<p>The Common Core State Standards for Mathematics (CCSSM) highlight the importance of students? conceptual understanding, mathematical reasoning, and problem solving in order to prepare students for college and careers. However, the success of this reform effort largely depends on how teachers actually design and implement instruction based on the goals of the new standards. In particular, teachers? concerns about the reform have a significant effect on this change and the implementation of reform curricula. While there exists an increasing amount of research on reform efforts, many questions still remain regarding the implementation of the CCSSM and teachers? concerns. The purpose of this qualitative dissertation study is to investigate the concerns teachers have regarding the implementation of the CCSSM and how those concerns relate to the enactment of their curricula. This study also explores how teachers address the mathematical shift of rigor defined in the CCSSM in connection with the tasks they use and types of questions they pose to their students. This research involved case studies of three Algebra 1 teachers. Each teacher was observed during three different lessons on linear/non-linear functions. Pre- and post-observation interviews were conducted both before and after each lesson was taught. In order to determine how teachers addressed the mathematical shift of rigor, three different analyses were conducted. Mathematical tasks in the teacher guided notes and lesson enactment were explored in conjunction with teacher questions and the use of the eight mathematical practices. Observations and interviews were used to examine how teacher concerns connect to their implementation of the CCSSM. In particular, cases based on the teacher interviews and videos were compared to one another to explore possible reasons why the teachers address the mathematical shift of rigor differently. Findings suggest a complicated relationship exists between teacher concerns and their intended and enacted curriculum. The results of this study showed that teachers at all stages of concern are not providing cognitively demanding lessons nor are they addressing the mathematical shift of rigor. Only during review lessons did two of the three teachers increase the cognitive demand of the tasks and questions they posed during the enactment of their curriculum. Regarding teacher concerns, different factors seem to take account for the complicated relationship between teacher concerns and their enacted curricula. First, the teacher with self concerns had a lack of content knowledge. This teacher was not able to adjust her intended curriculum, followed the textbook closely and had a difficult time addressing student misconceptions. The teacher with management concerns tended to express her students? low abilities in doing mathematics. This teacher thus focused on student ability, only slightly modified the intended curriculum and provided only low cognitive demand tasks and questions. Finally, the teacher with impact concerns had a high interest in student learning. This teacher was able to alter her intended curriculum based on student questions and misconceptions. However, her tasks and questions remained at a low cognitive demand for two of the three lessons. This study has implications for curriculum developers and professional development providers, as well as teachers and school administrators to help ensure the success of reform curriculum.

This study investigated factors that influence mathematics teachers’ use of dynamic mathematics software (specifically GeoGebra) for teaching and learning. Since society is so intertwined with technology, Keitel (1997) argues that it is becoming easier to find technological solutions for problems rather than to search for non-technological solutions. This could also hold true for teachers who need to adjust to teaching mathematics with the aid of resources such as Information and Communication Technology (ICT) in a changing society. One of the key trends reported on in the 2013 Higher Education edition of the 2013 NMC report is that the teacher’s role keeps on changing because of the ever increasing amount of resources available to students via the Internet (Johnson, Adams Becker, Cummins, Estrada, Freeman & Ludgate, 2013). In order to explore factors that influence mathematics teachers’ use of GeoGebra for instruction, a quantitative research design was used. Participants in the study were members of the V.A.W. These participants were purposefully selected since the organisation regularly has training workshops on GeoGebra and most of the organisation’s members were therefore familiar with GeoGebra. In order to obtain as large a response as possible, a website link to an e-survey, as well as an invitation to participate in the study were e-mailed in addition to hard copies of the survey which were distributed and collected. Multiple regression analysis was used to investigate the influence of the four UTAUT (Unified Theory of Acceptance and Use of Technology) constructs, namely Performance Expectancy (PE), Effort Expectancy (EE), Social Influence (SI), and Facilitating Conditions (FC) (independent variables) on teachers’ intention to use GeoGebra (dependent variable). Correlation statistics was used to establish whether correlations between the four UTAUT constructs and teachers’ intention to use GeoGebra existed, and if it did, how significant the correlations were – between each item on the survey as well as each UTAUT construct on teachers’ intention to use GeoGebra. This study found that the combination of Performance Expectancy, Effort Expectancy and Social Influence explained 30% of the variance in respondents’ intention to use GeoGebra. On its own however, only Social Influence was found to be a direct determinant of a respondent’s intention to use GeoGebra, with Performance Expectancy and Effort Expectancy not being significant predictors by themselves of respondents’ intention to use GeoGebra. Facilitating Conditions were not found to directly influence whether or not people actually used GeoGebra. Teachers’ intention to use GeoGebra was found to predict the actual use of GeoGebra for teaching and learning. / Dissertation (MEd)--University of Pretoria, 2015. / Science, Mathematics and Technology Education / Unrestricted

