text:
Stéphane Sirejacob
email: stephane.sirejacob@etu.univ-paris-diderot.fr

Françoise Chenevotot-Quentin
email: chenevotot.francoise@neuf.fr

Brigitte Grugeon-Allys
email: brigitte.grugeon-allys@u-pec.fr

PÉPITE Online automated assessment and student learning: The domain of equations in the 8th grade

Keywords: diagnostic assessment, Information and Communication Technology (ICT), elementary algebra; equations, student's profile, teaching suggestions

Pépite" is an online automated assessment tool for elementary algebra for students in secondary education (12-16 years-old) in France. Pépite was initially developed for students at the end of compulsory schooling in . At CERME9, we presented its transfer at different school levels and illustrated it with the design of Pépite test for grade 8 th students. Information provided by Pépite allows identifying students' consistent reasoning and calculation in order to organize teaching corresponding to students' learning needs. In this paper, we focus on the use of Pépite test for grade 8 th students to learn the domain of equations. We defined an epistemological reference of the algebraic domain that allows us not only to build the tasks selected for the test and to analyze students' responses but also to propose suitable courses adapted to students' learning needs.

Context of the study

This paper deals with the issue "Digital assessment of and for learning" of TWG16 "Students Learning Mathematics with Technology and Other Resources". Since the 1990s, our team has developed several multidisciplinary projects [START_REF] Delozanne | Vers un modèle de diagnostic de compétence[END_REF][START_REF] Grugeon-Allys | Diagnostic et parcours différenciés d'enseignement en algèbre élémentaire[END_REF] concerning the design, development and use of online tools for diagnostic assessment and student learning. One of these tools, named "Pépite", is relevant for learning elementary algebra for students of secondary education (12-16 years) in France. We have disseminated Pépite online tool on platforms1 largely used by teachers and students.

In this paper, we deal with the use of Pépite online assessment for learning the domain of equations for grade 8 th students. First, we revisit the theoretical foundations of Pépite online assessment. Then, we illustrate it with Pépite assessment for grade 8 th students in France (13-14 years). We specify both the didactical model and the computer model that automatically generates generic tasks, analyses students' work and provides descriptions of students' profiles. Finally, we discuss the potentialities of Pépite online assessment to propose suitable courses adapted to students' learning needs for the domain of equations for grade 8 th .

The theoretical and methodological framework

In the educational system, assessment is a complex issue. Usually, assessment results are generated from standardized and psychometric models. Studies highlight the strengths and limitations of such approaches to make instructional decisions (Kettelin-Geller & Yovanoff, 2009). To identify the features of appropriate online assessment for learning, we have chosen both a cognitive and epistemological approach and also an anthropological approach, the potentialities of which are described in [START_REF] Grugeon-Allys | Diagnostic et parcours différenciés d'enseignement en algèbre élémentaire[END_REF].

Epistemological and cognitive approach

Designing a test requires the selection of a set of tasks that enables the assessment to be realized. We agree with Vergnaud who stated, "Studying learning of an isolated concept, or an isolated technique, has no sense" (Vergnaud, 1986, p. 28). Furthermore, Vergnaud introduced a strong assumption: dialectics between genesis of a student's knowledge and mathematical knowledge structure. Beyond a quantitative analysis of responses, we have to define a qualitative didactical analysis (based on a collection of students' responses to the tasks) to identify the type of procedures and knowledge used by students in solving the tasks. To provide descriptions of a student's consistent reasoning, it is necessary to define a reference for modelling the mathematical competence, in a mathematical domain, at a particular school grade.

Anthropological approach

The epistemological approach is not sufficient so as to take into account the impact of the institutional context on students' learning. According to the anthropological approach, mathematical knowledge is strongly connected to the institutions where it has to live, to be learnt and to be taught; it is strongly connected to mathematical practices (curricula, etc.). [START_REF] Chevallard | L'analyse des pratiques enseignantes en théorie anthropologique du didactique[END_REF] analyses knowledge in terms of praxeology, that is to say in terms of type of tasks, techniques used to solve these tasks, technological discourses developed in order to produce, explain and justify techniques, and last, theory that justifies technological discourses.

A reference epistemological praxeology for algebraic knowledge

For a given mathematical domain, we defined a reference epistemological praxeology [START_REF] Garcia | Mathematical modelling as a tool for the connection of school mathematics[END_REF]) that makes it possible to create an a priori design that describes features of an appropriate assessment. For algebraic knowledge, such reference is based on results from didactics of algebra [START_REF] Chevallard | Le passage de l'arithmétique à l'algébrique dans l'enseignement des mathématiques au collègedeuxième partie : perspectives curriculaires : la notion de modélisation[END_REF][START_REF] Artigue | Teaching and Learning Algebra: approaching complexity trough complementary perspectives[END_REF][START_REF] Kieran | Learning and teaching algebra at the middle school through college levels[END_REF]. In its tool dimension [START_REF] Douady | The interplay between different settings: Tool-object dialectic in the extension of mathematical ability-Examples from elementary school teaching[END_REF], there are tasks for generalizing, modelling, substituting, proving. In its object dimension, there are tasks focused on calculus with algebraic expressions (calculating, substituting a number for a letter, developing, factorizing) or equations (solving). This reference makes it possible to define appropriate technology for an intelligent and controlled algebraic calculus, based on equivalence of algebraic expressions and the dialectic between numeric and algebraic treatment modes.

The domain of equations for grade 8 th students

The three following types of tasks are specifically related to equations (we will give some precise examples later about our experimentations):

-Modelling and putting a problem into equation (tool dimension). These tasks motivate the production of an equation in order to solve modelling problems and require semiotic conversions [START_REF] Duval | Registres de représentation sémiotique et fonctionnement cognitif de la pensée[END_REF].

-Solving an equation using an algebraic technique; proving that two equations are equivalent (object dimension). These tasks use the concept of equivalence and require transformational activity [START_REF] Kieran | Learning and teaching algebra at the middle school through college levels[END_REF].

-Testing if a number is a solution of an equation; identifying the degree of an equation (object dimension). These tasks are based on substitution and polynomial properties.

Features of Pépite online assessment

The Pépite online diagnostic assessment is based on a reference epistemological praxeology of the algebraic domain, both for designing tasks and for analyzing responses to the test. We will base this on the Pépite test for grade 8 th students.

The didactical model

Pépite test

The test (targeting 13-14 years old students) is composed of 10 diagnostic tasks and 22 individual items covering the types of tasks defined below (Table 1). The tasks may be multiple-choice or openended items (Figure 1).

Responses analysis: the multidimensional model of algebraic assessment

At the local assessment level (for each task), students' responses are not only evaluated as correct/incorrect, but also according to their technological discourse, that justifies the techniques. The analysis concerns validity of response (V) and seven dimensions: meaning of the equal sign (E), algebraic writings produced during symbolic transformations (EA), numerical writings produced during symbolic transformations (EN), use of letters as variables (L), algebraic rationality (J), connections between a semiotic register to another (T) and skills with negative and decimal numbers (N) (Table 2) [START_REF] Grugeon-Allys | Réguler l'enseignement en algèbre élémentaire : une approche multidimensionnelle[END_REF]. We code the responses with assessment criteria, which depend on knowledge and reasoning involved in the techniques 2 . We illustrate the multidimensional model of algebraic assessment on the task "Proof and calculation program" (Figure 1). In order to solve this task, two a priori strategies are possible: an arithmetic strategy using a number or an algebraic strategy mobilizing a variable. Several incorrect techniques can illustrate an arithmetic strategy (Table 3) according to the rules used to translate or transform numeric expressions. Algebraic strategy may be incorrect (J3) if the conversion rules (T3 or T4) or algebraic transformation rules (EA3 or EA4) are inadequate (Table 4).

Solutions

Reasoning 2 Contrary to usual practices in assessment, we do not attribute a code by technique for each task. This would lead to a multiplicity of codes on various tasks and would be unusable for a cross analysis on all the tasks of the test.

Student's profile, groups and differentiated strategies

The Pépite diagnostic assessment proposes both individual and collective assessment. The individual assessment, at the global assessment level (on a set of tasks), builds the student's profile which aims to identify features of algebraic knowledge and skills for the seven dimensions. The collective assessment locates a student on a scale with four components: skill in Algebraic Calculus (CA), skill in Numerical Calculations (coded CN), Use of Algebra for solving tasks (UA) and flexibility in Translating a semiotic register to another (TA). For each of those four components, different technological levels and appropriate benchmarks have been identified [START_REF] Grugeon-Allys | Diagnostic et parcours différenciés d'enseignement en algèbre élémentaire[END_REF]. Regarding to a class, students are divided into three groups according to their skill in Algebraic Calculus: CA1 (group A) -reasoned and controlled calculation preserving the equivalence of expressions -, CA2 (group B) -calculation based on syntactic rules often in blind, not always preserving the equivalence of expressions -and CA3 (group C) -meaningless and non-operative calculation. Therefore, for a given learning objective, it is possible to assign tasks to each group, depending to didactical variables related to the associated technological levels [START_REF] Delozanne | Vers un modèle de diagnostic de compétence[END_REF][START_REF] Pilet | Bridging diagnosis and learning of elementary algebra using technologies[END_REF].

