Proceedings of the Day Conference held at University of Nottingham, March 1997 and University of Oxford, June 1997
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Contents
1 Making Connections Through Active Graphing
Janet Ainley, Elena Nardi and David Pratt
Mathematics Education Research Centre, University of Warwick
Active graphing has been proposed as a spreadsheet-based pedagogic approach to support young children’s construction of meaning for graphs, particularly as a tool for interpreting experiments. This report discusses the initial planning and early data analysis arising in a detailed study of the active graphing approach. In this study, we hope to gain a deeper understanding of the children’s actions and the associated mental processes observed during active graphing.
2 ‘Learning Mathematics is Like … ‘View of Tutors and Students Beginning a Distance-Taught Undergraduate Course
Barbara Allen and Christine Shiu
Centre for Mathematics Education, The Open University
Long running distance taught courses often create a community of experience and practice among those who have participated in their presentation. As the au launches new undergraduate courses in entry-level mathematics we ask what factors shape and influence that practice? This report gives a snapshot of work-in-progress on these questions, as offered and discussed at our BSRLM session.
3 Working With 3-D Shapes – Poleidoblocs
Julia Anghileri and Sarah Baron
Homerton College, Cambridge
With the current emphasis on literacy and numeracy in the primary school there needs to be concern that the mathematics curriculum does not become distorted to the extent that fundamental ideas other than number become neglected Using Poleidoblocs, the coloured building bricks that have been consistently among the resources in many classrooms since their introduction in the 1950s, we have looked at the development of ideas that relate their use to the requirements of the National Curriculum beyond number. Characterisingfree play and analysing children’s responses to particular tasks has enabled us to jdentify developments in understanding of relationships among shapes and some ways they may be used.
4 Two Types Of Mental Arithmetic And The Empty Numberline
Meindert Beishuizen
Dept of Education, Leiden University, The Netherlands
The empty numberline (ENL) was introduced as a new model in Realistic Mathematics Education (RME), after discussions in the Netherlands how best to improve the basic skills up to 100. Studies like our first National Evaluation Test (1987) pointed to a possible imbalance between two types of mental arithmetic: much emphasis on mental strategies may diminish practice in mental recall of basic number facts. In 1990 the Freudenthal Institute (Utrecht University) set out the publication of a more balanced view, inc01]Jorating Leiden research into addition and subtraction up to 100. This background of the empty numberline might be relevant to British discussions today about (mental) maths teaching. A summary is given of the outline and the research outcomes of the experimental ENL-program in Dutch 2nd grades/British Year3. Apartfrom the positive cognitive results the ENL-model also stimulated pupils’ own recordings of mental steps. Effects of a short experiment in a British Year3/4 class are briefly mentioned.
5 Interpretations of a Classroom Vignette or What Does Reading About Someone Else’s Theories-in-Action Do For You?
Laurinda Brown and Alf Coles
Graduate School of Education, University of Bristol
This paper briefly tells the story, through four critical stages, of the developing complexity of our theories-in-action (SchOn, 1991) as teacher-researchers over a period of 18 months. These theories-in-action are related to the ways in which teacher and student purposes (Brown and Coles, 1996) act as organisingfoci through which intuitive ways of knowing (Bruner 1974, Fischbein 1982, Gattegno 1987) are accessed. The parallels between our learning, as teacher-educator and teacher, and the learning of our students are marked. We share this journey to illustrate a way of working which we value for our own learning but ask the question ‘what is it that the readers of such research accounts learn?
6 A Conjuring Atmosphere In 11-16 Mathematics Classrooms: Factors Affecting Its Creation And Maintenance
Steve Byatt
Ludlow Church of England School
From a standpoint that school mathematics lessons should allow children to function as mathematicians, it is argued that crucial processes are “generalising” and “stressing and ignoring” as participants search for “seeing the general in the particular”. 1 This, and a broadly social-constructivist perspective, provided the writer with an image of the classroom within which conjecturing and discourse are not only routine, but essential. My early focus was to explore the dynamics of this social arena. Interviews with 40 or so children suggested that peer pressure was hugely influential. Many of the children interviewed reported that much of what they did and said was based upon the reaction, both actual and predicted, of other children. Other areas to emerge lfrom these interviews were: teacher behaviour; classroom discipline. Almost three years on, I am currently in the process of re-examining the image of the “conjecturing classroom” that I began with.
