__Proposed
Schedule and Abstracts__

**Tuesday 8th July 12:00 – 13:00**

Welcome and introduction to TSG12 on Geometry Fou-Lai
Lin and Alain Kuzniak

Presentations

*Theme: Teaching and Learning Geometry :
Abilities and conceptions*

Liu Xiaomei: Study on the
characteristics of development rules of pupil's spatial abilities

__Abstract.__ Spatial sense, as the basis of
the spatial imagination development, should be one of the main training aims of
the geometry curriculum during the primary education or even the compulsory
education. The mathematics curriculum reform of basic
education in

We designed the test paper (it includes four part, symmetry, rotation, orientation
and orthogonal views) and selected more than 500 pupils of grade2-grade6
(about age 7-age 11) from several schools to take the test. After the analyses
of the test results, we have some conclusions as below:

·
We
find the development level of pupil’s spatial sense of the grade2-grade3
(age7-age8) is comparative unanimous, the differences of the development level
of the pupil of grade4-6grade (age9-age11) are not very great. Therefore, in
the process of spatial sense development, we may consider age7-age8 and
age9-age11 as two development stages.

·
All the pupils of
grade2- grade6 can reach the requirement of the first level of the test
content, the grade2-grade3 pupils can reach the second level in some aspects,
and the grade4-grade 6 pupils can reach the level

·
The
spatial sense development has not apparent differences in sex. The spatial
sense of all the pupils from grade2-grade6 which were tested does not exist any apparent differences male and female pupils.

·
The
spatial sense development has a close positive relationship with the score of
mathematics. The spatial sense test results between the pupils with good
mathematics score and the pupil with poor mathematics score is quite

·
Teaching
interference in time is very necessary, if providing the pupils of lower grade
with some proper teaching interference materials, there would be better
development in spatial sense. If otherwise, the spatial sense development of
the pupil would be restrained and even it would be the weak-side which can not
be compensated. But to understand the concept, usually it is proper at the age
of 9-10 when the pupil in grade4.

Based on the study result
mentioned above, we propose some suggestions to how to arrange the related
contents and material of geometry curriculum for primary school.

Walter Whiteley, Margaret Sinclair. Stewart Craven, Lily Moshe, Anna Dutfield,
and Melissa Seco. Helping elementary teachers
develop visual and spatial skills for teaching geometry.

__Abstract.__ We will present findings from our
research into the development of visual and spatial skills related to the
teaching of geometry. The study involved 18 practicing elementary teachers,
who, over three day-long sessions, worked on tasks involving transformational
geometry concepts. The tasks focused on the use of physical,
and dynamic models to explore 2- and 3-D relationships. Observations and work
samples showed that participants had significant difficulties in regard to:
inquiry, interpretation, communication, and connections. However, we were
encouraged by participant successes in re-thinking misconceptions, and in developing
communication and representation skills. Based on our results, we believe that
given the right sequence of tasks designed to support adults’ thinking and
learning, the opportunity to work with a variety of materials,
and the appropriate amount of time, practicing teachers can develop a richer
background in the domain of geometry.

Brenda Strassfeld: High school
mathematics teacher's beliefs about the teaching of geometry.

__Abstract.__ There continues to exist
a dilemma about how, why and when geometry should be taught. Also an intervention case study was conducted
with one teacher. This paper describes
the case of Rose who had been one of the 520 respondents to a questionnaire
that was used to gather data about high school mathematics teachers' beliefs
about geometry and its teaching with respect to its role in the curriculum, the
uses of manipulatives and dynamic geometry software
in the classroom, and the role of proofs.
A three factor solution emerged from the data analysis that revealed *a disposition towards activities, a
disposition towards an appreciation of geometry and its applications* and* a disposition towards abstraction*. These results enabled classification of
teachers into one of eight groups depending on whether their scores were
positive or negative on the three factors.
Knowing the teacher typology allows for appropriate professional
development activities to be undertaken.
This was done in the case study of
Rose where techniques for scaffolding proofs were used as an intervention for
Rose who had a positive disposition towards activities and appreciation of
geometry and its applications but a negative disposition towards
abstraction. The intervention enabled
Rose successfully to teach her students how to understand and construct proofs.

