domain: math shs

text:
Aisling Twohill
email: aisling.twohill@dcu.ie

Shape patterning tasks: An exploration of how children build upon their observations when asked to construct general terms

Keywords: Algebraic thinking, generalisation, shape patterns, reasoning, task-based interview, group interaction

. Shape patterns provide a context within which children may identify structure, and construct general terms. Within my research I presented children with shape patterns and observed their interactions as they discussed the structure of the patterns and sought to construct general terms. In this paper, I discuss the elements of the patterns to which they attended, and how their focus supported some children in constructing general terms, while other children experienced more challenge in their engagement with the patterns. Specifically I focus on children who observed both numerical and figural aspects, and I examine the varying levels of success they experienced.

Introduction

 [START_REF] Kaput | What is algebra? What is algebraic reasoning[END_REF]defined generalising, representing generalisations, and syntactically guided reasoning on generalisations as core aspects of algebraic thinking. Providing a context for functional thinking, shape patterning tasks may facilitate children in engaging with these core aspects. [START_REF] Strømskag | A pattern-based approach to elementary algebra[END_REF] defines a shape pattern as a sequence of terms, composed of 'constituent parts', where some or all elements of such parts may be increasing, or decreasing, in quantity in systematic ways. While a limited number of terms of a shape pattern may be presented for consideration, the pattern is perceivable as extending until infinity. In order to construct a general term for a shape pattern, children must "grasp a regularity" in the structure of the terms presented, and generalize this regularity to terms beyond their perceptual field (Radford, 2010, p. 6). In seeking to construct general terms for shape patterns, children's opportunities for success improve when they attend to both the spatial and numerical aspects of the structure of the pattern [START_REF] Radford | The progressive development of early embodied algebraic thinking[END_REF]. In seeking to extend a pattern, children identify the position of pattern elements by attending to the spatial structure, and in order to quantify such elements they must attend to the relevant quantities within the pattern terms presented.

In the Irish context, the Primary School Mathematics Curriculum (hereafter PSMC) includes an 'algebra' strand, and there are content objectives prescribed for teachers to address in their teaching, from when children commence primary school at four or five years of age (Government of Ireland, 1999). The curriculum does not, however, include the construction of general terms in any context, and beyond the first two reception years the patterns prescribed are strictly numerical in nature. During task-based group interviews, I asked children to construct general terms for each of three shape patterns. In this paper I will explore the tendency of children participating in my research to attend to the spatial and numerical aspects of pattern structures, as they worked towards the construction of general terms. Of the children who attended to both numerical and spatial elements, some children experienced success in applying their observations in order to construct general terms, whereas some children seemed to experience a challenge in translating their observations into meaningful understanding of the structure of the pattern.

The construction of general terms

Building upon the core aspects of algebraic thinking outlined by [START_REF] Kaput | What is algebra? What is algebraic reasoning[END_REF] and mentioned above, [START_REF] Blanton | Elementary children's algebraic thinking[END_REF] identified four key practices of algebraic thinking as: a) generalising, b) representing, c) justifying, and d) reasoning with generalisations, emphasising that these practices must focus upon structures and relationships. [START_REF] Blanton | Elementary children's algebraic thinking[END_REF] also presented the following three content domains within which children may apply the key practices of algebraic thinking: a) generalised arithmetic, b) equations, and c) functional thinking.

In particular, [START_REF] Blanton | A learning trajectory in 6-year-olds' thinking about generalizing functional relationships[END_REF] highlighted the relevance of functional thinking to the algebraic thinking of young children, stressing the connection between functional thinking and the four key practices of algebraic thinking highlighted above. [START_REF] Blanton | A learning trajectory in 6-year-olds' thinking about generalizing functional relationships[END_REF] emphasised the role of functional thinking in young children's algebraic thinking by stating that functional thinking includes generalisations of co-varying quantities and their relationship; representations of these relationships, and reasoning with the relationships in order to predict functional behaviour. As functional thinking, and specifically shape patterning, is absent from the PSMC, it is therefore highly improbable that most children attending Irish primary schools will have engaged with activities designed to develop functional thinking. [START_REF] Rivera | Formation of pattern generalization involving linear figural patterns among middle school students: Results of a three-year study[END_REF] identified two approaches utilised by children in seeking to construct general terms for geometric patterns. Some children in Rivera and Becker's longitudinal study focused solely, or primarily, on numerical aspects of the terms provided, and sought to use the numerical patterns observed in order to identify a commonality, extend the pattern and construct a general term. Such children the authors described as adopting a 'numerical' mode of generalising. In comparison, Rivera and Becker considered a child to have adopted a 'figural' approach if he/she used figural aspects of the pattern such as the shape of terms, or the position of elements, both within the term and relative to each other. A figural approach may include attention to numerical elements, but not in a manner that supersedes the child's perception of the spatial aspects of the pattern structure.

Along with observations of the structure of terms, children may observe various relationships within patterns which support their understanding. A 'recursive' strategy involves an examination of the mathematical relationship between consecutive terms in a sequence, and if using an 'explicit' strategy, a child identifies a rule for the relationship between a term and its position in the pattern [START_REF] Lannin | Generalization and justification: The challenge of introducing algebraic reasoning through patterning activities[END_REF]. Such observations work in tandem with children's numerical or figural approaches, and in this paper I will refer to children's explicit, or recursive thinking when relevant.

Radford asserted that children's constructions of generalities may be factual, contextual or symbolic. Factual generalisations involve instantiating a general structure to specific terms, whereby children do not express a generalisation as applicable to all terms, but apply an "operational scheme" which allows them to calculate a value for particular terms (p. 82). Many of the children involved in this research project applied factual generalisations when they described the near and far terms for the patterns, as in the example of Emily for Pattern 3 when she said "You'd need a hundred and twelve horizontal and then fifty-seven vertical". Emily applied her understanding of the 2n horizontal poles, and n vertical poles in order to calculate the number of poles for this far term. Equally she could apply this thinking to any term of this pattern, and could therefore be said to have factually generalised, even though there is no abstraction evident in her expression, and she is describing a specific term.

Contextual generalisations, by comparison, involved the consideration of non-specific terms (ibid.). While contextual generalisations are not completely abstract, or general to all terms, they indicate a distancing from the specific, whereby children may make reference to "the next term" or to a generic term. As an example, Grace could be said to have constructed a contextual generalisation for Pattern 1 when she said "whichever number it is at the top it will just be one more than it, and at the bottom". Symbolic generalisations involve the abstract expression of disembodied mathematical objects, wherein children express the algebraic concepts with no reference to the method of their calculation, or to any specific term. Throughout this paper I make use of 'n' to refer to a general term number, but all such expressions were generated by me, as representations of children's verbal utterances, constructions and gestures. The children were not prompted to construct symbolic generalisations.

Methodology

In seeking to explore children's constructions, I facilitated their engagement with the patterns in groups of four in a 'task-based interview' setting. [START_REF] Goldin | A scientific perspective on structured, task-based interviews in mathematics education research[END_REF] emphasised that task-based interviews, involving individuals or groups, have become an essential tool within mathematics education research, as the goal of mathematics education has moved from the transmission of disconnected facts, to the development of children's rich understanding and "internal constructions of mathematical meaning" (p. 524). In order to explore children's complex understandings, and to observe their approaches to the solution of problems, it is necessary to adopt a research approach designed specifically for this purpose. In this way, it may be possible for educators and policy makers to assess whether the application of progressive approaches within classrooms are working to develop children's mathematical understanding and robust problem-solving skills.

