Boundary objects in mathematics education and their role across communities of teachers and researchers in interaction.

25 July 2021

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Boundary objects in mathematics education and their role across communities of teachers and researchers in interaction. Publisher: Published version: DOI:10.1163/9789004419230_009 Terms of use: Open Access

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BOUNDARY OBJECTS IN MATHEMATICS EDUCATION AND THEIR ROLE

ACROSS COMMUNITIES OF TEACHERS AND RESEARCHERS IN

INTERACTION

Robutti, Ornella; Aldon, Gilles; Cusi, Annalisa; Olsher, Shai; Panero, Monica; Cooper, Jason; Carante, Paola; Prodromou, Theodosia

Keywords: teacher professional development, boundary crossing, boundary object, communities of teachers, meta-didactical transposition

ABSTRACT

Studying interactions across communities of mathematics teachers and educational researchers in professional development poses theoretical challenges. The authors suggest that it may be productive to view such encounters as the meetings of two professional communities that have different, possibly conflicting perspectives on the theory and practice of mathematics education. Drawing on the notions of boundary objects and boundary-crossing, the authors propose a framework for how teachers and researchers may learn from and with each other through joint work on common objects, through which they can explicate, reflect upon and modify their perspectives. Through the analysis of three educational projects from different countries - Italy, France and Israel - the authors describe ways in which the structure of boundary objects supports various aspects of dialogical learning, thus providing an initial frame for the proposed theoretical constructs.

RESEARCH INTEREST, AIM AND FOCI

Mathematics teachers and mathematics education researchers have many professional meeting points. Teachers play a central role in mathematics education, and at the same time participate in research conducted by researchers. Moreover, these two communities also meet and collaborate in professional development (PD) programmes, quite often conducted by researchers, especially in the context of the development and implementation of new curricula, instructional tools, or new teaching practices.

As teachers and researchers belong to different professional communities, they may have different perspectives on the collaborative work, due to different objectives of their involvement in the communities, whereby researchers may be more concerned with academic and theoretical aspects of the collaboration, while teachers might tend to focus on didactical implications.

Our research is with teachers and not on teachers, since teachers and researchers work collaboratively as co-designers of instructional resources (e.g., collaborative design of problem-solving lessons (Wake et al., 2016)). The learning in such collaborative settings contributes both to teachers’ professional development and to accumulation of academic knowledge and theories. However, analysing the learning of mathematics teachers and researchers within collaborative environments is a challenging task (Robutti et al., 2016). Our goal is to develop a framework aimed at both contributing to academic understanding of the challenge of productive collaboration between mathematics teachers and educational researchers, and advising practice in suggesting how to make the most of these collaborations. These two aspects - theoretical reflection and practical design - could be considered as the result of the work on problems of common interest, organized around boundary objects for both communities. We choose to frame collaborative activity between teachers and researchers as boundary encounters. The notions of boundary encounters mediated by objects is not new to the field of mathematics teachers’ professional development. Nolan, Horn, Ward and Childers (2011) have investigated the role of assessment tools as boundary objects in the preparation of novice teachers, Sztajn et al. (2014) have investigated mathematics professional development as design for boundary encounters, Johnson, Severance, Leary and Miller (2014) have investigated mathematical tasks as boundary objects in design-based implementation research, Cobb, McClain, Lamberg and Dean (2003) highlighted the role of district leading teachers’ pacing-guide as a boundary object in interactions with practicing teachers, and Kynigos and Kalogeria (2012) have investigated the role of planned teaching scenarios as boundary objects in the preparation of teacher educators. We draw on this notion of boundary, focusing on ideas of collaboration across communities. Our aim is to examine the theoretical constructs of boundary-encounter, boundary-object and boundary-crossing in the

specific context of collaboration between mathematics teachers and researchers. We start with an exposition of the general framework, state meta-level research questions, describe our methodology, and demonstrate the affordances of this methodology in three representative collaborative projects taking place in different European contexts. The final discussion will synthesize the ideas that emerge in the analysis and show our new contribution in the field.

THEORETICAL FRAMEWORK

There are several ways to model teachers' and researchers' actions and interactions: we chose Meta-Didactical Transposition - MDT (Arzarello et al. 2014; Robutti, 2018), because it is a useful model to describe the interactions among teachers and researchers as members of two different communities working together. Particularly, it is a model that supports a dynamic description of the praxeologies of a community of teachers involved in professional development designed and conducted by researchers, showing their evolution over time. According to Chevallard (1999), praxeology describes a human activity in terms of a practical part or praxis - how to do this? - and a theoretical part or logos - why doing this?. The practical part involves two components: task and techniques to solve it, and the theoretical part involves two other components: technical justification and theoretical support. Referring to teachers, the human activity may be teaching, in which case the praxeology is called didactical, or it may be working and learning in professional development, in which case it is called meta-didactical. The MDT model highlights the process by which praxeological components that are initially external are developed and used, for example by teachers in professional development, becoming internal to them. The model helps researchers in highlighting this process in terms of changes in teachers’ praxeology components.

We refer to the definition of “boundary objects”, originally proposed by Star and Griesemer (1989) within an ethnographic context and developed in different field of research (Carlile 2004, Tsui & Law 2007, Wegner 2010, Akkerman & Bakker 2011), including mathematics education (Kynigos, 2012; Rasmussen et al., 2009; Goos & Bennison 2017; Adler, 2017). A boundary object (henceforth BO) allows different communities to work together without preliminary consensus, due to its “interpretive flexibility” (Star, 2010). Boundary is not intended as a line of demarcation, but rather as a “metaphorical place” where different communities can act and possibly interact and create: “[...] common objects form the boundaries between groups through flexibility and shared structure – they are the stuff of action” (Star 2010, p.603). This interpretive flexibility allows both communities to find an interest in studying and using the BOs. Nevertheless, as Star warns: “The two other aspects of boundary objects, much more rarely cited or used, are: (1) the material/organizational structure of different types of boundary objects and (2) the question of scale/granularity” (Star, 2010, p. 602).

In this chapter, we consider the structural components of BOs and the actions taken on knowledge at the boundary as tools explaining the evolution of praxeological components from external to internal. We connect this evolution to the changes occurring in the BO itself, considered (following Star, 2010) as a dynamic object. We realize that teachers and researchers, when working together, rarely focus on the whole object at the boundary but more often on one of its components. Thus, our interest shifts from the whole BO to how it is made up of structural components. This kind of investigation is important not only from the researchers’ perspective but also from the designers’ perspective, and not only designers of tools but also of teaching interventions.

