CERME2 Proceedings - Mathematik, TU Dortmund

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European Research in Mathematics Education II Mariánské Lázně, Czech Republic Charles University, Faculty of Education February 24 - 27, 2001

Proceedings Edited by Jarmila Novotná

Prague 2002

European Research in Mathematics Education II Mariánské Lázně, Czech Republic Charles University, Faculty of Education February 24 - 27, 2001

CERME 2 is closely linked to the Research Project Cultivation of mathematical thinking and education in European culture. Since all papers and other presentations here are presented in English, which is not usually the first language of the presenters, the responsibility for spelling and grammar lies with the authors of the papers themselves.

Internet-Version (pdf-file): http://www.pedf.cuni.cz/svi/vydavatelstvi © Charles University, Faculty of Education, Prague, 2002 ISBN 80-7290-075-7

Contents

CONTENTS E. Cohors-Fresenborg: A young European society establishing a tradition for scientific conferences ……………………………………………………. WORKING GROUP 1: Creating experience for structural thinking ……….. M. Hejný, G.H. Littler: Introduction to WG1. Building structures in mathematical knowledge ……………………………………………….. M. Hejný: Creating mathematical structure …………………………….. J. Kratochvílová: Building the infinite arithmetic structure …………..... N. Malara: From fractions to rational numbers in their structure: Outlines of an innovative didactical strategy and the question of density C. Marchini: Instruments to detect variables in primary school ………... H. Meissner: Procepts in geometry ……………………………………... B. Pedemonte: Relation between argumentation and proof in mathematics: cognitive unity or break? ……………………………….... B. Schwarz, R. Hershkowitz, T. Dreyfus: Emerging knowledge structures in and with algebra …………………………………………... M. Singer: Thinking structures involved in mathematics learning ……... N. Stehlíková, D. Jirotková: Building a finite algebraic structure …….... P. Tsamir: Intuitive structures: The case of comparisons of infinite sets.. WORKING GROUP 2: Tools and technologies in mathematical didactics ... K. Jones, J.-B. Lagrange, E. Lemut: Introduction to WG2. Tools and technologies in mathematical didactics ……………………………….... J. Ainley, B. Barton, K. Jones, M. Pfannkuch, M. Thomas: Is what you see what you get? Representations, metaphors and tools in mathematics didactics ……………………………………………………………….... M. Cerulli: Introducing pupils to theoretical thinking: The case of algebra …………………………………………………………………... P. Gallopin, L. Zuccheri: A didactical experience carried out using, at the same time, two different tools: A conceptual one and a technological one ………………………………………………………………………. A. Hošpesová: What brings use of spreadsheets in the classroom of 11years olds? …………..…………………………………………………... J.-B. Lagrange: A multi-dimensional of the use of IC technologies: The case of computer algebra systems ………………………….............. J. A. Landa H., S. Ursini: Mediation of the spreadsheet: Composition of the argument ……………………………………………………….......... T. Lingefjärd, M. Holmquist: Mathematics, technology and examination in distance education ……………………………………………………. C. Mogetta: Struggling to prove motion: From dynamic perception to static theory ……………………………………………………………...

