MATHEMATICS LEARNING TRAJECTORIES: CLASS,CAPITAL AND CONFLICT

This is the story of Marie and Edward as they approach the time of transfer from the primary to the secondary school. They both consider themselves to be successful mathematicians and have shared common classroom experiences throughout their time in the primary school. However, as they approach this critical relocation point it becomes clear that their future mathematical careers are set on two distinct trajectories. This paper explores the impact of the class-formed family habitus on them as learners of mathematics, and describes how family transferred cultural capital propels one of them whilst notions of conflict threaten to restrict the progress of the other.     

RESEARCH IN MATHEMATICS EDUCATION: SOME ISSUES AND SOME EMERGING INFLUENCES

In this article, I introduce a typology of forms of algebraic thinking. In the first part, I argue that the form and generality of algebraic thinking are characterised by the mathematical problem at hand and the embodied and other semiotic resources that are mobilised to tackle the problem in analytic ways. My claim is based not only on semiotic considerations but also on new theories of cognition that stress the fundamental role of the context, the body and the senses in the way in which we come to know. In the second part, I present some concrete examples from a longitudinal classroom research study through which the typology of forms of algebraic thinking is illustrated.     

THE BIOGENETIC LAW AND ITS INFLUENCE ON THEORIES OF LEARNING MATHEMATICS

Reference to the history of mathematics has been widely used in discussions of the development of curricula, the problems students have in learning mathematics, and the development of concepts in the individual. This paper examines the background to the "Biogenetic Law" [1] which has influenced so much thinking in educational theory, and the use of the "principle of parallelism" where individual development is claimed to mirror the historical development of the subject matter. Interpretations of the history of mathematics which are used to justify these claims are examined, and the universality of the supposed concepts is challenged. Questions are raised about some of the fundamental tenets of Piagetian epistemology.     

MATHEMATICS TEACHERS' CHOICES OF EXAMPLES THAT POTENTIALLY SUPPORT OR IMPEDE LEARNING1

The main goal of the study reported in the paper is to highlight some features of teachers' choice of examples in the mathematics classroom, in reference to their potential contribution or impediment to student learning. Specific examples from classroom observations are presented and analyzed accordingly, pointing to the complexity of considerations required for desirable choices. The discussion addresses the potential implications of the findings to mathematics teacher education programmes.     

SOLVING DYNAMIC WILDLIFE RESOURCE OPTIMIZATION PROBLEMS USING REINFORCEMENT LEARNING

ABSTRACT. An important technical component of natural resource management, particularly in an adaptive management context, is optimization. This is used to select the most appropriate management strategy, given a model of the system and all relevant available information. For dynamic resource systems, dynamic programming has been the de facto standard for deriving optimal state‐specific management strategies. Though effective for small‐dimension problems, dynamic programming is incapable of providing solutions to larger problems, even with modern microcomputing technology. Reinforcement learning is an alternative, related procedure for deriving optimal management strategies, based on stochastic approximation. It is an iterative process that improves estimates of the value of state‐specific actions based in interactions with a system, or model thereof. Applications of reinforcement learning in the field of artificial intelligence have illustrated its ability to yield near‐optimal strategies for very complex model systems, highlighting the potential utility of this method for ecological and natural resource management problems, which tend to be of high dimension. I describe the concept of reinforcement learning and its approach of estimating optimal strategies by temporal difference learning. I then illustrate the application of this method using a simple, well‐known case study of Anderson [1975], and compare the reinforcement learning results with those of dynamic programming. Though a globally‐optimal strategy is not discovered, it performs very well relative to the dynamic programming strategy, based on simulated cumulative objective return. I suggest that reinforcement learning be applied to relatively complex problems where an approximate solution to a realistic model is preferable to an exact answer to an oversimplified model.     

EVIDENCE INFORMED POLICY AND PRACTICE: METHODOLOGICAL ISSUES INVOLVED IN CONDUCTING A SYSTEMATIC REVIEW IN MATHEMATICS EDUCATION

This paper focuses on a systematic review of studies on the impact of the pedagogy of the National Numeracy Strategy (England) on primary school pupils’ confidence and competence. Findings from the review are that gains in competence may be a result of the specification of the curriculum rather than changes in pedagogy, that interactive whole class teaching as envisioned has not been fully realised, and that low attaining pupils may be disadvantaged by the way in which whole class teaching is being interpreted. In conducting the review there were some tensions involved in judging the reliability, validity and generalisability of the findings of the included papers, and in judging the weight of evidence in terms of the review question. This paper will address these tensions, describe this particular review process and discuss the findings.     

AN INVESTIGATION INTO THE BALANCE OF PRESCRIPTION,EXPERIMENT AND PLAY WHEN LEARNING ABOUT THE PROPERTIES OF QUADRATIC FUNCTIONS WITH ICT

This paper will draw on research being developed within the Teaching and Learning strand of the ESRC InterActive Education: Learning in the Information Age project which is examining the ways in which new technologies can be used in educational settings to enhance learning. It will focus on the learning and understanding of quadratic functions using a graphical software package and includes a discussion of how the structuring of the activities influences the nature of the learning environment. The potential affordance that the software provides for experiment and play is highlighted in relation to the teacher's role in shaping the tasks to effectively make good use of this mode of learning.