Likert-scale questions to all students in the urban planning project (n=62). During the two years, ten high school students' mathematical experiences while investigating urban planning projects in their own neighborhoods were explored through interviews, observations, and an examination of artifacts (eg. presentations and worksheets) in order to develop the case studies. Findings indicate that real-world mathematical tasks that are mediated by professional technologies influence both students' mathematical self-efficacy and understanding. Student self-efficacy was impacted by causing a shift in students beliefs about their own mathematical ability by having students interest increase through solving mathematical tasks that are rooted in meaningful, real-world contexts; students' belief that they can succeed in real-world mathematical tasks; and a shift in students' beliefs regarding the definition of `doing mathematics'. Results in light of mathematical understanding demonstrate that students' increased understanding was influenced by the ability to use multiple representations of data, making connections between the data and the physical site that was studied and the ability to communicate their findings to others. Implications for informal and formal learning, use of GIS in mathematics classrooms, and future research are discussed. / Thesis (PhD) — Boston College, 2013. / Submitted to: Boston College. Lynch School of Education. / Discipline: Teacher Education, Special Education, Curriculum and Instruction.

Thesis (M.Ed. (Mathematics Education)) -- UNiversity of Limpopo, 2013 / In this study, I presented the exploration of Mpho’s growth in mathematical understanding. Mpho is a grade 10 mathematics learner. To fulfil such, a qualitative research method was employed. I explored her growth in understandings in the context of co-ordinate geometry, exponents, and functions. Data generation, management and representation were guided by the notion of teaching experiments. Analysis was done through mapping learner’s growth of mathematical understanding using Pirie-Kieren’s (1994) model. Findings suggest that learner’s growth in mathematical understanding can be observed, mapped and improved with the aid of probing questions.

Thesis (MEd. (Mathematics Education)) -- University of Limpopo, 2013 / In this study, I presented the exploration of Mpho’s growth in mathematical understanding. Mpho is a grade 10 mathematics learner. To fulfil such, a qualitative research method was employed. I explored her growth in understandings in the context of co-ordinate geometry, exponents, and functions. Data generation, management and representation were guided by the notion of teaching experiments. Analysis was done through mapping learner’s growth of mathematical understanding using Pirie-Kieren’s (1994) model. Findings suggest that learner’s growth in mathematical understanding can be observed, mapped and improved with the aid of probing questions.

This thesis reports on a longitudinal study of students’ understanding of decimal notation. Over 3000 students, from a volunteer sample of 12 schools in Victoria, Australia, completed nearly 10000 tests over a 4-year period. The number of tests completed by individual students varied from 1 to 7 and the average inter-test time was 8 months. The diagnostic test used in this study, (Decimal Comparison Test), was created by extending and refining tests in the literature to identify students with one of 12 misconceptions about decimal notation. (For complete abstract open document)

In our study, the problem area is that concept teaching in mathematics is too low to be able to achieve mathematical understanding. Mathematics also has two different parlance ​​that make teaching difficult and create a lack of clarity about which one of these parlances to use at which occasion and how these parlances ​​should be interpreted and how they are linked.  We have conducted an interview study with semi-structured interviews. The respondents consist of 16 teachers who were selected through our inclusion- and exclusion criteria. The material was transcribed, coded, and categorized, and we used the socio-cultural perspective as a theoretical framework to analyse the material.  One conclusion we can draw from the results of this interview study is that the teachers in this study express that they are aware that cooperative learning is beneficial for students' understanding of concepts, but that organizational problems such as lack of planning time, too few educators and too large classes make it is difficult to conduct a developing concept teaching. This can be a reason why the textbook takes up a lot of space in the classroom. Another conclusion we can draw is that the teachers in this study express that they have subject knowledge and that their pedagogical knowledge at certain times is not sufficient, especially when it comes to supporting students who need extra support and students with Swedish as a second language. Another conclusion we can draw from the results is that the students have difficulty transferring the concrete to the abstract in mathematics. This may be because the language used by the students is the everyday one, which does not correspond to what the students encounter in the textbooks and on tests. As the students get the smallest speaking space, they also rarely get the chance to use any of the language. The use of everyday language is not advocated in the socio-cultural perspective, but can be questioned when students do not encounter the correct language in everyday life, and can become an element in the teaching that can not relate to everyday life.

Thesis (M. Ed. (Mathematics)) -- University of Limpopo, 2022 / The purpose of this study was to document my enactment of the five key strategies of assessment for learning in my mathematics classroom to account for learners' mathematical understanding. I used a constructivism teaching experiment methodology to explore learners' mathematical activities as they interacted in the classroom. Twenty-five learners from my Grade 10 mathematics class took part in the study. Data were gathered through classroom observations, written work samples from learners, and the teacher's reflective journal. My enactment of the five key strategies enabled learners to participate in classroom discussions, collaborate with their peers, and use self-assessment tools while engaging in classroom interactions. The major findings revealed that, through my enactment of the five key strategies, learners developed conceptual understanding, procedural fluency and strategic competence of the concepts taught. In addition, practices such as the development of lesson plans detailing how the five key strategies will be enacted in the classroom, use of comment – only feedback for grading learners’ work, creating a conducive learning environment to allow the use of peer and self-assessment allowed for a meaningful enactment of assessment for learning in my classroom. Strategies four and five, whose primary goal is to encourage learners' participation in the lesson, were critical in promoting learners' mathematical understanding.

