Below is a list of papers by Keith Jones on various topics to do with the
teaching and learning of geometry. Where possible, these are provided as
full-text (usually in pdf format) - otherwise, copies of most of the papers are
available from:
Keith Jones, School of
Education, University of Southampton, Highfield, Southampton, SO17 1BJ, UK
e-mail: d.k.jones@soton.ac.uk
For a full list of all Keith Jones' papers (without abstracts), please click here.
Jones, K. Mackrell, K. & Stevenson, I. (2010), Designing digital technologies
and learning activities for different geometries. In Celia Hoyles and Jean-Baptiste
Lagrange (Eds), Mathematics Education and Technology: Rethinking the Terrain
(ICMI Study 17). New York: Springer. [Chapter
4, pp47-60] ISBN: 9781441901453
Click here for
full chapter in pdf format.
Abstract: This chapter focuses on digital technologies and geometry education, a
combination of topics that provides a suitable avenue for analysing closely the
issues and challenges involved in designing digital technologies for learning
mathematics, while, at the same time, recognising that the use of such
technologies can and does shape the mathematical activity of the user. In
revealing these issues and challenges, the chapter examines the design of
digital technologies and appropriate forms of learning activities for a range of
geometries, including Euclidian 3D and co-ordinate geometry and non-Euclidean
geometries such as spherical, hyperbolic and fractal geometry. This analysis
reveals the decisions that designers take when designing for different
geometries on the flat computer screen. Such decisions are not only about the
geometry but also about the learner in terms of supporting their perceptions of
what are the key features of geometry.
Jones, K. (2010), Linking geometry and algebra in the school mathematics
curriculum. In Z. Usiskin, K. Andersen & N. Zotto (Eds) Future Curricular
Trends in School Algebra and Geometry. Charlotte, NC: Infoage. [pp203-215]
ISBN: 9781607524724
Click here for
full chapter in pdf format.
Abstract: This chapter focuses on the linking of geometry and algebra in the
teaching and learning of mathematics - and how, through such linking, the
mathematics curriculum might be strengthened. Through reviewing the case of the
school mathematics curriculum in England, together with examples of how the
power of geometry can bring contemporary mathematics to life in the classroom,
the chapter argues for greater concinnity in the mathematics curriculum,
especially in terms of the harmonious/purposeful reinforcement of mathematical
thinking through the linking of geometry and algebra.
Ding, L. & Jones, K. (2009), Instructional strategies in
explicating the discovery function of proof for lower secondary school students.
In: Fou-Lai Lin, Feng-Jui Hsieh, Gila Hanna & Michael de Villiers (Eds),
Proceedings of the ICMI study 19 conference: proof and proving in mathematics
education. Taipei, Taiwan: National Taiwan Normal University. Vol 1,
136-141. ISBN: 9789860182101
Click here for
full article in pdf format.
Abstract: In this paper, we report on the analysis of teaching episodes
selected from our pedagogical and cognitive research on geometry teaching that
illustrate how carefully-chosen instructional strategies can guide Grade 8
students to see and appreciate the discovery function of proof in geometry.
Fujita, T., Jones, K. & Kunimune, S. (2009), The design of
textbooks and their influence on students’ understanding of ‘proof’ in lower
secondary school. In: Fou-Lai Lin, Feng-Jui Hsieh, Gila Hanna & Michael de
Villiers (Eds), Proceedings of the ICMI study 19 conference: proof and
proving in mathematics education. Taipei, Taiwan: National Taiwan Normal
University. Vol 1, 172-177. ISBN: 9789860182101
Click here for
full article in pdf format.
Abstract: In this paper we report on our analysis of textbooks commonly used
for teaching students about proof in geometry in lower secondary school in
Japan. From our analysis we found that, as expected from the curriculum
specification, deductive reasoning is prominent in Japanese textbooks. Yet the
way that proof and proving is presented in these textbooks shows geometry as a
very formal subject for study, one that omits to illustrate convincingly for
students the difference between formal proof and experimental verification. As
such, we argue that an improvement in textbook design is likely to involve
providing students with more effective instructional activities so that they
appreciate more fully the notion of ‘generality of proof’.
Jones, K., Kunimune, S., Kumakura, H., Matsumoto, S.,
Fujita, T. & Ding, L. (2009), Developing pedagogic approaches for proof:
learning from teaching in the East and West. In Fou-Lai Lin, Feng-Jui Hsieh,
Gila Hanna & Michael de Villiers (Eds), Proceedings of the ICMI study 19
conference: proof and proving in mathematics education. Taipei, Taiwan:
National Taiwan Normal University. Vol 1, 232-237. ISBN: 9789860182101
Click here for
full article in pdf format.
Abstract: In our work we focus on learning from the teaching of proof in
geometry at the lower secondary school level across countries in the East and in
the West. In this paper we summarize selected findings from a series of
classroom-based experiments carried out over an extended period of time. By
extracting key findings from our research, we show how we are identifying good
models of pedagogy and using these to develop new pedagogic principles that are
intended to help secondary school students not only to know ‘how to proceed’
with deductive proof, but also to understand more fully why such formal proof is
necessary to verify mathematical statements.
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
Click here for full
article in 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), 88-93. ISSN: 1463-6840
Click here for full
article in pdf 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
Click here for full article in pdf 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
Click here for full article in pdf
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.
Fujita, T. and Jones, K. (2008), The process of
re-designing the geometry curriculum: the case of the Mathematical Association
in England in the early 20th Century. Paper presented to Topic Study Group 38
(TSG38) at the 11th International Congress on Mathematical Education
(ICME-11), Monterrey, Mexico, 6-13 July 2008. 19pp
Click here for
full article in pdf format.
Abstract: This paper examines a key period of change in geometry teaching in
England. Our focus is the character and nature of the recommendations of the
geometry report of the UK Mathematical Association in 1902. We analyse
historical documents of the Mathematical Association using a theoretical
framework developed from Cooper’s model. Our analysis shows that the character
and recommendations of the Mathematical Association report was influenced by
various factors including: that the Mathematical Association members still
respected the traditional Euclidean approach to geometry as a basis for school
geometry; that the academic and power resources available to the Mathematical
Association at the time were not sufficient for a complete change from the
traditional approach; that conflicts between the various members of the
Mathematical Association prevented a complete consensus; and that the climate
outside the teaching committee of the Mathematical Association was not ready for
radical reform at that time.
Christou, C., Sendova, E., Matos, J-F., Jones, K.,
Zachariades, T., Pitta-Pantazi, D., Mousoulides, N., Pittalis, M., Boytchev, P.,
Mesquita, M., Chehlarova, T. & Lozanov, C. (2007), Stereometry Activities
With DALEST. Nicosia, Cyprus: University of Cyprus. ISBN: 9789963671212
[also published in Bulgarian, ISBN: 9789963671267, Greek, ISBN: 9789963671205,
and Portuguese, ISBN: 9789963671427]
Abstract: Spatial visualization is an important skill that deserves further
instructional attention. One way to improve students’ spatial visualization and
reasoning abilities is to provide learning activities that exploit the
possibilities of exploring the properties of 3D objects in appropriately
developed dynamic and interactive computer applications. The material presented
in this book relates to software applications developed as the main outcome of
the EU-funded project entitled DALEST: Developing an Active Learning
Environment for Stereometry. The book provides some background information on
the project, an overview of the functions of the software developed within the
framework of this project, plus some classroom activities for use with the
software.
