CME 2010 - Children's Mathematical Education

	Author	Title	Abstract
PLENARY LECTURES
1	Günter Krauthausen/University of Hamburg, Petra Scherer/University of Bielefeld Germany	Natural Differentiation in Mathematics (NaDiMa) Theoretical Backgrounds & Selected Arithmetical Learning Environments	Coping with heterogeneity and motivation of all students, independent from their capabilities, is one of the great challenges in primary school, especially for teaching and learning mathematics. The international project NaDiMa investigates the opportunities and the implementation of so called ›substantial learning environments‹ that allow a kind of ›natural differentiation‹. The crucial point with this kind of differentiation is that all students work together – on the same content, in accordance to their individual level. For learners, this natural differentiation should contribute to a deeper mathematical understanding as well as to the development of general learning strategies that could lead to a higher motivation. The NaDiMa project is meant as a contribution to the theoretical concept formation as well as to empirical data and consequences for differentiation and motivation. During the talk the theoretical background, the project objectives and realizations from the German team (arithmetical learning environments) as well as project results will be presented.
2	Stefan Turnau Poland	Theory of Didactic Situations from the Polish perspective	The Theory of Didactic Situations was initiated in the early seventies by Guy Brousseau, and has been developed since mainly in France. Among many theories modeling the process of mathematics learning and teaching it is "in my view" the one closest to the reality of the mathematics classroom. It helps to analyze and describe the phenomena of interactions between teacher and students, as well as students and mathematical knowledge. It also provides sort of guide for designing the teaching process or chain of didactic situations so as acquiring the knowledge aimed at to be guaranteed. The theory classifies typical didactic situations pointing those that lead not to proper learning of mathematics, yet being very common in mathematics classrooms. The one probably most common is the so called situation of institutionalization , which develops the view of mathematics as a system of laws and rules like the legal system. The theory is rather oblivious outside France, probably because it’s application for planning the teaching process, or didactic engineering , is very demanding for the teacher, requiring deep and large knowledge, as well as much more preparatory work than a teacher is able to afford.
3	Pessia Tsamir and Dina Tirosh Tel-Aviv University Israel	Defining And Proving With Teachers: From Preschool To Secondary School	This paper describes our work with K teachers' on definitions of triangles, with elementary school teachers on definitions of parity, and with secondary school teachers on proofs, validating or refuting elementary number theory statements. We illustrate and examine the Pair Dialogue approach that we use in professional development programs when working with teachers on their knowledge needed for teaching. Here we focus on issues related to teachers' mathematical knowledge.
4	Maarten Dolk The Netherlands	Teachers Practices Supporting Classroom Conversations
RESEARCH REPORTS
1	Jenni Back University of Plymouth Marie Joubert, University of Bristol United Kingdom	Professional Development For Teachers Of Mathematics: Engaging With Research And Student Learning	Drawing on data from a national project in England, researching effective professional development for teachers of mathematics to students of all ages (NCETM 2009), the paper considers the relationship between professional development, educational research and students’ learning. It considers three case studies of professional development for elementary and early childhood practitioners and explores how they made use of research findings and considered student learning in the ways in which they worked with teachers. The teachers’ responses to these ways of working are presented through a mixture of evidence taken from classroom observation and interview and the possible long term implications of the professional development are assessed.
2	Jiří Bureš, Hana Nováková, Jarmila Novotná Charles University in Prague, Faculty of Education Czech Republic	Devolution As A Motivating Factor In Teaching Mathematics	The paper presents a part of a broader research developed in the frame of the Theory of didactical situations in mathematics and focusing on how to understand, study and improve students’ school culture in case of problem solving in mathematics education. It studies the influence of the process of devolution and problem posing on students’ approach to problem solving in mathematics and on their motivation to transform their traditional approaches to it into more active and more "mathematical" ones. The five-stage organization of the developed didactical situation is described with respect to the role of devolution in each of the stages.