Sixteen children (with a mean age of 9.6 years) participated in the task-based interviews, when they were presented with three patterns (see Figure 1). When presented with the patterns, the children were asked to describe what they observed, to extend the pattern to subsequent or previous terms, and to construct near and far terms, as factual generalisations. The children were provided with concrete materials, and asked to construct pattern terms during their discussions. Having considered far terms, I asked the children to describe "any term" in the pattern to facilitate their articulation of a contextual generalisation. At the commencement of each interview, and at regular intervals throughout, the children were encouraged to work collaboratively, by constructing terms together, by sharing their ideas, whether they agreed or disagreed, and by asking questions of each other. The children did not receive any instruction before participating in the interviews. Seeking to explore the reasons for children's strategy choice, as well as the strategy chosen, the research approach I considered most appropriate was phenomenological, as a phenomenological approach to research seeks to explore a phenomenon where it occurs, and acknowledges the many factors which influence how the phenomenon plays out within the given setting [START_REF] Creswell | Qualitative inquiry and research design: Choosing among five approaches[END_REF]. In exploring the children's thinking, I not only analysed the children's actions, but I also sought to analyse the contributing factors that impacted on the strategies children employed. [START_REF] Goldin | A scientific perspective on structured, task-based interviews in mathematics education research[END_REF] advises that "by analysing verbal and nonverbal behaviour or interactions, the researcher hopes to make inferences about the mathematical thinking, learning or problem-solving of the subjects" (p. 518, my emphasis). In seeking to explore children's mathematical constructions, I was conscious throughout that my inferences from children's comments were approximations of their true meaning. As Van [START_REF] Van Manen | Researching lived experience: Human science for an action sensitive pedagogy[END_REF] attests "a good phenomenological description is an adequate elucidation of some aspect of the lifeworld" (p. 27, my emphasis) and while I sought to unpick as best I could how and why children thought about the mathematical tasks, I posit that it is not possible to feel a sense of completion, or closure, in relation to the children's thinking, but rather that interpretation is ongoing [START_REF] Postelnicu | The figurative method: A bridge from numerical to quantitative reasoning[END_REF].

In exploring the children's verbal utterances, I coded comments as referring to figural or strictly numerical aspects of a pattern. [START_REF] Rivera | Formation of pattern generalization involving linear figural patterns among middle school students: Results of a three-year study[END_REF] state that adopting a figural approach is to "figurally apprehend and capture invariance in an algebraically useful manner (p. 356). When coding children's statements, I identified a statement as indicating a figural approach if it included reference to the position of an object within a term, by using words such as 'top', 'bottom' or the deictic 'there' along with an associated gesture [START_REF] Radford | Algebraic thinking and the generalization of patterns: A semiotic perspective[END_REF]. An object in this context referred to a square, a diamond, a tile, a line, or any item which formed a constituent part of a term. Comments were deemed to indicate a numerical approach if no reference was made to the position of objects within a term. The term 'growing' was used regularly by children, and required some thought with regards to whether it indicated a figural approach. Typically, when mentioning growth children were referring to a sense of the terms' shape growing in size, that is "selectively attend[ing] to aspects of sameness and difference among figural stages", but I could not assume that this was always the case (Rivera & Becker, 2011, p. 356). Rather, it was necessary to attend to some term, or deictic within a child's comment, and to seek to determine the referent, which would indicate whether the child was referring to the shape as growing, or the quantity of constituent elements.

Findings and Discussion

In seeking to explore the strategies used by children, and the reasons underpinning their strategy choice, I firstly considered an overview of each child's approach across all three patterns. As the length of this paper does not allow for a complete explication of the approach of each child, Figure 2 presents an overview of my interpretation of the children's thinking, as it pertained to the balance between numerical and figural observations. In collating and analysing this data, I referred to the comments children made in the context of the exchanges they participated in. I also referred to fieldnotes made during the interviews, photographs of the children's constructions, and the children's drawings and jottings which I had retained as artefacts of their thinking during the interviews. I had video recordings for three of the four interview groups, and audio recording for the fourth, as not all participants of this group had assented to video recording.

Wholly numerical: all comments referred to numerical aspects of patterns. Alex Largely numerical, some comments focused on figural aspects, or observed figural aspects but not in a manner which seemed to support an understanding of the structure of the pattern.

Daniel, Luigi

Largely figural, but experienced confusion, or remained quiet during large parts of the interview. In some cases confusion was due to 'loyalty' to numerical aspects.

Cherry, Orla, Danny, Fiona, Jay

Largely figural, and comments indicated the use of numerical aspects to gain a strong understanding of a pattern's structure.

Ciaran, Grace, Emily, Arina, Jane, Wyatt, Christopher

Lily Rose

Exclusively figural: all comments referred to figural aspects of patterns. As can be seen from Figure 2, rather than demonstrating a dichotomy between children who approached all patterns figurally and children who approached all patterns numerically, this small group of children (n=16) span a continuum from children who made comments focusing largely on numerical aspects, through children who commented on both numerical and figural aspects in meaningful ways, to children who referred largely to figural aspects. In considering the aspects children were attending to, I sought to distinguish between children's observations which supported their thinking, and observations which they made and didn't build upon or apply in order to construct a general term. Later in this paper I will further explore how two of the children applied their observations of figural aspects.

While attending to figural aspects may have supported some children in constructing general terms, figural observations did not lead inevitably to generalisation. In order to compare children's tendency to observe figural aspects with their success in constructing factual or contextual generalisations, I generated a scoring rubric for children's progress towards the construction of a general term. A score of 0 indicated no progress, a correct extension of the pattern scored 1, some description of general terms scored 2, factual generalisation scored 3 and a contextual generalisation scored 4. I generated a total score for each child and calculated the mean score for each cluster of children identified in Figure 2 above. I found that the cluster of children identified in Figure 2 as making "largely figural" observations while applying numerical observations succeeded well, achieving a mean score of 8.6, where a score of 9 would equate to, for example, factual generalisation of all three patterns. In contrast the group of children who also made many figural generalisations, but expressed some confusion fared considerably less well, achieving a mean score of 3.2, where a score of 3 would equate to extending each pattern correctly, but not making any progress in describing a general term.

To explore what other factors may have impacted on children's progress, I chose to contrast the thinking of two children, Cherry and Arina, who worked together, and who both articulated figural observations, but made strikingly dissimilar progress. Arina achieved a score of 9 overall, while Cherry achieved a total score of 2. In this section I will discuss the girls' observations of Pattern 3, a 'fences' pattern presented in Figure 1 above. Arina had demonstrated strong figural thinking on two previous patterns, and succeeded in describing factual generalisations for both. When the children began their deliberations about the pattern discussed here, Arina remained very quiet, making few comments, but succeeded in constructing the 56th term using an explicit approach. She didn't verbally articulate her thinking enough to confirm whether this factual generalisation was based upon a numerical or figural mode of generalising, but she could be seen on the video footage counting up in twos to quantify the horizontal posts, and adding on a number equivalent to one more than the term number. By comparison, Cherry's verbal articulations of her thinking indicated that she observed within each fence one panel containing 4 posts, and every other panel containing three posts. When finding the number of posts needed for the nine-panel fence, she used a recursive approach, counting on six posts from the seven-panel fence she had constructed with match-sticks. I interpreted Cherry's perception of this pattern as including figural aspects, as she referred regularly to 'posts' and grouped the posts into groups of three or four, as appropriate. However, Cherry's approach to this pattern was dominated by a counting strategy, an analysis of which is beyond the scope of this paper. This counting strategy did not support Cherry in constructing far generalisations, and in seeking to construct a far generalisation, Cherry drew 56 panels, and began to count the number of posts needed.

Conclusion

Rivera and Becker (2011) suggest that when children only attend to numerical aspects of a pattern, they are grasping the commonality within the structure of the pattern at a superficial level. In analysing the comments made by the children in this research study, it may be seen that one cluster of children attended to figural elements but did not succeed in generalising. I would suggest therefore that some children who attend to figural aspects of the pattern, may persist with a limited and superficial understanding of the structure of the pattern, and that a figural perspective may not lead inevitably to successful construction of a general, or generic term. Other perceptions of the pattern structure seemed to be required in tandem with observations of both figural and numerical aspects.

In the examples given here, Arina's explicit approach supported her thinking, and in marrying an explicit approach with observations of both numerical and figural aspects, Arina grasped the structure of the pattern and successfully constructed a factual generalisation. In contrast Cherry demonstrated a consistent tendency to use counting as a strategy in seeking to quantify the number of elements of components of far terms of this pattern. While she could describe the figural structure of the pattern, her thinking seems to have been hindered by difficulties she encountered in seeking to conceptualise an expression which would allow her to calculate the number of posts without counting them. Even though this was the third pattern, and other children in her group had described explicit approaches during the interview, Cherry might not have made sense of the explicit thinking articulated by others. Equally, limitations in her multiplicative thinking may have contributed to this, and restrained her from exploring the explicit relationship between each term and its position in the pattern. Further research is merited into the interplay between the many aspects of patterns to which children may attend in seeking to grasp the structure.