The computer model

An iterative process between educational researchers, computer scientists and teachers was used to design and test different Pépite prototypes in order to improve the didactical model. We defined the conceptual IT model of classes of tasks, which allows the characterizing of equivalent tasks [START_REF] Delozanne | Automatic Multi-criteria Assessment of Open-Ended Questions: a case study in School Algebra[END_REF]. The software PépiGen [START_REF] Delozanne | Automatic Multi-criteria Assessment of Open-Ended Questions: a case study in School Algebra[END_REF] automatically generates the tasks and their analyses, at different grade levels. It uses Pépinière, a computer algebra system, to generate anticipated student correct or incorrect answers (according to the a priori analysis). Pépite automatically calculates a student's profile as well as profiles for groups of students. According to a learning objective defined by the teacher, Pépite generates tasks adapted to the related technological levels [START_REF] Grugeon-Allys | Diagnostic et parcours différenciés d'enseignement en algèbre élémentaire[END_REF].

Results and discussion

The information provided by Pépite diagnostic assessment allows the teacher to identify students with close profiles in algebra. Then, Pépite automatically generates differentiated routes corresponding to these algebraic profiles. As mentioned above, these routes were designed on the basis of a reference epistemological praxeology.

Differentiated routes for learning equations

Three differentiated routes were created concerning equations. The first route "Motivating the production of an equation and solving it with an equation solver" motivates the production of an equation. It includes tasks like "equalizing two calculation programs" (see the example below). Students have to solve them using an equation solver. The second route "Algebraic resolution of an equation" requires technologies for solving equations by algebraic methods (by using the concept of equivalence of equations). In the last route "Algebraic resolution of problems that lead to an equation", tasks that require a problem to be expressed as an equation and then solved, such as, "equalizing two perimeters of dynamic figures", are proposed.

We give now two examples of tasks for the first route. The first one aims to introduce equations and to highlight the inadequacy of arithmetic techniques to solve problems of first degree. As we can see, thanks to a thoughtful choice of the didactic variables, this task prevents arithmetic strategiesbecause of the presence of the unknown in both calculation programs -or "trial and errors" methods because the solution of this problem which is The second task is differentiated (Table 6) to take into account students' algebraic activity and makes the students work on semiotic conversions (from the representation register of algebraic writing to the representation register of calculation program). Differentiation relies on didactical variables: the left member of the equation for group A is a product and solving the equation needs to use the distributive property, while the equation for groups B and C do not require it to be solved. Moreover, the multiplication sign is used for groups B and C to suggest that one or more multiplications are expected in the expression.

For group A For groups B and C

Write a problem with two calculation programs that correspond to the equation 2(𝑥 + 7) = 5 -3𝑥.

Write a problem with two calculation programs that correspond to the equation 2 × 𝑥 + 7 = 5 -3 × 𝑥. We now present the results of research carried out in 2016 with a mathematics teacher we will call M2. M2 has been working in a REP establishment (high-priority education network) for three years. We chose him because he is not an expert. After an observation phase (6 hours) of his teaching practices, we proposed a whole teaching sequence to him on equations that takes into account the main epistemological aspects of the reference epistemological praxeology. M2 was free to adapt this sequence to his practices; however, both teacher and researcher have worked together to plan the implementation in the class.

M2 is required to introduce equations in his grade 8 th class. First, his 20 students (14 years old) completed the Pépite test. Then, they were been divided into three groups A, B and C. Only one student belonged to group A (reasoned and controlled calculation preserving the equivalence of expressions). The others students belonged to groups B (15 students who can calculate correctly expressions but without using semantic rules) and C (4 students who do not understand the calculus on algebraic expressions). M2 proposed to his students the three routes mentioned above, in the same order. Due to the fact that most of his students were in group B (15/20), M2 chose to give the same tasks to the whole class. After working on the three routes, the 20 students completed a written test on equations. We chose to focus on two tasks from this test to present our results. The first task was about solving three first-degree algebraic equations. Depending on the equation they solved, 7 to 11 students among the 20 students found the correct solutions. We particularly studied how many students used an algebraic technique. We observed that 17 out of 20 students solved the equations using the equivalence of equations. Even if they did not find the right solution, they had a strategy and transformed the equations in order to "eliminate" the unknown; they respected the concept of equivalence to do so. For the second task, equalizing two calculation programs (as presented above in table 5), 11 out of 20 students succeed for putting the problem into an equation.

Discussion

The Pépite assessment tool, based on an epistemological reference of the algebraic domain, allows the teacher to identify students' consistent reasoning and calculation in order to plan differentiated courses adapted to grade 8 th students' learning needs for the domain of equations. The mathematics routes tested in our experimentation seemed to have effects on the students' technological level: most of them used algebraic techniques to put a problem into equation. But this experimentation only concerns one teacher. So, in the ERASMUS + project "Advise me" which has just started in September 2016, we aim to carry out a larger scale research study.

We intend to validate these results for the field of arithmetic of integers for grade 3-4 pupils. [START_REF] Grapin | Étude de la validité de dispositifs d'évaluation et conception d'un modèle d'analyse multidimensionnelle des connaissances numériques des élèves de fin d'école[END_REF] carried out a multidimensional model of assessment for this new domain in elementary school. She defined an epistemological reference of arithmetic of integers to design an assessment tool in order to define pupils' profiles and to highlight the epistemological aspects of arithmetic to work according to pupils' learning needs. She organized an experimentation to study the evolution of pupils' profiles according to differentiated routes adapted to students' learning needs for the domain of arithmetic of integers. Data analysis is underway.

  obtained by successive trials. Algebraic techniques are necessary. For groups A, B and C Program A Program B Alex and Brenda choose the same start number. Alex tests the calculation program A and Brenda tests the program B. Then, Alex and Brenda find the same final result. Which start number did they choose? for motivating the production of and equation and solving it with an equation solver

Table 1 : Organization of the 8 th grade level test in terms of types of tasks Exercise 6: Proof and calculation program

 1 A student says to another student: "You will always find the same result if you take a number, you add 6 to that number, you multiply the result by 3, you subtract triple the initial number". Is this statement true for any number? Justify your answer.

	Types of tasks	Number of items Test items
	Calculus	4 / 22	7.1 / 7.2 / 8.1 / 8.2
	Producing numerical expressions 1 / 22	5
	Producing algebraic expressions	2 / 22	3.1 / 6
	Translation or recognition	14 / 22	1.1 / 1.2 / 1.3 / 2.1 / 2.2 / 2.3 / 3.2 / 4.1 / 4.2 / 9.1 /
			9.2 / 9.3 / 9.4 / 10
	Problem solving in different	1 / 22	6
	mathematics frameworks
	Justification
	Result
	The statement is true for any given number: true or false?

Figure 1: Example of generalization task

Table 2 : The multidimensional model of algebraic assessment (partial view)

 2

	Assessment dimensions	Assessment criteria
	Validity of response	V0: No answer
		V1: Valid and optimal answer
		V2: Valid but non optimal answer
		V3: Invalid answer
		Vx: Unidentified answer
	Algebraic writings produced	EA41: Incorrect rules make linear expressions a²->2a
	during symbolic transformations	EA42: Incorrect rules gather terms
		…
	Connections between a semiotic	T1: Correct translation
	register to another	T2: Correct but non optimal translation
		T3: Incorrect translation taking into account the relationships
		T4: Incorrect translation without taking into account the relationships
		Tx: No interpretation

Table 3 : A priori analysis for arithmetic strategies

 3

Table 4 : A priori analysis for algebraic strategies

 4

Table 6 : Task for working on semiotic conversions solver Experimentation in a grade 8 th class

 6

Pépite tools are available on LaboMep platform (developed by Sésamath, a French maths' teachers association): http://www.labomep.net/ and on WIMS environment (an educational online learning platform spanning learning from primary school to the university in many disciplines).Thematic Working Group 15Proceedings of CERME10

Proceedings of CERME10


domain: math shs

text:
Henrik P Bang

Niels Grønbaek
email: gronbaek@math.ku.dk

Claus R Larsen

Out-and in-sourcing, an analysis model for the use of instrumented techniques

We outline a model for analyzing the use of ICT-tools, in particular CAS, in teaching designs employed by 'generic' teachers. Our model uses the business economics concepts "out-" and "insourcing" as metaphors within the dialectics of tool and content for the planning of teaching. Outsourcing is done in order to enhance outcomes through external partners. The converse concept of in-sourcing refers to internal sourcing. We shall adhere to the framework of the anthropological theory of the didactic, viewing out-and in-sourcing primarily as decisions about the technology component of praxeologies. We use the model on a concrete example from Danish upper secondary mathematics to reveal what underlies teachers' decisions (deliberate or spontaneous) to incorporate instrumented approaches.

