Bernard R. Hodgson

On direct products of automaton decidable theories

Abstract This paper is concerned with connections between two different ways to prove decidability results: methods of automata theory and methods related to products of algebraic systems. Some basic automaton-theoretic conditions on relational structures are presented which give preservation theorems for decidability under certain direct products.

The political context

People have studied, learned and used mathematics for over four thousand years. Decisions on what is to be taught in schools, and how, are ultimately political, influenced by a number of factors including the experience of teachers, expectations of parents and employers, and the social context of debates about the curriculum. The ICMI study is posited on the experience of many mathematics teachers across the world that its history makes a difference: that having history of mathematics as a resource for the teacher is beneficial.

An arithmetical characterization of NP

Abstract Adleman and Manders ( Proc. 17th IEEE Symposium on Foundations of Computer Science, 1976 ) obtained a near characterization of NP using Diophantine predicates with quantifier bounds. We characterize NP as the class of sets definable by a polynomial predicate preceded by a string of quantifiers where each universal variable has a bound of the form p (| n |) and each existential variable has a bound of the form 2 q (| n |) for input n and polynomials p and q .

The Mathematical Education of School Teachers: Role and Responsibilities of University Mathematicians

The preparation of mathematics teachers for the primary and secondary school is a multifaceted task. It spans various periods of the teachers’ life, encompassing their experiences first as pupils, then as undergraduate students and finally as professionals learning from their own action and from in-service activities. Although relatively short, the formal part of this preparation is definitely crucial. Besides the need to include actual teaching practice as early as possible (i.e., working with pupils in real classrooms), three components of this preparation can be identified, which reflect the belonging and interests of the university educators responsible for the formal education of teachers: mathematics itself, didactics of mathematics and psychology of learning. The focus of this paper is on the first of these components. Mathematicians have a major and unique role to play in the education of teachers — they are neither the sole nor the main contributors to this complex process, but their participation is essential. Maybe this will be seen as a truism, at least in connection with the preparation of secondary school mathematics teacher. But I wish nonetheless to present here some comments about the context in which this role can and should be played. I also want to support the view that mathematicians should take part in the education of primary school teachers. I see such an involvement as important because of the perspective on mathematics itself mathematicians can bring to student teachers. Moreover, I believe this involvement can be a source of gratifying and stimulating mathematical moments for the mathematicians themselves. In the final part of this paper, I will briefly suggest a few examples of mathematical topics which, from my experience, nicely illustrates the richness of the mathematical content pertaining to student teachers, both of the primary and secondary level. But first I want to examine some aspects of the role and responsibilities of mathematicians in the preparation of schoolteachers, in particular from an historical perspective.

A normal form for arithmetical representation of N P-sets☆

It is shown that in the arithmetical characterization of the class N P previously given by the authors (Theoret. Comput. Sci. 21 (1982), 255–267), called EEBA form, all but one of the bounded universal quantifiers can be eliminated. This shows that EEBA sets do not form a hierarchy with respect to quantifier alternation. An application of the main result yields a transformation of the Polynomial Time Hierarchy of Meyer and Stockmeyer into the Diophantine Hierarchy of Adleman and Manders.

The roles and needs of mathematics teachers using IT

Teacher education stands as critical among the various issues pertaining to the integration of information technology into secondary mathematics education. Remarkable improvements have indeed taken place recently both with respect to the material aspects of IT (hardware, management, accessibility, maintenance) and the availability of high quality educational software. Successful use of the computer depends essentially on the quality of the teachers and of their pedagogical agenda. Renewed and diversified roles confront teachers in an IT educational environment. Pre-service and in-service teacher education must help all of them to modify their attitudes and develop the new competencies in mathematics, informatics and the didactics essential for them to fulfil their pedagogical mission.

Whither the Mathematics/Didactics Interconnection? Evolution and Challenges of a Kaleidoscopic Relationship as Seen from an ICMI Perspective

I wish in this lecture to reflect on the links between mathematics and didactics of mathematics, each being considered as a scientific discipline in its own right. Such a discussion extends quite naturally to the professional communities connected to these domains, mathematicians in the first instance and mathematics educators (didacticians) and teachers in the other. The framework I mainly use to support my reflections is that offered by the International Commission on Mathematical Instruction (ICMI), a body established more than a century ago and which has played, and still plays, a crucial role at the interface between mathematics and didactics of mathematics. I also stress the specificity and complementarity of the roles incumbent upon mathematicians and upon didacticians, and discuss possible ways of fostering their collaboration and making it more productive.

International Organizations in Mathematics Education

Although the history of internationalization in mathematics education goes back more than a century, the last few decades have witnessed a notable acceleration in the establishment of bodies aiming at grouping together members of the community. The purpose of this chapter is to survey international or multinational organizations created to support and enhance reflection and action about the teaching and learning of mathematics at various levels of education systems, worldwide or in some specific regions of the world. The oldest and best-known international organization in mathematics education is the International Commission on Mathematical Instruction, but there are many others established over the years, serving different purposes and covering various aspects of the field. Focussing on those connected to research in mathematics education, this chapter highlights the diversity thus encountered in connection with the aims of these organizations, their functioning, or the specific niche they occupy in the mathematics education landscape.

Artefacts and Tasks in the Mathematical Preparation of Teachers of Elementary Arithmetic from a Mathematician’s Perspective: A Commentary on Chapter 9

The main focus of this chapter is the mathematical preparation of primary school teachers in relation to the teaching of elementary arithmetic. Comments are offered, from the perspective of a mathematician involved in the education of pre-service teachers, on the importance and variety of artefacts (often of a cultural and historical nature) that can be used to support the learning of whole number arithmetic, as well as on the role played by mathematical tasks in fostering the ‘mathematical message’ that may be conveyed through the artefacts. I will discuss concrete examples taken from an arithmetic course created by my department specifically for prospective primary school teachers, intending so doing to illustrate a crucial observation about artefacts and tasks, namely, that they form an inseparable pair in the teaching and learning of mathematics. This chapter addresses the mathematical preparation of primary school teachers. The learning of arithmetic by actual primary school pupils is not an immediate aim of the work we do with our student teachers, but we claim that many of the artefacts and tasks discussed in our arithmetic course (and in this chapter) can be transferred to pupils – but of course with a necessary adaptation to young children new at such notions.

ICMI 1966–2016: A Double Insiders’ View of the Latest Half Century of the International Commission on Mathematical Instruction

This paper concentrates on the latest five decades of the International Commission on Mathematical Instruction. We had the privilege of occupying leading positions within ICMI for roughly half the period under consideration, which has provided us with a unique standpoint for identifying and reflecting on main trends and developments of the relationship between ICMI and mathematics education. The years 1966–2016 have seen marked trends and developments in mathematics teaching and learning around the world, at the same time as mathematics education as a scientific discipline came of age and matured. ICMI as an organisation has not only observed these developments but has also been a key player in charting and analysing them, as well as in fostering and facilitating (some of) them. We offer, here, observations, analyses and reflections on key issues in mathematics education as perceived by us as ICMI officers, and as influenced by ICMI.