Infinitely many cyclic solutions to the Hamilton-Waterloo problem with odd length cycles

It is conjectured that for every pair $(\ell,m)$ of odd integers greater than 2 with $m \equiv 1\; \pmod{\ell}$, there exists a cyclic two-factorization of $K_{\ell m}$ having exactly $(m-1)/2$ factors of type $\ell^m$ and all the others of type $m^{\ell}$. The authors prove the conjecture in the affirmative when $\ell \equiv 1\; \pmod{4}$ and $m \geq \ell^2 -\ell + 1$.Comment: 31 page

Shifting sands: students' understanding of the roles of variables in 'A' level mathematics

The move from GCSE to 'A' level mathematics in schools in England and Wales can be, at least partly, characterised by an increase in abstraction and sophistication associated with the introduction of more variables. I have described as 'second variable' situations those in which letters were used in roles that went beyond the single unknown value or the dependent and independent variable, for example - the equation y = mx + c as a general equation of a straight line - the use of (a, b) to describe a general point. This thesis attempts to describe the experiences of students in meeting these situations. The data which I present are drawn from a variety of sources, but come chiefly from a year spent teaching and observing two 'A' level mathematics classes in two different schools. Other sources are my own mathematical work and that of colleagues, in particular two groups of teachers with whom I met to discuss my research. I also refer to my teaching of other groups of students. My conclusions - distinguish between structural and empirical generalisation - identify the shifts in the roles of literal symbols which take place in the solving of some types of problem - describe how stereotyping affects students' treatment of literal symbols and assists in or interferes with solving of problems - list some components of 'second variable thinking'. My research method is qualitative rather than quantitative. It draws on my teaching experience and makes a virtue of the subjectivity of both the researcher and the reader. I offer a number of mathematical exercises to the reader and intend that he should draw on his experience of these exercises in interpreting the thesis. I expect the validity of the thesis to be judged by its coherence and by its capacity to inform the future practice of myself and of readers

Inquiry in University Mathematics Teaching and Learning. The Platinum Project

The book presents developmental outcomes from an EU Erasmus+ project involving eight partner universities in seven countries in Europe. Its focus is the development of mathematics teaching and learning at university level to enhance the learning of mathematics by university students. Its theoretical focus is inquiry-based teaching and learning. It bases all activity on a three-layer model of inquiry: (1) Inquiry in mathematics and in the learning of mathematics in lecture, tutorial, seminar or workshop, involving students and teachers; (2) Inquiry in mathematics teaching involving teachers exploring and developing their own practices in teaching mathematics; (3) Inquiry as a research process, analysing data from layers (1) and (2) to advance knowledge inthe field. As required by the Erasmus+ programme, it defines Intellectual Outputs (IOs) that will develop in the project. PLATINUM has six IOs: The Inquiry-based developmental model; Inquiry communities in mathematics learning and teaching; Design of mathematics tasks and teaching units; Inquiry-based professional development activity; Modelling as an inquiry process; Evalutation of inquiry activity with students. The project has developed Inquiry Communities, in each of the partner groups, in which mathematicians and educators work together in supportive collegial ways to promote inquiry processes in mathematics learning and teaching. Through involving students in inquiry activities, PLATINUM aims to encourage students‘ own in-depth engagement with mathematics, so that they develop conceptual understandings which go beyond memorisation and the use of procedures. Indeed the eight partners together have formed an inquiry community, working together to achieve PLATINUM goals within the specific environments of their own institutions and cultures. Together we learn from what we are able to achieve with respect to both common goals and diverse environments, bringing a richness of experience and learning to this important area of education. Inquiry communities enable participants to address the tensions and issues that emerge in developmental processes and to recognise the critical nature of the developmental process. Through engaging in inquiry-based development, partners are enabled and motivated to design activities for their peers, and for newcomers to university teaching of mathematics, to encourage their participation in new forms of teaching, design of teaching, and activities for students. Such professional development design is an important outcome of PLATINUM. One important area of inquiry-based activity is that of „modelling“ in mathematics. Partners have worked together across the project to investigate the nature of modelling activities and their use with students. Overall, the project evaluates its activity in these various parts to gain insights to the sucess of inquiry based teaching, learning and development as well as the issues and tensions that are faced in putting into practice its aims and goals

Inquiry in University Mathematics Teaching and Learning

The book presents developmental outcomes from an EU Erasmus+ project involving eight partner universities in seven countries in Europe. Its focus is the development of mathematics teaching and learning at university level to enhance the learning of mathematics by university students. Its theoretical focus is inquiry-based teaching and learning. It bases all activity on a three-layer model of inquiry: (1) Inquiry in mathematics and in the learning of mathematics in lecture, tutorial, seminar or workshop, involving students and teachers; (2) Inquiry in mathematics teaching involving teachers exploring and developing their own practices in teaching mathematics; (3) Inquiry as a research process, analysing data from layers (1) and (2) to advance knowledge inthe field. As required by the Erasmus+ programme, it defines Intellectual Outputs (IOs) that will develop in the project. PLATINUM has six IOs: The Inquiry-based developmental model; Inquiry communities in mathematics learning and teaching; Design of mathematics tasks and teaching units; Inquiry-based professional development activity; Modelling as an inquiry process; Evalutation of inquiry activity with students. The project has developed Inquiry Communities, in each of the partner groups, in which mathematicians and educators work together in supportive collegial ways to promote inquiry processes in mathematics learning and teaching. Through involving students in inquiry activities, PLATINUM aims to encourage students` own in-depth engagement with mathematics, so that they develop conceptual understandings which go beyond memorisation and the use of procedures. Indeed the eight partners together have formed an inquiry community, working together to achieve PLATINUM goals within the specific environments of their own institutions and cultures. Together we learn from what we are able to achieve with respect to both common goals and diverse environments, bringing a richness of experience and learning to this important area of education. Inquiry communities enable participants to address the tensions and issues that emerge in developmental processes and to recognise the critical nature of the developmental process. Through engaging in inquiry-based development, partners are enabled and motivated to design activities for their peers, and for newcomers to university teaching of mathematics, to encourage their participation in new forms of teaching, design of teaching, and activities for students. Such professional development design is an important outcome of PLATINUM. One important area of inquiry-based activity is that of “modelling” in mathematics. Partners have worked together across the project to investigate the nature of modelling activities and their use with students. Overall, the project evaluates its activity in these various parts to gain insights to the sucess of inquiry based teaching, learning and development as well as the issues and tensions that are faced in putting into practice its aims and goals

Proceedings of the Seventh Congress of the European Society for Research in Mathematics Education

International audienceThis volume contains the Proceedings of the Seventh Congress of the European Society for Research in Mathematics Education (ERME), which took place 9-13 February 2011, at Rzeszñw in Poland