Introduction

Has use of computers in schools resulted in better education? With the steadily growing take-up of technology throughout the world, this question is as important as ever. The role and importance of technology has undergone phases from initial excitement to, more recently, a mixture of cautious optimism, moderate skepticism, and the stance that the use of computers might forfeit the true values of educational discipline. A recent, rather extensive international report (OECD, 2015) indicates countries' improvements in learning by a number of measures against their investment in ICT. The foreword summarizes the implications for educational policy:

Mere embracement of ICT in itself is at best harmless. Access to ICT does not automatically improve learning, "The results also show no appreciable improvements in student achievement in reading, mathematics or science in the countries that had invested heavily in ICT for education".

In person teacher-learner contact is essential, "One interpretation of all this is that building deep, conceptual understanding and higher-order thinking requires intensive teacher-student interactions, and technology sometimes distracts from this valuable human engagement".

There is a need for alignment of technology and learning: "Another interpretation is that we have not yet become good enough at the kind of pedagogies that make the most of technology; that adding 21st-century technologies to 20th-century teaching practices will just dilute the effectiveness of teaching".

A deeply rooted trust in the progressive power of ICT (but with a somewhat unimaginative scope to traditional learning material), "Why should students be limited to a textbook that was printed two years ago, and maybe designed ten years ago, when they could have access to the world's best and most up-to-date textbook?". A tall order on teachers to meet the expectations (by pedagogy rather than by subject discipline), "Perhaps most importantly, technology can support new pedagogies … it is vital that teachers become active agents for change" (OECD 2012).

Other meta-studies [START_REF] Higgins | The Impact of Digital Technology on Learning: A Summary for the Education Endowment Foundation[END_REF]MBUL, 2015) point to similar conclusions, "There is at most a weak positive correlation between bulk use of computers and learning outcome."

In contrast, there are numerous reports on very fruitful and insight-giving use of computers [START_REF] Böhm | The Case for CAS[END_REF][START_REF] Heid | Theories for thinking about the use of CAS in teaching and learning mathematics[END_REF][START_REF] Nabb | CAS as a restructuring tool in mathematics education[END_REF]. These are often the result of computer focused teaching designs that are part of didactic research or teaching development, carried out by dedicated teachers. Therefore, the question is not how much, but how, about what, and by whom.

The Danish landscape

In the OECD report Denmark is ranked second in use of computers. From the mid 1990's there has been rapidly growing CAS-use in Danish high schools, starting with graphing calculators and accelerated through the extensive use of PC's from around 2005. The situation now is that most high schools use Maple, TI-Nspire, Geogebra, and/or a CAS-tool specially developed for Danish high schools (WordMat, a CAS engine integrated within Microsoft Word). Students bring their own PC to the classroom, and use of PC is required at examinations. Initially, the transformation was carried through by progressive and CAS-curious teachers, many of whom were inspired by reform pedagogic ideas that supported a shift from abstract mathematics towards applications and more intuitive conceptual understandings. There was (and is) also an element of believing in diffusion: If CAS helps advanced (university) students to solve advanced problems, we might as well use CAS to help less advanced students solve less advanced (but to them difficult) problems. The educational system eagerly supported the development. Mathematics has been a vehicle for use of PC in other subjects, and examinations using CAS could (possibly) help more students achieving higher levels in math. Since 2005, CAS has gone from being a tool for enthusiastic teachers to a tool for everyone, including teachers with less interest and less competency in CAS. There has been no essential change in the standard curriculum (only minor ones allowing time for, say, 𝜒 2 -test)and standards for CAS use have not been introduced. On the contrary, the curriculum endorses the use of CAS in mathematical modeling and concept building, but without any indication of how, and in connection with what topics, to carry this out. In this landscape, many teachers have developed templates that students are allowed to use in exams, and the preparation of students to use these has become an important activity during normal lessons. Students of teachers, who for one reason or another disfavor such, may find templates on the internet or borrow from friends. Most of such templates have little epistemic value and a rather narrow pragmatic value in the sense of [START_REF] Artigue | Learning of mathematics in a CAS environment: the genesis of a reflection about instrumentation and the dialectics between technical and conceptual work[END_REF] towards solving (standard) problems. With CAS at the national tests these tendencies of trivialization are even more pronounced, as problems must be formulated to be equally solvable on different CAS platforms, Denmark's extended use of computers in education reflects of course a trend in society but is also as described above to a large degree the result of explicit educational policies. Hence, a teacher has to find his/her pathway through the affordances, constraints, possibilities etc. stipulated by official guidelines, curriculum and instruction plans. As indicated, successful use of computers does seemingly not scale up (MBUL, 2015). In order to understand the reasons for this better we propose an analysis model to help understand teachers' decisions on use of computers in mathematics teaching.

Theoretic framework

As our proposal aims at elucidating teaching in an institutional context we find the anthropological theory of the didactics (ATD, [START_REF] Chevallard | L'analyse des pratiques enseignantes en théorie anthropologique du didactique[END_REF] well suited. We start by briefly recalling the most important concepts of ATD that we shall use. Mathematics as an enterprise (educational as well as scientific) is seen as human activity composed of two blocks each with two parts, a praxis block comprising types of problems, tasks, with techniques to accomplish these and a theory part comprising technology and theory. Together such two blocks are termed a praxeology (praxis + logos). Tasks are the immediate goals of the activity, i.e. finding the slope of a graph of a function at a given point. A task can be accomplished by several techniques, i.e. plotting the graph on a computer, zooming in on the point in question until the graph appears linear and reading off the slope. The technology part concerns the discipline discourse of the technique and its relation to the tasks, i.e. the scope and limits of computer rendering of graphs (in relation to variation of functions). The theory part is a discourse on the technology part and its relation to the praxis block, i.e. on the concept of linear approximation that the sketched approach leads and on how it is related to a larger body of mathematical knowledge and practice, for instance that of the theory of differentiation.

We would like to stress a couple of points. A given task can be unfolded in many praxeologies. To choose, detail and organize such unfoldings is the essence of teaching design. Any praxeology has underlying praxeologies, i.e. praxeologies aimed at slope of a linear function, and is itself related to/part of other praxeologies. A praxeology always comprises all four parts. This is one key point of the analysis in [START_REF] Barbé | Didactic Restrictions on the Teacher's Practice: The Case of Limits of Functions in Spanish High Schools[END_REF].

Praxeologies take place within organizations of mathematical practice and knowledge. In ATD such organizations are formed by two components, a mathematical body consisting of a totality of objects, concepts, statements, interrelationships, procedures, etc., termed the scholarly knowledge and an institution of society within which this body is taught, manifesting possibilities and constraints for acquisition of learning. The passage from scholarly knowledge to its institutional version (which has more components than indicated) is in ATD conceptualized as didactic transposition.

In (Artigue, 2002, p. 271) it is noted that didactic transposition in its first version was described with respect to rather traditional scholarly mathematics. In order to underline a computerized setting Artigue has singled out the term instrumented techniques.

The model

In ATD the teacher is considered as the director of the learners' didactical processes [START_REF] Barbé | Didactic Restrictions on the Teacher's Practice: The Case of Limits of Functions in Spanish High Schools[END_REF], that is, responsible for the establishment of relations between learners and organizations of knowledge within institutions. We shall take this a step further, viewing it as production of learning outcomes through production activities, the praxeologies. For this production, the teacher has at his/her disposal a palette of resources, typically in terms of techniques (along with their theoretical block) to solve tasks. The 'employees' who use the techniques towards the production are the students.

This setup is very similar to a business economic model of the production of a corporation. In order to enhance the outcome a corporation director makes sourcing decisions on the allocation of resources. In modern terminology, one speaks of outsourcing, insourcing, backsourcing 1 , "outsourcing involves allocating or reallocating business activities (both service and/or manufacturing activities) from an internal source to an external source" (Schniederjans, Schniederjans, & Schniederjans, 2005, p.3). Insourcing can be viewed as an allocation or reallocation of resources internally within the same organization. Any business activity can be outsourced or insourced (dichotomy), but this decision is crucial to the success of the corporation. The basic idea of outsourcing is old, essentially, it is the dictum 'buy or make'. However, in the last few decades, outsourcing has grown almost explosively. A main reason for this is the development of ICT. But outsourcing is risky. It is reported (Schniederjans et al., 2005, p. 12) that half of all outsourcing agreements fail due to lack of appropriate analysis, and the necessity of strategic planning has become evident. There seems to be general acceptance (Schniederjans et al., 2005, p. 9) that such starts with an analysis to identify the strengths of the corporation, in terms of core activities ('core competencies' in [START_REF] Schniederjans | Outsourcing and Insourcing in an International Context[END_REF]. Loosely, a core activity is what the corporation does better than its competitors and possible outsourcing providers. Core activities must be insourced, non-core activities are candidates for outsourcing and a balanced decision to achieve the strategic goals must be made. Key advantages of outsourcing of inspiration for didactic equivalents are: focus on core activities, gain of outside technology, enhancement of capacity and lower cost, whereas some key disadvantages are loss of control, increased costs, negative impact on employees' morale and difficulties in managing relationships with outsourcing provider.

