Introduction

Researchers in mathematics education are beginning to characterize how the students for primary school teacher learn the knowledge of Teaching Mathematics necessary to teach (and Krainer Llinares, 2006). Some of the approaches to the learning of knowledge needed to teach mathematics are based on perspectives located that considered inseparable the context where learning takes place and the way in which this is acquired (Hiebert et al. , 2007; Hiebert, Gallimore and Stigler, 2002; Wilson and Berne, 1999). A focus of interest in this type of research is located in the analysis of the resolution of professional tasks in learning environments where specially designed for master students can interact and make use of theoretical information of Didactics of Mathematics that is considered useful for teaching mathematics (Bairral, 2007; Llinares and Valls, 2007; Llinares, Valls and Roig, 2008; McGraw et al. , 2007; Tsamir, 2008).

From this perspective, the knowledge needed to teach mathematics relates to the capacity to identify and assign meaning to aspects of the practice (Martínez, 2005; Penalva, Escudero and beard, 2006; Van Is and Sherin, 2002) during the resolution of professional tasks related to the teaching of mathematics (diagnosis, educational planning and management) (Llinares, Valls and Roig, 2008). In the contexts where it gives this learning, the interactions of the participants They have appeared as a relevant factor, on having favored the incorporation of information of Didactics of the Mathematics in his structures of thought (Azcárate, Rodríguez and Rivero, 2007; Cos, Valls and Llinares, 2005). In this respect, some interventions in the programs of formation have incorporated technological tools to promote the communication and interaction between the students, questions of investigation being generated on the influence of these contexts in the learning processes (Llinares and Olivero, 2008; Mousley, Lambdin and Koc, 2003). A question of investigation that appears is to manage to know the effects that on the learning produce the interactions on line of students for teacher when they solve professional tasks and that demonstrate in the content of his speeches. Other one of the areas of interest centres on the characteristics of the formative contexts. From this perspective, some investigations underline the importance of the contexts in which the students for teacher solve tasks and how they influence these contexts in the construction of necessary knowledge to teach (Callejo, Valls, and Llinares, 2007; Llinares and Krainer, 2006; Morris, 2006; Tsamir, 2007; Valls, Callejo and Llinares, 2008). The results of these investigations have had implications in the design of different formative experiences of the professorship that promote the instrumental use of contents of Didactics of the Mathematics structured in environments of learning happened by the communication on line (Cos, Valls and Llinares, 2005; García et to., 2006; Callejo, Llinares and Valls, 2008; Llinares and Valls, 2007; King, Penalva and Llinares, 2006).

A specific content for the students for teacher constitutes it the knowledge of the characteristics of the arithmetical elementary problems (pae): categorization of the problems, levels of difficulty and possible strategies of resolution (Castro, Rich and Castro, 1995; Carpenter, Moser and Romberg, 1982; Puig and Cerdán, 1988; Socas, Hernández and It Nongives, 1998). In relation with the pae of structure aditiva, the investigation in mathematical education has identified different ways of categorizing the problems that involve the addition and the subtraction (Vergnaud, 1997; Verschaffel and Of Court, 1996). A categorization widely spread of these problems considers the semantic relations between the present quantities in the terms of reference, which has led to distinguishing three basic categories: problems of change (some event changes the value of an initial quantity), of combination (a particular set and two subsets disjuntos of him relate between yes) and of comparison (two quantities and the difference are compared between them) (Carpenter, Moser and Romberg, 1982; Verschaffel and Of Court, 1996).

