Ng Wee Leng

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.

The purpose of this study is to investigate the effects of an Ancient Chinese Mathematics Enrichment Programme (ACMEP) on pre-university students’ attitudes towards mathematics. A total of 37 students took part in this study and a mathematics attitudes scale was administered prior to and after the ACMEP which consisted of 4 sessions of one hour duration each, held once a week, conducted during curriculum time. Results of statistical analysis revealed that the post-ACMEP mathematics attitudes survey overall mean score was higher (t = 1.99, df = 31, p = 0.06) than that of the pre-ACMEP mathematics attitudes survey, with an effect size of 0.40. Overall, the conduct of the ACMEP appeared to improve pre-university students' attitudes towards mathematics.

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.

A design experiment was conducted to examine the role of the TI-Nspire, the latest graphing calculator from Texas Instruments, in teaching and learning calculus. This paper reports details on, and preliminary results of, the design experiment involving the design and conduct of a TI-Nspire Intervention Programme for an intact class of thirty-six secondary four students (15-16 years) from a secondary school in Singapore. Use of the TI-Nspire was integrated into teaching and learning Calculus concepts with the aid of the TI Navigator, a wireless classroom network system that enables instant and active interaction between students and teachers. Mathematics attitudes surveys and structured interviews were administered to assess the effects of the use of the TI-Nspire on students’ attitudes towards mathematics. It was found that appropriate use of graphical, numerical and algebraic representations of Calculus concepts using the TI-Nspire could enable the subjects to better visualize the concepts and make generalizations of relevant mathematical properties. Results of paired samples t-tests and interviews with students suggest that there the use of the TI-Nspire has a positive effect on students’ confidence in and perceived usefulness of mathematics.

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 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.

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.

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.

Teachers’ judgements of the cognitive demand of mathematical assessment items have implications for the nature of students’ learning experiences. However, existing taxonomies for classifying cognitive demand are often not customised for pre-university or A-level mathematics teachers to use. This paper reports on the development and field-testing of a cognitive demand instrument specifically for helping A-level mathematics teachers to sharpen their judgements of the cognitive demand of A-level mathematical assessment items. Fourteen A-level Mathematics teachers from a junior college and an Integrated Programme school in Singapore participated in this study, where they rated the cognitive demand of assessment items on selected Pure Mathematics topics from the national A-level examinations using the cognitive demand instrument. The instrument was found to be useful to the teachers in providing them with an awareness of how the cognitive demand of A-level mathematical assessment items could be understood through the dimensions of complexity, abstractness, and strategy.

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.