Towards an understanding of the shaping of researchoutcomes by contextual issues: reflections on the contributionsof the ReMath project

Lulu Healy & Ana Isabel Sacristán

Published online: 23 January 2014# Springer Science+Business Media Dordrecht 2014

Abstract In this paper, we react to Lagrange and Kynigos’s contribution to this special issue,in which they present some of their experiences from two cross-case studies in the ReMathproject (associated respectively with the use of two dynamic digital artefacts, Casyopée andCruislet). They described how the project, which involved teams from three different countries(France, Italy and Greece), working in different conditions and relying on different theoreticalapproaches, was structured in order to address the role played by contextual characteristics. Weconsider how the ReMath approach of using cross-studies, and especially the inclusion of across-analysis phase, served to identify particularities associated with designing andimplementing mathematics learning scenarios using digital tools. In their discussion of theprocess of re-contextualising, one of the factors which appears to have been particularlyinfluential in shaping research activities was theoretical framework and they show how thisfactor both influences and is influenced by institutional and cultural characteristics. To furtherexplore some of the other important contextual factors they pinpoint, such as the role andengagement of teachers, student characteristics and issues related to the empirical settings, wereflect upon the processes of re-contextualisation that we could identify in our own research.

Keywords Digital technologies . Context . Cross-experimentation and analysis . Networkingtheories

One of the fundamental challenges associated with any attempt to conceptualise the role ofcontext in mathematics learning lies in deciding what we wish the term “context” to refer to.We might define context as the set of the “interrelated conditions in which something exists oroccurs”,1 but this does not emphasise the relationships between the context and the some-thing—that is, the mediating role of context. So perhaps, it would be better to use a definition

Educ Stud Math (2014) 85:423–435DOI 10.1007/s10649-014-9538-2

1Merriam-Webster online dictionary

L. Healy (*)Post-Graduate Programme in Mathematics Education, Anhanguera University of São Paulo, São Paulo,Brazile-mail: lulu@baquara.com

A. I. SacristánDepartment of Mathematics Education, Cinvestav-IPN, Mexico City, Mexico

which stresses this factor, with context becoming the “situation within which something existsor happens, and that can help explain it”.2 Of course, this definition is not terribly precise andcan be operationalised in a multitude of manners, so it is very timely to pursue the goal ofseeking how to conceptualise both theoretical and methodological means for investigating howthe situation (context) impacts upon a mathematical learning activity (something) with digitaltools and influences its efficiency in engendering learning. We see this as the aim underpinningLagrange and Kynigos’s article in this special issue.

Our role in this paper is to reflect upon the approach that was developed as part of theReMath project in order to better understand the influence of context on the outcomes of theirresearch and to overcome what they term the contextual limits. As we offer this reflection, webelieve we should make clear from the outset that while we are outsiders in relation to thisproject, both of us have worked extensively in contexts in which the research activity has been(and is) shaped by Papert’s constructionist perspective (Papert, 1980, 1991)—one of thetheoretical frames used in the ReMath project—and inevitably our reactions will be influencedby this fact. We have also both worked both in and outside of our birth countries; we share theexperience of working in Europe (more precisely in London), and of working in Latin America(Brazil and Mexico, respectively), so to a certain extent, we might argue our own researchprogrammes have involved a sort of cross-experimentation, in that we have worked with thesame digital tools with the same (or at least very similar) educational aims in very differentphysical locations, structured by different institutional and societal constraints, as well as withquite different groups of learners. From these experiences, it takes very little to convince us ofthe fundamental shaping of research activity by the situation in which it occurs, and there canbe little doubt that illuminating the relationships which exist between context and technolog-ical integration is a concern of mathematics educators worldwide. Julie et al. (2010), forexample, describe and contrast the situations in four countries (Russia, Hong Kong, Vietnam,South Africa) and one region (Latin America). They noted that the complex issue of integrat-ing digital technologies into meaningful mathematics learning requires concerted efforts toaddress a host of mitigating factors. We see the contribution of Lagrange and Kynigos, and theactivities of the ReMath project described in other papers in this special issue, as a significantstep in this direction.

