Teaching

Teaching Experience

Mathematics for Science students

For much of my teaching career, I have taught either 1st year units for Science and Economics students or more advanced 3rd and 4th year subunits for mathematics students. For example, between 1978-1981, I was in charge of M150, a low-level 1st-year mathematics unit which included students from areas as diverse as Architecture and Zoology, and which at one stage had the largest enrolment at UWA. It comprised M155 (elementary calculus) and the other half was MS155 (elementary statistics). As lecturer-in-charge of this multi-stream service unit, I had to persuade colleagues that the standard method of teaching the subject was inappropriate and that a more practical approach would attract more students. That is, my aim was to present not only the rudiments of the theory but also illustrations of its usefulness: thus, assignments contained routine problems as well as real-world examples, all of them paired so that students could complete one (with a tutor's guidance if necessary) and then reinforce their knowledge by attempting another similar to the first. In addition, since no suitable text was available for the calculus half of the unit, I prepared and distributed 100pp of typed notes for the benefit of students. Much later these attracted the interest of a publisher who invited me to write a mathematics text for biology students: unfortunately, more pressing commitments have prevented me from completing that work.

Anecdotal evidence suggests that my efforts in M150 early in my career were well-received: indeed, some students who entered the unit with little interest in the subject gained sufficient confidence during the year to enrol for Mathematics 260, a 2nd year unit intended mainly for students in the behavioural and biological sciences, and this became the fastest growing unit in mathematics in the late 1970s.

Click here to see some student comments on M150 in the 1970s

In 1st semester 1998, I returned to being the coordinator of M150/152/155 and a lecturer for M155 (150 students in one of 2 streams). For this, I prepared 2-pages of general information, and 2-pages with a lecture schedule, course objectives and recommended reading, as well as student guides for the mid-semester test and the final exam, and detailed guides for markers of such assessment. I also revised the lecture notes and the list of 160 problems on calculus (and its applications) that I used in the 1970s. However, the department was, by then, short-staffed and under-funded, so tutorials were replaced by Calmaeth, that is, computer-aided learning software developed by a colleague specifically for M155. This meant that I also had to cope with many bugs and crashes in the new sysytem and inform its developer of my recommended changes; and I needed to coordinate the marking of 'workbooks' submitted by students as partial assessment of their Calmaeth work. In addition, the course content had barely changed since the 1970s, but the students were weaker in the 1990s, so I argued for major changes to be made to the design and content of M155.

Mathematics for Engineering students

In 2nd semester 2008, I coordinated M1010, a 1st year unit primarily for engineering students (enrolment 540) and I taught both of its calculus streams. For this, I prepared a 5pp General Information handout, and revised the Lecture Schedule and Unit Outline used in 1st semester 2008. As pdf files, these and other documents were placed on my academic website separate from the School's official website for M1010. This was so the content would be more clearly delineated, and easier to access via one bookmark. In addition, I revised and retyped some calculus lecture notes used by other staff in 2001-2007. These were extended in a major way and placed on my website. In the end, they contained

109pp of lecture notes in 5 sections (and subsections) with 130 worked examples,
12 Applications and 24 Explorations: the former illustrated how ideas covered in a section may be applied; and the latter invited (better?) students to explore further ideas related to topics covered in a section (none of the Applications or Explorations were examinable),
38 Bewares: ie, warnings about misunderstandings, wrong ideas, bad algebra, etc,
7 Bad vs Good Approaches to the solution of certain problems, intended to contrast common mistakes by students with correct arguments required by markers,
a 23pp Review of Algebra available on the web as an Appendix to one of James Stewart's calculus texts,
a 3pp end-of-semester summary of the calculus covered in M1010,
a Trial Calculus Exam to indicate the possible content and standard of the calculus half of the final exam.

At the start of the notes, I included the details for some additional reading (suitable books in the library, texts freely available online) as well as ideas on how to study calculus and why it is relevant for different careers. The notes also contained a few historical comments, relevant quotes about mathematics, and simple exercises (similar to preceding worked examples) for students to attempt.

