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Nicolina A. Malara, Giancarlo Navarra University of Modena and Reggio Emilia, Italy September 21–26, 2014, Herceg Novi, Montenegro 12 th International Conference of The Mathematics Education into the 21 st Century Project The Future of Mathematics Education in a Connected World Generalization questions at early stages: the importance of the theory of mathematics education for teachers and students

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In the socio-constructive teaching, maths teachers have the responsibility to: create an environment that allows pupils to build up mathematical understanding; make hypotheses on the pupils' conceptual constructs and on possible didactical strategies, in order to possibly modify such constructs. This implies that teachers must not only acquire pedagogical content knowledge but also knowledge of interactive and discursive patterns of teaching. The socio-constructivist approach 2 September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)

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In a constructivist teaching perspective the teachers need: to be offered chances, through individual study and suitable experimental activities, to revise their knowledge and beliefs about the discipline and its teaching, in order to overcome possible stereotypes and misconceptions, to become aware that their main task is to make students able to give sense and substance to their experience and to construct a meaningful knowledge by interrelating new situations and familiar concepts. The teachers’ needs 3 September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)

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The actual answers to these needs are extremely complex in the case of classical thematic areas, such as arithmetic and algebra, which suffer from their antiquity, and the teaching of which is affected by the way they historically developed. In the traditional teaching and learning of algebra the study of rules is generally privileged, as if formal manipulation could precede the understanding of meanings. The general tendency is to teach the syntax of algebra and leave its semantics behind. The state of the art 4 September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)

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We believe that the mental framework of algebraic thought should be built right from the earliest years of primary school when the child starts to approach arithmetic, by teaching him or her to think of arithmetic in algebraic terms. In other words, constructing their algebraic thought progressively, as a tool for reasoning, working in parallel with arithmetic. This means starting with its meanings, through the construction of an environment which informally stimulates the autonomous processing of that we call algebraic babbling. Our hypothesis on the approach to algebra 5 September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)

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Algebraic babbling can be seen as the experimental and continuously redefined mastering of a new language, in which the rules may find their place just as gradually, within a teaching situation which: is tolerant of initial, syntactically “shaky” moments, stimulates a sensitive awareness of the formal aspects of the mathematical language. Algebraic Babbling 6 September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)

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While learning a language, the child gradually appropriates its meanings and rules, developing them through imitation and adjustments up to school age, when he/she will learn to read and reflect on morphological and syntactical aspects of language. We believe that a similar process has to be followed in order to make pupils approach the algebraic language, because it allows them to understand the meaning and the value of the formal language and the roots of the algebraic objects. An example of algebraic babbling The babbling metaphor 7 September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)

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Pupils have translated algebraically the verbal sentence “The number of finger biscuits is 1 more than twice the number of chocolate cookies”. 8 September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy) 1×2 a+1×2 (a = number of finger biscuits) fb+1×2 a×2+1 fb+1×2=a fb=ch+1×2 a=b×2+1 a×2+1=b (a = number of chocolate cookies) (a–1)×2 Please, reflect on the pupils’ sentences. An example of algebraic babbling (10 years)

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Pupils have translated algebraically the verbal sentence “The number of finger biscuits is 1 more than twice the number of chocolate cookies”. 9 September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy) 1×2 a+1×2 (a = number of finger biscuits) fb+1×2 a×2+1 fb+1×2=a fb=ch+1×2 a=b×2+1 a×2+1=b (a = number of chocolate cookies) (a–1)×2 Pupils are going to discuss on the correctness of the paraphrases expressing in many different ways the same sentence. An example of algebraic babbling (10 years)

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10 Example – USUAL BEHAVIOR September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy) It is rare that a teacher devotes significant attention to the linguistic aspects of the language of mathematics both for semantic and syntactical aspects. Usually, he/she does not promote students' reflection on the interpretation of formulas as linguistic objects in themselves and as representations that objectify processes of solving problem situations. The teacher does not encourage the meta- cognitive and meta-linguistic aspects in the teaching of mathematics.

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Example – A PRE-ALGEBRAIC PERSPECTIVE The teacher has to: interpret each pupils’ writing and understand their underlying ideas, Make the pupils interpret the writings and assess their efficacy, reflecting on their correcteness and fitness to the verbal sentence, discuss on the equivalences or differences among the writings and select the appropriate ones. 11 September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)

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Example – A PRE-ALGEBRAIC PERSPECTIVE The teacher must be able to: act as a participant-observer, i.e. keep his/her own decisions under control during the discussion, trying to be neutral and proposing hypotheses, reasoning paths and deductions produced by either individuals or small groups; predict pupils’ reactions to the proposed situations and capture significant unpredicted interventions to open up new perspectives in the development of the ongoing construction. This is a hard-to-achieve baggage of skills 12 September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)

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Our studies make us aware of the difficulties that teachers meet as to the design and management of whole classroom discussions. They highlight how, in the development of discussions, teachers: do not make pupils be in charge of the conclusions to be reached tend to ratify the validity of productive interventions without involving pupils. Our beliefs 13 September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy) We believe that a careful reflective analysis of class processes is needed if one wants to lead a teacher to get to a productive management with pupils.