They begin by highlighting three points in relation to the use of digital technologies and thelearning of mathematics: (1) It is an area which has attracted substantial international researchand has offered new perspectives on mathematics teaching and learning. (2) Serious attemptsto insert computers into the school system have been made in many, many countries across theworld, with massive investments in financial and human resources. And (3) the impact onschool practices has been resoundingly limited—regardless of the context. This last pointseems to us really very significant and worthy of consideration. Indeed, it is this verydisjunction between effort and impact which seems to sustain the paper, with the hypothesisof the authors being that one of the contributing factors is the relative lack of attention that hasbeen paid to contextual issues involved in the designing, implementing and researching ofdigital tools within mathematics classrooms.

At first sight, the use of an outcome that seems to be universal to suggest that more attentionshould to be given to contextual issues might seem counterintuitive—the limited impact ofdigital technology in mathematics teaching seems to transcend the specific contextual issues ofthe implementing countries. Might it not be argued that there is some factor, common to all theattempts of integrating technology into the mathematics curriculum, which constrains itsimpact on mathematics teaching and learning? We could even ask whether or not the limited

2 Oxford online dictionary

424 L. Healy, A.I. Sacristán

impact of digital tools is something specific to (school) mathematics education or whether it isone that relates equally to other curriculum areas. This is not a question that is explored in thepaper and hence we mention it only in passing here, but another question that did concern us aswe attempted to understand their approach was whether the relationships between the threekinds of research outcomes highlighted in Lagrange and Kynigos’s paper (interpretation ofparticipants’ behaviour; digital media, scenarios and their uptake; and theoretical frameworks)can be treated in the same ways—or, to put it another way, are the same theoretical tools andmethodological procedures necessarily appropriate for investigating relationships betweencontext and the mathematics learning processes that occur as learners interact with digitaltools; investigating the relationships between context, the design of digital media and thespread of their use in mathematics education practices; and investigating the role of context ontheories on mathematics education?

Lagrange and Kynigos offer a particular approach developed to attempt to illuminate allthese different relationships. They claim that, while insufficient attention is given to theinfluence of contextual factors, it cannot be assumed that the outcomes that emerge in aparticular research context will necessarily apply beyond it. Taken in its extreme, this claimpoints to an almost insurmountable gap between research and practice. In this sense, themethodology they describe might not only serve to operationalise the impact of contextualfactors, but also to bridge gaps between different institutional communities. Before discussingthe research methods they employed in more detail, we briefly consider attempts to concep-tualise context in mathematics education research.

1 Conceptualising context

Above, we mentioned the challenge associated with defining context. Given that it is such abroad concept, it is not surprising that it has been treated in a number of different ways withinmathematics education. The work of Jean Lave was seminal in challenging the relatively staticnotions of learning transfer that characterised research in the field of mathematics education inthe 1970s and 1980s. She focussed on how mathematical performance and choice of mathe-matical procedures are not consistent across contexts but, rather, are intimately connected tothe situations in which they occur (Lave, 1988). Nunes, Schliemann and Carraher (1993)confirmed this discontinuity, showing how the mathematical success of the participants in theirstudy was radically different in and outside of the school setting: with performance in schoolsubstantially inferior to performance in street mathematics. One consequence of results such asthese was an increased interest in the situated nature of cognition and how mathematicalthought cannot be separated from the context in which it occurs—a view which leadsessentially to rejecting the notion of learning advances as decontextualisation.

This aspect of the situated cognition agenda has been taken up by some of the researchersworking with digital technologies in mathematics education, with terms such as “situatedabstraction” (Noss & Hoyles, 1996) and “situated proofs” (Sacristán, 1997; Moreno-Armella& Sriraman, 2005) serving to illustrate the role of particular tools of an expressive media inconstraining and affording particular actions and practices. In this sense, digital tools arefrequently confounded with context: the tool being seen as, to some extent, the context forlearning and, furthermore, as one which brings innovative media and new infrastructures fordoing and expressing mathematics. Nowhere, perhaps, is this more evident than within theconstructionist perspective, with its central premise that learning “happens especially felici-tously in a context where the learner is consciously engaged in constructing a public entity,whether it’s a sand castle on the beach or a theory of the universe” (Papert, 1991, p. 1).

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Perhaps, paradoxically, this attention to tool as context may be one of the factors that hascontributed to the lack of attention to (other) contextual issues that have affected bothtechnology use in mathematics classroom and the nature of research into such practiceshighlighted by Lagrange and Kynigos: When the digital artefact is seen as the central,innovative element of the learning scenarios to be investigated, perhaps it is not surprisingthat there is a tendency for other contextual issues to become overshadowed.