In the didactic version, the client is a didactically transposed knowledge organization along with the teacher(s) to direct the didactic processes. The outsourcing provider is an external knowledge organization, typically within a CAS. In the business model, external/internal refers to ownership. For our purpose the fundamental feature of ownership is that it allows for control of processes, i.e. outsourcing implies loss of control. We shall take this as the defining property. Hence outsourcing a mathematical activity means allocating it to a resource at the price of giving up control of processes.

A blunt example could be a teacher encouraging students to find solutions to homework on the internet; a more elucidating example is employment of instrumented techniques in the form of blackbox applications of CAS. As pointed out, any activity can be outsourced or insourced, that is full praxeologies, be it punctual, local or regional, or just parts of praxeologies, typically the praxis block. To be more precise, the starting point of CAS outsourcing is typically a problematic task to be solved by the outsourcing provider's technique thereby creating a transformed or new praxeology. We stress that a CAS such as Maple is not solely a provider. To the extent that a teacher exercises control over CAS processes, these are considered insourced. Outsourcing to CAS is a more restrictive concept than mere use of CAS. (Teacher control must be distinguished from student control as the latter is the result of the first, and perhaps of other competencies, acquired without the influence of the teacher.)

1 Backsourcing means reallocating tasks from external sources to internal. This could be in order to regain control of the production process but could also be imposed by outside regulatives. In an educational context such could be new stipulations of use of CAS at national tests.

Thematic Working Group 15

Proceedings of CERME10

A simple example (an object of many controversies) illustrates the concepts. Arithmetical computations require a careful analysis of what are core activities that accordingly should be insourced. Depending on (long-term) learning goals, these could be the systematics of paper and pencil algorithms, skills of mental arithmetic with "nice numbers", etc. On the other hand, multiplication of many-digit numbers is hardly a core activity and is therefore a candidate for outsourcing to calculators2 . This does not mean that tasks, which can be solved by mental computation, should not be insourced by calculator techniques. The point is that the core activity of mental computing may give control also of some calculator computations. Note that a calculator praxeology is completely different from its non-instrumented equivalents, for instance its theory part may involve representation of numbers in a finite memory.

The very decision to use CAS (or other instrumented techniques) involves, regardless of its specific use, outsourcing. The teacher has no control of the coding that underlies the CAS, the syntax, the defaults, the library of routines, etc. Most CAS-tools are developed with teaching in mind, at least partly. Perhaps most importantly, the CAS design may have didactic intensions, which the teacher may surrender to if not disable. Maple's 'clickable math' is a good example of this. The relationship between non-instrumented mathematics and computerized mathematics resembles that of a strategic partnership with mutual outsourcing. This relationship is dialectic in nature. The potential of CAS in mathematical praxeologies needs non-instrumented mathematics to be redeemed.

There is of course nothing new in the very idea of strategic planning. Mathematical activity has at all times involved use of 'non-controlled' components and didactic considerations have always had this as a condition. The modern aspect of CAS is the magnitude of impact, calling for a much clearer elaboration of such planning. The addition of the concepts of out-and insourcing to ATD offers a model for reflection on crucial choices between instrumented and non-instrumented praxeologies on basis on insight in the CAS-tool and in possible mathematical activities. On one hand, the model gives a framework for investigation of ordinary teachers' undertakings and perhaps more importantly, of what is not undertaken. On the other, it provides a strategic planning scheme for the teacher cf. (Schniederjans et al., 2005, Figure 1.3), where II+III are the crux of the matter: I.

Establishment of content and learning goals of the mathematical organization to be taught II.

Detailed analysis of subject matter and activities of possible praxeologies. III. Identification a. Core activities b. Non-core activities IV.

Sourcing decisions a. Core activities are insourced b. Non-core activities are candidates for outsourcing.

How do teachers decide on what is a core activity? The dialectics of pragmatic and epistemic value [START_REF] Artigue | Learning of mathematics in a CAS environment: the genesis of a reflection about instrumentation and the dialectics between technical and conceptual work[END_REF] seems inevitable, but is not directly reflected in the dichotomy of out-and insourcing. The computational power of several thousand digits, obviously to be outsourced, may have epistemic value in relation to approximation by decimal expansions. The pragmatic value of graphing of polynomials may be an asset of outsourcing in order to study whether polynomials have desired properties, which are considered epistemic of certain mathematical models. In other praxeologies graphing, by hand or by CAS, may be insourced.

Methodology for prospective work with the model

We aim at a fully-fledged model to give a general description of ordinary teachers' implementation of CAS and through this, an insight in the scaling-up question mentioned previously. Our first step is to analyze a rather extensive material of reports on teaching designs with CAS, succeeded by reflections on further development, modification and refinement of our model. These reports are produced by project participants at Center for Computer based Mathematics Education (CMU3 ), University of Copenhagen, as the last step of a reflective practitioner process. The mission of CMU is to support use of CAS in Danish high schools respecting core mathematics qualities in order to reverse the trivialization tendencies described above. The only condition for participating is a moral subscription to this mission. Thus, teachers have been free to choose subject, CAS platform (within CMU's coaching expertise), design of teaching, etc. This first round of analysis data is collected in contemplation of dissemination, rather than evidencing answers to research questions, but in a systematic way that allows for a grounded theory approach. 4Having an elaborated model, we intend a large-scale investigation on Danish mathematics high school teachers' choices and rationale for outsourcing to CAS.

A sketch of an analysis: a praxeology of finding derivatives

The so-called 3-steps method of finding the derivative, 𝑓′(𝑥 0 ), of a function is the canonical approach to differentiation in Danish high schools, explicitly mentioned in official guidelines. We recall: (1) With ∆𝑦 = 𝑓(𝑥 0 + ∆𝑥) -𝑓(𝑥 0 ) form the fraction ∆𝑦 ∆𝑥 ;

(2) reduce the fraction to make lim (if it exists). These are tasks in three praxeologies with rather separate theory blocks involving algebra, topology and geometry. In a CMUproject (Differentialregning, tretrinsreglen), the teacher, in the sequel L, wants to improve on students' understanding of the method by CAS-outsourcing "to give the students hand-on experience of secant and tangent slope and limits through experimentation with CAS sheets" (our transl.

). An outsourcing strategy like this is rather common in Denmark. L has worked on the teaching problematics of the 3steps method for many years 'without really understanding why students find it so difficult' (our translation from Danish taken from the project report). This time L starts with a thorough analysis of prerequisites ending in 12 points. L decides to use CAS in the case of 𝑓(𝑥) = 𝑥 2 on three points of the 12: '(5) computing slope of a straight line; ( 7) understanding what tangent and slope are; ( 12) understanding (the concept of) limit' (our transl.). A few observations: (A) L is by his very wish to understand reasons for learning difficulties led to in-and outsourcing considerations. (B) There is a tendency to regard pragmatic and epistemic values as separate features: ( 5) is pragmatic and ( 7

Further use and development

L is an example of a teacher with neither desire nor reputation to be a front-runner, but navigating resourcefully and dedicatedly under post-modern circumstances of mathematics teaching. Our observations (A), … (E) apply to many teachers (CMU, 2015; CMU, 2016) so the sketch of an out-/in-source analysis of L's project is testimony that our approach may have potential for shedding light of the kind of decisions, with shortcomings and potentials, that ordinary teachers make. The business metaphor seems confluent with natural praxis of resource considerations, thus providing a framework for large-scale investigations much similar to studies of business economics forces that govern trade and production. One may fear a risk of introducing yet another business corporation model to education. Outsourcing is growing in business due to the incitement of fierce competition. While perhaps tempting, a flat educational interpretation of this is misleading. The rooting in a business model is motivated within a local or regional mathematical organization through its level of didactic co-determination. Even though mathematics may be seen as a productive force, learning outcome is not a commodity. It cannot be bettered simply through optimization tactics.

Our sketch has focused on director decisions, i.e. the teacher's planning. Further development must include employees, i.e. students, that is, the last step of the didactical transposition: matter taught  matter learned.

