I wrote this a few years ago. I’m not sure if I posted here before but it’s topical again, so here goes.

Finally it is being recognised that we have *two* Leaving Cert Maths courses; Mathematics (Project Maths that is) and Applied Mathematics. The latter is not really applied mathematics at all but very rudimentary mathematical physics. It is mainly devoted to mechanics and contains the sort of material one would meet in a physics course in the first year of university. Technically it is actually ‘theoretical physics’ but such a term is probably a bit grand given that most people associate the term ‘theoretical physics’ with the really fundamental physics of the world of elementary particles. The continued existence of the Applied Mathematics course in its current form on the Leaving Cert is a little bit of a mystery to me.

The fundamental philosophy behind the Project Maths concept was that by turning mathematics into a form of applied mathematics, it would become more attractive to students. Furthermore, by teaching maths in a more active way, understanding would be enhanced. But true applied mathematics is an advanced subject. Naturally enough, it involves applying mathematics to all sorts of physical, chemical, biological, social and financial systems. To become a good applied mathematician you need to acquire excellent mathematical technique and develop an ability to formulate real world problems as mathematical problems. This is a very advanced skill that takes many years to develop, mainly through constant practice. After a while, you begin to see real world problems in mathematical terms and this is a skill that is beyond the capabilities of most science and engineering graduates, never mind school-leavers. It is completely unrealistic to expect secondary school students to be able to look at a previously-unseen system, write equations to represent that system and have the technique to solve those equations. Mathematical modelling, for that is what we are talking about, is too hard. We need to dampen our expectations considerably.

In addition, the ‘old’ maths course was much too wide-ranging and was really a little bit of a mish-mash of the very abstract and the very practical. The sheer breadth of the material made it inevitable that students would learn by rote.

So, what I would have done is discontinue the two subjects, Mathematics and Applied Mathematics and replace them with Pure Mathematics and Applicable Mathematics. The term, ‘applicable’, is used increasingly to describe the content of a course or book or journal in which the core subject matter is routinely used, or has the immediate potential to be used, in real world applications. The purpose is not so much to concern the student or reader with the actual process of applying the material, but in developing the techniques that will ultimately provide the basis for such applied work.

I would leave the syllabus of the Pure Mathematics course up to the mathematicians but, as an engineer, it seems to me that pure mathematics is characterised by an emphasis on *proof*. Pick up an engineering book and you will see lots of equations but you will rarely see the word ‘theorem’ or ‘proof’ and even rarer still, the word ‘lemma’. Pure Mathematics is made up almost entirely of theorems and lemmas. Unfortunately, Pure Mathematics is uniquely singled out for criticism as being of no practical value and on that basis it is suggested that it is unattractive for young people and generally a waste of time. Subjects like classics or religious education or art or any of numerous minor languages, or even aspects of the English syllabus, avoid such scrutiny and criticism. It seems that someone can always construct a valid reason for studying any subject but pure mathematics. But it would be very sad indeed if the there was no subject on the Leaving Cert list that offered students a glimpse into one of the greatest achievements of mankind, namely the creation (discovery?) of the highly complex, interconnected and relentlessly logical world of mathematics. While I believe that mathematics of any form is a minority taste, it does not mean that it will not be studied at Leaving Cert by significant numbers.

For me, the Applicable Mathematics course would be dominated by the sort of mathematics that is studied and used by engineers, physicists and computer scientists. The emphasis in Applicable Mathematics would be on learning the mathematics that is applied routinely and there would be very little consideration given to the proof concept. Subjects like calculus, differential equations, linear algebra (matrices), probability and statistics and basic numerical methods could be covered. No doubt there would be some overlap with the content of the Pure Mathematics course but that would not be a problem if there was a sufficiently different approach used in each subject.

It goes without saying that, as with all education, both of these mathematics courses should be taught in a way that is as interesting as possible and that promotes problem solving and actual thinking rather than rote learning. But that could be said about every single subject on the Leaving Cert list. Uniquely, however, mathematics has been singled out for the ‘Project’ approach.

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I was asked by The Irish Times to sit paper II of the project maths syllabus this summer. I didn’t like it much, the paper was almost entirely devoted to practical examples of the use of probability and statistics. Most of the challenge was understanding the way the questions were put, and there was very little use of basic mathematical tools such as matrices and vectors on either paper.

It sounds to me that what you’re calling ‘applicable maths’ is really mathematical methods. It would certainly make for a very nice and useful course. I’m not sure you would need a ‘pure’ maths paper, could leave that for college!

Mathematical Method would probably be a better name alright. I think thought that some ‘pure maths’ would still be worthwhile. I seem to remember doing some pure maths (in 1980!) that could be classified as Number Theory and it was fairly accessible – and interesting. Other things like vectors, sets, calculus proofs etc might have an audience at second level. Maybe this kind of thing is too hard though.