In turn, every category is subdivided depending on the mystery in the problem. For example, a problem of change can change as if the mystery is the initial quantity, the quantity of change or the final quantity. In addition, distinctions have been established between problems of change or comparison, as it is the direction of the produced change or the relations of comparison that are established (increasing or diminishing). Other one of the studied aspects of the pae of structure aditiva refers to his level of difficulty or to the strategies of resolution used by the pupils (Carpenter, Moser and Romberg, 1982; Verschaffel and Of Court, 1996). The difficulty of the PAE of structure aditiva is in the habit of assuming to his semantic structure and to the place that occupies the mystery in the problem (Vermilion and Olive, 1987). As for the strategies of resolution of these problems, Carpenter, Moser and Romberg (1982) thought that, in general, the children shape directly the action or relation described in the problem and, with the time, his knowledge uses on the inventories (the count) and to remember numerical concrete facts as more efficient abstractions of the direct modeling to solve the problems. From this perspective, the knowledge on the characteristics of the PAE is considered to be relevant to teach mathematics in Primary Education and, therefore, It has started being considered to be an object of learning for the students for teacher. In this respect, some investigations on the learning of the characteristics of the pae of structure aditiva with teachers in exercise (Carpenter et to., 1989; Fennema et to., 1996) they identified the relations that these establish between the knowledge of the development of the mathematical thought of the children in relation with the pae of structure aditiva and his practice instruccional The results of these investigations indicate that the knowledge of the information about the different types of problems, the strategies used by the children of primary to solve them and how they evolve these strategies it was producing changes in the practice instruccional of the teachers. These conclusions are important from the point of view of the teachers' initial formation in all that that they provide the base in order that the students for teacher develop his knowledge with major extent (Carpenter, Fennema and Franke, 1996). Nevertheless, little it is known on the way as the students for teacher they learn the characteristics of the pae of structure aditiva during his universitary education. Especially, in Spain there have developed studies centred on, at least, three areas of interest. On the one hand, the learning of these topics has been analyzed from the perspective of the "changes" in the conceptions of the students for teacher on the characteristics of the pae of structure aditiva during his participation in formative designed ad hoc experiences (Cos and Valls, 2006). On the other hand, the learning has related to the aptitude to identify the underlying structure of different types of pae of structure aditiva with negative numbers (Dark and García, 2004) and of multiplicative structure (Castro and Castro, 1996), emphatically in the difficulty that represents this task for the students for teacher. And finally, there has linked itself the learning of the characteristics of the pae with the possible relations that the students construct for teacher between the theory proceeding from the Didactics of the Mathematics and the evidences that observe in the content of the problems that they are proposed them (García et to., 2006). These investigations have revealed the difficulties that the students have for teacher in providing with meaning to the semantic characteristics of these problems.

In these contexts, the introduction of alternative spaces of learning in the programs of formation allows to generate questions on the learning of these ideas on the part of the students for teacher. Our investigation has as aim characterize the way in which a group of

students for teacher of primary learns to conceptualize the education of the resolution of pae of structure aditiva of a stage by means of the use of theoretical information in contexts of communication on line.

THE PROCESS OF LEARNING TO TEACH MATHEMATICS

The placed perspectives of the learning allow to understand the education of the mathematics as a practice that can be learned and understood (Llinares, 2004). From this perspective, to learn to teach mathematics is understood as a process of providing with meaning to the professional experience. According to Wenger (2001), this process of providing with meaning to the professional experience places in social contexts of negotiation of meanings, where there takes place the participation happened by instruments of the practice. The instruments of the practice are any way, already be material or symbolic, that seeks to promote the action of the subjects that realize specific tasks (Llinares, 2004; Verillon and Rabardel, 1995).

Bearing these theoretical references in mind, we think that the students for teacher can provide with meaning to the education of the resolution of problems in primary when they take part in environments of learning designed to analyze the didactic potential of the arithmetical problems. The context of analysis of these problems creates conditions in order that the students for teacher use theoretical information from different domains: knowledge of the mathematics, knowledge of the school mathematics and pedagogic specific knowledge of the mathematics (Llinares, Valls and Roig, 2008; King, Penalva and Llinares, 2006). To the beginning of the formative situations, the decisions and actions of the students for teacher can be influenced by previous knowledge on the way of proposing pae children of primary (Azcárate and Castro, 2006). Nevertheless, this knowledge can transform in a knowledge more professionalized by means of the progressive integration of the information of Didactics of the potentially useful Mathematics for the analysis of the problems. This transformation is understood as a process of carrying out the theoretical information incorporated into the formative contexts in which own tasks of a teacher are solved of primary and meanings are negotiated.