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F. Olivero, O. Robutti: An exploratory study of students’ measurement activity in a dynamic geometry environment …………………………… A. Routitsky, P. Tobin: Graphic calculators: Use in mathematics in Victorian secondary schools ………………………………………..…... B. Schwarz, R. Hershkowitz: Production and transformation of computer artifacts: Towards construction of meaning in mathematics ………….... POSTERS AND ABSTRACTS OF ADDITIONAL PAPERS LINKED WITH WG2 …………………………………………………………………. L. Bilousova: Development of intellectual skills of the pupils with computer technologies ………………………………………………… T. Byelyavtseva: Student’s projects are a tool for the formation of investigating skills …………………………………………………….... O. Chumak: Basic structurization and interactive algorithmization in mathematical education ……………………………………………….... D. Leder, C. Scheriani, L. Zuccheri: The mathematics of the boys/girls: exchange of experience among boys/girls of the same age …………….. C. Pellegrino, L. Zuccheri: A video about mathematics ………………... S. A. Rakov: Mathematical packages as a tool of a constructive approach in mathematical education …………………………………………….... C.-M. Chioca: What kind of obstacles may be expected in the simultaneous learning of mathematics and computer software? ……….. D. Kontozisis, J. Pange: Using cooperative learning to teach primary mathematics to AD/HD children in a computer-based environment ….... WORKING GROUP 3: Theory and practice of teaching from pre-service to in-service teacher education ………………………………………................ F. Furinghetti, B. Grevholm, K. Krainer: Introduction to WG3. Teacher education between theoretical issues and practical realization………….. N. Climent, J. Carrillo: Developing and researching professional knowledge with primary teachers ……………………………………..... K. Krainer: Investigation into practice as a powerful means of promoting (student) teachers’ professional growth …………………….. A. Kuzniak, C. Houdement: Pretty (good) didactical provocation as a tool for teacher’s training in geometry ………………………………..... P. Marshall: A study of primary ITT students’ attitudes to mathematics J. P. da Ponte, H. Oliveira: Information technologies and the development of professional knowledge and identity in teacher education ………………………………………………………………... M. Tzekaki, M. Kaldrimidou, X. Sakonidis: Reflections of teachers’ practices in dealing with pupils’ mathematical errors …………………..

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Contents

ABSTRACTS OF ADDITIONAL PAPERS LINKED WITH WG2 ……..... A. N. Ilína, O. A. Ivanov: Simulators in mathematics teacher education .. T. Oleinik: Development of critical thinking …………………………… WORKING GROUP 4: Social interactions in mathematical learning situations …………………………………………………………………….. G. Krummheuer: Introduction to WG4. The comparative analysis in interpretative classroom research in mathematics education …………… J. Back: Some numbers are straight and some are round: Considering meaning and focus in classroom talk ………………………....………… R. Barwell: Narrative orientation in the construction and solution of word problems by English additional language (EAL) learners of mathematics…………………………………………………………...… A. J. Bishop, P. C. Clarkson, G. E. FitzSimons, W. T. Seah: Studying values in mathematics education: Aspects of the VAMP project ……..... B. Brandt: Classroom interaction as multi-party-interaction: Methodological aspects of argumentation analysis …………………..… R. Hedrén: Learning in mathematics during group discussions of some rich problems ………………………………………………………….... S. Maury, S. Stephan: Solving an algebra problem in a triadic situation in tenth grade ……………………………………………………………. T. Rowland: Pragmatic perspectives on mathematics discourse ………... G. Sensevy, A. Mercier, M.-L. Schubauer-Leoni: A model for examining teachers’ didactic action in mathematics, the case of the game “RACE TO 20” ………………………………………………………………….. H. Steinbring: Forms of interactive construction of new mathematical knowledge ……………………………………………………………..... WORKING GROUP 5: Mathematical thinking and learning as cognitive processes …………………………………………………………………….. L. Bazzini, P. Boero, R. Garuti: Algebraic expressions and the activation of senses ……………………………………………………... E. Cohors-Fresenborg: Individual differences in the mental representation of term rewriting ………………………………………… S. Hershkovitz, P. Nesher, J. Novotná: Cognitive factors affecting problem solving ……………………………………………………….... M. Maracci: Drawing in the problem solving process ………………..... I. Schwank: Analysis of eye-movements during functional versus predicative problem solving …………………………………………...... WORKING GROUP 6: Assessment and curriculum ……………………….. O. Björkqvist: Introduction to WG6. Assessment and curriculum ……... T. Assude: Elements on evolution of official curriculum in France. The case of inequalities in the "Collège" level ……………………….....

