Ng, Wee Leng

In the light of the recent focus on the use of technology in education, a Crucial Factors in the Integration of ICT Survey (CFS) was developed to examine mathematics teachers' perceptions of the degree of importance pertaining to the key factors influencing the integration of Information and Communications Technology (ICT) into the classroom. The purpose of this study was to field-test this instrument which measures secondary school mathematics teachers' perceptions of the positive impact of the following factors on ICT implementation as identified by a review of the literature: (a) usefhlness and worthiness of technology, (b) support from various departments in the school, (c) availability and accessibility of technology, (d)professional development opportunities in technology, (e) leadership, planning and implementation of technology, and ( f ) partnerships with external organisations. In this study, a total of 60 pre-service mathematics teachers were surveyed. The results show that these pre-service teachers rate both professional development opportunities and the availability and accessibility of technology as the most influential factors among the six in determining the extent to which they will employ ICT in the classroom. However, all factors are deemed important in determining whether teachers utilised technology in the classroom.

The purpose of this study was to examine the effects of a calculus course using the flipped classroom model on undergraduate students’ achievement in mathematics which was measured by their scores on three quizzes, a test, and a final written examination, as well as their overall scores. The scores of a total of 58 second year students, comprising 17 students in the experimental group and 41 students in the control group, enrolled in a university degree programme in Singapore were analysed retrospectively using analysis of covariance (ANCOVA) so as to control for initial differences. The experimental group comprised students who took the flipped calculus course in the August 2016 semester while the control group comprised students who took the same calculus course taught using a lecture-tutorial approach in the August 2013 semester. Results of ANCOVA show that after controlling for initial differences the experimental group scored statistically significantly higher in the test but lower in the final examination than the control group.

The main objective of this thesis is to define a nonabsolute integral measure theoretically. More precisely we define an integral of the Henstock type, called the H-integral, on measure spaces with a locally compact Hausdorff topology that is compatible with the measure. Relevant results pertaining to the H-integral are established.

In Chapter 1, we define the H-integral and derive the properties that are fundamental to an integral. We describe in Section 1.1 how certain objects in the space are chosen to be generalised intervals and relate the definition to some concrete examples. The H-integral is defined in Section 1.2 and we prove that it includes the well-known Kurzweil-Henstock integral [18] on the real line. The basic properties that hold true for the H-integral, in particular, the Henstock's lemma and the monotone convergence theorem, are derived in Section 1.3.

Chapter 2 aims to relate the H-integral to known integrals. In Section 2.1, we define the M-integral, which is a McShane-type integral, and prove that a function is M-integrable if and only if it is absolutely H-integrable. The domains of H-integration and M-integration are also extended to measurable sets. Subsequently in Section 2.2 we establish the equivalence between the M-integral and the Lebesgue integral. we also show that a function H-integrable on an elementary set is Lebesgue integrable on a portion of the elementary set. In Section 2.3 we establish the fact that the H-integral includes the Davies as well as the Davies-McShane integral defined by Henstock in [13]. This is done by establishing the equivalence between the Lebesgue integral and the Davies as well as the Davies-McShane integral. The conclusion here is that if a function is measurable then the absolute H-integral, the M-integral, the Lebesgue integral, the Davies integral and the Davies-McShane integral and the Davies-McShane integral are all equivalent.

Further results of the H-integral are given in Chapter 3. We begin by proving in section 3.1 that H-integrable functions are measurable and proceed to give a necessary and sufficient condition for a function to be H-integrable. We also prove that the H-integral is genuinely a nonabsolute one by constructing an example which is H-integrable but not absolutely H-integrable. Two concepts very relevant to the H-integrability, namely the generalised absolute continuity and equi-integrability, are introduced in Section 3.2 and some results involving these concepts are proved. Section 3.3 is devoted to proving the convergence theorems of the H-integral. We start with the proofs of the equi-integrability theorem and the basic convergence theorem and illustrate how the mean convergence theorem can be proved with the aid of the two former theorems. The controlled convergence theorem is proved in s few lemmas and by applying the basic convergence theorem.