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The project promotes a revision of the teaching of arithmetic in relational sense an early use of letters to generalize and to codify relationships and properties a reshaping of teachers’ professionalism (knowledge, beliefs, behaviors, attitudes, awareness) through sharing processes of theoretical questions connected to practice. 1998 - 2014 14 September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)

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The scientific setup of ArAl Project is illustrated in the Theoretical Framework the Glossary the Units The Glossary constitutes in many aspects the theoretical heart of the project. It was conceived with the aim of aiding teachers in their approach to theory, through the clarification of specific conceptual or linguistic constructs in mathematics and in maths education. 15 September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)

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Methodology The development of the project is based on a net of relationships involving: the university researchers as maths educators the teachers-researchers as tutors the teachers the pupils. 16 September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)

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The cycle of teachers’ mathematics education 17 September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy) Classroom activity Joint reflection among teachers, tutors, maths educators Development of theoretical framework, methodologies, materials leads to influences

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Transcript Other commentators Teacher E-tutor A teacher records a lesson, sends its Commented Transcript (CT) to an E-tutor. The E-tutor comments the CT, sends the new version to other members of the team. The other members write their comments too. The CT so reached becomes a powerful tool for teschers’ reflection and learning. The Multicommented Transcripts Methodology 18 September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)

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The cycle of teachers’ mathematics education 19 September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy) Classroom activity Joint reflection among teachers, tutors, maths educators Development of theoretical framework, methodologies, materials leads to influences MTMClass Episodes influence

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We show now a set of classroom episodes which testify: the effects of the joint work among teachers, tutors and maths educators which make teachers embody theoretical results addressing a new classroom practice; the achievement of new believes, of a new language and of new ways of acting in the classroom; the pupils’ conceptualizations and attitudes towards a relational and pre-algebraic vision of arithmetics. The classroom episodes 20 September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)

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21 The pupils are reflecting on: 5+6=11 11=5+6 Piero observes: “It is correct to say that 5 plus 6 makes 11, but you cannot say that 11 'makes' 5 plus 6, so it is better to say that 5 plus 6 'is equal to’ 11, because in this case the other way round is also true”. Example 1 (8 years) September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy) What can we say about Piero’s sentence?

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22 The pupils are reflecting on: 5+6=11 11=5+6 Piero observes: “It is correct to say that 5 plus 6 makes 11, but you cannot say that 11 'makes' 5 plus 6, so it is better to say that 5 plus 6 'is equal to’ 11, because in this case the other way round is also true”. Example 1 (8 years) September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy) Piero is discussing the relational meaning of the equal sign.

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23 Example 1 (8 years) – USUAL BEHAVIOR September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy) Teachers and pupils 'see' the operations to the left of the sign ‘=‘ and a result to the right of it. In this perspective, the ‘equal’ sign expresses the procedural meaning of directional operator and has a mainly space-time connotation (left right, before after). The task “Write 14 plus 23” often gets the reaction ‘14+23 = ’ in which ‘=‘ is considered a necessary signal of conclusion and expresses the belief that a conclusion is sooner or later required by the teacher. '14+23' is seen as incomplete. The pupils suffer here from lacking or poor control over meanings.

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24 Example 1 (8 years) – A PRE-ALGEBRAIC PERSPECTIVE September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy) When shifting to algebra, this sign acquires a different relational meaning, since it indicates the equality between two representations of the same quantity. Piero is learning to move in a conceptual universe in which he is going beyond the familiar space-time connotation. To do this, pupils must ‘see’ the numbers on the two sides of the equal sign in a different way; the concept of representation of a number becomes crucial.

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25 Miriam represents the number of sweets: (3+4)×6. Alessandro writes: 7×6. Lea writes: 42. Miriam observes: "What I wrote is more transparent, Alessandro’s and Lea’s writings are opaque. Opaque means that it is not clear, whereas transparent means clear, that you understand.” Example 2A (9 years) September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy) What can we say about Miriam’s sentence?