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J. Lukács, K. Tompa: About the reform of mathematics examination in Hungary …………………………………………………………………. J. Törnroos: Mathematics textbooks and students’ achievement in the 7th grade: What is the effect of using different textbooks ……………..... WORKING GROUP 7: The role of metaphors and images in the learning and understanding of mathematics …………………………………….……. B. Parzysz: Introduction to WG7. Working together on metaphors and images …………………………………………………………………... G. Chartier: Using «GEOMETRICAL INTUITION» to learn linear algebra ………………………………………………………………….. M. Maschietto: The transition from algebra to analysis: The use of metaphors in a graphic calculator environment ………………………… J.-C. Régnier, M. Priolet: Teachers’ use of semiotic registers ………..... E. Robotti: Verbalization as a mediator between figural and theoretical aspects …………………………………………………………………... L. Rogers: From icons to symbols: Reflections on the historical development of the language of algebra ………………………………... POSTERS M. Barešová: Sample of pedagogical communication in mathematics lesson ……………………………………………………………………. C. Green: The effects of type of support on children’s thinking when tackling mathematical investigations …………………………….. M. Kaslová: Theory and practice of teaching from pre-service and inservice teacher education - Phenomena of in-service practice training … M. Kubínová: Improving teachers’ beliefs about mathematical education J. Molnár: Euler’s theorem ……………………………………………... M. Panizza, J.-P. Droughard: Reasoning Process and Process of Control in Algebra …………………………………………………….... J. Perný: Space imagination on the cube ……………………………….. V. de M. Santos: Generating knowledge and meaning to teach mathematics …………………………………………………………….. B. Sarrazy: Study of anthropo-didactic functions and cognitive effects of interactions in three contrasted teaching contexts ………………….... E. Swoboda: The atomic analysis of the conceptual fields: Similarity (A case study) …………………………………………………………... V. Sýkora: Semiotic representations in the process of construction of mathematical concept …………………………………………………… A. Ulovec: The role of image schemata in the development of new cognitive objects ………………………………………………………... AUTHORS …………………………………………………………………..

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Introduction

A YOUNG EUROPEAN SOCIETY ESTABLISHING A TRADITION FOR SCIENTIFIC CONFERENCES CERME 2 was the second conference of the new society ERME, i.e. the European Society for Research in Mathematics Education. CERME 1 was held in Haus Ohrbeck (near Osnabrück, Germany) in August 1998. The Programme Committee under the chair of Barbara Jaworski established a new culture for scientific conferences in the field of mathematics education in Europe. Work was mainly done in working groups. The spirit of communication, co-operation and collaboration was meant to be a characteristic trademark for this new series of European conferences on research in mathematics education. CERME 2 was held in Mariánské Lázně from February, 24th to 27th 2001 in the Czech Republic. The Programme Committee decided to follow in the footsteps of CERME 1 and to plan CERME 2 again in a style of collaborative group work. The intention was that each group would engage in scientific debate with the purpose of deepening mutual knowledge about topics, problems and methods of research in this field. The scientific programme consisted mainly of this group work. This time even plenary lectures were complete omitted. CERME 2 was organised by the following Programme Committee: • • • • • • •

Elmar Cohors-Fresenborg (chairman, Germany) Christer Bergsten (Sweden) Tommy Dreyfus (Israel) Barbara Jaworski (United Kingdom) Maria Alessandra Mariotti (Italy) Jarmila Novotná (Czech Republic) Julianna Szendrei (Hungary)

Setting up the Groups The Programme Committee first had to consider the topics for the groups. The final list was a result of a process in which the continuation of working groups from CERME 1 and new proposals from PC members or members of the ERME-Board were involved. Eventually, themes were agreed upon and group leaders were sought for the 7 groups. First, the PC invited group co-ordinators, acknowledged experts, each having research interest and expertise in the topic of the group. In several cases, the colleagues asked could not accept the invitation, therefore, other decisions had to be made. In a second step, the PC, together with the group co-ordinators, looked for group leaders with the aim that – depending on the estimated size of the group – 2 to 4 group leaders from different countries should form the leading team. A balance of nations was sought in the group 7

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leadership. Of course, not every person invited was able to accept. So some compromises on this balance had to be made. The groups chosen and the group coordinator and group leaders who finally participated were as follows: Theme 1 2 3 4 5 6 7

Building Structures in Mathematical Knowledge Tools and Technologies in Mathematical Didactics Theory and Practice of Teaching -from Pre-service to In-service Teacher Education Social Interactions in Mathematical Learning Situations Mathematical Thinking and Learning as Cognitive Processes Assessment and Curriculum The Role of Metaphors and Images in the Learning and Understanding of Mathematics

