Purposes of Mathematic Education Glenn Mason-Riseborough (7/4/1997) There are many purposes of mathematics education at secondary school level. These purposes may be best broken down into two main subgroups -- an individual’s purposes and society’s purposes. An individual’s reason for studying mathematics is likely to be entirely selfish. The person may study mathematics simply because they find it enjoyable. Alternatively, a student may study mathematics because in many cases a certain level of mathematics qualification is needed in order to get into tertiary courses or vocations. Related to this, a student may realise that he/she needs certain skills or a knowledge base that is best obtained through studying mathematics. From a teaching perspective, mathematics education provides jobs for educators and mathematicians. From the perspective of society, a knowledge of mathematics can increase productivity in the workforce. This leads to greater technological and scientific advances, and this (arguably) leads to a better society as a whole. Due to space constraints in this essay, I will cover only two of the purposes stated above. The first purpose relates to promoting enjoyment of mathematics in individuals. The second relates to increasing mathematical education in order for society to benefit as a whole. What can a teacher do to promote enjoyment of mathematics in the classroom? Perhaps if we were to ask people why they ‘like maths’ or ‘hate maths’, we would get as many different responses as people surveyed. The enjoyment of mathematics seems to be an extremely subjective experience. We all respond differently when we are confronted with learning mathematics, and yet mathematics in itself is a completely objective, culture- free, emotion-free, time-invariant world (see Penrose (1989) pp 426-429 for an argument on mathematics as a platonic reality). Therefore, it seems that it is the act of realising mathematics in our world that creates the subjectivity. It is only when mathematics is communicated (to oneself through cognition or others) that it is put into this cultural, subjective context. It is only at this point that mathematics seems to fall into a ‘like maths/hate maths’ dichotomy. From this, we can see that when we talk about ‘mathematics education’, it is the education, not the mathematics that creates the subjectivity. Often it seems to be the case that the amount a student enjoys a subject is directly proportional to the amount the student likes his/her teacher in that subject. Students and teacher form a relationship in the classroom, and this relationship can bias students’ attitudes to the subject. If the teacher responds positively to the students, then the students respond positively to the subject. This can be seen from a behavioural perspective. The student becomes classically conditioned via second-order conditioning, into a specific way of responding to mathematics (see for example Sternberg (1994) pp 237-249 for a discussion of classical conditioning). What can a teacher do about this? The obvious answer would be to get the students to like them (the teacher), and enjoy coming to the classroom. This is often easier said than done. The specific way in which the teacher does this is going to vary between cultures, and individuals within a culture. In New Zealand (and other western countries) there seems to be a trend towards a more ‘user-friendly’ teacher attitude. In the past the teacher ruled the classroom with an iron hand. The slightest deviation by the students from what was expected was punished by caning. Today the teacher is seen as a ‘facilitator.' Most teachers try to be approachable, and questioning by the students is actively encouraged. A link can also be made between enjoyment and a student’s level of understanding. If the student does not understand what is being taught, then it is unlikely that s/he will like the subject. Also, a distinction should be made between instrumental understanding and relational understanding. ‘Instrumental behaviour is behaviour that is maintained by the environmental consequences that it produces’ (Jones, 1996). When this definition is applied to understanding, it can be seen that instrumental understanding contains very little or no cognitive processing -- the student is simply following ‘rules without reasons’ (Skemp, 1976). Conversely, relational processing involves ‘knowing both what to do and why’ (Skemp, 1976) (cognitive processing involved). The reason for this distinction is that I am suggesting that enjoyment comes from relational but not instrumental understanding. I am making this suggestion for two reasons. Firstly from a purely subjective, personal perspective. In my studies in the past I have had to prove certain mathematical problems. Sitting down and working through these formally requires relational understanding. I must know what I am doing and why. On the other hand, writing out a proof in an exam after previously memorising it is instrumental understanding -- no reasoning is required. For me the former situation is enjoyable, while the latter is not. My second reason is based on how computers work. Computers work algorithmically, they follow rules dictated by software and