Zachariades, T., Jones, K.,
Giannakoulias, E., Biza, I., Diacoumopoulos, D. & Souyoul, A. (Eds) (2007),
Teaching Calculus Using Dynamic Geometric Tools. Southampton, UK: University
of Southampton. ISBN: 9780854328840 [also published in Bulgarian, ISBN:
9789604660063, and Greek, ISBN: tba]
Abstract: An issue for many students studying mathematics in upper secondary
education or college, as international research demonstrates, is comprehending
the concepts of Calculus/Analysis. In attempting to improve teaching, the
combination of the dynamic nature of the concepts of Calculus/Analysis and its
historic roots in geometry, leads to the suggestion that teaching may be aided
by the use of dynamic geometry software. The material presented in this book is
one outcome of the EU-funded project entitled CalGeo: Teaching
Calculus/Analysis with the use of dynamic geometry tools. The didactic
activities presented in the book are informed by the CalGeo framework and
address the introduction of concepts and the teaching of Calculus/Analysis
theorems with the use of dynamic geometry software. Each activity offered in the
book consists of one or more worksheets for students, accompanied by guidance
for the teacher.
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
Click here for
full article in pdf 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.
Zachariades, T., Pamfilos, P., Christou, C., Maleev, R. and Jones, K. (2007),
Teaching introductory calculus: approaching key ideas with dynamic software.
Paper presented at the CETL–MSOR Conference 2007 on Excellence in the
Teaching & Learning of Maths, Stats & OR, University of Birmingham, 10-11
September 2007.
Click here for
full article in pdf format.
Abstract: While, commonly across the world, selected key ideas of the
Calculus are introduced to students in the final years of schooling, and are
thence built upon as students take a full course in Analysis at University,
there remains much to learn about how best to introduce such ideas and how to
develop and expand the ideas at University level. This paper reports on the work
of a European-funded project involving four countries in which the potential of
dynamic software was exploited in the teaching of topics such as infinite
processes, limits, continuity, differentiation and integration. Amongst the
approaches adopted in the project, problem-solving situations were developed
through which students, while their knowledge may initially be inadequate, could
approach intuitively the central mathematical notion in ways that are consistent
with formal mathematical definitions. Amongst the implications of the project,
in terms of the debate about what is suitable preparation for students embarking
on a course of analysis at University level, are that it might be useful to
think in terms of two categories of learning activity – the first is introducing
student to relevant concepts and the second focuses on the teaching of theorems.
These two categories entail a different design of learning activity.
Fujita, T.
and Jones, K. (2007),
Learners' understanding of the definitions and hierarchical classification of
quadrilaterals: towards a theoretical framing, Research in Mathematics Education,
9, 3-20. ISSN: 1479-4802
[journal volume also available as a book ISBN: 0953849880]
Click here for
full article in pdf format.
Abstract: Defining and classifying quadrilaterals, though an established
component of the school mathematics curriculum, appears to be a difficult topic
for many learners. The reasons for such difficulties relate to the complexities
in learning to analyse the attributes of different quadrilaterals and to
distinguish between critical and non-critical aspects. Such learning, if it is
to be effective, requires logical deduction, together with suitable interactions
between concepts and images. This paper reports on an analysis of data from a
total of 263 learners. The main purpose of the paper is to present a theoretical
framing that is intended to inform further studies of this important topic
within mathematics education research. This theoretical framing relates
prototype phenomenon and implicit models to common cognitive paths in the
understanding of the relationship between quadrilaterals.
Ding, L. and Jones, K. (2007), Using the van Hiele theory to analyse the
teaching of geometrical proof at Grade 8 in Shanghai, China. In D.
Pitta-Pantazi & G. Philippou (Eds), European
Research in Mathematics Education V (pp612-621). Nicosia, Cyprus: University
of Cyprus. ISBN: 9789963671250
Click here for
full article in pdf format.
Abstract: The data reported in this paper come from a study aimed at
explaining how successful teachers teach proof in geometry. Through a careful
analysis of a series of lessons taught in Grade 8 in Shanghai, China, the paper
reports on the appropriateness of the van Hiele model of ‘teaching phases’
within the Chinese context. The analysis indicates that though the second and
third van Hiele teaching phases could be identified in the Chinese lessons, the
instructional complexity of, for example, the guided orientation phase means
that more research is needed into the validity of the van Hiele model of
teaching.
Christou, C., Jones, K., Pitta-Pantazi, D., Pittalis, M., Mousoulides, N.,
Matos, J. F, Sendova, E., Zachariades, T. and Boytchev, P. (2007) Developing
student spatial ability with 3D software applications. Paper presented at the
5th Congress of the European Society for Research in Mathematics Education
(CERME), Larnaca, Cyprus, 22-26 Feb 2007. 10pp.
Click here for
full article in pdf format.
Abstract: This paper reports on the design of a library of software
applications for the teaching and learning of spatial geometry and visual
thinking. The core objective of these applications is the development of a set
of dynamic microworlds, which enables (i) students to construct, observe and
manipulate configurations in space, (ii) students to study different solids and
relates them to their corresponding nets, and (iii) students to promote their
visualization skills through the process of constructing dynamic visual images.
During the developmental process of software applications the key elements of
spatial ability and visualization (mental images, external representations,
processes, and abilities of visualization) are carefully taken into
consideration.
Christou, C., Jones, K., Mousoulides, N. & Pittalis, M. (2006), Developing
the 3DMath dynamic geometry software: theoretical perspectives on design,
International Journal of Technology in Mathematics Education, 13(4),168-174. ISSN: 1744-2710
Click here for
more details about the article.
Abstract: This paper reports on the theoretical perspectives underpinning the
design of a 3D geometry software environment called 3DMath. The idea of 3DMath
is to develop a dynamic three-dimensional geometry microworld, which enables
students to construct, observe and manipulate geometrical figures in 3D space,
and to focus on modelling geometric situations, and enable teachers to help
students construct their understanding of stereometry. During the developmental
of 3DMath, the key elements of visualization (mental images, external
representations, and the processes and abilities of visualization) were
carefully taken into consideration. The aim of this paper is to illustrate how
the design of the 3DMath software was informed by these key elements of
visualization, as well as by theories related to the philosophical basis of
mathematical knowledge and to semiotics. Thus, the paper describes how the
features of the software are designed to enhance the elements of visualization,
and to satisfy the characteristics of instructional techniques that are
appropriate to these theoretical perspectives.
Ding, L. & Jones, K. (2006), Students’ geometrical thinking development at
Grade 8 in Shanghai, In Novotná, J., Moraová, H., Krátká, M. & Stehlíková, N.