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26 Miriam represents the number of sweets: (3+4)×6. Alessandro writes: 7×6. Lea writes: 42. Miriam observes: "What I wrote is more transparent, Alessandro’s and Lea’s writings are opaque. Opaque means that it is not clear, whereas transparent means clear, that you understand.” Example 2A (9 years) September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy) Miriam reflects on how the non-canonical representation of a number helps to interpret and illustrate the structure of a problematic situation.

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27 Example 2A (9 years) – USUAL BEHAVIOR September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy) Traditionally, in the Italian primary school, pupils become accustomed to seeing numbers as terms of an operation or as results. This leads, inter alia, to see the solution of a problem as a search for operations to be performed. The prevailing view is that of a procedural nature: the numbers are entities to be manipulated. The pupils are not guided towards reflection, through the analysis of the representation of the number, on its structure. Actually teachers rarely explain that…

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28 Example 2A (9 years) – A PRE-ALGEBRAIC PERSPECTIVE September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy)... each number can be represented in different ways, through any expression equivalent to it: one (e.g. 12) is its name, the so called canonical form, all other ways of naming it (3×4, (2+2)×3, 36/3, 10+2, 3×2×2,...) are non canonical forms, and each of them will make sense in relation to the context and the underlying process. As Miriam observes, canonical form, which represents a product, is opaque in terms of meanings. Non canonical form represents a process and is transparent in terms of meanings.

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29 Example 2A (9 years) – A PRE-ALGEBRAIC PERSPECTIVE September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy) Knowing how to recognize and interpret these forms creates in the pupils the semantic basis for the acceptance and the understanding, in the following years, of algebraic writings such a-4p, ab, x 2 y, k/3. The complex process that accompanies the construction of these skills should be developed throughout the early years of school. The concept of canonical/non-canonical form has for pupils (and teachers) implications that are essential to reflect on the possible meanings attributed to the sign of equality. Let us see example of these skills:

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30 Example 2B (11 years) - A PRE-ALGEBRAIC PERSPECTIVE September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy) The pupils have the task of representing in mathematical language the sentence: “Twice the sum of 5 and its next number.” When the pupils’ proposals are displayed on the whiteboard, Diana indicates the phrase of Philip and justifies her writing: “Philip wrote 2×(5+ 6 ), and it is right. But I have written 2×(5+ 5+1 ) because in this way it is clear that the number next to 5 is a larger unit. My sentence is more transparent”. What can we say about Diana’s sentence?

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31 Example 2B (11 years) - A PRE-ALGEBRAIC PERSPECTIVE September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy) The pupils have the task of representing in mathematical language the sentence: “Twice the sum of 5 and its next number.” When the pupils’ proposals are displayed on the whiteboard, Diana indicates the phrase of Philip and justifies her writing: “Philip wrote 2×(5+ 6 ), and it is right. But I have written 2×(5+ 5+1 ) because in this way it is clear that the number next to 5 is a larger unit. My sentence is more transparent”. Diana is emphasizing the relational aspects of the number made evident by its non-canonical form.

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32 Example 3A (10 years) September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy) The task for the pupils is: ‘Translate the sentence 3×b×h into natural language’. Lorenzo reads what he has written: “I multiply 3 by an unknown number and then I multiply it by another unknown number.” Rita proposes: “The triple of the product of two numbers that you don’t know” Lorenzo observes: “Rita explained what 3×b×h is, whereas I have told what you do.” What can we say about Lorenzo’s sentence?

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33 Example 3A (10 years) September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy) The task for the pupils is: ‘Translate the sentence 3×b×h into natural language’. Lorenzo reads what he has written: “I multiply 3 by an unknown number and then I multiply it by another unknown number.” Rita proposes: “The triple of the product of two numbers that you don’t know” Lorenzo observes: “Rita explained what 3×b×h is, whereas I have told what you do.” Lorenzo captures the dichotomy process-product. Another example:

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34 Example 3B (two teachers) September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy) Rosa and Viviana are two teachers of one of our groups. They are discussing on a problem concerning the approach to equations using the scales: “There are 2 parcels of salt in the pot on the left, and 800 grams in the pot on the right”. Rosa explays her task: “How heavy is the salt?” Viviana observes: “It would be better to write: Represent the situation in mathematical language in order to find the weight of a packet of salt”. Please, comment Rosa’s and Viviana’s sentences.

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35 Example 3B (two teachers) September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy) Rosa and Viviana are two teachers of one of our groups. They are discussing on a problem concerning the approach to equations using the scales: “There are 2 parcels of salt in the pot on the left, and 800 grams in the pot on the right”. Rosa explays her task: “How heavy is the salt?” Viviana observes: “It would be better to write: Represent the situation in mathematical language in order to find the weight of a packet of salt”. Rosa and Viviana are reflecting on the dialectics representing/solving.

