David Burghes

Control and optimal control theories with applications

Systems dynamics and differential equations Transfer functions and block diagrams State-space formulation Transient and steady state response analysis Stability Controllability and observability Multivariable feedback and pole location Introduction to optimal control Variational calculus Optimal control with unbounded continuous controls Bang-bang control Applications of optimal control Dynamic programming Appendices: Partial fractions Notes on determinants and matrices Solutions to problems.

Tiering at GCSE : is there a fairer system?

This paper outlines the difficulties with the current system of tiering in GCSE Mathematics exams, indicating both the problem of equality across tiers of the same grade and a particular problem in Mathematics, which with three tiers has a foundation tier on which it is not possible to gain a C grade.The authors outline a different system in which all candidates take basic papers on which all grades from G to C can be awarded and then an extension paper is available for the higher grades. This system has been tried out with a sample of pupils from a number of schools and a non-operational award was made. The authors describe the results of this and discuss potential difficulties with this system. Comparisons between the systems are made, showing that the established system of tiering is inherently unstable, while the proposed system, though not perfect, has much to commend it.

Entwicklung und Erprobung eines Tests zur „mathematischen Leistungsfähigkeit“ deutscher und englischer Lernender in der Sekundarstufe I

ZusammenfassungIm Rahmen von empirischen Untersuchungen zum Lehren und Lernen von Mathematik haben wir einen Test entwickelt (“Potentialtest”), der die “mathematische Leistungsfähigkeit” von 13/14jährigen Jugendlichen in England und Deutschland für Vergleichszwecke messen soll. Im vorliegenden Beitrag beschreiben wir die Entstehung des Tests sowie Resultate der Durchführung des Tests bei 1036 englischen und deutschen Lernenden. Die Resultate werden unter Berücksichtigung von — aus unseren früheren Fallstudien bekannten — Charakteristika des Mathematikunterrichts in beiden Ländern interpretiert.AbstractIn the context of empirical investigations into the learning and teaching of mathematics, we have developed a test (“Potential Test”) which is to measure the “mathematical potential” of 13/14-year-olds in England and Germany for the purpose of comparisons. In this paper we will describe the construction of the test as well as results of testing it with 1036 English and German pupils. The results will be interpreted with regard to characteristic features of mathematics teaching in both countries, known from our previous case studies.

The introduction of discrete mathematics into the school curriculum

Some of the changes in teaching mathematics in secondary schools in the UK are considered, with particular emphasis on the introduction of discrete mathematics. The development of resource material for younger pupils is described, together with a review of new options in discrete maths in A‐level courses. Some possible developments in this area of mathematics are also discussed, with the suggestion that it is the mathematics behind new technology that is more fundamental than the actual use of new technology, to help with teaching and learning of mathematics.

Response to Key Issues Raised in the Post-14 Mathematics Inquiry.

This article is a detailed response to the issues raised by the Post-14 Mathematics Inquiry in the UK. It aims to debate some of the central issues in mathematics teaching in the UK, including recruitment and retention of mathematics teachers, the curriculum content, national assessment, teaching resources (including ICT) and national strategies and policy (including inspection). Throughout, we have tried to base our recommendations on evidence and experience from the many teachers and tutors we work with as well as on our own experience. We have not hesitated to make what could be seen as controversial recommendations, but we believe a fundamental rethink of education policy and practice is needed if mathematics teaching and learning is to improve. We have also considered the impact that a proposed ‘National Centre for Excellence in Mathematics Education’ might have on the situation, although we doubt that it can have a marked long-term impact in the current UK situation.

The interface between mathematics and design and technology in secondary schools

ABSTRACT This article suggests that there are important connections between design and technology and mathematics curricula and that these are not fully made in the United Kingdom National Curriculum. Using three examples, the authors show how there are real opportunities in design and technology teaching for incorporating mathematical learning, but that all too often these are lost because they are not made explicit. Again, the authors argue that too much mathematics teaching is dominated by abstract investigation rather than practical problem‐solving. They conclude by suggesting that teaching should transcend subject boundaries and be concerned with real‐life situations.

New technology: an example of the potential to influence the mathematics curriculum

The mathematics behind the design and use of bar codes is outlined. The paper makes the point that while recent technological developmens are being used to enhance the way mathematics is taught, there is another equally important aspect of new technology. This is the mathematics needed to develop the technology which could influence future mathematics curricula in schools and colleges. This aspect is explained by looking closely at the mathematics used in bar code design.

Chapter 6 – Controllability and observability

In this chapter, we study the controllability and observability concepts. Controllability is concerned with whether one can design control input to steer the state to arbitrarily values. Observability is concerned with whether without knowing the initial state, one can determine the state of a system given the input and the output. The study of controllability and observability for linear systems essentially boils down to studying two linear maps: the reachability map L r which maps, for zero initial state, the control input u(·) to the final state x(t 1); and the observability map, L o which maps, for zero input, from the initial state x(t 0) to the output trajectory y(·). The range and the null spaces of these maps respectively are critical for their studies.

Interactive video for mathematics teaching

In this article, we outline the design of an interactive video (IV) disc currently being produced for mathematics teaching. It is centred on the simulation of running a school disco, and involves decision making with economical and financial implications. The paper also explores further possible uses for IV in mathematics teaching.

The use of discrete mathematics in the teaching of mathematics

In this article, we show how discrete mathematics could play a valuable role in encouraging an investigatory approach in mathematics teaching in schools. Six examples are given which show how practical problems can be readily turned into mathematical problems, which can be easily tackled by all pupils. In many cases, the best solution will not at first be found, but at least intuition and numerical skills will provide a solution. This contrasts with other mathematical investigations where it is difficult for many pupils to make any progress at all.