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Developing Geometrical Reasoning in the Secondary School
This project was supported by an award from the Qualifications and Curriculum Authority (as part of it's work on developing the curriculum) and was carried out by the Southampton/Hampshire Group of mathematicians and mathematics educators (see list of members below).
The aim of the project was to develop and trial some teaching/learning materials for use in schools that focus on the development of geometrical reasoning at the secondary school level.
The report on the project illustrates how it is possible to enable pupils from all ages and attainments within the lower secondary (Key Stage 3) curriculum to participate in mathematical reasoning, given appropriate tasks, teaching and classroom culture. The study suggests that it is appropriate for all teachers to aim to develop the geometrical reasoning of all pupils, but equally that this is a non-trivial task. Obstacles that need to be overcome are likely to include uncertainty about the nature of mathematical reasoning and about what is expected to be taught in this area among many teachers, lack of exemplars of good practice (although this report attempts to address this by providing a range of lesson descriptions), especially in using transformational arguments, lack of time and freedom in the curriculum to properly develop work in this area, an assessment system which does not recognise students oral powers of reasoning, and a lack of appreciation of the value of geometry as a vehicle for broadening the curriculum for high attainers, as well as developing reasoning and communication skills for all students.
The findings of the project were incorporated into the QCA report Developing Reasoning through Algebra and Geometry (see below for details).
Project Report
Brown, M., Jones, K. & Taylor, R. (2003), Developing Geometrical Reasoning
in the Secondary School: outcomes of trialling teaching activities in
classrooms, a report to the QCA. London: QCA. ISBN: 0854328092
Download complete report here.
Download Executive summary here.
Conference Papers and Workshops
Brown, M., Jones, K., Taylor, R. & Hirst, A, (2004), Developing Geometrical
Reasoning. In: Ian Putt, Rhonda Faragher & Mal McLean (Eds),
Proceedings of the 27th Annual Conference of the
Mathematics Education Research Group of Australasia (MERGA27), 27-30 June
2004, Townsville, Queensland, Australia. Vol 1, pp127-134. ISBN: 1920846042
Click here for
full paper in pdf format.
Workshop on "Developing Geometrical Reasoning"
10th International Congress on Mathematical Education,
Copenhagen, Denmark, 4-11 July 2004.
Jill Barton, Jo Lees & Peter Ransom
QCA publication
(incorporating findings from the
project report, together with those from several other projects examining the
development of mathematical reasoning in various contexts)
QCA (2004), Developing Reasoning through
Algebra and Geometry. London: QCA. ISBN 1 85838 550 4
Available for download here.
The Southampton/Hampshire Group
The members of the Southampton/Hampshire Group are:
Jill Barton, Cams Hill School, Fareham
Margaret Brown, King's College London/School Mathematics Project
(member of QCA advisory Group)
Ann Hirst, University of Southampton
Keith Hirst, University of Southampton (member of the group to
March, 2003)
Keith Jones, University of Southampton
Carol Knights, Applemore College, Dibden Purlieu, Southampton
Jo Lees, Crestwood Community School, Eastleigh
Paul Morris, Brune Park Community School, Gosport
Adrian Oldknow, Chichester University College (member of QCA
Advisory Group)
Peter Ransom, The Mountbatten School, Romsey
Ron Taylor, Mathematics Inspector for Hampshire (member of QCA Advisory Group)
Related CRME Projects
DALEST: Developing an Active Learning Environment for Stereometry
(3D geometry)
This project is supported by a grant from the EU, award:
224269-CP-1-2005-1-CY-MINERVA-M
CalGeo: teaching calculus using dynamic geometric tools
This project is supported by a grant from the EU, award:
11892-CP-1-2004-1-GR-COMENIUS-C21
Primary Trainee Teachers' Spatial Subject Knowledge and their Classroom
Performance
This project is part-supported by an award from the UK Teacher Training
Agency.
Cross-national
Study of the Generation of Geometrical Knowledge in Lower Secondary School
Mathematics
This project is being carried out in collaboration with the University of
Glasgow, and the Universities of Shizuoka and Kumamoto, Japan. The project is
supported by a University of Southampton School of Education Scholarship and ORS
award 2005037007
Developments in the Geometry Curriculum in the UK and Japan
This project was supported by a full-time research student with funding
from Japan.
The
Mediation of Learning in Dynamic Geometry Environments
This project was
part-supported by grant A94/16 from the University of Southampton
Research Fund.
Manipulatives in Teacher Education in Geometry
This project was supported by grant number 201535/93 from the Brazilian
Ministry of Education Conselho Nacional de Desenvolvimento Científico e
Tecnológico (CNPq)
Indicative bibliography
Atiyah, M. (2001), Mathematics in the 20th Century: Geometry versus Algebra, Mathematics Today, 37(2), 46-53.
Chicago School Mathematics Project staff, (1971), The CSMP Development in Geometry, Educational Studies in Mathematics, 3(3-4), 281-285.
Fielker, D. S. (1981). Removing the shackles of Euclid, Mathematics Teaching, 95, 16-20; 96, 24-28; 97, 321-327. [Reproduced as a booklet by ATM , 1983]
Healy L. and Hoyles C. (1998), Justifying and Proving in School Mathematics: Summary of the results from a survey of the proof conceptions of students in the UK. Research Report, Mathematical Sciences, Institute of Education, University of London.
Hiebert, J et al (1997), Making Sense: Teaching and learning Mathematics With Understanding. Portsmouth, NH: Heinemann.
Jones, K. (2000), Critical Issues in the Design of the Geometry Curriculum.
In: Bill Barton (Ed), Readings in Mathematics Education. Auckland:
University of Auckland, pp 75-91.
Click here for
full article in pdf format.
Küchemann, D. & Hoyles, C. (2002). The Quality of Students’ Reasons for the Steps in a Geometric Calculation, Proceedings of the British Society for Research into Learning Mathematics, 22(2), 43-48.
Royal Society (2001), Teaching and learning geometry 11-19. London: Royal Society.
van Hiele, P.M. (1986). Structure and Insight: a theory of mathematics education. Orlando FL: Academic Press.
Page updated 08 September 2006
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