Maike Braukmüller
email: maike.braukmueller@westermanngruppe.de

Angelika Bikner-Ahsbahs

Dirk F Wenderoth

Wenderoth Towards

Towards linking teaching, technology and textbooks

Keywords: Delphi technique, textbooks, digital tools, anthropological theory of the didactic

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Although digital tools are available for everyday classroom practice, researchers as well as teachers are questioning how the use of digital tools can improve teaching and learning [START_REF] Hillmayr | Digitale Medien im mathematischnaturwissenschaftlichen Unterricht der Sekundarstufe[END_REF]. Despite from the digital tool itself, two main factors are responsible for students' success: the teachers' integration of digital tools into classroom practice and a supplementary use of digital and traditional resources [START_REF] Hillmayr | Digitale Medien im mathematischnaturwissenschaftlichen Unterricht der Sekundarstufe[END_REF]. Teachers' integration requires change or innovation of current teaching practices. Research on innovation of teaching and learning shows that teachers are more likely to use digital tools if they address teachers' needs. Moreover, fruitful innovation in schools should rather be an incremental process building on current teaching practices than a revolutionary break (Bikner-Ahsbahs & Doff, 2019). These teaching practices are highly influenced by textbooks: on the one hand, textbooks specify the mathematical content; on the other hand, they shape the didactic style that teachers apply in their teaching [START_REF] Valverde | According to the Book[END_REF]. Thus, textbooks represent teaching practices as well as traditional resources. Taking this into consideration, to provide the potential for integrating a digital tool for learning in classroom practice, already the development of the tool should follow a user oriented design that supports students' learning and furthermore takes into account the textbook and teachers' practices and needs.

In the research and development project Multimodal Algebra Learning (MAL), the overall goal is to develop a digital algebra learning system (MAL-system) that supports the students' transition from arithmetic to algebra, however, this paper is restricted to linear equations. With the design of the MAL-system we want to overcome the disadvantages of using either digital or physical manipulatives by integrating physical manipulatives in a digital environment [START_REF] Reinschlüssel | Multimodal Algebra Learning: From Math Manipulatives to Tangible User Interfaces. i-com[END_REF]. The subproject MAL textbook focuses on (A) integrating teachers' perspectives into the design process of the MAL-system and (B) providing strategies for using the MAL-system facilitated by textbooks. This will be achieved by answering the following research questions:

(1) What kind of teaching practices do teachers consider when confronted with the concept of the MAL-system?

(2) What kind of needs for teaching do teachers address for the design of the MAL-system?

(3) What kind of criteria for accepting or rejecting the MAL-system do teachers express? (4) Which principles for integrating the MAL-system into textbooks can be extracted from ( 1)-( 3)?

This paper focuses on the methodical design of the MAL textbook study based on a Delphi study. It will show that this choice is fruitful to inform research and development of digital tools with teachers' practices. These practices are conceptualized in the subsequent theoretical framework.

Theoretical Framework

In order to consider established teaching practices in the design of the MAL-system and, beyond that, to develop guidelines to include the MAL-system into established textbooks, the theoretical framework should address institutionalized forms of teaching and learning. This is possible by considering a theory that captures these teaching practices, the Anthropological Theory of the Didactic (ATD). The ATD [START_REF] Bosch | Introduction to the Anthropological Theory of the Didactic (ATD)[END_REF]) models human activities by the concept of praxeology. According to ATD, human activity splits up into a practical and a theoretical component. The former is broken down into a set of types of tasks and a technique that tells how these tasks are or might be carried out; the latter is broken down into technology and theory. The technology explains how and why a technique works. The theory comprises basic assumptions and views supporting and justifying the technology. In many cases, it is hard to reveal the theory because it is usually taken for granted. The practical component is often referred to as the knowhow, whereas the theoretical component is the knowledge behind, in the sense of raison d'être. The set of all praxeologies of one person is called her praxeological equipment. In general a person's praxeological equipment emerges out of previous and current institutional settings including the people surrounding her.