Following my usual custom (see above under Mathematics for Science students), I distributed weekly 'worksheets' on calculus in which problems were 'paired' so that students could complete one and then reinforce their knowledge by attempting another similar to the first. In addition, each worksheet contained at least one simple Application and/or Exploration stated more briefly than than those in my lecture notes (see above).

Click here to see some student comments on M1010 (Calculus) in 2nd semester, 2008

Finally, some students nominated me for 'lecturer of the year' at UWA and I sincerely thank them for their consideration. As a result, I was voted the top lecturer at UWA in 2008 and 19th best among 6420 nominations across all Australian universities. To see this, click on Lecturer of the year, and then on 2008 Results and look for my name halfway down the page, and then put your cursor over View top 10.

In 2nd semester 2010, I taught two streams of the Linear Algebra (LA) half of M1010 (enrolment 600). For this, I revised the Lecture Schedule and Unit Outline which had been used for LA in previous years. As pdf files, these and other documents were placed on my academic website separate from the School's official website for M1010. This was so the content would be more clearly delineated, and easier to access via one bookmark. In addition, I prepared 106pp of personal lecture notes which supplemented the so-called First-Year Linear Algebra Notes which had been used by other lecturers for over 10 years. In the end, my lecture notes contained

5 sections with over 100 worked examples (gaps were left in the working of some examples, to encourage students to attend lectures and to participate more actively in those lectures),
7 Applications and 6 Explorations,
30 Bewares: ie, warnings about possible misunderstandings, incorrect arguments, etc.

My website for LA in 2010 also contained

a 23pp Review of Algebra so students could revise basic ideas about school-level algebra,
a 12pp end-of-semester summary of the linear algebra covered in M1010,
solutions for all Tests on LA during each semester in the years 2008-2010: this was so students could see what they might expect in the current semester's Tests and prepare for them accordingly; also they provided further avenues for revision at the end of semester,
a Trial Linear Algebra Exam to indicate the possible content and standard of the LA half of the final exam.

At the start of my notes, I included references to some text-books (one with a solutions manual) which were available in the library. The notes also contained a few historical comments, relevant quotes about mathematics, and simple exercises (similar to preceding worked examples) for students to attempt.

Following my usual custom, I distributed weekly 'worksheets' on linear algebra in which problems were 'paired' so that students could complete one and then reinforce their knowledge by attempting another similar to the first. In addition, each worksheet contained at least one simple Application and/or Exploration stated more briefly than than those in my lecture notes (see above). Or, it included an historical anecdote with a related diagram which explained the diverse cultural background for some topics involving linear algebra.

Also, my notes and worksheets contained a few cartoons related to mathematics, to break the tedium or frustration which many students feel when studying this subject.

Click here to see some student comments on M1010 (LA) in 2nd semester, 2010

Mathematics for Economics students

Between 1986-1991, I was in charge of Mathematics 170, a compulsory 1st year unit on calculus and finite mathematics for Economics students which I presented with an ethos similar to that for M150: namely, with a modicum of theory aimed entirely at the solution of realistic problems in economics. This unit had a large enrolment of Asian students on the UWA campus and was an integral part of UWA's twinning programme with Sunway College in Malaysia. Again, written evidence suggests that such students and the college staff responded well to my efforts on their behalf, both as lecturer and as coordinator of the unit.

Projects for mathematics students

At more advanced levels, I have taught courses on different facets of modern algebra and discrete mathematics, both for students at UWA and for staff in South East Asia. In addition, at UWA between 1974-1983, I developed a "project form of assessment" for my 3rd year units that was well-received both by students and by colleagues at other Universities. In brief: students were invited to complete at most two projects (ie, open-ended problems or major essays) on some topic related to the course; and, to guide them in their choice, a list of over 200 projects was prepared, each with a short paragraph to introduce the topic plus a few references to prompt further investigation (all done before the advent of word processors and databases, and before enrolments dropped from 60 to 20).

When I returned to teaching 2P1 (Abstract Algebra) in 1996 after a gap of several years, I provided students with three assignments, each of which counted towards a student's final mark and included an open-ended problem to encourage independence and initiative among the students. That is, I introduced projects into formal assignment work, albeit in a very restricted way, and again students expressed their enthusiasm for being allowed to investigate mathematical ideas on their own.

