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1Peter Appelbaum, Arcadia University, USASense and representation in elementary mathematicsTwo concepts are central to contemporary mathematics education theory and practice: the support of sense-making by pupils, and support for developing facility with representations. This presentation problematizes and recasts both of these concepts by framing learners as artists - creators and producers - within a curriculum that usually wants them to "consume and use" instead. A common assumption is that mathematics curriculum is content that represents and interprets. Applying work of Sontag that argued against representational art, we can generate new forms of learning activities where artists evoke parody, abstraction, decoration, and non-art in ways that make mathematics vibrant and relevant to several of our conference themes.
2Milan Hejný Charles University, Prague, Czech RepublicScheme oriented educational strategy in mathematicSchema is understood as the memory structure that incorporates clusters of information relevant to comprehension. It gets embedded in a person's mind by repeated stay in a certain kind of environment (one's house, school, shopping centre). Scheme oriented mathematics education is described, discussed and illustrated on the primary level.
3João Pedro Da Ponte Research Group DIFMAT Centro De Investigação Em Educação E epartamento De Educação Faculdade De Ciências Da Universidade De LisboaInvestigating mathematics: A challenge for students, teachers, and mathematics education researchersI start this paper by considering different conceptions of mathematics and showing how exploring and investigating are central to the mathematical activity. Next, I present several examples of students investigating mathematics in the classroom that illustrate important aspects of an exploratory approach to mathematics teaching and learning. Such as approach does not depend only on the nature of tasks, but requires an analysis of the roles of teachers and students, the communication patterns in the classroom as well as the overall organization of content and processes in meaningful mathematics teaching units. I indicate that this kind of teaching is rather demanding and refer three main conditions to help teachers in developing professionally to carry it out: collaborating, researching their own practice and getting involved in the professional community. I close with a brief discussion on the relationships of investigating, teaching, and learning. I argue that as students explore and investigate mathematics, teachers need to investigate their own practice in professional collaborative settings.
4Zbigniew Semadeni Institute of Mathematics, University of Warsaw Factors hampering independent mathematical thinkingThe talk will concentrate on various obstacles which hamper the implementation of the idea of independent mathematical thinking and on some ways of overcoming them.
1Jenni Back And Tony Beauchamp National Centre For Excellence In Teaching Mathematics And King's College London, Independent Research ConsultantChange: what change? Considering whether a national initiative has had an impact on the quality of classroom talk in primary mathematics This paper will consider the transcripts of two mathematics lessons in a primary school as 'telling cases' (Mitchell 1984) which illustrate the nature of communication in primary mathematics classrooms in the United Kingdom at present. Using qualitative analysis and a theoretical framework developed as part of an earlier study, the quality of the talk is examined and the nature of the social and mathematical dimensions of the talk analysed. Quantitative analysis explores the proportions of the talk that are attributed to the pupils and to their teachers and comparisons are made with earlier findings. The examples are selected to throw light on the quality of mathematical communication and the nature of participation in the classroom talk by the teachers and pupils involved.
2Birgit Brandt , Johann W. Goethe-Universität, Germany Konstantinos Tatsis University of The Aegean, GreeceAnalysing interaction processes with Jigsaw during Mathematics Lessons in Elementary SchoolThe interaction processes that stem from the Jigsaw cooperation form can be analysed by decomposing the recipients' roles based on their interactional status and their interpersonal speech acts. In this paper we elaborate on the influence of an individual's participation form for the ongoing interaction process.
3Ivona Grzegorczyk California State University Channel Islands And Educational Games Research InstituteDevelopment of abstract thinking in mathematics in computer enviroment This paper is summarizing ongoing research into the use of computer games as a tool for developing abstract thinking in a mathematics classroom. We will discuss the results of three educational gaming environments supplementing mathematics instruction for teenagers that were designed and student tested not only for improvement of content knowledge, but also for innovative thinking and abstraction development. The study shows that a carefully chosen digital learning environment stimulates a higher-level thinking in the majority of players.
4Ján Gunčaga Department Of Mathematics Pedagogical Faculty Catholic University In Ruzomberok SlovakiaPreparing of pupils for notion of limitsIn this article we present our results concerning qualitative and quantitative research carried out at the St Andrew secondary school in Ruzomberok in October 2007. The research was aimed at the relationships between input and output factors in the teaching process. In the article we describe the results of input test.