To describe such a structure, we use and develop the metaphor of the paradigm of object-oriented programming. Languages that support object-oriented programming typically exploit the concepts of “inheritance” and “encapsulation”, considering each object as an instance of a class (a container). We will use these terms in a similar sense: this model is organized around objects which are gathered in sets of objects, the class, which is the whole set including objects that inherit from the general properties of the class. The following example illustrates the concepts of class, inheritance and encapsulation. Tables, chairs, cookers, etc. are objects of the class “furniture” and within the object “table” we can consider kitchen tables, bedside tables, garden tables and so on. Each of the object inherits properties from the class (each of them being movable for example) and is encapsulated within a “greater” object (a bedside table being for example of the same “nature” as a table in the sense that there is a tray on which it is possible to lay something). Viceversa, analyzing a bedside table allows to infer some of the properties of all the pieces of furniture, such as movability.

Using this metaphor in the case of BOs means considering a BO as a class or a container of different encapsulated objects that inherit the main properties of the container. Reciprocally, working on a component of the object one may infer general properties of the container. Thus, inheritance can be seen both in a top-down movement where encapsulated components inherit properties of the container and in a bottom-up movement where properties highlighted for components can be inferred for the main object.

Furthermore, BOs exist if and only if someone acts on them. It is through the actions on knowledge at the boundary that the boundary can evolve, causing also an evolvement of the understanding of the BO (or one or more of its components). The boundary object is a means that will serve to explain praxeologies of the different actors.

The boundary indeed is not fixed, and might pass through three different interactional forms that are different approaches to sharing and understanding knowledge at the boundary (Carlile, 2004). Interactions can develop at a syntactic, semantic or pragmatic level, according to the way in which knowledge and BO understanding are shared and evaluated by the different interacting actors. Such encounters, where actors of different communities interact on knowledge and perspectives related to BOs, are called boundary encounters. Interactions occur at a syntactic level when differences and dependencies between actors are acknowledged, and the language to communicate on the BO needs to be shared. At this phase knowledge and perspectives are transferred in order to facilitate shared discussions on the BO, allowing actors to agree on the identification of the BO’s structural components and on the vocabulary to point out its properties.

When the joint work on the BO highlights some differences of perspective, which appear unclear or ambiguous, a translation of knowledge and perspective is needed. Translation affects the meaning of the objects and occurs at a semantic level of the discourse. Developing common meaning of the BO or of some of its components may be important to address interpretive differences across boundaries. In some cases, translation is not just a matter of translating different meanings, but of negotiating interests and making trade-offs between actors. Following Granger (1976, p.377) in a semiotic perspective: "we introduce the opposition "syntax/semantic" by means of an "ontological" criterion. Semantics refers to the property of a sign relating to its possible reference to an "individual". Syntactics refers to a property that relates to the possible reference of a sign to a relationship between signs".

Interactions occur at a pragmatic level when translation is not sufficient and common meanings and knowledge need to be invested in practice at least in one of the communities involved. It implies an effort not only to learn about what is new, but also to transform current domain-specific knowledge accordingly. Transformation develops at a pragmatic level of interaction, where the actors share and create knowledge related to use. It means that at least one of the communities integrates new pieces of knowledge and interpretations that have been shared or created in the interactions.

Transfer, translation and transformation of knowledge and perspectives at the boundary are made possible by the interpretative flexibility of the object. Moreover, transfer and translation highlight the boundary of the object, while transformation leads actors to represent current and more novel forms of understanding of the BO, causing it to evolve. This sharing of knowledge and understanding of the BO (or of one or more of its components) potentially creates learning opportunities for all the involved actors. Akkerman and Bakker (2011), in their review of the literature on boundary objects, detect four potential learning mechanisms which can be activated and supported by activities on BOs:

• identification: coming to know what the practices of the different involved communities are about in relation to one another;

• coordination: creating cooperative and routinized exchanges between practices; • reflection: expanding one’s perspectives on the practices;

• transformation: collaboration and co-development of (new) practices and perspectives.

In this chapter, we focus in particular on the learning mechanism of transformation (in the sense of Akkerman & Bakker, 2011), which can be surely influenced by transformation activities (in the sense of Carlile 2004) on the BO, but not exclusively. Transfer and translation activities can also foster and support transformation as a learning mechanism.

The learning mechanism of transformation develops through several steps.

• Confrontation, when different actors working on the BO are confronted with some lack or problem that forces them to seriously reconsider their current practices and possible interrelations.

• Recognizing a shared problem space, which we intend as the boundary where the different actors can encounter, discuss and work on the object.

• Hybridization, when the actors engage in a creative process in which a hybrid cultural form emerges from the encounter and dialogue of their respective practices.

• Crystallization of what is created, which actually becomes part of the respective practices of the involved actors.

• Maintaining uniqueness of the intersecting practices.

It is important to highlight the depth of the learning mechanism of transformation, which in passing through these phases, “involves real dialogue and collaboration between ‘flesh-and-blood partners’ at either side of the boundary” (Engeström et al., 1995, p. 333). More than the other learning mechanisms, it leads to profound changes in practices. According to Carlile, domain-specific knowledge turns out to be transformed at a pragmatic level. New techniques and justifications of such techniques are developed to solve tasks related to the BO. In specific circumstances, especially when the actors’ tasks are the same even though they come from different communities, the learning mechanism of transformation may even lead to the creation of a new, in-between practice (Akkerman & Bakker, 2011), which is what the MDT model defines a “shared praxeology”1 in the sense that both communities understand and agree on techniques and, in some cases, also on their theoretical justifications. Even when this sharing of praxeologies is only ideal, and the goals and tasks of the different involved communities remain distinct, each actor’s praxeologies can evolve making some external element become internal to his/her own praxeologies (Arzarello et al. 2014; Prodromou et al., 2018). This evolution of praxeologies is thus simultaneously a process of professional development and a product of a learning activity made possible by the joint work of subjects (teachers, researchers) on a BO.

RESEARCH QUESTIONS AND RESEARCH METHODOLOGY

The focus of this chapter is on the role of BOs in the context of the collaboration between teachers and researchers within mathematics education research projects. Specifically, the analysis we propose is aimed at, on the one hand, characterizing the idea of BO in the context we are studying and, on the other hand, highlighting the learning mechanisms that can be developed thanks to the interaction between different communities working on BOs. With regards to this second aspect, our hypothesis is that there is an interrelation between the evolution of BOs and the evolution of teachers’ and researchers’ praxeologies. Therefore, characterizing BOs is a starting point for studying their evolution and the impact that this evolution has in terms of teachers’ and researchers’ learning.

