Alan J. Bishop

Second international handbook of mathematics education

There is much debate within mathematics teacher education over ways in which professional and academic foci could be made to complement each other. On the one hand, teachers’ craft knowledge is emphasized, mainly as this relates to the particular and local level of teaching; on the other hand, the importance of academic subject knowledge cannot be denied. In this chapter the focus will be on how to blend and balance the two through activities in which teachers learn from other teachers, particularly the co-learning of teachers and teacher educators. It will discuss professional relationships, reflective practice, community building, and research in practice. Examples of research-based programs involving lesson study (LS) and the Learner’s Perspective Study (LPS) have moved the relevant research in this area to yet another level, in which theory and practice are combined. Projects such as these and others from diverse parts of the world will be presented and discussed.This chapter seeks to provide an integrating theoretical framework for understanding the somewhat disparate and disconnected literatures on “modelling” and “technology” in mathematics education research. From a cultural–historical activity theory, neo-Vygtoskian perspective, mathematical modelling must be seen as embedded within an indivisible, molar “whole” unit of “activity.” This notion situates “technology”—and mathematics, also—as an essential part or “moment” of the whole activity, alongside other mediational means; thus it can only be fully understood in relation to all the other moments. For instance, we need to understand mathematics and technology in relation to the developmental needs and hence the subjectivity and “personalities” of the learners. But, then, also seeing learning as joint teaching–learning activity implies the necessity of understanding the relation of these also to the teachers, and to the wider institutional and professional and political contexts, invoking curriculum and assessment, pedagogy and teacher development, and so on. Historically, activity has repeatedly fused mathematics and technology, whether in academe or in industry: this provides opportunities, but also problems for mathematics education. We illustrate this perspective through two case studies where the mathematical-technologies are salient (spreadsheets, the number line, and CAS), which implicate some of these wider factors, and which broaden the traditional view of technology in social context.

MATHEMATICS EDUCATION IN ITS CULTURAL CONTEXT

This paper presents the results of a series of analyses of educational situations involving cultural issues. Of particular significance are the ideas that all cultural groups generate mathematical ideas, and that ‘Western’ mathematics may be only one mathematics among many. The values associated with Western mathematics are also discussed, and various issues raised by these analyses are then presented.

Spatial Abilities and Mathematics Education – A Review

This paper examines the interface between two sets of psychological constructs. One set concerns the visual/spatial field and the other, mathematics education. But rather than do a conceptual analysis or report an empirical investigation, I have chosen to review the different research emphases which have contributed to knowledge in this area, and to describe what they can, or cannot, offer the mathematics educator. I show that this literature does contain many potentially fruitful ideas and approaches which we in mathematics education can use either in the classroom, or to guide our own research efforts. Also it will be seen that the goals of research psychologists may lead them in a direction which is away from our concerns, and that therefore we must exercise caution and keen judgement in selecting those ideas and approaches which will enable us to develop our own field.

Mathematical Knowledge: Its Growth Through Teaching

In the first BACOMET volume different perspectives on issues concerning teacher education in mathematics were presented (B. Christiansen, A. G. Howson and M. Otte, Perspectives on Mathematics Education, Reidel, Dordrecht, 1986). Underlying all of them was the fundamental problem area of the relationships between mathematical knowledge and the teaching and learning processes. The subsequent project BACOMET 2, whose outcomes are presented in this book, continued this work, especially by focusing on the genesis of mathematical knowledge in the classroom. The book developed over the period 1985-9 through several meetings, much discussion and considerable writing and redrafting. Our major concern was to try to analyse what we considered to be the most significant aspects of the relationships in order to enable mathematics educators to be better able to handle the kinds of complex issues facing all mathematics educators as we approach the end of the twentieth century. With access to mathematics education widening all the time, with a multi- tude of new materials and resources being available each year, with complex cultural and social interactions creating a fluctuating context of education, with all manner of technology becoming more and more significant, and with both informal education (through media of different kinds) and non- formal education (courses of training etc. ) growing apace, the nature of formal mathematical education is increasingly needing analysis.

Values in Mathematics Teaching — The Hidden Persuaders?

Values are at the heart of teaching any subject, but are rarely explicitly addressed in the mathematics teaching literature. In particular, research on values in mathematics education is sadly neglected. This chapter addresses these gaps by drawing together the various research and theoretical fields that bear upon the values dimension of mathematics education. It begins with a theoretical reflection on the distinctions between values, beliefs and attitudes, and continues with reviewing the literature relating teachers’ values to their decisions and actions in the classroom. Moving to the limited research on values in mathematics education, there is discussion of values in the increasingly researched area of socio-cultural aspects of mathematics education. The second half of the chapter is devoted to issues regarding research approaches to studying values in our field, and presents two projects, one based in Monash University, Australia and the other at the National Taiwan Normal University in Taipei. The first project focused on the relationship between teachers’ intended and implemented values, and the second explored teachers’ values as constituting their pedagogical identities. Implications of this research for teachers’ professional development are drawn, and the chapter finishes by outlining the research difficulties inherent in this area, and offers a set of challenges designed to carry the research agenda forward.

