Papers of the BSRLM Geometry Working Group
Forsythe, S. & Jones, K. (2009), Tasks that support the
development of geometric reasoning at KS3, Proceedings of the British Society
for Research into Learning Mathematics, 29(3), 103-108.ISSN: 1463-6840
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pdf format.
Abstract: Students at Key Stage 3 (ie aged 11-14) in English schools are
expected to learn the definitions of the properties of triangles, quadrilaterals
and other polygons and to be able to use these definitions to solve problems
(including being able to explain and justify their solutions). This paper
focuses on a pair of Year 8 students (aged 12-13) working on a task using
dynamic geometry software. In the research, the children investigated triangles
and quadrilaterals by dragging two lines within a shape (ie the diagonals of a
quadrilateral, or base and height of a triangle) and noting the position and
orientation of the lines which gave rise to specific shapes. Following this, the
students were asked to use what they had found in order to construct specific
triangles and quadrilaterals when starting with a blank screen. While the
research is currently ongoing, and is using a design research methodology, the
evidence to date is that the task has the potential to scaffold students’
thinking around the properties of 2D shapes and hence support the development of
geometric reasoning.
Sinclair, N. & Jones, K. (2009), Geometrical reasoning in the
primary school, the case of parallel lines, Proceedings of the British
Society for Research into Learning Mathematics, 29(2), pp tbc. ISSN:
1463-6840
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format.
Abstract: During the primary school years, children are typically expected to
develop ways of explaining their mathematical reasoning. This paper reports on
ideas developed during an analysis of data from a project which involved young
children (aged 5-7 years old) in a whole-class situation using dynamic geometry
software (specifically Sketchpad). The focus is a classroom episode in which the
children try to decide whether two lines that they know continue (but cannot see
all of the continuation) will intersect, or not. The analysis illustrates how
the children can move from an empirical, visual description of spatial relations
to a more theoretical, abstract one. The arguments used by the children during
the lesson transcend empirical arguments, providing evidence of how young
children can be capable of engaging in aspects of deductive argumentation.
Jones, K., Lavicza, Z., Hohenwarter, M., Lu, A., Dawes, M.,
Parish, A. & Borcherds, M. (2009), Establishing a professional development
network to support teachers using dynamic mathematics software GeoGebra,
Proceedings of the British Society for Research into Learning Mathematics,
29(1), 97-102. ISSN: 1463-6840
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format.
Abstract: The embedding of technology into mathematics teaching is known to be a
complex process. GeoGebra, an open-source dynamic mathematics software
that incorporates geometry and algebra into a single package, is proving popular
with teachers - yet solely having access to such technology can be insufficient
for the successful integration of technology into teaching. This paper reports
on aspects of an NCETM-funded project that involved nine experienced teachers
collaborating in developing ways of providing professional development and
support for other teachers across England in the use of GeoGebra in
teaching mathematics. The participating teachers tried various approaches to
better integrate the use of GeoGebra into the mathematics curriculum
(especially in geometry) and they designed and led professional development
workshops for other teachers. As a result, the project initiated a core group
which has started to be a source of support and professional development for
other UK teachers of mathematics in the use of GeoGebra.
Jones, K., Fujita, T., Clarke, N. & Lu, Y.-W. (2008), Proof
and proving in current classroom materials, Proceedings of the British
Society for Research into Learning Mathematics, 28(3), 142-146. ISSN:
1463-6840
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format.
Abstract: Research across many countries reports that teaching the key ideas of
proof and proving to all students is not an easy task. This paper reports on the
session of the BSRLM Geometry Working Group which examined current classroom
material from the UK with the intention of uncovering the ‘opportunities for
proof’ in geometry that are provided by such material. To carry out such an
analysis three analytical frameworks are compared. Two of the analytical
frameworks, while placing proof and proving in a wider context of learners’
mathematics, may not fully uncover the detail of proof and proving. The third
analytical framework, while permitting a detailed analysis of explicit proof and
proving, may not fully account for textbooks that devote most space to
discussions of proof and proving and/or contain problems that implicitly provoke
proof. This comparison reveals some of the complexity of textbook analysis and
suggests that further work is needed on a suitable analytical framework.