(Eds.), Proceedings 30th Conference of the International Group for the
Psychology of Mathematics Education (PME30), vol 1, p382. [extended
abstract]
Click here for
full article in pdf format.
Abstract: The main aim of this study is to investigate geometry teaching at
the lower secondary school level in Shanghai, with particular attention to the
relationship of the teaching/learning phases organized by teachers with
students’ thinking levels demonstrated in classrooms and examination papers at
Grade 8 (students age 14). Analysis of data from the pilot study suggests that
an essential teaching strategy used by the Chinese teachers was mutually
reinforcing visual and deductive approaches in order to develop students’
geometric intuition in the learning of deductive geometry.
Fujita, T. and Jones, K. (2006) Primary trainee teachers’ understanding of
basic geometrical figures in Scotland. In, Novotná, J., Moraová, H., Krátká, M.
and Stehlíková, N. (eds.), Proceedings 30th Conference of the International
Group for the Psychology of Mathematics Education (PME30). Prague, Czech
Republic, vol 3, pp129-136.
Click here for
full article in pdf format.
Abstract: Whilst teachers’ mathematics knowledge is known to play a
significant role in shaping the quality of their teaching, much less is known
about the nature and extent of that knowledge, how it develops, and how such
development can be supported through initial teacher training and continuing
professional development. Earlier research has indicated that pre-service
(trainee) primary teachers’ subject knowledge of geometry is amongst their
weakest knowledge of mathematics. This paper reports on an analysis of geometry
subject knowledge data gathered in Scotland from undergraduate pre-service
primary teachers, focusing on their ability to define and classify
quadrilaterals. The results indicate that many trainee primary teachers have
relatively poor command of these aspects 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. ISSN: 1463-6840
Click 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 BSRLM 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. (2006), Primary trainee teachers’ knowledge of
parallelograms, Proceedings of the British Society for Research into Learning
Mathematics, 26(2), 25-30. ISSN: 1463-6840
Click here for
full paper in pdf format.
Abstract: Considerable research has indicated that amongst the factors which
make the most significant contribution to high student achievement in
mathematics is secure subject knowledge on the part of the teacher as this
underpins an approach to mathematics in which topics are seen as part of a
coherent set of related ideas, with clear progression and links to previous and
future learning. This paper reports part of the findings from a study of trainee
teachers’ knowledge of basic geometrical figures, in particular focusing on what
knowledge they have of parallelograms and how they use this knowledge to solve
geometrical problems. The findings indicate that only a minority of trainee
primary teacher have a fully sophisticated knowledge of parallelograms and of
how to use the properties of parallelograms to solve relevant problems.
Ding, L. and Jones, K. (2006) Teaching geometry in lower
secondary school in Shanghai, China, Proceedings of
the British Society for Research into Learning Mathematics,
26(1), 41-46. ISSN: 1463-6840
Click here for
full paper in pdf format.
Abstract: This paper reports on a study of geometry teaching at the lower
secondary school level in Shanghai, China. Through an analysis of data from
observing a variety of Year 9 (Grade 8) lessons, and utilising data from the
students' performance in school examinations, the study suggests that teachers
in this region of China use classroom strategies that attempt to reinforce
visual and deductive approaches in order to develop students' thinking in the
transition to deductive geometry education.
Edwards, J. and Jones, K. (2006),
Linking geometry and algebra with GeoGebra, Mathematics Teaching,
194, 28-30. ISSN: 0025-5785
Click here for the article in
pdf format. [article also reproduced, with permission, in the Wellington (NZ)
Mathematics Association newsletter, issue 4, 2006, pp12-14]
Abstract: GeoGebra is a software package and is so named because it combines
geometry and algebra as equal mathematical partners in its representations. At
one level, GeoGebra can be as a dynamic geometry system like other,
commercially available, software. But this is only part of the story. Another
window (the algebra part of GeoGebra) provides an insight into the
relationship between the geometric aspects of figures and their algebraic
representations. Here each equation or set of coordinates can be edited in the
algebra window and the figure instantly changes. What is more, an equation (or a
function) can be typed into the space at the foot of the GeoGebra
interface and the corresponding geometric representation will appear in the
geometry window. Perhaps utilising GeoGebra could inspire a change from
regular forms of enrichment/ extension activity to things that need high level
thinking, and things that pupils may find themselves wanting to follow-up
outside school lessons.
Jones, K. (2005), Research on the use of dynamic geometry software: implications
for the classroom. In: J. Edwards & D. Wright (Ed), Integrating ICT into the
Mathematics Classroom. Derby: Association of Teachers of Mathematics. pp
27-29. ISBN: 1898611408 or 9781898611400 [the chapter is a reprint of MicroMath, 18(3),
18-20].
Click here for the article in
pdf format.
Abstract: This short user-review summarise the research that has
investigated the use of dynamic geometry software (DGS) in the
teaching and learning of mathematics. Overall, the research has
found that DGS cannot provide a self-contained environment and
that the software itself does not necessarily mean that students
will learn geometry theory. Research also suggests that it can
take quite a long time for the benefits of using DGS to emerge
but that this investment is worthwhile in developing
students' knowledge of geometry. The sorts of tasks that
students tackle, the form of teacher input and the general
classroom atmosphere are all important factors.
Jones, K. (2005), Research bibliography on the use of dynamic geometry
software. In: D. Wright (Ed), Moving on with Dynamic Geometry. Derby:
Association of Teachers of Mathematics. pp 159-160. ISBN: 1898611394 or
9781898611394 [the chapter is a reprint of MicroMath, 18(3),
44-45].
Click here for
the article in pdf format.
Abstract: This bibliography lists research that has
investigated the use of dynamic geometry software (DGS) in the
teaching and learning of mathematics. The bibliography is not
intended to be exhaustive; rather it includes the major studies
across the range of research that has been published.
Jones, K. (2005), The
shaping of student knowledge: learning with dynamic geometry software. Paper presented at the Computer Assisted Learning Conference 2005
(CAL05), Bristol, 4-6 April 2005.
Click here for
full paper in pdf format.
Abstract: The focus of this paper is a software genre usually referred to as
'dynamic geometry' because of the ability of the user to dynamically manipulate
geometrical figures created with the software tool. Using data from a
longitudinal study of 12-13 students' use of dynamic geometry software, the
focus of the analysis is on the interpretations the students make of geometrical
objects and relationships when using this form of software. The analysis
suggests that the students' mathematical reasoning is shaped by their
interactions with the software in that their ability to explain geometrical
facts and relationships evolves from imprecise, 'everyday' expressions, through
reasoning that is overtly mediated by the software environment, to mathematical
explanations of the geometric situation that transcend the particular tool being
used. Such findings suggest that curriculum initiatives that encourage the use
of dynamic geometry software are appropriate but that the incorporation of such
software into classroom practices is unlikely to be straightforward.
Ding, L., Fujita, T. and Jones, K. (2005), Developing geometrical reasoning in
the classroom: learning from highly experienced teachers from China and Japan.