Algebra tiles and the MAL-system

The MAL-System is developed based on algebra tiles [START_REF] Dietiker | Making Connections: Foundations for Algebra[END_REF]. Algebra tiles can be used for modeling linear and quadratic expressions and equations with integers. A small square represents the number one, a big square represents x², and a rectangle with the side lengths of the two squares represents the variable x. The three types of tiles usually have three different colors on one side and are all red on the other side. The red side indicates a "negative sign" (Figure 1 a). For modeling a linear equation a so-called equation mat with two distinct sides, left and right, is needed. By laying out the appropriate tiles, an equation is modeled (Figure 1 b). The equation can be solved by taking away or adding the same tiles on both sides and dividing both sides into the same number of equal sets. A subtraction zonean area where all the tiles within are supposed to be subtractedcan be introduced; either to mark the difference of sign and arithmetic operator or to express subtraction without negative Tiles (Figure 1 c). As the focus in this paper is on linear equations only, x²-tiles are not addressed in this paper.

Methodology and Method

In Germany, textbook authors are primarily in-service teachers. Therefore, they are familiar with established teaching practices in their schools and many of the teachers' needs for teaching. As these teachers author a certain textbook series in a team, they are also influenced by the teaching practices provided by this textbook series. Hence, their praxeological equipments are affected by their experience as teachers as well as textbook authors. This awareness of relevant praxeologies makes these teachers experts for the MAL textbook study to inform the design of the MAL-system through established teaching practices and relevant needs for teaching algebra. In order to gain main directions of consensus for adapting the MAL-system to these practices and needs, an expert study in the style of a Delphi study is conducted.

According to various definitions, any Delphi study comprises an expert group communication with several iterations and guided feedback to achieve consensus on a given question [START_REF] Linstone | The Delphi Method -Techniques and Applications[END_REF]. The MAL textbook study conducts three iterations combining group discussions and questionnaires building on one another (Figure 2). Three groups of authors, working on books for three different performance levels are involved (Math Alpha: comprehensive school (covers and combines lower secondary education, secondary education and high school education); Math Beta: special school; Math Gamma: high school). These groups are selected because they cover a wide range of school types at different performance levels in Germany. The first iteration includes group discussions of the teams Math Alpha and Math Beta. Due to organisational reasons the team of Math Gamma did not participate. The topics are algebra learning, digital learning, gamified learning, and the MAL-system, introduced by a concept video. The second iteration includes all experts to answer an online questionnaire readdressing the topics from the first iteration by open and closed items. For example, the high relevance of the balance model to the experts came up in both discussions. As this was unexpected, the questionnaire asked for advantages and disadvantages of it within the whole expert group. In the third iteration, comprised statements based on the answers in the second iteration are evaluated by the experts. According to the example with the balance model, this iteration reveals tendencies on the central advantages and shows the willingness to make compromises concerning disadvantages.

The group discussions are audio taped, transcribed and analysed in a sequential way [START_REF] Przyborski | Gesprächsanalyse und dokumentarische Methode[END_REF] based on ATD concepts to reconstruct the textbooks' and teachers' praxeologies, identify teachers' needs and principles for textbook integration. The data set of the second iteration is analysed by theory-driven content analysis [START_REF] Mayring | Qualitative Content Analysis: Theoretical Background and Procedures[END_REF] to capture individual praxeologies and needs based on the authors' experiences in school. The final data set is analysed by descriptive statistics to identify quantified relevancies of the praxeological elements from the iterations before.

Results

This section shows the three steps of data analysis applied to reconstruct the praxeological equipment of the expert authors. Presenting the results, we focus specifically on the balance model since the teachers have shown a strong commitment to it throughout the three iterations. Praxeological elements that came up while comparing algebra tiles and the balance model in the first iteration are presented. As the experts could hardly detach from the balance model while judging algebra tiles, the second iteration asked separately for advantages and disadvantages of the two models. The categorized range of experts' opinions extracted from data analysis is then presented. From the third iteration, the final evaluation of selected (dis-) advantages is presented.