For 2P1 in 1997, I also introduced an element of "journal writing" into the assignment work: ie, in addition to the open-ended problems used in the 1996 assignments, students were asked to write about their experience of the unit, what they had read in the text, and which topics they found especially difficult. Again, weaker students found that this was a good way of gaining confidence in abstract algebra, without being penalised for expressing their opinion of the subject.

Click here to see some student comments on projects and on 2P1

Writing course for honours students

I was the first person in the department to deliver a course aimed at preparing honours students to write their 50pp dissertation. It began in January 1992 when, in response to a request from the Head of Department, I provided written guidelines for mathematics students on how to write their honours dissertations. Subsequently, in 1993, this became one of the handouts in a "writing course" for the same students. After a considerable amount of background reading, this course was presented via 4 lectures, 4 tutorials and 4 assignments and included:

the 7pp "Guidelines for writing a mathematics dissertation",
a 3pp handout on plagiarism (an edited excerpt from a book by an eminent historian of mathematics): its purpose was to stress the importance of suitable acknowledgement in students' dissertations,
a 2pp excerpt from the Proc. Royal Soc. Edinburgh and the Math. Proc. Camb. Philo.s Soc. illustrating what editors expect by way of well-written mathematics,
4 handouts (18pp in all) covering the lecture content of the course,
carefully selected excerpts from past dissertations, collated into two and three page handouts (with a balance in each between pure and applied mathematics and mathematical statistics) on which students were expected to comment via a written report, as partial assessment for the course,
a 4pp handout comprising a radically altered version of a research paper, on which students had to write brief editorial comments aimed at improving the paper's overall exposition.

Here is a web site that includes information on writing and presenting mathematics.

Linear algebra

When I took over the linear algebra part of M200 (then called 2LA) in 1993, I instigated a major revision of the handouts and assignments so they were more informative and more relevant. These were further amended in 1994 so that by then students had

an information sheet and course outline, plus an annotated list of recommended books that would help them to understand certain features of linear algebra and to appreciate some of its diverse applications,
a weekly 'assignment' (so-called) consisting of problems in two sections, A and B: the first providing the absolute minimum amount of work to be done by students (ie, two problems: one theoretical in character but requiring no more than a few lines of argument, and the other being purely computational); and the second giving further practice in routine problems, developing additional low-level ideas in linear algebra, or outlining a realistic application of the theory. Solutions to Section A problems were provided via the usual avenues (ie, Closed Reserve in MPSL), whereas those for Section B problems were only available to tutors. The purpose of this arrangement was to encourage students to ask staff for help with the latter in tutorials and practice classes, thereby improving student participation in the course. Note also that each assignment sheet began with an indication of which material in the text students should be reading in that week.
a short-answer, non-computerised mid-semester test worth 10% of the final mark plus two non-formal assignments to be marked by tutors: the aim being to provide feedback to students on their progress and to minimise the level of copying that had been reported,
a 2pp handout at the end of semester providing excerpts from past linear algebra exams which indicated clearly which questions were relevant for the current content of the course and which was discussed briefly in lectures,
2pp handouts with instructions and advice regarding the mid-semester test and the June exam,
all administrative handouts were on coloured paper whereas problem sheets were on white, so students could clearly distinguish the two in their files.

Click here to see some student comments on M200 (Linear Algebra)

Matrix algebra

In 2nd semester 2003, I taught 2MA2, a second (but low-level) course on matrix algebra and its applications, intended primarily for 2nd year students in science. As required (and as is my custom), both the lecture notes and the 'Worksheets' (ie, weekly sheets of problems) contained numerous applications of matrix algebra, as well as routine practice in calculating various things (in total, 70pp of notes with 90 worked examples, and the worksheets contained mainly 'paired' problems on routine calculation; and everything, including complete solutions to odd-numbered problems, was accessible on the internet).