The instrumentalización of the information of Didactics of the Mathematics implies a process of reificación and of participation in a certain community of practice (Wenger, 2001). The duality between the process of reificación and the participation, according to Wenger (2001), can allow the development of the comprehension of the experience of teaching mathematics in terms of the theoretical information that joins the structures of professional knowledge of the students for teacher in the shape of conceptual instruments. The conceptual instruments are understood as concepts and theoretical constructions, Originated by the investigation in Didactics of the Mathematics, which there allow to interpret and to understand the situations of education and learning of the mathematics, facilitating the capture of decisions as for the way of treating the above mentioned situations to achieve aims instruccionales (García et to., 2006; Lin, 2005; Llinares, 2004).

In this investigation, to characterize the learning of a group of students for teacher of primary, we use the idea of instrumentalización of the theoretical ideas of Didactics of the

Mathematics, understood as the relations that the students establish for teacher between a semantic categorization of the pae of structure aditiva of a stage, the levels of difficulty and the strategies of resolution (Castro, Rich and Castro, 1995), as a way of determining useful criteria for the task of planning the education. The use of theoretical information during the analysis of the potential of the pae of a stage in the planning of the education is a manifestation of the transformation of the theory in conceptual useful instruments in the practice of teaching mathematics (Eraut, 1996; García et to., 2006). In this respect, we assume that the instrumentalización of the information of Didactics of the Mathematics can take place in social contexts of negotiation of meanings, demonstrating by means of the interactions produced by the students for teacher when they solve a professional task, since it is the planning of the education (King, Penalva and Llinares, 2006).

From this perspective, in contexts happened by the communication on line, we understand the interactions as the product of discursive activities of negotiation of meanings concerning shared topics of discussion (Barberá, Badia and Mominó, 2001). This way, in the spaces of debate on line, the participation of the students for teacher acquires the form of speeches in those who express arguments to themselves and there are negotiated meanings assigned to the classification and resolution of the pae of structure aditiva of a stage and the difficulties of learning derived before the task of planning the education. These speeches can manage to reflect the different uses of the information of Didactics of the Mathematics in the resolution of tasks of planning (King, Penalva and Llinares, 2006) and, therefore, they can indicate features of the learning of the students for teacher.

The previous theoretical references make arise questions of investigation relative to the way how the students for teacher construct arguments in you debate on line when they analyze pae of structure aditiva of a stage from a semantic perspective with the aim to plan the education. In this investigation, we appear the following question: ¿How do students carry out the theoretical information of Didactics of the Mathematics for teacher when they analyze pae of structure aditiva of a stage before the task of planning the education in a context on line?

To give response to this question i) we identify the different uses that the students do for teacher of the theoretical information when they analyze a collection of pae of structure aditiva, and ii) we describe the different situations of negotiation in a context on line, produced to give response to the task of the planning of the education.

CHARACTERISTICS OF THE ENVIRONMENT OF LEARNING ON LINE

This investigation places in the application of an environment of learning on line in a program of teachers' formation of Primary Education. The environment of learning was designed to approach the didactic analysis of the problems and the resolution of problems in primary using a methodological approximation b-learning. The structure of the environment of learning was consisting of four parts, a part presencial and three you divide on line. Each of the parts in which the environment of learning was structured receives the name of "session" to indicate that it was constituted by an aim of learning, a series of tasks, a space of interaction and a channel of delivery of documents on the part of the students for teacher (Valls, Llinares and Callejo, 2006). Every session had a variable duration in number of days (from 8 to 15 days).