Hohenwarter, M. and Jones, K. (2007), Ways of Linking Geometry
and Algebra: the case of GeoGebra, Proceedings of the British Society
for Research into Learning Mathematics, 27(3), 126-131. ISSN: 1463-6840
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format.
Abstract: This paper discusses ways of enhancing the teaching of mathematics
through enabling learners to gain stronger links between geometry and algebra.
The vehicle for this is consideration of the affordances of GeoGebra, a form of
freely-available open-source software that provides a versatile tool for
visualising mathematical ideas from elementary through to university level.
Following exemplification of teaching ideas using GeoGebra for secondary school
mathematics, the paper considers current emphases on geometry and algebra in the
school curriculum and the current (and potential) impact of technology (such as
GeoGebra). The paper concludes by raising the implications of technological
developments such as GeoGebra for the pre-service education and inservice
professional development of teachers of mathematics.
Jones, K., Fujita, T. & Ding, L. (2006), Informing the
Pedagogy for Geometry: learning from teaching approaches in China and Japan,
Proceedings of the British Society for Research into Learning Mathematics,
26(2), 109-114.
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here for full paper in pdf format.
Abstract: An authoritative report into the teaching and learning of geometry
argued, amongst other things, that the most significant contribution to
improvements in geometry teaching are to be made by the development of good
models of pedagogy, supported by carefully designed activities and resources.
This meeting of the Geometry Working Group provided an opportunity to consider
approaches to the teaching of geometry developed in China and Japan and to
review what research might have to contribute to developing new pedagogic
approaches.
Fujita, T. and Jones, K. (2002), Opportunities
for the Development of Geometrical Reasoning in Current Textbooks
in the UK and Japan, Proceedings of the British Society for
Research into Learning Mathematics, 22(3), 79-84.
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here for full paper in pdf format.
Abstract: Developing a good model of the school geometry
curriculum continues to be one of the most important tasks in
curricular design in mathematics. This paper reports on an
initial analysis of current best-selling textbooks used in lower
secondary schools in Japan and the UK (specifically England and
Scotland). The analysis indicates that, following the
specification of the mathematics curriculum in these countries,
Japanese textbooks set out to develop students' deductive
reasoning skills through the explicit teaching of proof in
geometry, whereas comparative UK textbooks tend, at this level,
to concentrate on finding angles, measurement, drawing, and so
on, coupled with a modicum of opportunities for conjecturing and
inductive reasoning. The available research suggests that each
approach has its own strengths and weaknesses. Finding ways of
capitalising on the strengths and mitigating the weaknesses could
prove helpful in formulating new curricular models and designing
new student textbooks.
Jones, K., Mooney, C. and Harries, T. (2002),
Trainee Primary Teachers' Knowledge of Geometry for Teaching. Proceedings
of the British Society for Research into Learning Mathematics,
22 (2), 95-100.
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here for full paper in pdf format.
Abstract: One outcome of the
implementation of the (UK) National Numeracy Strategy at the
primary school level is the privileging of the teaching and
learning of number. Yet, as the recent Royal Society report on
geometry stresses, it is important to begin the developing of
spatial thinking and reasoning at this level. This report reviews
what trainee primary teachers might need to know about geometry
in order to teach the geometry component of the mathematics
curriculum effectively and confidently. Some initial findings are
given from research which suggests that, in the UK, geometry is
the area of mathematics in which trainees perform most poorly in
initial baseline tests and have the least confidence to teach.
Hence it is the area in which trainees need to make most progress
if they are to gain qualified teacher status.
Jones, K. and Fujita, T. (2001), Developing a
New Pedagogy for Geometry. Proceedings of the British Society
for Research into Learning Mathematics, 21(3), 90-95.
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here for full paper in pdf format.
Abstract: Major improvements in the teaching and learning of
geometry will only come, argues a recent report from the Royal
Society and Joint Mathematical Council, through the development
of a completely new pedagogy for geometry. This report examines
existing models of pedagogy for geometry and considers what
research might have to contribute to the development of new
approaches. New pedagogic approaches for geometry need to give
greater emphasis to work in 3-D, incorporate the effective use of
computer technology, especially dynamic geometry software, and
focus on discursive methods of engagement and methods of
assessment so that the pressure on pupils is not solely to rote
learn.