In, Bosch, M. (ed.) European Research in Mathematics Education IV.
Barcelona, Spain: ERME, pp727-737. ISBN: 8461132823
Click here for full paper in pdf
format
Abstract: International comparative research in mathematics education has found,
perhaps unsurprisingly, that teachers are a key influence on pupil learning.
Given that the development of pupils’ capability in geometrical reasoning
continues to be an issue of considerable international concern, this paper
reports an analysis of lower secondary school lesson suggestions offered by
expert teachers from China and Japan (countries selected because they represent
some interesting similarities and contrasts). The analysis indicates some
striking similarities between suggested lessons, but some noteworthy
differences. Both these may be related to the educational context in which the
lesson suggestions are presented.
An earlier version of this paper is available as:
Jones, K., Fujita, T. and Ding, L. (2005), Teaching geometrical reasoning:
learning from expert teachers from China and Japan, Paper presented at the 6th
British Congress on Mathematical Education (BCME6), Warwick, March 2005.
Published version available as: Jones, K., Fujita, T. and Ding, L. (2005), Teaching
geometrical
reasoning: learning from expert teachers from China and Japan, Proceedings of
the British Society for Research into Learning Mathematics, 25(1),
89-96. Click here for
the paper in pdf
format.
Christou, C., Pittalis, M., Mousoulides, N., & Jones, K. (2005), Developing
3D dynamic geometry software: theoretical perspectives on design. In F. Olivero
& R. Sutherland (Eds), Visions of Mathematics Education: Embedding Technology
in Learning; Proceedings of the 7th International Conference on Technology and
Mathematics Teaching. Bristol, UK, 26-29 July 2005. Vol 1 pp 69-77. ISBN:
0862925592 or 9780862925598
Click here for
full article in pdf format.
Abstract: This paper reports on the theoretical perspectives underpinning the
design of a 3D geometry software environment called 3DMath. The idea of 3DMath
is to develop a dynamic three dimensional geometry microworld, which enables (i)
students to construct, observe and manipulate geometrical figures in 3D space,
(ii) students to focus on modeling geometric situations, and (iii) teachers to
help students construct their understanding of stereometry. During the
developmental of 3DMath, the key elements of visualization (mental images,
external representations, and the processes and abilities of visualization) are
being carefully taken into consideration. The aim of this paper is to illustrate
how the design of the 3DMath software is informed by these key elements of
visualization, as well as by theories related to the philosophical basis of
mathematical knowledge and to semiotics. Thus, the paper describes how the
features of the software are designed to enhance the elements of visualization,
and to satisfy the characteristics of instructional techniques that are
appropriate to these theoretical perspectives.
Jones, K. (2005), Using Logo in the teaching and learning of mathematics: a
research bibliography, MicroMath, 21(3), 34-36.
Click here for full article in pdf format.
Abstract: This review suggests that students working with Logo, by creating
and interacting with objects that are visible, quantifiable, and adhere to
conventional mathematics, build connections between spatial and
numeric/algebraic thinking. Using Logo can help students make mathematics more
concrete, while simultaneously supporting algebraic formalisation of actions as
Logo 'procedures'. Working with Logo affords students opportunities to try out
ideas and modify plans, elements that are key to mathematical problem solving.
Students can make and test conjectures, a vital component of mathematical
reasoning.
Jones, K., Fujita, T. and Ding, L. (2004), Structuring
mathematics lessons to develop geometrical reasoning: comparing
lower secondary school practices in China, Japan and the UK.
Paper presented at the Symposium on Comparative Studies in
Mathematics Education at the British Educational
Research Association Annual Conference, University of
Manchester, 15-18 September 2004.
Click here for
full article in pdf format.
Abstract: Achievement in mathematics continues to be a
crucial factor in the success of school systems around the world.
As a result, this area of the curriculum has been the subject of
considerable international comparative research, mostly focussed
on pupil achievement but also examining teaching methods,
curricula, and so on. In all this, and perhaps unsurprisingly,
the central role of teachers, and how they structure their
lessons, has emerged as a key factor in pupil learning. A number
of projects have examined the structure of mathematics lessons,
either to typify individual lessons in specified countries, or as
an attempt to describe the variety of lesson structures used by
particular teachers in particular countries over a sequence of
lessons. To date there has been little comparative work
specifically on how teachers structure mathematics lessons to
develop geometrical reasoning despite the issue of how to improve
geometry teaching being of considerable international concern.
This paper reports early data from a larger comparative study
that includes the analysis of classroom teaching materials. This
paper compares suggestions about how teachers might structure
geometry lessons in lower secondary school in three countries,
China, Japan, and the UK (specifically England), chosen because
they represent some interesting similarities and contrasts. The
analysis focuses on the background to the suggestions available
to teachers, in particular where approaches are similar and where
they diverge. What the implications might be for student
achievement in geometry in the three countries is identified as
an area for future research.
Ding, L. and Jones, K. (2004), The Structure of
Mathematics Lessons: researching the development of geometrical
reasoning in lower secondary schools in China. Paper
presented at the European Society for Research in Mathematics
Education Summer School, Podebrady, Czech Republic, August
2004. 6pp.
Abstract: Following a period of declining emphasis,
especially in Western countries, geometry is re-emerging as a key
component of the mathematics education at school level. As a
result, the ways in which teachers might develop their
students' geometrical reasoning is the subject of
considerable international debate. Yet despite international
comparative studies, such as TIMSS, there has been little
comparative work specifically on how mathematics teachers
structure lessons to develop geometrical reasoning. This paper
identifies key questions about the structure of lessons that are
intended to develop geometrical reasoning and proposes an
approach to examining such lessons in lower secondary schools in
China
Fujita, T., Jones, K. and Yamamoto, S. (2004), Geometrical
intuition and the learning and teaching of geometry. Paper presented at the Topic
Group on Research and Development in the Teaching and Learning of
Geometry, 10th International Congress on Mathematical Education,
(ICME-10), Copenhagen, Denmark.
Click here for
full paper in pdf format.
Abstract: Intuition is often regarded as essential in the
learning of geometry, but how such skills might be effectively
developed in students remains an open question. This paper
reviews the role and importance of geometrical intuition and
suggests it involves the skills to create and manipulate
geometrical figures in the mind, to see geometrical properties,
to relate images to concepts and theorems in geometry, and decide
where and how to start when solving problems in geometry. Based
on these theoretical considerations, we illustrate a range of
student tasks that we argue should contribute to developing
students' geometrical intuition.
Fujita, T., Jones, K. and Yamamoto, S. (2004), The role of
intuition in geometry education: learning from the teaching
practice in the early 20th Century. Paper presented at the Topic Group on the History
of the Teaching and the Learning of Mathematics, 10th
International Congress on Mathematical Education, (ICME-10),
Copenhagen, Denmark..
Click here for
full paper in pdf format.