The first iteration

In the first iteration, the teams Math Alpha (12 participants) and Math Beta (5 participants) took part in a group discussion each. The researcher gave an introduction on algebra tiles and the experts were invited to work with them. Afterwards, the experts were asked if they can imagine using algebra tiles in class. Both groups immediately refused to do so, giving the same technological 1 arguments. Similar to expert A10 2 : "[the balance model] is just easier for the kids to understand" (#236-237) 3 , teacher expert B2 said: "the balance model is much easier" (#76-77) and narrowed to "more illustrative" (#78) by B3. Another technological argument that came up in both groups in the beginning is that algebra tiles are "too abstract" (A10, #238) or rather "very abstract" (B3, #82). 1 The reconstructed praxeological elements are highlighted in italics. 2 The experts from Math Alpha are named AXX and those from Math Beta BXX.

3 (#236-237) refers to the lines in the original transcript that is not shown here due to the limited space.

Consensus

In the discussion of the Math Alpha team, A1 then mentioned that "the problem of the balance model is clearly that I cannot express negative numbers with it" (#239). This was a turning point in the discussion towards algebra tiles, validated by three more experts. A1 continued to point at a technique based on algebra tiles: "one can slide the […] xs that cancel out each other together" (#242-243). This was justified by the technology that "this […] also always helps" (#243). At this point, the Math Alpha group stopped comparing algebra tiles and the balance model.

The process in the Math Beta group was slightly different. B6 presented a "counter technology" concerning two different techniques. One was working with the balance model on the iconic level; the other was working with algebra tiles on the enactive level. Based on the technological argument that the enactive helps in real life and the iconic does not, B6 saw an advantage of algebra tiles versus the balance model. Based on that, B5 suggested a didactic technique combining the two models. She started with the core technology: "the equal sign […] means the balance" (#125-126) that was substantiated by the technology that the pupils should interpret the equal sign as a relational and not as an arithmetical sign. Based on that, her didactic technique related to the balance model began with using it, "this [equal sign] could maybe be introduced with the balance" (#126-127). But then B5 suggested to use algebra tiles "in the beginning really only using additive elements" (#128-129), before involving the subtraction zone or negative tiles. This was justified by the didactic technologies that the concrete action of laying out tiles is helpful and the subtraction zone "is a demanding conception" (#132).

This analysis shows that the experts' praxeological equipments on teaching linear equations is highly bound to the use of the balance model. The experts could hardly think of didactic techniques involving algebra tiles that fit to their theoretical arguments and moreover they could hardly detach their theoretical arguments from the balance model. Even the positive statements on algebra tiles were mostly connected to weaknesses of the balance model. To reveal theoretical arguments detached from comparing the two models we asked for advantages and disadvantages of the two models separately in the second iteration.

Second iteration

In the second iteration a questionnaire was sent to 37 experts. 20 of them answered (5 Math Alpha, 4 Math Beta, 11 Math Gamma) the questionnaire. The answers on the advantages and disadvantages of the balance model are summed up in categories in Table 1.

Advantages Disadvantages

Balance model We focus on the complementary findings in this iteration. As expected, the experts mentioned the core technology from the first iteration as an advantage: the balance model illustrates equivalence very well. In contrast, this is a disadvantage of algebra tiles, e. g. one expert from Math Alpha stated "[the] equivalence is taken for granted […]". This technology is connected to the technological argument: the balance model is intuitive for the students whereas algebra tiles are not.

The other way round, an advantage of algebra tiles is that negative numbers and quadratic equations can be illustrated. In particular, the lack of expressing negative numbers was mentioned as a disadvantage of the balance model several times. This also already occurred in the discussion of Math Alpha in the first iteration as a technique to favor algebra tiles.

The third iteration offers further clarity about the experts' preferences by rating the findings above.