7 Values in Mathematics Education: What is Planned And What Is Espoused?
Lim Chap Sam and Paul Ernest
University of Exeter
This paper seeks to explore values that are explicitly and implicitly documented in the Malaysian school mathematics curriculum and to compare them with mathematics teachers’ perceptions of what values are appropriate to be taught through mathematics. The study finds that what is planned only partially matches what is espoused by the teachers in the sample.
8 Cypriot Children’s Interpretation On a Piagetian Task On Volume
Anastasia Evangelidou
Shell Centre, University of Nottingham
The present study investigates the ideas of Cypriot Primary School children, of age ten to twelve years, on Volume and Capacity. The research is still on going but up to now findings suggest that children’s answers are similar to those described by Piaget in his work on Volume. Here we present the answers of eight pupils who were involved in the study. Only three of the tasks used are described. In these particular tasks some of the children’s answers were very similar to those used by Piaget to demonstrate the absence of conservationof ‘volume occupied’.
9 Polysemous Estimation Words in the Classroom: Comprehension And Task Performance
Michael A. Forrester and Christopher Pike
Dept of Psychology, University of Kent
Linguistic aspects of mathematics education are viewed as problematic for young children acquiring mathematical skills. This study examined the relationship between the comprehension of estimating words and phrases (e.g.roughly, between, guess) and children’s estimations abilities. Sixty-four children ( aged 6-7; 8-9; 10-11) completed an estimation word comprehension test before undertaking estimating tasks in mathematical and non-mathematical contexts. The results indicated that children across this age range find estimation words easier to understand when accompanied with number or measurement expressions. Comprehension of specific estimation words varied considerably and was significantly associated with only two of twenty-four estimation tasks. These findings highlight the need for caution when interpreting the relationship between mathematics discourse and mathematical ability itself.
10 Who Is Being Sensible About Calculators
Derek Poxman, Consultant, and Jane Duffin
University of Hull
This paper has two parts each written by one of the authors. One of us (DF) presents a shortened version of a Review of International Research on the Availability and Use of Calculators in Schools 5-14 (hereinafter called the Review), and the other (JD) presents her experiences of the Calculator Aware Number Curriculum (CAN) with primary children and with innumerate undergraduate students at the Hull University Numeracy Centre. The Review was one of two projects on calculators funded by SCAA and undertaken preparatory to the production of a discussion document to be published by SCAA in March, 1997.
11 Tales Of Power: Foucault in the Mathematics Classroom
Tansy Hardy
University of Nottingham
Power and how teachers talk about it has been a recent interest for me. In particular I have been looking at the power relation between mathematics teachers and descriptions of mathematics teaching and the school mathematics curriculum. My intention is to develop an account of how that power relation works. I have found that the theories of Michel Foucault on the operation of power within groups and institutions provide me with tools with which I can develop such an account. I have used those ‘tools’ to give me a way of looking at teaching and learning interactions, teachers’ talk about how they plan their work and how they view their practice. I have used these tools as a way of highlighting what might have been previously unexamined areas for me.
12 Researching Geometric Thinking In Out Of School Contexts
Zlatan Magajna
University of Leeds and University of Ljubljana
We consider the use of school maths in out-oi-school context. First, the notion of a context of an action is clarified and distinguished from some other notions of context. We apply Saxe’s four parameter model to describe the context of an action and argue that the accomplishment of the mathematical goals that emerge during an activity is a necessary condition for the use of school maths in out-of school¬settings. We wonder if this condition is also sufficient and claim that a CAD setting may be a suitable context for researching this question.