Figure 1 :

 1 Figure 1: The three patterns presented to children.

Figure 2 :

 2 Figure 2: An overview of my interpretation of the children's thinking, as it pertained to their focus on numerical and figural aspects of patterns

  Figure 3 presents an illustration of how Arina may have been quantifying the number of posts for terms in this pattern. 5th Term in the Pattern: n=5. 2n horizontal Posts and n+1 Vertical Posts Total number of posts: 2n + (n+1) = 3n+1

Figure 3 :

 3 Figure 3: A representation of Arina's perceived structure of the Fences pattern.

Proceedings of CERME10


Tarek Djouad
email: tarek.djouad@gmail.com

Alain Mille
email: alain.mille@univ-lyon1.fr

Trace based system in TEL systems: theory and practice

Keywords: Activity traces, Trace based system, Human learning indicator activity, Indicators engineering, Technology Enhanced Learning systems

We present in this paper an easier way to manage activity traces and to compute human learning indicators activities in Technology Enhanced Learning TEL systems. We review our research work related to Trace based system TBS and we explain how we use TBS to develop new and generic model to represent the indicator life cycle from its creation to its reuse. This paper presents the underlying theory and how this theory is implemented to compute human learning indicators activities available for use with any other learning platform, provided the TBS can access the learning platform traces.

Introduction

Most Technology Enhanced Learning (TEL) systems use activity traces to make diagnoses during the learning process in order to propose appropriate assistance for learners, tutors or teachers. Some TEL systems can be used by designers or teachers to set up these diagnoses and describe the information to be collected at the design step [START_REF] Ouali | The Strategic Organization of the Observation in a TEL system: Studies and First Formalizations[END_REF], but in most cases, both the data collection and diagnostic processes are embedded into the code and the IT developers are needed for designing and modifying them. In practice, therefore, it is very difficult to personalize the tracing and diagnostic processes, and, consequently, in most cases distance education systems offer very little in the way of personalization and remediation services during learning activities. In the field of online education research, researchers and analysts are also very interested in activity traces and are increasingly equipping themselves with tools to build interesting data from application logs, and then treat them as activity traces before carrying out treatments to understand current learning processes [START_REF] Georgeon | Supporting activity modelling from activity traces[END_REF][START_REF] Ji | Exploiting activity traces and learners reports to support self-regulation in project based learning[END_REF][START_REF] Lebis | Approche narrative des processus d'analyses de traces d'apprentissage : un framework ontologique pour la capitalisation[END_REF]. Learning analytics is now a big issue for researchers [START_REF]Learning analytics and knowledge Conference[END_REF] and it is very important for the efficiency of the research to be able to get clear information on the semantics of the logs, which remains currently a big challenge. In order to overcome these difficulties, we have been working for several years to consider activity traces as knowledge objects available to designers, teachers, tutors, learners, researchers and, in general, to those involved in learning processes. Since activity traces are constructed to analyze, diagnose and understand expected or feared learning phenomena, we have also developed representations of learning indicators and their computation processes, based on traces of modelled activities. These new implementation versions of Trace based system we propose in this paper are intended to compute human learning indicators activities. In this paper, we present the state of our current research and its latest developments, illustrated with practical examples while recalling the theoretical foundations of the modelled activity trace. We describe on the one hand the formalization of the approach and on the other hand the developed and experienced software environments. We conclude by showing the strong potential of this approach for professional, individual and opportunistic learning activities, in particular with regard to the ethical aspects of e-learning.

Activity traces in TEL systems

Most of learning platforms produce activity traces. These traces are saved in different formats: Logs files, Databases, video and audio files, etc. Several works propose to use and to exploit these traces in TEL systems. For example, Betbeder [START_REF] Betbeder | Environnement audiographique synchrone : recueil et transcription pour l'analyse des interactions multimodales[END_REF], Guraud [START_REF] Guéraud | Suivi de classe á distance : propositions génériques et expérimentation en électricité[END_REF] and Dyke [START_REF] Dyke | Tatiana, un environnement daide à l'analyse de traces dinteractions humaines[END_REF] define experiments to collect multimodal data in order to analyze traces. Mazza [START_REF] Mazza | Monitoring an online course with the GISMO tool: A case study[END_REF], France [START_REF] France | Visualisation et règulation de l'activitè des apprenants dans un EIAH tracè[END_REF] and Cram [START_REF] Cram | A complete chronicle discovery approach: application to activity analysis[END_REF] visualize traces in real time and provide a feedback to teachers and learners on their own activities. Ferraris [START_REF] Ferraris | Modèlisation de scènarios dapprentissage collaboratif pour la classe[END_REF] and Voisin [START_REF] Voisin | Une approche conduite par les modèles pour le traçage des activités[END_REF] propose model driven engineering approaches to understand learning scenarios using trace. As activity traces represent very important knowledge containers, it is meaningful to provide specific environment for managing them as such. Several frameworks have been developed in order to manage activity traces [START_REF] Santos | Helping the tutor to manage a collaborative task in a web-based learning environment[END_REF][START_REF] Avouris | Beyond logging of fingertip actions: analysis of collaborative learning using multiple sources of data[END_REF] as for helping the designers to integrate them in the target application [START_REF] Bouhineau | Dècrire, valuer et Partager des Donnèes, Opérateurs, Processus danalyse et Résultats d'études sur l'apprentissage Humain avec Ordinateur[END_REF] or for helping researchers and analysts to understand the learning process in a batch way [START_REF] Dyke | A model for managing and capitalising on the analyses of traces of activity in collaborative interaction[END_REF][START_REF] Toussaint | Proposition d'un Framework de Traitement de Traces pour l'Analyse de Connaissances Perceptivo-Gestuelles[END_REF]. These approaches are very useful for managing and requesting traces as data, but do not consider traces as knowledge containers per se. In this paper, we will focus on one knowledge based approach for collecting, representing, managing, requesting and transforming activity traces. As far as we know, this approach is original and we will illustrate in which extent this approach allows new usages in the context of Technology Enhanced Learning systems.

Trace based system

The Trace based system or TBS is proposed and implemented [19] by the TWEAK [20] research group to manage modelled activity traces or m-traces [START_REF] Settouti | A Trace-Based System for Technology-Enhanced Learning Systems Personalisation[END_REF][START_REF] Zarka | TStore: A Trace-Base Management System using Finite-State Transducer Approach for Trace Transformation[END_REF][START_REF] Cordier | Trace-Based Reasoning-Modeling interaction traces for reasoning on experiences[END_REF]. An m-trace in TBS is a structured object: the trace model and the corresponding trace instance in the form of a sequence of observed elements or obsels. Each instance obsel part of an m-trace is temporally situated by a time stamp and satisfies the trace model part of the m-trace. TBS proposes explicit transformation operators to be applied to a set of m-traces (transformation sources) in order to obtain other transformed m-traces (transformation targets). All m-trace obsels are represented by structured information resulting from a transformation operation using source m-trace obsels. Each m-trace is the result of some transformation of a lower level m-trace, except for the lowest level, directly built from an observation process constructing the primary m-trace. TBS proposes three steps for using m-traces (Figure 1): 4 Case studies: indicator computations using TBS

We propose in this section some case studies of how to use TBS in real learning situations to compute human learning indicators activities. The new developed systems we propose here are based on TBS and their goal is to compute indicators activities using modelled traces. We will illustrate two systems we built: Trace based indicator management system TB-IMS, and multi-agents indicator computation system IC-MAS. But first we will illustrate the latest implemented KTBS: a kernel trace based system which is a concrete implementation of TBS.