5Eszter Herendiné-Kónya F. Kölcsey Reformed Teacher Training College, Debrecen, HungaryGeometrical transformations and the concept of cyclic orderingIn this paper we describe a research on the connection between geometrical transformations and orientation. We discuss the particulars of the thinking process and typical difficulties connected to the field of geometrical transformations. We pay special attention to the problem of cyclic order.We investigate pupils' competence in primary school, especially in Grade 2 (age 7-8).
6Edyta Jagoda Charter School Nr 1, Rzeszow, PolandBuilding the concept of line symmetry on the planeIn his paper I have took up analysis of the key processes characteristic to the levels of reasoning in geometry. I have discussed particular of the thinking processes and their functioning by 12- 16 year old students with indication to the concept of line symmetry on the plane.
7Michaela Kaslová UK V Praze, Pedf - Kmdm, Czech RepublicExpression of quantity - language of children 5-7 years oldExpressing quantity is traditionally associated with numbers. That is why it is important to examine how quantity is perceived and expressed by children? Does specification of quantitative information depend only on child evolution? What instruments are used by children to express quantity of an object?
8Ivana Kovárová The Mathematical Institute, Faculty of Science, Pavol Jozef Šafárik University in Košice, SlovakiaApproaches to solving processes of one fuzzy problem by primary school childrenWe have focused on description of different pupils' ways of solving the fuzzy problem and to monitor development of mathematical thinking depending on the children's age. This is based on the process of solving the problem. Results of this experiment are obtained by the method of detailed analysis of pupils' solutions.
9Renata Kozieł University of Silesia, Katowice Faculty of Ethnology and Sciences of EducationTeaching computer-assisted mathematics to primary studentsThe subject of the research report is the natural pedagogical experiment which was to show an innovative didactic situation and changes in arithmetic knowledge and skills learnt by first grade elementary students.
10Graham H Littler University Of Derby, UK

Darina Jirotková Charles University, Prague, Czech Republic
Primary school pupils' misconceptions in numberA group of four universities from the UK, Czech Republic, Israel and Italy collaborated to find common misconceptions in number across the four countries. The paper cites some of the misconceptions founding all the countries and looks in detail at a specific task which the authors developed together with the results gained from the task over the whole primary range of pupils and beyond.
11Bożena Maj University of Rzeszow,
Developing creative mathematical activities during lessons of mathematicsThis paper presents a research project conducted among mathematics teachers. The aim of the project was to improve the teachers' ability to develop creative mathematical activities. For that purpose, diagnostic activities, workshops and lessons' observations were organized. The results show a considerable improvement in the teachers' ability and their attitude towards mathematical activities.
12Ema Mamede University Of Minho, PortugalFocusing on children's early ideas of fractionsThis paper describes children's understanding of quantities represented by fractions in quotient, part-whole and operator situations. The studies involve two samples of first-grade children, aged 6 and 7 years from Braga, Portugal. These children were not taught about fractions before. Two questions were addressed: (1) How do children understand the equivalence of fractions in quotient, part-whole and operator situations? (2) How do they master the ordering of fractions in these situations? Quantitative analysis showed that the situations in which the concept of fractions is used affected children's understanding of the quantities represented by fractions; their performance in quotient situations was better than their performance in the other situations.
13Carlo Marchini Mathematics Department University Of Parma, Italy

Anne Cockburn, School Of Education And Lifelong Learning University Of East Anglia,
Norwich - U.K
Teaching practices revealed through arithmetic misconceptionsThe aim of our research was to investigate whether primary school pupils (7-11 year old) can distinguish between simple arithmetic errors and misconceptions. Data from a specifically designed task are presented. We demonstrate the importance of teachers' practice in the development of pupils' reflexive thinking.
14Carlo Marchini, Paola Vighi. Mathematics Department of University of Parma, Italy

Ewa Swoboda University of Rzeszów, Poland
How to reveal geometrical independent thinking in the lower primary yearsIn School Year 2005/2006, we extended, in Italy, a part of a research initiated by Ewa Swoboda in Poland. Our research used simple tools but our results indicate that certain geometric aspects are present in the mind of pupils before a formal teaching of geometry.