Conceptualisations of context in mathematics education research generally have evolvedsubstantially over the past 30 years. In the 1980s and early 1990s, context was commonlyassociated principally with aspects of the task, and a distinction drawn between tasks involving“everyday” contexts and the “decontextualised”, “abstract” tasks of traditional school mathe-matics (Boaler, 1993). The picture today is far more complex—context is viewed as a dynamicevolving entity, interrelated with a diversity of elements at many different levels, including, forexample, discursive practices and situated identities (Ewing, 2005), pedagogy (Sullivan,Zevenbergen, & Mousley, 2003), classroom norms (Yackel & Cobb, 1996), institutionalsettings (Cobb, McClain, Lamberg, & Dean, 2003; Cobb, Dean, Lamberg, Visnovska, &Zhao, 2011) and the like. Indexing the web of contextual factors simply stresses the complex-ity of attempting to understand how they affect the behaviour of the various actors ineducational settings, hence we identify with Lagrange and Kynigos’s desire to go beyondany such indexing and to develop means of reliably assessing how the context of any learningactivity influences the results associated with it.

Returning to the issue of research and practice, a deeper understanding of the relationshipsbetween research outcomes and contexts in which learning and teaching occurs in the presenceof digital technologies may also be useful for research related to attempts to introduce andevaluate any innovation in mathematics classrooms. Regardless of whether it involves digitaltools or not, any innovative approach is vulnerable to association with conflicts with existingteaching practices and institutional agendas and is likely to involve the introduction of newartefacts that need to be appropriated by those involved in implementing the innovation. In thissense, we would suggest that the idea of the re-contextualisation of research outcomes, by bothcarrying out similar interventions in different settings and engaging in multiple analyses of thedata obtained, may not only contribute to the body of results related to the teaching andlearning of mathematics with digital technologies, but more generally to our knowledge abouthow context impacts on either resistance or appropriation of new artefacts and practices inmathematics classrooms. We now turn to consider more explicitly the approaches to context inthe ReMath project that are described by Lagrange and Kynigos.

2 Approaches to context in the ReMath project

The ReMath project, described by Lagrange and Kynigos, attempts to network a variety ofdifferent theoretical frameworks in order to build a concrete and common base for itstheoretical and empirical activities. There are a number of elements of the project which apriori favour systematic comparisons of contextual factors. These include the range oftheoretical perspectives adopted by different team members; the bringing together of re-searchers from different countries with different national education systems; the participationof research teams with different histories of, and approaches to, working with (different)teachers; the involvement of students likely to have followed different trajectories of mathe-matics learning and experienced different approaches and access to technology use; and thepossibility of using the same tool in different educational institutions (schools and colleges), inwhich different mathematical curricula are implemented, different pedagogic approaches are

426 L. Healy, A.I. Sacristán

employed, different equipment and resources and different assessment regimes implemented.Last, but not least, are the different digital tools designed for use in the project. This list ofcontextual issues indicates the complexity of the situations under consideration and themultiple levels from which context might be analysed.

Amongst all these differences, it is interesting to note that it was possible for the researchersin all the different teams to adopt a common methodology: that of design research. AsLagrange and Kynigos point out, design research involves intervention in real-life (rather thanlaboratory) contexts and has the double aim of understanding the processes of mathematicslearning while building innovative learning situations to support engagement in these process-es. In order that the ReMath researchers could begin to concentrate specifically on issues of acontextual nature, they engaged in the methods they term “cross-experimentation” and “cross-analysis”. Cross-experimentation involved the design and analysis of learning activities usingthe same digital tool by two different teams in different countries, while cross-analysis refers tothe practice where one team analysed another team’s data through the lens of its owntheoretical framework.