Coordinator Milan Hejný (CZ)

Further group leaders Graham Littler (UK) Pessia Tsamir (IL)

Keith Jones (UK)

J-Bapt. Lagrange (F)

Fulvia Furinghetti (I)

Barbro Grevholm (S) Konrad Krainer (AT)

Götz Krummheuer (D)

Gérard Sensevy (F)

Inge Schwank (D)

Pearla Nesher (IL)

Ole Björkvist (FIN) Klára Tompa (H) Bernard Parzysz (F) Nuria Gorgorio (E)

The team of group leaders organised a process of reviewing the delivered research papers. In some groups, the reviewing process was done among the group leaders, in others, group leaders identified other suitable and competent reviewers. As an outcome of this reviewing process, papers were accepted or rejected or a proposal was made to transfer the paper into another group where it would fit the topic better. The reviewers were asked to give supportive comments on the papers; even in the case of acceptance, the writers of the papers should receive support to improve the paper. The intention was that these accepted papers should form the first step of the scientific debate of CERME 2. The decision in CERME 1 that there should be no “oral delivery” of a paper within the group at the conference was identified as a good decision by most of the participants. Due to the fact that it takes some time to become acquainted with this new style of scientific conference, these goals were not fully achieved in the first conference. Therefore a new attempt was made with CERME 2 to promote this new idea of preparing and executing a scientific conference as a forum of scientific debate. The change of scientific traditions and the establishment of new styles need a lot of staying power. It has to be understood more as a process than as a breaking point. Considering this, CERME 2 has made a big step forward.

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Introduction

Publication of Conference Proceedings During a meeting at the end of the conference, the members of the old and the newly elected ERME-Board discussed ways of how to improve the scientific level of European research in mathematics education and what role the publication of the outcome of the conferences could play. Publication was regarded as a third step in the scientific debate: The first step consists of writing, reviewing and re-writing of papers. The second step consists of debating the scientific themes within the working groups; here, the accepted papers are merged into the debate because they have been read by the members of the groups before the meeting. The third step consists of a new process of rewriting and reviewing the papers, as a result of the scientific debate, for publication in the proceedings. It was decided that only those papers which had finally been accepted by the team of the group leaders for each group should be published in the conference proceedings. We are glad that we can now present the results of this intensive scientific debate to the public. We take the opportunity to thank, very warmly, Jarmila Novotná as head of the local organising committee for the marvellous work which she and her team did to draw nearer to the common goal of promoting European research in mathematics education. We also thank the many unknown helpers, not only before and during the conference, but on the occasion of writing the preface also in terms of all the technical support. As a result, an interesting and stimulating research publication can now act as a catalyst in strengthening and enriching the research in our growing discipline. Elmar Cohors-Fresenborg on behalf of the CERME 2 Programme Committee, December 2001

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WORKING GROUP 1 Creating experience for structural thinking

Group leaders: Milan Hejný Graham Littler Pessia Tsamir

Working Group 1

INTRODUCTION TO WG1 BUILDING STRUCTURES IN MATHEMATICAL KNOWLEDGE Milan Hejný1, Graham H. Littler2 1

Charles University in Prague, Faculty of Education, Czech Republic [email protected] 2 University of Derby, UK [email protected]

Our brain does not store individual pieces of knowledge separately. Some of the pieces are joined together to form a linkage and there are many such linkages in our brain. Some linkages are inter-linked by common piece(s) of knowledge. We understand the structure of knowledge to be the web of all these linkages. The cumulative understanding of the structure of knowledge considers all mathematical knowledge to be divided into large linkages such as geometry, algebra, logic.... A learning process in this view is understood to be adding a new piece of knowledge onto one of the already existing linkages. The generic approach to the structure of knowledge considers the structure to be dynamic, in which each new piece of knowledge causes a reorganisation of the existing structure. A form of stock-taking process takes place. Does the new piece of knowledge join one or more of the already established linkages? Does it cause two or more linkages to join? Is it a completely new piece of knowledge which cannot join any existing knowledge? This process means the restructuring of the already existing mental structure. The majority of the contributions to WG1 were based on the generic approach to the building of mathematical structures and in all of these, the research interest was focussed on those phenomena which play an important role in this process. A broad variety of topics were presented in the papers – algebra, arithmetic, 2D and 3D geometry, set theory, logic, argumentation, concept creation and research methods.

