Influences on mathematical ideas and on the learning of mathematics

At the first level, there are clearly influences regarding mathematical concepts and skills. Before children come to school, their parents will often have taught them to count, to begin to measure, to talk about shapes, time, directions etc. Neither will this kind of parent's talk and activity cease when the children begin school. Within the immediate adult community particular knowledge, often related to mathematics, will be shared, as Nunes documents in the next chapter. The newspapers and other media reports involving money, charts, tables, percentages will all inform and educate the learner, complementing and supplementing the information and ideas being learnt in school. Now, with sophisticated calculators and home computers becoming more widely available, the young mathematical learner may well be racing ahead of the formal mathematics curriculum and will often outstrip the teacher's knowledge in some specific domains. Equally the sources for mathematical intuition are frequently images from society, rather than from within school. Popular images and

beliefs, for example, about statistical and probabilistic ideas seem not only to come from society but also seem to be relatively impervious to formal educational influences. (Tversky and Kahnemann, 1982; Fischbein, 1987).

Geometrical images will come from physical aspects of the environment, although what is important is how the individual interacts with that environment. Thus, once psychologists thought that living in a 'carpentered environment' (with straight sided houses, rectangular shapes, right angles etc) was very important for learning geometry and for developing spatial ability. Now we know that what is important is how individuals interact with their environment. So for example, to an urbanised European the desert of central Australia may seem to be devoid of anything which could aid spatial development, but the spatial ability of the Aborigines who live and work there is known to be exceptional because of what they have to do to survive there (Lewis, 1976). This is equally the case with Polynesian navigators (Lewis, 1972) and Kalahari nomads (Lea, FME, 1990). interactions with the physical environment of a society undoubtedly give rise to many geometrical images and intuitions. The other major source of societal influence on the learner's knowledge of mathematical ideas is the language used. The relationships between language and mathematics are of course extremely complex, and there is no space here to cover all the ground which has in any case already been analysed by others (Pimm, 1987; Zepp, 1989; Durkin and Shire, 1991). From the perspective of this chapter the two most important aspects for mathematics educators to be aware of seem to be:

- the fact that mathematics is not language-free,

- not all languages are capable of expressing the mathematical concepts of MT culture.

The first point may seem obvious, but it has profound implications. Mathematical knowledge, as it is developed in any society relates to the language of communication in that society. As has already been shown, the mathematics curriculum in many countries has been based largely on the Western-European model and it has a certain cultural, and therefore, linguistic basis. Though this basis is an amalgam of different

languages, the principal linguistic root is believed to be Indo-European (Leach, 1973). That particular 'shorthand' omits the important Greek and Arabic connections in the development of universally applicable mathematics, and we should perhaps consider the Indian-Greek-Arabic-Latin chain as being its original language base. From this base, Italian, Spanish, French, German, and English developed its language repertoire during the 17th, 18th and 19th centuries, and it is probably the case that nowadays English is the principal medium for international mathematical research developments. This is an important problem for any country, researcher, or student, for whom English is not the first or preferred language. In relation to the second point, in many countries of the world there are several languages used, but for national and political reasons, one (or some) are specifically chosen as the national language(s). It is likely that not all the languages being used in a society will necessarily be capable of expressing the concepts and structures of the MT culture - this will largely depend on the roots of the language. European-based languages, those with Indian roots, and the Arabic family of languages appear to have the least structural differences, although there are always particular vocabulary gaps as international mathematics develops.

Other languages, in rural Africa , Australasia, and those used amongst indigenous American peoples, are being studied and demonstrate their difficulties in expressing both the structures and the vocabulary of the MT culture’s version of mathematics (see, for example, Zepp, 1989 and Harris, 1980). This is not of course to say that these languages are incapable of expressing any mathematical ideas - they will certainly be capable of expressing the mathematical ideas which their cultures have devised. This is the important linguistic

relationship with ethno mathematics - and another reason for seeking to create more independent societally-based mathematics curricula rather than relying on the model from MT-based societies. Thus in this way the societal language(s) can reinforce the societal mathematics which can offer the bases for alternative curricula.