Trace based indicator management system TB-IMS

In order to demonstrate concretely how to compute indicators activities using TBS, we developed an effective method and its implementation [START_REF] Djouad | Observing and Understanding an On-Line Learning Activity: A Model-Based Approach for Activity Indicator Engineering[END_REF] to carry out the entire life cycle of indicators activities. We recall that in TEL systems an indicator is a mathematical variable with a set of values [START_REF] Dimitracopoulou | State of the Art on Interaction Analysis for Metacognitive Support and Diagnosis[END_REF]. In our work, we propose an indicator as composed of a model descriptor and a set of instances, and we propose four steps to describe and compute an indicator instance using a TB-IMS. Each indicator instance is computed using these four steps:

-Step1: To compute a new indicator, we propose to associate it with a set of empty m-traces. These m-traces will be instancied later for evaluating the indicator formula to compute indicator instances, -Step2: We associate a transformation sequence to the empty indicator. The transformation sequence related to these m-traces describes how to move from a primary m-trace (described by its model) to a specific indicator (described by its formula), -Step3: Collects data from the learning environment and builds the primary m-trace instance, -Step4: Finally, we execute the transformation sequence, from the primary m-trace instantiated in step 3 to the indicator defined in step 1.

Indicator computation and m-trace transformation are managed by the prototype we developed: Trace based-indicator management system TB-IMS [START_REF]Trace based indicator management system[END_REF]. Figure 3 shows the TB-IMS collector module, the transformation module, the equation editor and the visualization result. In this example we compute the proportion between chat messages and private messages related to 'user15' according to a time interval T. Computation is possible without coding in machine. The system saves in its databases the indicator, its transformation, intermediate m-traces and the primary m-trace, which allows the indicator to be reused to compute new other indicators.

A multi-agent system to compute human learning indicators activities

We propose also to use Multi-agents technology to compute indicators activities based on TBS [START_REF] Djouad | A multi-agents system to compute human learning indicators activities based on model-driven engineering approach[END_REF]. The indicator computation-multi agents system IC-MAS [START_REF]Indicator computation-multi agent system[END_REF] we propose allows creating, updating, deleting, reusing and sharing indicator. These agents are important in order to:

-Minimize the indicators computation complexity. If the indicator computation is difficult (Example: the case of a high level indicator defined by sub-indicators) then our system reuses existing indicators to compute new one. We use information like transformation sequence, computation rule and category to perform this computation, -The need to compute a same indicator in different contexts is very high. Several users attempt to compute the same indicator with closest time intervals. The system creates for each new computation a list of agents that will minimize the indicator computation time. We propose these following agents to build our system:

-The collecting agent uses raw data issued from learning platforms (Example: database, Log files, etc.) to select useful data used to compute indicator. The collecting agent is guided by variables which are listed in the computation rule to optimize the collecting step. Once completed, it stores the collecting results in an m-traces base as a primary m-trace and keeps the collecting history to reuse it later. This agent aims to prepare things for having a better performance in a future collecting step, -The transformation agent transforms a primary m-trace created by the collecting agent. Each transformed trace becomes a variable labelled according to the name of the transformed trace and takes its value from the number of instances present in the transformed m-trace. This agent uses transformation operators such as filtering, merging, matching, etc. to transform m-traces, -The computation agent executes the computation rule using variables created by the transformation agent and saves it in the indicator base, -The interface agent displays indicators using graphical views (Pie charts, Bar graphs, Histograms, etc.). It also retrieve information provided by teachers to define indicators, -The coordinator agent coordinates and synchronizes the different agents to provide a consistent functioning of the system. Figure 4 explains an example of indicator computation using IC-MAS.

Conclusion

Our research was an opportunity to work on the consolidation of collaborative work around the theory of modelled traces. Indeed, we were able to experiment concretely with its potential to formalize and represent the notion of indicator as a structured computer object that provides the knowledge necessary for its elaboration. This knowledge structure integrates the modelled source trace as described in the learning environment, the set of logical and temporal transformations necessary to allow the calculation of indicators whose semantics are thus made completely explicit. This explicit representation is proposed in the form of an operational IT structure, adaptable by designers or teachers to new situations without requiring reprogramming, but requiring only a modification of the description of the mobilized knowledge. For researchers, the availability of a basic semantics for activity traces makes it possible to integrate this knowledge into the parameterization of Learning Analytics tools, but also to design specific and documented experiments to serve the object of their research. Finally, and these are the latest developments under way in our research, the ability to explain activity traces opens up very promising solutions in terms of ethics of use of personal data by involving learners and teachers in controlling the use of activity traces during learning. Learners and tutors can even, during the activity itself, suggest new ways of interpreting their learning behavior to make easier their task or to build new knowledge about human learning processes.

1 .

 1 Users, as teachers/tutors or learners, use learning platforms. These platforms provide raw data as a source of observation. TBS connects to learning platforms, collects raw data and uses these data to build a primary m-trace (model and instance). This primary m-trace is then saved in an m-trace base, 2. TBS uses this primary m-trace and transforms it into other transformed mtraces according to the semantics of these transformations. The transformed m-trace is saved in the m-trace base. In turn, these transformed m-traces can be transformed again into other transformed m-traces. Starting from one primary m-trace, a transformation graph is progressively built and saved for providing explicit explanations of any transformation, i.e. providing the semantics of any m-trace in the m-trace base, 3. Moreover, the m-trace base can be used by any assistant to manage indicators, allow indicator computation, provide smart visualization, etc.

Fig. 1 .

 1 Fig. 1. Trace based system used by our research team to manage m-traces.

4. 1

 1 Short presentation of the used kernel trace based system (KTBS) Kernel trace based system[START_REF] Ktbs | [END_REF] is an implementation of TBS. KTBS considers an activity trace as a model and a set of obsels (m-trace). Is allows to create mtraces bases, m-traces models, m-traces obsels, transformed m-traces, etc. Each obsel has a set of attributes like type, value, time-begin, time-end, etc. KTBS stores modelled traces as RDF ontology and manipulates them using a set of operators like: filter, fusion, SPARQL query, etc. It proposes to useJSON [25],REST [26] and TURTLE [27] to describe m-traces. Figure2presents an example of m-trace model in KTBS using Turtle syntax.

Fig. 2 .

 2 Fig. 2. Example of creating a model in KTBS using Turtle syntax.

Fig. 3 .

 3 Fig. 3. Computing the Proportion indicator using TB-IMS.

Fig. 4 .

 4 Fig. 4. Computing the Proportion indicator using IC-MAS.


text:
Ioannis Papadopoulos

Nafsika Patsiala

'Father Woodland': A learning environment to facilitate the development of algebraic thinking

Keywords:

In this paper, the contribution of the use of the "Father Woodland" learning environment in Grade-3 students' algebraic thinking is examined. Four types of thinking were identified indicating a progressive movement towards the use of symbolic language that seems to have a rather developmental character. In their solutions the students induced rules for solving equations that will later be introduced formally to them.

Introduction

Given the central position of algebra in the secondary curriculum which led to a separation between arithmetic and algebra with the first being the main focus of elementary mathematics curriculum, mathematics educators try to cope with the challenge of managing the transition from arithmetic to symbolic algebra. Numerous researchers admit that this separation deprives children of powerful schemes of thinking in the early grades and makes it more difficult to learn algebra in the later years (e.g., [START_REF] Kieran | Learning and teaching algebra at the middle school through college levels[END_REF]. One way to address this issue might be to study the impact of certain learning environments in the students' development of algebraic thinking. [START_REF] Papadopoulos | Using mobile puzzles to develop algebraic thinking[END_REF] working with a mobile puzzle environment found that sixth graders exhibit a progressive movement towards algebraic thinking. In this context, we try to explore the potential contribution of another specific learning environment called 'Father Woodland' in young students' algebraic thinking that would allow us to identify certain steps in this movement from arithmetic to algebra. This is based on two of the algebra goals specified by NCTM (2000) standards, i.e., (i) represent and analyze mathematical situations and structures using algebraic symbols, and (ii) use mathematical models to represent and understand quantitative relationships. Therefore, we try to examine whether this environment facilitates the achievement of these goals through the identification of the types of thinking that the students followed in order to cope with the given tasks.

