Professor Simon Mukwembi

**Methods of Teaching and Supervision**

Prior to joining UKZN, I have taught a wide variety of courses from service courses such as mathematics for food science and nutrition, and engineering mathematics to purely mathematical modules such as calculus, analysis, algebra, linear algebra, linear mathematics, number theory and graph theory. Then the class size varied from 7 to 450 students. Since my promotion to Senior Lecturer, at UKZN, I have taught the following courses: MATH130 Introduction to Calculus, MATH140 Calculus and Linear Algebra, MATH144 Operations Research, MATH236 Discrete Mathematics with Applications, MATH239 Applied Finite Mathematics, MATH105 Augmented Quantitative Methods, MATH707 Graph Theory (Hons). Class sizes vary from 4 to 212 students. I attach here my weekly in-class time table and class sizes for 2012, Semester 2 (Annexure 2A). In each undergraduate course, apart from consultation times, I have 4 periods of lectures and 4 periods of tutorials per week; each period being a 45 minute session. For each honours course, I have 1 period of tutorials and 3 periods of lectures per week. I always hold 2 consultation hours per week for every course. In addition, I inform my students that they are welcome to come to my oce for further consultation at other times for as long as I do not have an immediate commitment. Typically, I spend from 5 to 20 hours per week depending on whether it is a rst, or a second year module or honours module, an augmented or a main stream module. MATH130, 140, 144 and 105 are rst year, undergraduate modules. MATH105 is an augmented course for management science and commerce students with poor educational backgrounds. The purpose of the augmented courses is to cover the syllabus of the corresponding main stream course and, in addition, to give supplementary material to bridge the gap between High School mathematics and university mathematics. These courses have students with very dissimilar learning capacities. Although MATH144 is a rst year module, it also attracts interested second and third year students and thus consists of students with dissimilar capacities to process information. MATH236 and 239 are second year undergraduate courses, whilst MATH707 is an honours course.

Over the years, I have learnt that teaching is a very complex task which can be equated to irrigation. Just like in irrigation, it is simply not enough to bring water where it is required. Flooding can destroy crops or drive nutrients away. In fact mastering the water resource requires the delicate management of knowing how much is already there, how much is needed and when, and finally, which are the optimal channels and regulation. My lecturing style is a blend of classical lecturing, the interactive approach and, where appropriate, the thesis method. My delivery is therefore a blend where strands are drawn when needed depending on the situation. First, irrespective of the lecturing style, I believe that the lecturer must ensure complete and full understanding of the subject matter prior to the lecture and that the lectures should always be well-delivered. I endeavour to meet this goal. To illustrate, in 2011 100 % of the MATH105 students found that I was always well prepared for my lectures, whereas 97% of the students felt that my lectures were clear and well-delivered. (Student evaluation reports for MATH236 (2010), MATH105 (2011), MATH707 (2012) and Postgraduate supervision evaluation report have been included as Annexures 2B, 2C, 2D and 2E respectively.)

In all my modules, I use scanned beforehand-written slides which are projected onto the screen by the data projector (Annexure F). My slides are concise, clear and user-friendly; I carefully choose different colours to highlight important concepts. The slides have been praised by students. In acknowledgement of the effective use of this tool, one of the students' comments reads:

*Lecture slides were clearly explained and easy to understand (Annexure 2C).*

For the augmented modules, I sometimes include, in my slides, exercises with no solutions. I then solve these exercises step-by-step during my lecture, encouraging students' full participation in the process with me being a facilitator. This method has proved fruitful for classes with students of varied levels of understanding, as it promotes not only learning of the subject matter, but also helps the students learn to think logically, learn problem-solving methods and improve their writing and communication skills as we interact.

To achieve the goal of contextualising my teaching, one fundamental aspect that I employ is to identify and understand the capacity of students to process information. This will help and inform in setting the pace and style of my lecture. In augmented courses, for example, I conduct lectures at a slower pace than usual, I use simpler examples etcetera which all contribute to a more relaxed and caring atmosphere. This ensures that students with poor educational backgrounds grasp the basic fundamental concepts better. I also interact with them more often. A comment in the augmented module MATH105 meant to provide the aspects of the module that the students found most useful reads as follows:

*The way Dr Mukwembi lecturing makes things clear and understandable (Annexure 2C).*

In MATH236, one student hinted:

*Mukwebi should teach other lectures his way of lecturing (Annexure 2B).*

Regarding my lecturing pace, one student in the same class wrote:

*A very organized, patient, respectful, wonderful lecturer.
You know he is really qualfied for this position and he don't need any more improvement.
If I was in charge of the Maths department I would raise his salary up
to the higher standard (Annexure 2B).*