Third iteration

In the third iteration, the 37 experts were again asked to answer a questionnaire. We received 19 answers (6 Math Alpha, 4 Math Beta, and 9 Math Gamma). Figure 3 These results emphasize the importance of the core technology substantiating the use of the balance model. More than 80% of the experts agreed that the balance model enriches teaching because it illustrates the equivalence very well. In contrast to that, about 50% of the experts rated algebra tiles as insufficient for their teaching because the intuitive access to equivalence is missing. Only 20% did not take this as a barrier for classroom teaching.

The strongest advantage of the algebra tiles over the balance model is the possibility to represent negative numbers. More than 50% of the experts evaluated this feature as enriching for their teaching, and only 10% disagreed on that. However, only one third of the experts evaluated the balance model as insufficient due the lack of negative numbers and as one half of them disagreed, the balance model was still rated positive on average.

All in all, the results show that the balance model is indispensable in current teaching practices, however, the lacking representation of negative numbers gives space for the algebra tiles to enter classroom practice.

Discussion

The didactic technique -teaching linear equations with the balance model -reconstructed in the previous section reveals insight into current teaching practices. Independent of the type of school, the balance model turned out to be the most relevant model for teachers when teaching linear equations. However, the experts were aware of the problems and limitations of the model, e. g. no negative numbers, special kind of balance is unknown to kids, limited possibilities to explore equations concretely, aspects also discussed in the literature (e.g. [START_REF] Vlassis | The Balance Model: Hindrance or support for the solving of linear equations with one unknown[END_REF]. Especially the missing representation of negative numbers points to a need in current teaching. The results from the third iteration suggest a complementary use of the two models allowing to address strengths of both models in a complementary way. We suggest introducing equations with the balance model by working enactively with the balance in class to make the students experience the equal sign as a relational sign of keeping two collections of weights in balance. The balance may then be readdressed in the MAL-system as a feedback symbol for equality when the students begin to work with algebra tiles. Starting only with positive tiles and additive equations, a balance in the middle, that is balanced or not (depending on the correctness of the actions carried out) instead of the equal or unequal sign is used.

As the authors theoretical and technological arguments on teaching linear equations are highly influenced using the balance model, it should not be replaced by the MAL-system. The balance model seems to be very fruitful for teaching in all the three types of schools in Germany since it provides intuitive access to the equal sign of equations and the equivalence relation. Addressing a current need like the representation of negative numbers which is missing in the balance model turns out to be a criterion for accepting the MAL-system. In addition, a careful integration of the MAL-system into textbooks could compensate the teachers' judgement of algebra tiles as being abstract. This could be done by creating a context that makes algebra tiles more accessible to students, for example by gamification or as an expanded digital environment for exploring problems. An important issue for increasing acceptance of the MAL-system is its correspondence to textbook praxeologies, as well.

Looking back to the methodology, the Delphi study with the three iterations has proven to be fruitful for investigating teachers' needs for teaching, their praxeologies and textbook praxeologies.

The discussions have led to three important insights: the teachers' strong commitment to the balance model; contrasting the balance model with Algebra Tiles has made the teachers sensitive to disadvantages of a given technique that is normally not questioned; it has shown that a negotiation process is needed to disclose hidden needs for teaching underlying the praxeologies. The second iteration has provided the possibility to readdress aspects not considered before like the balance model. The final questionnaire supplies a quantitative overview, showing possible differences according to the types of schools. In conclusion, the three steps of the Delphi study exactly fit the purpose of the MAL textbook study. As they are not tied to the topic of algebra, they unlock potential for further research on the role of textbooks in reform processes with a wider scope.

Figure 1 a

 1 Figure 1 a: algebra tiles; b: equation with negative tiles; c: equation with subtraction zone

Figure 2 :

 2 Figure 2: Procedure of the MAL textbook study

  shows the evaluation of the four complementary statements: (1) The balance model enriches my teaching because it illustrates the equivalence very well. (2) The algebra tiles are insufficient for my teaching because they do not illustrate the equivalence. (3) The use of algebra tiles would enrich my teaching because it can represent negative numbers. (4) The balance model is insufficient for my teaching because it cannot represent negative numbers.