But language issues are extremely complex, particularly from a societal perspective, and the political and social conflicts which different language use can cause, can seem to be of a different order from those which should concern mathematics educators. Nevertheless so much damage has been done to cultural and social structures in many countries by assuming the universal validity of MT-based mathematics that we cannot ignore the language aspects of this cultural imperialism (see Bishop, 1990). If countries, and societies within countries, are to engage in the process of cultural reconstruction then the language element in relation to informal, non-formal, and formal mathematics education is critical. A final point concerning the informal and non-formal influences on mathematical ideas is that they have a cumulative effect. They build up into an image of 'mathematics' as a subject itself. For example, we have already noted that it is projected as being an important and prestigious subject in both industrial and developing societies and is thereby projected as being essentially a benign subject. Little mention is publicly made of its extensive association, through fundamental research, with the armaments industry, with espionage and code breaking, and with economic and industrial modelling of a politically-partial nature. Little public debate occurs about the questionable desirability of fostering yet more mathematical research to make our societies yet more dependent on even more complex mathematically-based technology (see however, Davis, 1989 and Hoyrup, 1989). The reaction of 'the media' to the examination question about costs of armaments shows the extent of public ignorance of these

matters.

 Equally mathematics in society is typified, and imagined by most people, as the most secure, factual and deterministic subject. There is little public awareness of the disputes, the power struggles, or the social arenas in which mathematical ideas are debated and constructed. Descartes' dream still rules the general societal image of mathematics. For example, in the study by Bliss et al. (1989) concerning children's beliefs about "what is really true" in science, religion, history and mathematics, the majority of children in England, Spain and Greece considered both mathematics and science to be truer than history or religion. As Howson and Wilson (1986) put it "Only in mathematics is there verifiable certainty. tell a primary child that World War 2 lasted for ten years, and he will believe it; tell him that two fours are ten, and there will be an argument" (p.12). At a second level the informal and non formal societal influences concern the learning of mathematics. We have already noted the beliefs about its difficulty and its motivations, but there are also more fundamental and significant beliefs about how mathematics is learnt.

Paralleling the popular image of mathematics as secure factual knowledge is the widespread belief that mathematical procedures need to be practiced assiduously and over-learnt so that they become routine, and that this should go hand-in-hand with the memorizing of the various conceptual ideas and their representations. Another popular belief concerns 'understanding' as being an all-o r-nothing experience, rather than a gradual increasing of meaning and constructed connections. Overall the popular image is of a received, objective, form of knowledge,  rather than of a constructed subject, running contrary to what we know from recent cognitive research. (Lave, 1988; Schoenfeld, 1987). There is also, as has been mentioned, the impression that because it is reputed to be a difficult subject, only some learners will be able

to make progress with it. Thus, rather like artistic or musical ability, young people perceive themselves as either having mathematical ability or not. The research on mathematical giftedness does in fact demonstrate that it clearly is a precocious talent, appearing early rather like musical giftedness, but one suspects that the image from society about mathematical ability is rather more broad in its reference than just to giftedness. 

The concept of 'ability' does seem to be a pervasive one in society outweighing 'environment' as the cause of achievement, particularly in the 'clear-cut' subject of mathematics. The accompanying belief is that one is fortunate to be born with this ability since it is (clearly!) innate. The pride of a parent on discovering that their child is a gifted mathematician is probably only clouded by the popular image from society of mathematical geniuses being slightly odd characters living in a remote and esoteric inner world and unable to socialise with

other people - a theme to which several contributors to the ICMI conference on 'Popularization' referred. Equally parents, although not necessarily rating their son's or daughter's abilities any differently, do apparently believe that mathematics is harder for girls and requires more effort to succeed in (Fox, Brody and Tobin, 1980). This belief undoubtedly helps to shape the feeling, prevalent in different societies, that mathematics is not such an important subject for girls to study. Fortunately many womens' groups are now hard at work dispelling this image, and IOWME (the International Organization of Women and Mathematics Education) has been particularly active. Recent research (Hanna, 1989) for example shows that the creation of a more positive and favourable image for girls who are mathematically able is apparently having some beneficial effects. These then are some of the most significant influences which society exerts on the learners of mathematics. The 'messages' the learners receive are many, and often conflict. Some are intentionally influential, while others are merely accidental. But they all help to shape important images and intuitions in the young learners' minds, which then act as the personal cognitive and affective 'filter' for subsequently experienced ideas. Let us then move on to consider how the learners make sense of, or cope with, these various societal influences which they experience. Are there any research ideas which can help us to understand and interpret the learners' situation?