  ) & (12) are (by L phrased as) aiming at epistemic value. (C) Some core activities are recognized as such and insourced, i.e. algebraic reduction of polynomial expressions such as ∆𝑦 for 𝑦 = 𝑎 𝑥 2 + 𝑏 𝑥 + 𝑐 -partly insourced to paper and pencil, partly to CAS. Other core activities are outsourced, i.e. use of sliders on the graph 𝑦 = 𝑥 + 2 to find lim 𝑥→0 𝑥 + 2, as last step in the 3-step method with 𝑥 replacing ∆𝑥; (D) non-core activities are not spelled out. What is it that sliders can do for secant-tangent considerations without sacrificing core activities? (E) Affective aspects are outsourced: 'CAS tools should … activate students and challenge their desire to … explore mathematical problems' (our transl.) From L's reflections, it appears that the outsourcing (D) is indirectly motivated by the textbook treatment of the subject. Textbooks rarely have core activity considerations, but rather bold instigations to CAS use. This risk of dilution of mathematical competency is pinpointed by the concept 'outsourcing core activities'.

Proceedings of CERME10

A business equivalent of the historically initial excitement about the freeing potential of calculators and the afterthought concerning (permanent?) loss of core activities: The reservation system of the flight company TWA was superior to those of its competitors, i.e. a core activity, but was outsourced in the 90's. TWA never regained its market share and went out of business in 2001.Thematic Working Group 15Proceedings of CERME10

The Danish Industry Foundation, Department of Mathematical Sciences at University of Copenhagen, The Danish Ministry of Education, and Maplesoft Inc. sponsor CMU.

For further details about the CMU material, we refer to the CERME 10 poster of TWG 15[START_REF] Bang | Coaching and engaging. Developing teaching with CAS in High School[END_REF] Thematic Working Group 15 Proceedings of CERME10


text:
Maha Abboud

Janine Rogalski
email: rogalski.muret@gmail.com

Real uses of ICT in classrooms: Tensions and disturbances in mathematics teachers' activity

Keywords: Teachers, technology, activity, tensions, disturbances, dynamic environment

This paper presents an extension of approaches of the teacher technology-based activity, articulating the Double Approach alongside with the Instrumental Approach within the overarching frame of Activity Theory. Tensions and disturbances are defined for analysing the dynamics of the teacher's activity when ICT tools are mediating both teacher's and students' activity. The approach is illustrated throughout a comparative study of two "ordinary" teachers using dynamic geometry.

Various tensions related to the temporal, cognitive and pragmatic dimensions were observed, differently managed depending on personal, material and social determinants. Tensions are inherent to the dynamics of the situation. Together with disturbances, they are lenses contributing to a finegrained analysis of teachers' activity.

Introduction

The activity of "ordinary" teachers integrating technology into teaching is constrained and depends on several determinants, namely personal, institutional and social. The work of researchers such as [START_REF] Ruthven | Towards a naturalistic conceptualisation of technology integration in classroom practice: The example of school mathematics[END_REF], [START_REF] Drijvers | The teacher and the tool: Instrumental orchestrations in the technology-rich mathematics classroom[END_REF] and [START_REF] Abboud-Blanchard | Teachers and technologies: shared constraints, common responses[END_REF] emphasize the need to study the practices of these teachers, often not technology experts and practicing in non-experimental conditions(i.e. ordinary practices). One of the aims of such studies is to better understand what happens in the classroom and thereby to address professional development issues [START_REF] Clark-Wilson | A methodological approach to researching the development of teachers' knowledge in a multi-representational technological setting[END_REF]. The aim of the present paper is to contribute to this research line by introducing two new theoretical concepts, tensions and disturbances. These concepts were developed within a model of instrumented activities of teacher and students and were actually used as complementary resources within ICT teacher education programs.

We consider the teacher as managing an "open dynamic environment" [START_REF] Rogalski | Y a-t-il un pilote dans l'avion?[END_REF], and we focus on both the relationship between the lesson preparation and its actual implementation (anticipation, adaptation); and also on the management of the inherent uncertainty within such an environment. Indeed, the use of technology adds a "pragmatic" dimension emphasizing the "open" character of the environment that constitutes the classroom activity. [START_REF] Monaghan | Teachers' Activities in Technology-based Mathematics Lessons[END_REF], stresses that this use leads to an increased complexity in teachers' practices and also that the uncertainties related to students' mathematical activities with technologies bring teachers to modify their objectives during the lesson in progress, leading them to focus on new "emergent goals". The concepts we introduce enable an Thematic Working Group 15 Proceedings of CERME10 analysis of the impact of the dynamics of students' interactions with technology tools on the management of the planned (by the teacher) cognitive route [START_REF] Robert | A cross-analysis of the mathematics teacher's activity. An example in a French 10th grade class[END_REF], and the possible divergences from this during the lesson. In this paper, we provide an example of the comparative analysis of the activity of two "ordinary" teachers' uses of dynamic geometry with their (6 th grade) students to describe the methodology and associated analytical tools and to highlight their usefulness. We selected this particular example from our research data as it is relatively easy to present in a short paper.

Theoretical and methodological approaches

The ergonomics theoretical perspective considers teaching as a case of dynamic management of the teaching environment [START_REF] Rogalski | Y a-t-il un pilote dans l'avion?[END_REF]. This environment is "open" as it contains many uncertainties due to the fact that the students' activity cannot be completely predicted and the teacher is often in an improvisation mode. The teacher's conceptions of the mathematical domain to be taught, and of the relation students have to it, are subjective determinants of his professional activity. These conceptions condition the "didactical process" he wants his students to follow i.e. the planned cognitive route, alongside the management of the processes developed during the lesson [START_REF] Robert | A cross-analysis of the mathematics teacher's activity. An example in a French 10th grade class[END_REF].

Although the didactic scenario is familiar, the students' diversity and the specific context of the class introduce a factor of uncertainty. This uncertainty is exacerbated when students are working with a technological tool as the teacher may encounter difficulties to control the tool's feedback due to students' manipulations and to identify their emerging interpretations. Teachers often have to deal with tensions due to the presence of the tool and its role in the student's activity, and also its interaction with the mathematical knowledge at stake. Following Rabardel's Instrumental Approach (2002), technological tools can be viewed from both the teacher's and the students' perspectives. In both cases, the subject-object interactions are mediated by the tool. As Rabardel states:

Beyond direct subject-object interactions (dS-O), many other interactions must be considered: interactions between the subject and the instrument (S-I), interactions between the instrument and the object on which it allows one to act (I-O), and finally subject-object interactions mediated by an instrument (S-Om). Furthermore, this whole is thrown into an environment made up of all the conditions the subject must take into consideration in his/her finalized activity (Rabardel, 2002, p.42-43).

Nevertheless, the object of teacher's activity is the students' learning, whereas the object of the students' activity is the content of the task given by the teacher; their instruments based on the same tool are thus different. Figure 1, presents how these two instrumented activities are articulated within the dynamics of class preparation.

Tensions and disturbances

In our approach, we depart from the way [START_REF] Kaptelin | Activity Theory in HCI: Fundamentals and Reflections[END_REF] introduced the terms "tension" and "disturbance" when presenting the concept of contradiction central in Engeström's framework of analysis for how activity systems develop [START_REF] Engeström | Quand le centre se dérobe : la notion de knotworking et ses promesses. (When the center does not hold: The concept and prospects of knotworking)[END_REF]. These terms appear in their familiar use; emphasis being put on the analysis of contradictions in activity systems as main learning sources.

We do not define tensions as conflicts or contradictions. In the teacher's activity tensions are manifestations of "struggles" between maintaining the intended cognitive route and adapting to phenomena linked to the dynamics of the class situation. Some of these tensions might be predicted by the teacher and so he/she plans how to manage them. Others are unexpected and constrain the teacher to make decisions, in situ, that direct his/her actual activity.

Disturbances are consequences of non-managed or ill-managed tensions that lead to an exit out of the intended cognitive route. Disturbances happen when a new issue emerges and is managed while the current issue is not completely treated or when the statement of a new issue is not part of the initial cognitive route.

We consider here only tensions and disturbances related to the local level of a class session; while some tensions are or might be managed at a more global level (i.e. over several sessions). Figure 2 illustrates how tensions can be related to different poles of the system of teacher-and-student activities; they can be shaped differently along three dimensions (previously introduced by Abboud-Blanchard (2014)): temporal, cognitive, and pragmatic.