A further indication of the good relationship between me and my class is given by the following comments:

*He was always approachable and if we had any problems he was
always available and willing to help (Annexure 2D).*

This view is not only limited to my undergraduate and honours students; but even my Ph.D. students conrm:

*Dr Mukwembi treated me very well during my PhD studies. Our relationship was cordial and
I could go for consultation to him anytime.
He was always happy to assist or give me advice on my project.
Sometimes I felt free to discuss with him my personal issues.
This enabling environment helped me to make good progress in my studies (Annexure 2E).*

To achieve my goal of creating a learning environment that is essential for all students, I always encourage questions, and pause in the lecture to answer them. In support of this, 88% of the 2011 MATH105 students felt that I encourage questions in lectures, and 91% thought that, in general, I have a good relationship with the class. To improve the out-of-class learning environment, I always, as indicated earlier on in this submission, hold 2 consultation hours per week for every course. For my Masters and Ph.D. students, I always set aside one afternoon for a meeting with each of them. In addition, I inform my students that they are welcome to come to my office for further consultation at other times for as long as I do not have other immediate commitments. In line with this, 97% of the 2011 MATH105 thought that I was available to oer guidance and support to students when needed. 97% and 91% of the 2011 MATH105 and 2010 MATH236 students, respectively, commented that I was approachable. One student pointed it out,

*What I find most useful was visiting Dr Mukwembi personal
in he's office for help in the course (Annexure 2C).*

Yet another way of achieving a fertile learning environment is to effectively use the available technology. On a daily basis, I post my lecture slides after the lectures on the course website, Moodle, to allow students access, through the WWW, to all course information and material (Annexure 2H). During lectures, this facility affords students the opportunity to concentrate only on understanding the lecture material, participate in class and reflect as they avoid the distractions which emanate from note-making. Further, I post on the website every useful information such as the course information sheet, past tests and exams, tutorials, appropriate solutions, results of tests etcetera. The course information sheet serves, among other things, as a record of the end products of the learning process, i.e., the outcomes. It also contains the syllabus, recommended books, tests information including dates and venues, assessment methods including how their duly performance mark will be calculated, lecture and tutorial time tables and my contact details. It is remarkable to note that 100% of the 2011 MATH105 class agreed that they found my method of posting learning material on the website helpful and very convenient. However, it is conceivable that posting more material on website may lead to poor lecture attendance since students would then just download the notes etcetera from the website and update themselves without necessarily attending lectures. Interestingly, my class attendance at lectures seems to indicate otherwise.

To achieve the goal of relating theory to practice, I have always worked with real world situations, pointing out links between the course material and other areas in the field centered at meeting the need for student development inline with globalization. One comment from the 2010 MATH236 reads as follows:

*Graph Theory, it is applicable in real life situations. Data encryption,
it has made me realize how data can be simply hidden (Annexure 2C).*

Some of my current students become highly motivated; they sometimes communicate with me using cryptography { a method, which I teach in MATH236, of sending confidential information. Over the years I have learnt that practical examples, if included in lectures, stimulate students to think in new ways. To justify, 84% of the 2011 MATH105 class, and 81% of the 2010 MATH236 class thought that as a result of studying my module, they learnt to think in new ways. The goal of relating theory to practice is not only limited to my undergraduate students. I also encourage it to my postgraduate students.

One major pitfall of the classical method of lecturing is that since the method is relatively easy to catalogue and recite, it is very tempting to grind on relentlessly and generate boredom rather than understanding. In my lectures I always face and watch my audience to make observations for feedback. I look at various indicators which in turn guide my lecture pace and actions. Some of the indicators are noise levels, type of questions asked and facial expressions of my audience members. These indicators allow me to gauge the students' level of interest, connection to the lecture and inform me to adjust accordingly. I use various methods to energize and remedy my lecture. I often allow a 10 minute break in case of double lectures. I sometimes make jokes that are related to the lecture content in order to stimulate, inspire and reconnect with students. More feedback is obtained in tutorial sessions, where with the help of tutors, I guide individual students in the process of solving problems. In tutorials our face-to-face interaction with each student allows us to identify areas of weaknesses and of strengths of each student. A corollary to all this is that I regard feedback as a niche aspect that informs my lecturing.

