I wanted certainty in the kind of way in which people want religious faith. I thought that certainty was more likely to be found in mathematics than elsewhere. But I discovered that many mathematical demonstrations, which my teachers expected me to accept, were full of fallacies.... Having constructed an elephant upon which the mathematical world could rest, I found the elephant tottering, and proceeded to construct a tortoise to keep the elephant from falling. But the tortoise was no more secure than the elephant, and after some twenty years of very arduous toil, I came to the conclusion that there was nothing more that I could do... (Russell 1956 pp.54-55)
For over two thousand years, mathematics has been dominated by the belief that it is a body of infallible and objective truth, far removed from the affairs and values of humanity (Ernest 1991 p.xi). This body of truth is seen as existing in its own right independently of whether anyone believes or even knows about it. Bloor (1973 p.43) argues that this belief in the independent existence of mathematical truth implies that mathematics is a realm, a bounded territory. Knowledge and the use of mathematics then requires two stages, access to the realm and then activity within it. The first stage is fallible. Hence discussion of the process of selection and education and the influences which promote or inhibit access to mathematical skills is possible. However what happens within mathematics itself is regarded as closed to discussion. This is seen as predetermined and certain. Therefore a mathematical calculation is the tracing out of what is already there, the calculation exists 'in advance'. It was this belief in the certainty of mathematics which allowed Kant to write:
We can say with confidence that certain pure a priori synthetical cognitions, pure mathematics and pure physics, are actual and given; for both contain propositions which are thoroughly recognised as absolutely certain...and yet as independent of experience. (Kant, 1783)
In more recent times there have been serious critiques of this belief in the certainty of mathematics, the belief that fundamentally mathematics exists apart from the human beings that do mathematics and that Pi is in the sky. However, as argued by Davis (1986 p.164), the reception given to opponents of this belief still ranges from coolness to indifference. We argue that this belief is not only deep in the psyche of mathematicians but also of learners and teachers and its influence still distorts mathematics education.
The certainty of mathematics has been under question. A growing number of mathematicians and philosophers are arguing that mathematics is fallible, changing and the product of human inventiveness (Ernest 1991). Others (Bloor 1973;Wittgenstein 1956) argue that rather than a calculation corresponding to an absolute truth , this truth is located in utility and the enduring character of social practice.
And of course there is such a thing as right and wrong...but what is the reality that 'right' accords with here? Presumably a convention, or a use, and perhaps our practical requirements (Wittgenstein 1956).
They argue that mathematics is not a body of truth existing outside human experience. It is a construct or an invention rather than a discovery, a collection of norms and hence social in nature.
Sociologists and mathematicians such as Ashley and Betebenner (1993) argue that philosophers have tried but failed to show how modern mathematics and science either pictured the world as it was or used a perfectly consistent, neutral meta-language. They suggest that mathematics did not develop in a cultural or social vacuum but rather that it reflects and magnifies cultural transformations. Hersh (1986 p.25) echoes Russell's regret at the loss of certainty but still argues against the attempt to root mathematics in some non-human reality and for the acceptance of the nature of mathematics as a certain kind of human mental activity. He suggests that the result would be a loss of some age-old hopes but a clearer understanding of what we are doing and why.
This attack on the certainty of mathematics led to the questioning of its neutrality. If mathematics is certain, if it reflects the God-like power of innate, transcendent human reason, if it is a body of absolute truth, and if the answers are already written, then it is independent. It must be neutral. However if mathematics is a social construct, an invention not a discovery, then it carries a social responsibility.
A proponent of this view, Joseph (1987 pp.22-23) suggests that the present structure of mathematics education is Eurocentric, being based upon four histographic pillars:-
Many writers (Joseph 1987; Anderson 1990; Bishop 1990) argue that the Eurocentric bias of mathematics infuses the subject with an elitist, racist and sexist bias. They argue that the belief in the certainty and neutrality of mathematics and science deprives these subjects of any cultural or social context. Hence mathematics and the natural sciences place no value upon the historical, cultural or political milieu within which they are located. Indeed mathematicians such as Pythagoras, Euclid, Cauchy-Riemann, Fourier, and Newton are cited as the source of western mathematics without any further reference to the times within which they lived or to the influences upon their work. They are abstracted from time and space and presented as if they and their work are timeless, complete and the absolute truth. This separation from culture and relevance makes mathematics inaccessible to those already alienated from society by educational disadvantage and by gender, race and class.