Abstract: Intuition is often regarded as essential in the
learning of geometry, but questions remain about how we might
effectively develop students' such skills. This paper
provides some results from analyses of innovative geometry
teaching in the early part of the 20th
century, a time when significant efforts were being made to
improve the teaching and learning of geometry. As examples, we
examine the tasks for students that can be found in
Treutlein's "Geometrical Intuitive Instruction"
(Germany) and Godfrey's geometry textbook (England). The
analyses suggest that educators at that time attempted to develop
students' intuitive skills through various practical tasks
such as drawing, measurement, and imagining and manipulating
figures, which could be useful for current geometry teaching. We
also identify different approaches taken to the development
mathematics teaching in Germany and England.
Jones, K. (2004), Book review: The Changing Shape of Geometry, edited by
Chris Pritchard, Mathematics in School, 33(4), 35-36. ISSN:
0305-7259
Click here for
more information on this review.
Abstract: This is a review of a book that sets itself the goal of celebrating
“the best of geometry in all its simplicity, economy and elegance". It does this
by collecting together more than 50 articles published in the Mathematical
Gazette and Mathematics in School over the past 100 years. With contributors
including a Nobel Laureate and a Pulitzer Prize winner all sharing their love of
geometry, this book impresses from a first look and never fails to amaze,
entertain, educate and inform.
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.
Abstract: This paper summarises some of the work on a project sponsored
by the UK Qualifications and Curriculum Authority to develop
teaching ideas that focus on the development of geometrical
reasoning at the secondary school level. The project explored
what is possible both within and beyond the current requirements
of the UK National Curriculum and the Key Stage 3 strategy, and
to consider the whole ability range. [full report online
below]
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
Click here for
full report in pdf format.
Download Executive summary here
(in pdf format).
Abstract: This report to the QCA details the work of a group
of mathematicians, mathematics educators, local authority
officers, and teachers on developing teaching ideas that focus on
the development of geometrical reasoning at the secondary school
level. 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.
Fujita, T. and Jones, K. (2003), The place of experimental tasks in geometry
teaching: learning from the textbook designs of the early 20th
century, Research in Mathematics Education,
5, 47-62. ISSN: 1479-4802
[also available as a book ISBN: 0953849848]
Click here for
full article in pdf format.
Abstract: The dual nature of geometry, in that it is a
theoretical domain and an area of practical experience,
presents mathematics teachers with opportunities and dilemmas.
Opportunities exist to link theory with the everyday knowledge of
pupils but the dilemmas are that learners very often find the
dual nature of geometry a chasm that is very difficult to bridge.
With research continuing to focus on understanding the nature of
this problem, with a view to developing better pedagogical
techniques, this paper examines the place of experimental tasks
in the process of learning geometry. In particular, the paper
provides some results from an analysis of innovative geometry
textbooks designed in the early part of the 20th
Century, a time when significant efforts were being made to
improve the teaching and learning of geometry. The analysis
suggests that experimental tasks have a vital role to play and
that a potent tool for informing the design of such tasks, so
that they build effectively on pupils' geometrical
intuition, is the notion of the geometrical eye, a term
coined by Charles Godfrey in 1910 as the power of seeing
geometrical properties detach themselves from a figure.
Fujita, T. and Jones, K. (2003), Interpretations
of National Curricula: the case of geometry in Japan and the UK.
Paper presented at the British Educational Research
Association Annual Conference, Heriot-Watt University, 10-13
September, 2003.
Click here for
full article in pdf format.
Abstract: This paper presents an analysis of how the geometry
component of the National Curricula for mathematics in Japan and
in one selected country of the UK, specifically Scotland, is
interpreted by textbook writers. 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 Scottish
textbooks tend, at this level, to concentrate on measuring,
drawing, finding angles, 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. An emerging issue is how the design of textbooks might
either build on, or neglect, students' intuitive skills when
they tackle geometrical problems.
Fujita, T. and Jones, K. (2003), Critical
review of geometry in current textbooks in lower secondary
schools in Japan and the UK, Proceedings of the 27th
Conference of the International Group for the Psychology of
Mathematics Education, Vol 1, p220 [extended abstract].
Click
here for the extended abstract 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 an initial
analysis of current best-selling textbooks for lower secondary
schools in Japan and the UK (specifically Scotland) using an
analytic framework derived from the study of the textbooks in the
"Trends in International Mathematics and Science Study"
(TIMSS). Our 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. (2003), Classroom implications of research on
dynamic geometry software. In: M. A. Mariotti (Ed), European
Research in Mathematics Education III. Pisa: University of
Pisa. ISBN: 8884921848 or 9788884921840s
Click here for full article in pdf format.
Abstract: This short user-review summarise the research that has
investigated the use of dynamic geometry software (DGS) in the
teaching and learning of mathematics. Overall, the research has
found that DGS cannot provide a self-contained environment and
that the software itself does not necessarily mean that students
will learn geometry theory. Research also suggests that it can
take quite a long time for the benefits of using DGS to emerge
but that this investment is worthwhile in developing
students' knowledge of geometry. The sorts of tasks that
students tackle, the form of teacher input and the general
classroom atmosphere are all important factors.
Jones, K. and Mooney, C. (2003), Making space for geometry in
primary mathematics. In: I. Thompson (Ed), Enhancing
Primary Mathematics Teaching and Learning. London: Open
University Press. pp 3-15. ISBN: 0335213758 or 9780335213757 [invited chapter]
Click here for
full article in pdf format.
Abstract: This chapter examines the structure and
recommendations of the UK National Numeracy Strategy (NNS) with
respect to the teaching of geometry at primary school. It looks
at ways in which the NNS recommendations might be best taken
forward and whether there are important aspects of geometry that
the Strategy has omitted or to which it has paid too little
attention. It suggests that, until spatial and visual thinking is
given greater status within the mental and oral segments of
primary mathematics lessons, and until more curriculum space at
primary level is devoted to geometry, children may well continue
to have insufficient opportunity to develop fundamental
visualisation and spatial reasoning skills that are so important
in an increasingly visual world.
Jones, K. (2003), Book review: Fractal Geometry, edited by
Jonathan M. Blackledge, Allan K. Evans and Martin J. Turner, Mathematics
Teaching, 183, p47. ISSN: 0025-5785
Click here for more information
on this review.
Abstract: This edited collection is wide-ranging, covering applications of
fractal geometry in aircraft design, finance, geology, digital image
compression, and cryptography. As a book aimed at researchers in fractal
geometry, this is a useful book for those with some expertise in the subject.
Mooney, C., Fletcher, M. and Jones, K. (2003),
Minding your Ps and Cs: subjecting knowledge to the
practicalities of teaching geometry and probability, Proceedings
of the British Society for Research into Learning Mathematics,
23(3), 79-84.
Click here for
full article in pdf format.
Abstract: knowledge as a The review of the implementation of
the UK National Numeracy Strategy by Ofsted (Nov. 2003) has
highlighted weak subject consistent feature in unsatisfactory
teaching. This study looks at the subject knowledge of generalist
primary trainees in the areas of geometry and probability and
their ability to apply their knowledge to problem solving tasks.