Figure 3 :

 3 Figure 3: Ratings on complementary statements (BM: balance model; AT: algebra tiles)

Table 1 : Categorized results of (dis-) advantages of balance model and algebra tiles

 1

		 illustrative	 no negative numbers or quadratic
		 intuitively accessible	equations
		 enactive and iconic working	 hard to illustrate fractions
		 builds on students' knowledge	 more enactive working needed
		 prepares new knowledge	 transition to symbolic level problematic
		 illustrates equivalence (transformations)	 students do not know the balance
	Algebra	 illustrative	 abstract
	tiles	 negative numbers and quadratic equations	 linking of area and term is difficult

Acknowledgment

The research reported on in this paper is part of the ongoing project "Multimodal Algebra lernen" (MAL), which is funded by the German Federal Ministry for Education and Research in the grant program "Erfahrbares Lernen" (experienceable learning) (grant number: 16SV7550K).


Mats Brunström Maria Fahlgren
email: maria.fahlgren@kau.se

Project in preparation -Connected classroom technology (CCT) to enhance formative assessment in mathematics education

Keywords:

come

Background

It is well established that formative classroom practice in mathematics has a positive impact on students' learning and development. The widely used theoretical framework for formative assessment (FA) by [START_REF] Black | Developing the theory of formative assessment[END_REF] identifies five key strategies and three agents (teacher, learner, peer). The strategies are:

1. clarifying and sharing learning intentions and criteria for success; 2. engineering effective classroom discussions and other learning tasks that elicit evidence of student understanding; 3. providing feedback that moves learners forward; 4. activating students as instructional resources for one another; and 5. activating students as the owners of their own learning (p. 8)

The literature on formative assessment has emphasized the potential of technology to enhance FA strategies in mathematics education [START_REF] Clark-Wilson | Emergent pedagogies and the changing role of the teacher in the TI-Nspire Navigator-networked mathematics classroom[END_REF]. The European project Formative Assessment in Science and Mathematics Education (FaSMEd) investigated different aspects of the use of digital technology to promote formative assessment. Besides the five key strategies and the three agents introduced above, the FaSMEd framework identifies a third dimension consisting of three technological functionalities: (a) sending and displaying, (b) processing and analysing, and (c) providing an interactive environment. Interactive environment refers to a learning environment in which interactions, both between user and machine and between classroom participants, are based on mathematical exploration activities [START_REF] Cusi | Promoting formative assessment in a connected classroom environment: design and implementation of digital resources[END_REF].

In the planned study, we will investigate educational use of the type of technology referred to as Connected Classroom Technology (CCT). [START_REF] Irving | The impact of educational technology on student achievement: Assessment of and for learning[END_REF] defines CCT as "… a networked system of personal computers or handheld devices specifically designed to be used in a classroom for interactive teaching and learning." (p. 16). To date, several studies, primarily conducted in the US, have demonstrated that CCT facilitates several FA strategies. This literature emphasizes how the 'sending and displaying' functionality of CCT can be used. For example, teachers could use CCT to access more comprehensive information about students' mathematical thinking which could be used to guide immediate adaptation of subsequent instructions [START_REF] Shirley | Connected classroom technology facilitates multiple components of formative assessment practice[END_REF]. Moreover, by facilitating the display of different student solutions, CCT can support the teacher in engaging students in whole-class discussions in which they receive feedback, both from the teacher and their peers [START_REF] Cusi | Promoting formative assessment in a connected classroom environment: design and implementation of digital resources[END_REF], providing a forum for reflection on their own responses in comparison to their peers. These activities relate to the third, fourth, and fifth FA strategy. [START_REF] Cusi | Promoting formative assessment in a connected classroom environment: design and implementation of digital resources[END_REF] argue that this could also be a way to activate the first FA strategy by focusing on criteria for success when discussing different solution strategies. Concerning the functionality, 'processing and analysing', CCT often includes a classroom response system (CRS) which processes student responses and presents statistics, e.g. in the form of charts (e.g. [START_REF] Clark-Wilson | Emergent pedagogies and the changing role of the teacher in the TI-Nspire Navigator-networked mathematics classroom[END_REF]. In relation to the 'providing an interactive environment', [START_REF] Shirley | Connected classroom technology facilitates multiple components of formative assessment practice[END_REF] describe how teachers have used CCT to collect and analyze student data received by laboratory-based investigations.