Chapter 4 is the most important part of this thesis. We generalise our work in [23] for the H-integral in this chapter. The main theorem, namely the Radon -Nikodym theorem for the H-integral, is proved in Section 4.1 with which we give a descriptive definition of the H-integral in Section 4.2. By imposing a different condition, a second version of the main theorem and subsequently a second descriptive definition of the H-integral are also given. The purpose of Section 4.3 is to report on our findings in [23]. some results corresponding to those we prove in Section 4.1 are given for the Euclidean space setting. We also show how some known results on the real line, for example, the fundamental theorem of calculus for the Kurzweil-Henstock integral, can be deduced.

The purpose of this study was to investigate how the use of TI-Nspire™ could enhance the teaching and learning of calculus. A conceptual framework for the use of TI-Nspire™ for learning calculus in a mathematics classroom is proposed that describes the interactions among the students, TI-Nspire™, and the learning tasks, and how they lead to the learning of calculus. A design experiment was conducted in a class of 35 students from a secondary school in Singapore. Use of TI-Nspire™ was integrated into the teaching and learning of calculus concepts in the classroom with the aid of TI-Nspire™ Navigator, a wireless classroom network system that enables instant and active interaction between students and teachers. It was found that the appropriate use of graphical, numerical and algebraic representations of calculus concepts using TI-Nspire™ enabled students to better visualize the concepts and make generalizations about relevant mathematical properties. In addition, the students were able to link multiple representations, especially algebraic and graphical representations, to improve their conceptual understanding and problem-solving skills. Six roles of TI-Nspire™ in classroom mathematical practice were identified from the findings of the experiment; TI-Nspire™ was used as an exploratory tool, graphing tool, confirmatory tool, problem-solving tool, visualization tool and calculation tool. This suggests that TI-Nspire™ is a multi-dimensional tool that supports mathematics learning. Overall, the findings of the study indicate that TI-Nspire™ is an effective tool to develop mathematical concepts and promote learning and problem solving.

In this paper, it is shown how the Banach-Steinhaus theorem for the space P of all primitives of Henstock-Kurzweil integrable functions on a closed bounded interval, equipped with the uniform norm, can follow from the Banach-Steinhaus theorem for the Denjoy space by applying the classical Hahn-Banach theorem and Riesz representation theorem.

We introduce the notion of an H-primitive being the limit of a sequence of absolutely continuous functions satisfying certain conditions and use it to formulate an alternative definition of the Henstock-Kurzweil integral on a closed bounded interval. Furthermore, the definition provides a characterisation of the primitive of a Henstock-Kurzweil integrable function.

In recent years, computer-based technology has enabled university lecturers to teach their courses using non-traditional pedagogies. One such pedagogy is the flipped learning model. As flipped learning is being used more frequently to teach undergraduate mathematics, instructors need to collect data to identify practices that work well to promote student mathematics achievement and favourable perceptions toward this new learning mode. This paper describes six different types of pre-class tasks for a flipped Linear Algebra II course in a Singapore university, such as short videos narrated by the instructor, synopses, summary sheets, worksheets of problems and activities, and online quizzes. The sample comprised 15 pre-service teachers, who had adequate to good mathematics backgrounds, and their participation in this project would prepare them to implement flipped learning in school mathematics in the future. On average, they spent about one hour to complete these weekly pre-class tasks, but the stronger ones reported spending less time on these tasks than the other students. Almost all the students rated very highly these tasks in terms of helping them to learn and enjoyment at mid-semester and end-of-course surveys. These perceptions had weak correlations with the course grade. Suggestions for practice and future research are discussed.

This paper reports on a study on a semester-long flipped university mathematics course (Calculus II) taught to a cohort of pre-service teachers enrolled in the Bachelor of Science (Education) programme at the National Institute of Education, which is the autonomous teacher training institute of Nanyang Technological University, Singapore. The current study is the second phase of a three-phase project which developed a comprehensive framework to guide the design of three stages of flipped learning activities: pre-class tasks; in-class interactions; and post-class consolidation. A mixed methods research design was used to collect quantitative and qualitative data over many occasions, through methods such as weekly surveys, to investigate students’ perceptions of flipped learning activities. Results of the study suggest that the students generally found the flipped learning activities helpful and enjoyable.