The Role of Heuristic Proof in Mathematics Teaching

With the great emphasis now being placed on the importance of ‘rigour’ in new mathematics programmes, many educators have been led to disparage intuition as the vitally important tool that it is in developing mathematical insights. Increasingly one sees evidence, even in technical schools, of pupils actually being discouraged from arriving at mathematical perceptions through unorthodox (and uncontrollable!) channels of analogy involving considerable divergent thinking or through consideration of physical models with which they are familiar.

As a mathematician, this pre‐occupation with ‘purism’ greatly disturbs the author. Mathematicians do not create through the formal apparatus ‐‐ they only apply formalism after ‘guessing’ results intuitionally. We have become overconcerned with the way the package is wrapped and less concerned with what is in it. Especially, the role of heuristic argument is widely misunderstood and misused in schools and colleges. The author hopes that this article will help to rectify this sorry state of affairs.     

Modelling of glioblastoma growth by linking a molecular interaction network with an agent-based model

In this work, a mathematical model of malignant brain tumour growth is presented. In particular, the growth of glioblastoma is investigated on the intracellular and intercellular scale.

The Go or Grow principle of tumour cells states that tumour cells either migrate or proliferate. For glioblastoma, microRNA-451 has been shown to be an energy dependent key regulator of the LKB1 (liver kinase B1) and AMPK (AMP-activated protein kinase) pathway that influences the signalling for migration or cell division.

We introduce a mathematical model that reproduces these biological processes. The intracellular molecular interaction network is represented by a system of nine ordinary differential equations. This is put into a multiscale context by applying an agent-based approach: each cell is equipped with this interaction network and additional rules to determine its new phenotype as either migrating, proliferating or quiescent.

The evaluation of the proposed model by comparison of the results with in vitro experiments indicates its validity.     

Comparing stage-scenario with nodal formulation for multistage stochastic problems

To solve real life problems under uncertainty in Economics, Finance, Energy, Transportation and Logistics, the use of stochastic optimization is widely accepted and appreciated. However, the nature of stochastic programming leads to a conflict between adaptability to reality and tractability. To formulate a multistage stochastic model, two types of formulations are typically adopted: the so-called stage-scenario formulation named also formulation with explicit non-anticipativity constraints and the so-called nodal formulation named also formulation with implicit non-anticipativity constraints. Both of them have advantages and disadvantages. This work aims at helping the scholars and practitioners to understand the two types of notation and, in particular, to reformulate with the nodal formulation a model that was originally defined with the stage-scenario formulation presenting this implementation in the algebraic language GAMS. In addition, this work presents an empirical analysis applying the two formulations both without any further decomposition to perform a fair comparison. In this way, we show that the difficulties to implement the model with the nodal formulation are somehow reworded making the problem tractable without any decomposition algorithm. Still, we remark that in some other applications the stage-scenario formulation could be more helpful to understand the structure of the problem since it allows to relax the non-anticipativity constraints.

     

Variations on a Result of Bressoud

The well-known Rogers-Ramanujan identities have been a rich source of mathematical study over the last fifty years. In particular, Gordon’s generalization in the early 1960s led to additional work by Andrews and Bressoud in subsequent years. Unfortunately, these results lacked a certain amount of uniformity in terms of combinatorial interpretation. In this work, we provide a single combinatorial interpretation of the series sides of these generating function results by using the concept of cluster parities. This unifies the aforementioned results of Andrews and Bressoud and also allows for a strikingly broader family of q-series results to be obtained. We close the paper by proving congruences for a “degenerate case” of Bressoud’s theorem.     

Number concepts with Number Worlds : thickening understandings

This study focuses on the nature of preservice elementary school teachers’ understandings of several concepts in elementary number theory that are evoked by a computer-based microworld called Number Worlds. In particular, the focus is on the concepts of factor, multiple and prime number. The notion of ‘thickness’ is examined with respect to understanding mathematical concepts and alludes to various attributes of Number Worlds that may be responsible for provoking different layers and depths of understanding, termed ‘thick understanding’.     

Local Spectral Properties of Typical Contractions on ℓp-Spaces

We study some local spectral properties of contraction operators on ?_p, 1 < p < ∞ from a Baire category point of view, with respect to the Strong* Operator Topology. In particular, we show that a typical contraction on ?_p has Dunford’s Property (C) but neither Bishop’s Property (β) nor the Decomposition Property (δ), and is completely indecomposable. We also obtain some results regarding the asymptotic behavior of orbits of typical contractions on ?_p.

     

A tale of mathematical myth-making: E T Bell and the ‘arithmetization of algebra’

Throughout E T Bell’s writings on mathematics, both those aimed at other mathematicians and those for a popular audience, we find him endeavouring to promote abstract algebra generally, and the postulational method in particular. Bell evidently felt that the adoption of the latter approach to algebra (a process that he termed the ‘arithmetization of algebra’) would lend the subject something akin to the level of rigour that analysis had achieved in the nineteenth century. However, despite promoting this point of view, it is not so much in evidence in Bell’s own mathematical work. I offer an explanation for this apparent contradiction in terms of Bell’s infamous penchant for mathematical ‘myth-making’.     

“Can you help me count these pennies?”: Surfacing preschoolers’ understandings of counting

ABSTRACT

Capturing the breadth and variety of children’s understanding is critical if studies of children’s mathematical thinking are to inform policy and practice in early childhood education. This article presents an investigation of young children’s counting. Detailed coding and analyses of assessment interviews with 476 preschoolers revealed understandings that would be overlooked by solely assessing the accuracy of their responses. In particular, many children demonstrated understandings of counting principles on a challenging task that were not captured by other, simpler tasks. We conclude that common approaches to capturing young children’s mathematical understanding are likely underestimating their capabilities. This study contributes to researchers’ understanding of what making sense of counting looks and sounds like for preschool age children (3–5 years), the development and relations among counting principles (one-to-one, cardinal, and patterns of the number sequence), and the affordances of challenging, open-ended tasks. We close by considering the implications of recognizing and building from what children know and can do for researchers, practitioners, and policymakers.     