In their analysis on how different institutions contribute to the moulding of the contextswhere research takes place, Lagrange and Kynigos distinguish between two kinds of set-tings—the empirical and the academic:

Context in empirical settings is characterized by teachers, students and other actorschosen to take part in empirical research, their relationship with the artifact(s) at stake,and between them, as well as policies and pedagogic norms at several levels fromschool, to curriculum and assessment regimes, and to national education systems and,etc. Context in academic settings includes the institutional and cultural environmentwithin which researchers choose to work, particularly the relationships to and expecta-tions of funding agencies, their positioning in relation to colleagues locally, nationallyand internationally as well as their relationships with teachers and school for empiricalresearch. (Lagrange & Kynigos, this issue)

Although the distinction between these two settings signals a different set of contextualfactors that mould and impinge upon the activities that occur within each, it also points to thereciprocal interrelationships between them: characteristics associated with the teachers and thestudents who participate in the study influence, for example, research outcomes, which in turninfluence characteristics associated with the academic setting and back again onto the empir-ical. Lagrange and Kynigos describe this as the “entangled circular relationship” betweenresearch outcomes and contextual characteristics. Perhaps another way of looking at thiswould be to suggest that, in design research, the settings investigated live neither entirely inthe research world nor in the world of the school—a kind of hybrid setting is created,simultaneously characterised by the different actors, researchers, teachers and students, whobring with them their own relationships with issues such as pedagogy, curriculum, assessment,theory, political policies, as well as a kind of temporary suspension or loosening of, at leastsome of, the usual institutional constraints. Although design research is classroom based, it isimportant to remember that as a methodology, it is theory driven and it could be argued,therefore, that the concerns of the academic setting have a certain primacy over the empirical.We believe that this is an important consideration in relation to considering the scope of theresearch outcomes and their applicability beyond the contexts in which they emerge and one towhich we will return later in the paper.

In the ReMath project, the methods were organised in three steps: the design of a didacticaldigital artefact (DDA) by each team; the cross-experimentation of each DDA by two teams(the designing team and another); and cross-analysis in which each team analysed the data of

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the other one. The procedures are well constructed to investigate the use of a same tool, withthe same (stated) educational goal, in different locations and settings—especially since,although the actual design of the tool appears to have occurred in one setting only, the designof the learning activities in which the tool use was embedded took place in each of that tool’simplementation contexts. Each research team was responsible for analysing the data emanatingfrom their teams’ activities, but because the pairs of studies were also subjected to a cross-caseanalysis, a unified description of both studies was possible. The result of using this method-ology is that the differences—and presumably the similarities—between implementations maythrow a new kind of light on the factors responsible for them. Actually, by far, the most strikingmediating factor, in the descriptions presented in the Lagrange and Kynigos paper, is, at firstsight, that of theoretical perspective. This factor influenced initial tool design, it influenced taskdesign, it influenced teaching practices during the implementation and it influenced theanalyses of the outcomes associated with the implementation. From a research point of view,and in the aim of interrelating theoretical frameworks, this seems extremely valuable—itcertainly shows how theoretical orientation impacts upon the context to be researched.Perhaps, though, there is a cost. Since this factor seems to be so strong, it is possible thatother factors were pushed off centre stage.

In our own work, we have also both been involved in experiments involving the design anduse of the same digital tools in different countries. In our cases, however, there were no parallelstudies mediated by alternative theoretical frameworks. Instead, the design process wasinformed, in particular, by the same underlying constructionist framework, and we may nothave been so conscious of how the constructionist vision behind the approaches we usedimpinged on all aspects of the activity; on the other hand, it may have focused our attention onother aspects.

For example, by reflecting on the results of using the same dynamic geometry activities inresearch projects in both England and Brazil (and also in Canada), we were able to identify anumber of differences in the engagement of both teachers and students in the different countries.For instance, in activities related to proof, it was clear that the English teachers and students were,on the whole, familiar and comfortable with the processes of conjecturing and experimenting; incontrast, for the Brazilian teachers, these processes were cited as a major motivation for bringingtechnology use into their classroom—a still rather remote possibility for most of them, given therestricted access they experience (Jahn & Healy, 2008). On the other hand, in activities involvingthe exploration of geometrical transformations, at the level of student interaction with thesoftware, we noted the emergence of very similar story lines (with object and image underreflection, for instance, appearing in narratives concerning lovers who meet on the axis ofreflection as the students sought to make mathematical sense of the on-screen activity).

A similar finding emerged in our studies of teachers and students interacting withdynagraphs (graphs in which both the x- and the y-axes are horizontally configured and thedragging of an input along the x-axis causes an output to move along the y-axis) in Brazil andCanada (Sinclair, Healy, & Sales, 2009), where some striking similarities were observed in thenarratives used to describe quadratic functions, discontinuous functions, the behaviour ofasymptotes and the like. A message here is that, in addition to looking at how differences incontextual factors change what occurs, it may also be interesting to consider what stays thesame. We should say that in neither country, dynamic geometry use was an integral part of themathematics curriculum being followed in any of the schools we worked with; this means thatthe lessons with the digital tools had a high novelty value, and connections with the usualschool curriculum were perhaps placed temporarily in a second plane.