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36 Examples 3A-3B – USUAL BEHAVIOR September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy) Rosa’s task is set in a ‘classical’ arithmetic perspective: she looks for the solution and emphasizes the search for the product. This way, the pupils learn that the solution of a problem coincides with the detection of its result and with the search of operations. The consequence of this attitude is that the information of the problem are seen as ontologically different entities and separated into two distinct categories: the data and what one needs to find. Pupils solve the problem by operating on the former and finding the latter.

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37 Examples 3A-3B – A PRE-ALGEBRAIC PERSPECTIVE September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy) Viviana’s task is set in an algebraic perspective: it induces a shift of attention from elements in play towards the representation of the relationships between them and on the process. She draws the pupils from the cognitive level towards the meta-cognitive one, at which the solver interprets the structure of the problem and represents it through the language of mathematics.

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38 Examples 3A-3B – A PRE-ALGEBRAIC PERSPECTIVE September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy) This difference between the attitude that favours solving (Rosa) and that which favours representing (Viviana) is connected to one of the most important aspects of the epistemological gap between arithmetic and algebra: while arithmetic implies the search for solution, algebra delays it and begins with a formal transposition of the problem situation from the domain of natural language to a specific system of representation.

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39 Example 4 (12 years) - A PRE-ALGEBRAIC PERSPECTIVE September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy) Thomas has represented the relationship between two variables this way: a=b+1×4 and he explains: “The number of the oranges (a) is four times the number of the apples (b) plus 1”. Katia replies “It's not right: your sentence would mean that the number of oranges (a) is the number of apples (b) plus 4 (1×4 is 4). You have to put the brackets: a=(b+1)×4”. Reflect on Thomas’s and Katia’s sentences.

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40 Example 4 (12 years) - A PRE-ALGEBRAIC PERSPECTIVE September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy) Thomas has represented the relationship between two variables this way: a=b+1×4 and he explains: “The number of the oranges (a) is four times the number of the apples (b) plus 1”. Katia replies “It's not right: your sentence would mean that the number of oranges (a) is the number of apples (b) plus 4 (1×4 is 4). You have to put the brackets: a=(b+1)×4”. Thomas and Katia are discussing the translation between natural and algebraic language and the semantic and syntactic aspects of mathematical writings.

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41 Example 4 (12 years) - USUAL BEHAVIOR September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy) Sentences in the mathematical language as, e.g., a=(b+1)×4, are generally seen from an operational point of view rather than an interpretative one. Students unaccustomed to reflecting on the meanings of the sentences written in algebraic language, in this case merely observe that “a=b+1×4 is wrong because there are no brackets”.

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42 Examples 4 - A PRE-ALGEBRAIC PERSPECTIVE September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy) Translating from the natural language to the mathematical one (and vice-versa) favors reflecting on the language of mathematics, that is interpreting and representing a problematic situation by means of a formalized language or, on the contrary, recognizing the problematic situation that it describes in a symbolic writing. Closely related to the act of representing is the issue of respecting the rules in the use of a (natural or formalized) language. In teaching mathematics, rules are generally ‘delivered’ to pupils, thus losing their social value of a support to the understanding and sharing of a language as a communication tool.

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43 Examples 4- A PRE-ALGEBRAIC PERSPECTIVE September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy) Pupils should be guided to understanding that they are acquiring a new language which has a syntax system and semantics. They internalize from birth that compliance to the rules allows communication, but it is highly unlikely that they will transfer this peculiarity to the mathematical language. In order to overcome this obstacle, we ask pupils to exchange messages in arithmetic-algebraic language with Brioshi, a fictitious Japanese pupil who speaks only in his mother tongue. This trick works as a powerful didactical mediator to highlight the importance of respecting the rules while using the mathematical language.

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44 Open questions September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy) When and how does the curtain on algebra begin to open? Which teachers’ attitudes can favour pre- algebraic thinking? Do you agree with the idea that algebra doesn’t follow arithmetic, but rather develops by intermingling along with it right from the first years of primary school? Which mathematics education should future teachers receive in order to improve their sensitiveness towards those micro-situations that allow to ‘see algebra within arithmetic’?

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45 Open questions September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy) Which is your position about these topics? Which kind of difficulties (cultural, social political difficulties) do you see about the spreading of this type of teaching in the classes? Which constraints? Which is the status of Early algebra in your country? Which are the teachers' dominant beliefs about algebra and early algebra in your country? Which importance do these questions have in the pre-service mathematics education of teachers?

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46 September 21–26, 2014, Herceg Novi, Montenegro, Workshop Malara-Navarra (Italy) Thank you

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