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The Grundlagen der Geometrie brought Euclid’s ancient axioms up to the standards of modern logic, anticipating a completely mechanical verification of their theorems. There are five groups of axioms, each focused on a logical feature of Euclidean geometry. The first two groups give us ordered geometry, a highly limited setting where there is no talk of measure or angle. From these, we mechanically verify the Polygonal Jordan Curve Theorem, a result of much generality given the setting, and subtle enough to warrant a full verification. Along the way, we describe and implement a general-purpose algebraic language for proof search, which we use to automate arguments from the first axiom group. We then follow Hilbert through the preliminary definitions and theorems that lead up to his statement of the Polygonal Jordan Curve Theorem. These, once formalised and verified, give us a final piece of automation. Suitably armed, we can then tackle the main theorem.

There is an extensive academic literature documenting students' pre-instructional interpretations of phenomena and events, showing that these interpretations are not consistent with 'energy' as used in physics. Furthermore, there is evidence showing that teaching does not always result in students' interpretations becoming consistent with teaching. This evidence has been collected across many decades and in many countries. This thesis presents a theoretical rationale for the design, implementation and evaluation of a teaching sequence aiming to promote conceptual understanding of the concept of energy to 15-16 years old physics students in Cyprus. Students' pre-instructional ideas about energy were investigated and were found to be consistent with those reported in the international literature. A design study was conducted with the aim of improving students' conceptual understanding compared to that might be expected from physics teaching usually conducted in Cyprus. Among the perspectives and the methods critically reviewed to inform the design, the perspective set out by Leach and Scott (2002) drawing on the concept of Learning Demand was chosen. Furthermore, the energy ideas proposed within the SPT11-14 project (2006) were used as a basis for the development of the theoretical framework of the teaching sequence. The research questions for the study were as follows: RQ1. What concepts of energy are used by a Cypriot cohort of upper high school students prior to teaching? RQ2. How do the conceptions and learning of the sub-cohort of Cypriot students taught through the research-informed approach compare with those following 'normal teaching' after instruction? RQ3. How do the understandings of the energy concept, of a small sub-group of students, develop during the lessons of the research-informed approach? The study involved thirty six Cypriot students in two classes of an urban upper secondary school. One of the classes acted as an intervention group, receiving the 'new' teaching sequence whilst the comparison group followed normal teaching. For addressing RQ1, data were collected through a pre-test administered to all participant students. Pre-test data and data collected through a post-test also administered to all participant students were used to test the comparative effectiveness of the teaching, thereby addressing RQ2. To address RQ3, data collected from a small number of experimental students through pre and post-test, two short-length diagnostic probes, two interviews and through an interview conducted with the experimental teacher were used. Analysis of data allowed for the following answers to each research question: RQ1: The students' pre-instructional views about the concept of energy were not within the current scientific beliefs; rather, these were within an alternative context. They were consistent with findings reported internationally in the literature. RQ2: The students' understandings of the concept of energy taught through the research-informed approach were significantly higher after instruction compared with those following 'normal teaching'. RQ3: The students' understandings of the concept of energy through the lessons of the research-informed approach were developed in a manner consistent with the intended aims of the teaching. The answers to the three research questions strongly suggest that the proposed teaching sequence for energy was found to be effective.

This study is the first attempt to develop, implement and evaluate a computer plus talk teaching sequence with the purpose of enhancing 15-16 years old students‘ understanding of chemical rate of reaction concept in Uganda. A total of 247 students aged 15-16 years from two High Schools participated in the study. The experimental class (C&TA) consisted of 108 students, 51 males and 57 females while the comparison class (NTA) consisted of 139, 73 males and 66 females. Based on recommendations in the literature that chemical rate of reaction is a difficult topic to teach and learn, I developed a research-based teaching sequence on chemical rate of reaction using a computer and talk approach, implemented this teaching sequence in Uganda and evaluated this teaching sequence by making comparisons with ‗normal‘ teaching. Research-based teaching is a novel approach to pedagogy in Uganda, so I had to train the experimental teacher for two weeks. The trainings were very important in ensuring that the teacher acquired working knowledge about C&TA prior to the actual implementation in the classroom. The study investigated whether the students who followed the C&TA had better understanding of the difficult areas (also called learning demands): (1) rate of reaction, (2) proper orientation of reacting particles, (3) the relationships between activation energy and chemical rate of reaction, (4) the effect of temperature and (5) the effect of concentration of reactants on chemical rate of reaction. I undertook a quasi-experimental study to assess the C&TA‘s impact. I analysed classroom interactions to inform discussion of what influenced its effectiveness, and inform decisions about whether the C&TA sequence was implemented consistently with its design. The statistical analyses of the post-test scores show that the experimental class (C&TA) students demonstrated better understanding across all the five difficult areas compared to the comparison class (NTA) students. The findings indicate that female students benefited (a little) more from the intervention than male students. The results show that the C&TA intervention had an effect of the same magnitude across the ability range. Further findings show that C&TA support teaching large classes and that it is possible for a teacher in Uganda to teach in a more interactive/ dialogic way with relatively little training on the communicative approaches. Evidence shows that aspects of the C&TA teaching sequence that were effective in supporting students‘ learning of chemical rate of reaction were: computer simulations and modelling, teaching goals, worksheets, social constructivist perspective on teaching and learning along with the communicative approaches. Further findings show varied benefits and challenges from using C&TA. The teacher and students perceived C&TA as a good method of teaching and learning. Indicating that the use of a computer and talk approach (C&TA) is a feasible alternative teaching approach to didactic teaching in science classrooms in Uganda. It also suggests that C&TA could be adopted for teaching and learning other subjects.

This study investigates how teachers in a rural Kenyan primary school use code-switching to help pupils to understand scientific concepts. The study takes place in a teaching and learning environment where English is the 'official' language of teaching beyond year three but where it is accepted by the education ministry that the English proficiency of most of the pupils may not be sufficient for them to fully access the science curriculum. It is, therefore, also accepted that teachers may use three languages, English, Kiswahili and Kigiryama, in the classroom, to explain scientific ideas. The study begins by considering some of the barriers to learning science concepts experienced by those who are taught in a second language. There is then an exploration of code-switching strategies as observed by other researchers many of which have been recorded within science lessons in other Kenyan primary schools. This study then uses observational and interview methods to investigate the actual use of code-switching within the lessons of standard four and five pupils at the target school. These observations, combined with those of other researchers are then used to produce a typology of codeswitching interventions commonly used by teachers during science lessons at the school at Mida Primary School. Following this a second literature review outlines a framework of the key features of explanation, clarifying how teachers use language to explain scientific concepts. The code-switching typology and this framework are then utilised to enable the analysis of six transcripts of science lessons at the target school. The analysis is carried out with the purpose of examining the contribution that each code-switching intervention makes to the meaning-making process. Some implications for teacher training are presented concerning how actual teacher behaviour, with respect of their use of code-switching, compares with what might be expected if trained for a planned use of code-switching when explaining scientific concepts. Also included in the study is a brief examination of other contextual factors that might affect the use of code-switching by teachers in science lessons and the implications that these might have on teacher training for the effective use of codeswitching. Finally, some suggestions for further research and an evaluation of the approach taken in this study are presented.