These are our main research questions:

1. How can we characterise the structure of BOs that are analysed in our examples? What kind of evolution of BOs can be described?

2. What are the possible dialogical learning mechanisms that could be highlighted in our examples? How can these mechanisms be described in relation to the evolution of BOs? How does the evolution of BOs impact the evolution of teachers’ and researchers’ praxeologies?

As stated above, we interpret boundaries as “metaphorical places” where different communities meet to work together on BOs and develop a dialogue to reflect on them. In tune with this idea, boundary crossing is interpreted as the process through which the collaborative work on the BOs lead both to their evolution and to the corresponding evolution of participants’ praxeologies.

To answer the questions of group 1, we present three examples that describe collaborative ways of working at the boundary and show different kinds of BOs. The data we collected are mainly audio and video-recordings of meetings between the participants of the different communities, materials produced during these meetings and reflections collected by participants (interviews, written logs, etc.). The data we analyse in this paper were selected, among all the collected data, because they highlight explicit discourses and reflections, between the participants of the different communities, about the BOs.

Our attention is on the discourses developed by the participants of the different communities when they work together on BOs. These discourses are analysed in Carlile’s framework, which is useful to characterize both the level of the discourses (syntactic, semantic, pragmatic) and the corresponding actions on the domain-specific knowledge connected to the use of BOs (transfer, translation and transformation). The evolution of BOs is described in relation to the actions that have provoked it by enlarging the space of shared understanding of the objects.

Moreover, we aim at highlighting also the impact that the actions have on the domain-specific knowledge in terms of learning of the actors involved in these processes. In order to answer the questions of group 2, we analyse this learning referring to the categories proposed by Akkerman & Bakker (identification, coordination, reflection, transformation), highlighting learning mechanisms as the result of specific discourses on BOs and corresponding actions.

1 _{We write shared praxeology in quotation marks to stress that we do not have to take for granted that praxeologies can be shared completely:}

TURIN EXAMPLE: MERLO ITEMS AS BOUNDARY OBJECTS

This example refers to MERLO (Meaning Equivalence Reusable Learning Objects), introduced by Etkind and Shafrir (2013) and developed by other researchers over the world (Arzarello, Robutti & Carante, 2015). This is a research and a teacher professional development project on the design and use of MERLO items as didactical tool. The example presented in this section refers to activities developed within a professional development course for in-service teachers: the second level two-year Master: “Professione Formatore in Didattica della Matematica” (Profession Educator in Mathematics Education), held from 2013 to 2015 at the University of Turin, for teachers of secondary school (lower and higher), having researchers as trainers. The general objective of the Master programme was the training of mathematics teacher educators and the construction of a professional model of teacher training included in the national curriculum (National Guidelines and Guidelines) and in the national evaluation of schools (reference framework for mathematics in INVALSI tests). A particular aim of the programme was the design and discussion of curricular materials for students (in line with institutional references as “Mathematics for the citizen”, m@t.abel or the National Scientific Degrees Plan).

The participants can be divided into two main communities: one of 5 researchers and one of 29 in-service teachers (selected from among applicants). The teachers were selected among 60 applicants according to the following criteria: experience in school (i.e. number of years in school, institutional roles of the teacher within his/her school), experience in the field of research (i.e. previous collaborations with researchers), experience in teacher education (i.e. previous involvement in PD programmes, particularly the national m@t.abel Project). The selection was made through curriculum vitae and interview. One of the activities in the master involved MERLO items. Both communities worked together on MERLO items, with the aim of analysing items produced by other researchers over the world, and designing new items contextualised in the Italian curriculum.

During the whole process the two communities met both face to face and at distance, through a Moodle platform. The activity of the Master involving MERLO items was organised around a set of meetings between at least one of the communities and a member of another community with the role of broker (Arzarello et al., 2014): 1) two 2-hour meetings between R1 (Israeli researcher) and the two communities; 2) meetings among teachers along with R2 (PhD student); 3) two 2-hour meetings between R2 and the other researchers of the Italian team; 4) a final meeting between the teachers and R2.

The collected data consists of: MERLO items designed by the teachers, video recordings of meetings, texts on the platform (uploaded files and interventions in the forum), namely the content related to the productions of items and the explanations and justifications of the steps they followed to design them and to discuss and reflect upon them. Here we view MERLO items as boundary objects, and the different meetings as moments in which the two communities worked and interacted at the boundary. We will show the role played by this work at the boundary in determining a modification in the design, structure, and underlying theoretical criteria of MERLO items, which is an evolution of the boundary object.

We first briefly introduce the main characteristics that MERLO items had before their modification thanks to the work on them and to the interactions between the two communities. The MERLO items were designed as tables of 3 columns and 2 rows, with one cell occupied by the task and the other 5 by different statements on mathematical concepts (Figure 1). The task for the solver of a MERLO item (first cell of the table in figure 1) was: “a) identify and mark the statements (2 or more) equivalent in meaning; b) write the reasons that guided the choice.”

Figure 1. MERLO item on fractions (A, B, C equivalent, D, E distractors)

One is the main statement that gives rise to the item, and is the target statement representing a mathematical concept in a certain register (graphical, symbolic, numerical, verbal - Duval, 2006).

The other 4 are or not equivalent in meaning to the target statement.

The task for the solver (Figure 1) consists in: a) identifying and marking the statements (2 or more) equivalent in meaning (and not marking the others); b) writing the reasons that guided the choice.

These characteristics of the MERLO items, and the theoretical basis on which they were conceived, were shared during the first meeting, when R1 (who played the role of broker between his Israeli community - focused on research - and the Italian communities of teachers and researchers) introduced MERLO items as a new didactical and methodological tool for teaching and assessing mathematics. He presented the framework with its two main criteria for design (Meaning Equivalence for correct answers and Surface Similarity for distractors), along with suggestions of opportunities for learning for both the researchers and the teachers.

The definitions of Meaning Equivalence and Surface Similarity that R1 transferred are:

“The term Meaning Equivalence designates a commonality of meaning across several representations. It signifies the ability to trans-code meaning in a polymorphous – one-to-many – transformation of the meaning of a particular conceptual situation, through multiple representations within and across sign systems. By Surface Similarity we mean same/similar words appearing in the same/similar order as in the target statement. To assess if students have understood a concept, you can show statements with similar words but different meaning: when students mark both of them, it means that they have not understood”.

