Introduction: Finding common ground beyond fragmentation

This chapter begins with an outline of the European context within which the twenty six research papers presented in this book emerged. A particularly important aspect of this context is Network 27 on Didactics, Learning and Teaching of the European Educational Research Association (EERA) which formed the core of the research community in which this work was developed over a five year period (2006-11). The next part of the chapter provides an overview of the six sections which make up the structure of the book as a whole. A discussion then follows of the clear continental divide with respect to didactics, learning and teaching in the European landscape which is based on the references used by the contributors to this book. This leads to a consideration of the historical origin of present-day didactics which can be traced back to a common heritage in the work of Jan Amos Comenius (1592-1670) in order to provide a platform in the search for common ground. In the section which then follows there is a discussion of the didactic triad as a tool for holding the complexity of teaching-studying-learning situations and this is considered in an expanded context in which classroom interaction in the school is placed within a wider societal context. Based on a review of the contributions to this book, the final parts of this chapter consider existing knowledge gaps between different national traditions and also identify themes that form the basis for building and extending common ground. The themes that have been identified through this process of synthesis relate to pedagogical content knowledge, learner knowledge, joint didactical action, curriculum research, the so called shift from teaching to learning, the philosophy of Bildung and its practical implications, links between theory and practice and the significant role of experimental schools. Finally these themes are proposed for consideration within the wider research, policy and practice community as the basis for future international co-operation that offer the potential to advance mutual understanding and common insights in this fiel

Exploring Periodic Orbit Expansions and Renormalisation with the Quantum Triangular Billiard

A study of the quantum triangular billiard requires consideration of a boundary value problem for the Green's function of the Laplacian on a trianglar domain. Our main result is a reformulation of this problem in terms of coupled non--singular integral equations. A non--singular formulation, via Fredholm's theory, guarantees uniqueness and provides a mathematically firm foundation for both numerical and analytic studies. We compare and contrast our reformulation, based on the exact solution for the wedge, with the standard singular integral equations using numerical discretisation techniques. We consider in detail the (integrable) equilateral triangle and the Pythagorean 3-4-5 triangle. Our non--singular formulation produces results which are well behaved mathematically. In contrast, while resolving the eigenvalues very well, the standard approach displays various behaviours demonstrating the need for some sort of ``renormalisation''. The non-singular formulation provides a mathematically firm basis for the generation and analysis of periodic orbit expansions. We discuss their convergence paying particular emphasis to the computational effort required in comparision with Einstein--Brillouin--Keller quantisation and the standard discretisation, which is analogous to the method of Bogomolny. We also discuss the generalisation of our technique to smooth, chaotic billiards.Comment: 50 pages LaTeX2e. Uses graphicx, amsmath, amsfonts, psfrag and subfigure. 17 figures. To appear Annals of Physics, southern sprin

Using materials from the history of mathematics in discovery-based learning

This paper reports on attempt to integrate history of mathematics in discovery-based learning using technology. Theoretical grounding of the idea is discussed. An exploratory environment on triangle geometry is described. It is designed to support and motivate students' activities in learning through inquiry. Conjectures about properties of Lemoine point and Simson line are produced and proved by students using e-learning textbook

Common Visual Representations as a Source for Misconceptions of Preservice Teachers in a Geometry Connection Course

In this paper, we demonstrate how atypical visual representations of a triangle, square or a parallelogram may hinder students’ understanding of a median and altitude. We analyze responses and reasoning given by 16 preservice middle school teachers in a Geometry Connection class. Particularly, the data were garnered from three specific questions posed on a cumulative final exam, which focused on computing and comparing areas of parallelograms, and triangles represented by atypical images. We use the notions of concept image and concept definition as our theoretical framework for an analysis of the students’ responses. Our findings have implication on how typical images can impact students’ cognitive process and their concept image. We provide a number of suggestions that can foster conceptualization of the notions of median and altitude in a triangle that can be realized in an enacted lesson