So we have outlined two incompatible views of mathematics. One is premised on certainty, neutrality, the peek into the mind of God. The other sees mathematics as a social construct and hence open to change, progress and development and as an unfinished project. These differing views lead to a fundamentally different approach to mathematics teaching and learning and hence different attitudes of students to learning mathematics.
The British education system still fails to provide a substantial proportion of the population with even basic mathematical skills (Cockcroft 1982). Even worse, it leaves many with an abiding dislike of the subject. We will now examine mathematics education in the context of the belief in the certainty and neutrality of mathematics for part of the explanation of this failure.
The concept of mathematics as a body of infallible and objective truth, whilst questioned by many mathematicians and philosophers, appears to be still widely held by society, teachers and students. An analysis of both the Cockcroft Report (1982) and a report by Her Majesty's Inspectorate which looked into the nature of mathematics teaching in Britain (1985), concludes that the mathematical approach taken in schools in Britain :
“....is that an absolutist view of mathematics is assumed.” (Ernest 1991 p.223)
This perception that mathematics is a certain and neutral subject clearly has a number of consequences for the teaching of the subject. Abstracted from any socio-political context, mathematics can be taught within the strictures of its own boundaries thus retaining for the pupils its mysticism and ritualistic nature. Certainly much work has been done since the introduction of mathematics into the mass education system to increase understanding of mathematics. However, just as certainly the history of mathematics is one of failure on a large scale for the students of mathematics. The Cockcroft Report notes that at the time the report was written 'about...one-third of the year group, leave school without any mathematical qualifications in 'O'level or 'CSE'(1982 p.56.)
It seems that despite calls for over one hundred years for an approach to mathematics that interests and stimulates children at school, mathematics is still a subject that confuses and alienates. A school inspector wrote in 1989 that she was horrified to find that at both primary and secondary level “nobody seemed to enjoy mathematics; not even the teachers” (Cross 1990 p.4)
Writers such as Rogers (1969), Dewey (1964) and Knowles (1980) argue that learners are self-directed beings who learn best when they perceive the relevance of knowledge to their lives, and when learning is related to problem solving. If mathematics is perceived as a fixed and unvarying body of truth independent of social concerns, then it is difficult to see any room for negotiation or where life experiences can be used in the learning process. If mathematics is neutral it has little to contribute to the learner's knowledge of themselves or their immediate world. All this contributes to a lack of motivation and hence a tendency to failure. As Thom writes:
In practice a mathematician's thought is never a formalised one...one accedes to absolute rigour only by eliminating meaning; absolute rigour is only possible in, and by, such destitution of meaning. But if one must choose between rigour and meaning, I shall unhesitatingly choose the latter. (Thom, 1973, pp202-203)
The Cockcroft Report (1982 p.71) suggested that there are three elements in mathematics teaching - facts and skills, conceptual structures, and general strategies and appreciation. The last is of interest to this paper. General strategies are defined as procedures which guide the choice of which skills to use and which knowledge to draw on. Crucially they enable a problem to be approached with confidence and with the expectation that a solution will be possible. With these strategies is associated an awareness of the nature of mathematics and attitudes towards it. An alternative approach to mathematics teaching can be developed by adopting an alternative view of the nature of mathematics. What follows are two examples of viewing mathematics, not as a certain, abstract, neutral discipline but as a human invention, a world of ideas created not by God but by human beings.