3	Paula Cardoso and Ema Mamede Sec. School Alberto Sampaio and CIFPEC - University of Minho Portugal	Analysing The Effects Of Situations On Fractions Learning Environments – The Case Of Quotient Situations	This paper describes a study on the effects of quotient interpretation on students’ understanding of the concept of fraction. An intervention program and pre- and post-tests were conducted with students between 11 and 12 years of age (N=84) from Braga, Portugal. A quantitative analysis showed that students improved their ideas about the equivalence and ordering of fractions after working with fractions presented in quotient situations; their performance on solving problems of naming, ordering and equivalence of fractions presented in part-whole and operator situations improved as well. Implications for rational numbers learning environments are discussed.
4	Despina Desli Democritus University of Thrace, School of Education Sciences in Pre-School Age Greece	Young Children’s Organization And Understanding Of Data In Everyday Mathematics Situations	This study examined how young children organize and interpret data that come from real-life situations presented graphically. Fifty Kindergarten and fifty Year 1 children were asked to construct and interpret picture graphs and symbolic graphs with two and four bars. Their performance in constructing the graphs was not affected neither by the age nor by the type of graphs. Differences in interpreting the data in the graphs were observed with the oldest children showing more complex reasoning. Similar reasoning across age groups was found in the different types of graphs. Educational implications are discussed in accordance to the need of using graphs as a tool to help children solve the range of mathematics problem they could encounter.
5	Barbora Divišová, Naďa Stehlíková Charles University in Prague, Faculty of Education Czech Republic	Ability To See In Geometry Or A Geometric Eye	The article deals with a special type of mathematical problems which we call (geometric) problems effectively solvable by insight (or PSI problems). The PSI problems have a quick geometric solution, nevertheless, pupils often attempt a much more complicated or even impossible algebraic solution. The text presents a theoretical framework (seeing in geometry, geometric eye, insight, schema) and results of related research. The methodology of our research is described and preliminary results of the pilot study discussed.
6	Elita Volāne, Elfrīda Krastiņa, Elga Drelinga Riga Education and Management Academy, Daugavpils University Latvia	Of Mathematics, Domestic Science And Technologies In Integrated Learning At Primary School	development and value acknowledgement create the necessity for acquiring the content of learning as a whole. Integrated learning is one of the means that make it possible. The present article regards the theoretical prerequisites of framing the content of learning in mathematics, domestic science, and technologies and the pilot research data testifying to the possibility of holistic learning of these subjects. The conceptual standpoints provide the basis for the teaching material “Practical Mathematics for Form 1”(2008) by the authors of the present article.
7	Francesca Ferrara, Ketty Savioli Dipartimento di Matematica, Università di Torino Italy	Acquiring A Sense Of Motion: Toward The Concept Of Function At Primary School	The research discusses a long-term teaching experiment in which some children have to study graphs representing motion data using technological devices. The analysis of two episodes, related to Benny’s behaviour, will show the relevance of perceptuo-motor activities in understanding processes, and traces of the hypothesized multimodal nature of mathematical learning.
8	Jodie Hunter University of Plymouth UK	Developing Early Algebraic Reasoning Through Exploration Of Odd And Even Numbers	Student transition from arithmetic to algebraic reasoning is recognised as an important but complex process. An essential element of the transition is the opportunity for students to make conjectures, justify, and generalise mathematical ideas concerning number properties. Drawing on findings from a classroom-based study, this paper outlines how odd and even numbers provided an appropriate context for young students to learn to make conjectures and generalisations. Tasks, concrete materials and specific pedagogical actions were important factors in students’ development of algebraic reasoning.
9	Edyta Juskowiak Adam Mickiewicz University, Poznań Poland	Graphic Calculator As A Tool For Provoking Students’ Creative Mathematical Activity	The article presents a part of a research, whose goal was to study and describe the ways of applying the graphic calculator by 14-year-old students when solving a specific kind of tasks. This article attempts to answer a question: What mathematical activities does the graphic calculator provoke?, with an emphasis on the manifestations of the students’ creative activities. A unique calculator program enabling recording as well as replaying the work with this particular tool has been applied during the research.