Tensions related to the cognitive dimension appear in the gap between the mathematical knowledge the teacher anticipated would be used during task performance and the knowledge that is actually involved when students identify and interpret feedback from the instrument. Tensions related to both the pragmatic and cognitive dimensions are produced by the illusion that mathematical objects and operations implemented in the software are sufficiently close to those in the paper-and-pencil context (we refer to [START_REF] Balacheff | Didactique et intelligence artificielle[END_REF] analysis of the "transposition informatique"). Tensions related to a temporal dimension are frequent in ICT environments and are linked to the discrepancy between the predicted duration of students' activity and the actual time needed to perform the task. Teachers are generally aware of such tensions; they often manage them by taking control of the situation, either by directly giving the expected answer or by manipulating the software themselves. Finally, a tension non-specific to ICT environment may concern the didactical contract: Students cannot identify the type of answer the teacher is expecting. ICT environments may amplify this type of tensions when students are uncertain of the goal of the activity i.e. is it about a mathematical object to manipulate with the software or about the use of the software itself.

Figure 2. Tensions and disturbances within the dynamics of class management

Illustrating the theoretical approaches through a comparative case study

We present how tensions and possible disturbances appear in the case of two teachers, Alan and Colin, using dynamic geometry software (Geoplan) with 6 th grade students to introduce the notion of perpendicular-bisector. They are both "ordinary" teachers who use technological tools occasionally, and willingly, in ways that are in line with the institutional expectations, that is to introduce students to an experimental approach. The two teachers designed the same cognitive route based on the succession of two tasks: moving several points (eight) on Geoplan screen in order to place each of them at the same distance from two fixed points, M &N, (ICT task) and then similarly drawing 8 points with the same condition in a paper-and-pencil context (p&p task). Each teacher's final goal was to: give the definition of the perpendicular-bisector as a set of points equidistant from two given points; and establish an efficient associated construction method using compasses. Alan's school is in a low-income socio-economic zone, while Colin is in a middle-class zone. Their working environments are different: Alan had access to a traditional classroom and a computer room that lacked either a video projection device or a black board, while Colin worked in a classroom equipped with laptops.

The sessions included in our analysis were video recorded by the teachers themselves. Our choice of data collection approach is to reduce as far as possible the impact of researchers on the teacher's and students' activity in the class. The analysis of the teachers' preparation documents and deferred interviews enable the identification of some personal and social determinants. We then compare the observed succession of episodes in the video alongside the planned cognitive route, to enable us to detect tensions and disturbances.

Results

A somewhat surprising result is that both Alan and Colin managed the session without temporal tensions despite a number of "unfavourable" material and social determinants. In Alan's case, these could have resulted in strong tensions, e.g. the time needed to move from classroom to computer room and students' prior cognitive difficulties. In fact, Alan took into account the social determinants of his class and the material constraints by anticipating and avoiding tensions that could have produce disturbances through a threefold organisation: temporal, pragmatic and cognitive. Indeed, Alan closely supervised his students and organised their activity by structuring the cognitive route as a succession of well-defined sub-tasks. This mode of guidance has been identified previously as a common approach that teachers use to manage such experimental approach in order to avoid students' erratic behaviour [START_REF] Abboud-Blanchard | Teachers and technologies: shared constraints, common responses[END_REF]. The rhythm of sub-task completion is also strictly planned and guided. This is probably linked to Alan's personal determinants that led him to establish strong routines to discipline students in all moments. Indeed, not all teachers with this type of students are able to establish such routines and to be at ease when implementing them. Colin avoided temporal tensions in quite different ways. He started by presenting the task both with ICT and p&p. During the session, he used the IWB for sharing elements of the task outcomes with the whole class. He particularly drew students' attention to where they should look on the screen, and by doing so, avoided some pragmatic and cognitive tensions. Colin's open attitude may be related to a personal determinant of "compliance" inherent in his relationship with his students.

Regarding tensions related to the cognitive dimension, an important result is the shared illusion of transparency: Implicitly, Alan and Colin took for guaranteed that after completing the ICT task, all students would have detected the existence of a straight line on which all equidistant points are situated. This was clearly not the case. This tension was not managed as indicated by the absence of any collective comment concerning the point of transition between the ICT task and the p&p one. For some students, this fact led to a divergence from the intended cognitive route: a "local" disturbance.

These students persisted conscientiously throughout the whole session to draw equidistant points without appreciating the notion of a straight line as the set of such points. A tension related to the didactical contract was also observed in both classrooms when, during the p&p task, some students tried to place the 8 points at the same positions they occupied on the screen. What may have triggered this students' interpretation of the task differs for the two teachers. Alan had introduced the p&p task, by saying "now we will do the same task but without the computer". However, for Colin, the computers were not shut down and thus students may have continued to refer to what they saw on the screen. During the session, both teachers succeeded in managing this tension by explaining the differences of the two situations.

Finally, a pragmatic tension that was not managed was an implementation issue that could be related to a shared belief among teachers that students are skilled with technology using a trial and improvement approach. At the beginning of the ICT task, for any given point (P) on the screen, the students could read the relative distances to the point M and N and, when moving this variable point, they could observe the numerical changes. However, dragging the point P to maintain the equality involved two degrees of freedom on the plane. Therefore, an efficient approach relied on both students' awareness of this constraint and their development of an adequate concept-based strategy to "maintain a constant dimension when moving along the other" or "anticipate the curve on which the point is moving" (as shown in [START_REF] Abboud-Blanchard | A lens to investigate teachers' uses of technology in secondary mathematics classes[END_REF]. In Colin's class, some students continued to drag points without any strategy even when the teacher asked them to engage in the p&p task. As a consequence, they could not easily be aware of the efficiency of using compasses instead of rulers when switching to the p&p context. For Alan's students, we can infer that some of them succeeded to place only few points with limited opportunities to notice their alignment.

Overall, we have identified a set of tensions in the activity of two ordinary teachers use of dynamic geometry and have illustrated the dimensions they may affect. By predicting many tensions, these teachers avoided some of these tensions by anticipating and organising the students' activity relative to the temporal, cognitive and pragmatic dimensions. The teachers managed others in situ mainly via individual interactions with, and support for the students. Their approach to classroom management depends on personal, material and social determinants. Nevertheless, the analysis also shows that even though they succeeded to maintain the essence of their intended cognitive goal, some unrecognised or ill-managed tensions led some students to diverge from the planned cognitive route.

Discussion and conclusion

In this paper, we have presented how and why we developed the notions of tensions and disturbances to analyse the dynamics of teachers' activity in ordinary contexts when they are using ICT in line with the institutional demands. We first schematised how ICT occupies two different positions as an instrumental tool for the teacher and for the students. The first schema is based on the postulate that there exists a crucial difference in the object of activity of the teacher (e.g. students' understanding/learning of the mathematical content involved in the task), and of the students, which is essentially to complete the task. Within the second schema, we added and defined several types of

Figure 1 .

 1 Figure 1. Teacher's and students' instrumented activities within the preparation phase We now consider the classroom environment and present how the two instrumental situations are articulated within the dynamics of class management, indicating possible tensions and disturbances.

Proceedings of CERME10

tensions that may appear within the teacher's activity along three dimensions: temporal, cognitive, and pragmatic, and at the level of the didactical contract. The cases of Alan and Colin are examples of ordinary, experienced teachers investigated within a larger study, who use technology regularly.

In addition, they are convinced that dynamic geometry enables students to make pertinent observations through the immediate feedback (this belief is widely shared by teachers). We identified a set of tensions: some of which were managed through anticipation; others in real-time (depending on different personal and social determinants); some were not detected or detected and not managed by the teachers. If teachers are able to identify and manage tensions, they can maintain the intended cognitive route for students or, when disturbances occur, modify this route to different effect. Several issues remain open for further research. First, the present study focused on the "local level" -an analysis of a specific classroom session. It will be necessary to extend our concepts and methodological tools to analyse tensions and disturbances at the more global level of a sequences of tasks on particular mathematical topics. Secondly, we analysed cases with "simple" ICT-based students' tasks; other cognitive and pragmatic tensions could appear when the tasks involve objects and operations that are more deeply modified by the "transposition informatique". Finally, at the deeper level considered in [START_REF] Artigue | Digital technologies: a window on theoretical issues in mathematics education[END_REF], what different kinds of tensions would teachers encounter when students are engaged in mathematical activity involving mathematical objects (an epistemic orientation) compared with more specific computer-based tasks (a pragmatic orientation). Finally, we consider that the concepts of tensions and disturbances have enriched the range of theoretical tools to study teachers' instrumental activity, in particular for the identification and analysis of critical aspects of the dynamics of this activity. We conjecture that this could also inform approaches to teacher education.


text:
Alison Clark-Wilson
email: a.clark-wilson@ucl.ac.uk

Gilles Aldon

Iveta Kohanová
email: iveta.kohanova@ntnu.no

Ornella Robutti

Introduction to the papers of TWG15: Teaching mathematics with resources and technology

Keywords: Technology uses in education, mathematics instruction, teaching mathematics, teacher competencies, teacher behavior, computer assisted instruction

Introduction

For the second successive CERME two groups addressed mathematics education research concerning technology. TWG15 focused on issues concerning teaching, teacher education and professional development, whereas TWG16 focused on students' learning with technologies and software and task design issues (see Drijvers,Faggiano,Gerianou & Weigand,Introduction to TWG16 in this volume).