Since I was told that the students were weaker than average, and less interested in mathematics than students in other courses, I decided to assess 2MA2 in part by an (individual) project, 6pp long worth 10% of the final mark, the aim being to get students involved in the course by relating matrix algebra to their other academic interests (there were also two formal Assignments worth 5% each and one Short Test worth 10%). For this, I distributed 9pp of guidelines, topics and references for possible projects, and many of my suggestions were 'applied' in nature. However, like for the Assignments, most students copied their work verbatim from the internet or a book, or from other students.

2MA2 became M2214 in 2004, after the department changed the content to being half matrix algebra and half abstract algebra, all at an elementary level. It was still intended for 2nd year students with average-ability, but now there was no time to cover any applications in any detail. Consequently, when I taught M2214 in 2006, I reduced my 2MA2 notes to 60pp with 75 examples, and provided another 60pp and 40 examples for the abstract algebra half of the course (but added over 20 major applications throughout the written notes as motivation). Also, my assessment no longer included any project work: there were four formal Assignments worth 2.5% each and two Short Tests worth 10% each (and again half the class copied their assignment work).

Abstract algebra

I have taught various courses on abstract algebra, discrete mathematics and number theory at 2nd, 3rd and 4th year level, but rarely anything on semigroup theory (my research area). Below I comment on a few of these courses.

In 2nd semester, 2001, I taught the first half of 4P5 (Commutative Rings & Galois Theory) at a lower level than 10 years before, to accommodate changes in student ability (enrolment: 4 students). For example, in my half, students were assessed in part by two assignments, each worth 10% and each including one open-ended problem. Also, all my lectures were typed in TeX (almost 40pp in total) and distributed to the students before each lecture, so they could read them before the lecture and then concentrate on further explanation in the lecture itself.

Also, in 2nd semester, 2001, I taught the first half of 3P5 (Groups and Symmetry): ie, a basic introduction to group theory and its applications, delivered at the level of 2P1 in 1996-97 (enrolment: 18 students); with one test and one assignment, each worth 10% and the latter including one open-ended problem; all my lectures were typed in TeX (30pp in total) and distributed to the students before each lecture, so they could read them before the lecture and then concentrate on further explanation and examples in the lecture itself (note: in this course, I had to allow for a few students who had no background in algebra at all, whereas most of the class had already done a course on ring and number theory, although there was little evidence of that).

Business statistics

During the last three decades, I have occasionally been asked to teach low-level statistics to 1st year students in the physical and social sciences: eg, Math Stats 155 (MS155) in ???, and Economics and Business Statistics 106 (EBS106) in ???.

In 2nd semester 1997, I assisted with SM155 (Stats and Modelling 155) but, to a large degree, I coordinated the unit by preparing lecture notes, problems and solutions, and ensuring all such material was sent in due time to Sunway College, Kuala Lumpur, in the absence of any of this being done by the lecturer-in-charge. This involved me in editing statistics notes and problems from M170, and preparing additional notes and examples, all at a level suitable for SM155 (ie, lower than the M150 course I taught two decades before); selecting worked examples and relevant exercises in the text for students to study (after the students in my stream had asked for such assistance); typing all such material in TeX and putting it on the web, and then re-editing it so it could be used for transparencies in lectures; advising the lecturer-in-charge on how to conduct a mid-semester test; and ensuring (by email and fax) that the lecturer at Sunway College was fully informed on course content and assessment.

My work on MS155 began indirectly in ?? when I was the coordinator of M150, in which the statistics half, MS155, was being taught by a statistician (who later became a Professor of Statistics at a university in WA). I redesigned the course and rewrote the problems, so the ideas were more comprehensible and easier to follow for the students, many of whom began the course with an inherent dislike for statistics. As a result, some students asked if they could enrol in a 2nd year statistics course, but were disappointed when they found that I would not teach it. When I returned to statistics in 2005 by teaching one stream in EBS106, I closely followed the lead of the course coordinator regarding the level and content of lectures. However, I also stressed motivation and application of the different topics presented.