Early algebraic thinking and 'Father Woodland' environment. [START_REF] Cai | A global dialogue about early algebraization from multiple perspectives[END_REF] do not limit algebraic thinking in earlier grades to simply mastering arithmetic and computational fluency but it goes deeper in identifying the underlying structure of mathematics which includes the development of particular ways of thinking, analysis of relationships between quantities, noticing structure, generalization, problem solving, justifying, proving and predicting. [START_REF] Cai | The development of students' algebraic thinking in earlier grades[END_REF], in a cross-cultural comparative study talk about multiple representations (pictures, diagrams, tables, graphs, and equations) that are used to represent functional relationships between two quantities and more specifically about 'pictorial equations' used to represent quantitative relationships providing thus a means for developing students' algebraic ideas. This raises the necessity to make the distinction between the external and internal representations in the sense of considering at a minimum configurations of symbols or objects external to the individual learner or problem solver (i.e., concrete materials, pictures/diagrams, spoken words, written symbols) that can be described mathematically and configurations internal to the individual (i.e., mental models and cognitive representations of the mathematical ideas underlying the external representations) respectively [START_REF] Goldin | Representation in mathematical learning and problem solving[END_REF]. Such internal representations are inferred from the way the students express their aspects of the process of mathematical thinking in their written responses. It seems that certain learning environments can be in favor of introducing young learners to these aspects of algebraic thinking [START_REF] Papadopoulos | Using mobile puzzles to develop algebraic thinking[END_REF]. In the current study a specific learning environment has been chosen. It is called 'Father Woodland' and is about a Czech fairy-tale figure owning a farm who organizes tug-of-war games among the animals living in the farm [START_REF] Hejný | Early conceptual thinking[END_REF]. The weakest animal is the mouse. Two mice are as strong as a cat, a cat and a mouse equal a goose and a goose and a mouse equal a dog (Fig. 1). The strength of each animal is represented by a picture and an icon (symbol) and the students are asked mainly to decide between two groups the stronger one or to add some animals to the weaker group in order to create two equivalent groups, or to reveal the identity of hidden animals so as to obtain equity. It is a rich environment. Hejny and his colleagues use it in a series of textbooks they produced. The relevant tasks within these textbooks are connected with the development -among others-of an early number sense, pre-concept of divisibility, the lowest common multiple and greatest common divisor as well as the solving of equations. [START_REF] Hejný | Early conceptual thinking[END_REF] used this environment with Grade 1-3 students focusing on how the environment facilitated the identification and acceptance of the association between animals and quantities. [START_REF] Marchini | Teachers' best practices using differentiation[END_REF], used also a modified version of the environment to fit in the Italian schools and worked with Grade-1 students focusing on how the variety of "ways for representing the same mathematical concept together with treatment inside a register and conversion between registers facilitate pupils' understanding and the construction of concepts" (p. 55). In this study we focus on the use of this environment as a way to smooth the transition from arithmetic to algebra (in the sense of using pictorial equations as a means or developing algebraic ideas, see [START_REF] Cai | The development of students' algebraic thinking in earlier grades[END_REF] by considering the various types of students' thinking that would show a progressive movement towards algebraic thinking.

Design of the study

Seventy 3 rd graders (8-9 years old) participated in the study. They were the total population of three classes from two primary schools and they represent a sample of an ordinary Greek primary school. They had no previous experience working with this kind of environments and they had not been taught any of the basic concepts of algebra such as equations or variables. When the students were introduced to the 'Father Woodland' environment, each tug-of-war game was presented using both the pictorial and symbolic representations of the animals. The whole study (part of a broader one) lasted five weeks. The students were initially introduced to 'Father Woodland' and then on a regular basis they were given tasks to solve individually. The whole project took part in parallel to the normal teaching and was not integrated in the content of their math lessons. This paper focuses on the first 10 tasks due to the limited number of pages. There are 3 collections of tasks. In the first, there were two groups of animals in each task and the students were asked to add a mouse to the weaker group in order to make both groups equivalent (Fig. 2). This demands comparison and relational thinking connected to the notion of equality as an equivalence. In the second, the tug-of-war game took place but one (or some) of the animals wore a mask. The students were invited to find the animal(s) behind the mask (Fig. 3, left and middle). The aim was to exploit relational thinking in the form of using alternative ways for representing the unknown quantity. Finally, in the third, the students were asked to create two equally strong teams using any combination of the farm animals (Fig. 3, right). The aim was to see whether the students exploit the experience gained before and how intuitive mathematical ideas are embedded in their creations. For each task, the students were asked to explain their answer in a separate textbox. During the study, no feedback was given to the students about their answers. The students' worksheets constituted the data for this study. These data were examined in order to identify evidence of early algebraic thinking and possible formal mathematical concepts, which are informally used in the students' answers. In the context of qualitative content analysis, inductive category development was used to organize the categories.

Results and discussion

After the data examination, the answers were categorized in four types. The criteria for this categorization were the ways students chose to express the underlying structure in each task (i.e., using pictures, words or symbols), the relationships among the given quantities (i.e., using the given or new (invented) relationships), and the mathematical information contained within the pictorial representation (i.e., identifying a basic unit, substituting animals with their equivalents, adding/subtracting the same quantity in both sides, etc.). The four types are: (i) using pictorial language, (ii) using words to express relationships, (iii) combining words and symbolic representations and (iv) using 'symbolic' language to express relationships. Obviously not all the students applied all the types. This is why it was decided to choose a proper sample of students to show the diversity of the approaches taken.

Type 1 -Using pictorial language

This type refers to the students who preferred drawing pictures in detail rather than using a symbolic representation (Task-E, Fig. 4). The answer is correct. The missing mouse must join the group on the left to get two equal teams but the reasoning is weak since it is limited to merely transfer the 'abstract' information into a more 'realistic' version and it lacks an explicit explanation of the 'underlying' thought. It seems that the student fails to shift the focus to the existing relationships between the values of the participating animals. This type was used by the majority of the students and it proved more convenient for most of them to express their solution of the problem. Actually, in this type, the students made a step forward by trying to use words for expressing relationships among the quantities as it can be seen in the next two examples. This choice in many of the cases was described by the students in detail revealing thus their line of thought. In Task-J, one student created two equivalent groups by placing 5 dogs on the left and 20 mice on the right (Fig. 5). His explanation was: "I thought that 5 dogs are as strong as 5 geese and 5 mice. But, 5 geese = 5 mice and 5 cats and 5 cats are as strong as 10 mice. So 20 mice". The student exploited all the default information given by the pictures in Figure 1, e.g., 1 d(og)=1g(oose)+1m(ouse), 1g(oose)=1c(at)+1m(ouse), and 1c(at)=2m(ice). Then, the whole process can be presented on a more formal way as 1d=1g+1m ⟹ 5d=5(g+m)[multiply both parts by the same number] ⟹ 5d=5g+5m [distributive property]⟹ 5d=5(c+m)+5m [substitute with equivalent] ⟹ 5d=5c+5m+5m ⟹ 5d=5x2m[substitute with equivalent] +10m ⟹ 5d=10m+10m ⟹ 5d=20m. It must be said that this is not explicitly outlined by the child. But this enables us to identify in the student's explanation the seeds of the mathematical reasoning described above. In the same way we will try to see the possible formal way of expressing the students' answers in the remaining part of the paper. Actually, we make inferences about students' internal representations on the basis of their production of external representations [START_REF] Goldin | Systems of representations and the development of mathematical concepts[END_REF].

The second example concerns Task-H which asked the students to identify the animal hidden behind the mask. The student's answer was (Fig. 6, left): "There is a cat hiding behind the mask. Because, a mouse and a dog are as strong as a cat and two mice. I figured it out because a mouse and a dog are as strong as 5 mice so behind the mask is a cat since 2 cats and 1 mouse are as strong as 5 mice too". The student's starting point was the right part of the equation and she chose the mouse as the building block to replace all the involved animals. It is necessary to mention here that the students did not restrict themselves on the given relationships that were given during the first session (Fig. 1). They were able, during the next sessions, to identify new relationships based on the given ones. This student made use of one of these invented relationships by claiming implicitly that a dog has the same strength with 4 mice. This claim results to the total amount of 5 mice in the right part. Given that a cat has the same strength with 2 mice, it means that there are 3 mice in the left part of the equation plus the hidden animal. Two mice are needed to obtain equality; therefore, a cat must be placed behind the mask. The strategy that led to successful solution here was: (i) choose the basic unit (e.g., the mouse), (ii) translate the picture to equation, (iii) substitute the dog by its equivalent number of mice and execute the operation, (iv) substitute the cat by its equivalent number of mice and execute the operation, (iv) find the unknown (x).