TWG15 engaged in work that was stimulated by contributions in the form of 19 research papers and 6 posters that had responded to the call, which had highlighted the following themes:

 The specific knowledge, skills and attributes required for efficient/effective mathematics teaching with technologies and resources.  The design and evaluation of initial teacher education and teacher professional development programmes that embed these knowledge, skills and attributesto include programmes that involve teachers' working and learning in online communities.  Theoretical and methodological approaches to describe the identification/evolution of teachers' practices (and of effective practices) in the design and use of technology and resources in mathematics education.  Theory and practice related to the formative/summative assessment of mathematical knowledge in a technological environment.

The work of TWG15 drew upon research from 17 countries: Australia, Austria, Denmark, Faroe Islands, France, Israel, Italy, Lebanon, Germany, Greece, Norway, Palestine, Spain, Sweden, Turkey, United Kingdom and USA.

TWG15 themes

The contributions to TWG15 were grouped according to the following themes: large-scale professional development through online courses; technology-mediated assessment of students' mathematical learning; establishing quality criteria for digital mathematics tasks; understanding teacher perspectives on technology use; in-service teachers' knowledge and practice; pre-service teachers' knowledge and practice; and the advancement of theories on technology use in mathematics education.

The TWG15 was organised as follows:

1. A research paper by Kimeswenger was selected to be the focus of a single TWG session as it highlighted a new issue for the TWG, which was the development of quality criteria to support the selection of (dynamic) digital resources for teaching mathematics. 2. Two symposia, to address the themes: on-line large-scale professional development courses; and digital assessment of students' mathematical learning. These included selected papers that were presented by the main author, followed by an invited reaction by one participant. 3. The remaining papers were grouped by theme and presented as individual short presentations by the main author, followed by individual reaction by another invited participant.

In all cases, the discussion was opened to the whole group (in small groups of 6-8 participants), which provided the opportunity for explicit links to be made with the topics of the poster submissions by both the TWG leads and participants, and to encourage all participants to share their own knowledge and experience during discursive work. Brief feedback from these small groups was collected at the end of each session.

Large-scale professional development: Online courses

The papers by Hohenwarter et al., Taranto et al. and Panero et al. focused on the design and early evaluation of three large-scale professional development online courses that had been designed for participants from Austria-Romania-Turkey, Italy and French-speaking countries respectively. Central to all three courses was the objective to offer practicing teachers an opportunity to develop their uses of technology in mathematics classrooms. These courses were described as either 'open online' (OOC) or 'massive open online' (MOOC), where the word 'massive' implied that there were no geographical boundaries nor limits to teachers' registration and participation, although the language of the course was a limiting factor.

The three courses used theoretical frames in different ways. Hohenwarter et al. adopted Koehler and Mishra's Technological, Pedagogical and Content Knowledge model (TPACK, Koehler & Mishra, 2005) to inform their course design. Panero et al. and Taranto et al. sought to network the theories of meta-didactical transposition [START_REF] Arzarello | Meta-Didactical Transposition: A Theoretical Model for Teacher Education Programmes[END_REF], documentational genesis [START_REF] Gueudet | Towards New Documentation Systems for Mathematics Teachers?[END_REF] and communities of practice [START_REF] Wenger | Communities of practice. Learning, meaning and identity[END_REF] to understand the collaborative work of teachers in the online context as seen through their productions.

The invited reaction given by Bretscher stimulated a discussion that raised the following issues: defining and understanding 'participation' within open online courses; specific design features of (M)OOCs for teachers of mathematics and the balance between technological and mathematical content; the appropriateness of (M)OOCs for the (large-scale) professional development of teachers; and how research methodologies might need be developed to assess the impact of (M)OOCs on teachers' classroom practices.

Assessing students' mathematical learning

The papers by Sikko et al., Chenevotot-Quentin et al. and Olsher & Yerushalmy were centered on the use of technology in classes by teachers and students, for activities and assessment. Even if referring to different school levels, they focused on the technology as a means to support teachers' assessment activities. In the first case, Sikko et al. presented the use of motion sensors in the Norwegian primary school classroom to support pupils' construction of meanings for functions and their graphs. This work was set in the context of a large European project (Formative Assessment in Science and Mathematics Education, FaSMEd) aimed at researching the use of technology for formative assessment. In the second case, Chenevotot-Quentin and colleagues showed the use of a technological tool for the assessment of lower secondary school students' learning of numbers and equations. Their technological tool is applied in a way that is consistent with an epistemological analysis of the topics and with the theoretical perspective of the Chevallard's Anthropological Theory of Didactics (ATD, [START_REF] Chevallard | La transposition didactique[END_REF]. Olsher & Yerushalmy presented a platform where students respond to geometrical tasks using a dynamic geometry environment, which are then classified within the platform according to their geometrical dynamicity. From the teacher's perspective, the three papers presented in this session engaged the participating teachers with professional considerations in diverse and deep ways: as designers of tasks for assessment, as teachers while teaching; and as observers of students. The papers highlighted the importance of teachers' and researchers' collaborative work in the design and evaluation of such resources for the classroom.

The discussion at the end of the presentation, stimulated by the invited reaction given by Yerushalmy focused on: the potential impacts of online formative assessment on teaching; the nature of online mathematics tasks and their formative/summative assessment; possible theoretical frameworks to support design and evaluation; automatization of students' responses and subsequent feedback to students/teachers; high stakes testing; and issues of design.

Quality criteria for digital mathematics tasks

The paper by Kimeswenger problematized the existence of online platforms that host many thousands of user-generated digital resources for teaching mathematics, which presents a particular challenge for (other) teachers as they seek to locate suitable resources that meet their individual requirements.

The author described a project in its early stages that seeks to develop a research-informed set of criteria to support different methodologies for user-review. The research focused on the views of 'experts' with respect to the existence of quality criteria alongside their personal descriptions of the 'educational value' of digital mathematical resources, concluding eight quality dimensions. This has led to an exploration of the possible correlation between resources that are highly rated as other users have decided that they have a 'high-quality author' and those that are identified by users as containing 'high quality material'.

As anticipated, the TWG15 participants were most interested by, and animated to discuss, the issues raised by this paper, given that many had themselves been involved in the design of open educational resources or worked alongside teachers to try to support them to make thoughtful resource selections. This discussion concerned: the authors of quality criteria and the mathematical cultures/content/values on which such criteria might be based; the role of a consumer-led approach (i.e., 'likes' by teachers?) or a community-led approach; and, given the vast number of available resources, the usefulness of new algorithms that might automatically score 'quality', based on developed criteria

Technology integration: Understanding teacher perspectives

The two papers by Abboud & Rogalski and Bretscher both addressed aspects of technology integration into 'ordinary' secondary mathematics classrooms in France and England respectively.

Whilst Abboud & Rogalski analyzed videos of lessons at distance using an 'ergonomic' theoretical approach [START_REF] Robert | A cross-analysis of the mathematics teacher's activity. An example in a French 10th-grade class[END_REF] that highlighted tensions and disturbances in the observed practice, Bretscher used classroom observation and interviews to research aspects of a teacher's mathematical knowledge for teaching with technology. These two papers instigated a critical discussion within the working group that was revisited several times during the conference as TWG15 sought to understand, and question the use of the word 'ordinary' to describe teachers (and their classrooms) within research studies. For some this referred to experienced teachers who are dependent on their own ability to (re)design lesson with technology (as in Abboud & Rogalski). For others, it referred to teachers who are required to adapt their teaching to their situation and institutional constraints in a world of changing digital tools. There was a general agreement that teachers who are involved in research studies/projects/communities concerning the use of technology in mathematics were rarely 'ordinary'. One helpful description that was offered described the set of teachers who were not yet aware of their own instrumental genesis with new technologies (or that of their students), which seemed to resonate with many of the researchers in the TWG. The TWG15 participants concluded that 'ordinary' was an unhelpful descriptor and this highlighted the importance that researchers describe teachers' contexts more fully (i.e. country, teacher background, school system, school curriculum, etc.) to enable deeper and more critical insight into each other's research settings.

The paper by Kolovou & Kynigos differed from the two previous papers by focusing specifically on the learning processes of the designers of dynamic digital resources to foreground students' and teachers' mathematical creativity, which is fully described in their paper. By focusing on a 'community of interest' (which included teachers) that had been formed around the design of a particular creative book (c-book), the authors show how the participants' learning was stimulated by the boundary objects [START_REF] Fischer | Distances and diversity: sources for social creativity[END_REF] in the design process.