This introduction to the MERLO framework in terms of two main criteria immediately resonated with the community of researchers, who recalled the idea of conversion by Duval (2006). Initially, the two communities of researchers and teachers accepted these criteria as they were presented, considering them totally in accordance with their theoretical references.

During this meeting, the discourse is initially at a syntactical level, when R1 intervenes to foster a transfer of the idea of “meaning equivalence” to the Italian communities. Then, when the community of researchers refer to the idea of “conversion” to interpret the concept of meaning equivalence, the discourse is developed at a semantic level, since one of the communities working at the boundary feels the need to carry out a translation.

In the following meetings, the teachers were engaged in workshops during which they solved and discussed MERLO items, working in small groups and interacting with researchers, and working together on the design of new MERLO items for their classes. In the excerpt below, the teachers and R2 were discussing the design of a MERLO item taking into account the definition of meaning equivalence proposed by R12_.

1 T1: If we start from a general case, then we can go to a particular case, but we cannot go back from the particular to the general case. Therefore, there is not a “logical equivalence”.

2 T2: The mathematical context creates problems, because a graphical representation of a parabola does not represent all the parabolas, but it represents only one. At school, we usually link an example with a concept.

3 T3: Of course, examples are indispensable. Examples are an essential part of our teaching practice and they must appear in MERLO items.

4 T2: We have to give an example when we want to evaluate if students have understood the various representations of a concept: a function, a table, a graph... You can give the parabola’s definition, the graph y=x2, a table with some values of x and y.

5 R2: The problem consists in finding a good formulation for the task: “equivalent” or “shared” meaning.

6 T2: “Share the same meaning”... I think it is good, isn’t it?

7 T3: “To share the same meaning” does not mean “to have the same meaning” or “to be equivalent”.

8 T2: For example, you can talk about the derivative concept in relation to the slope of a tangent or in relation to the applications in physics. They “share the same meaning”.

Although the community of teachers began designing MERLO items according to the criteria introduced in the first meeting, they gradually became aware that Meaning Equivalence can be used in different senses when teaching and learning mathematical concepts, and that the theoretical definition proposed by R1 is not sufficient to cover all the didactical needs (Lines 1 to 5). It seemed necessary to negotiate meanings of Meaning Equivalence, to include examples of equivalence considered in a larger sense than the logical/epistemological one.

2In the three examples the numbers of the quotes in the transcripts are not consecutive because the transcripts refer to different moments within meetings between participants of the different communities or to different meetings.

According to Akkerman and Bakker (2011), we can interpret this as a confrontation with some lack - or problem - that forces the intersecting worlds to seriously reconsider their current practices and the interrelations. This confrontation had a consequence in the creation of a new sense of Meaning Equivalence, born in the community of teachers: different statements may only share a mathematical meaning. For share teachers intend that it can be not only a complete correspondence in the meaning but also a partial overlapping or a logical consequence. For example: a general case shares meaning with a specific case (Line 4), a theorem or a property shares meaning with a consequence of it or an application of it (Line 8). What we can see here is a process of hybridization according to Akkerman and Bakker (2011): given a certain problem space, practices that are able to cross their boundaries engage in a creative process in which something hybrid—that is, a new cultural form—emerges. In this process of hybridization, ingredients from different contexts are combined into something new and unfamiliar: teachers are not able to figure out the new idea of task they have created, being aware that the old one is not acceptable any more. A product of this process was exactly the change in the students’ task, according to different senses that the members agreed to use for the Meaning Equivalence criterion (Lines 6, 7, 8).

We can interpret the exchange between R2 and the community of teachers at the pragmatic level of praxeologies: trying to apply the theoretical definition of Meaning Equivalence for the creation of tasks for their classes, the teachers recognized a dissonance and felt the need to re-interpret the idea of Meaning Equivalence according to their own didactical practice, where recognizing examples and non-examples is an important facet of understanding. As a result, we can highlight a transformation (in Carlile’s sense) of the novel form of domain-specific knowledge on which the creation of MERLO items is based.

In a subsequent meeting R2 shared with the other researchers of the Italian team the discussion results on the application of the concept of meaning equivalence within the community of teachers.

The feedback from the teachers was an occasion for the researchers to reflect on the issue from a theoretical point of view and to delve deeper into the study. The community of researchers recognized the problem faced by the teachers and highlighted the need for giving new theoretical basis to the design of MERLO items: the theoretical interpretation through “conversion” is not sufficient to explain the concept of “sharing the same meaning”. This led to a search for new theoretical elements within research literature, which enabled the community to identify two new elements to be added within the theoretical frame on which the design of MERLO items is based. The first theoretical element is the notion of “sphere of practice” (Kilpatrick, Hoyles & Skovsmose, 2005), which enables to highlight the complexity of meaning in mathematics education. A sphere of practice can be characterised expressing the rules, routines, priorities, values, and actions which are attached to it. Meanings are constructed in spheres of practice. The second theoretical element comes from Frame Semantics (Fillmore, Johnson & Petruck, 2003), according to which word-meanings must be described in relation to semantic frames – schematic representations of the conceptual structures and patterns of beliefs, practices, institutions, images, etc. that provide a foundation for meaningful interaction in a given community. This is an excerpt of the discussion that developed within the community of researchers when they shared these theoretical elements as a new basis on which the definition of meaning equivalence (in the sense of “sharing the same meaning”) could be grounded.

40 R3: I think that the notion of sphere of practice is very useful to support our idea of meaning and equivalence of meaning in a pragmatic sense, because it depends on the “practice”: the mathematical logical/epistemological content is not enough to embrace completely the meaning.

41 R2: Thus, the teaching practice in the classroom acquires relevance.

42 R4: Well, but this is not enough if we want to create a solid theoretical background to the criterion of Meaning Equivalence.

43 R3: The theory of Frame Semantics makes stronger the background thanks to contributions from semantics. According to this perspective, the meaning of words is impossible to be defined in an absolute way, because many other aspects have to be considered (beliefs, practices, institutions, images, etc.).

44 R2: We can enlarge the discussion moving from words to mathematical representations, maintaining the fundamental basis.

45 R4: In this way, we have a semantic interpretation that can complete the theoretical background with the pragmatic view.

46 R2: I remember that the teachers suggested to use the expression “share the same meaning” as task in MERLO items. What do you think?

