Conjecture, proof, and sense in Wittgenstein's philosophy of mathematics

One of the key tenets in Wittgenstein’s philosophy of mathematics is that a mathematical proposition gets its meaning from its proof. This seems to have the paradoxical consequence that a mathematical conjecture has no meaning, or at least not the same meaning that it will have once a proof has been found. Hence, it would appear that a conjecture can never be proven true: for what is proven true must ipso facto be a different proposition from what was only conjectured. Moreover, it would appear impossible that the same mathematical proposition be proven in different ways. — I will consider some of Wittgenstein’s remarks on these issues, and attempt to reconstruct his position in a way that makes it appear less paradoxical.

The 'algebra as object' analogy: a view from school

Treating algebraic symbols as objects (eg. “‘a’ means ‘apple’”) is a means of introducing elementary simplification of algebra, but causes problems further on. This current school-based research included an examination of texts still in use in the mathematics department, and interviews with mathematics teachers, year 7 pupils and then year 10 pupils asking them how they would explain, “3a + 2a = 5a” to year 7 pupils. Results included the notion that the ‘algebra as object’ analogy can be found in textbooks in current usage, including those recently published. Teachers knew that they were not ‘supposed’ to use the analogy but not always clear why, nevertheless stating methods of teaching consistent with an‘algebra as object’ approach. Year 7 pupils did not explicitly refer to ‘algebra as object’, although some of their responses could be so interpreted. In the main, year 10 pupils used ‘algebra as object’ to explain simplification of algebra, with some complicated attempts to get round the limitations. Further research would look to establish whether the appearance of ‘algebra as object’ in pupils’ thinking between year 7 and 10 is consistent and, if so, where it arises. Implications also are for on-going teacher training with alternatives to introducing such simplification.

Transition of pupils from Key Stage 2 to 3 deemed gifted and talented in mathematics: an initial study

In this article Geoff Tennant and Dave Harries report on the early stages of a research project looking to examine the transition from Key Stage (KS) 2 to 3 of children deemed Gifted and Talented (G&T) in mathematics. An examination of relevant literature points towards variation in definition of key terms and underlying rationale for activities. Preliminary fieldwork points towards a lack of meaningful communication between schools, with primary school teachers in particular left to themselves to decide how to work with children deemed G&T. Some pointers for action are given, along with ideas for future research and a request for colleagues interested in working with us to get in touch.

Issues in language and development in multilingual settings: a case-study of knowledge exchange and teacher education in South Africa

The quality of a country’s human-resource base can be said to determine its level of success in social and economic development. This study focuses on some␣of the major human-resource development issues that surround the implementation of South Africa’s policy of multilingualism in education. It begins by discussing the relationship between knowledge, language, and human-resource, social and economic development within the global cultural economy. It then considers the situation in South Africa and, in particular, the implications of that country’s colonial and neo-colonial past for attempts to implement the new policy. Drawing on the linguistic-diversity-in-education debate in the United Kingdom of the past three decades, it assesses the first phase of an in-service teacher-education programme that was carried out at the Project for Alternative Education in South Africa (PRAESA) based at the University of Cape Town. The authors identify key short- and long-term issues related to knowledge exchange in education in multilingual societies, especially concerning the use of African languages as mediums for teaching and learning.

The Mathematics of Control Theory

This text contains papers presented at the Institute of Mathematics and its Applications Conference on Control Theory, held at the University of Strathclyde in Glasgow. The contributions cover a wide range of topics of current interest to theoreticians and practitioners including algebraic systems theory, nonlinear control systems, adaptive control, robustness issues, infinite dimensional systems, applications studies and connections to mathematical aspects of information theory and data-fusion.

Mathematics as a tool in the Life Sciences

Teaching mathematics to students in the biological sciences is often fraught with difficulty. Students often discover mathematics to be a very 'dry' subject in which it is difficult to see the motivation of learning it given its often abstract application. In this paper I advocate the use of mathematical modelling as a method for engaging students in understanding the use of mathematics in helping to solve problems in the Biological Sciences. The concept of mathematics as a laboratory tool is introduced and the importance of presenting students with relevant, real-world examples of applying mathematics in the Biological Sciences is discussed.