hardware. Computers could therefore be said to have instrumental but not relational understanding. Since computers have no concept of enjoyment, enjoyment is not associated with instrumental understanding. My own reason for undertaking mathematics at secondary school was primarily because I enjoyed solving problems (relational understanding). My secondary school education was at a large, middle class, primarily white (and Asian) school. I was in the top streamed class for three of five years (there was no streaming for the other two years). Looking back, in the streamed classes I think we spent a larger than normal proportion of our time in mathematics doing problem solving activities. Our teachers allowed us time to work on these problems and solve them for ourselves. The other two years are something of a blur to me -- I did not learn much and I was extremely bored. Thus for me, during the streamed years the purpose (enjoyment) was achieved for me, during the other years it was not. For New Zealand as a whole, this purpose would be working in some classrooms and schools and not in others. Enjoyment is very much dependant on the style of teaching and the personality of the individuals involved. What increased mathematical knowledge will benefit society? The answer to this question clearly depends on the society one is talking about. Each society has its own unique needs and requirements and what is important to one society may be superfluous to another. For example Isaac Newton developed (amongst other mathematical techniques) differential calculus in response to a perceived need to calculate the behaviour of moving objects. Up until this point (in recorded history) there had been no need for this branch of mathematics. Indeed, in some societies even today this knowledge is not deemed necessary. However, to any spacefaring society (which Western society is increasingly becoming), this knowledge is integral to space travel. Having said above, that we cannot generalise for all societies the specific mathematical skills needed, we can still see commonalities in all societies. A universal need in any society is the need for its members to communicate. ‘Mathematics provides a means of communication which is powerful, concise and unambiguous’ (Cockcroft). This does not mean that people should converse in mathematical symbols and equations. On the contrary, people should continue to talk in natural languages, but be aware of the inherent ambiguity involved. Learning mathematics in general increases this awareness and thus benefits society, although the specific knowledge taught in different societies may differ. New Zealand is seen as a bicultural society. What this means is that the Maori society perceive different objectives (this essay has chosen to distinguish between purposes and objectives as defined by Niss) for themselves than other New Zealanders. This view has led to the introduction of two curricula -- one for Maori the other for non-Maori. Each curriculum addresses the required objectives as seen by their authors. Both sets of authors want to see New Zealand and New Zealanders benefit, they simply see different ways of achieving this. When teachers enter their classrooms, they probably have many purposes in the back of their minds that they want to emphasise. These purposes and the context in which they are placed are more than likely going to influence the style and content of the lesson. This essay looked at the problem a teacher faces promoting enjoyment of mathematics. Bottom line, the teacher has to be likeable and the content (relationally) understandable. How this is achieved will depend on the context the teacher finds him/herself in. The other purpose discussed was that of increasing mathematical knowledge to benefit society. Communication is a major part of any society and increasing communication skills is going to benefit all concerned. Societies in general, however, have unique needs and the specific objectives will differ between societies. These purposes mentioned are by no means exhaustive, and the classifications of purposes stated at the beginning of this essay is probably completely arbitrary. All these purposes are intricately connected and the distinction between them becomes extremely grey. For example, the purpose of benefiting society may be a superset of many other purposes such as promoting enjoyment (creating a happier society), increasing communication skills, developing skills for economic or technological growth, or developing general mental capabilities in individuals for everyday use. References: Cockcroft, W. H. Mathematics Counts. Report of the Committee of Enquiry into the Teaching of Mathematics in Schools. London: Her Majesty’s Stationery Office. Jones, B. (1996). The Psychology of Learning. Unpublished Lecture Notes for Paper 461.109, University of Auckland. Niss, M. Goals as a Reflection of the Needs of Society. In R. Morris (ed.). Studies in Mathematical Education, Vol. 2. Paris: UNESCO. Penrose, R. (1989). The Emperor’s New Mind. Oxford University Press. Skemp, R. R. (1976). Relational Understanding and Instrumental Understanding. In Mathematics Teaching. Sternberg, R. J. (1994). In Search of the Human Mind. Harcourt Brace.