In order to meet the changing needs of today’s students and society, instructors need to adapt to new teaching methods. The purpose of this case study is to explore the integration of technology into teaching at a mathematics department at a South African University. Questionnaires were completed by staff lecturing undergraduate mathematics and both quantitative and qualitative data were collected in this survey. Selected interviews were conducted with respondents to obtain richer data. The study shows that half of the staff members feel that chalkboards are more suitable than technology for teaching mathematics. This finding supports the idea of a strong subject culture. Age did not emerge as a factor for preference of either technology or the chalkboard, although gender, academic qualification and teaching qualification did. Subject culture is strongly rooted under the male members of staff, while female staff felt more positive towards the use of technology for teaching. The higher up in the ladder of academic qualifications, the stronger the belief in the chalkboard for teaching. Teaching qualification indicated a preference for technology for teaching. Use of chalkboards decreased significantly over the past ten years, while the use of modern technologies has increased accordingly. Teaching of large groups has necessitated the use of technology. A shift in attitude towards technology use in teaching is perceived. There is a trend of moving towards using new technologies. The study showed that the majority of teaching staff make limited use of the LMS. The use of other technologies as a learning tool for students was found to be limited amongst staff. Teaching staff at this department do integrate technology into their teaching, and therefore practise blended teaching. However, many of the benefits offered by technology are underutilised and the use of technology does not necessarily lead to improved learning. / Dissertation (MSc)--University of Pretoria, 2015. / Mathematics and Applied Mathematics / Unrestricted

The aim of this research study is to investigate how undergraduate mathematics can be en- hanced through research and how this enriched content can be exploited for the enrichment of academically stronger students. The study o ers a unique blend of mathematical and educa- tional research. In the rst part of the study a problem stemming from teaching the undergraduate topic of complex numbers, namely on how to represent the zeroes of functions, particularly polynomi- als, is researched. The notion of sibling curves ([51], [52]) o ers an elegant and natural way to represent the zeroes of a polynomial, which is explored and expanded in this thesis. A library of sibling curves for well-known functions is developed and presented. Signi cant research results are that every polynomial of degree n has n sibling curves. This result gives a more geometric interpretation of the roots of a polynomial than the Fundamental Theorem of Algebra. I then focus on quadratic polynomials with complex coe cients and prove that, although the siblings are not always parabolas, the two sibling curves are always congruent and that they lie on a hyperbolic paraboloid determined by the coe cients of the polynomial. Some of these results are reported on in [115]. The second part of the study centres on utilising the researched knowledge on sibling curves for student enrichment. A group of rst year students were guided through a number of designed activities using an inquiry-based learning approach to explore polynomials, complex numbers and ultimately sibling curves. Implementation of the programme as well as experiences are reported on, following a research approach of evaluation research. Student as well as facilitator experiences, discoveries and learning curves were captured in order to analyse perspectives. Re- sults show that there is a need to stimulate and challenge academically strong undergraduate students. The study further shows that all the participants of this enrichment programme ben- e ted from this experience. The students were engaged with the work and had the opportunity to delve deeper into the mathematical topic while sharpening their problem solving skills. I, as facilitator, had the opportunity to interact closely with academically strong students and experience their needs rst hand, which added a new dimension to my teaching. This research also demonstrates how enrichment programmes can be a vehicle to expose enriched content to academically strong students. The dual value of the study is that it adds not only to the knowledge base of complex number theory, but also to the body of reported experiences on student enrichment in undergraduate mathematics teaching. It is envisaged that research ndings reported on in this study will lead to an increased focus on student enrichment at tertiary level. This study exposes this element of teaching academically strong students and o ers possible avenues of challenging these students. / Thesis (PhD)--University of Pretoria, 2015. / Mathematics and Applied Mathematics / PhD / Unrestricted