Anderson, horrified at the high failure rate at all levels of mathematics by non-whites in the United States (Commission on Professionals in Science and Technology 1986), developed a non-Eurocentric approach to mathematics teaching that by jettisoning the absolutist approach endeavours to ensure the relevance of mathematics and hence make it accessible to all (1990). Anderson has employed this approach with a full spectrum of college students, but principally Black and Latino adults (young and old). The early sessions are discussions on the historical, cultural, and socio-political implications of mathematics. At all stages an emphasis is placed on the role of other races and cultures in the development of the subject. The importance of mathematics to real people in real life is drawn out by regular class discussions of current issues in the social and natural sciences, the development of technology and job market skills. The emphasis is on the quality of mathematics knowledge rather than the quantity, thus reducing the time pressure. Anderson claims that this approach leads to students having a more positive, self-assured attitude about themselves successfully doing mathematics.
The Department of Adult and Continuing Education at the University of Exeter has completed a year long research project to see if mathematics acts as a barrier to access to higher education to adult returners. On the basis of data gathered from a national survey of Access courses (Benn and Burton 1993), it became clear that, as with Anderson's work, Access mathematics tutors were succeeding with groups who had low levels of general education and very low levels of earlier achievement and confidence in mathematics. Access courses are targeted at those groups traditionally under-represented in higher education namely women, ethnic minorities, unemployed and those from working class backgrounds. For these groups to succeed, their attitudes and approaches to mathematics have to be fundamentally changed. They need to see the subject not as one more absolute, unyielding barrier but as social construct, a tool pliable to their bidding. This is done by breaking down the concept of mathematics as a body of infallible and objective truth and giving ownership and control of the subject to the students. Access mathematics tutors reported that the main elements of their teaching is encouragement and understanding; that tutors need to be patient and remove the often difficult and disabling pressures of time and that Access mathematics needs to be taught in context and have a relevancy to real life and other parts of the course. They note that the involvement of students and tutors in free discussion and dialogue in a supportive atmosphere helps students develop confidence.
Initial contact with mathematics staff was seen as important, with clear and friendly pre-course counselling essential. If possible pre-course assistance and\or workshops should be available. Students should be given an honest indication of the work involved. It is reassuring for students if maths phobia and the reasons for it are discussed early in the course. The methodological approaches recommended are open access workshops, flexible learning tutorial packs, self-help groups and a modular approach with one-to-one support most frequently mentioned even in these financially constrained days.
There is an urgent need to build confidence by showing that it is acceptable to be wrong and by placing the emphasis on methods rather than answers; to develop a positive attitude to mathematics by encouraging students to take ownership of mathematics by messing around with, exploring and enjoying numbers. The survey showed that students were coming onto the Access courses very worried by the mathematics component but the techniques outlined gave them a sense of confidence and control.
This paper has discussed two incompatible views of mathematics, that of a body of infallible and objective truth rooted in the belief in the essential certainty and neutrality of the subject and that of mathematics as a social construct.
The danger of regarding mathematics as a God-given, absolute subject is that it may, and arguably has, lead to an absolutist pedagogy which ensures that mathematics remains a collection of rules and facts to be remembered, a subject that has a mystique which makes it accessible only to a chosen few. It remains a subject that seems to have very little relevance to life outside of the classroom, but where success or failure has implications for a persons self or moral worth (see Buxton 1981).
This pedagogical approach has had limited success when the whole body of students in Britain is considered. Its failure is even more marked with groups that consistently underachieve in our education system, groups such as ethnic minorities, the working class and girls or women. As has been illustrated earlier, practitioners in the field who are teaching these groups have developed alternative approaches. They set mathematics in a historical, cultural and socio-political environment and they ensure a more relevant syllabus set in the context of every-day life. They ensure mathematics is seen like other disciplines as a negotiated journey, a quest and a voyage of discovery.
The main result is an increase in student motivation with subsequent increase in success. This practice, though perhaps pragmatic rather than theoretical in origin, reflects the view of philosophers such as Wittgenstein that mathematics far from being a body of truth is in fact a collection of norms. Far from a peek into the mind of God, it is not even supported by a tortoise. And, most interestingly, this practice appears to work.
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