Type 3 -Combining words and symbolic language

This type starts -as a first step towards symbolic language-to combine words and symbols to show relationships between the participating animals. This is one answer for Task-I: "I figured it out because becomes (equals) plus . So there is hiding behind the mask".

The student in the first half of her answer used the word 'becomes' to denote the equality between cat and mice (Fig. 6, right). But, in the second half she used the sign of '=' to denote again the relationship between mice and geese (left part) and dogs (right part). Firstly, she substituted the cat (c) with 2 mice (m). Now, the left part consists of two identical sub-groups (a goose and a mouse per subgroup) which if substituted by their equivalence in terms of dogs reveal the identity of the unknown. So, starting from the left part:

2𝑔 + 𝑐 = 𝑥 + 𝑑 𝑐=2𝑚 ⇒ 2𝑔 + 2𝑚 = 𝑥 + 𝑑 [𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑒 𝑤𝑖𝑡ℎ 𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡] ⇒ 2(𝑔 + 𝑚) = 𝑥 + 𝑑 [𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑣𝑒 𝑝𝑟𝑜𝑝𝑒𝑟𝑡𝑦] 𝑔+𝑚=𝑑 ⇒ 2𝑑 = 𝑥 + 𝑑[𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑒 𝑤𝑖𝑡ℎ 𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡] ⇒ 𝑥 = 𝑑 [𝑠𝑢𝑏𝑡𝑟𝑎𝑐𝑡 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦].
The knowledge that 2 mice plus 2 geese is the same as 2 dogs -which is based on the given relationship that a mouse plus a goose is the same as a dog-reveals an implicit understanding of the above mentioned distributive property.

Type 4 -Using symbolic language to express relationships

The last type used by the students abandons the use of words and the reasoning is mainly symbolic.

The first example is an answer from Task-F dealing with the animal behind the mask. The student started with the left group, using a symbolic expression to show its substitution by a dog (

). Then, based on this expression she wrote another one to show the solution ( , see Fig. 7 left). The first claim of the student seems arbitrary but if seen carefully it makes use of known relationships in order to obtain new ones. The left group represents the sentence m+m+c. Given that m+c=g(oose) the sentence becomes m+g which, according to the given relationships, equals with a dog. Then, it is obvious that what is needed in the right part of the equation is a mouse. The second example shows a solution for Task-C. This solution is considered more elaborated since the student made use of the sign of the required operation to show the equivalence. This is indicative of understanding both the operation that takes place and the correct use of the sign of this operation. The missing animal is the mouse that must join the left group to get equality (Fig. 7, right). The sum of the strength of the three mice equals the strength of the goose. Again, this is a relation different than the given ones and it is important that so young students exhibit this ability, to use and combine given situations in order to get new ones. So, it is interesting to follow the thought of the student. Two mice equal with a cat. Then, a cat and a mouse have the same strength with a goose. Consequently (transitivity) three mice are equivalent with a goose. Expressing relationships using this symbolic language to solve a problem constitutes an important step towards the development of algebraic thinking. Besides, all the answers show an explicit focus of the students to the underlying structure of each equivalence in order to reach a solution.

It is intersting to examine now the findings of this study in the light of a previous one. [START_REF] Papadopoulos | Using mobile puzzles to develop algebraic thinking[END_REF] in their study based on the use of mobile puzzles with 6 th graders, distinguished mainly four types of students' thinking (translating the picture to equality expressions, using words to show the relationship, using symbolic language to show the relationship, and combination of more than one of the previous types). This means that there is a match between the types of thinking in these two studies and this stregnthens the possible positive contribution of puzzle-like learning environments to the development of young students' algebraic thinking. The feeling from the first study (no numerical data available) was that the order of these types is rather developmental in the sense that types 3 and 4 are more advanced (and thus less frequent), and are met at the end of the project (an indication that they are connected with the accumulated experience). One could attribute this finding to the teachers' appreciation of the symbolic answers instead of textual ones. However, this is not the case, since the teachers were not involved in the project and in the meanwhile the students did not receive any feedback about their answers. In this study an effort was made to get arithmetical evidence that would shed light on this issue. Table-1 confirms that Types 3 and 4 are indeed the less frequent ones. The low frequency of Type-1 was expected since the pictorial language was already included in the statement of the tasks.

Table-2 presents the distribution of the last two types across the range of the tasks. As it can be seen to a great extent the number of instance for each Type is increased as we move towards the last tasks indicating that these types are connected with the accumulated experience.

Conclusions

The findings of our research indicate that the 'Father Woodland' environment might contribute to the development of students' algebraic thinking. The four types of thinking mirror the rules induced by the students in order to solve the posed problems. Starting from certain external representations of equality sentences the students made an attempt to express their internal representations through the shift from pictorial to symbolic language. Obviously, this is not all that matters with the development of algebraic thinking with young children. However, it cannot be considered trivial. The students had to add or remove the same animal (quantity) from both sides, to substitute certain animals with their equivalence, isolate the unknown animal (variable) trying to maintain the same strength between both groups of animals applying at the same time the distributive law or transitivity. Despite that lack of explicit knowledge about operations and relations hinders a good approach to algebra [START_REF] Gerhard | How arithmetic education influences the learning of symbolic algebra[END_REF], it seemed that there were instances of an implicit knowledge of certain rules for solving equations which will be later introduced formally to the students. However, it still remains to answer questions like: In what way the transition from the animal symbols back to the arithmetical or algebraic equations will be possible? Additionally, the findings support the developmental character of these types of thinking when the students use puzzle-like learning environments aiming to support algebraic thinking. However, this does not mean that some students did not occasionally move backwards to previous types of thinking. This is in itself a significant finding we aim to explore further since the relatively small number of participants does not allow to generalize our findings.

Figure 1 :

 1 Figure 1: Equivalences in the Father Woodland environment

Figure 2 :

 2 Figure 2: The first group of activities

Figure 4 :

 4 Figure 4: Use of drawings Type 2 -Using words to express relationships

Figure 5 :

 5 Figure 5: Use of mouse as the basic unit

Figure 6 :

 6 Figure 6: Use of words (left) and combination of symbols and words (right) to explain relationships If we translate the reasoning of the student into its formal version, considering x the unknown, we obtain the following series of equations. 𝑥 + 𝑐 + 𝑚 = 𝑚 + 𝑑 𝑑=4𝑚 ⇒ 𝑥 + 𝑐 + 𝑚 =

Figure 7 :

 7 Figure 7: Usage of symbolic representations (left and right)

Table 1 : Frequency of Types 1-4

 1

		Type-1	Type-2	Type-3	Type-4
	Tasks A-E	5 (1.6%)	253 (83%)	14 (4.6%)	33 (10.8%)
	Tasks F-I	8 (3.74%)	157 (73.36%)	24 (11.22%)	25 (11.68%)
	Task J	2 (2.94%)	56 (82.35%)	8 (11.77%)	2 (2.94%)

Table 2 : Distribution of Types 3-4 across tasks

 2

		Task-A	Task-B	Task-C	Task-D	Task-E	Task-F	Task-G	Task-H
	Type-3	3	2	4	3	3	7	7	7
	Type-4	5	6	3	8	10	8	8	8

Proceedings of CERME10


text:
Alik Palatnik
email: umapalatnik@gmail.com

Boris Koichu
email: boris.koichu@weizmann.ac.il

Four-component decomposition of sense making in algebra

Keywords: Algebraic sense-making, problem-solving, project-based learning, integer sequences

This article presents a case in which a pair of middle-school students attempts to make sense of a previously obtained by them position formula for a particular numerical sequence. The exploration of the sequence occurred in the context of two-month-long student research project. The data were collected from the students' drafts, audiotaped meetings of the students with the teacher and a follow-up interview. The data analysis was aimed at identification and characterization of events and algebraic activities in which the students were engaged while making sense of the formula. We found that the students' conviction, by the end of the project, that the formula "makes sense" emerged when they justified the formula, checked its generality, discovered a geometry mechanism behind it, and found that it came to cohere with additional formulas. The findings are summarized as a suggestion for a four-component decomposition of algebraic sense making.