15A. W. Merlin, N. I. Merlina, Chuvash State University after I.N. Ulyanov, Russia. Mathematics Faculty Mathematics analyses and differentiated equations, Methods of teaching MathematicsSelf-organization and TRIZ in teaching Mathematics in the creative work with gifted childrenDuring the latest years active searching of innovative forms and methods of teaching both beginners, middle aged and senior pupils are conducted. It's done so in order to stimulate the pupils ' intellectual growth. The orientation on the child's inner experience is one of the central ideas in the field of school pedagogies. At present there has appeared a new trend: a TRIZ pedagogic was founded in our country at the end of the 80s last century as a scientific and pedagogical branch. The theory of solving invention problems was its basis and it was founded by the Russian (more exactly Soviet) school of G.S. Altschuller.
16Daria Muzyczka Wielogłowy Group of SchoolsThe differences in mathematical thinking of children in the in-school and out-of-school situations.The report refers to the preliminary diagnosis of the difficulties that children have while adapting their mathematical skills acquired at school in real-life situations. As the primary school students and their parents stated, they rarely find school knowledge useful in real-life situations. Simultaneously, they do not try to use their out-of school experience in math classes. The problem was also noticed while analyzing the results of the competence tests conducted annually in the sixth class of primary schools. Thus we can observe a divergence between the "school" and "life" mathematics, which results in the students' unwillingness to learn, their lack of effectiveness and problems at tests and external exams.
17Ioannis Papadopoulos University of Patras, GreeceDeveloping problem solving strategies via estimating the area of irregular shapesIn this paper we are interested in the work of 6th graders (11-12 years old) when they face non-standard tasks with the area of irregular shapes. These shapes add to the students an extra level of difficulty when they include curved lines in their boundaries. We record the strategies the students decided to apply in order to overcome these difficulties. We further analyze their work for whether there is evidence the strategies are primarily due to the computer environment or the use of paper and pencil.
18Gabriela Pavlovičová - Valéria Vasková Constantine The Philosopher University In Nitra, SlovakiaPerceptions of numbers of from 5 to 6 aged childrenIn this article we deal with observation of children's perceptions of number. We investigate process in which to numerical information the conceptions of numbers are assigned. The numerical information is word three and the child's drawing is used to mediate the numerical conceptions of children. The experiment was realized with the children in the kindergarten. We analysed drawn children's conceptions of number 3 and created concept map from those drawings.
19Marta Pytlak, University of Rzeszów, PolandConnections - as a fundamental element to constructing mathematical knowledge (exemplify of one pre-algebra task)Polish primary school do not talk about algebra a lot. Teaching mathematics on this level of education is oriented on arithmetic. In my paper, I would like to present a part of results from my research carried out in a primary school among fourth grade students. This research concerns discovering the regularity, which leads to algebraic reflection
20Filip Roubíček Institute of Mathematics of The Academy of Sciences of The Czech Republic, PragueThe pupils' interpretation of mathematical writingsCommunication in the teaching of mathematics is distinguished by using various systems of semiotic representation. Their usage is often given by customs or conventions in mathematics. The acquirement of rules how to form, interpret, and use them properly plays an important role in communication and cognitive processes. The problems in communication tend to be caused by not only different mathematical images or communication contexts but also non-accepting of conventional mathematical writing by the pupils or insufficient acquaintance of the pupils with rules of their usage. Some cases of pupils' inventional representations are presented and analysed.
21Maria De Fátima Sardinha; Pedro Palhares; Fernando Azevedo LIBEC/CIFPEC, Institute Of Child Studies, University Of Minho, PortugalProblem stories in the education for numeracy and literacy"Problem stories" are a teaching strategy for problem posing. This work is about the analysis, through an interdisciplinary approach, of the implementation of this teaching strategy that joins Mathematics with Portuguese Language and promotes an education for numeracy and literacy. We present and discuss some studies stressing the concern, at an international level, like the Organisation for Economic Co-Operation and Development (OECD) showed in the Programme for International Student Assessment (PISA), on the levels of numeracy and literacy of the whole population, as well as on the development of competencies. We also present and discuss definitions central to the understanding of this teaching method, particularly of problem posing and solving, and about the construction of stories.