The information of this investigation comes from the texts elaborated by the students for teacher during his participation in the second part on line (session on line 2). This session had as aim that the students for teacher were learning to identify the semantic structure of the problems additives, the levels of difficulty and the strategies of resolution of these problems on the part of the pupils of Primary Education. In this session the students for teacher had to realize i) a task, which will be described later, and take part in ii) a debate on line for the communication of the ideas and the establishment of agreements in relation with the joint resolution of the task. This debate was thought as a space that was favoring the interaction between the students for teacher. The development of this part of the environment of learning was carried out across the web platform of the university, to which the students for teacher could have access by means of a personalized key of access (password) (1appears) 1 appears: Window of introduction to the session on line 2 of the environment of learning

The proposed task was consisting of analyzing six pae of structure aditiva proceeding from a collection elaborated before by the own students (I square 1) in the first one it divides on line (session on line 1) of the environment of learning. The analysis had to be realized bearing in mind the theoretical information that was providing them in different documents. The task had three paragraphs:

a) To classify the problems of the collection using semantic criteria,

b) To arrange them according to the difficulty that they present, justifying the decisions,

c) To indicate for every problem two possible strategies that the children would use to solve them before using the algorithms of the operations (" the accounts ").

I square. 1 Problems of the collection elaborated by the students in the session on line 1

Nº Terms of reference of the problem1 Hello, I am Sara. I have a problem, can you help me? My brother and I celebrate together our

birthday and my mom wants to know how many children we have invited. Here you have our lists of guests: Guests of Sara: Pepe, Maria, Jose and Laura. Guests of the brother of Sara: It was Shining, Toni, Laura, Sandra and Carmen. How many children will come to our house?

2 In a forest there has been a fire and all the trees have burned. Maria, Alberto and Stela are very sad because already they cannot play there and have decided to return to plant some trees. Maria plants 4, Alberto plants 3 and Stela plants also 3. Do all the trees plant in total?

3 The bus that goes up to Holy Pola works out empty of the station. In the first stop 9 persons rise. In the second stop they raise 12 more. How many passengers it is now in the bus?

4 In the corralito there were 15 hens. When the farmer returned to the corralito, it saw that 6 hens had escaped. How many hens stay in total in the corralito?

5 White of egg is 15 years old and Pedro has 11. How many years it has White of egg more than Pedro?

6 To observe how many points the children must obtain to gain every prize.

Carmen has 49 points, he lacks how many points to obtain the scooter?

The information provided to the students for teacher was consisting in: i) a semantic categorization of the pae of structure aditiva of a stage: I change (increasing and diminishing), combination (total and part) and comparison (increasing and diminishing); ii) the levels of

difficulty in the resolution of these problems, and iii) the strategies commonly used by the pupils of Primary Education to solve them: modeling, count and use of numerical facts (Castro, Rico and Castro, 1995; Carpenter, Moser and Romberg, 1982). The proportionate information was allowing to the students for teacher to realize the analysis of the problems beyond informal and intuitive criteria.

To solve the task, the students for teacher took part in a debate on line that was allowing them to interact with his companions, in order to achieve agreements in relation with the analysis of the problems from the information of the Didactics of the Mathematics. This debate on line was kept opened for 11 constant days.

METHODOLOGY

PARTICIPANTS

The information of this investigation comes from 9 students for teacher (7 women and 2 men) of a total of 53 students in his first year of the program of formation. 9 students were forming a part of one of two groups (G1) in that 53 students were divided for teacher to take part in the environment of learning on line. These students, before took part in the session on line 2 of the environment of learning, had taken part in the session presencial (40 hours) and in the session on line 1 (13 days it activates).

ANALYSIS

The information of this investigation corresponds to the participations of 9 students for teacher in the debate of the session on line 2. The picture 2 gathers the total number of messages issued by the students in the debate and his temporary distribution.

I square 2 Number of contributions for every day of debate in the session on line 2

Days of active debate

1st

2nd 3rd 4th 5th 6th 7th 8th 9th 10th 11th Total

Number of contributions

4 9 11 1 6 4 7 19 6 20 0 87

The realized analysis assumes that the theoretical information joins progressively in the discussion and his use demonstrates in the content of the speeches (the contributions) of the students for teacher during the debate on line. There is analyzed the content of the speeches produced in the debate on line considering the dimension epistémica (Barberá, Badia and Mominó, 2001), in that there is born in mind how they construct arguments in correspondence with the use of the theoretical proportionate information and his relation with the empirical contributed evidences.