10	Vida Manfreda Kolar, Tatjana Hodnik Čadež University of Ljubljana, Faculty of Education Slovenia	Didactic Material As A Mediator Between Physical Manipulation And Thought Processes In Learning Mathematics	The use of didactic material in mathematics classes has an important role in the formation of mathematical thinking. This article focuses on some problems related to the use of didactic material in teaching and learning mathematics from the aspect of associating physical manipulation and thought processes. The article presents the results of an empirical study that aimed to determine whether the views on the issue of didactic material in teaching and learning mathematics depend on the status of respondents. The answers of teachers and students to a set of questions explored the influence of teachers’ practical experience on their attitudes towards the role of didactic material in mathematics classes.
11	Eszter Herendiné-Kónya and Margit Tarcsi Ferenc Kölcsey Reformed Teacher Training College Hungary	Preparation For And Teaching Of The Concept Of Area	In this study the preparation and the teaching of the concept of area as well as the introduction of area standard units and problems related to the computation of area are presented. An experiment conducted in class 5 and 6 is shown, in which we focused on various activities, patterns of work and the application of area computation in everyday life. The efficiency of the experiment was measured on the basis of pre-tests and post-tests.
12	Bozena Maj University of Rzeszow Poland	Analyzing Mathematics Students’ Lesson Plans: Focusing On Creative Mathematical Activities	This paper presents an analysis of mathematics students’ lesson plans. The students were participating in a series of workshops focused on creative mathematical activities. The aim of the lessons was to develop some kinds of such activities for the pupils. The analysis of these lesson plans was made in order to examine the students’ – future mathematics teachers’ – ability to plan and organize the work of their students in such a way that they can have the opportunity to undertake different kinds of creative mathematical activities. The results of our analysis have shown that most students can design a lesson which fulfils the initial aim and their work revealed some aware actions to the direction of creativity.
13	Carlo Marchini and Jenni Back Mathematics Department of University of Parma, Italy University of Plymouth, United Kingdom	Teachers’ Best Practices Using Differentiation	Some theoretical reflections about differentiation originate from the analysis of protocols realized during an experiment involving a treatment of arithmetic in grade 1, on the basis of a semantic environment designed by Hejný et al. (2006). This environment together with the teacher’s practice drives naturally towards differentiation by outcome and its acceptance by the pupils as a motivating tool.
14	Beata Matłosz Thomas Kelly High School, Chicago, Illinois USA	The Effect Of Intelligence Type On Learning Algebra	The author applies the theory of multiple intelligences (MI) while teaching algebra to below-average 14-year-old students (scoring between the 25th and 45th percentiles on the Illinois Standards Achievement Test), hypothesizing that using didactic methods based on dominant intelligence types would enhance students’ learning. The author administered 2 MI tests to students, supplemented by a questionnaire for parents and 4 student assessments to identify learning success. Students showed improvement in assessments, particularly when the dominant intelligence type was linguistic. Thus, mathematics education seems to depend on students’ problem-solving predispositions and how the teacher organizes instruction and selects mathematical tasks.
15	Antonella Montone, Michele Pertichino Department of Mathematics, University of Bari Italy	Mathematics In Kindergarten: Grown-Up Things	Teaching mathematics in kindergarten gives the opportunity to interpret the adult world through children’s eyes, rejecting the idea of a child as a small adult. In this paper we intend to suggest opportunities and ways to carry out this view of scientific education aiming to integrate imagination with a scientific approach and to solve adults’ problems from a child’s point of view.
16	Ioannis Papadopoulos Hellenic Ministry of Education Greece	Using Simple Arithmetic Calculators As A Diagnostic Tool On Place-Value	In this study 3^rd graders are coping with tasks relevant to place value using simple arithmetic (broken) calculators. Main aim of the study is to signify the potential usage of calculator by the teacher as a diagnostic tool that could reveal misconceptions of the students or limited understanding of the concept of place value.
17	Gabriela Pavlovičová, Júlia Záhorská Constantine the Philosopher University in Nitra, Slovakia	The Impact Of Formative Assessment On Pupil´S Academic Achievement At The Elementary School	In this article we deal with the question of the evaluation of the pupil’s learning results. We observed the impact of formative assessment on pupil’s learning outcomes which they achieved in our research. We present the results of an experiment in which we stepped into the teaching process by formative assessment of teaching materials prepared by us. Experimental and comparison group consisted of 14-15 years old students of secondary school.