A focus on pre-service teachers

The papers in this theme offered different approaches to pre-service teachers' training and the different interpretations of their required knowledge about technology.

Prodromou investigated the usefulness of a flipped classroom approach in tertiary education in Australia. The theoretical frame was that of the 'four pillars' that define a flipped classroom, which take account of the flexible environment; a shift in the learning culture; intentional content; and the role of educators (Flipped Learning Network, 2014). The analysis of an experiment with pre-service teachers was presented with a particular focus on the role of the lecturer in a flipped classroom. The study by Herrelko tracked the implementation of technology in a mathematics methods course for pre-service teachers in the USA. The method, based on the Apple Classroom of Tomorrow framework (ACOT, [START_REF] Dwyer | Apple classrooms of tomorrow[END_REF], sought to describe the necessary conditions for the development of pre-service teachers who are knowledgeable about instructional technology. Baya'a et al. focused on pre-service teachers' TPACK [START_REF] Koehler | What happens when teachers design educational technology? The development of Technological Pedagogical Content Knowledge[END_REF] and provided analyses of the impact of teacher preparation courses that had been shown to develop teachers' TPACK.

The main questions that these presentations highlighted, and were discussed by the TWG, are linked to understanding the pre-service teachers' perspectives in the design and implementation of mathematics with technology. Leading on from this, there is a need for deeper understanding of the required technological, mathematical, pedagogical, and epistemological knowledge that is essential for future teachers in order to prepare them to use digital tools effectively in their teaching.

A focus on in-service teachers

This theme concerned in-service teachers, their professional engagement in the various activities related with teaching: planning lessons; using technologies; working in communities; orchestrating different devices in laboratory activities for students; balancing laboratory activities and more traditional teaching. The presenters of the papers showed various aspects of the ways that teachers work with technologies, both related to their teaching practices and to the design and management of educational materials.

Tamborg shared research on the use of a platform in Denmark, Meebook, for planning mathematics lessons in accordance with the teachers' pre-determined learning objectives, teaching approaches and curriculum. The framework used for the study is the 'instrumental approach' [START_REF] Gueudet | Towards New Documentation Systems for Mathematics Teachers?[END_REF], along with the 'documentational approach' (Gueudet et al., 2012) to describe the teachers' collective processes in their use of the platform to plan their lessons. Kayali's study centered on an investigation into the uses of mathematics education software by English secondary mathematics teachers, to understand why some software is used more or less than others, in which ways and for which reasons. Again, the instrumental and documentational approaches are the adopted frames alongside the 'teaching triad' [START_REF] Jaworski | Investigating Mathematics Teaching: A Constructivist Enquiry[END_REF], for the collection and analysis of data on teachers' considerations when implementing tasks in mathematics lessons. Zender and colleagues showed a motivating way to support students' learning with technologies outside of German classrooms using MathCityMap, a geo-located application for smartphones, which is used as an instrument for a range of situated mathematical tasks. The collaborative professional development of teachers in Lebanon on the use of GeoGebra in mathematics classes was the theme introduced and discussed by Kasti et al.. They based their research on the frame of Valsiner's three zones [START_REF] Valsiner | Culture and the development of children's action: A theory of human development[END_REF] and the TPACK theory [START_REF] Koehler | What happens when teachers design educational technology? The development of Technological Pedagogical Content Knowledge[END_REF], using questionnaires and interviews to investigate how GeoGebra is introduced in various mathematics activities.

Advancing theories on technology use in mathematics education

The papers by Gustafsson and Grønbaek et al. both focused on advancing theories on technology use in mathematics education. Gustafsson investigated the potential of Ruthven's Structuring Features of Classroom Practice framework ( 2009) as a tool to analyze empirical data to conceptualize and probe teachers' rationales for technology integration in the mathematics classroom. Gustafsson's results showed that, whilst the framework captures most aspects of their rationales, it did not fully encompass teachers' justifications with respect to their students' attitudes and behaviors. Hence, he suggested the addition of a new (sixth) structuring feature that relates to teachers' craft knowledge of the use of technology to manage different types of student behaviors or attitudes.

Grønbaek et al. suggested the addition of the concepts of out-and in-sourcing to [START_REF] Chevallard | La transposition didactique[END_REF] Anthropological Theory of the Didactics. These concepts, taken from the field of business economics, are used as metaphors within the dialectics of tool and content in the planning of teaching. Their addition offers a 'production model' to support teachers' reflection on crucial choices between instrumented and non-instrumented praxeologies. This highlights the need for teachers to be able to identify core activities (with potential for in-sourcing) and non-core activities (candidates for outsourcing) based on the learning goals and the possible praxeologies when planning their use of technology in mathematics lessons.

TWG15 participants' reflections

During the final TWG session, the participants were invited to reflect upon, and record on paper, their own insights and learning during the conference. A textual analysis of the 31 responses highlighted the following aspects of knowledge exchange:

 Broader appreciation of theoretical frames and their uses: exposure to new theories; consideration of the limitations of theories; discussions of networking theories; and reflections on personal interpretations and applications of theories.  Deeper understanding of international contexts: theoretical traditions; institutional constraints (i.e. curriculum and examinations); the rejection of the concept of an 'ordinary' teacher, which seems difficult to define or to establish a common meaning across countries.  Widening of knowledge on emerging themes: opportunities afforded for large-scale teacher development through MOOCs and OOCs; the need for 'quality' criteria for digital resources to support their selection/uses by teachers; and the role of technology within formative and summative assessment that relates strongly to the mathematics curriculum, its values and traditions.

Some participants reflected upon the unintended consequences of the division of the technology group for the two different foci (teachers and students), highlighting that an opportunity was missed to explore 'How to develop a good framework to capture the interplay between mathematics content, technology (tools/resources), teachers and students/learners?'.

Finally, a few participants commented on the collegiality of TWG15, highlighting they had also been 'inspired' as they learned about the CERME spirit: 'humility -and how to express both orally and in writing with humility'.

Conclusions

The broad range of papers and posters presented at CERME 10 highlighted the diversity of research interests in the participating countries. However, many common concerns prevail. The design and implementation of programmes and courses for future and practicing teachers was one such challenge. Debate over the exact content of such courses in order to address the knowledge and skills to integrate technology into future mathematics teaching practice was paramount, alongside the modes of delivery and the integration of teachers' classroom experimentations. This highlighted the different views and perceptions of the simplicity and complexity for teaching mathematics with digital tools and the dilemma between technology appearing to make a teacher's role easier (e/g/ by automatically marking students' productions), whilst at the same time introducing new teaching challenges (e.g. by introducing new representational forms and related interactions). The TWG discussed ways to face such challenges through the development of research-informed teachers' collaborative professional development models that integrated coaching, face-to-face and online communities, often conducted or sustained over a time period of years, rather than months, drawing on the outcomes of the recent ICMI Survey on this theme [START_REF] Robutti | ICME international survey on teachers working and learning through collaboration: June 2016[END_REF].

A common challenge is the scaling of such professional development models, for which (M)OOCs might offer some solution, although substantial research is needed to evaluate the 'best designs' to respond to the many different cultural contexts and requirements. The networking of theories proposed by Panero et al. and Taranto et al. provided some potential theoretical and methodological tools to this effect.

The topic of the automated assessment of students' digital work led the group to question deeply the nature of mathematical activity (and mathematics itself) that warranted such assessment. This raised a general concern over the ease with which closed mathematical questions can be posed and digitally assessed and the much greater technical challenge to design automated assessment that privileged mathematical processes such as reasoning, justification and proof. Looking ahead to CERME11, TWG15 concluded the following questions, which might inform individual and collaborative research efforts over the next two years:

 Which theoretical frames and methodological approaches focus on aspects of the collaborative work of researchers and teachers within the context of the use of technology for teaching mathematics?  What approaches might be fruitful to raise teachers' awareness of the mathematicalpedagogical decisions concerning the design and use of technology for learning and its assessment?  How do we create opportunities and approaches that support teachers to appreciate and plan for the process of students' instrumental genesis?  In the design of technology-focused professional learning for mathematics teachers (pre-and in-service), what is the balance between professional needs across generic technologies and mathematics-specific needs? and how can this be achieved?

  The poster by Recio and the two papers by Olsher & Yerushalmy and Chenevotot-Quentin et al. respectively contributed greatly to this debate. TWG 15 critiqued advancement in theories concerning teachers' uses of technology in mathematics education. In particular, the notions of 'tensions and disturbances' as a theoretical construct to support analyses of teachers' practices (Abboud & Rogalski), alongside extensions to Ruthven's 'Structuring Features of Classroom Practices' (Bretscher & Gustafsson, and the poster by Simsek & Bretscher).

Proceedings of CERME10