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The "Mathematics For All" Movement

With increasing numbers of students attending elementary and secondary schools around the world, the issue whether the traditional Western mathematics curriculum (Capital M Mathematics), with its tightly sequenced formal approaches to number, measurement, geometry, algebra and trigonometry, is still appropriate for all learners in all cultures in all countries has been brought into question. Writers on ethnomathematics such as d'Ambrosio (1985, 1989) have argued that in the past, school mathematics has been an élitist affair, especially suited to the preparation of middle-class males for prestigious professions such as engineering, accounting, and the natural sciences. In this sense it has been a value-laden, effective sieve used for the selection of future leaders.

What is needed, d'Ambrosio (1985) has argued, is a totally new approach whereby different mathematics curricula are developed, always with the specific needs of groups of potential learners in mind. From this point of view, mathematics education has been recognised as having a strong political agenda (Mellin-Olsen, 1987). Critical theorists such as Frankenstein (1989) reject the idea that mathematics is a culture-free phenomenon, and in so doing point out that the sanitised, abstract statements in Western mathematics curricula are but one way of interpreting the world among many competing discourses. What is needed, they say, is a "Mathematics for all" approach, in which curricula are developed which make sense to individual learners now. What is not needed is a curriculum which sacrifices the majority for the future needs of a tiny minority who will proceed to study major studies in mathematics in universities - although, of course, these students should have the opportunity to study courses which meet their immediate and future needs. From the perspective of the "mathematics for all" goal, national, centrally prescribed mathematics curricula are not likely to be satisfactory (Damerow, Dunkley, Nebres, & Werry, 1984).

The title of Clements's (1989) book, Mathematics for the Minority, captures the same sentiment - specifically, that in the past, Western school mathematics has catered for the needs of a minority. Elsewhere we have argued (Ellerton & Clements, 1989a) that the effect of such an approach has been that the main lesson learned by most school leavers after many years of being forced to study mathematics at school, was that they could not do it. Nevertheless, despite strong anthropological evidence to the contrary (Bishop, 1988; d'Ambrosio, 1985; Harris, 1991), there is a strong feeling around the world that mathematics is a culture-free phenomenon (Ellerton & Clements, 1989b).

In the last paragraph, we summarised some researchers' attempts to place mathematics education research within a framework which takes into account the cultural and in particular the political forces which surround any mathematics education enterprise. The approach to educational research which attempts to paint a holistic picture, rather than to separate elements of a culture, has, of course, been borrowed from anthropology, and there can be little doubt that in the past decade, anthropological approaches to mathematics education research have been increasingly adopted.

In anthropological research the researcher attempts to map the complex web which defines cultures and sub-cultures. Clearly, qualitative methods are more appropriate than quantitative, although when appropriate, the latter can be used. At the macro level, mathematics education researchers such as Harris (1991) and Watson (1989) have studied the mathematical systems which have developed within various cultural groups, and have been interested in how young children acquire knowledge and understanding of these systems. At the micro level, some mathematics education researchers have been especially interested in the sub-cultures within a cultural setting in which mathematics teaching and learning occurs.

Researching the Discourses of Mathematics Classrooms

An important example of a sub-culture at the micro level is the mathematics classroom in a formal Western education setting, and in the 1980s there was a large amount of research aimed at disentangling and understanding complex classroom processes. As the French mathematics educator, Brousseau (1983), has argued, children not only bring to classrooms their own mathematical knowledge, but also their knowledge of the ways a particular classroom and a particular teacher operate. Brousseau coined the expression "didactical contract" to describe the way in which pupils create the rules of mathematical classrooms and in so doing define the parameters within which teachers operate on a day-to-day basis. As Balacheff (1986), another French researcher, has argued, many students in mathematics classrooms are engaged in a game of convincing their peers and their teacher that they know what they are talking about, and that for the students this is often more important than their engagement with the mathematics itself.