18	Marta Pytlak University of Rzeszow Poland	Social character of learning mathematics and building individual web of cognitive connections	Social character of learning plays an import_ant role in the process of learning mathematics. It can support the process of building student's mathematical knowledge. This paper present how students create their own web of cognitive connections during the work with the tasks concerning generalization and how the social character of learning influent on this individual web.
19	C. Sabena, L. Bazzini, C. Strignano Department of Mathematics, University of Torino Italy	Imagining A Mysterious Solid: The Synergy Of Semiotic Resources	Recent studies have pointed out the significance of perceptuo-motor and embodied activities in mathematics learning. Assuming such a viewpoint, we use the theoretical notion of ‘Action, Production and Communication Space’ (Arzarello, 2008) and the practical tool of ‘semiotic line’ (described in this report) to analyse the cognitive and semiotic dynamics that occur during mathematical lessons. We discuss the case of a classroom activity introducing fourth grade children to 3D geometry. In group-work with the supervision of the teacher, children are asked to imagine a mysterious solid, i.e. a solid composed by the minimum number of equilateral triangles. We use the semiotic line to study the role and synergy of different semiotic resources used by the children and the teacher. In particular, gestures and gazes are analysed in order to identify their contribution to the cognitive activity (mainly imagining), as well as the communicative and the didactical ones.
20	Mirosława Sajka Pedagogical University of Krakow Poland	Pre-service Mathematics Teachers’ Understanding of the Basic Symbolism of Functions	This article concerns pre-service mathematics teachers’ understanding of particular symbols related to functions. A variety of different interpretations of the symbol h(x–3) and the condition h(x) = h(x–3) were disclosed in the context of one problem which was given to students of Mathematics at the Pedagogical University in Krakow. The ways of interpreting the symbols and conditions brought about different ways of solving the problem but also misunderstandings. Four kinds of the sources of difficulties are identified: firstly, the intrinsic ambiguities in the mathematical notation, secondly, the restricted context in which some symbols occur in teaching and the limited choice of mathematical tasks at school, thirdly, teachers idiosyncratic interpretation of mathematical tasks, and finally, some unrealized false teachers’ convictions. All of them should be taken into account within the framework of mathematics teachers’ training as well as within the general mathematics teaching and learning process.
21	Ewa Swoboda, Edyta Jagoda University of Rzeszow, Charter School Nr 1, Rzeszow Poland	Various Intuition of the Point Symmetry (From the Polish School Perspective)	For acting in a geometrical world it is crucial to possess the ability of leading reasoning which is based on the dynamic concepts representations. In our approach we try to built the theoretical framework of creation isometries. In this article we are focused on the development of intuitional understanding of the point symmetry.
22	Konstantinos Tatsis, Bozena Maj University of Western Macedonia, Greece University of Rzeszow, Poland	Pre-Service Mathematics Teachers’ Strategies In Solving A Real-Life Problem	The paper presents the investigation of pre-service mathematics teachers’ behaviour and solution strategies for a real-life problem based on the context of a post office situation. Our students’ initial reactions included frustration and discomfort, probably because of their lack of experience. Finally, no student presented a complete solution, namely the mathematical model used by the ticketing machine in the post office. However, some students managed to reach a partial solution. The basic action undertaken by the vast majority of students was the visual representation of the situation described in the problem.
23	Konstantinos Tatsis University of Western Macedonia Greece	Pre-Service Teachers’ First-Time Creations Of Open-Ended Problems	Open-ended problems are said to enhance students’ creativity by offering them multiple solutions and solution paths. Thus, teachers are expected to have some experience in such problems. This paper describes a part of an instructional series for pre-service teachers, who were – among other activities – asked to create their own open-ended problem based on a given phrase. Few working groups completed the task. The problems were analysed from a mathematical and linguistic point of view and the results of the analysis show that despite their lack of experience these few pre-service teachers created interesting problems by including everyday and complex data; however, the complexity and subjectivity were eventually interpreted negatively by their colleagues.