Mathematical methods

When I took over the mathematical methods part of M200 (then called 2MM) in 1993, I felt the problems used previously were poorly phrased and inadequate in number, and they lacked motivation. So for this course also, I undertook a major revision of the handouts so that students had

a course outline and a list of recommended reading,
a weekly 2pp sheet of problems (and their solutions) emphasising the computational and applicable features of the subject and indicating which problems were to be handed in,
an interactive computer program in the computer laboratory enabling students to visualise how a Fourier series approximates a given function and how the associated Gibbs phenomenon occurs,
a 2pp handout with instructions and advice regarding the November exam,
all administrative handouts were on coloured paper whereas problem sheets were on white, so students could clearly distinguish the two in their files.

History of mathematics

When I returned to teaching 3H1 (History of Mathematics) in 1994 after a gap of several years, I revised my handouts for the course so that students had

a 4pp handout providing information on the course, how it would be assessed overall, and how the essay in particular would be assessed,
a 16pp handout entitled "Essay Guidelines" comprising an annotated bibliography of the primary and secondary literature relevant to the course and the essay in particular: the aim of this was to help students to find information on the course itself as well as references for their essay,
2pp handouts of "literature searches" and "critical commentaries": students were assigned to prepare written responses to particular items contained therein,
a 3pp handout on plagiarism (an edited excerpt from a book by an eminent historian of mathematics): its purpose was to stress the importance of suitable acknowledgement in students' essays,
a "Specimen Exam", indicative of the 1994 June exam, was distributed to students at the end of semester,
a 29pp compendium of 35 possible essay topics, each with a brief description and pertinent references, was placed in the Closed Reserve of the Mathematics Library. This was bound with high-quality essays completed by past students to show current students what was expected. The folder also contained specimens of past "literature searches" and "critical commentaries" to help students in that area as well.

For more information on the history of mathematics, see Fred Rickey's Home Page

Technology in mathematics teaching

In March 1996, I was invited to be a beta-tester (the only one in Australia) for a Macintosh version of Scientific Word (SW) and Scientific Workplace (SWP). The first aims to imitate most features of Word and allow the user to prepare high-quality mathematical documents, large and small, without knowing anything about TeX; and the second includes SW as well as a simplified interface with Maple and Mathematica so the user need never know the language and commands of those computer algebra packages in order to apply them. The controller of the beta-list has commented that my "input on the beta-list has been the most consistently helpful of anyone's ..." and my "participation in the beta-list is most appreciated". SWP has the capacity to dramatically change the teaching and learning of mathematics as well as the preparation of students for careers in engineering and science. Its possibilities could include the following, some of which are already being explored in America and Britain.

The basic features of SW are as easy to learn as the corresponding ones for Word, and little extra effort is required to produce TeX-quality mathematics. This means students could submit all their assignment work in typed format, and thereby acquire the skill to write coherent documents for others to read. Much talk has been devoted to the literacy and communication skills of engineering and science graduates: SW provides a natural environment in which those skills could be fostered.
Lecturers could find it is better to prepare their notes in SW than to handwrite them and then reproduce them on a blackboard. The alternative will be to run through them displayed on an overhead projector (taken directly from a notebook computer or from previously printed transparencies) while students do nothing but listen, knowing that such notes will be accessible later in a computer laboratory.
So far, most efforts at promoting the use of Maple and Mathematica in lower-level course work have failed, mainly because students are reticent to learn the language and commands of those packages. Indeed, to pose a reasonable problem using them often means half the problem must be devoted to explaining how to access and use the package; and that detracts from the problem's main purpose: namely, the underlying mathematics. SWP bypasses such complications, so these packages become as easy to use as an everyday calculator. This could mean a radical change in the nature of the problems and projects given to students, inviting them to become more involved in the exploration of basic mathematical ideas.

A long-term project which I am planning aims to develop teaching and learning materials which use SW and SWP. In the end, I hope to see students writing more coherent mathematics and reporting their work in a more readable fashion; using the computer more frequently for word-processing and practical tasks; gaining more confidence and independence in their approach to mathematics; and improving their knowledge and appreciation of certain aspects of mathematics; and staff adopting more innovative and effective ways of teaching mathematics.

For more information on SW and SWP, see MacKichan Software