47 R4: I think it is in line with the discussed theoretical contributions. I like this new formulation of the task.

48 R3: I would specify to mark statements that share the same “mathematical” meaning. And that’s it.

We can interpret the theoretical discourse that developed in the community of researchers again at a pragmatic level: the researchers’ practice had to face the problem of defining new theoretical elements that represent the new foundation for MERLO items. In this step the transformation (in Carlile’s sense) of domain-specific knowledge is more evident, because work of researchers on the boundary object led to the identification of a new way of conceiving MERLO items: this new theoretical basis does not only include conversions between registers, as in the first design (Duval, 2006), but also other theoretical perspectives, able to support the whole set of practices typical of teaching/learning activity in a class (Fillmore et al., 2003; Kilpatrick et al., 2005; Arzarello et al., 2015; Carante, 2017). These theoretical reflections were then shared with the community of teachers in the final meeting between R2 and the teachers.

As a result of these actions on boundary objects, a new version of the students’ task, based on the idea of “sharing of meaning”, intuitively formulated by teachers and then supported from a theoretical point of view by the intervention of researchers in the discourse, has been developed (see Figure 2). The result is tangible in a new formulation of the task, which constitutes the final product as MERLO item with a modified task - that stems from teachers’ didactical needs - along with an enlarged theoretical background - that comes from researchers’ perspective. Lines 46-47-48 show the crystallization (Akkerman and Bakker, 2011) of what has been created in the previous steps by teachers: the new task in terms of sharing meaning - that was hybrid at the boundary - is considered to be embedded in practice, and also justified in theory, adding new frames to those presented by R1 at the beginning. Together, the two communities of teachers and of researchers arrive at a final (shared) product of their joined and connected work, crystallising it for the future in a stable version of item. The teachers of the Master took their degree of teachers’ educators in mathematics education and then were invited to attend seminars and training workshops in Italy. Among them, they were invited to a national summer school organised by CIIM (Commission for Mathematics Education in Italian Mathematical Union), to attend a workshop of 4 hours for teachers from all over Italy. In this workshop, not only did they recognise themselves as a team called “gruppo MERLO”, but they also demonstrated that they had completely embraced and retained the ideas developed in the master regarding the sharing of meaning and its didactical sense (http://www.umi-ciim.it/wp-content/uploads/2016/09/Robutti_2.pdf).

Figure 2. MERLO item on inverse proportionality: new formulation of the task (A, B, C equivalent, D, E distractors)

LYON EXAMPLE: FORMATIVE ASSESSMENT AS A BOUNDARY OBJECT

FaSMEd (Formative Assessment in Science and Mathematics Education) was a European project aimed at “researching the use of technology in formative assessment classroom practices, in ways that allow teachers to raise students’ achievement in mathematics and science” (FaSMEd, Document of Work). The French team focused in particular on the teachers’ ways of appropriating and augmenting available technology to activate formative assessment (henceforth FA) strategies in the classroom with technology. Therefore, our focus was on

the sharing of resources in the classroom;

• the orchestrating role of the teacher in the classroom;

• the FA key-strategies (Black & Wiliam, 2009) activated in the classroom and their dynamics.

From January 2014 to December 2016 the FaSMEd project involved in France 18 teachers working in different school grades.

During the first year of the project, the researchers’ team met the teachers to inform them about key aspects of the project and to discuss the technology that they were using or that they wished to use in their classroom. Then, the first months of the school year 2014/2015 were devoted to focus the lessons on FA and to introduce and appropriate the technology in the classrooms.

During the second year of the project, researchers and teachers met regularly to co-design cycles of FA lessons that mainly consisted of: a test for the students, analysis of the results, feedback and remediation in the classroom and then again a test for the students. Several observations in the classroom were conducted. In the first months of the school year 2015/2016, researchers and teachers met to fill in the project websites (https://ife.ens-lyon.fr/fasmed/ and https://microsites.ncl.ac.uk/fasmedtoolkit/) with descriptions and suggestions drawn from the implemented lessons and to re-design and re-implement some FA cycles.

During the last year of the project, researchers and teachers met to evaluate the entire FaSMEd experience, to share suggestions about technology and FA drawn from personal experiences, and to generalize them in order to advise other teachers.

In a design-based research perspective, teachers and researchers collaborated to:

• Design the setting and the FA cycle with technology that the teachers implemented and the researchers observed in the classrooms;

• Analyse collaboratively the results of the implemented practices;

• Refine tasks, solutions, methodologies and design principles coming back to previous phases of the work. • Teachers’ and researchers’ meetings were more frequent during the second year of the project, devoted to

implementation and observation in the classrooms. Such meetings generally comprised: • cluster meetings with all teachers participating in the project;

• design/analysis or discussion meetings with groups of teachers from the same school; • informal discussions with the individual teacher before and after the lesson.

Interactions were mainly about the evolving conception of FA for each participant in the project and about the functionalities of technology which supported each teacher in implementing FA strategies. We compared similar strategies with different technologies or different strategies with the same technology in different contexts and at different levels. Each teacher was asked to write a description of his/her implemented lessons, to detect key-moments of FA and explain how the technology helped him/her. Researchers organized a cross-reading of the material and read it themselves. Then, after revision, the resources were made available on the French website (https://ife.ens-lyon.fr/fasmed/).

The interactions between teachers and researchers in the FaSMEd project revolved around the concept of FA, its principles and implementation. FA is the boundary object on which the different actors have different perspectives: the teachers a more professional eye, and the researchers a more theoretical interest. However, they found a common space where ideas and practices could be shared around this object.

With this example, we want to show how the BO is progressively accessed through the inferences made on its components. In the following, we consider three short excerpts from the meetings that we held throughout the project. In the first (March 2015) we discussed two specific components of FA as a boundary object-container, which are

multiple choice tests (MCT) and clickers. Two primary teachers and two researchers were involved. Teachers focused on MCT and in particular on the test about fractions that the students had just taken with clickers. They wanted to propose to the students the same test in an open written form, as they were used to doing before, in order to see if there is any difference between the two modalities of assessment. Their usual didactic praxeology, before starting the FaSMEd project, to face the task of assessing students’ understanding entailed a mainly summative assessment with open-ended questions in written form. This technique was justified by the teachers’ experience with written summative assessment and impressions on the use of MCT.