Visualising and Mathematics in a Pre-Technological Culture

Earlier in this article I said that I was not going to discuss trategies for educational development in Papua New Guinea, although as you can probably infer, I find that problem both fascinating and formidable. My concern here was to offer you some data which, I hoped would contrast in various ways with the data you would normally meet. But how successful have I been? Leaving aside those readers who work in Papua New Guinea or similar cultures, consider how different this data is. Even in technologically developed societies and cultures do we not find some of these problems — sometimes with adults, certainly with children? Could I give as a general description of those people “those who have not yet been inducted into the mathematicians culture” or sometimes even “those who have chosen not to enter it”? Perhaps if we consider mathematics education as a form of cultural induction we would realise both the enormity of the task and the range of influences that can be brought to bear. We would for example not only consider problems like “What are the skills necessary to be a successful mathematician?” but also others like “What is the value of entering the mathematicians world?” and “Why do we consider it to be so important?” If we do consider mathematics to be problem-solving par excellence, then we should also recall that it is only one approach to problem-solving and it can be seen by ‘outsiders’ as a very strange business. (As another example, ask yourself why you spend a long time looking for a quick solution). Even if we feel we know what the values of learning mathematics are we then face problems such as how do we transmit those values? What do we know about the role of the teacher as a cultural transmitter, as an example, as a model for imitation? And there are many other questions. Mathematics education has powerful cultural and social components. Perhaps we should give them the attention which we have already given to the psychological components.

EDUCATING STUDENT TEACHERS ABOUT VALUES IN MATHEMATICS EDUCATION

Current recommendations for changing mathematics education ignore the crucial role of values in those changes. Values have also been ignored in research into affective issues and it seems from current research that they are only addressed in implicit ways in the mathematics classroom. This chapter offers an analysis of values in mathematics teaching and develops some frameworks for creating values-related activities with student teachers. It bases these ideas on a current research project based in Australia.

Mathematical Values in the Teaching Process

The fact that mathematics is a part of every pupil’s school curriculum is rarely questions — it is indeed difficult to find a country where mathematics is not a significant part of the curriculum. Yet we also know that in most countries many pupils continue to experience a great deal of difficulty with the subject. Mathematics educators the world over are continually searching for ways to improve this situation. “Maybe the pupils don’t enjoy it enough” it is said, so there are strategies to make mathematics more fun. Or “Maybe they don’t see the point of it enough”, so the search is on for more ‘relevance’. Research is continually directed at this problem, the possibilities and probabilities are weighed and assessed, and the debate goes on.

Mathematics Education and Student Values: The Cultivation of Mathematical Wellbeing

This chapter argues that school mathematics involves more than just the “performance” of students and that “working mathematically” means far more than being good at a specified set of skills, as well as more than being able to show mastery of various conceptual structures. It suggests that experienced teachers understand that the wellbeing of many students can diminish when they are asked to engage with mathematics learning. Underlying such engagement and hence mathematical performance is a command of a specific language that holds the conceptualizing process together. Moreover, of particular importance for this chapter are the values, and their language, that are embedded within mathematics and its pedagogy and how they can be invoked to enable better engagement, improved student wellbeing and consequently better performance.

From the Few to the Many: Historical Perspectives on Who Should Learn Mathematics

Today we take for granted that everybody should be offered the opportunity to learn mathematics. However, it was not until well into the 20th century that “mathematics for all” became an achievable goal. Before then, the geographical location of schools in relation to children’s homes, the availability (or non-availability) of teachers capable of teaching mathematics, parental attitudes to schooling, economic circumstances of families, and social and psychological presuppositions and prejudices about mathematical ability or giftedness, all influenced greatly whether a child might have the opportunity to learn mathematics. Moreover, in many cultures the perceived difference between two social functions of mathematics—its utilitarian function and its capability to sharpen the mind and induce logical thinking—generated mathematics curricula and forms of teaching in local schools which did not meet the needs of some learners. This chapter identifies a historical progression towards the achievement of mathematics for all: from schooling for all, to arithmetic for all, to basic mathematics for all; to secondary mathematics for all; to mathematical modelling for all; and to quantitative literacy for all.