Introduction

Sense making has long been a focal concern of the mathematics education research community (e.g., [START_REF] Kieran | Learning and teaching algebra at the middle school through college levels: Building meaning for symbols and their manipulation[END_REF]NCTM, 2009). NCTM (2009) recognised sense making as a means to know mathematics as well as an important outcome of mathematics instruction. To review, NCTM (2009) refers to sense making in mathematics "as developing understanding of a situation, context, or concept by connecting it with existing knowledge" (p. 4). Nevertheless, NCTM (2009), as well as many additional mathematics education publications, is rather inexplicit as to what sense making comprises of and how it occurs. Moreover, it has been broadly acknowledged (e.g., [START_REF] Schoenfeld | Reflections on problem solving theory and practice[END_REF] that empirically-based knowledge about the processes involved in sense making, as well as knowledge about the processes involved in learning through mathematical problem solving, is insufficient.

The case presented in this article occurred with two 9 th graders, Ron and Arik (pseudonyms) who participated in the Open-Ended Mathematical Problems project, which was conducted by the authors (an abridged version of [START_REF] Palatnik | Sense making in the context of algebraic activities[END_REF]. The initial part of Ron and Arik's project lasted for three weeks and resulted in an insight solution to the problem of finding a position formula for a particular sequence. This part is analysed elsewhere [START_REF] Palatnik | Exploring insight: focus on shifts of attention[END_REF]. The insight gained was celebrated as an important highlight of the project. The students told us, however, that they found the formula "by chance" and that it did not make sense for them. As a result, making sense of the obtained formula became an explicitly chosen goal and the main theme of the second part of the students' project. This part had lasted for four weeks and ended when the students succeeded, in quite an idiosyncratic way, to make sense of the formula.

The goal of our study was to discern the activities and processes involved in the sense making effort. Specifically, we pursued the following research questions:

In which events and algebraic activities were the students engaged while attempting to make sense of a formula? What were some of the processes involved in the students' explicitly expressed conviction, by the end of the exploration, that the formula "makes sense"?

Theoretical background

In empirical studies, the notion of sense making frequently denotes ways by which learners of mathematics act upon a particular entity in the context of particular mathematical activity. The expression "to make sense of…" is attributed in different studies to such entities as proofs, instructional devices, concepts, solution methods and problem situations (e.g., [START_REF] Smith | A sense-making approach to proof: Strategies of students in traditional and problem-based number theory courses[END_REF][START_REF] Rojano | Intertextuality and sense production in the learning of algebraic methods[END_REF].

The idea of algebra as an activity was elaborated by [START_REF] Kieran | The changing face of school algebra[END_REF][START_REF] Kieran | Learning and teaching algebra at the middle school through college levels: Building meaning for symbols and their manipulation[END_REF]. Kieran identifies three types of activities in school algebra: generational, transformational, and global/meta-level activities, and argues that each type has special affordances to meaning construction. The generational activity involves the forming of the objects of algebra (e.g., algebraic expressions or formulas) including objects expressing generality arising from geometric patterns or numerical sequences. The transformational activity includes various types of algebraic manipulations. Transformational activity can involve meaning construction for properties and axioms on which the manipulations rely. A related point is highlighted by [START_REF] Hoch | Structure sense versus manipulation skills: an unexpected result[END_REF], who proposed the notion of structure sense, which is related to algebraic manipulations and aspects of symbol sense [START_REF] Arcavi | Developing and using symbol sense in mathematics[END_REF] in relation to friendliness with symbols as tools, an ability to switch between attachment and detachment of meaning, and an examination of the meaning of symbols. Finally, [START_REF] Kieran | Learning and teaching algebra at the middle school through college levels: Building meaning for symbols and their manipulation[END_REF] argues that meaning construction is associated with global/meta-level mathematical activities (e.g., problem solving, working with generalizable patterns) in a sense that "these activities provide the context, sense of purpose, and motivation for engaging in the previously described generational and transformation activities" (p. 714). It is essential for the forthcoming analysis that when the learners are engaged in a global/meta-level activity, they can carry it out in a variety of ways, and the decision to use the algebraic apparatus arises as learners' choice.

Treatment of sense making as an inseparable part of mathematical thinking makes the MGA model of creating mathematical abstractions [START_REF] Mason | Mathematical abstraction as the result of a delicate shift of attention[END_REF] particularly important for our study. The main operational categories of the model are Manipulating, Getting-a-sense-of, Articulating (hence MGA). The MGA model elaborates on the processes of creating abstraction as a helix, in which each cycle includes its own, local, sense-making act. Briefly, the model presumes that manipulating familiar mathematical objects (M) leads to the formation of a sense of generality or regularity based on properties of these objects (G), and then to the articulation of that general property or regularity (A), which in turn forms new objects for further manipulations. [START_REF] Mason | Mathematical abstraction as the result of a delicate shift of attention[END_REF] suggested that the driving force behind the process of creating abstractions is the gap between expected and actual results of manipulations.

To summarize, in our study we adapt NCTM's (2009) perspective on sense making, and elaborate on it in an algebraic context. Our theoretical framework is built upon the idea of algebra as an activity [START_REF] Kieran | The changing face of school algebra[END_REF][START_REF] Kieran | Learning and teaching algebra at the middle school through college levels: Building meaning for symbols and their manipulation[END_REF] and on analytical apparatus of [START_REF] Mason | Mathematical abstraction as the result of a delicate shift of attention[END_REF] model of mathematical thinking known as Manipulating -Getting-a-sense-of -Articulating (MGA).

Method Learning environment, participants and the mathematical context

The Open-ended Mathematical Problems project, in the context of which the case of Ron and Arik took place, is being conducted, since 2010, in 9 th grade classes for mathematically promising students. The learning goal of the project is to create for students a long-term opportunity for developing algebraic reasoning in the context of numerical sequences. It is of note that 9 th graders in Israel, as a rule, do not possess any systematic knowledge of sequences; this topic is taught in the 10 th grade.

The project is designed in accordance with the principles of the Project-Based Learning (PBL) instructional approach (e.g., [START_REF] Blumenfeld | Motivating project-based learning: Sustaining the doing, supporting the learning[END_REF]. Specifically, the organizational framework of the project is as follows. At the beginning of a yearly cycle of the project, a class is exposed to 8-10 challenging problems. The students choose one problem and work on it in teams of two or three. They work on the problem at home and during their enrichment classes. Weekly 20-minute meetings of each team with the instructor (the first author) take place during the enrichment classes. When the initial problem is solved, students are encouraged to pose and solve follow-up problems. At the end of the project, all teams present their results to their peers. Then 4-6 teams, chosen by their classmates, present their work at a workshop at the Technion -Israel Institute of Technology, attended by academic audience (for more details see [START_REF] Palatnik | Learning through long-term mathematical research projects[END_REF].

Ron and Arik chose to pursue the Pizza Problem (Figure 1) which is a variation of the problem of partitioning the plane by n lines (e.g., [START_REF] Pólya | Induction and analogy in mathematics[END_REF]. Using [START_REF] Kieran | Learning and teaching algebra at the middle school through college levels: Building meaning for symbols and their manipulation[END_REF] terminology, we expected the PBL environment and the Pizza Problem in particular to afford students to be engaged with generational and transformational activities in the context of a global/meta-level activity. In this way, the students were provided with opportunities for developing algebraic sense-making and wewith an opportunity to study their sense-making effort.

Data sources and analysis

We audiotaped and transcribed protocols of the weekly meetings with Ron and Arik (eight 20minute meetings), collected written reports and authentic drafts that the students prepared for and updated during the meetings (more than 40 pages) and interviewed the students by the end of the project. These data were used to create a description of the students' exploration and for dividing it into events.

In accordance with the presented above methodological principles for exploring sense making, we discerned the activities the students chose to be engaged in: proving, generalizing, pattern-seeking, Every straight cut divides a pizza into two separate pieces. What is the largest number of pieces that can be obtained by n straight cuts?

A. Solve for n = 1, 2, 3, 4, 5, 6.

B. Find a recursive formula for the n th term of the sequence.

C. Find a position formula for the n th term of the sequence.

Thematic Working Group 03

Proceedings of CERME10 and question-generating. We also applied the MGA model to trace mathematical objects manipulated by the students in a sequence of activities potentially contributing to sense-making.