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CREATING MATHEMATICAL STRUCTURE Milan Hejný1 Charles University in Prague, Faculty of Education, Czech Republic [email protected]

Abstract: Structuring mathematics belongs to the most complex of long-term cognitive processes. Some results which we gained whilst studying this phenomenon will be presented in this contribution which is aimed at general ideas (research methods, the anatomy of the process, different ways of structuring), and two other contributions aimed at non-standard structures: the finite algebraic structure of “restricted arithmetic” and the infinite arithmetical structure of “triads”.

1. The aim of the paper In this paper we would like to give a general view of our understanding of what creating a mathematical structure means and how it can be studied. Our study is influenced by ideas of L. Kvasz (1998), and P. Vopěnka (2000), based on longterm experimental education and the research of D. Jirotková, J. Perný, J. Perenčaj, B. Rozek, E. Swoboda, M. Tichá and is closely connected to the research of N. Stehlíková, J. Kratochvílová presented in our WG1.

2. Internal mathematical structure (IMS) The idea of a mathematical structure was profoundly explained in the Bourbaki’s famous Architecture of Mathematics. Our aim is to investigate how a mathematical structure is created in an individual’s mind. To avoid the danger of confusing the two different readings of the term mathematical structure, we will add the adjectives external and internal to distinguish the two. The differentiation corresponds to Bolzano’s idea which was elaborated by Karl Popper into the idea of three worlds (Popper, Lorenz, 1994). For the world of individual minds – the second of Popper’s worlds – we use the word internal, and for his third world – the world of culture – we use the word external.

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The research has been supported by the grant VZ J13/98/114100004, Kultivace matematického myšlení a vzdělanosti v evropské kultuře. 14

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There are several possibilities of how the term internal mathematical structure (hereinafter IMS) can be understood. In his study, van Hiele (1986) discusses two different ways of how structure can be characterized: “For Piaget there are three characteristics: 1. Structure has a totality, 2. Structure is achieved by transformations, 3. Structure is autoregulating. … In structural psychology (Gestalt psychology) there are four important properties that govern structure: 1. It is possible to extend a structure. Whoever knows a part of the structure also knows the extension of it. 2. A structure may be seen as a part of a finer structure. … 3. A structure may be seen as a part of a more-inclusive structure…. 4. A given structure may be isomorphic with another structure…” Piaget’s characteristics are too general; we have used them as an inspiration. The four properties of Gestalt psychology helped us to understand the problem, but they are only focused onto a part of the building process of IMS. Moreover, the second sentence in property 1 is, according to our experience, not true. In our opinion, the most important phenomenon of IMS is its connectedness. Bell (1993), when discussing psychological principles that underlie designing teaching, starts with connectedness which he characterizes by the statement - a fundamental fact about learned material is that richly connected bodies of knowledge are well retained; isolated elements are quickly lost. The same idea with the stress on a constructivistic approach is expressed by Hiebert & Carpenter (1992, p. 66, 67): We propose that when relationships between internal representations of ideas are constructed, they produce networks of knowledge…. These two ideas, construction and connectedness, give the core of our approach to IMS. In our understanding, IMS is a dynamic set of networks with different pieces of knowledge like ideas, concepts, facts, relations, examples, solving strategies, arguments, algorithms, procedures, hypotheses, … as centroids of these networks. IMS binds all these networks together and equipped this set with an organization. Networks may be structured like vertical hierarchies, or, they may be structured like webs. … A mathematical idea or procedure or fact is understood if it is part of an internal network. … The degree 15

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of understanding is determined by the number and the strength of the connections Hiebert & Carpenter (1992, p. 67). In our long term experimental teaching, we observed that a student’s ability to built IMS is a deep characteristic of his/her cognitive style and profoundly depends on the autonomy of his/her approach to mathematics and on his/her structuring appetite i.e. the desire to create different graphs, tables, lists and overviews in order to get understanding of how ‘all these facts are connected together’. Our comparative analyses of how IMS is built in the traditional transmisive vs. the constructivistic teaching showed that the crucial role in building IMS is played by concepts. Conceptual knowledge is knowledge that is rich in relationships. A unit of conceptual knowledge is not stored as an isolated piece of information; it is conceptual knowledge only if it is part of a network. Hiebert & Carpenter (1992, p. 78). In the transmissive teaching the stress is put on how to deal with it; in the constructivistic teaching the stress is put on what it is (see example 1 below).