I consider my methods of teaching to be up to the task. To illustrate, the course MATH144 has become more and more popular amongst students for instance in 2006 it had 36 students whereas when I first taught it in 2007 the number of students increased to 41, and in 2008 it went up to 49. In 2012 it had 104 students. One of my colleagues commented that,

*In Operations Research I, student registration has increased significantly since
Dr Mukwembi took over the course, not because it is a \soft option", but because he
has the gift of presenting Mathematics so that even the weaker students can follow
his explanations and see the relevance of the material to real-life situations. (Annexure 4A).*

Turning to postgraduate supervision, I strongly believe that postgraduate supervision leads to the generation of new knowledge and stimulates research. Since my promotion in 2010, I have graduated 2 Ph.D. students, P.Y. Ali (with P.A. Dankel- mann, 2012) and J.M. Morgan (with H.C. Swart, 2013). Moreover, I supervised 2 Postdoctoral students, T. Vetrik and P.Y. Ali. Currently, I am supervising a total of 8 postgraduate students, S. Munyira, J.P. Mazorodze, V. Mukungunugwa and P. Mafuta (Ph.D., with A.G.R. Stewart, all registered at the University of Zimbabwe), S.W. Mgobhozi (Ph.D., with E. Chikodza), B.T. Fundikwa (MPhil, with A.G.R. Stewart), F.J. Adewusi (MSc, with P. Winter) and T.M. Shumba (MSc, with B.G. Rodrigues).

I believe that an important determinant for quality research supervision is that you need to be an effective researcher. My approach to postgraduate supervision is that I should be the director which involves determining the topic, facilitating, advising, guiding and teaching my student the research techniques among other things. Choosing an ideal topic for a research student involves not only selecting an interesting topic for the student, but also a topic that is manageable within the time constraints and a topic that contains scope for original work. Admittedly, it is rather hard to know in advance if a certain topic will have these three qualities. At this stage of my career, I now have the experience to ascertain this. This was confirmed by one of my graduated student who remarked,

*Dr Mukwembi is quite knowledge of research on \distances in graphs", the topic of my project.
The problems that he suggested for my project were nice and solvable.
The methods which we used for solving the problems were appreciated by both the
examiners of my PhD thesis and reviewers of the articles we
had submitted to journals. I honestly admire his knowledge of the subject. (Annexure 2E).*

This was further confirmed by one of the external examiners for J.M. Morgan's Ph.D. thesis, which was accepted by all the three examiners without any corrections, who wrote,

*Not only is the quality of this thesis first-rate, so too are the exposition,
organization and motivation. Not only should Mrs Morgan be commended
for her accomplishment but Dr Mukwembi and Professor Swart are to be
commended for their superior supervision of this work (Annexure 2I).*

As a result of my experience in supervision, I have learnt that problems faced by research students are enormous; they find the research process daunting, quite arduous, intimidating, intellectually very taxing and sometimes confusing. I always attempt to remedy this by first assuring the student on the onset that even the most well known scholars started o as learners with very little knowledge about the research process. Further, I also share my own research experiences, both bad and good, with the student. In support of this, one of my Ph.D. graduated student remarked,

*Since Dr Mukwembi loves his field his enthusiasm is contagious.
He loves research and challenge. Whenever he saw me laundering,
he'd make a concrete input & get me over the blockage (Annexure 2E).*

Real work begins with a discussion on the roles and responsibilities of supervisors and students as per our university's requirements. I then give my student published research articles to study. The aim of this exercise, apart from allowing the student to identify his or her area of interest, is to expose the student to various research techniques in the literature. This is normally followed by seminars in which the student reports on the articles read. At an appropriate stage, I assign a research question that is closely related to the surveyed articles to the student to work on and we meet regularly to deliberate, brain storm and thereby advising the student of possible methods that could be used to solve the problem. Once problems have been solved, the student then works on writing the work up concisely with my thorough guidance.

I always give my students feedback on submitted work promptly. Indeed one of my graduated Ph.D. student had this to say,

*His feedback was always in less than 24hours!
He also immediately acknowledges receipt of work & gave an expected response time.
The feedback was incredibly encouraging, all the time (Annexure 2E).*

Prompt return of students' work contributes to continuous and smooth ow of re- search work. Finally, but not least, I always ensure that my own research, and my students' research result in quality work. In support of this, one of the examiners for J.M. Morgan's commented,

*Several results obtained in this work have been published, accepted for publication or submitted
for publication in well-known, quality journals in the area of graph theory (Annexure 2I).*

Adding, for P.Y. Ali's thesis, Professor Rob Slotow wrote,

*You and your supervisors are to be congratulated on an excellent piece of work.
Two examiners passed the thesis as is, and the third examiner only some minor
typographical changes. This is an unsual occurrence and you are to be
commended on this. (Annexure 2G).*

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