At the primary school level, the understanding of multiplication and division is critical, and division is a particularly challenging concept for most learners. I argue that the concept of division can be grasped more readily if learners learn to regulate their own mental processes intentionally; and that metacognitive strategies can be cultivated. Learners’ cognitive development and their propensity for visual imagery at age 11 to 12 years provide an opportunity to mediate such a strategy while learning mathematics concepts. Visualisation gives learners access to what I call “the virtual space of the mind”, where they can reconstruct an external life situation as an internal reality, upon which they can act mathematically. Based on a review of the relevant literature, the use of visualisation in problem solving (division in particular) at this developmental age was found to be under-explored, which gave rise to the need to develop a useful and feasible instructional design. Design Research was applied, as its iterative, cyclic nature allows the researcher to work through phases while developing various prototypes of the design. The fourth prototype of this design was tested with sixteen Grade 6 participants in the classroom setting of an English-medium primary school in Gauteng Province, South Africa, where the strategy was mediated for division in money-, area- and rate contexts. The research enabled the identification of specific design principles to underpin a final design for the mediation of visualisation as a metacognitive strategy in learning multiplicative concepts. / Thesis (PhD)--University of Pretoria, 2016. / Science, Mathematics and Technology Education / Unrestricted

The National Council of Teachers of Mathematics (1989) has identified problem solving as a major goal of school mathematics. Arithmetic word problem solving is difficult for children. The primary cause of this difficulty is not computational, as once believed, but representational. Children have difficulty understanding and representing the information in the problem. The purpose of this study is to design, implement, and evaluate a constructivist instructional approach to help children be successful arithmetic word problem solvers. It is a three week meaning-based approach to problem writing implemented by the teacher in a third grade classroom in a college laboratory school. The approach has children working collaboratively to author their own word problems. Children write math "stories" based on their everyday experiences. The children then write different types of math stories, along the lines of the typology similar to that proposed by Riley, Greeno, & Heller (1983). Children next explore how these math stories can be turned into problems by deriving the many questions that can be asked from any one story, making it into several problems. Subsequent instruction introduces the idea of multi-step, multi-type story problems. The instructional approach is guided by the important underpinnings in constructivist theory of the need for discourse, collaboration, and knowledge construction. This dissertation is an empirical study, qualitative and descriptive in nature. My field notes, videotapes, and audiotapes of each day's session, and the children's oral and written work provide the raw data for the study. The schematic knowledge necessary to understand arithmetic word problems and Riley, Greeno, and Heller's word problem typology (1983) serve as the theoretical frameworks for the analysis of the data. The data show that children construct the schematic knowledge necessary to understand word problem structure across problem types, knowledge they did not have at the outset of the study. The stories and problems the children create collaboratively and the questions and discussions the children and the teacher pursue together in the spirit of mathematical discourse demonstrate that this approach holds promise as a basis for robust, meaning-based instruction in arithmetic word problem solving.

