apakah guru matematika perlu menggunakan teori pembelajaran?

Do Teacher of Mathematics Need Theories?

Educational issue are rarely clear cut. An individual teacher may hold very firm views on a particular issue in mathematical education, but must at the same time accept that very different, even completely contrary, views may be held by a collegue in the same school. Examples are not hard to find. In recent years the availability of pocket calculators has sparked off discussion and controversy about how and when calculators should be used. If young children are allowed to use them will they ever learn their multiplication tables? Could sensible use of calculators enhance understanding? A  variety of different kinds of structural apparatus exists for helping children  to acquire the concepts of elementary number. Is such apparatus essential?  Wich is the best? Some teachers believe that mathematics should be a silent activity with each of the children always producing their own work, but other teachers value  discussion between pupils. Is discussion important for all or do some pupils  opt cut and so learn nothing? Deciding on appropriate mathematics for older, low-attaining pupils has always been a problem : is social arithmetic the best answer or is it seen as irrelevant by the pupils? The debate about the place of calculus has continued throughout much of this century. Is to there a place for calculus  before the sixth form or is it conceptually too difficult for all but a very few? These are only a small selection of the many issues which whould lead to debate and disagrecement.

                In accepting a particular viewpoint, or in taking sides on a particular issue, it could be said that a teacher  has accepted a theoretical position. Throughout any day in school we are adopting particular ploys and using particular methods because we believe they work. Such limited theories are based on experience, intuition and perhaps even on wishful thinking. They may be helpful, they may, on the other band be dangerous. For Example, is it dangerous introduce division of fractions in the primary school? It  migh be if, in not understanding, children become frustrated  and anxious and come to reject mathematics as a meaningful and worthwhile activity. It appears that the job of teaching cannot be done without accepting theoretical views, though such theories, it whould undoubtedly be claimed , should be based finaly on empirical evidence. In this sense it appears that we do need theories as a basis for decision-making in the classroom.

                However, although  teachers do need  to adopt and practise theories in their daily work it is not unusual to find many who are sceptical or even disparaging about the value of large-scale  theories. Major theories which might enlighten the teaching learning process are dismissed as irrelevant. It is possible that such theories are rejected without being given serious consideration. For example, it might not be appreciated by those of us who use and value structural and other apparatus that the invention of the equipment might have been prompted by acceptance  of a particular learning  theory. One of the earlier kits was devised by Stern (1953) because here believe in Gestalt theory demanded that such apparatus was available for children. Of course, it is possible a radically different teaching style.

                A theory should be based on observation of children’s behaviour in learning situation. Subsequently, the theory shoul enable us to explain what we see in school and also to take appropriate action. In this rense our theory explains, and could ever predict, phenomena. Hopefully, with sufhcient data on which to construct hypotheses , our theory might present a systematic view  of phenomena whilst at the same time remaining relatively simple to graps. The large-scale, general theories which are  sometimes rejected by teacher have usually been based on a systematic view extrapolated from a much wider range of even and situations than any one individual can have experienced and contemplated. The view which underlies this book is that education is too important for us to be able to dismiss as irrelevant theories of learning which attempt to do what has just been described. Child (1986) explained it by saying , “ inovation and soeculation in learning... are more likely to succeed when they are informed by sound theoretical frameworks”.

                 One major problem is that there can appear to be a large number of conflicting  or contradictory general theories in existence. Historically, two major kinds of theory have been developed, referred to here  as ‘ behaviourist’ and ‘cognitive’ these two certainly do confict, though recent work has attempted to reconcile some aspect. Within these two vey different schools of thought there have been variation and amandements throughout this century. It is perhaps more important to think first about the major differences between the two and not to worry about differences within or in any overlap which might be thought to exist between them. The major differences can be explained by referring  to situation in learning mathematics.

                It is very important that children come to an adequate undesrtanding of place-value. At acertain stage in the education of young children it would be resounable to ask them ‘four hundred and twenty-seven as a number . Some children would write

   40027.

  Others 4027.

      or even 400207.

And these would not be only answers otfered from within the class. Most children it is honed, whould correctly write

427

But the incorrect responses, however view, would require remediation. How should remedial action be taken ? How should the children be taught the concepts in the first place?