3. How can IMS be studied? Five of the research methods which we have elaborated for the investigation of concept creation (Hejný et al, 1990, 28-34) are listed below. The methods were focused to networks of particular ideas, concepts and procedures, hence they could be, and were used in analyzing IMS. Explain some idea (prime, area, fraction, triangle, Pythagorean theorem) to your a) classmate, b) younger friend. We found out that if the explanation retained the formal school approach, the quality of the student’s understanding of this idea is low, and, as a rule, is more procedural than conceptual and the network of this idea, as a part of IMS, is usually poor. Example 1. An eight-year-old boy asked his older sister Alice what ‘per cent’ was. She showed him formulas: percentage = 100 rate/base, b = 100r/p, r = pb/100. He was not satisfied and said: “I did not ask you how you can calculate it, I would like to know what it is”. The girl was not able to answer this question which was focused to the concept ‘per cent’. Alice’s knowledge was rather procedural and formal. Her three-parts knowledge of ‘per cent’ is isolated.

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Describe a concept in a non-standard way. A student is asked to define a ‘circle’ without the concept of ‘distance’, or a ‘median of a triangle’ without the concept of the ‘midpoint of a segment‘ or a ‘L.C.M’ without the concept of ‘multiple’. Example 2. University students, future teachers, were asked to define a ‘prime’ without the concept of divisor. Their answer was “It is not possible, the definition of a ‘prime’ requires the concept of division or multiplication”. The same task was perfectly solved by seventh graders: “A number n is not prime, if you can arrange n pebbles into the rectangular shape. If you can not do that and the only shape is the line of n pebbles, n is prime”. Students used the linkage between concepts ‘area’ and ‘multiplication’ to find a new definition of a prime. Use a non-standard notation in a counting procedure. E.g. a student is asked to use ‘↑’ for addition and ‘↓’ for subtraction; e.g. 5↑2 = 7, 4↑(–3) = 1, 6↓1 = 5, 2↓(-1) = 3. We observed that fifth graders deal with this notation better than eight graders who exhibit a strong tendency to rewrite the arrow notation into the usual one. When asked to explain the symbol 5↓↓2, the majority of eight graders take it as 5 – (-2) = 7, for majority of fifth graders it was just the confirmation of the subtraction (5 – 2 = 3) and for some of them it was the doubled subtraction (5- 2 – 2 = 1). One these students gave the following argument: he wrote ‘5↓↑2’ and said “I had 5 crowns, I lost 2 crowns and I found 2 crows, so I have 5 crowns; in your case I lost 2 crowns twice, so I have just 1 crown”. A more demanding task is counting in Roman numerals (LXI times CIL = ?). Solve a problem with restricted instruments. E.g. find the midpoint of the side of a given rectangle if you only have one straight-edge (not a ruler), or use a calculator with a 12-digit display to multiply two 10-digit numbers. Derive one idea from an other. E.g. derive the idea of subtraction from the idea of addition via ‘count on’ calculation. Derive the formula for the area of a trapezoid from the formula for the area of a triangle. Derive the formula for sin (α-β) from the formula for sin (α+β).

