High School Material

Contemporary Mathematics

Text

Introduction to Contemporary Mathematics is the main text in a course for selected Years 11 and 12 college/high school students, which has been running since 2006. There are four topics, two each year:

The philosophy of the course is discussed in the Introduction to the above text.

Videos

Here are some videos I made for the course. The first set is a short introduction and motivation for the course. Sets 2–5 develop RSA cryptography from scratch, with essentially no prerequisites. They follow the indicated sections from Chapter 2 in the above textbook, omitting material that is not relevant to RSA.

  1. Introduction and Overview (6 videos, 21:58 minutes)
  2. Number Systems (Section 2.1) (one video, 5.27 minutes)
  3. Factorisations, Primes and Properties (Section 2.3) (30 videos, approx 120 minutes)
  4. Modular Arithmetic, Fermat’s Little Theorem (Section 2.4) (17 videos, approx 78 minutes)
  5. RSA Cryptography: Algorithm, Theorem, Examples (Section 2.5) (16 videos, approx 82 minutes)

The above lectures are those in an EdX mini-mooc style course, with various small questions along the way. (When you first click the link you may be prompted to sign up for an edge.edx.org account. This is necessary even if you already have an edx.org account. The platform is essentially the same in both cases).

Final Days

The last student intake for the ANU Extension Program was 2024 and the course will terminate at the end of 2025.2

When and if the environment improves, I believe it would be possible to have a similar program in mathematics and computer science, with an introduction to quantum computing and A.I. as topics. It would be a lot of work to develop such a program, but well worthwhile!

Workshops for Teachers

One spin-off from the Introduction to Contemporary Mathematics course was a series of one day workshops for mathematics teachers in the ACT system, introducing the teachers to a variety of applications of mathematics.

To obtain an idea of what was done, and what could be done in future workshops, here are the promotional flyers.

1962 High School Exams

Until 1966 there were 5 years of high school in New South Wales, the standard age of completion was 17 years, and the final set of (public) exams were known as the Leaving Certificate Examinations. It was not uncommon to complete the Leaving Certificate at the age of 16. Most students, however, left school at the Intermediate Certificate level, after 3 years of high school, and the standard age of completion was then 15.

From 1967, students completed an additional year of high school, so the standard age of completion was raised to 18. The new system was known as the Wyndham Scheme. Students typically studied a broader range of subjects than in pre-Wyndham years, but arguably the mathematics/physics/chemistry subjects were not studied in any greater depth despite the additional year of schooling.

Here is a History of Mathematics Examinations in New South Wales between 1788 and 2010!

Below are the 1962 exam papers for Mathematics I and II Pass Level (which essentially correspond to Mathematical Methods in the current Australian curriculum) and for Mathematics I and II Honours Level (which essentially correspond to Specialist Mathematics).

Note: It is normally the case that exams are much harder if you have not actually done the relevant course and lots of similar problems. So don’t be discouraged by looking at the following!

Mathematics I Pass Level 1962.
Mathematics II Pass Level 1962.
Mathematics I Honours Level 1962.
Mathematics II Honours Level 1962.

The Mathematics II Honours paper was the most difficult of the two honours level papers, and was similar in style and content to the Cambridge and Oxford entrance examinations for students specialising in mathematics. Here are solutions and comments for the 1962 paper.

Mathematics II Honours Level 1962, questions, solutions, comments.
(Note: Question 7 contains an error.)

  1. The official discoverers of RSA cryptography are three mathematicians Ron Rivest, Adi Shamir and Len Adleman, who published the method in 1977. However, another mathematician Clifford Cocks, working for the British security agency, actually discovered the full method in 1973. For bureaucratic security reasons it was classified and not published, even though the British thought it did not have any use! ↩

  2. Over breakfast in China in 2004, Ian Chubb (ANU vice chancellor/president at the time) and John Stanhope (Australian Capital Territory [ACT] chief minister) decided it would be appropriate and advantageous to the ACT community to allow gifted Year 11 and 12 students to do – and gain credit for – first year university mathematics, physics or chemistry courses. For appropriately capable students this was possible for physics and chemistry, particularly since high school studies in those two subjects were not prerequities for the university courses. For mathematics this was not the case.

    The alternative was to devise a program for students with demonstrated mathematical ability to expose such student to some aspects of twentieth century mathematics which did not require an extensive background. In a series of meetings with senior mathematics teachers from the majority, if not all, of the Year 11 and 12 mathematics programs in the ACT, the teachers and I (as MSI representative) agreed to a course which led to the text linked above. The course was taught by a series of selected ACT teachers, with whom I met on a regular basis as needed. The ACT Board of Secondary School Studies approved the course as one that could count to a student’s Higher School Certificate.

    However, “the review of ANU Extension identified that the program is costly to run and does not achieve the equity targets [my emphasis] that are set out in the University’s Strategic Plan” according to an ANU Managing Change proposal 19 March 2024.

    Personal Opinion: The ANU is indeed financially constrained for a variety of reasons. But regarding the “equity targets” – the program was always intended for suitably qualified students. To admit less qualified students would be demoralising for these students, and to lower the standards of the programs would defeat the purpose of providing an early bridging between high school and university for gifted students. The equity issues need to be addressed in early primary and high school programs. It is not fair to students to put them in programs for which they are not prepared, nor should should such programs be cancelled if they are too high a level for most students.** ↩