                If our theoretical view is  that the children learn thought practising to produce the correct response to a given stimulus, then we should give them more practice. Such an approach might incorporate the use of apparatus, but the fundamental intention is to give practice. In this approach there might well be the underlying assumption that we are there to feed information  and knowledge into the mind of the child. In an extreme form the approach might be reffered to as rote learning.

                If however we believe  that children learn  through making sense of the world themselves, we would wish them to discover the essential relationship through interaction with an appropriate environment. Thus we might well provide structural and other apparatus and devise activities and experiences, allowing exploration of the stucture of the situation. It would, of course, be necessary to ensure that the notation emerges as being logical and efficient, so some teacher intervention is inevitable. In this way understanding would grow from within, as it were. Any attemp to hasten the child by injecting rote methods might not only be unsuccessful, it might persuade the child that mathematics is meaningless and worthy only of rejection.  

                It should be stressed that these two contrasting approaches are not intended to explain fully the difference between particular behaviourist and cognitive approaches, they are merely intended to illustrate how possible interpretations might manifest themselves in mathematics lessons. It would be wrong to tie rote learning too closely to the behaviourist approach  and by implication suggest that it has no place within a cognitive approach. There is, after all the eclectic view that children do need to develop their own understanding from within, but that there might be a very firm place for practice, and even perhaps for some element of rote learning.

                It is unfortunate that conflicting theories and variation on a common theme might lead some teachers to reject them all. Some conflict is, after all only to be expected within a discipline with a very short history. It is sometimes forgotten that the so called ‘pure’ science have been the subject of many battles over many hundreds of years. Even now disagreements can still exist. Scientific theories are continually being modified, elaborated and simplified and from time to time, radically new ideas are produced.  In the world at large decisions have  to be made and they are madeon the basis of existing theoretical views. Not all such decisions ultimately turn out to have been correct. Particular theories of learning raight also be wrong or might need cualification or amandement. But the formulation of a theory and the observation of it in action are both part of the process through which we improve our understanding. We can learn more about the learning process if we are prepared to encourage the formulation of  theories and then test out those which appear most likely to help.

Learning is a mental activity. We might therefore understand more about learning if we knew more about the functioning of the brain as a processor of information. The brain  receives  information, interprets it, stores it, transfomrs it, associates it with other information to create new information and allows information to be recalled. In recent years considerable attention has been accorded to the information –processing aspect of learning theories and this has led to conciderable interest in goes on inside the brain. It has been known for many years that different learning activities take place in different parts of the brain though that very simple statement unfortunately glosses over complexities which are certainly beyond our scope for the present.

The relationship between the chemistry of the brain the never impulses which are generated and learning are, likewise, too complex for this book. It must be clear, however,  that might understand much more about learning, as an aspect of psychology, when we understand more about the workings of the brain as an aspect of psychology.

                One of the traditional justifications for teaching mathematics is taht it teaches logical thinking. Unfortunately, the logic of mathematics is not necessarily the same as the logic of any other  sphere of human intellectual activity. The argument therefore stands or falls on the theory that the ability to think logically in mathematics is a transferable skill and can be put into practice outside mthematics. This assumption has been known in the past as ‘transfer of training’. Shulman(1970) said. Transfer of training is the most important single concept in any educationally relevant theory of learning. There is no doubt that the former view that studying geometry or Latin made one a better logical thinker is now completely discredited. Nevertheless, some lateral transfer must be possible lateral implying the transference of skill one domain to the achivement of a parallel skill in another domain (though ‘parallel’ is not easy to define in this contex), for without it learning would be extremely slow and would be limited to what had actually been encountered in the course of instruction.

                There is no general agreement about the extent to which lateral transfer can take place in mathematics. There have been psychologist and learning theorists who have expressed the view that broad transfer can take place that ideas and strategies can be transferred within a discipline and perhaps even outside. Thus it might be believed that mastery of the idea of balance as a physical property using weigh-scales and weights can be transferred and applied to the solution of linear equations and might even be transferable to studies of balance in nature and balance in economics. It might also be believed that learning how to prove results in Euclidean or any other sort of geometry would be transferable to proof in other branches of mathematics to proof in other disciplines such as science  and even to proof in a court of law . Other psychologists have believed that transfer only occurs to a very limited extent perhaps only to the extent that identical elements occur. This latter view probably carries more conviction than the former at the present time. Some transfer must be possible, but it will probably be limited and might depend on the conditions under which learning takes places. It is certainly not wise to assume that transfer of skills will occur when teaching mathematics.