Another example worth mentioning is a Mexican/British study (Rojano, Sutherland, Ursini,Molyneux, & Jinich, 1996; Sutherland, Rojano, Mochón, Jinich, & Molyneux, 1996;

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Molyneux-Hodgson, Rojano, Molyneux, Sutherland, & Ursini, 1999), where a group ofstudents in each country (in Mexico and in the UK) was introduced to the use of spreadsheetsas a mathematical modelling tool, something neither of the groups had experienced before.The study analysed how the different cultures influenced mathematics practices, and cleardifferences were observed between the two groups in their preference for external represen-tations (more preference towards the algebraic in Mexico and more towards the graphical inthe UK), in their understanding of the kind of answers they were expected to produce and inthe way they conceived the role of mathematics in the practice of science. Moreover, quitesubstantial differences in the types of teaching methods and approaches to mathematics wereobserved, with a top to bottom teaching approach in Mexico that contrasted with the oppositein the UK. The researchers of this study point out that whereas there may be invariants (such asobstacles to algebraic thinking), which go across cultures, there are also many ways in whichprevious school experience determines the ways in which students approach mathematicalproblems. At the end of the study, though both groups made progress in their mathematicalreasoning, the same approach helped the students in each country in different ways. Theauthors add that more attention needs to be given to what is meant, in a particular schoolculture, by a specific notion (such as that of “school algebra”) and that it is necessary to takeinto account the teaching approaches.

While the above examples compare analogous interventions or studies taking place indifferent countries and therefore in different cultural contexts, we have also taken part inresearch projects involving different types of subjects; these can be considered as involving usin processes not dissimilar to the cross-experimentation used in the ReMath project, withcross-studies taking place in another kind of empirical setting: the subject context.

For instance, Sacristán (1997) carried out an experiment attempting to create meanings forthe infinite through computer-based explorations, with four groups of subjects—pairs ofstudents of three different age groups and a fourth group of teachers as subjects—with theintent of comparing the characteristics emerging in each group. Although the details of thatresearch are beyond the scope of this paper, one finding is that, whereas each of the groups ofstudents approached the explorations with an open mind, albeit with different mathematicalbackgrounds, the teachers approached the tasks with a “learned” attitude that constrained theirfreedom to explore, created expected pre-conceptions of the results of the explorations andtherefore limited how we had conceived the tasks.

Another case comes from research undertaken in the ongoing Brazilian-based project“Towards a more inclusive mathematics education”.3 This project aims to design and inves-tigate learning scenarios in which students with different disabilities work with multisensorytools (both digital and material) to explore and appropriate different mathematical concepts(see, for example, Healy, 2012; Fernandes, Healy, Martins, Rodrigues, & Souza, 2011). Thetools and tasks that compose the learning scenarios also have been (and are being) used bystudents without disabilities. Certain characteristics of the students investigated are different—some do not see with their eyes, some use their hands rather than their mouths as tools forspeaking and some face particular challenges related to cognitive processes such as memoryand attention. This means that is it possible to focus our analyses on how the particularitiesassociated with such characteristics differently mediate the mathematical meanings that thelearners express. In turn, it has also led to the identification of lacunas in the theoreticalframeworks we had previously used. For example, our work with blind learners alerted us to

3 This project was funded by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (project no.23038.019444/2009-33). Lulu Healy also received a grant from Conselho Nacional de DesenvolvimentoCientífico e Tecnológico.

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the need to attend closely to the gestures they produced with their hands and to how spokenexpressions of mathematical properties and relationships are frequently accompanied orpreceded by visual–gestural expressions (Healy & Fernandes, 2011). Attention to how math-ematics is communicated through visual–gestural means is becoming ever more central to ouranalyses, and to the theoretical framework we are building, as we attempt to better understandhow mathematics learning is mediated in contexts in which Libras, the Brazilian signlanguage, is the main language for communication (Fernandes & Healy, 2013).

