Preservice Secondary School Mathematics Teachers\u27 Current Notions of Proof in Euclidean Geometry

Much research has been conducted in the past 25 years related to the teaching and learning of proof in Euclidean geometry. However, very little research has been done focused on preservice secondary school mathematics teachers’ notions of proof in Euclidean geometry. Thus, this qualitative study was exploratory in nature, consisting of four case studies focused on identifying preservice secondary school mathematics teachers’ current notions of proof in Euclidean geometry, a starting point for improving the teaching and learning of proof in Euclidean geometry. The unit of analysis (i.e., participant) in each case study was a preservice mathematics teacher. The case studies were parallel as each participant was presented with the same Euclidean geometry content in independent interview sessions. The content consisted of six Euclidean geometry statements and a Euclidean geometry problem appropriate for a secondary school Euclidean geometry course. For five of the six Euclidean geometry statements, three justifications for each statement were presented for discussion. For the sixth Euclidean geometry statement and the Euclidean geometry problem, participants constructed justifications for discussion. A case record for each case study was constructed from an analysis of data generated from interview sessions, including anecdotal notes from the playback of the recorded interviews, the review of the interview transcripts, document analyses of both previous geometry course documents and any documents generated by participants via assigned Euclidean geometry tasks, and participant emails. After the four case records were completed, a cross-case analysis was conducted to identify themes that traverse the individual cases. From the analyses, participants’ current notions of proof in Euclidean geometry were somewhat diverse, yet suggested that an integration of justifications consisting of empirical and deductive evidence for Euclidean geometry statements could improve both the teaching and learning of Euclidean geometry

The Mean-Median Map

PhD ThesisThe mean-median map enlarges a nite (multi)set of real numbers by adjoining to it a new number such that the mean of the enlarged set is equal to the median of the original set. An open conjecture states that, starting with any nite set, the sequence of the new numbers generated by iterating this map stabilises, i.e., is eventually constant. We approach this problem from a new perspective, that of a dynamical system over the space of nite sets of piecewise-a ne continuous functions with rational coe cients, de ning the map pointwise. We develop a theory for the dynamics in the neighbourhoods of the local minima of the limit function |the limit of the generated sequence| establishing its local shapes and symmetries. We also show that the conjecture can be veri ed in exponentially many neighbourhoods simultaneously by computing a single dyadic rational orbit of a variant of the map. We then study a common pre-stabilisation behaviour of rational orbits, and construct a family of initial sets for which stabilisation can be delayed arbitrarily. Finally, using our theory, we extend the existing computational results by over two orders of magnitude. The results reveal that the total measure of the regular neighbourhoods is far from full, suggesting the existence of a region with a new, presently unexplained, dynamical behaviour

Creativity and divergent thinking in Geometry education

The teaching of geometry has been neglected at the expense of other disciplines of mathematics such as algebra in most secondary schools for Africans in South Africa. The research aimed at establishing the extent to which creativity and divergent thinking enhance the internalisation of geometry concepts using the problem-based approach and on encouraging learners to be creative, divergent thinkers and problem solvers. In the research, Grade 7 learners were guided to discover the meaning of geometric concepts by themselves (self-discovery) and to see concepts in a new and meaningful way for them. This is the situation when learners think like the mathematicians do and re-invent mathematics by going through the process of arriving at the product and not merely learn the product (axioms and theorems), for example, discover properties of two- and three-dimensional shapes by themselves. Furthermore, learners were required to use metaphors and analogies, write poems, essays and posters; compose songs; construct musical instruments and use creative correlations in geometry by using geometric shapes and concepts. They tessellated and coloured polygons and pentominoes in various patterns to produce works of art. Divergent thinking in geometrical problem solving was evidenced by learners using cognitive processes such as, amongst others, conjecturing, experimenting, comparing, applying and critical thinking. The research was of a qualitative and a quantitative nature. The problem-based approach was used in teaching episodes. The following conclusions and recommendations were arrived at: * Geometric shapes in the learner's environment had not been used as a basis for earning formal geometry. * Second language learners of mathematics have a problem expressing themselves in English and should thus be given the opportunity to verbalize their perceptions in vernacular. * Learners should be made to re-invent geometry and develop their own heuristics/strategies to problem solving. * Learners should be trained to be creative by, for example, composing songs using geometric concepts and use geometric shapes to produce works of art, and * Activities of creativity and divergent thinking should be used in the teaching and learning of geometry. These activities enhance the internalisation of geometry concepts. Groupwork should be used during such activities.Educational StudiesD. Ed. (Didactics