Jones, K. and Rodd, M. M. (2001), Geometry and
Proof, Proceedings of the British Society for Research into
Learning Mathematics, 21(1), 95-100.
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here for full paper in pdf format.
Abstract: Is Euclidean geometry the most suitable part of the
school mathematics curriculum to act as a context for work on
mathematical proof? This paper examines some of the issues
regarding the teaching and learning of proof and proving
specifically in relation to Euclidean geometry. Evidence is
reviewed which suggests that giving mathematical explanation a
higher profile in the classroom should help teachers connect with
students' reasoning and guard against the students
experiencing learning to prove as no more than a ritual
determined by the teacher.
Jones K. (2000), Teacher Knowledge and
Professional Development in Geometry, Proceedings of the
British Society for Research into Learning Mathematics, 20(3),
109-114.
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full paper in pdf format.
Abstract: The successful teaching of geometry depends on
teachers knowing a good deal of geometry and how to teach it
effectively. This report provides a review of what is known about
teacher knowledge in geometry, how the knowledge develops and how
this knowledge development can be supported by professional
development .The available evidence suggests that attention could
usefully be paid both to the initial and continuing education of
teachers of mathematics in terms of their background and
understanding of geometry.
Clausen-May, T., Jones, K., McLean, A. and
Rollands, S. (2000), Perspectives on the Design of the Geometry
Curriculum, Proceedings of the British Society for Research
into Learning Mathematics, 20(1 & 2), 34-41.
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here for full article in pdf format.
Abstract: The question of how to construct an appropriate
geometry curriculum is a long-standing one. A recent estimate
suggests that there are more than 50 geometries. This creates a
fundamental problem in devising a geometry curriculum: there are
just too many interesting things to include so some decision has
to be made as to what to include and what to exclude. This report
features three perspectives on the issue of the design of the
school geometry curriculum.
Mogetta, C., Olivero, F. and Jones K. (1999),
Designing Dynamic Geometry Tasks that Support the Proving
Process. Proceedings of the British Society for Research into
Learning Mathematics,19(3), 97-102.
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here for the complete report in pdf format.
Abstract: A major challenge for mathematics education is to
find ways in which proof in geometry has communicatory,
exploratory, and explanatory functions alongside those of
justification and verification. Ongoing research is suggesting
that providing students with tasks which state "prove
that..." might actually inhibit students' capacity
for proving. In contrast, open tasks which favour a dynamic
exploration of a statement and encourage the use of
transformational reasoning may allow students to reconstruct, in
terms of properties and relationships, all the elements needed in
the proof. In this report we consider the transforming of closed
problem into open ones and discuss the use of dynamic geometry
software, such as Cabri, in such a process.
Mogetta, C., Olivero, F. and Jones K. (1999),
Providing the Motivation to Prove in a Dynamic Geometry
Environment. Proceedings of the British Society for Research
into Learning Mathematics,
19(2), 91-96.
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here for the complete report in pdf format.
Abstract: The use of dynamic geometry software may provide
opportunities to improve the teaching and learning of
mathematical proof within the context of plane geometry. Yet, it
seems, if the approach to proving continues to emphasise a
standardised linear deductive presentation, little improvement in
student conceptions may result. This paper considers the design
of geometrical tasks that could provide the motivation to prove.
Chronaki, A. (with Jones, K.) (1999), Language
Use and Geometry Texts.Proceedings of the British Society for
Research into Learning Mathematics, 19(1), 95-100.
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here for the complete report in pdf format.
Abstract: Recent research suggest that with classroom tasks
that combine spatial experiences, mathematising, and
communicating, pupils may reveal the nature of their own spatial
images and personal language in describing these spatial
contexts, and experience the use of formal terminology in making
accurate descriptions of their observations and constructions.
This report focuses on issues of language use involved in
geometry activities when particular emphasis is placed on
encouraging pupils' practice of informal and formal
mathematical vocabulary.
McLeay, H., O'Driscoll-Tole, K. and Jones K.
(1998), Using Imagery to Solve Spatial Problems. Proceedings
of the British Society for Research into Learning Mathematics, 18(3), 83-88.
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here for the complete report in pdf format.
