One of the questions every mathematics teacher is regularly asked is “When are we ever going to use these things?” This is probably the hardest question to answer in a satisfactory manner! We talk about the applications of mathematics in physics, engineering and computer science.
Since I deal with many IT students, my favourite answer is “Google! The Page Rank algorithm, which is used by the Google search engine, is a highly successful application of graph theory and matrices!” That catches the interest of a few. But what about the students who are not interested in pursuing one of these areas? How can we make mathematics more relevant to those who want careers in finance, economics and the social sciences?
Of course we can, many should be thinking. Experts in these fields know that mathematics and quantitative analysis are of fundamental importance. But are our students, especially those who study mathematics at post-secondary level, being given the opportunity to hear about these applications of mathematics?
Mathematics is a compulsory subject throughout primary and secondary education – it is considered, and rightly so, a core subject throughout these highly important formation years, and a pass at Ordinary level is fundamental in order to continue one’s studies at post-secondary and tertiary level.
After secondary school, students have to choose which combination of Advanced and Intermediate level subjects to pursue, keeping in mind the University degree they would like to read for. Of course these include Mathematics in two different forms, the classic Pure Mathematics and Applied Mathematics, offered both at Intermediate and Advanced level.
Pure Mathematics provides a basis of the main important areas in mathematics, including calculus, vector and matrix analysis, numerical methods and complex numbers. It is a necessary choice for students who wish to follow a career in Architecture, Engineering, IT, Accounting or Physics.
Applied Mathematics is essentially a course in Mechanics and is highly recommended, mainly at Intermediate level, for students interested in Engineering and Physics.
And this is where I feel there is a lacuna in Mathematics education at this level. The syllabi, both at Advanced and Intermediate level, in fact specify that this is a course in mechanics. But what about all the other forms of Applied Mathematics?
The qualification of Applied Mathematics with the description “(Mechanics)” rightly seems to imply that there are other forms of Applied Mathematics: Discrete Mathematics, highly relevant to Computer Science and even to Operations Research; Financial Mathematics, for anyone interested in Finance and Economics; Quantitative Analysis and Statistics, of primary importance to any Social Science such as Psychology, Sociology, Management. But at present there is still no provision for these forms of Applied Mathematics at this level.
If we look at the current British educational system, which is very often the main model which Malta follows, these extremely important topics have been provided for at this level for many years. The British Advanced (A) level and Advanced Supplementary (AS) level system works on the modular concept. Students can choose from a number of modules, including Pure Mathematics, Statistics, Mechanics and Decision Mathematics, and these serve as building blocks for the qualification required – three modules to achieve an AS level certificate, and six modules to complete an A-level certificate in the subject.
There are compulsory core Pure Mathematics modules, and then students are free to “mix and match” units which give them a flavour of the various forms of Applied Mathematics – of course, one can always choose six modules in Pure Mathematics, which leads to the traditional Pure Mathematics A-level.
Other options include A/AS-level Mathematics, which can include Statistics, Mechanics and Decision Mathematics modules, as well as A/AS-level Statistics. Statistics and Quantitative Analysis are central to many areas, including physical sciences, life sciences, social sciences and financial subjects and, in my experience, many students at tertiary level struggle with these subjects, particularly especially since they would often have stopped studying any form of numeracy at age 16.
Looking back at the local examinations, the provision of Statistics at Intermediate level (and possibly A-level) as part of the Matriculation certificate is surprisingly missing, particularly since Statistics and Operations Research are now available at Bachelor and Master of Science level at the University.
Decision Mathematics modules include areas of operations research and graph theory and its applications, and are of particular interest to computer science students, as well as to anyone interested in logistics and operations management.
Going back to the British system, there is another more recent AS-level option called the Use of Mathematics, specifically intended for “non-mathematicians”, and the emphasis here is on modelling and communicating real-life situations using mathematics.
Perhaps similar options at Intermediate Level will lead to more students taking up some form of Mathematics at post-secondary level, facilitating their tertiary level studies and making mathematics more relevant in today’s competitive world.
There has been much debate on post-secondary Mathematics education in the UK recently. Studies have shown that more students have been taking A-level Mathematics in some form. Yet studies commissioned by the Education Ministry, among others, have reported that, compared to other countries, the proportion of students studying mathematics beyond the age of 16 is much lower than that in other OECD countries.
I quote from a report published by ACME (Advisory Committee on Mathematics Education “A major problem in England, together with Wales and Northern Ireland, is the two-year gap for most young people between the end of GCSE and the start of university or employment, during which the large majority do no mathematics. This is the most striking and obvious difference between the mathematics provision here and that in other comparable countries. It is not just a case of students missing out on two years of learning mathematics, serious though that is, but of their arriving at the next stage of their lives having forgotten much of what they did know.”
This is comparable to the situation in Malta, where the education system is quite similar, and many students who do not feel competent enough in Mathematics do not choose any form of Mathematics as part of their Matriculation certificate.
According to the Matriculation Certificate Report for 2011, 25 per cent of all students chose Pure/Applied Mathematics at A-level, and 24 per cent chose Pure/Applied Mathematics at Intermediate level. It would be interesting to carry out a study which would indicate which University courses these students pursued, as well as reviewing the course choices by students who did not choose any form of mathematics as part of the Matriculation certificate, in order to establish whether they would have benefited had they chosen a numerical subject.
In the education sector, a lot of emphasis is currently being given to the National Curriculum Framework, which covers the compulsory education years. An entire document is dedicated to science education, entitled “A Vision for Science Education in Malta”.
The main targets of the reform in this area include an increase in the number of students who take up science courses at post-secondary and tertiary level, as well as an improvement in Malta’s performance in international education surveys in the science area, specifically mentioning the TIMSS (Trends in International Mathematics and Science Study) and PISA (Programme for International Student Assessment) urveys in which Malta has recently started participating.
In the PISA 2009 results, which measure mathematical literacy for 15-year-olds in 74 countries, Malta classified in the 40th position, with an average score below the OECD average, and below that of most EU countries.
This does not bode very well for science education in general, because it is widely accepted that strong mathematical literacy is essential for studying science at post-secondary and tertiary level.
For these reforms to bear the desired fruit, I believe one must also view the wider picture of post-secondary education, particularly, the years between the end of compulsory education and the start of a university degree, and have a more cohesive and seamless system up to the age of 18/19. And it should be a system which takes into consideration the teaching of mathematics within the wider context of the teaching of science, something which “A Vision for Science Education in Malta” fails to do.
I hope this article will contribute to a debate about mathematics education in the post-secondary phase. Mathematics in its pure form is beautiful to some, a night-mare to others. It is undeniable that, in today’s world, numerical literacy of some form is very important – perhaps opening up access to new professions, without ever killing the classical ones which are already available.
For many courses at tertiary level it is today becoming more important for students to attain a higher level of numerical literacy before they move on to become part of the labour force. With the government’s vision of making Malta a centre of excellence in IT and Finance, it is crucial that our future graduates have a deep understanding of mathematics, and the capabilities to apply it to these and similar areas.
Ms Zarb lectures in Mathematics and Statistics at St Martins Institute of IT.