The study goes on the raise the question, "is a profound
understanding of fundamental mathematics (PUFM)(Ma, 1999)
possible for generalist teachers?"
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. ISSN: 1463-6840
Click here for
full article 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.
Fujita, T. and Jones, K. (2002), The bridge between practical
and deductive geometry: developing the "geometrical
eye". In: A. D. Cockburn and E. Nardi (Eds), Proceedings
of the 26th Conference of the International Group for the
Psychology of Mathematics Education, Vol 2, 384-391, UEA, UK.
Click here for
full article in pdf format.
Abstract: The dual nature of geometry, as a theoretical
domain and an area of practical experience, presents mathematics
teachers with the opportunity to link theory with the everyday
knowledge of their pupils. Very often, however, learners find the
dual nature of geometry a chasm that is very difficult to bridge.
With research continuing to focus on understanding the nature of
this problem, with a view to developing better pedagogical
techniques, this paper reports an analysis of innovative geometry
teaching methods that were developed in the early part of the
20th Century, a time when significant efforts were being made to
improve the teaching and learning of geometry. The analysis
suggests that the notion of the geometrical eye, the
ability to see geometrical properties detach themselves from a
figure, might be a potent tool for building effectively on
geometrical intuition.
Jones, K. (2002), Issues in the teaching and learning of
geometry. In: Linda Haggarty (Ed), Aspects of
Teaching Secondary Mathematics. London: Routledge. pp
121-139. ISBN: 0415266416 or 9780415266413. [invited chapter]. ISBN: 0-415-26641-6
Click here for
full paper in pdf format.
Abstract: This chapter analyses a range of key issues in the
teaching and learning of geometry. These include the nature of
geometry, why geometry is important in the curriculum at school
level and beyond, what geometry can be included at the school
level, the aims of teaching geometry, and how geometry can be
best taught and learnt. The chapter addresses the use of
information and communication technology in geometry education
and concludes that the twenty-first century is one where spatial
thinking and visualisation are vital areas for education. The
chapter includes a number of tasks related to the issues
addressed and an appendix giving details of resources.
Jones, K. and Fujita, T. (2002), The design of geometry
teaching: learning from the geometry textbooks of Godfrey and
Siddons, Proceedings of the British Society for Research into
Learning Mathematics, 22(1&2), 13-18. ISSN: 1463-6840
Click here for
full paper in pdf format.
Abstract: Deciding how to teach geometry remains a demanding
task with one of major arguments being about how to combine the
intuitive and deductive aspects of geometry into an effective
teaching design. In order to try to obtain an insight into
tackling this issue, this paper reports an analysis of innovative
geometry textbooks which were published in the early part of the
20th Century, a time when significant efforts were being made to
improve the teaching and learning of geometry. The analysis
suggests that the notion of the geometrical eye, the
ability to see geometrical properties detach themselves from a
figure, might be a potent tool for building effectively on
geometrical intuition so as to provide a bridge into deductive
geometry.
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
(1&2), 95-100. ISSN: 1463-6840
Click 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., (2002), Geometry in the A-level mathematics
curriculum, Occasional Papers in Science, Technology,
Environmental and Mathematics Education. Southampton:
University of Southampton, pp 4-5.
Abstract: Currently there is little opportunity to study
geometry after the age of 16. The aim of this short paper is to
contribute to discussions of what geometry might be suitable for
inclusion in the specification for A-level mathematics.
Jones, K., (2002), Research on the use of dynamic geometry software: implications for the classroom, MicroMath,
18(3), 18-20.
Click
here for full article in pdf format.
Abstract: This article summarise the research that has
investigated the use of dynamic geometry software (DGS) in the
teaching and learning of mathematics. This review is not intended
to be exhaustive, rather the research is categorised under three
main headings: interacting with the software, designing
teaching activities and learning to prove. Overall,
the research has found that DGS cannot provide a self-contained
environment and that the software itself does not necessarily
mean that students will learn geometry theory. Research also
suggests that it can take quite a long time for the benefits of
using DGS to emerge but that this investment is worthwhile in
developing students' knowledge of geometry. The sorts of
tasks that students tackle, the form of teacher input and the
general classroom atmosphere are all important factors.
Jones, K., (2002), Research bibliography: dynamic geometry software, MicroMath, 18(3), 44-45.
Click
here for full article in pdf format.
Abstract: This bibliography lists research that has
investigated the use of dynamic geometry software (DGS) in the
teaching and learning of mathematics. The bibliography is not
intended to be exhaustive; rather it includes the major studies
across the range of research that has been published.
Jones, Keith (2001) Spatial thinking and visualisation. In, Teaching and
Learning Geometry 11-19. London, UK: Royal Society, 55-56. ISBN: 085403563X
Click here for full article in pdf format.
Abstract: The provision of mathematics curriculum that encourages students to
develop their powers of spatial thinking and visualisation, as important
components of their geometrical reasoning, is seen as a key area for development
in mathematics education. This short article reviews the nature of spatial
thinking and visualisation, both in mathematics education more generally and in
geometry in particular, illustrating how both forms of thinking are vital to
mathematics.
Jones, K. (2001), Learning geometrical concepts using dynamic geometry
software. In: Kay Irwin (Ed), Mathematics
Education Research: A catalyst for change. Auckland, New
Zealand: University of Auckland. [invited paper].
Click
here for full article in pdf format.
Abstract: Dynamic geometry software promises direct
manipulation of geometrical objects and relations. This paper
reports aspects of a research study deigned to examine the impact
of using such software on student conceptions. Analysis of the
data from the study indicates that, while the use of dynamic
geometry software can assist students in making progress towards
more mathematical explanation (and thereby provide a foundation
on which to build further notions of deductive reasoning in
mathematics), the 'dynamic' nature of the software influences the
form of explanation, especially in the early stages.
Jones, K. (2001), Boxed into a corner: teaching definitions is
a tricky business. Times Education Supplement, 19 January 2001; p14 [invited article for the mathematics curriculum special].
ISSN: 0040-7887
Abstract: The (UK) National Numeracy Strategy includes the
term 'oblong' in the framework for teaching mathematics to pupils
aged 8 and over. This article takes issue with this
specification. It argues that the term "oblong" is not
mathematical and that its use in mathematics lessons both makes
it more difficult for pupils to appreciate that squares are
rectangles and undermines the key idea of understanding
generalisation in mathematics (when "weaker" concepts
are used to generalise towards 'stronger' concepts rather than
exclude them).
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. ISSN: 1463-6840
Click 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, 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. ISSN: 1463-6840
Click 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.
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. ISSN: 1463-6840
Click
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.
Jones, K. (2000), Providing a foundation for deductive
reasoning: students' interpretation when using dynamic geometry
software and their evolving mathematical explanations. Educational
Studies in Mathematics, 44(1-3), pp 55-85.
Click
here for full article in pdf format.