This anthropological approach to mathematics education research has been particularly strong in Europe. In Germany, Bauersfeld's (1980) article, "Hidden dimensions in the so-called reality of a mathematics classroom," provided insight into the directions of much of the research that would be carried out by German mathematics education researchers in the 1980s. Bauersfeld (1991) and Voigt (1992) are among many German researchers who have also been using anthropological methods to research the culture of mathematics classrooms. Voigt's (1985) important article on "Patterns and routines in classroom interaction," showed, through analyses of the discourse patterns of a range of mathematics classrooms, that even teachers who were consciously attempting to use constructivist methods, tended to impose their own ways of thinking about mathematics on the children to a greater extent than they would like to have admitted.

The use of anthropological methods in research into mathematics classrooms has not, however, been confined to Europe. In the United States of America, for example, Paul Cobb (1990b) has written extensively about the need to use anthropological research methods to investigate mathematics classroom environments, and Cobb and co-workers at Purdue University, in the United States have reported a number of detailed studies which reveal the nature and effects of discourse in elementary mathematics classrooms (see, for example, Cobb, Yackel and Wood, 1988, 1992). There is evidence, too, of collaborative mathematics classroom research projects, involving researchers from different countries, being mounted (see, for example, Krummheuer and Yackel, 1990). Easley and Easley (1992), who used discourse analysis techniques to compare discourses in elementary mathematics classrooms in the United States and in Japan, conjectured that the main reason why so many Japanese children perform so well in mathematics may lie in the different interaction patterns which characterise the mathematics classrooms of the two countries. Researchers from the University of Chicago (see, for example, Stigler and Baranes, 1988), who have used both discourse analysis and statistical analysis in their comparative studies of mathematics classrooms in the United States, China, Japan and Korea, have reached similar conclusions. Stigler and Baranes (1988) commented:

This emphasis in Japan on verbal discussion of, and reflection upon, mathematical topics - so rarely found in American elementary school classrooms - is also evidenced in other ways in our observations. For example, the most common means of publicly evaluating student work in the Japanese classrooms was to ask a student who obtained an incorrect answer to a problem to put his work on the board, and then to discuss, with the entire class, the process that led to the error. The most common form of evaluation in the American classrooms, by contrast, was simply to praise a student who answered the problem correctly. By focusing on errors, Japanese teachers have a natural basis on which to build a discussion: Praies, on the other hand, functions as a conversation stopper. (p. 296)

Research of this type points to the fact that although most teachers of mathematics are not always conscious of recurring patterns of discourse and behaviour in their classes, it is these patterns which, in fact, have the greatest influence on the extent and quality of mathematics learning which takes place.

Situated Cognition Research

Situated cognition research, which combines anthropological and statistical research methodologies and, depending on the researcher, has also called upon critical theory, has continually challenged the thinking and broadened the perspectives of mathematics educators in the 1980s and 1990s. As noted previously in this paper, mathematics is now regarded by many mathematics education researchers as a socially constructed body of knowledge, and situated cognition research has emphasised the importance of taking into account the cultures of learners when developing mathematics curricula (Brown, Collins, & Duguid, 1989).

Ideally, the Mathematical (capital "M") skills, concepts and relationships that children are asked to learn in schools should be linked not only with any capital M Mathematics they already know, but also with their personal worlds (Watson, 1987). If curricula and teaching methodologies are such that capital "M" mathematics is presented in highly formalised ways which do not take account of the backgrounds of the learners, then cognitive links are not likely to be established, and what the children write in mathematics examinations will represent nothing more than mere rote knowledge and skills. While this may assist the children to except pass examinations - and it may not even do that if the examinations require more than rote recall and a mechanical use of skills - it will be of no practical utility.

On this point the recent "situated cognition" research in mathematics education, by researchers such as Carraher (1988), Lave (1988), and Saxe (1988), has suggested that people in all classes and walks of life are capable of performing quite complex mathematical operations provided that the context in which the mathematics is presented links with the learners' personal worlds. Situated cognition research has revealed, for example, how street children in Brazil who are failing school mathematics (Carraher, 1988; Saxe, 1988), are actually capable of performing quite difficult mental calculations quickly and accurately in their normal out-of-school roles as candy sellers in the streets.