**Wednesday 9th July 12:00 – 13:30 **

*Theme: The question of the proof*

AnnaMarie Conner: Argumentation
in a Geometry Class: Aligned with the Teacher’s Conception of Proof

__Abstract.__ While a straightforward relationship
between argumentation and proof is not universally accepted, it is reasonable
that argumentation, as a tool for student learning, would be instrumental in
the teaching of proof in high school geometry classes. This study examines the
argumentation in one student teacher’s high school geometry classes and
suggests a possible relationship between the observed argumentation and the
student teacher’s conception of proof. Aspects of the student teacher’s
conception of proof could be seen in how she supported argumentation in her
classes.

Aiso Heinze,
Stefan Ufer, Ying-Hao
Cheng, and Fou-Lai Lin. Geometrical Proof Competency
and its Individual Predictors – a Taiwanese – German study

__Abstract.__
Differences in learning outcomes between students from “Eastern” and “Western”
cultures are well documented and subject to research in mathematics education
for several years now. In particular, the fact that students from

*Theme: Instruments' use*

Jesús Hernandez Pérez. Dinamic geometrie software and Escher techiques
in la

__Abstract.__ In the School Year 2006-07 our
Institution took part in an European Comenius Project
called *Partners in Patterns. *Our work
was related to the elaboration of patterns generated by plane movements of
simple structures called minimum motives. We took as models the

D.
Goreve,

__Abstract.__ We decided to examine what our
students do with Dynamic Geometry Environments (DGE) when they become exposed to them. We
were also interested in the impact of computer usage on the development of a
student's insight, hoping to find ways to improve this impact as a result of
our research.

We
chose construction problems in geometry as a subject matter in the research
layout, mainly for two reasons: almost no knowledge of majority of our students
in this type of problems, and intrinsic qualities of DGE that render their
usage for this type of problems natural and useful.** **

To characterize the level of a student's geometrical
expertise, we used Van Hiele (VH) based tests to
assess their level according to the Van Hiele theory
of geometric development (Usiskin,1982), and Barabash-Guberman (BG) tool to assess the depth of a
student's insight at each level (Barabash, Guberman, 2006).** **We also based upon some findings of
our previous research (Gorev, Gurevich,
Barabash 2004).

The
research question we have posed was:

**What
are the different ways of using DGE when facing construction problems on different
insight levels (1 – 4) for students who are on lower VH levels (0 – 2), and for
students who are on higher VH levels (3 – 4)?**

We have found sound differences
between students classified into lower VH levels and those in higher levels. We
discovered numerous subtle differences concerning the inter-relations between
the insight level of the problem, the VH and insight level of a student, the
stage of the construction and a student's argumentation.

We
can say that the impact of DGE usage became visible in the manner of dealing
with complex construction problems. At the same time, inductive work culture
must be created. Most students had not previously dealt with justification
based on work in a computerized environment, and most of them had never been
exposed to this type of reasoning as valid reasoning. For them, visual
reasoning was not considered to be analytical.

We
suggest some impacts on teaching based on out results and observations.

Francesca Martignone: Mathematical
machines laboratory activities

__Abstract__: The Laborator of Mathematical Machines (MMLab:
www.mmlab.unimore.it) at the Department
of of Mathematics in

In recent years the MMLab research group (that includes academic researchers,
university students, teachers, and members of the association ‘Macchine Matematiche’)
coordinated by M.G. Bartolini Bussi
and M. Maschietto, carried out various activities
with the Mathematical Machines, namely: laboratory sessions aimed at classes of
students in secondary schools and groups of university students (an average of
1300-1500 secondary students a year come their mathematics teacher to
experience hands-on mathematics laboratory); long-term teaching experimentsin primary and secondary schools (Bartolini Bussi & Maschietto, 2008); workshops at conferences (national and
international) and exhibitions. Therefore, the MMLab
carries out both didactical research and the popularization of mathematics.

As the didactical research concerns,
the studies in the MMLab deal with the role of tools
in mathematics teaching and learning; these studies are framed by the
theoretical framework of semiotic mediation within a Vygotskian
perspective (Bartolini Bussi
and Mariotti, in press). Therefore, the activities
with the Mathematical Machines are included in the approach to mathematics by
means of cultural artefacts which evoke history and
require direct handling. In fact, the use of Mathematical Machines can bring
students closer to a historical (Barolinni Bussi, 1996) and physical dimension that is tangible thanks
to the construction of mathematical concepts. This richness in aspects,
mobilized by the activities carried out with the Mathematical Machines, has
opened interesting research perspectives, which are being developed. In this
articles are presented: the laboratory activities carried out in the MMLab, a long-term teaching experiment in a secondary
school classroom (grade 7) and a brief description of the researches in
progress.