In the above cases, the differences and similarities in the interactions of the different studentgroups with the same tools, and the mathematical meanings which emerge during thoseinteractions, provided another means of looking beyond particular contexts. In these cases,rather than institutional variables, it could be said that we are controlling factors associatedwith how different learners experience the world. This we believe is another way of openingnew windows onto the research outcomes described by Lagrange and Kynigos, as well as thecharacteristics which shape them.

On one hand, these different research projects have highlighted some of the complexitiesinvolved in re-contextualising research outcomes associated with the design and implementa-tion of technology-integrated learning scenarios and how, to some extent, any re-contextualisation process is likely to involve some re-tuning of the designs used: it is rare thatan intervention designed for one context can be imported “as is” to another. On the other hand,the concerns raised by Lagrange and Kynigos alert us to the fact that the frameworks emergingfrom our own research do not yet explicitly encompass contextual issues of a more institutionalnature. With this in mind, in the next section, we return to the ReMath project’s examplesdiscussed by Lagrange and Kynigos, focusing on their attempts to examine contextualcharacteristics and how these impacted on the implementation and outcomes of the DDAsby the different teams, leading to, in particular, their considerations of institutional constraintsand school cultures.

3 The cultural contextual dimension

In Lagrange and Kynigos’s analyses of the stages of the design experiments conducted—inboth cross-experimentation and cross-analyses—a contextual characteristic that seems todominate is one that can be understood as cultural. We are conceiving this cultural character-istic as something that includes institutional constraints, school characteristics and theoreticalframeworks of each school setting, influencing all levels of actors: institutions, teachers,researchers and students. These elements of a school culture can differ from a tool designer’sculture and theoretical orientations.

This difference is first evident in the epistemological profiles of each of the digital tools andhow they were conceived by each of the teams. Here, the role of theoretical leaning is alreadyevident: the tool which differs most from standard representations of school mathematics andfrom curriculum structures is the one conceived by constructionist researchers—the Cruislettool. In fact, it is claimed that the designers of that tool and associated tasks recognised thatthey “took some distance from the traditional structure of the mathematics curriculum”. Incontrast, Casyopée was conceived as compatible with the French curriculum and its evolutionand meant to be more easily implemented and adopted by institutions, teachers and students.

In this, we see a marked cultural difference in how tools are conceived, not only in theirepistemology, but also for their uptake in the different school settings. The very conception ofthe Cruislet tool with its difference from the standard representations valued in schoolmathematics curricula in France (where “every session has to be understood by the participants

430 L. Healy, A.I. Sacristán

as a visible step in the curriculum progression”), turned out to be a major issue, mediating allaspects of the research activity for members of the French-based DIDIREM research team.

In describing the considerable challenges of organising experimentation in France, witha tool that was conceived in a different culture, a number of contextual factors arementioned. In particular, institutional constraints led them to conduct the empiricalresearch outside of mathematics lessons, in special workshops and multidisciplinaryprojects. If we reflect back on the results from the work of Lave (1988) and Nunes,Schliemann, and Carraher (1993) cited earlier, this change in setting in itself would beexpected to bring substantial changes to the mathematical practices adopted by thestudents. They may not have actually associated interacting with this particular artefactwith doing school mathematics at all.

In Greece, the same institutional constraints were not experienced—despite the fact that theCruislet artefact also differed from the demands of the mathematics curriculum in that country.This makes the case study a very interesting one. What light then does the approach to contextdeveloped by the ReMath researchers throw on this issue? Once again, the different theoreticalstances adopted by teams in France and Greece, respectively, receive much attention fromLagrange and Kynigos. The Cruislet artefact was created by the Greek team and its designshaped by the constructionist perspective of the team members. For that research team, thedistance of its representations from the mathematics curriculum was not seen as an obstacle;instead, the activities they developed were intended to bring quite new approaches to thenotion of function—not possible with more traditional, non-digital tools. In contrast, thetheoretical approaches adopted by the French team, and especially Chevallard’s anthropolog-ical theory of didactics, created a gap with the French mathematics curriculum that had to beovercome, or at least compensated for, in the tasks created for the experimentation. Here, thereis strong evidence as to how, in relation to the academic setting, the theoretical stance proved tobe an extremely influential contextual characteristic.

