Proceedings of the Day Conference held in Loughborough, May 1995
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Contents Vol 15 No 2
1 Cognitive acceleration through mathematics education: an analysis of the cognitive demands of the national curriculum and associated commercial schemes for secondary mathematics
Mundber Adhami, David C. Johnson and Michael Shaver
Centre for Educational Studies. King’s College. University of London
CAME aims to contribute to mathematics education by distinguishing between intervention (learning activity aimed at increasing intellectual development) and instrUction (utilising pupils’ present competence to process mathematical concepts and procedures). This is being realised by providing exemplary lessons i1l1nathematics embodying teaching and class-management skills which researclll1as shown to accelerate intellectual development. The need for such lessons is further supported through an analysis of the ‘thinking’ demands implied by the Sltatements of Attainment in the National Curriculum Matl1ematics (1991) and the nature of the questions/exercises included in some widely used commercial schemes. The analyses suggest that for many pupils the level of thinking implied, as defined in terms of Piagetian levels, provides little opportunity for tile development of ‘formal reasoning’ throughout the period of secondary schooling.
Janet Duffin and Adrian Simpson
School of Mathematics,. University of Hull
This paper builds on a previous one, presented at AMET/BSRLM. We have continued to work on the meaning of understanding since then, making use of the comments of others and looking further at the literature. We have recently come to see the core of our research as anempting to look at the leamer as if from the inside – as if we were the leamers themselves. However, as teachers and observers of the learning of others, we have been attempting to link our thinking about the internal processes of the learner with their possible, observable external manifestations. It is the nature of, and the distinction, between the internal and external characteristics together with an examination of them as consequences of our theory which we explore in this paper.
College of St. Mark and St. John Plymouth
This paper describes a discussion of, and participation in, an aspect of the on-going analysis of transcripts of conversations with year ten students. This is one of a number of devices intended to elicit validity for the interpretation of these conversations. One narrow aspect of the interpretation is identified: the mathematical content ‘as experienced by the students.’ Mathematics content is categorised into five domains of knowledge as articulated by HMI (DES 1985), facts skills, conceptual structures, general strategies and personal qualities. The discussion focuses upon definitions of these five domains provided by the author and their application to the transcript of a short conversation between two students working on enlargement and scale factors.
Una Hanley and Tansy Hardy
Manchester Metropolitan University
Our ITT courses are permeated by the notion of students articulating their experience and knowledge of teaching, their concerns and awareness in order for them to develop a reflective attitude. We have been working with students on using personal writing as a device to enable them to reflect upon and analyse their practice. Students have produced written reflections about course sessions, school experiences, their reading, their concerns throughout the year. We are concerned about the role we play in their progression in this aspect of the course: whatt strategies can we use to get students to consider and reconsider their writing? In what ways do students position themselves within their writing and how can we respond to enable them to be perceive other possibilities and ‘move them on’? We want to share our experiences to date, discuss what the indicators of students ‘reflecting’ might be and consider the implications of this for the ways we choose to respond to their writing.
5 Linking the visual and the symbolic: a microgenetic analysis of students’ evolving approaches to generalising problems
Lulu Healy and Celia Hoyles
Department of Mathematics, Statistics and Computing Institute of Education, University of London
In this paper, we describe our attempts to adopt a methodology consonant with a theory of learning based on constructivism and ‘making of connections’. The methodology is being developed as part of the research project Visualisation, Computers and Learning]. In this project we have been investigating student approaches to a sequence of algebraic problems presented with visual information. We are documenting the trajectory of visual and symbolic approaches, attempting to identify the form in which they occur, why they occur and if they are inter-connected. The sequence of activities includes work on the computer and we are exploring if and how interactions with the software connect with other approaches. To illustrate our methodology, we present data from two stwJents who worked through the problems in different IIUlthenu:Jtical settings.
6 Researching the Learning of Geometrical Concepts in tbe Secondary Classroom: problems and possibilities
University of Southampton
Researching the learning of geometrical concepts in the secondary classroom presents both problems and opportunities. The specification of the geometry curriculum, the need to concretise abstract geometrical objects for classroom activities, the role of the teacher and the need to reconsider geometrical notions from different viewpoints are all factors which affect the acquisition of geometrical concepts by pupils. These factors Cll1I provide problems for the researcher. Yet there are also significant opponunities both to influence policy decisions and to contribute to both theoretical and practical debates regardin, the teaching and learning of geometry.
Svlvia Johnson and Susan Elliott
Sheffield Hallam University
Academic support in Mathematics arose from a demand by students and staff for extra help. It has taken two distinct forms. An open access drop in Centre – Maths Help – has been open for 1.5 hours each day of the student term. In addition Susan Elliott has mounted a number of short courses in specific topics at the request of either students or staff. These have lasted anything from half a day to two hours a week for six weeks. This report summarises an evaluation of support activity during the calendar year 1994. It highlights the difficulty of providing targeted support to those most in need, points to a significant lack of confidence amongst even those students following highly mathematical courses, and emphasises the need within the University sector for skilled diagnostic teaching more commonly found in the school sector.
Christine Lawson and Clare Lee
Chipping Norton School and Oxford University
This is a report on a small research project carried out jointly by a Mathematics teacher (Clare) and an English teacher (Christine). We began this research because we were both interested in the role of language in the learning process. We wanted to explore how students use language in order to think through concepts, express and communicate their learning. We wanted to consider their use of language in both oral and written form. The format of pupils working through a mathematical problem gave us a vehicle for this exploration.
Mercedes A. McGowen. Phil DeMarais and Carole Bernett
William Rainey Harper College, Palatine, Illinois
This article reports the results of a longitudinal study which compared the perfor¬mance of college students taught introductory algebra using non-traditional materials to the performance of students taught using traditional materials and methods and their success in subsequent courses. The non-traditional materials focused on the concept of function, with skills taught in context and the use of graphing calculators integrated throughout the course.
Homerton College, Cambridge
This is a theoretical paper. in the quasi-empiricist tradition. It considers mathematical generalisation in the well-trodden context of inductive reasoning. Much of the vast literature on inductive reasoning belongs to the philosophy of science; by contrast, mathematics is supposed to be essentially deductive. Such a view flies in the face of mathematical heuristic. Questions about how we come to know things in mathematics and why we believe things for which our evidence is at best partial, are usefully examined within an inductive framework.
The Open University
In February 1994 the Open University launched a part-time distance taught PGCE course for prospective primary and secondary teachers. In order to support the development of IT competence every student was loaned an Apple Macintosh complete with a printer. Clarisworksl and a modem. In February J 995 they were sent software for the computer conferencing system. FirstClass This paper describes the way in which students have made use of this medium to increase the opportunity for communication with other students on the course. tutors and the course team. The focus in particular is on the experience of secondary mathematics students.
University of North London
I have been investigating what mathematics Primary B.Ed. students feel it is important for them to study to be competent teachers in the classroom, and Ihe thinking behind their choices. This survey was intended 10 explore the implications of this for the restructured six-subject B.Ed. degree.
Nene College, Northampton
I ask teachers how they know what children know and can do in mathematics. The complexities of interpreting my interview data parallel the complexities of interpreting children’s mathematics. In this paper I summarise some of the issues discussed in the seminar.
University of Bristol, School of Education
This group met for the second time at Loughborough with a focus of thinking through the methodology/methods issue. Anne Watson gave a presentation ‘I think I’ve got a methodology’ which was followed by a critical commentary from Leone Bunon.