This raises the question whether school mathematics should be generated by societal needs and aspirations rather than be an appendix to it. Or to put this in the form of another question, can mathematics curricula be properly constructed by armchair theorists remote from the action? We believe that it is an important lesson of history that the answer to this question is "No," yet the existence of national curricula and assessment systems in mathematics in many countries would suggest that education administrators and politicians have failed to learn it.

Two Examples of Learners for Whom Capital M Mathematical Knowledge Was Useless Except, Perhaps, for the Purpose of Passing Examinations

Why should mathematics education researchers be turning towards anthropological methods of investigation? The answer is that they now realise that so much of what has become acceptable practice and behaviour in standard mathematics lessons in many countries (though not necessarily all countries - see Stigler and Baranes, 1988) does not help children to understand mathematics. Indeed, the sub-culture associated with many school mathematics programs has been destructive in the sense that it has given even apparently successful learners a false sense of security - they have been persuaded that they have understood something when, in fact, they have not. Anthropological investigations can lay bare the unsatisfactory aspects of what goes on in mathematics classrooms, and can provide data which will assist mathematics educators to define what constitutes quality mathematics learning environments.

We now provide two classroom examples which illustrate that how a narrow emphasis on drill and practice, on getting right answers, can in fact be educationally useless. We would wish to comment, at the outset, that with both examples the teachers were capable, sincere and dedicated persons with the best interests of their students at heart. We believe that most experienced observers would regard them as "good teachers." Also, we would assert that the type of outcome which is revealed is typical of what is achieved in most mathematics lessons conducted around the world.

Example 1- multiplying with fractions. Clements and Lean (1988) reported research in which a whole class of Grade 5 children had been drilled in how to multiply fractions. Before Lean and Clements interviewed the children the class had devoted considerable time to finding answers to questions like: "Find the value of 7/11 x 792." The children tackled such tasks in a mechanical way: for example, in the particular case mentioned they tended to divide 11 into 792, "cancel" the 792 and put 72 in its place (if they got the division correct), multiply 72 by 7 and, if they got the multiplication correct, write down 504 as their answer. They checked their answers at the back of the book, put ticks or a crosses depending on whether they were right or wrong, and then went on with the next multiplication involving fractions.

When the children were interviewed it became clear that not one child in the class had any idea of what 7/11 might mean. They did not know why they had divided 11 into 792, except that this was what the teacher had told them to do, and that this was how you got the right answer. No child was able to make up an example in real life where someone might want to find out the value of 7/11 x 792. When shown 12 stones, and asked to give 1/4 of them to the interviewer, quite a few of the students who had got 7/11 x 792 correct gave the interviewer 4 stones!

Example 2- the area of a triangle. Recently one of the authors watched a Grade 8 lesson, in an Australian classroom, on "the area of a triangle." The lesson began with the teacher reminding the students that in the previous lesson they had learned how to find the area of a rectangle. When the teacher then asked the revision question,"How do you get the area of a rectangle?" no one volunteered an answer, so he asked a particular student for the answer. The student, looking somewhat embarrassed, stammered, "180?"

Ten minutes later, after the teacher had shown the class how any triangular region could be regarded as half a rectangular region, the formula A = (b x h)/2 was written on the board and then a worked example of how this could be used was provided by the teacher. The students were then told to turn to a page in their textbook on which there were 20 exercises which had exactly the same structure as the worked example (the length measures of the base and the altitude changing for each exercise). The students quickly got down to work doing the exercises, and seemed to be pleased that they were getting correct answers. But when some of the children were subsequently interviewed it became clear that they could not relate the calculations they had carried out with any property of the triangles. They had merely followed a rule in order that they would get correct answers.

Constructivism Politics

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