4. The research on the building of IMS A researcher who wants to study IMS and its creation faces at least one serious difficulty. The object of the study is a long-term process and if studied ‘in vitro’ it gives only particular results. To get a complex understanding of how an IMS is built we have to study it ‘in vivo’. 17

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In this respect, a great help for our research was our ten years of experimental teaching in a primary school (years 1975-1980 - fifth to eighth grades, years 1984-1989 third to eighth grades) during which we not only gained valuable experience, but also kept detailed pedagogical diaries which profoundly facilitated the understanding the problems. Generally speaking, each new piece of mathematical knowledge can be regarded as a part of building an IMS. However, we are going to restrict our attention to those mental activities, which bring new linkage between already existing and/or new pieces of knowledge. We saw such a linkage in example 2. In our experimental teaching, we identified many different kinds of such building IMS-steps. Probably the most useful were: 1. the appearance of a question which concerns mathematical structure (e.g. What it is? Why does it work? How are these two ideas linked together?) 2. the appearance of a strategic mathematical problem2 in an individual’s mind 3. finding the linkage between two or more already existing pieces of knowledge 4. finding the connection between new and existing knowledge 5. introducing some organization into already existing knowledge 6. the extension (generalization) of a piece of knowledge 7. looking for new non-standard solving strategies 8. the need to give an explanation of disharmony if it appears 9. the reorganization of an already existing structure 10. finding the linkage between two or more structures 11. using abstraction to create a new, more abstract structure 12. organizing the whole structure in a clear, simple written form To elucidate some of these building IMS-steps, we will illustrate them. The illustrations come from the author’s experimental teaching. We have to add that there was no difficulty in observing these IMS-steps, since each student who was involved in such an activity had a strong need to discuss his or her problem and investigation with the teacher. This will be clearly seen in example 4. More detailed illustrations of the presented points will be given in the above contributions of the author’s colleagues. 2

As an (internal) strategic we mean such a problem which survives in an individual’s mind for a long time. In example 3 we will see an illustration: finding strategy for the two-pile REMOVE game with the characteristic 4 was the strategic problem for Ben for the period of one year. In the history of mathematics (external) strategic problems were e.g. the trisection of an angle, the Fermat problem, etc. 18

Working Group 1

5. Linkage between pieces of knowledge Example 3. A die was on the table and three four graders, Adam, Ben and Cid, were asked what number of spots were on the bottom face of the die. All three used the strategy of the missing number – they found out that sides 1, 2, 4, 5 and 6 could be seen, hence the missing number had to be 3. We repeated the experiment three more times and then I rolled two dice. This time, Ben immediately gave the correct answer for both dice. He explained his strategy: “Here, the top number is 1, so the bottom number must be 6. The top and the bottom numbers create a pair. It is 2-5, or 1-6, or 4-3.” Adam and Cid grasped this idea. Neither of boys found the pattern “the sum of two opposite numbers is 7”, or even the rule “bottom number = 7 – top number”. Next day, we did the same experiment with an octahedron. First, all boys applied the missing number strategy. Then Adam took the octahedron and looked at it carefully. He tried to remember all four pairs of opposite numbers. When the die was rolled for the second time, Adam immediately said the bottom number. His answer was correct. Adam said: “You know it is Ben’s trick; pairs are (he pointed to the corresponding sides of the die) 3 and 6, 1 and 8, 7 and 2, and 4 and 5; you know, it is now easy”. Ben said that Adam had a good memory. Adam answered that the task was not so difficult, since in each pair “you have one small and one large number”. A couple of days later, I played the same game with Adam and Ben. Cid was not in school. This time, the die was a dodecahedron, a solid both boys were familiar with. Before we started the game, Adam asked me if he could look at the die. Ben said that it was not fair but then he changed his mind and started to write a list of opposite numbers on this die. The list was not completed when Ben exclaimed: “Thirteen, it is thirteen, it must be thirteen, the sum of each pair is thirteen.” He was very happy and proud of this investigation. After a while, Adam agreed with this idea. He added that it was seven for the hexahedron and nine for the octahedron. So I rolled the dodecahedron and both boys used the strategy “count on to 13” (top number + ? = 13). After that, I prepared an icosahedron die. This time I put the numbers on the faces randomly. The rule “the sum of opposite numbers is 21” was not true. Some half an hour later, I asked the boys to play the game with the icosahedron. They agreed and Ben asked me, how many numbers there were on the solid. I answered twenty and rolled the die. The top number was 12. Ben said: “So the bottom number must be eight.” I hesitated: “Are you sure?” Adam said “Sure”, but he stopped for a moment and corrected Ben’s answer: “It’s nine, you have one more.” I repeated my hesitation: “Are you sure?” “Oh yes, nine for certain, sure,” said Adam and Ben agreed. So I carefully took the solid and showed the bottom number to boys. It was 3. The boys were surprised and Adam took the 19