Abstract: A key issue for mathematics education is how
children can be supported in shifting from "because it looks
right" or "because it works in these cases" to
convincing arguments which work in general. In geometry,
forms of software usually known as dynamic geometry environments
may be useful as they can enable students to interact with
geometrical theory. Yet the meanings that students gain of
deductive reasoning through experience with such software is
likely to be shaped, not only by the tasks they tackle and their
interactions with their teacher and with other students, but also
by features of the software environment. In order to try to
illuminate this latter phenomenon, and to determine the
longer-term influence of using such software, this paper reports
on data from a longitudinal study of 12-year-old students'
interpretations of geometrical objects and relationships when
using dynamic geometry software. The focus of the paper is the
progressive mathematisation of the student's sense of the
software, examining their interpretations and using the
explanations that students give of the geometrical properties of
various quadrilaterals that they construct as one indicator of
this. The research suggests that the students' explanations
can evolve from imprecise, "everyday" expressions,
through reasoning that is overtly mediated by the software
environment, to mathematical explanations of the geometric
situation that transcend the particular tool being used. This
latter stage, it is suggested, should help to provide a
foundation on which to build further notions of deductive
reasoning in mathematics.
Jones, K. (2000), Critical issues in the design of the school
geometry curriculum. In: Bill Barton (Ed) (2000), Readings
in Mathematics Education. Auckland, New Zealand: University
of Auckland. pp 75-90. [invited paper]
Click here for
full article in pdf format.
Abstract: The fundamental problem in the design of the
geometry component of the mathematics curriculum is simply that
there is too much interesting geometry, more than can be
reasonably included in the mathematics curriculum. The question
that taxes curriculum designer is what to include and what to
omit. This paper does not seek to resolve the disagreements over
the geometry curriculum as, given the nature of the problem, such
an endeavour is unlikely to be successful. Rather, the aim is to
identifying and review critical issues concerning the design of
the geometry curriculum. These issues include the nature of
geometry, the aims of geometry teaching, how geometry is learnt,
the relative merits of different approaches to geometry, and what
aspects of proof and proving to accentuate.
Jones, K. (2000), Critical issues in the design of the school
geometry curriculum. Invited presentation for the Topic Group
on Geometry at the 9th International Congress on
Mathematical Education (ICME9), Tokyo, Japan, August 2000.
Click here [or
click here for
an extended version of the paper (in pdf format)]
Abstract: Designing the geometry component of the mathematics
curriculum is probably the most difficult topic to get right. The
fundamental problem is that there is just too much interesting
geometry that could be included. This leads to a range of
critical issues including just what geometry to include and how
to prevent learners experiencing geometry as an incoherent
jumble. Also critical are issues of teacher knowledge of
geometry.
[see above article on extended version available in pdf
format]
Jones K. (2000), Teacher knowledge and professional
development in geometry, Proceedings of the British Society
for Research into Learning Mathematics, 20(3),
109-114. ISSN: 1463-6840
Click here for
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.
Jones, K., Gutiérrez, A, and Mariotti, M. A. (2000), Proof in
dynamic geometry environments, Editorialship of a special issue
of Educational Studies in Mathematics. 44(1&2),
pp 1-3.
Click
here for full article in pdf format.
Jones K. (1999), Student interpretations of a dynamic geometry
environment. In: Inge Schwank Ed, European
Research in Mathematics Education. Osnabrueck, Germany:
Forschungsinstitut für Mathematikdidaktik, pp 245-258. ISBN: 392538653X or
9783925386534
Click
here for full article in pdf format.
Abstract: It seems that student interpretations of
computer-based learning environments may result from the
idiosyncrasies of the software design rather than the
characteristics of the mathematics. Yet, somewhat paradoxically,
it is because the software demands an approach which is novel
that its use can throw light on student interpretations. The
analysis presented in this paper is offered as a contribution to
understanding the relationship between the specific tool being
used, in this case the dynamic geometry environment
Cabri-Géomètre, and the kind of thinking that may develop as
a result of interactions with the tool. Through this analysis a
number of effects of the mediational role of this particular
computer environment are suggested.
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. ISSN: 1463-6840
Click
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. ISSN: 1463-6840
Click
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. ISSN: 1463-6840
Click
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.
Hoyles, C. and Jones, K. (1998), Proof in dynamic geometry
contexts. In: C. Mammana and V. Villani (eds), Perspectives
on the Teaching of Geometry for the 21st
Century. Dordrecht: Kluwer, pp121-128. ISBN: 0792349903 or 9780792349907
Click here for
full article in pdf format.
Abstract: Proof lies at the heart of mathematics yet we know
from research in mathematics education that proof is an elusive
concept for many mathematics students. The question that we
are now asking is whether the introduction of dynamic geometry
systems will improve the situation - or will it make the
transition from informal to formal proof in mathematics even
harder? How far will innovative teaching approaches with
computers assist pupils in developing a conceptual framework for
proof and in appropriating proof as a means to illuminate
geometrical ideas or will computer use be seen to replace any
need for proof?
Jones, K. (1998), Deductive and intuitive approaches to
solving geometrical problems. In: C. Mammana and V.
Villani (eds), Perspectives on the Teaching of Geometry for
the 21st Century. Dordrecht: Kluwer, pp78-83. ISBN:
0792349903 or 9780792349907
Click here for
full article in pdf format.
Abstract: Approaches to the teaching and learning of a chosen
topic in geometry can be located somewhere between what are
sometimes perceived as two extremes. One such extreme is
characterised as "intuitive", the other as "formal" or "axiomatic". There seems to be a
number of ways of looking at the relationship between these two
positions.
Jones, K. (1998), The mediation of learning within a dynamic
geometry environment. In: A. Olivier and K. Newstead
(Eds), Proceedings of the 22nd
Conference of the International Group for the Psychology of
Mathematics Education. Stellenbosch, South Africa: University
of Stellenbosch, Volume 3, pp96-103.
Click
here for full article in pdf format.
Abstract: Computer-based learning environments promise much in
terms of enhancing mathematics learning. Yet much remains unclear
about the relationship between the computer environment, the
activities it might support, and the knowledge that might emerge
from such activities. The analysis presented in this paper is
offered as a contribution to understanding the relationship
between the specific tool being used, in this case the dynamic
geometry environment Cabri-Géomètre, and the kind of
thinking that may develop as a result of interactions with the
tool. Through this analysis I suggest a number of effects of the
mediational role of this particular computer environment
McLeay, H., 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. ISSN: 1463-6840
Click
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. ISSN: 1463-6840
Click
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. ISSN: 1463-6840
Click
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), Children learning to specify geometrical
relationships using a dynamic geometry package. In: E.
Pehkonen. (Ed), Proceedings of the 21st
Conference of the International Group for the Psychology of
Mathematics Education. Finland, University of Helsinki,
Volume 3, pp121-8.
Click
here for full article in pdf format.