Self, role and character: developing a professional identity as a drama teacher

The paper reports on research into what may have influenced trainees on four post-graduate teacher training courses in England to become specialist drama teachers rather than pursue careers in the world of professional entertainment. In doing so it raises questions regarding the value of considering teaching as a performing art. The paper goes on to explore how drama trainees regard an understanding of performance, and an ability to both use and demonstrate performance techniques, as integral to their role as subject specialists. The subsequent discussion examines how a drama teacher’s professional identity may be seen as being made up of the three inter-connected elements, self, role and character. Thus, while all teaching may be considered to involve some elements of performativity , this paper suggests that, for the drama specialist, an understanding of what constitutes ‘performance’ has a particular importance. One conclusion drawn from the research is that recognising the place of performance in their practice may result in experienced teachers of drama regarding themselves as artists whose art is teaching drama; another is that recognising the different ways in which adopting a role may involve performance could be of value to all teachers and teacher educators.

An Examination of the factors that shape the subjectivities of female art students in Initial Teacher Training

The school subject of Art and the profession of the primary school teacher are gendered female and both are considered low status within the field of Education and other professional areas of society. A number of sociological studies have examined the impact of gendered socialisation and habitus on females’ career choices and various educational initiatives have been put in place over the years to encourage females to select subjects and/or pursue career paths normally associated with males. Yet Art and primary school teaching continue to be a popular choice with middle class girls. Based on a critical ethnographic study of female BAED Art students, who are training to be primary school teachers, this study is an examination of the many factors, historically and contemporaneously that have shaped and continue to shape the subjectivities of females and frame their aspirations and ambitions. Within this discourse significant aspects of the history of Art and Art Education that have contributed to and influenced the construction of the female artist, and their consequent impact on artistically talented females’ personal identity as artists, are also examined.

The MaPS project: mapping teacher conceptions of musical development

The chapter starts from the premise that an historically- and institutionally-formed orientation to music education at primary level in European countries privileges a nineteenth century Western European music aesthetic, with its focus on formal characteristics such as melody and rhythm. While there is a move towards a multi-faceted understanding of musical ability, a discrete intelligence and willingness to accept musical styles or 'open-earedness', there remains a paucity of documented evidence of this in research at primary school level. To date there has been no study undertaken which has the potential to provide policy makers and practitioners with insights into the degree of homogeneity or universality in conceptions of musical ability within this educational sector. Against this background, a study was set up to explore the following research questions: 1. What conceptions of musical ability do primary teachers hold a) of themselves and; b) of their pupils? 2. To what extent are these conceptions informed by Western classical practices? A mixed methods approach was used which included survey questionnaire and semi-structured interview. Questionnaires have been sent to all classroom teachers in a random sample of primary schools in the South East of England. This was followed up with a series of semi-structured interviews with a sub-sample of respondents. The main ideas are concerned with the attitudes, beliefs and working theories held by teachers in contemporary primary school settings. By mapping the extent to which a knowledge base for teaching can be resistant to change in schools, we can problematise primary schools as sites for diversity and migration of cultural ideas. Alongside this, we can use the findings from the study undertaken in an English context as a starting point for further investigation into conceptions of music, musical ability and assessment held by practitioners in a variety of primary school contexts elsewhere in Europe; our emphasis here will be on the development of shared understanding in terms of policies and practices in music education. Within this broader framework, our study can have a significant impact internationally, with potential to inform future policy making, curriculum planning and practice.

Student teachers’ perceptions of reading and the teaching of reading: the implications for teacher education

This article reports on research which identified perceptions of reading and the teaching of reading held by trainee teachers and the impact on my provision as a teacher educator. It found that students’ past and present experiences of learning to read and being a reader influenced their perceptions of what reading is and of what it means to teach reading. As a teacher educator, I am not able to give students long experience of seeing children becoming readers, but I am able to give them richer experiences of reading in personally and culturally relevant contexts. This has implications for the nature of subject knowledge required by a student teacher of reading and the curriculum and practice of teacher education.