Ryan Smith, Karen Hollebrands, Kathleen Iwancio,
and Irina Kogan. The Affects of a Dynamic Program for Geometry on College Students’
Understandings of Properties of Quadrilaterals in the Poincaré
Disk Model.

__Abstract__. Prior research on students’ uses of technology in the context of
Euclidean geometry has suggested technology can be used to support students’
understandings of properties of geometrical objects. This research study
examined the ways in which students used a dynamic geometry tool, NonEuclid, as they examined properties of quadrilaterals in
the Poincaré Disk model. Eight students enrolled in a
college geometry course participated in a series of task-based interviews, one of
which focused on quadrilaterals. The Van Hiele levels
were used to characterize students’ understandings of properties of
quadrilaterals and the ways in which students used the technology when
reasoning at each of these levels was identified.

**Friday 11th July 12:30 – 13:30**

*Theme: Teaching and learning geometry in elementary school*

Francisco Olvera, Gregoria
Guillén and Olimpia Figueras: Teaching geometry in elementary school.

__Abstract.__ This present
document contains an analysis of the teaching practice of two elementary school
teachers in service when they teach solid geometry in their classes. It is part
of an ampler research project (Olvera, 2007) in which
by means of the constitution of groups of teachers, it is intended to develop
strategies for planning teaching activities for introduction of solid geometry
in Mexican elementary schools.

In one
first stage a Local Theoretical Model (Filloy y
cols., 1999) referred to solid geometry teaching and learning in elementary
school was elaborated. For it, we were analyzed, on the one hand, works about
solid geometry teaching and learning (e. g. Guillen,
1997) and works developed in the line of teacher formation and/or thought (e.
g. Da Ponte and Chapman, 2006). Another hand, diverse
curricular documents for elementary school was analyzed. This allowed to need
the teaching model subscribed for solid geometry in Mexican elementary school
and to delimit the criteria to make analysis that displayed here.

In
order to characterize the teaching practice of teachers class situations were
observed and individual interviews were made. With it could be determined
beliefs, conceptions and knowledge in relation to the different phases in which
teacher work moments are define, before, during and after teaching. As it
leaves from the results obtained in the study is possible to emphasize: i) Solid geometry knowledge that teachers have, influences
on the classes planning. ii) The use that they are conferred to solid geometry
is poor, as forms that are in the environment or as forms that can be modeled.
iii) The evaluation of the knowledge learned by their students is made from
their written productions. iv) The participant
teachers face difficulties to improve teaching that they distribute.

It is
possible to stand out the influence that text book had in the different phases
from the educational work. In agreement with the results obtained until this
moment, the convenience feels of facilitating teachers’ formation in solid
geometry, if we want that the actual situation of teaching this subject in
elementary school changes.

Jiří
Vaníček. Procedural
Part of geometrical objects in concepts and strategies of primary school

__Abstract.__ The article concerns the research of the
geometry perception teaching at primary school pupils by means of computer
application involving turtle graphics. Chidren had to
pass several tasks to fill a given figure in the application called Encircler. They had to encrypt so-called procedural part of
the shape and use the turtle program to encircle it. They were forbidden from
filling the whole figure only by paving without using the turtle program.

The course of research tries to catch in what manner and what strategy a pupil divides the figure into parts created by procedural way and how to discover the symmetry of the shape parts and its reflection in the written program, what help gives the elementary geometrical shapes from which the figure is possible to compose. About 200 pupils in some Czech schools in the classes from three to five grades were tested. The results of the main tests are commented in this paper.

Deborah Moore-Russo. Teachers exploring and
re-forming the concept of parallel.

__Abstract.__ The concept of parallel is vital in the
study of mathematics and has been critical in the historical development of
geometry. This paper reports on a
qualitative research study that examined how preservice
and inservice secondary mathematics teachers explored
and re-formed the concept of parallel through a series of individual and small
group activities involving extensive, collaborative discussion and definition
creation. Results show that 1)
definition creation was a novel, highly beneficial activity for the teachers
who initially produced uneconomical, incomplete, and even incorrect definitions
for parallel and that 2) some secondary mathematics teachers, with mathematics
undergraduate degrees, struggled to develop a rich sense of what it means to be
parallel.