Abstract: In order to understand the learning taking place
when students use a dynamic geometry package such as Cabri-Géomètre,
a particular focus for study needs to be on the learning mediated
through employing such a resource. In this paper I describe
how one pair of 12 year old students begin learning how to
specify geometrical relationships in Cabri. I argue that,
while Cabri provides certain elements of the mathematical
language necessary for the articulation of relevant mathematical
ideas, significant aspects must be provided by the teacher.
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. ISSN: 1463-6840
Click
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.
Gorgorió, N. and Jones, K. (1997), Cabri i visualització, Biaix,
10, 21-23. [in Catalan - a version of the invited paper presented to The Topic Group on The Future of
Geometry, 8th International Congress on Mathematical Education (ICME8),
Seville, Spain, 14-21 July 1996].
Click
here for full article (in pdf format) presented at ICME8, in English).
Abstract: The advent of powerful computer graphic packages has
coincided with renewed interest in all forms of visual
representation in mathematics. As a result, we need to be clear
about what we mean by the visual processing necessary to solve
mathematical problems involving visual phenomena. Visual
processing involves the ability to mentally manipulate and
transform visual representations and visual imagery. This paper
describes three different elements of the visualisation process:
crude visualisation, visualisation as the reading of visual
information, and visual processing. Illustrations are given of
how the use of a dynamic geometry package such as Cabri-Géomètre
both needs and contributes to developing visualisation in all
these three senses.
Gorgorió, N. and Jones, K. (1996), Elements of the
Visualisation Process within a Dynamic Geometry Environment,
Invited paper presented to The Topic Group on The Future of
Geometry, 8th International Congress on Mathematical Education,
Seville, Spain, 14-21 July 1996.
Click
here for full article in pdf format.
Abstract: The advent of powerful computer graphic packages has
coincided with renewed interest in all forms of visual
representation in mathematics. As a result, we need to be clear
about what we mean by the visual processing necessary to solve
mathematical problems involving visual phenomena. Visual
processing involves the ability to mentally manipulate and
transform visual representations and visual imagery. This paper
describes three different elements of the visualisation process:
crude visualisation, visualisation as the reading of visual
information, and visual processing. Illustrations are given of
how the use of a dynamic geometry package such as Cabri-Géomètre
both needs and contributes to developing visualisation in all
these three senses.
Jones, K. (1996), Coming to know about "dependency"
within a dynamic geometry environment, Proceedings of the 20th
Conference of the International Group for the Psychology
of Mathematics Education. Valencia, Volume 3, 145-52.
Click
here for full article in pdf format.
Abstract: The ability to define relationships between objects
is one of the most powerful features of a dynamic geometry
package such as Cabri-Géomètre. In this paper I document
how one pair of 12 year old students begin to come to know about
this form of functional dependency within this particular
computer environment. I suggest that this process of coming to
know about dependency may be understood as an interweaving
between the "voices" of the students and the teacher
within the socially organised activity taking place in the
classroom.
Jones, K. (1996), Acquiring abstract geometrical concepts: the
interaction between the formal and the intuitive. Proceedings
of the Third British Congress on Mathematics Education,
Manchester, pp239-46.
Click
here for full article in pdf format.
Abstract: The acquiring of formal, abstract mathematical
concepts by students may be said to be one of the major goals of
mathematics teaching. How are such abstract concepts acquired?
How does this formal knowledge interact with the students'
intuitive knowledge of mathematics? How does the transition from
informal mathematical knowledge to formal mathematical knowledge
take place? This paper reports on a research project which is
examining the nature of the interaction and possible conflict
between the formal and the intuitive components of mathematical
activity. Details are presented of an initial study in which
mathematics graduates, who could be considered to have acquired
formal mathematical concepts, tackled a series of geometrical
problems. The study indicates the complex nature of the
interaction between formal and intuitive concepts of mathematics.
The plans for the next stage in the research project are
outlined.
Edwards, J. and Jones, K. (1996), Book review: Dynamic Geometry, edited by
Ronnie Goldstein, Hilary Povey and Peter Winbourne, Micromath,
12(3), 40-41.
Click here for the review in
html format.
Abstract: With the current and continuing attention on the
teaching of number, it is refreshing to welcome this useful
publication on geometry from NCET. Its aim is to provide
"ideas, suggestions and approaches" to using dynamic
geometry software in secondary mathematics based on the
experiences of eight teachers from four very different schools
who had access to such software over a term or so. In its twenty
pages there are a wealth of ideas and practical suggestions in this booklet.
Jones, K. (1995), Dynamic geometry contexts for proof as
explanation. In: L. Healy and C. Hoyles (Eds), Justifying
and Proving in School Mathematics. London, Institute of
Education, 142-54.
Click here for
full article in pdf format.
Abstract: Providing a mathematics curriculum that makes proof
accessible to school students appears to be difficult. This paper
describes work carried out in a secondary school mathematics
class in which students worked on tasks designed to enable them
to experience the necessity of certain geometrical facts
that are true in Euclidean geometry. In these tasks, the students
were asked to construct figures using the dynamic geometry
package Cabri-Géomètre such that each figure was
invariant when any basic object used in the construction was
dragged. It is argued that working on these tasks provided
the students with suitable experiences to enable them to explain why
these geometrical facts are necessarily true. The changing
quality of the students' mathematical analysis suggests that
working on suitable tasks with a dynamic geometry package may
allow some students to develop an appreciation of proof as
explanation.
Jones, K. (1995), Geometrical reasoning, Proceedings of the British
Society for Research into Learning Mathematics, 15(3), 43-47. ISSN: 1463-6840
Click here for
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 study 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, Proceedings
of the British Society for Research into Learning Mathematics, 15(3), 41-42. ISSN: 1463-6840
Click
here for the complete report in pdf format.
Abstract: This report of a meeting of the BSRLM 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.
Jones, K. (1995), Researching the learning of geometrical
concepts in the secondary classroom. Proceedings of the
British Society for Research into Learning Mathematics, 15(2),
31-34. ISSN: 1463-6840
Click
here for the complete report in pdf format.
Abstract: This article illustrates the ways in which 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
can provide problems for the researcher. Yet there are also
significant opportunities both to influence policy decisions and
to contribute to both theoretical and practical debates regarding
the teaching and learning of geometry.
Jones, K. (1994), On the nature and role of mathematical
intuition. Proceedings of the British Society for Research
into Learning Mathematics, 14(2), 59-64. ISSN: 1463-6840
Click here for the article in
pdf format.
Abstract: This paper discusses some of the issues surrounding the current
notion of mathematical intuition and how these might be researched.
Jones, K. (1993), Researching geometrical intuition. Proceedings
of the British Society for Research into Learning Mathematics, 13(3),
15-19. ISSN: 1463-6840
Click here for the article in
pdf format.
Abstract: This paper describes an exploratory study of the
nature and role of geometrical intuition in the solving of
geometrical problems. The results of the study suggest that
geometrical intuition plays an important role in the critical
decisions that problem solvers make when tackling geometrical
problems.
Homepage: Keith Jones
Collaborative Group for Research in Mathematics Education: CRME
For details of the BSRLM geometry working group, click here.
Page updated 27 October 2010