22Ewa Swoboda, University of Rzeszów, Joanna Synoś, Public Kindregarten No 17 in Rzeszów. Danuta Pluta, Public Kindregarten No 17 in Rzeszów,
Various manipulation functions in solving geometrical tasks by 4-6 years old childrenIn educational studies of mathematics, the role of manipulation is highlighted. The action is a base for learning an early arithmetic. Manipulation in learning geometry is an argumentative topic, because of different theoretical bases for creation of geometrical concepts. Some theories underline a great importance of visual information in forming the first level of understanding geometry. Such approach is present in works of P. Vopěnka or M. Hejný. From our former experiments it results, that children are able to act in their early years in the geometrical world. Assuming that visual information gives the first stimulus for creation of geometrical concept, we undertook the experiment to observe the role of manipulation in early geometry.
23Margit Tarcsi Ferenc Kölcsey Reformed Teacher Training College, Debrecen, HungaryActivities in the preparation for and establishment of the concept of length and perimeterIn this paper an experiment is described, which focuses on the contribution of the various activities and devices to the preparation for and establishment of the concept of perimeter. The formation of the concept of rectangles and squares and their interrelationship are also analysed. The aim is to demonstrate to what extent and how the various methods such as whole class teaching, group and pair work help learners to gain knowledge, to develop their thinking and creativity in this area.
24Konstantinos Tatsis, Sonia Kafoussi And Chrysanthi Skoumpourdi University Of The Aegean, GreeceDiscussing on the fairness of probabilistic games: The creation of a discursive community with kindergarten childrenIn the present paper we provide an analysis of the verbal interactions that took place during the realisation of two activities in a kindergarten classroom. Focusing on discussions about the fairness of the games played we observed the strategies that children used to justify their opinion and monitored the development of their intuitions concerning the fairness of a game.
25Jiří Vaníček University Of South Bohemia In České Budějovice, Faculty Of Education, Czech RepublicProcedural and paving way of building-up of geometrical object in concepts and strategies of primary school pupilsThis presentation concerns geometry perceptions of primary school pupils in computer applications involving turtle graphics. The main goal was to analyze which strategies children use by discovering which part of a given shape they are capable of drawing by applying an algorithm built from commands of several given types. The application called Obkreslovačka (Tracer) has been created for assessment in schools. Pupils' strategies and procedures were analyzed, with particular attention to how they divide a figure into parts within their turtle procedures, and how strategies and procedures depended on the specific geometrical shapes involved. This presentation reports on a set of tests of about 11-years-old pupils in 28 South Bohemian schools.
1Anne Cockburn, Eleanor Cockerton, Ralph Manning & Paul Parslow-Williams University Of East Anglia, U.K.Thinking to the future: prospective teachers encouraging children's mathematical thinking.This workshop is a practical exploration into the lessons of two prospective primary teachers half way through their teacher preparation course. The aim is to develop the quality of their education and, in turn, enhance the mathematical experience and prospects of children in the future.
2Ronit Hoffmann, Ronith Klein Kibbutzim College Of Education, Technology And Art, IsraelPromoting thinking through a modern approach for problem solvingThe purpose of the workshop is to introduce the participants to a unique way of problem solving involving algorithmic thinking and the use of computers in the mathematics classroom. In the workshop we will focus on problems for pre-service teachers for the elementary school. The participants will write algorithms and excel programs for their implementation.
3Silva Kmetič The National Education Institute SloveniaMathematics through investigations and problem solvingThe workshop target is to present an idea how mathematical thinking can be developed through investigations and problem solving and still covering traditional curricula. The workshop session will include the activities with starting points derived from known explanatory proof, from the standard textbook exercise and from real life situation. We will discuss about feelings, motivation, safety, confidence and competence, posing questions, extending problems and we will share our experiences. Good problem solver need to be willing to explore, to be persistent and tolerant to certain amount of frustration. For these characteristics learning environment need to be safe and simulative what would be also discussed.