European Research in Mathematics Education II

die in his hands. He observed it for a while and said: “You know, this is a false die, we cannot play that game with it.” So I asked Ben and Adam to correct the numbers on the false icosahedron which they did successfully. They knew that for the correctly numbered (point symmetrical convex) polyhedron the sum of numbers on parallel faces must be n +1, where n is the number of faces. They also knew that this rule holds for all “suitable” solids and that a pyramid is not a suitable solid, since it has no parallel faces. In the above experiment, we identify the boys’ six discoveries: a) the opposite-side numbers create pairs; you have to remember these pairs (hexahedron, octahedron and dodecahedron) b) in each pair, there is one small and one large number – this helps the memory (octahedron) c) the rule: the sum of each pair of the opposite numbers in the dodecahedron is thirteen (it is an invariant) d) the rule holds for the hexahedron and the octahedron and therefore should be true for each shape, particularly for the icosahedron e) if the rule fails in the icosahedron, the die is false f) generalisation of the rule to all suitable polyhedrons. Three of these ideas can be classified as steps of the building of IMS. The observations b) and c) paves the way for the discovery d) - the rule which links six pieces of knowledge, six pairs 1-12, 2-11, 3-10, 4-9, 5-8 and 6-7. At the same time, the rule can be regarded as the principle which introduces an organisation into the set of six pairs. The idea e) transforms the octahedron rule for the hexahedron rule and the octahedron rule. The decision f) proves that boys generalised these rules to “all suitable polyhedrons” and took it as the criterion of the correctness of a die. 6. Linkage between structures Two examples illustrate students discovering of the linkage between structures. Example 4. In the fifth grade, we used to play several REMOVE games. Students discovered the strategy for some of them. However, the two-pile REMOVE3 with the characteristic 4 was too difficult for them. They found the strategy for small m, n, but the general strategy was not found.

3

Two piles of m and n stones are given. Two players remove in turn either any number of stones from one pile or k stones from each pile, where 1 ≤ k ≤ 4. The player removing the last stone wins. 20

Working Group 1

A year later, in the sixth grade, we introduced the set of REACH THE CORNER4 games. The strategy of such a game can be easily visualised: each critical5 square of the board will be marked. To do this we start with marking the terminal square (1,1) which is critical. Then we cross all squares from which the critical square (1,1) is accessible by one move: {(1,y), 1 4 ! x2 > + 2 there is evidence that the graphical sign recalls the procedure adopted for solving equations, which is stored in memory, but is not appropriate in this case. Here the sign activates a familiar procedure, which is valid in situations like x2 = 4, which resembles x2>4 as far as sign is concerned. In this case (x2 >4 ! x > +2) there is a distorted relationship between sign and denotation. In this perspective, we can also include the errors which identify the equivalence of equations (inequalities) with their algebraic transformability. The student activates a known procedure without checking its applicability. We

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could speak of routine procedures, which in this case are not the first step towards solution, but rather a repetition of routine mechanisms. At this point two main questions arise: 1. Which introduction of equations and inequalities should be adopted to avoid (or at least to limit) such kinds of errors 2. Which control should be applied in the procedure. As far as point 1 is concerned, we believe that the first approach to equations and inequalities is fundamental. In our view, equations and inequalities should be treated simultaneously and not in sequence. In fact, introducing firstly equations and secondly inequalities usually implies that solving procedures valid for equations, remain predominant also for inequalities. As a consequence inequalities are considered as a sort of “pathologic equations” and treated as such. The link between sign and denotation is totally distorted. Moreover, we suggest that equations and inequalities should initially arise from problems of modeling. Coming to point 2 (the problem of control), Maurel and Sackur (1998) suggest that procedures for solving inequalities can be controlled in two ways: by using a different representation register and by assigning specific numerical values to check the validity of the given inequality. In addition, Chiappini (1998) points out the emergence of difficulties in keeping a close control of the symbols >.