Aim and study design

The aim of the proposed research and development activity is to develop design principles to guide formative teaching activities using CCT:

 principles for designing tasks that have good potential to make use of CCT facilities  guiding principles for teachers in monitoring, selecting and displaying student solution(s)

to use as a base for dialogues  guiding principles for teachers in their choice of interaction type (e.g. group discussion, student presentation, whole-class discussion) in different situations  guiding principles for teachers in the orchestration of various digital resources to enhance an interactive classroom in which all students are engaged These principles will be refined through successive cycles of design-based experimentation, across a range of teaching contexts. In each cycle, the current version of the principles will be made operational in curriculum materials and teacher guides designed to support the use of formative teaching processes within a particular course unit. Through testing across a range of contexts in each cycle, and across a range of topic units over successive cycles, the principlesand classroom techniques through which they can be operationalisedwill be refined.

The intervention to be developed and evaluated, then, consists of (a) the system of design principles to guide formative teaching activities using CCT, and (b) the operationalisations of these principles in the curriculum materials and teacher guidance for the (different) topic-specific course units designed in each cycle.


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

Richard Noss

Piers Saunders Celia Hoyles

Laura Benton

Laura Benton Key

Key Factors for Successfully Embedding a Programming Approach to the Primary Maths Curriculum at Scale

Keywords: Primary mathematics, computer science, teacher professional development, coding, programme evaluation

come   L'archive ouverte pluridisciplinaire

Introduction

In 2014 in England, the previous information and communications technology (ICT) primary curriculum was replaced by a new computing curriculum specifying mandatory content (but no pedagogical guidance) for the teaching of computer programming to all pupils (Department for Education, 2013). Some factors that contributed to Logo and other early programming initiatives not fulfilling their potential have been described in earlier work [START_REF] Benton | Building mathematical knowledge with programming: insights from the ScratchMaths project[END_REF], a key factor being identified by [START_REF] Noss | Windows on mathematical meanings: Learning cultures and computers[END_REF] as the importance of fostering a sense of teacher understanding and ownership of any programming innovation.

There have been significant technological developments since this early teaching of computer programming, with a number of block-based languages such as Scratch now freely available and widely used. These environments have helped to address some of the difficulties of mastering programming syntax, but there remains the challenge of ensuring that teachers first appreciate why they are introducing programming as part of mathematics -and then have opportunities to develop appropriate skills to teach programming.

The ScratchMaths (SM) 2-year intervention1 aimed to develop the mathematical knowledge of pupils (aged 9-11 years) through programming. The SM approach was to select and design activities around core computational ideas that would then be used as vehicles to explore specific mathematical concepts and promote mathematical reasoning. For example, the concept of variable is developed by enabling pupils to first explore the Answer block, initially on its own, but then in association with its companion block Ask in the context of a short script that draws polygons. Pupils encounter the limitations of the Answer block when confronted by a situation that needs the answer to contain two different values simultaneously, stimulating a need to introduce the new idea of Variable. This approach enables parts of computing to be taught within, or as a supplement, to mathematics lessons. SM initially adopted a design research methodology to produce pupil and teacher materials along with professional development (PD) to support the teachers to exploit the powerful ideas of computational thinking as a way to engage pupils in mathematical thinking (see [START_REF] Benton | Bridging Primary Programming and Mathematics: Some Findings of Design Research in England[END_REF] for a detailed account of the design of the study). SM is also being independently evaluated by another university through a randomized control trial (RCT, see Education Endowment Fund, 2016 for the detailed research design of the RCT). The first year's content (Y5 in England) focuses primarily on the computing curriculum and developing pupils' programming skills. In the second year (Y6), pupils utilise these programming skills to explore key mathematical ideas within the primary mathematics curriculum (place value, polygons and ratio and proportion). 111 English primary schools (6300 pupils) were recruited to the project by open invitation through networks such as the National Association for Advisers in Computing Education (NAACE) and regional information events. The schools were randomly assigned to the control and treatment groups at the school level. Schools did not pay to participate in the project and the teaching materials are freely available from the ScratchMaths website under creative commons licensing (www.ucl.ac.uk/scratchmaths).