The application of a pilot pull planning system to construction projects

A new planning system was introduced as a pilot within a large UK construction company. The system, which attempts to address some of the problems of construction “front end” planning, is investigated relative to Lean Construction and specifically the Last Planner system. The purpose is to see if it can be used as the basis for applying a lean planning model which the company intends to introduce and test through a research project. The existing system is seen to have strengths in terms of goodwill and commitment from the participants but is still fundamentally linked to the schedule pushed traditional approach to planning which is seen to be unsuccessful. An attempt to use “first run studies” to produce high quality planning and performance information was partly successful and indicated possibilities for future implementation. Further work is needed to fully develop the application model and training in the fundamentals of the system will be needed to improve performance

Cabri's role in the task of proving within the activity of building part of an axiomatic system

We want to show how we use the software Cabri, in a Geometry class for preservice mathematics teachers, in the process of building part of an axiomatic system of Euclidean Geometry. We will illustrate the type of tasks that engage students to discover the relationship between the steps of a geometric construction and the steps of a formal justification of the related geometric fact to understand the logical development of a proof; understand dependency relationships between properties; generate ideas that can be useful for a proof; produce conjectures that correspond to theorems of the system; and participate in the deductive organization of a set of statements obtained as solution to open-ended problems

Improving QED-Tutrix by Automating the Generation of Proofs

The idea of assisting teachers with technological tools is not new. Mathematics in general, and geometry in particular, provide interesting challenges when developing educative softwares, both in the education and computer science aspects. QED-Tutrix is an intelligent tutor for geometry offering an interface to help high school students in the resolution of demonstration problems. It focuses on specific goals: 1) to allow the student to freely explore the problem and its figure, 2) to accept proofs elements in any order, 3) to handle a variety of proofs, which can be customized by the teacher, and 4) to be able to help the student at any step of the resolution of the problem, if the need arises. The software is also independent from the intervention of the teacher. QED-Tutrix offers an interesting approach to geometry education, but is currently crippled by the lengthiness of the process of implementing new problems, a task that must still be done manually. Therefore, one of the main focuses of the QED-Tutrix' research team is to ease the implementation of new problems, by automating the tedious step of finding all possible proofs for a given problem. This automation must follow fundamental constraints in order to create problems compatible with QED-Tutrix: 1) readability of the proofs, 2) accessibility at a high school level, and 3) possibility for the teacher to modify the parameters defining the "acceptability" of a proof. We present in this paper the result of our preliminary exploration of possible avenues for this task. Automated theorem proving in geometry is a widely studied subject, and various provers exist. However, our constraints are quite specific and some adaptation would be required to use an existing prover. We have therefore implemented a prototype of automated prover to suit our needs. The future goal is to compare performances and usability in our specific use-case between the existing provers and our implementation.Comment: In Proceedings ThEdu'17, arXiv:1803.0072

Evaluating megaprojects: from the “iron triangle” to network mapping

Evaluation literature has paid relatively little attention to the specific needs of evaluating large, complex industrial and infrastructure projects, often called ‘megaprojects’. The abundant megaproject governance literature, in turn, has largely focused on the so-called ‘megaproject pathologies’, i.e. the chronic budget overruns, and failure of such projects to keep to timetables and deliver the expected social and economic benefits. This article draws on these two strands of literature, identifies shortcomings, and suggests potential pathways towards an improved evaluation of megaprojects. To counterbalance the current overemphasis on relatively narrowly defined accountability as the main function of megaproject evaluation, and the narrow definition of project success in megaproject evaluation, the article argues that conceptualizing megaprojects as dynamic and evolving networks would provide a useful basis for the design of an evaluation approach better able to promote learning and to address the socio economic aspects of megaprojects. A modified version of ‘network mapping’ is suggested as a possible framework for megaproject evaluation, with the exploration of the multiple accountability relationships as a central evaluation task, designed to reconcile learning and accountability as the central evaluation functions. The article highlights the role of evaluation as an ‘emergent’ property of spontaneous megaproject ‘governing’, and explores the challenges that this poses to the role of the evaluator