                The learning difficulties which one observes as a teacher of mathematics raise many other questions for which one might seek an answer from theories. For example, although reflection on our own experience should suggest to us that learning cannot be achieved in a hurry, some children appear to learn incredibly slowly. What determines the rate of learning? Some children  make very rapid progress a few even make astounding progress given the opportunity to learn at their rate rather than the class rate. Is it possible to accelerate the learning of mathematics for more pupils or even for the majority of pupils and, if so how? At the moment it seems that for many children it is not a matter of whether they can learn mathematics more quickly : rather it is a question of why they apper to takein hardly anything at all. Is it that mathematical ability is a peculiar aptitude possessed by only a few?

                Individual differences are very significant in many spheres of human activity. Some of us are barred from particular occupations because of physical characteristics, like being too small, too overweight, or having poor eyesight. Many of us who have become teachers of mathematics because of an apparent aptitude and a liking for the subject would not have been able to become teachers of other subject, like English or history. Amongst internasional athletes some are good only at running, others at jumping events and yet others at throwing events. Individual differences might be inportant even within mathematics. Hadamard (1945), in discussing mathematics, drew attention to great differences in the kind of mathematical aptitude which individuals has displayed. In the classroom it might be that different learning environments and different teaching styles are needed for different pupils, which would present very great teaching problems in the sense that any individual teacher also presumably has preferences which are in accord with only a proportion of the pupils. Any acceptable theory which enables us to understand individual differences would be very valuable.

                One interpretation of the evidence of what children appear to learn and appear to have difficulty with is that there are serious stumbling-blocks in the logical structure of mathematics. With many young children, the ideas of place-value appear to present hurdles which cause frequent falls. With slightly older children the introduction of algebraic notions causes problems for which some pupils, in later life, never forgive us. There are mathematical ideas, like ratio and rate, which continue to cause difficulty for many throughout adult life, even though they are extremely important ideas. It is possible to survive in life without understanding the implications of a fall in the rate of inflation, but it is a pity that so many adults have learning difficulties with mathematics. So what is it about particular aspects of mathematics, algebra and rate of change, which makes them so difficult? When we analyse the structure of mathematics in order to devise the optimum teaching sequence, how do we allow for the fact that the logical order of topics might fail us for psychological reasons?

                A major complexity in learning any subject is the relationship with language learning. At a surface level the effects may be observed when a child cannot do the mathematics because the particular language used is not understood. There are many examples of peculiar language and of familiar words used in different or very specific ways in mathematics. At a deeper level, to understand the language is to understand the concept which a particular word symbolizes. More fundamental still is the relationship between language and learning. Does language merely enable one to comunicate leaning that has already taken place? Is language the vehicle which enables us to formulate our ideas and manipulate them to crate new meaning? Is it that language development is inextricably tied to overall cognitive development and cannot be thought of as a separate entity?

                It has been suggested earlier that the learning environment might be an important factor in promoting the understanding of mathematics. It might, therefore, be postulated that the richer the environment the more efficient the learning, but to some extent that begs the question. What constitutes a rich learning environment in a subject which is basically a creation of the human mind and in which the aim is to enable abstract argument to take place through the manipulation of symbols? The belief that young children must be allowed and encouraged to interact in a very active manner with physical or concrete materials is a theoretical stance suggested through experience of teaching young children (though not all young children are provided with such an environment). If we accept this and provide an environment rich in equipment and learning materials for young children, how soon can we wean them from it? Do we need to do anything for older children in, for example, coming to terms with algebra? Or should we not be attempting algebra until the pupils can manage without concrete apparatus? When can children begin to learn only from exposition and from books?

                These are some of the many aspect of mathematics learning for which we might seek answer. Many of the theoretical viewpoint expressed in subsequent parts of this book do attempt to account for questions raised above. It has already been suggested that teachers need theories, hence major theories of all kinds and from many sources are included within the discussion of particular questions. First, however, we take a look at the problems from the child’s point of view, to see what is being learned and how thoroughly. We investigate the empirical evidence on which, it has been declared, theories might be formulated.