In this dissertation, I propose that network theory offers a useful frame for informing mathematics education. Mathematical understanding, like the discipline of formal mathematics within which it is subsumed, possesses attributes characteristic of complex systems. As the techniques of network theorists are often used to explore such forms, a network model provides a novel and productive way to interpret individual comprehension of mathematics. A network structure for mathematical understanding can be found in cognitive mechanisms presented in the theory of embodied mathematics described by Lakoff and Nez. Specifically, conceptual domains are taken as the nodes of a network and conceptual metaphors as the connections among them. Examination of this metaphoric network of mathematics reveals the scale-free topology common to complex systems. Patterns of connectivity in a network determine its dynamic behavior. Scale-free systems like mathematical understanding are inherently vulnerable, for cascading failures, where misunderstanding one concept can lead to the failure of many other ideas, may occur. Adding more connections to the metaphoric network decreases the likelihood of such a collapse in comprehension. I suggest that an individuals mathematical understanding may be made more robust by ensuring each concept is developed using metaphoric links that supply patterns of thought from a variety of domains. Ways of making this a focus of classroom instruction are put forth, as are implications for curriculum and professional development. A need for more knowledge of metaphoric connections in mathematics is highlighted. To exemplify how such research might be carried out, and with the intent of substantiating ideas presented in this dissertation, I explore a small part of the proposed metaphoric network around the concept of EXPONENTIATION. Using collaborative discussion, individual interviews and literature, a search for representations that provide varied ways of making sense of EXPONENTIATION is carried out. Examination of the physical and mathematical roots of these conceptualizations leads to the identification of domains that can be linked to EXPONENTIATION.

No description available.

Thesis (M. Ed. (Mathematics Education)) -- University of Limpopo, 2017 / The purpose of this study was to explore the role of folding back in enriching grade 10 students’ reasoning in geometry. Although various attempts are made in teaching and learning geometry, evidence from several research studies shows that most learners struggle with geometric reasoning. Hence, this study came as a result of such learners’ struggles as shown in the literature as well as personal experiences. The study was a constructivist teaching experiment methodology that sought to answer the following research questions: How does folding back support learners’ interaction with geometric reasoning tasks during the lessons? How does a Grade 10 mathematics teacher use folding back to enrich student reasoning in geometry? The teaching experiment as a research design in this study was found useful in studying learners’ geometric reasoning as a result of mathematical interactions in their learning of geometry. Therefore, it should be noted that the teaching experiments were not conducted as an attempt to implement a particular way of teaching, but rather to understand the role of folding back in enriching learners’ reasoning in geometry. As a referent to the teaching experiment methodology, the participants in this study were 7 grade 10 mathematics learners’ sampled from a classroom of 54 learners. These seven learners did not necessarily represent the whole class in accordance with the purpose of the study. This requirement was not necessary in determining rigour in teaching experiments. Instead interest was on “organising and guiding [teacher-researchers] experience of learners doing mathematics” (Steffe & Thompson, 2000, p. 300). Furthermore, the participants were divided into two groups while working on the learning activities. Participants were further encouraged to share ideas with each other as they solved the learning activities. Data was collected through video recording while learners were working on mathematical learning activities. The focus was on the researcher-teacher – learners and learners-learners interactions while working on geometric reasoning learning activities. Learning activities and observations served as subsets of the video data. Learners were encouraged to share ideas with each other as they v solved the learning activities as recommended by Steffe and Thompson (2000). Likewise, in order to learn the learners’ mathematics, the researcher could teach and interact with learners in a way that encourage them to improve their current thinking (Steffe & Thompson, 2000). In analysing data, the study adopted narrative analysis. The researcher performed verbatim transcription of the video recordings. Subsequently, information-rich interaction for each mathematical learning activity, where folding back was observed was selected. The selections of such information-rich interactions were guided by Martin’s (2008) framework for describing folding back. The main findings of the study revealed that in a learning environment where folding back takes place, learners’ reasoning in geometry is enriched. The researcher-teacher’s instructional decisions such as discouraging, questioning, modelling and guiding were found to be effective sources through which learners fold back. The findings also revealed that learners operating at different layers of mathematical understanding are able to share their geometry knowledge with their peers. Moreover, the findings indicated that in a learning environment where folding back takes place, learners questioning ability is enriched. Based on the findings of the study, the recommendations were that Mathematics teachers should create a learning environment where learners are afforded the opportunity to interact with each other during geometry problem solving; such is a powerful quest for folding back to take place. / Research Chair Developmental Grant at the University of Limpopo