Ng, Wee Leng

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.

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 (CBT) has enabled university lecturers to teach their courses using non-traditional pedagogies. One such pedagogy is the flipped learning model. Under this model, students learn the basic content on their own using pre-class tasks and then come to class to engage in more challenging work such as solving difficult problems. CBT can play two important roles in flipped learning, namely to deliver learning materials efficiently and to assess student achievement effectively. This paper describes how these two roles were applied to a flipped Linear Algebra II course in the National Institute of Education (Singapore), taken by a group of student teachers (n = 15) over a 12-week period from January to April 2018. Their perceptions of flipped activities were gathered using weekly surveys, mid-semester survey, end-of-course survey, and end-of-course interviews. They generally agreed that flipped learning using CBT was helpful and enjoyable. As flipped learning becomes more common among university lecturers in Asian countries, it is beneficial to share experiences of utilising CBT to promote active learning of mathematics among university students.

Computer algebra systems or software that can manipulate mathematical objects numerically, symbolically and graphically are poised to change the way teachers teach and students learn mathematics. In this paper, to address this change, the development of the Computer Algebra System Attitude Scales, through adapting a widely used computer attitude scale and writing new items, is described. A field test of this instrument in assessing the attitudes of 50 pre-service teachers toward computer algebra system (CAS) upon completion of a CAS-related module requirement of their teacher training programme. The results of the field testing are also discussed.

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.

Anecdotal evidence suggests that many pre-university students (Years 11 and 12) and some mathematics teachers in Singapore have several misconceptions in solving equations of the form f(x) = f⁻¹(x) and that the appropriate use of the graphing calculator (GC) has the potential to correct these misconceptions. This paper analyses a group of prospective teachers' responses to a diagnostic test and pre- and post-diagnostic test surveys which not only revealed various inappropriate uses of the GC in solving the equation f(x) = f⁻¹(x) but also illustrated pedagogical roles the GC could play in correcting students' misconceptions about functions and the understanding of the concepts of functions. The paper also discusses the implications of these findings on classroom practices and pedagogical strategies pertaining to the use of the GC at the pre-university level.

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.