24	Lenka Tejkalová Charles University in Prague Czech Republic	Mathematics And Language Integrated Learning – Identifying Teacher Competences	Teaching a content subject through a foreign language is an educational trend of growing importance, which proves to increase motivation and improve learners’ attitude to the content subject. This report presents an ongoing research in the field of teacher training for mathematics and foreign language integrated teaching, aiming at shifting the focus towards mathematics and the specifics of mathematics teacher training.
25	Dorota Turska, Ryszarda, Ewa Bernacka Department of Psychology, Maria Curie–Sklodowska University Poland	Conduct Of Male Teachers Of Mathematics In The Perception Of Female And Male Pupils. A Lower Secondary School Perspective	The purpose of this article is to determine the behaviour of male teacher of mathematics in the perception of Polish male and female pupils lower secondary school. The results obtained suggest that female pupil, relative to male students, assess input and output in a less favourable manner. Female lower secondary school pupils taught by male teachers give a lower grade to the actual course of the lesson. In the perception of female pupils (as compared to male pupils), a lesson of mathematics becomes less effective and less frequently used to enhancing abilities of all pupils. Maybe the belief that "mathematics is a male domain" works both as described for stereotype (a teaching point of view) and autostereotype (point of view of female pupils).
26	Paola Vighi Mathematics Department, University of Parma Italy	Proportional Reasoning And Similarity	The main hypothesis of this work is the following: the approach to proportional reasoning would be better starting from geometrical concepts, in particular from similarity of figures. So, we prepared some activities with the aim to promote the comparison between numbers and the individuation of the fourth number after three numbers, working on similar figures. We analyze the results.
27	Jenny Young-Loveridge & Judith Mills University of Waikato, Hamilton New Zealand	"Without Maths We Wouldn’t Be Alive": Children’s Motivation Towards Learning Mathematics In The Primary Years	This paper presents data on the beliefs, values, and attitudes of 9- to 11-year-olds towards their mathematics learning. The children were very positive, optimistic, persistent, and highly motivated to learn mathematics. These findings are considerably more positive than those found with children five years earlier. Possible reason for the shift is that a decade of reform in mathematics education has had time to impact on classroom practices. Another possibility is that the present cohort differed by including more children from high SES communities who were average to high achievers in mathematics for their year level.
28	Bożena Rożek Pedagogical University of Krakow Poland	Structures of 2d Arrays Identified in Children's Drawings	The research presented in this paper concerns the understanding of a certain type of a regular two-dimensional array of elements by pupils by 10-13 years old children. Two-dimensional arrays which are characterised by different structures there are for example: rectangular structure end a slant one. The objective of the study was to answer the following questions: Are children able to notice two different structures in the same array? Perceiving the features of a slant structure by children proved to be considerably more difficult than drawing a rectangular structure. The disproportion in the degree in which the drawings reflected these particular types of regular arrays is significant and important. It suggests that in the case of children of this age the structures of two-dimensional arrays are not formed yet but still undergo the process of being constructed.
29	Anna K. Zeromska Pedagogical University of Krakow Poland	The perimeter and the area of geometrical figures – how do school students understand these concepts?	The paper presents some results of a research focused on the following questions: a) How do primary and secondary school students understand the perimeter and the area of geometrical figures? b) Do the students know that these two concepts are independent from each other? The research reveals an unexpected result: the higher level of education students are the less they are conscious of the aforementioned independency. The question is why?
30	Marianna Ciosek Pedagogical University of Krakow Poland	How could we help a student who knows, although makes mistakes?	The presentation concerns the use of algebra by lower secondary school students. The way in which a student tried to solve a few word problems will be shown in detail. The way was long and not free from mistakes, but ended with a success. An analysis of the student's beheviour reveals, that what did cause his difficulties was not transforming verbal conditions of a problem into a system of equations - as we could expect - but...something else.
