Proposed Schedule and Abstracts

Tuesday 8th July 12:00 – 13:00

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


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 China begun at the early of this century has made great change in the goal and contents of the geometry curriculum of compulsory education. Developing the spatial abilities and spatial sense should be the main aim in geometry curriculum. It hopes that this study can investigating the development rules, characteristics and levels of the spatial sense development of the pupils, so we can provide more evidences that the textbooks and teaching can do best for developing students spatial abilities and spatial sense.


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 3 in some aspects.

·         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 East Asia outperform their Western peers when solving complex mathematical tasks, though the Chinese teaching tradition focuses mainly on memorization and practicing simple procedure, was considered as a surprising result. In our research on geometric proof we studied the differences between individual proof competency of students from these two cultures (exemplified by Germany’s and Taiwan’s cultures). Results of an empirical quantitative study with students from grade 9 show, as expected, that Taiwanese students have a higher geometric proof competency than German students. Moreover, based on a deeper statistical analysis we hypothesize that the higher proof competency can be explained by a better quality of basic knowledge on geometric concepts and procedures.


Theme: Instruments' use

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

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 Alhambra’s nazari mosaics which are part of our cultural heritage. For that purpose we have used freeware dynamic geometry software (SGD) called Geogebra which was well accepted by our students. On the other hand our pupils had been working out the same patterns using manual techniques based on Escher’s drawings. It was a marvellous surprise, apart from the beauty that they discovered, when they found that the two approaches led to the same patterns.


D. Goreve, I. Gurvich, M. Barabash. Solving construction problems of different levels of insight: how students of various levels profit from the computerized environment?  

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: at the Department of of Mathematics in Modena, is a research centre for the teaching and learning of mathematics by means of instruments. The name comes from the most important collection of the Laboratory, containing more than 200 Mathematical Machines: that is, working reconstructions of many mathematical instruments taken from history of geometry (e.g., pantographs, curve drawers…)

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 12Some 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 Japan, the program Constitution of wide-angle mathematical basis focused on knots (Program leader Akio Kawauchi) runs as one of the National Programs, the “21st 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