4Tatyana Oleinik, Nikolay Ivashenko, Andrey Prokopenko, UkraineProblems of formation learning environmentWe discuss on major problems and ways of Ukrainian pedagogical innovation which based on democratic transformations in education. Our special study is formation learning community (environment) of successful learners.
5Monique Pijls & Dain De Kramer University of Amsterdam, Graduate School of Teaching and LearningStudents discussing their mathematical ideas: group-tests and mind-mapsIn an explorative research project, teachers experimented with new ideas to make their students discuss (i.e. Show, explain, justify and reconstruct their work) their mathematical ideas with each other. Two kind of special tasks were developed: group tests and mind maps. Also, the role of the teacher was studied in order to evoke discussions between students. In this workshop, tasks and results will be presented, experienced and discussed.
6Malka Sheffet Ronit Bassan-Cincinatus Kibbutzim College of Education, Israel.Percentages are not another name for fractions. Workshop dealing with percentages.One of the commonly used mathematical subjects in every day life, and in sciences, is percentage. Therefore, it is our duty, as educators, to take care that our students will possess a well-founded understanding of this subject.
7Lambrecht Spijkerboer APS-Institute, The Netherlands.Designing mathematics walksThere is a lot of mathematics to be found in the streets. The world around us provides many opportunities to come up with 'mathematical' tasks based on everyday situations. Realistic problems lead to authentic presentations of questions to be tackled by pupils of various levels. In maths classes in school, pupils are confronted with descriptions of everyday situations. They have read and realise what is the situation in order to use their mathematical knowledge to solve those everyday situations. These descriptions of every day situations are not needed in mathematics walks, because you meet the situation and that makes the questions more obvious. Moreover, pupils are encouraged by the discovery that maths knowledge and skills actually help them to cope with many real-life problems. In mathematics walks we aim to bring pupils into daily-life situations in which they see that you can recognise, use and discover mathematics. Maths turns out to be versatile, interesting, enjoyable and sometimes surprising. Reason enough to get out of the classroom and take a walk in the school's surroundings.
8Marie Tichá Institute of Mathematics As Cr, Praha, Czech RepublicDeveloping teachers' subject didactic competence: case of problem posingIn the workshop, problem posing will be discussed as one of the ways of empowering subject didactical competence (pedagogical content knowledge). The work will start with examples of problems posed by student teachers and elementary teachers.
1Jenni Back, Els De Geest, Ros Sutherland, Marie Joubert And Christine HirstDeveloping the teaching of mathematics: looking at Continuing Professional Development (CPD) initiatives
2Jana Cachova Pdf Uhk, Czech RepublicThe method of incomplete pictures and pupils' ideas
3Barbora Divišová Charles University in Prague, Faculty of Education, Czech RepublicSome Geometric Problems Which Help to Overcome a Pupil's Tendency to Use Formulas
4Agata Hoffmann Institute of Mathematics, University of Wrocław, PolandIs it possible to teach our pupils to think independently?
5Emmanuel Jayko Jaiyeola, Department 0f State For Education (Dose) The GambiaProcess of change of teaching on ratio and proportion by making aware of a knowledge acquisition model: case study
6Alicja Michalska, Primary And Junior High School Complex in Bieździedza, Poland Lidia Zaręba, Pedagogical University of Cracow, PolandFrom research on understanding a letter symbol by students of a junior high school
7Filip Roubíček Institute of Mathematics of The Academy of Sciences of The Czech Republic, PragueLanguage structures of pupils within problem posing and problem solving
8Naďa Stehlíková Charles University In Prague, Faculty of Education. Czech RepublicHow do practising teachers use projects in their teaching of mathematics?
9Aleksandra Urbańska Pedagogical University of Krakow, PolandOn the numerical competence of six-year-old children
10Michaela Ulrychová Charles University In Prague, Faculty of Education, Czech RepublicDevelopment of functional thinking
11Valéria Vasková, Gabriela Pavlovičová Constantine The Philosopher University In Nitra, SlovakiaChildren's models of number 3 at the age of 6 - 7
12Vighi Paola Mathematics Department Of University Of Parma, ItalyFrom a caterpillar to a butterfly: a learning project for kindergarten
13Iryna Zarichna Lviv Regional Scientific-Methodological Institute Of Education, UkraineOn mathematical education of 6 years old children in Ukraine

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