In the issued messages, there were identified different ideas expressed by the students for teacher in the shape of phrases who constituted the units of meaning (Strijbos et to., 2006).

For example, the figure 2 shows three units of meaning contained in the same message produced by one of the students for teacher.

It figures 2: Units of meaning identified in a message issued by the student 3

According to the structure (Student 3 - 11:00:21 29/04/27)

US 1 I agree with the classification that you have done (for it I am not going to repeat it).US2 Also I agree in that the problem 3 is of increasing change since we meet an action

described in the TIME and the initial quantity (0 passengers) is increasing and the final quantity changes.

US3 Also I would like to comment on the doubts to you that I have had in the problem 6. I agree in that it is of increasing change but I about a beginning thought that I was of COMPARISON, since Carmen has to compare the points that it has and the points for the scooter. Since basically it has to compare both points I praised myself in a beginning for this structure. But it is true that in the question of the problem appears the mystery of " quantity of change ". So I agree with you. A greeting

US. p: Unit of important number p; with p = {1, 2, 3}

Every unit of meaning, considered as a unit of analysis, was codified taking as a reference four categories of use of conceptual instruments proposed by García et to. (2006), which describe different uses of the theoretical information of Didactics of the Mathematics on the part of the students for teacher in contexts of initial formation (I square 3).

This analysis allowed to identify uo to a point the learning process of some students for teacher on having showed how his contributions were moving from rhetorical categories to categories of instrumentalización.

In addition, the speeches generated in the debate on line were organized by means of conversational chains understood as a sequence of interactions between the students for teacher that they debate on the same topic (Llinares and Valls, 2007). For example, a topic of discussion for the students can be " the semantic category of the pae of structure aditiva ".

I square 3 Categories of use of conceptual instruments

Category Description of the categoryCategory 1. Arguments from previous knowledge

The student for teacher realizes a "natural" description of some aspects of the pae of structure aditiva from his previous experience.

Category 2. Rhetorical use of the theoretical ideas

The student for teacher makes "rhetorical" use of the theoretical ideas provided without establishing relations between these ideas and the empirical evidences contained in the problems of the collection: actions, structure and / or present quantities in the problems.

Category 3. Beginning of the instrumentalización of the theoretical ideas

The student for teacher begins to do instrumental use of a semantic categorization of the pae of structure aditiva of a stage, the levels of difficulty and / or the strategies of resolution of these problems, relating them to the empirical evidences contained in the problems of the collection.

Category 4. Integrated use of

The student for teacher makes integrated use of the theoretical ideas provided to analyze the problems of the collection, relating them at all

the theoretical ideas like conceptual instruments

time to the empirical evidences contained in these problems.

Every conversational chain was represented in a graph that gives information about the subjects that take part in the negotiation, how these use the theoretical ideas to argue, the relation that they establish between the content of the produced speeches and the time in which it takes place (it figures 3).

The analysis of the participations of the students for teacher in the debate developed in three stages. In the stage 1, meetings were carried out between the investigators to provide with meaning to every category of use of conceptual instruments from the assembled information. The investigators analyzed together a sample of the messages to assign the describers. The generated criteria were applied after independent way by each of the investigators to the total of the analyzed messages. The discrepancies were discussed and the capture of final decision was agreed by consensus. Hereby, every unit of analysis was codified in one of four categories of use of conceptual instruments. In the stage 2, were considered to be the different conversational chains. Every chain was represented graphically and there was identified the topic that produced the interaction. In the stage 3, was considered in a global way the instrumental use of the theoretical information in the different conversational chains to describe how the instrumentalización took place.

Figures 3 graphical Representation of a conversational chain (C1) centred on the semantic category of the pae of structure aditiva