*Theme: Questioning the curricula*

Wang Linquan. Chinese reformation of
mathematics curriculum in geometry

__Abstract:__
Geometry is one of the most important components of Chinese school
mathematics. Since 2001, there is a large reformation of Mathematics has been
happen not only in the content of School Mathematics, but also in the
philosophy of teaching, learning and activities outside the class. Close
attention is also given to embed the Mathematics thinking methodology on the
process of instruction and problem solving. Based on the New Curriculum, we are
trying to reform the instruction of geometry. We are still challenged by the
new standards of middle school and high school mathematics. We have made some
progress, a number of problems have been encountered.

Aim
of teaching geometry:

Chinese
educators try to help students set up a solid foundation, acquire experience of
abstract from, classify or move to figures, recognize their characteristics, to
find the location of an objects. We also try to help students to master basic
knowledge and essential skills. Chinese teachers pay attention to cultivate students abilities of spatial sense and ability of
geometrical intuition which consist of the basic factors.

Structure
of geometry in School Mathematics

According
to the importance of the knowledge and ability of a student’s learning, the
requirements of teaching geometry are divided into four levels.

Level
1: Know something (elementary level)

Level
2: Correctly understand (developing level)

Level
3: Master the topic and be able to apply it (advanced level)

Level
4: Master the topic very well, and be able to apply it skillfully (highest
level).

Different emphases of students’
abilities

Most teachers pay attention to
improving students mathematical ability, but we have our own emphases. We stress
improving students’ ability in mathematical thinking. In our opinion, students
should be able to:

observe, compare, analyze, synthesize,
abstract and generalize;

mathematical reasoning using induction,
deduction, and analogy methods;

express their own thoughts or points of
view logically and exactly;

discover mathematics relationships
according to suitable concepts and using methods in their own levels.

Current Adjustment of the content
of School Geometry

We simplify the system of
geometrical axioms and theorems, reduce the requirement of geometrical
reasoning and proving, just as

simplify the system of Geometrical axioms
and theorems. reduce the complexity for the problem
solving.

increase emphasis to the activities of practice.;To increase emphasis to the geometrical
transformation

New Advancement of teaching
Geometry in high school.（Grade
10 to 12）Some challenges and contradiction
we are confronted with in the reformation.

Tomoko
Yanagimoto: An Approach to
Teaching Knot Theory in Schools

__Abstract__. Knot
Theory has been now studied actively world-wide. One of the reasons is
that Mathematical Knot is so simple,
but it has many
unsolved problems. Another reason is that it is known to be related to scientific research
fields, such as Genome DNA.

In ^{st} Century COE Program”. Our program “Teaching
Knot Theory in Schools” is served as a part of this
program. This report
outlines the educational significance of teaching Knot Theory in elementary school. An experimental research of this paper suggests that
Knot Theory is useful as an effective teaching material on the purpose of
improving the pupils’ sense of space.

**Saturday 12th July 12:00 – 13:30**

*Theme: Teacher Preparation and Development*

Helena Boublil. The Challenge: Didactic
of Geometry to Prospective Teachers

__Abstract__. This research is centered on the appropriation of geometrical and
didactical knowledge within the initial preparation of primary teachers. It
seeks to coordinate the didactic preparation and the geometrical preparation in
such a way that it will enhance the teaching capabilities and skills of future
teachers.

This research has a structure which brings together the objectives
of geometrical preparation (the development of visualization, language and
reasoning) and of the didactic formation (the development of didactic knowledge
allowing to intervene in the context of teaching of geometry). The different geometrical and didactical
knowledge were put into the context of questions, problems and didactical
situations. (Brousseau, 1998). They draw on the various types of geometrical
activities (observation, manipulation, construction, problem solving…)
incorporating into them the concepts, approaches and employment didactical
means. These activities encompass the multiplicity and coordination of the
“registers of representation” (Duval, 1995) and aims at the progress of geometrical
preparation according to “levels of geometrical thinking” (van Hiele, 1959/1985).

The methodology of didactic engineering (Artigue,
1988) was employed as a means of organizing the "didactic engineering of
teaching formation" which guides the functions of different
"engineering of geometrical concepts" within the realm of the
training-formation. According to the theoretical framework developed, we have
analyzed the role of “training situations” in the geometrical and didactical
evolution of 116 students, future teachers. We evaluated the functions of
geometrical concepts in various situations as well as student’s participation
and decision making processes in practical works of teacher training formation
(didactical questions exams and essay on design and planning of teaching
situations). We connect the decision making processes to** **the didactic
and geometrical knowledge studied within the framework of the course
objectives.

Final group discussion