Abstract: This report focuses on the use of imagery to solve a
range of spatial problems. The research projects reviewed in this
report offer some insight into the range of strategies used by
solvers of spatial problems and point to relationships between
spatial and verbal skills.
Jones, K. and Bills, C. (1998), Visualisation,
Imagery and the Development of Geometrical Reasoning. Proceedings
of the British Society for Research into Learning Mathematics,
18(1-2), 123-128.
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here for the complete report in pdf format.
Abstract: This report focuses on some aspects of the nature
and role of visualisation and imagery in the teaching and
learning of mathematics, particularly as a component in the
development of geometrical reasoning. Issues briefly addressed
include the relationship between imagery and perception, imagery
and memory, the nature of dynamic images, and the interaction
between imagery and concept development. The report concludes
with a series of questions that may provide a suitable programme
for research and lays the foundation for further work of the
BSRLM geometry working group.
Jones, K. (1998), Theoretical Frameworks for
the Learning of Geometrical Reasoning. Proceedings of the
British Society for Research into Learning Mathematics, 18(1-2), 29-34.
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here for the complete report in pdf format.
Abstract: With the growth in interest in geometrical ideas it
is important to be clear about the nature of geometrical
reasoning and how it develops. This paper provides an overview of
three theoretical frameworks for the learning of geometrical
reasoning: the van Hiele model of thinking in geometry,
Fischbein's theory of figural concepts, and Duval's
cognitive model of geometrical reasoning. Each of these
frameworks provides theoretical resources to support research
into the development of geometrical reasoning in students and
related aspects of visualisation and construction. This overview
concludes that much research about the deep process of the
development and the learning of visualisation and reasoning is
still needed.
Jones, K. (1997), A Comparison of the Teaching
of Geometrical Ideas in Japan and the USA. Proceedings of the
British Society for Research into Learning Mathematics, 17(3), 65-68.
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here for the complete report in pdf format.
Abstract: The release of a videotape of typical geometry
teaching in Japan and the US allows a comparison to be made of
the teaching methods typically employed. While the typical US
lesson emphasised skill acquisition, the typical Japanese lesson
focused on the solving of complex problems through pupil
exploration and presentation.
Ochepa, I., Tafewa, A., Winbourne, P., Hudson,
B. and Adhami, M. (1996), Motivating the Learning of Geometry: a
report of the geometry working group, Proceedings of the
British Society for Research into Learning Mathematics,
Sheffield, pp55-57.
This report looks at two cases of learning and
motivation for learning in geometry - the use of local artefacts
in learning (in this case, traditional mud hut design in Nigeria)
and the geometry of frieze patterns.
Jones, K. (1995), Geometrical Reasoning, Proceedings
of the British Society for Research into Learning Mathematics, 15(3), 43-47.
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the complete report in pdf format.
Abstract: The ICMI study conference on Perspectives on the
Teaching of Geometry for the 21st Century took place in Italy
in September 1995. This paper reports on the discussion of one of
the conference working groups which considered geometrical
reasoning. Four main themes are covered: visual reasoning,
geometrical reasoning in context, the meaning of proving in
learning geometry, and assessing the range of reasoning ability
in geometry. There was general agreement at the conference that
more research is necessary in order to effectively address the
wide range of issues that were discussed.
Jones, K. (1995), Contexts for Teaching
Geometry: a report of the geometry working group, Proceedings
of the British Society for Research into Learning Mathematics, 15(3),
41-42.
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here for the complete report in pdf format.
Abstract: This report of a meeting of the geometry working
group considers what it means to educate someone geometrically
and what are useful contexts in which to consider geometry. An
undergraduate unit on symmetry is described and this leads to the
discussion of context. The relationship between geometry and
algebra is briefly mentioned.
Contact the founder and co-ordinator of the
BSRLM Geometry Working Group:
Keith Jones, School of
Education, University of Southampton, Highfield, Southampton,
SO17 1BJ, UK
e-mail: dkj@soton.ac.uk
BSRLM Geometry Working Group homepage
For more publications by Keith Jones on the teaching and learning of geometry (with abstracts), click here.
Homepage: Keith Jones
University of Southampton Collaborative Group for Research in Mathematics Education: CRME
Page updated 02 February 2010