1 R1: It can be something to propose in a second moment, maybe. 2 R2: To leave some question open.

3 R1: To leave some question open.

4 T1: And it’s true, this is something we wondered and that could have been interesting, it’s to give them the test in a written form, to see the difference... Is there any difference, do we observe any difference? […]

5 T1: Does technology do this? Does it lead to different answers? This is the question we had. 6 R1: Uhmm, uhmm

7 T1: Just to have,… why not?

8 R2: Here the difference is between a MCT and an open question. 9 T1: Yes… otherwise we should present it…

10 T2: If they have memorized all the [MC] options, they will still encounter difficulties in a written [open] test.

11 R1: No, it is the same question you asked, there is a different process between an open question and a choice among three options that you have to…

12 R2: But it has nothing to do with technology. 13 R1: Yes, it has nothing to do with technology. 14 T2: No, it is something else.

15 T1: It is the modality of assessment.

16 T2: We need to be aware that we don’t ask them the same thing.

The interactions began at the syntactic level (lines 1-13), with an unclear relationship between the two components: MCT and technology (clickers). Teachers focused on technology and its effect on students’ answers. Although the researcher’s attempted (line 8) to distinguish MCT, as a modality of assessment and technology, as one of the possible means of assessment, teachers associated ‘assessment with technology’ with ‘MCT’ and ‘written assessment’ with their usual way of assessment that is ‘open-ended questions’. The need to clarify what the different actors are talking about triggered an action of transfer on the components (MCT and technology) of the boundary object. The syntactical use of MCT and technology in the discourse between teachers and researchers is now shared: MCT can be proposed with or without technology. Thus, the discourse moves onto a semantic level. The teachers’ concern is specified (lines 14-16): it is not related to the use of technology but rather to the modality of assessment, that can be with multiple choice or open-ended questions. Teachers focus on the meaning of the objects they are acting on. In terms of learning mechanisms, according to Akkerman and Bakker (2011), we can identify a process of confrontation followed by recognizing a shared problem space as the beginning of a transformation of practices. Researchers took into account the teachers’ idea to propose open-ended questions and the teachers distinguished and compared MCT and open-ended questions: for all the actors it was beneficial to confront this component of FA and to reconsider their respective practices.

R1 then shifted the attention to the pragmatic level “What I would rather like is that we focus on how we can ensure that students who didn’t succeed somewhere, how we can get them to succeed, you see”.

This intervention, focused on formative aspects of the modalities of assessment, encouraged the teachers to explain in detail the technical device they wanted to experiment with for assessing students’ understanding of fractions. The technique consisted of a cycle: to propose a MCT for students; to create level groups, according to the results, for proposing remediation tasks; to propose a MCT again focusing on specific questions. The justification of this praxeology relies on the teachers’ experience and knowledge about the different students’ abilities and attitudes, and on the necessity to interpret the results of the MCT in a formative way. The interactions with the researchers, who

constantly highlight the formative aspects of the teachers’ planned work, lead them to make some reflections on their device, justifying explicitly some choices made in a conscious way and relating it to FA principles. We can observe this in the following excerpt, where teachers were reflecting on the remediation tasks to propose.

122 T1: We highlight the difficulties, I think. Are we proposing something that can put them in a situation enabling them to succeed next time?

123 R1: Yes, yes. Exactly.

124 R2: Otherwise it’s just a training.

125 T1: Because otherwise they are going to do exactly the same… 126 T2: That’s why for choosing the… I tried to vary, you see.

127 T1: But it’s fine that we have… and that we go back and forth among what we observed in the exercises they failed globally and that we say, well, maybe we didn’t make them cope with enough situations…

128 R1: … allowing them to understand this… 129 T2: Exactly.

130 T1: This representation, yes.

In this moment of the meeting, when teachers and researchers discuss the experimental praxeology, the discourse is at a pragmatic level, and the action on the components of the boundary object is a transformation in the sense of Carlile (2004) aiming at grounding didactic techniques on FA principles. In terms of learning, we have a hybridization (Akkerman & Bakker, 2011) of the FA cycle (MCT-analysis-remediation-MCT) as a didactic technique. Differently from the initial praxeology, teachers’ technique integrates feedback that is an essential component of FA.

At the beginning of the second year of intervention in the classroom, teachers and researchers met again and the teachers presented a FA cycle implemented in another classroom on another concept: numbers and operations. So the praxeology leaning on the hybrid experimental technique and a beginning of internalization of FA principles seems to be crystallized in the teachers’ practice. In the following excerpt we return to the same components of FA with a deeper awareness. Notice that the distinction between MCT and technology, as components of the boundary object, is now established, and the teachers and the researchers recall and discuss the use of MCT (lines 72-83) and then the role of technology (lines 138-146) as two distinct objects in their practices. This is a sign that, for both teachers and researchers, transfer and translation succeeded in creating a boundary space where both language and meaning are shared.

72 T1: The fact that there is a choice, then, they [the students] said that it allowed them to... well I think this is related to the MCT, that it discards some possibilities. It also allows them to filter a little bit the possible answers 73 R2: The possible answers.

74 T1: Exactly.

75 R1: This is interesting. The fact that they have a choice. 76 T1: It guides them.

77 R1: They are, yes, they aren’t lost. They have a… yes yes

78 T1: With respect to what is expected, I think, it allowed them to… 79 R1: Yes.

80 T1: Delimit a little bit.

81 T2: And moreover they felt more skilful when they answered, I think. 82 R1: Yes.

83 T2: It’s true. They dare to answer.

The discourse focuses on the use of MCT, at a pragmatic level, especially on the benefits for students to have a choice. Teachers and researchers discuss metacognitive effects of the use of MCT on students’ work and attitudes. Specifically, they discuss what is effectively “easier” for students in the new technical device in terms of modalities of assessment.

139 T1: And then it is funny what Mél said: “But then I didn’t need to calculate in my mind” I found it surprising [...] As if the clicker, she told me: “But I didn’t need” but I told her “But then how did you answer?” 142 R2: It isn’t a calculator!

143 T1: It isn’t a calculator, it isn’t the tool that calculates for you and so it’s, for some of them, it’s surprising because they had the impression that there was a magic side.

144 R2: Hum.

145 T1: And they forgot that eventually they were anyway [...] calculating in their mind.

146 R2: This is the magic side but if after you work on it in terms of skills, it gives them the idea: “Well, at the end, it was not that magic, indeed, I thought!”

The discourse focuses on the students’ use of clickers and, in general, on the role of technology. Teachers and researchers discuss metacognitive effects of the use of clickers on students’ work and attitudes. In particular, they discuss what is effectively “easier” for students in the new technical device when they use clickers to answer. Teachers’ didactic praxeology to assess students’ understanding used the same technique (MCT-analysis-remediation-MCT) justified by the teachers’ experience and reflection on metacognitive effects for students in terms of FA. Since the teachers once more implemented the hybrid technique experimented in the previous year in a totally spontaneous way, we can detect a beginning of crystallization of it in the teachers’ praxeologies. For the researchers, in turn, it becomes evident what is actually possible to implement in the classroom, reflecting on general suggestions that the research can give to practice in terms of FA.