13 Interactions Between Visualisation And Symbolic Senses
Ian Malabar and Dave Pountney
School of Computing and Mathematical Sciences, Liverpool John Moores University
Research at Liverpool John Moores University into effective use of computersff.T. in mathematics teaching has included the influence that enhanced computer visualisation techniques has on pupils’ learning, and in particular how this visualisation interacts with symbolic representation and manipulation skills to the benefit of the latter, if at all. A research experiment which attempts to quantify the influence of visualisation in the learning about graphs and functions is outlined. This involves the design of an interactive package which links graphical and symbolic representations in such a way that it reinforces students’ conceptual understanding of functions, and at the same time enhances their ability to visualise.
14 Coping With The Requirements For Rigour: The Novelty Of University Mathematics
Elena Nardi
University of Warwick
Given that the first-year mathematics undergraduates enter the university with A -level experiences that do not include any formal introduction to proving, the encounter with their course’s requirements for rigour is deeply problematic. Based on the findings from a doctoral research project on the novice mathematician’s learning difficulties in the encounter with mathematical abstraction, this paper samples a number of these difficulties: extracts of conversations between Oxford tutors and their students are presented and discussed from the point of view of highlighting the difficulty of switching from informal (school) to formal (university) mathematical thinking.
15 Aspects Of Practising In School Mathematics
Adrian J. Pinel
Chichester Institute of Higher Education
This preliminary paper briefly outlines an analysis of a range of purposes of practising mathematics wIthin a school context, defining three main objectives of practising. A more detailed analysis and discussion of the research into these objectives and the author’s categorisation of practising will form the basis of a paper to be submitted to BSRLM later in 1997. The current paper reviews the historical context, as waves of educational thinking and practice have spread nationally over the past forty years. Within this review, such key aspects as ‘practical activity’, ‘pace ” ‘investigation’ and ‘schemes’ are briefly discu.’lsed. In the second part, some illustrative indications are given of the problems of engendering change, and of the kind of assumption held by the author. The notions of ‘productive practising’ as developed in Germany and of ‘green practising’ as developed by the author are introduced and illustrated, then related briefly to the author’s categorisation of practising.
16 What Can Semiotics Offer Mathematics Education?
Adam Vile
South Bank University
There is developing interest in semiotics as a theoretical perspective in mathematics education. This paper examines semiotics from both a theoretical and practical perspective in order to begin discussion as to how and why semiotics may be a useful perspective for mathematics educators to adopt, or at least consider. A connection is drawn between the work of Vygotsky and Peirce and theoretical development provides the possibility of a consistent philosophical position that transcends Cartesian dualism and offers a new way of seeing. Then semiotics is examined practically from the point of view qualitative methodology. Finally suggestions are made about the role of theory in mathematics education and the possible role of semiotics as one of those theories.
17 Investigating Children’s Intuitive Understanding Of Number Operations By Formalising Their Mental Strategies
David Womack
University of Manchester
Research has shown that young children’s intuitive view of addition is non-commutative. This paper describes a 14 week study of a small group of 5 and 6 year old children inventing and using their own symbols to ask and answer questions about numbers in a ‘tranformationally rich subculture. In a ‘stepping stone’ scenario, symbols referred to changes (transformations) of position rather than aggregations of objects. No reference was made to traditional mathematical terms such as addition, subtraction, equals etc but the aim was to develop a pedagogy whereby the children’s intuitive system could eventually be ‘grafted’ onto the conventional system of arithmetic.
18 Working Group on Social Research in Mathematics Education
Convenors: Peter Gates and Tony Cotton
Nottingham University
This was the third meeting of the group and it had been decided at the previous meeting in London to focus part of the discussion on the paper presented to BERA ~ Jo Boaler titled “Setting, Social Class and the Survival of the Quickest”l. In fact this paper provided the stimulus for a discussion that took all of the session. Attention focussed particularly on several extracts from the paper. There are collected in an appendix to this report.
19 Semiotics And Mathematics Education Working Group
Paul Ernest, Exeter University
Adam Vile, South Bank University
This meeting consisted of three short presentations by members of the working group illustrating, through various perspectives, ways in which the semiotic lens can be applied to mathematics teaching and learning. The aim of the session was to explore a variety of semiotic points of view and to look for commonalties and differences in theoretical perspectives and methodologies with the intention of constructing some shared understanding of the various semiotic terminology and registers in the context of mathematics education research. Brief reports of the sessions are included below.