In terms of the contextualisation of the empirical settings, although aspects of thetheoretical approaches adopted by each team might account for differences in the scenariosthat were designed for experimentation and for the analysis of the interactions whichoccurred, other cultural and contextual characteristics are needed to explain other aspectsof the implementations. Why, for example, was it possible to organise a long experimen-tation in the Greek research site, which presumably occurred in the mathematics classes ofthe student participants, while in France, only short experimentation in specific circum-stances was possible? Among several explanations given is the fact that Logo program-ming, which was central in the Educational Technology Lab in Athens (ETL) Cruislettasks, obeys a general pattern of the work of the ETL team and presumably is alsosomething familiar for schools, teachers and students in Greece or at least the schoolinvolved in the experimentation; whereas, conversely, it is not embedded in the DIDIREMculture nor in the French school culture in general.

The apparent ease of the Greek team in organising extensive experimentation with theCruislet tool seems to be associated with local factors, such as the team’s relation with thesenior teacher involved in the implementation, who was in fact a teacher educator trainedat ETL and thus had a history of working with the researchers and seems to holdconstructionist views. It is claimed that the intervention, based on constructionist mathe-matics, was informally respected by the school due to the status of the senior teacher. Anddespite the fact that ETL is involved in a wide-scale teacher education project of theministry, there was no obligation, nor encouragement, from above to engage in suchactivities. This contrasts with the country-wide educational constraints that are evokedas explanatory factors in the French case.

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4 The teacher dimension

This brings us to one of the most central aspects of the empirical settings: the teacher. Differentteachers, with different views about mathematics, about mathematics learning and abouttechnology use, probably impact (locally speaking at least) upon the practices that emergeas much as the technological tools. We might ask whether there is something that identifiesteachers who are willing to experiment with digital tools that distinguish them from theircolleagues. Are there some aspects concerning teachers’ preparation or other experiences (inpre-service or in-service) that make them more or less prone to work with digital tools ingeneral or even with specific kinds of digital tools? Can we identify some teachers whoseexperiences in learning mathematics, or learning to teach mathematics, narrow the distancebetween practices involving digital tools and practices which do not?

We would not want to suggest that there is such a thing as a standard mathematics teacher—in the way that there are standard representations of mathematics—but building understand-ings of the teacher’s impact on the learning situation, or on the teacher’s role, may lead to somekind of contextualising profile that could represent a useful tool for analysis, especially asconcerns the spread (or non-spread) of the use of digital tools in mathematics classroom. In thelate 1980s and early 1990s, an attempt in this direction was made by Hoyles, Noss andSutherland (1991), who investigated a group of teachers who participated in an in-serviceeducation course that involved the design, implementation and analysis of microworlds formathematics learning. As part of the analyses, a series of “caricatures” were made to classifyteachers according to different types. The factors that defined these caricatures representedaspects of the teachers’ previous experiences and beliefs, which appeared to be important indetermining the kinds of technology use they valued and the kinds of activities they werewilling to carry out during the project.

Of course, teachers’ views can change, and participating in a research project like ReMathappears to have been a change-provoking experience for some of the teachers—and apparentlyespecially so in the case of teachers who have strong long-term relationships with the designand research teams and are familiar with their respective theoretical frameworks, as was thecase in two of the three teams presented in the paper. For these teachers, it would seem thatengaging in the challenge of experimenting with a tool designed with a framework unfamiliarto them, and then participating in the cross-analyses of the data collected by the other team,challenged their views of epistemology and pedagogy, perhaps also increasing their awarenessof how their teaching practices have been influenced by institutional constraints.

The first case study, involving the Casyopée artefact, provides a case in point. In the theoryof semiotic mediation, adopted by the Italian team, but not familiar to the French teachers, acritical aspect of a didactic situation is assigned to the collective classroom discussions, givingthe teacher a prominent role. A significant finding associated with the cross-analyses in thiscase study was the recognition by the teachers that the theoretical stance of their team had ledthem to privilege a-didactical phases of student interaction with the digital artefact and tounderestimate the complexity of the formularization and institutionalization of the knowledgeexpressed during these interactions. From a research point of view, this result suggests that thereflections upon research frameworks used in an “outside” context stimulate revision of thetheoretical tools used by the “insiders”.

