This paper is a study on some upper bounds of the Mertens function, which is often considered somewhat of a ``mysterious" function in mathematics and is closely related to the Riemann Hypothesis. We discuss some known bounds of the Mertens function, and also seek new bounds with the help of an automated conjecture-making program named CONJECTURING, which was created by C. Larson and N. Van Cleemput, and inspired by Fajtowicz's Dalmatian Heuristic. By utilizing this powerful program, we were able to form, validate, and disprove hypotheses regarding the Mertens function and how it is bounded.

Benzenoids are graphs representing the carbon structure of molecules, defined by a closed path in the hexagonal lattice. These compounds are of interest to chemists studying existing and potential carbon structures. The goal of this study is to conjecture and prove relations between graph theoretic properties among benzenoids. First, we generate conjectures on upper bounds for the domination number in benzenoids using invariant-defined functions. This work is an extension of the ideas to be presented in a forthcoming paper. Next, we generate conjectures using property-defined functions. As the title indicates, the conjectures we prove are not thought of on our own, rather generated by a process of automated conjecture-making. This program, named Cᴏɴᴊᴇᴄᴛᴜʀɪɴɢ, is developed by Craig Larson and Nico Van Cleemput.

This paper describes some results of a research study on conjecturing and proving in a dynamic geometry environment (DGE), and it focuses on particular cognitive processes that seem to be induced by certain uses of tools available in Cabri (a particular DGE). Building on the work of Arzarello and Olivero (Arzarello et al., 1998, 2002; Olivero, 2002), we have conceived a model describing some cognitive processes that may occur during the production of conjectures and proofs in a DGE and that seem to be related to the use of specific dragging schemes, in particular to the use of the scheme we refer to as maintaining dragging. This paper contains a description of aspects of the theoretical model we have elaborated for describing such cognitive processes, with specific attention towards the role of the dragging schemes, and an example of how the model can be used to analyze students’ explorations.

This study is comprised of two parts: an unauthorised biography of the South African musician David Kramer, as well as a reflective look at the process of writing this biography. In this regard the following aspects were looked at closely: finding an appropriate style, biography versus propaganda, conjecturing, the bilingual nature of the text, problems of research, ethics, influences, make-believe, approach to the subject, intertextuality, and fictionalisation. The central question of the biography is to highlight the success of a fellow Worcester (the author’s hometown) boy. The central research questions of the thesis are the fictionalisation of the nonfiction text, intertextuality, and the question of a text written in both English and Afrikaans. With regard to the aforementioned fictionalisation, a biographical text is classified as “nonfiction”, because it deals with a real person and real events. However, a text such as David Kramer – an unauthorised biography presents an alternative perspective, in that the narrative often moves into fiction, or “creative nonfiction”. Written texts are traditionally divided into two fields: fiction or nonfiction. Nonfiction is deemed to be fact, truth, whereas fiction is the fruit of an author’s imagination. But perhaps the notion of truth versus untruth is too limited, and one should include the words “objectivity” and “subjectivity”. Some texts incorporate both elements, be they newspaper editorials which are mostly opinion, advertisements which are highly subjective, or biographies such as Taraborrelli’s Madonna – An Intimate Biography, which often reads as a novel. This doctoral thesis looks at David Kramer – an unauthorised biography, which is at times “faction”, to illuminate the sections where the text fell somewhere between fiction or nonfiction. In attempting this exercise, intertextuality was useful in two ways. Firstly, to ground the text in a reality the reader could believe, as it brought “real” things to the text, such as song lyrics, photographs, et cetera, all things which brought some credibility to the truth of the text, and secondly to place the events being described in a certain timeframe. The use of English and Afrikaans in the biography was to reflect that Kramer uses both languages in his songs, and furthermore, to give an idea of the South Africa at the time of Kramer’s early success: the divides of English/Afrikaans, white/black, liberal/conservative. / Thesis (PhD)--University of Pretoria, 2011. / Unit for Creative Writing / Unrestricted

This paper describes some results of a research study on conjecturing and proving in a dynamic geometry environment (DGE), and it focuses on particular cognitive processes that seem to be induced by certain uses of tools available in Cabri (a particular DGE). Building on the work of Arzarello and Olivero (Arzarello et al., 1998, 2002; Olivero, 2002), we have conceived a model describing some cognitive processes that may occur during the production of conjectures and proofs in a DGE and that seem to be related to the use of specific dragging schemes, in particular to the use of the scheme we refer to as maintaining dragging. This paper contains a description of aspects of the theoretical model we have elaborated for describing such cognitive processes, with specific attention towards the role of the dragging schemes, and an example of how the model can be used to analyze students’ explorations.

Researchers conceptually recommend inquiry-based learning as a necessary means to alleviate the problems of learning but this study has embarked on practical implementation of inquiry-based facilitation and learning in Euclidean Geometry. Inquiry-based learning is student-centred. Therefore, the teaching or monitoring of inquiry-based learning in this study is referred to as inquiry-based facilitation. The null hypothesis discarded in this study explains that there is no difference between inquiry-based facilitation and traditional axiomatic approach in teaching Euclidean Geometry, that is, H0: μinquiry-based facilitation = μtraditional axiomatic approach. This study emphasises a pragmatist view that constructivism is fundamental to realism, that is, inductive inquiry supplements deductive inquiry in teaching and learning. Participants in this study comprise schools in Tshwane North district that served as experimental group and Tshwane West district schools classified as comparison group. The two districts are in the Gauteng Province of South Africa. The total number of students who participated is 166, that is, 97 students in the experimental group and 69 students in the comparison group. Convenient sampling applied and three experimental and three comparison group schools were sampled. Embedded mixed-method methodology was employed. Quantitative and qualitative methodologies are integrated in collecting data; analysis and interpretation of data. Inquiry-based-facilitation occurred in experimental group when the facilitator probed asking students to research, weigh evidence, explore, share discoveries, allow students to display authentic knowledge and skills and guiding students to apply knowledge and skills to solve problems for the classroom and for the world out of the classroom. In response to inquiry-based facilitation, students engaged in cooperative learning, exploration, self-centred and self-regulated learning in order to acquire knowledge and skills. In the comparison group, teaching progressed as usual. Quantitative data revealed that on average, participant that received intervention through inquiry-based facilitation acquired inquiry-based learning skills and improved (M= -7.773, SE= 0.7146) than those who did not receive intervention (M= -0.221, SE = 0.4429). This difference (-7.547), 95% CI (-8.08, 5.69), was significant at t (10.88), p = 0.0001, p<0.05 and represented a large effect size of 0.55. The large effect size emphasises that inquiry-based facilitation contributed significantly towards improvement in inquiry-based learning and that the framework contributed by this study can be considered as a framework of inquiry-based facilitation in Euclidean Geometry. This study has shown that the traditional axiomatic approach promotes rote learning; passive, deductive and algorithmic learning that obstructs application of knowledge in problem-solving. Therefore, this study asserts that the application of Inquiry-based facilitation to implement inquiry-based learning promotes deeper, authentic, non-algorithmic, self-regulated learning that enhances problem-solving skills in Euclidean Geometry. / Mathematics Education / Ph. D. (Mathematics, Science and Technology Education)