Microteaching: some research studies and research questions

An attempt has been made to present an overview of the current research work done in the area of ‘Microteaching’. It also incorporates a list of current research resources for the use of research workers in this area.

The study has led to some pertinent research questions which need further experimentation and investigation in the field of microteaching for their resolution.     

Insights into Dedekind and Weber’s edition of Riemann’s <Emphasis Type="Italic">Gesammelte Werke</Emphasis>

In this paper, I present and analyse Dedekind’s and Weber’s editorial work which led to the edition of Riemann’s Gesammelte Werke in 1876. With several examples, I suggest that this editorial work is to be understood as a mathematical activity in and of itself and provide evidence for it.     

Kaplansky’s “infinite abelian groups” and its two test problems

Abstract

Kaplansky’s Infinite Abelian Groups, originally published in 1954, has recently been republished. In recognition of its continuing significance after 65?years, the authors illustrate some important ideas and techniques described in the book. In particular, a class of groups is defined for which there are affirmative answers to the author’s two famous “Test Problems.”     

An Attempt to Provide Appropriate Mathematical Knowledge for Students of Economics

Mathematics courses for students of the social sciences have a vital role to play in providing relevant mathematical backgrounds for increasingly mathematical subjects.

The course described in this article has been designed for those first year degree students reading economics at University College, London, who have not passed G.C.E. A level? ? General Certificate of Education at Advanced level. mathematics. The aims and structure of the course are described, indicating those features which are thought to be of particular importance. The author reports her impression of the impact of the course on one group of students. The students’ own views about the course have also been collected and are summarized here, together with their more general views about mathematical education. Much of the analysis is thought to be relevant to all those involved in the mathematical education of social scientists.

     

Optimality in Social Choice

Marengo and Settepanella (2010) have developed a geometric model of social choice when it takes place among bundles of interdependent elements, showing that by bundling and unbundling the same set of constituent elements an authority has the power of determining the social outcome. In this article, we will tie the model above to tournament theory, solving some of the mathematical problems arising in their work and opening new questions which are interesting from both a mathematical and social choice point of view. In particular, we will introduce the notion of u-local optima and study it from both a theoretical and a numerically probabilistic point of view; we will also describe an algorithm that computes the universal basin of attraction of a social outcome in O(M ³log M) time (where M is the number of social outcomes).     

Normalising geometrical figures: dynamic manipulation and construction of meanings for ratio and proportion

Enlarging-shrinking geometrical figures by 13 year-olds is studied during the implementation of proportional geometric tasks in the classroom. Students worked in groups of two using ‘Turtleworlds’, a piece of geometrical construction software which combines symbolic notation, through a programming language, with dynamic manipulation of geometrical objects by dragging on sliders representing variable values. In this paper we study the students’ normalising activity, as they use this kind of dynamic manipulation to modify ‘buggy’ geometrical figures while developing meanings for ratio and proportion. We describe students’ normative actions in terms of four distinct Dynamic Manipulation Schemes (Reconnaissance, Correlation, Testing, Verification). We discuss the potential of dragging for mathematical insight in this particular computational environment, as well as the purposeful nature of the task which sets up possibilities for students to appreciate the utility of proportional relationships.     

New opportunities for encouraging higher level mathematical learning by creative use of emerging computer aided assessment

This article defines ‘higher level mathematical skills’ and details an important class: that of constructing instances of mathematical objects satisfying certain properties. Comment is made on the frequency of higher level tasks in undergraduate work. We explain how such questions may be assessed in practice without the imposition on staff of an onerous marking load. Included are examples which have been implemented on a free computer aided assessment system. Lastly we report an investigation of students’ reactions to these questions and discuss their design and impact.     

Studying advanced mathematics in England: findings from a survey of student choices and attitudes

The UK Government has set a goal that the “vast majority” of students in England will be studying mathematics to the age of 18 by the end of the decade. The policy levers for achieving this goal include new Core Maths qualifications, designed for over 200,000 students who have achieved good grades at the age of 16 but then opt out of advanced or A level mathematics. This article reports findings from a cluster-sampled survey of over 10,000 17-year-olds in England in 2015. Participants’ views on post-16 mathematics are presented and discussed. The main finding is that they are strongly opposed to the idea of compulsory mathematical study, but are less antithetical to being encouraged to study mathematics beyond 16. We consider how attitudes vary by gender, prior attainment, study patterns and future aspirations. The article considers the implications of these findings in the current policy landscape.     

The life and work of Sophie Germain

We discuss the life of Sophie Germain, her struggle to educate herself and to win acceptance within the French mathematical community, and her contributions to number theory and to the theory of elasticity.

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Received: 23 January 2003, Accepted: 25 March 2003, Corespondence to: Gareth Jones     

Nouvelles extensions des équations intégro-différentielles non linéaires de Volterra

Summary In the framework of the authors' research papers devoted to studies concerningmathematical systems with mixed structures and in particular to studies on nonlinear integro-differential equations of Volterra and Picone types [5], [14]–[15], an initial value problem concerning a new extension of Volterra's integro-differential equations is considered and the existence, the unicity and the stability of its solution is proved.

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A M. le Professeur Wolfgang Gröbner pour son 75-e anniversaire

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The research reported in this paper was supported in part by the National Research Council of Canada through the University of Alberta by Grant NRC-A4345.