Solar Seismology from Space. a Conference at Snowmass, Colorado

The quality of the ground based observing environment suffers from several degrading factors: diurnal interruptions and thermal variations, atmospheric seeing and transparency fluctuations and adverse weather interruptions are among the chief difficulties. The limited fraction of the solar surface observable from only one vantage point is also a potential limitation to the quality of the data available without going to space. Primary conference goals were to discuss in depth the scientific return from current observations and analyses of solar oscillations, to discuss the instrumental and site requirements for realizing the full potential of the seismic analysis method, and to help bring new workers into the field by collecting and summarizing the key background theory. At the conclusion of the conference there was a clear consensus that ground based observation would not be able to provide data of the quality required to permit a substantial analysis of the solar convection zone dynamics or to permit a full deduction of the solar interior structure

Vagueness in mathematics talk

The Cockcroft Report claimed that "mathematics provides a means of communication which is powerful, concise and unambiguous". Such precision in language may be a conventional aim of mathematics, particularly when communicated in writing. Nonetheless, as this thesis demonstrates, vagueness is commonplace when people talk about mathematics. In this thesis, I examine the circumstances in which vagueness arises in mathematics talk, and consider the practical purposes which speakers achieve by means of vague utterances in this context. The empirical database, which is considered in Chapters 4 to 7, consists almost entirely of transcripts of mathematical conversations between adult interviewers (including myself) and one or two children. The data were collected from clinical interviews focused on a small number of tasks, and from fragments of teaching. For the most part, the pupils involved in the study were aged between 9 and 12, although the age-range in Chapter 7 extends from 4 to 25. I draw on a number of approaches to discourse associated with 'pragmatics' -a field of linguistics - to analyse the motives and communicative effectiveness of speakers who deploy vagueness in mathematics talk. I claim that, for these speakers, vagueness fulfills a number of purposes, especially 'shielding', i. e. self-protection against accusation of being wrong. Another purpose is to give approximate information; sometimes to achieve shielding, but also to provide the level of detail that is deemed to be appropriate in a given situation. A different purpose, associated with a particular form of vagueness (of reference), is to compensate for lexical gaps in pursuit of effective communication of concepts and ideas. I show, in particular, how speakers use the pronouns 'it' and 'you' in mathematics talk to communicate concepts and generalisations. Some consideration is given to the intentions of 'expert speakers of mathematics when they deploy vague language. Their purposes include some of those identified for novices. Teachers also use vagueness as a means of indirectness in addressing pupils; this strategy is associated with the redress of 'face threatening acts'. My thesis is that vagueness can be viewed and presented, not as a disabling feature of language, but as a subtle and versatile device which speakers can and do deploy to make mathematical assertions with as much precision, accuracy or as much confidence as they judge is warranted by both the content and the circumstances of their utterances. I report on the validation and generalisation of my findings by an Informal Research Group of school teachers, who transcribed and analysed their own classroom interactions using the methods I had developed