Findings: Ron and Arik make sense of the obtained formula

We present here four main events that occurred during students' sense-making pursuit.

Event 1: Choosing new goals

The following conversation took place just after the students presented their solution of the Pizza Problem to the instructor:

Instructor:

Now you have a lot of work to do, and this is great. First of all, you see that the formula works. Now we have to think why it works, and try proving that it works.

Ron (to Arik):

Write it down. "Why it works, and prove that it works" (laughs), it is interesting! Ron accepted instructor's suggestion. In his words: "When we have a formula, but don't know its meaning, it is not interesting. If we knew how the formula is constructed, we would know it 100%. We got it by chance. So we do not know what it means." In addition, both students proposed to explore a more general problem, that of plane partitioning (see Figure 2). The students also suggested additional objects to explore: the points of intersection of the cutting lines with and within a circle representing a pizza and the number of segments on the cutting lines.

Event 2: Simultaneous exploration of several sequences and first manipulation with a formula

Having chosen the above goals, the students started making sketches and counting: segments within the circle, closed and open parts of the plane and points of intersection of the cutting lines, for different numbers of lines (see Figure 3a-c). As a by-product, the students noticed that the sum of the first n odd numbers also equals n 2 . They also began exploring the connections between different sequences (see Figure 3d-3e). In particular, Ron noticed that the differences between the corresponding terms of the sequences form a sequence 0, 1, 2, 3... (see columns X,Y at Figure 3d). To obtain an explicit formula for the sequence 2, 5, 9, 14 … (the numbers of intersections of the cutting lines with and within the circle), Ron adjusted the formula 3c) in the following way: "I thought it would be like the previous formula, but it did not fit. So I got rid of 1 and added n [to the right side of the formula], and it was right."

1 2 2 + n n = P n  into the formula n n n X=   2 2 (Figure

Event 3: Producing an explanation of why the target formulas worked

The wish to understand why the formula returns the maximum number of pieces was a repeated theme in weekly meetings with the instructor. The students eventually answered this query in the following way. After exploring of new drawings Ron and Arik realized that the maximal number of pieces is obtained when a new cutting line crosses all the previous lines in new points. As a result, the students concluded that a new cutting line added n new intersection points to the existing configuration of lines. For the students, it was an explanation of why the formula

n n n X=   2 2
returned the maximum number of the intersection points. They further asserted that this idea also explained, for them, why the target formula 1 2 2 + n n = P n  returns the maximum number of pieces.

Event 4: "Proving" the target formula

As mentioned, the need to prove the correctness of the formula for the Pizza Problem was an additional driving force for the students. First, Ron suggested: "We thought of a way to prove it [the position formula]…[in order to do so, we wanted] to connect all the formulas we had, every table we've made… may be it will give us the formula, then we will know that it is a true formula indeed.

Then we'd have a proof". Ron and Arik built upon the following inference: for any number of cuts, the sum of the number of open and closed pieces (see Figure 2) equals the overall number of pieces into which a plane is divided. They explored the sequences for open and closed pieces. The number of open pieces for n cuts, 2n, was easy for them to find and explain: adding a new cutting line adds exactly two open pieces to the drawing. For the closed pieces the students empirically (i.e., by counting on the drawings) obtained a sequence 0, 0, 1, 3, 6 for 1, 2, 3, 4 and 5 cuts, respectively. They perceived it as "quite close" to the target sequence (2, 4, 7, 11, 16…) and began manipulating the target formula (

1 2 2 + n n = P n 
) in a way similar to adjustment in Event 2. Eventually Ron and Arik obtained the correct expression  

1 2 2 2    n n
. The last piece of the puzzle came when Ron and Arik and their classmate with whom they consulted devised and realized the following plan. Since the

formula 1 2 2 + n n = P n 
represents the total numbers of pieces and since Ron and Arik have obtained the formulas for the numbers of closed and open pieces, the three formulas should match. After several unsuccessful attempts, Ron and Arik implemented this idea and algebraically connected the three formulas. In their final presentation, they showed a slide with the following transformations:

  2 2 2 4 2 4 4 2 1 2 2 2 2 2              n n n n n n n n n ;   2 2 1 2 1 2       n n n n
. This and validation of all three formulas by means of Excel tables were presented as "the proof" of the target formula, and the formula itself was treated as "making sense" by the students.

Discussion

The four-weeks-long exploration of two 9 th grade students working on a particular project has been presented. The answer to the first research question (about events and algebraic activities in which the students were engaged while attempting to make sense of the previously obtained position formula) straightforwardly follows from the above exposition. Briefly, the students were engaged in generational and transformational activities in the context of the global/meta-level activities of explaining to themselves why the formula worked and of proving the formula. It is of note that Ron and Arik's persistence to make sense of their formula is unusual. We suggest two circumstances contributing to the emergence of the students' self-imposed sense-making goal. First, the students' activities were organized and driven by their interest to a particular mathematical phenomenon and not merely to generation of some patterns (cf. [START_REF] Hewitt | Train spotters' paradise[END_REF], for train spotters metaphor). Second circumstance is the organizational setting of the project, which was in accordance with project-based learning instructional approach [START_REF] Blumenfeld | Motivating project-based learning: Sustaining the doing, supporting the learning[END_REF]. In such an environment the students had a chance to get used to the long-term, open-ended explorations, to the high level of expectations and to having room and time to spend with a problem.

Our second research question concerned the processes involved in student sense making. To answer the query "why the formula works" the students examined the geometric mechanism behind the formula. In the course of generational activity the students experimented with concrete drawings (i.e., drawings with 4-6 cutting lines), which apparently served as a visual tool to reveal a generic process that occurs when a line is added to a system of n existing lines. Accordingly, the multi-stage process of abstracting, at each stage of which an MGA cycle occurred, seems to be the central process underlying the why-part of the students' sense-making effort.

The query "how to prove the formula" turned to be the thorniest part of the project. The students addressed this query when they succeeded to show how the target formula came to cohere with two geometrically related formulas. These formulas were obtained by means of exploration of the connections between the sequences chosen by students. The connections were found in the process that featured counting on the drawings, pattern-sniffing in the tables and manipulating the previously obtained formulas by adjusting them. Eventually, the target formula was inserted in a cloud of related formulas, which did not exist when the students began the sense-making pursuit.

Thus, the process of generating a cloud of formulas and checking it for coherence seems to be an important process in the proving part of the students' sense-making effort (cf. Rohano, Filloy & Puig, 2014, for sense making by connection of a new mathematical text to a system of texts). It is of note that the coherence was achieved not only among various objects, but also by means of a coherent exploration strategy.

As argued, Ron and Arik constructed meaning of the target formula in a sense-making process consisting of sequence of generational and transformational algebraic activities in the overarching context of global, meta-level activity, long-term problem solving. In this sense-making process, the students: (1) formulated and justified claims;

(2) made generalizations, (3) found the mechanisms behind the algebraic objects (i.e., answered why-questions); and (4) established coherence among the explored objects. We now take the liberty of formulating this summary as a proposal for a fourcomponent decomposition of sense making (see Figure 4). The aspects of generalizing, justifying and search for mechanism in sense making are in line with the main attributes of symbol sense [START_REF] Arcavi | Developing and using symbol sense in mathematics[END_REF] as well as findings about the role of generalizing and justifying in meaning construction (e.g., [START_REF] Lannin | Generalization and justification: The challenge of introducing algebraic reasoning through patterning activities[END_REF][START_REF] Radford | Layers of generality and types of generalization in pattern activities[END_REF]. However, establishing coherence has not yet been considered as part of sense making.

The four-component decomposition elaborates NCTM's (2009) definition of sense making in the following way. First, it presents sense making as a conjunction of processes. Second, it highlights the potential of algebraic activities to provide students with means to make sense of algebraic objects.

Figure 1 :

 1 Figure 1: The Pizza Problem

Figure 2 :

 2 Figure 2: Division of the plane: there are three closed (hatched) and eight open pieces

Figure 3 :

 3 Figure 3: The strategy employed in Events 1 and 2

Figure 4 :

 4 Figure 4: Four aspects of an algebraic sense making through algebraic activities

Proceedings of CERME10