Our analysis underlines that it is possible to act on the boundary object viewed as a container, by acting on some of its components. In our case, actions on MCT and clickers led teachers and researchers to design a FA cycle. Moreover, there is a continuum relating transfer activities, translation activities and transformation activities on the components of the boundary object that allows to infer general properties of FA. At the beginning of the first boundary encounter we were in a space between translation and transformation, and the latter became stronger in the boundary encounter of the second year of the project. Transformation action on the boundary object, in the sense of Carlile (2004), is a necessary transformation of knowledge related to uses but not sufficient for a complete transformation as a learning mechanism, in the sense of Akkerman and Bakker (2011). The latter entails several processes, starting from confrontation of meanings and uses and coming out with a beginning of crystallization of praxeologies, passing through a hybridization of techniques and justifications.

HAIFA EXAMPLE: TOOLS FOR CURRICULAR DESIGN AS A BOUNDARY OBJECT

The Edumap project is motivated by teachers’ changing relationship with learning resources. As teachers come to act more substantially as co-designers of the enacted curriculum, supplementing textbooks with additional internet-based resources, or even constructing learning sequences from scratch, they are faced with the challenge of maintaining instructional coherence. This requires sensitivity to didactic nuances of learning resources, and tools that help teachers make use of such sensitivity.

We are addressing this need by means of two complementary browser-based tools. One is a tagging tool for associating metadata with web-based resources (figure 3), and the other is a dashboard for browsing and navigating

collections of tagged resources (figure 4). The tools make use of prescribed categories of metadata – designed by researchers and modified in cycles of research/redesign (see Cooper & Olsher, 2018, for a description of the categories). Our current research focuses on affordances of these tools for the professional development of teachers, in helping them develop a curricular discourse that is sensitive to didactical nuance, which is necessary for informed

Figure 3. Partial snapshot of a browser extension tool for tagging web-based resources

selection of resources, and in applying this sensitivity to searching for learning resources. We have been conducting small-scale interventions with practicing and pre-service teachers, first in educational settings – graduate courses comprising 5-25 mathematics teachers – and later in authentic classroom settings – using tools to tag and to reflect upon enacted teaching sequences.

Figure 4. Dashboard representing balance of a collection of resources

Our work has revealed a boundary – discontinuities between researchers' and teachers' perspectives on curricular design and task selection. Activity on this boundary is mediated at three levels (A. syntactic, B. semantic, C. pragmatic) through interactions with three nested boundary objects.

A. At the syntactic level, a vocabulary of instructional design is transferred between researchers and teachers as categories of metadata, which attend to two notions of instructional coherence (see Cooper & Olsher, 2018). To attend to the coherence-of-design (Pepin, Gueudet, Yerushalmy, Trouche, & Chazan, 2015), we included categories such as mathematical skills invoked in the resource (e.g. modeling, inferring), mathematical objects (e.g. functions, equations, triangles), actions with and on these objects (e.g. adding/composing functions, manipulating equations) representations of mathematical objects (e.g. graphic, symbolic, numeric, verbal), and type of media (e.g. text, dynamic-sketch, video). To attend to the coherence-in-use (ibid.) of enacted learning sequences, we included categories such as role in the sequence (e.g. opening task, practice, assessment) and class arrangement (e.g. whole class, individual work). This framework of categories is transferred from researchers to teachers, and is negotiated when teachers request new categories such as duration, grade and difficulty level, and general ranking.

B. At the semantic level, when teachers use tools for tagging and for searching in a tagged collection, they engage in translation when they try to make sense of categories of metadata. Faced with the need to decide which categories of metadata to use as filters, they described their dilemma: “We were left with a lot of tasks [after initial filtering], so we started to think what else is important for us in assessment”, and “we understood that we must prioritize [categories of metadata]”. This activity brought them to be explicit about their considerations for task selection – considerations that normally remain tacit – and to translate these considerations into the researchers’ vocabulary. This involved two aspects of learning through reflection: perspective making (Boland and Tenkasi, 1995) - making explicit one’s understanding and knowledge of a particular issue, and “reflexively [giving] access to implicit and unstated assumptions” (p. 364), and perspective taking (ibid.) - “taking of the other into account, in light of a reflexive knowledge of one’s own perspective” (p. 362). Through discussions, researchers engaged in translation when analyzing and try to make sense of teachers’ use of these categories.

Example 1. Teachers struggled to express their ideas in terms of metadata

1 T: Opening the topic [of integral], we were interested in a numeric representation of the problem… not symbolic like you see on exams and in regular problems. But we want them to perform mathematical operations on [the functions].

2 R: Can you say something about the difference between the mathematical operations you expect in opening a new topic, practice and assessment?

3 T: I don’t know these terms. 4 R: Of course. In your own words.

5 T: For opening the topic: calculating numbers. In practicing the topic, they’ll use more equations, algebraic technique

.

Following the teacher’s expression of didactic considerations “in his own words”, the researcher helped him express these ideas in terms of representations (numeric/symbolic) and operations on functions (modify/analyse). This translation was a joint effort, mediated by the researcher, providing opportunities for both teachers and the researcher to reflect on their perspectives.

Example 2. Managing without “difficulty level

”

6 T: Preparing practice, I need [tasks] to be in accordance with the stage I’m at. There’s no regard [in categories of metadata] for the level of difficulty.

7 R: It’s very difficult to rate levels of difficulty when we don’t know what the students know. 8 T: But there are some that are just practicing calculations.

9 R: We have that. It’s the type of mathematical operation [“modify”].

Teachers often think of tasks in terms of their difficulty, a category that the researchers preferred to avoid. When “unpacked”, some aspects of difficulty may be translated into the researchers’ terms, e.g. “modification” operations may, in some contexts, be a translation of “easy”.