Using Genetic Algorithm to solve Median Problem and Phylogenetic Inference

Genome rearrangement analysis has attracted a lot of attentions in phylogenetic com- putation and comparative genomics. Solving the median problems based on various distance definitions has been a focus as it provides the building blocks for maximum parsimony analysis of phylogeny and ancestral genomes. The Median Problem (MP) has been proved to be NP-hard and although there are several exact or heuristic al- gorithms available, these methods all are difficulty to compute distant three genomes containing high evolution events. Such as current approaches, MGR[1] and GRAPPA [2], are restricted on small collections of genomes and low-resolution gene order data of a few hundred rearrangement events. In my work, we focus on heuristic algorithms which will combine genomic sorting algorithm with genetic algorithm (GA) to pro- duce new methods and directions for whole-genome median solver, ancestor inference and phylogeny reconstruction. In equal median problem, we propose a DCJ sorting operation based genetic algorithms measurements, called GA-DCJ. Following classic genetic algorithm frame, we develop our algorithms for every procedure and substitute for each traditional genetic algorithm procedure. The final results of our GA-based algorithm are optimal median genome(s) and its median score. In limited time and space, especially in large scale and distant datasets, our algorithm get better results compared with GRAPPA and AsMedian. Extending the ideas of equal genome median solver, we develop another genetic algorithm based solver, GaDCJ-Indel, which can solve unequal genomes median prob- lem (without duplication). In DCJ-Indel model, one of the key steps is still sorting operation[3]. The difference with equal genomes median is there are two sorting di- rections: minimal DCJ operation path or minimal indel operation path. Following different sorting path, in each step scenario, we can get various genome structures to fulfill our population pool. Besides that, we adopt adaptive surcharge-triangle inequality instead of classic triangle inequality in our fitness function in order to fit unequal genome restrictions and get more efficient results. Our experiments results show that GaDCJ-Indel method not only can converge to accurate median score, but also can infer ancestors that are very close to the true ancestors. An important application of genome rearrangement analysis is to infer ancestral genomes, which is valuable for identifying patterns of evolution and for modeling the evolutionary processes. However, computing ancestral genomes is very difficult and we have to rely on heuristic methods that have various limitations. We propose a GA-Tree algorithm which adapts meta-population [4], co-evolution and repopulation pool methods In this paper, we describe and illuminate the first genetic algorithm for ancestor inference step by step, which uses fitness scores designed to consider co- evolution and uses sorting-based methods to initialize and evolve populations. Our extensive experiments show that compared with other existing tools, our method is accurate and can infer ancestors that are much closer to true ancestors

Improving the problem-solving potential (PsP) of highly-able transition year students through participation in a mathematics intervention

There is widespread agreement in education about the existence of a cohort of students with ability or potential above that of their peers. In this thesis, they will be referred to as “highly-able” students, and the focus will be on those students whose high ability is in mathematics. Contrary to popular belief, highly-able students have additional educational needs to be catered for. In Ireland, classroom differentiation remains the sole in-school measure available, despite post-primary mathematics education undergoing large-scale changes over the past decade. In the mid-2000s, research highlighted the performance of Ireland’s top students in international mathematics assessments as an area of concern, yet these have failed to improve and, in some cases, have actually disimproved. When the additional needs of highly-able students are not met, they are at risk of negative traits of perfectionism, under-achievement, behavioural problems in class, and so on. This research focussed on addressing the additional educational needs of highly-able students in Ireland by targeting an improvement in their mathematical Problem-solving Potential (PsP), a newly-designed triad construct with three influencing factors: problem-solving skills, mindset, and mathematical resilience. To facilitate the development of potential, external resources and supports are needed to act upon and encourage the abilities and traits of an individual. In order to achieve that, a mathematics intervention was designed. The intervention utilised collaborative problem-solving as the pedagogical approach, following the Collaborative Problem-solving (CoPs) model specifically designed for this research, to outline the problem-solving process in a group dynamic. Although it is sometimes presumed that highly-able students prefer to work alone, research has found that they are willing collaborators with like-minded peers. The intervention was implemented across six cohorts of students, accessed through the Centre for Talented Youth Ireland, over a three-year period, and resulted in an increased PsP for 97% of participants