We see some resonance of this finding with the aspects of our own research. Many of theprojects in which we have worked have also involved long-term collaborations with practicingteachers. Indeed, the project of designing inclusive mathematics learning scenarios for studentswith and without disabilities was initially motivated by teachers, who, concerned about howthey might include students who had previously received their mathematics education in

432 L. Healy, A.I. Sacristán

specialist school in their own classrooms, sought collaboration from the research community.In this project, the methodology was a little different to that developed in the ReMath project,in that it attempted to combine aspects of design research with methods drawn from co-generative inquiry, a kind of participatory action research. In a co-generative study, the inquiryprocess is linked to action in a given context and involves participants in re-signifying thepractices within this context. This brings another kind of research outcome into play—and onethat is specific to the context involved. It also means that neither the academic nor the empiricalsetting has, a priori, primacy over the other. We have found that by bringing the differentperspectives of teachers and researchers to the process of analysing data associated with learningin the presence of digital tools can result in the development of our professional identities relatedto both these roles and, by focussing on how the hybrid settings we investigate differ from theregular classroom situation, also throws light onto institutional constraints such as expectedpedagogies and classroom organisation, curriculum and assessment (Healy & Santos, 2014).Although this method differs from the ReMath approach, the construction of tools for dataanalysis which serve both the teaching and research communities seems to us another way ofcontributing to the re-contextualisation of the results we obtain in our hybrid research settings.

5 Final remarks

This article has presented some of our reflections on the ReMath approach to developtheoretical and methodological tools aimed at enhancing our understandings of the relation-ships between context and research outcomes, in the particular case of using digital artefacts inthe teaching and learning of mathematics. We have argued that the construct that is mostpresent in their analyses is that of theoretical framework—and, generally speaking, it appearsin the form of contextual characteristic rather than research outcome. They present a mostconvincing case to illustrate the extent to which research activity is fundamentally shaped bythe theoretical orientation of those conducting it. While in itself this is not surprising, we doagree that the inclusion in the research project of the cross-analysis phase is a promisingstrategy for identifying particularities associated with designing and implementing mathemat-ics learning scenarios using the same digital tools that depend on the theoretical orientation ofthe researchers involved.

In terms of tools and scenarios, one of the factors highlighted as an issue by Lagrange andKynigos is the distance, or even mismatch, that can exist between the conception of thedynamic digital artefacts developed for the ReMath project and country-based institutionalconstraints, views of the curriculum structures and other cultural characteristics. Clearly, thisdistance also impacts on teachers’ reactions to the tools. In Brazil, for example, we have foundthat, at least in terms of initial reactions, there is a tendency amongst the mathematics teacherswe have worked with to give more legitimacy to digital tools in which the representationclosely resembles what they recognise as school mathematics (Healy, 2006). It may only bewhen they experience such tools in action in a situation for which they have recognised a needfor changes in established patterns of practice that they are prepared to accept the validity ofalternative forms of doing and expressing mathematics (Healy & Santos, 2014).

What, then, is the message for those of us researching mathematics learning with digitaltechnologies? Or for those seeking explanations for the limited impact of these technologies onthe world’s mathematics classrooms? It might be tempting to suggest that if we wantconservative institutions to sanction the use of something new, then we should minimise itsdistance from the old. Or that if we want teachers to integrate a new artefact into their teachingpractices, then the changes it engenders should not perturb these practices too much.

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From a constructionist point of view, however, such a conclusion is somewhat disturbing. Ifwe are satisfied with the status quo, why introduce new technologies at all? If what we want isfor students to learn paper-and-pencil mathematics, and progress through a curriculum de-signed with static representation in mind, then should we really be spending so much moneyon installing digital resources?

Many of us, though, are not satisfied with things as they are; we believe that newmathematical knowledge and practices, and new ways of communicating about them, willemerge from the new digital technologies and dynamic representations and that these can leadto a more learnable, a more inclusive, mathematics. This cannot be separated from contextualfactors, but also cannot be completely dictated by them. Papert (2006) argued that it makeslittle sense to use digital tools to serve a curriculum that was created without them, offering thefollowing analogy to ridicule the situation: “We’d never have had aeroplanes…if we hadconstrained the new transportation to follow the schedules of the sailboats…but that’s what weare doing in our schools” (Papert, 2006).

One hope then is that, by better understanding how contextual factors influence learningwith—and the spread of—digital tools in mathematics classrooms, we might be better placedto influence those with some control over these factors, raising awareness of those involved inall levels of mathematics education that the time is ripe for embracing rather than avoidingdigital representations that are distant from conventional ones and in modifying those othercurricula factors that constrain us to the current status quo.

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