The purpose of this study was to explore at-risk (Native Taiwanese) children's habits of mind, applying Vygotsky's ZPD theory in learning habits of mind in math. Workshops were used to teach pairs of siblings' habits of mind. The study was conducted with 62 subjects and 62 siblings or older peers in two elementary schools. Each pair was randomly assigned into either the experimental or the control group. Siblings who were in the experimental group participated in the workshops to receive training that could help the experimental subjects to learn habits of mind. A pretest and a posttest were given to assess their habits of mind in math. Analysis of data revealed no significant differences between experimental group and control group in the pretest. In the posttest, there were significant differences between experimental group and control group in the areas of patterning, describing, and visualizing, but there was no significant difference in the “experimenting” condition. In summary, the data shows that patterning is easy to learn, visualizing comes next, describing is more difficult, and experimenting is the most difficult. All of these habits of mind can be learned through applying Vygotsky's ZPD theory and using sibling workshop, but there is a need for the students and siblings to have extensive time to practice.

abstract: Research shows that the subject of mathematics, although revered, remains a source of trepidation for many individuals, as they find it difficult to form a connection between the work they do on paper and their work's practical applications. This research study describes the impact of teaching a challenging introductive applied mathematics course on high school students' skills and attitudes towards mathematics in a college Summer Program. In the analysis of my research data, I identified several emerging changes in skills and attitudes towards mathematics, skills that high-school students needed or developed when taking the mathematical modeling course. Results indicated that the applied mathematics course had a positive impact on several students' attitudes, in general, such as, self-confidence, meanings of what mathematics is, and their perceptions of what solutions are. It also had a positive impact on several skills, such as translating real-life situations to mathematics via flow diagrams, translating the models' solutions back from mathematics to the real world, and interpreting graphs. Students showed positive results when the context of their problems was applied or graphical, and fewer improvement on problems that were not. Research also indicated some negatives outcomes, a decrease in confidence for certain students, and persistent negative ways of thinking about graphs. Based on these findings, I make recommendations for teaching similar mathematical modeling at the pre-university level, to encourage the development of young students through educational, research and similar mentorship activities, to increase their inspiration and interest in mathematics, and possibly consider a variety of sciences, technology, engineering and mathematics-related (STEM) fields and careers. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics for the Life and Social Sciences 2020

The Hispanic population in the United States experiences many challenges in education that have placed them behind their Euro-American counterparts in terms of achievement. These challenges are associated with socioeconomic status and family structure, educational expectations, cognitive skills, and low-quality schooling in the elementary grades. The purpose of this study was to examine how Hispanic students construct science learning in an argument-based inquiry classroom. This research constitutes a qualitative case study grounded in a sociocultural constructivist framework. Data was collected using a variety of qualitative techniques, including nonparticipant observations, analysis of semi-structured interviews, audio recordings, transcription, and observations. The focal participants of this study are three Latino/Hispanic students, two in fifth grade and one in fourth grade. Findings indicated that the two aspects of an argument-based inquiry approach impact students learning in science under diverse conditions. Students also encounter particular challenges while they are involved in this learning context.

In this study a model for the design of mathematical software was developed and tested. The model, which is based upon current cognitive theories of learning, was used to design Transformer, a computer learning environment (CLE) for exploring transformational geometry. In a pilot study, the software was used in middle-school classrooms and in in-service workshops, and then refined for use in the current study which tested the design model qualitatively. Two cycles of in-depth interviews were conducted with each of twelve high-school students, who used the CLE to solve geometric problems. Transcripts from videotapes of student work on two problems were extracted and subjected to a protocol analysis. Analysis revealed patterns of misconceptions and patterns of problem-solving approaches by the students, which led to suggested improvements in software design model and in the CLE. The most common misconceptions, confusions regarding reflections and rotations, led to redesigning display features including placement of mirrors and animation of rotation. Many participants found the use of both physical manipulatives and transparent screen overlays a significant problem-solving aid. It was concluded that CLEs which incorporate the use of such manipulatives would be easier to understand for many students. The model was expanded to include design principles of making the CLE accessible to a greater number of students. Implications of the research for mathematics education and for software design are discussed and suggestions for further testing of the model presented. Appendices include a typical protocol analysis of one student's work on a selected problem.