C. At the pragmatic level, structured activities in professional development brought into play the communities’ praxeologies. Such activities included individual tagging of a chapter from an e-textbook followed by a group discussion, co-constructing a tagged collection of classroom tasks in a particular topic, and using the dashboard to construct a skeleton learning sequence from a tagged collection, comprising three tasks – an opening task, practice, and assessment. These activities brought into play the parties’ goals and values regarding mathematics education. Sharing perspectives at this level not only encouraged deeper reflection on the communities’ perspectives, it also has the potential to transform teachers’ instructional praxeologies and researchers’ praxeologies as developers of instructional tools. Examples of reflection at the pragmatic level:

Searching for instructional resources, teachers used the category of representations (verbal, numeric, graphic, symbolic) to distinguish among didactic roles (opening, practice, assessment)

10 R: Why graphic representation?

11 T1: Opening a new topic… it’s something they’re seeing for the first time… for these specific students, it can help if they see it graphically.

12 T2: We want to ask “why”, and often the answer comes from the visual.

Hence, graphic representation is appropriate for opening, considering specific student needs.

Practices that emerged in professional development suggest how the tools may eventually transform teachers’ work. Hybridization – one of the processes of transformation (Akkerman and Bakker, 2011) – was prominent, and may be particularly productive in drawing on teachers’ and researchers’ perspectives to create something fundamentally new.

Example 4. Combining “informed” and “idiosyncratic” search practices

Teachers’ practices of selecting supplementary tasks are based on available search engines; teachers review the first (i.e. most popular) results of their search by keywords, and select the most appropriate according to tacit considerations that may never be articulated. From the researchers’ perspective, the didactically-sensitive dashboard should replace this practice. Yet the practice that emerged in professional development was a hybridization: filtering according to metadata until 3-4 tasks remained, reviewing them, and selecting from among them based on idiosyncratic considerations. The sense in this practice was described by teachers:

“We thought we need to reduce to 1 task, but later we understood it’s better to be left with 3-4 and to check them… there’s the human factor, I like it or I don’t like it, I need to look at the task, does it suit me as a teacher?”

Example 5. Emergent goals

The tools were designed in order to help teachers select learning resources that fit their values and goals. Yet some teachers saw using the tools as an opportunity for professional development:

“We wanted to arrive at things that would surprise us… [would] changed our original thinking”.

“It can change your style of teaching. From a state where you’re set in your ways about how to teach, you see something that someone else thought was suitable, and you say wow. This is very positive, because you want to refresh your instruction, with regard to considerations for selecting tasks”.

Such “wow” moments occurred, for example, when teachers who thought of video resources for opening a new topic (“it doesn’t rely on prior knowledge”) discovered that others had tagged them as practice, or even as assessment. Reflecting on how this might be done, they eventually came to extend their instructional repertoire, and selected video resources for new purposes.

DISCUSSION AND CONCLUSION

In this chapter we have focused on the role played by BOs in fostering collaborative work developed by teachers and researchers involved in PD programmes, and we have analysed the effects of this joint work in terms of evolution of the BO and of dialogical learning mechanisms that can be highlighted.

The first set of RQs focused attention on the structure of BOs and on the analysis of their evolution as an effect of the actions developed by the members of the teachers’ and researchers’ communities.

We interpreted BOs in our examples through the metaphor provided by the paradigm of object-oriented programming (focusing, in particular on the properties of “encapsulation” and “inheritance”), with the aim of

clarifying “the material/organizational structure of different types of BOs” at different “scale/granularity” (Star 2010, p. 602).

Our examples enable, on the one hand, to highlight a complex structure of BOs, and, on the other hand, to verify our hypothesis that the actors’ interactions with BOs are necessary conditions for the BOs to exist and evolve. Moreover, in all the examples we have focused on specific structural components of each BO, which inherit properties of the BO itself, and we highlight how the actions on these components foster a deeper understanding of the BO and extend the space where teachers and researchers meet.

The BO presented in the Turin example (henceforth MERLO BO) can be intended as a container of singular specific components: the formulation of the task, the statements A, B, C, D, E, the relationships among them (in the example of Fig.1, A, B, C equivalent; D and E distractors), and the theoretical criteria to design MERLO items (Meaning Equivalence and Surface Similarity). The MERLO BO exists due to teachers’ actions on it, working on the design of Merlo items through the construction of representations and tasks for the students. The first effect of these actions is the evolution of two of the components of the MERLO BO, highlighted by a new formulation of the task (the term “meaning equivalence” is substituted with “share the same mathematical meaning”, attending to a didactical point of view and not only to a strictly logical-epistemological one) and consequently, an enlargement of the possible representations to be included in the different statements (influencing the “statements” component). The discussion developed when the new version of the task was analysed by the community of researchers. This had further impact on another component of the MERLO BO: the theoretical criteria for designing MERLO items, in fact, are interpreted in tune with the new formulation of the task, enlarging the reference. This example enables us to stress the importance of the role played by the evolution of some components of the BO in fostering the evolution of other components and the subsequent enlargement of the boundary of the BO. The enlargement of the boundary of the BO is not documented in all our examples because the evolution of the BO and the collaboration between researchers and teachers are at different stages.

In the Lyon example, formative assessment plays the role of an abstract BO in the interactions between teachers and researchers within the FaSMEd project. Nevertheless, the different actors rarely act on the whole object, but rather they act on some of its structural components: formative assessment strategies, using data collected through particular modalities of assessment (e.g. MCT) and tools (e.g. clickers and related software). The analysed episodes highlight first the distinction and then the relationships established between these structural components. Translation actions on such components allow the different communities to agree on shared meanings for them: teachers reconsider the formative use and interpretation of MCT in their experimental practice with clickers, and researchers validate the implementability of formative assessment principles, and the conditions for their implementation in the classroom, strengthening their foundation. Through this action of translation from both the teachers’ and the researchers’ sides, the boundary enlarges and, as a consequence, the teachers’ experimental technique comes to be more grounded on formative assessment theories. This example highlights, therefore, how the properties of the BO could model the evolution of the meaning given to some of its components.

In the Haifa example, a 3-tiered nested BO is described, aligned with the three levels of discourse (Carlile, 2004). At the syntactic level, categories of metadata (keywords) are incorporated in tools, which act as BOs at the semantic level. Professional development activities (tagging resources, searching in a tagged corpus), in which the tools are used, act as a BO at the pragmatic level. These boundary objects evolve over time through action and interaction. At the syntactic level, categories of metadata are added, removed, refined or renamed as they are discussed among teachers and researchers. This syntactic evolution has immediate impact on the tools, which automatically adapt themselves to changes in the nature of tagged metadata. At the pragmatic level, teachers’ practices of searching and sequencing instructional resources develop through interaction with the tools. Additionally, researchers modify or add PD activities, shifting from pragmatics of balancing a curriculum to its organization on a timeline. These evolutions occur slowly, yet even in short time spans, where the BOs themselves remain static, the nature of the