WORKSHOPS
1	Jenni Back and Jodie Hunter University of Plymouth UK	Tasks And Big Ideas In Supporting Teachers To Develop Professionally	Although most European countries have well established programmes for the initial education of their teachers, provision to support their continuing professional development once they are practising as teachers within schools is more variable. It is often up to the individual teacher to pursue their own professional development according to their own agenda. This workshop will draw on some of the findings about provision of professional development for elementary teachers from the RECME project which studied 30 different professional development initiatives with a mathematical focus in England. It will also reflect the leaders’ experiences of involvement in school based professional development, conferences for teachers, and workshops both for NRICH (www.nrich.maths.org.uk) and CIMT (www.cimt.plymouth.ac.uk).
2	Agata Hoffmann Institute of Mathematics, University of Wroclaw Poland	Perpetum Mobile – Mathematics As a Motivational Factor Of Teaching-Learning Mathematics?	Teachers have to create a process of teaching-learning mathematics and they all experience difficulties with it. So, they need to motivate their students to do that. I will try to determine whether mathematics itself could be useful in it.
3	Uldarico Malaspina Jurado Pontificia Universidad Católica del Perú	Searching For Optimal Solutions With Children	In this workshop, we will work on several problems which can be presented to children between 8 and 16 years old and which challenge to obtain optimal solutions. These problems, which are not usually included in primary and secondary education, are particularly important to motivate children to do mathematics and to teach these pupils considering their experiences, their intuitive approximations, and their different reactions in the classroom.
4	Silva Kmetič The National Education Institute Slovenia	Computers Help Math Learning	The workshop is going to be dedicated to the key learning situations such as pattern observation, connection discovery, learning with the help of dynamic pictures, problem situation and data investigation as well as to the learning with feedback supported by different software. We are going to focus on both the productive knowledge and the development of ICT maturity in math education.
5	Andriy Kovalchuk, Vyacheslav Levitsky, Igor Samolyuk, Valentyn Yanchuk Zhytomyr State Technological University, Ukraine	Formulator Tarsia: A Software Tool For Creating Learning Activities	This paper introduces a software tool that helps teachers to create, print out, save and exchange puzzle-like handouts in a form of jigsaws, dominoes, rectangular and matching cards, etc. for later use in a class. The main indent is to provide a teacher with a tool to experiment with various styles of application an "active" learning approach while teaching mathematics.
6	Beata Matłosz, Mirosława Sajka Thomas Kelly High School, Chicago, USA Pedagogical University of Krakow, Poland	Applying The Theory Of Multiple Intelligences	The purpose of the workshops is to apply Howard Gardner’s theory of multiple intelligences to the teaching of mathematics. In light of this theory, the participants will analyze a video clip from a math class and create mathematical activities best suited to students of a particular intelligence type, based on the students’ profiles.
7	Carol Murphy University of Exeter UK	Dialogue And Arithmetic: Defining The Dialogic Space And Analysing The Learning	The workshop intends to share ongoing analysis of video and transcript data that has been collected as part of the ‘Talking Counts’ project at the University of Exeter. The project was funded by Esmee Fairbairn and involved my colleagues, Ros Fisher and Rupert Wegerif. The project was designed around the principles of Exploratory Talk (ET) developed at the University of Cambridge. The challenge we had set ourselves was to examine how young, lower attaining children (aged 6 to 7) engaged in collaborative tasks and how dialogue supported their learning in arithmetic.
8	Ketty Savioli, Francesca Ferrara, Luciana Bazzini Dipartimento di Matematica, Università di Torino Italy	Analysing Children’s Understanding From a Semiotic Stance	This research presents a tool to analyse, from a semiotic perspective, some children’s processes of understanding during mathematical problem solving. The tool under consideration is called timeline, and it is the result of work made by a large research group, in which both primary school teachers and university researchers take part. It arises from many analyses carried out on primary school learners that solve various problematic situations, being filmed by a videocamera. The analyses use the videos, and the written productions of the children. They principally focus on the gestures, the words, and the inscriptions that are used in the activities
POSTERS
1	Ján Gunčaga Catholic University in Ružomberok, Faculty of Education Slovakia	Number Systems
2	Tatyana Oleinik, Svetlana Dozenko, Irina Tamogskala Skoworoda Pedagogical Univesity Ukraine	Webquest as Means of Increase of Students’ Motivation