The primary research question for the RCT conducted by the external evaluators was "What has been the effect of the intervention on the development of pupils' mathematical skills as measured by a randomised control trial?" However, in parallel to this the authors of this paper sought to research how the curriculum materials were used by teachers to seek to understand its effects. The research reported here concerns the latter study.

Theoretical background of ScratchMaths and its pedagogical design

The SM design was framed by constructionist theory whereby pupils would engage with the mathematical ideas by building programs to explore them. (See [START_REF] Benton | Bridging Primary Programming and Mathematics: Some Findings of Design Research in England[END_REF]. This constructionist approach was operationalized in a pedagogical approach structured by five key ideas, called the '5Es framework' derived from the theoretical underpinning of constructionism [START_REF] Papert | Mindstorms: Children, computers, and powerful ideas[END_REF] and described in [START_REF] Benton | Bridging Primary Programming and Mathematics: Some Findings of Design Research in England[END_REF], see also [START_REF] Benton | Building mathematical knowledge with programming: insights from the ScratchMaths project[END_REF]. These five unordered constructs are: Explore: Pupils learn from computer feedback. Pupils should have opportunities to explore different ways of dealing with constraints and ambiguity as well as investigating their own and others' ideas and debugging different types of errors. Through this exploration pupils should be encouraged to take control of their own learning and understand the reasons behind different outcomes.

Explain: Pupils use different modes of communication to articulate learning and the reasoning behind choices of approach. Pupils should have opportunities to explain their own ideas as well as answer and discuss reflective questions from the teacher and peers. Pupils should be encouraged to use the programming language as a 'tool to think with' and to support explanations of key mathematical ideas.

Envisage: Pupils predict outcomes of their own and others' programs with specific goals prior to testing out on the computer. Pupils should be given opportunities to consider program goals and the outcomes of different strategies before conducting their own exploration in Scratch. This should be balanced with activities which allow discovery through exploration.

Exchange: Pupils develop ideas through interacting and comparing with others. Pupils should have opportunities to share and build on others' ideas as well as being encouraged to both justify their own solutions and understand another's perspective on a problem. bridgE: Pupils make links between contexts beyond the Scratch programming environment and the mathematics domain by explicit re-contextualization and reconstruction within the language of mathematics.

Our overall aim in developing this framework was to chart a course for pupils in which they could begin to express mathematical ideas in Scratch and to provide guidance for teachers on the pedagogical strategies (which the English National Curriculum does not stipulate) that could lead to successful implementation of the SM intervention.

Professional Development

Prior to teaching each year of the SM intervention, teachers were offered two full days of professional development, spaced a few months apart. During these sessions, the teachers were introduced to Scratch and the SM curriculum content. The 5Es framework guided the design of these highly interactive sessions, with teachers given the opportunity to participate in activities that incorporated exemplars of these different pedagogical strategies. After experiencing for themselves the pedagogical approach, as modelled by the PD leaders, teachers were also explicitly introduced to the approach and the definitions of each of the constructs.

Curriculum Materials

The learning objectives of all activities within the SM curriculum are focused towards one or more of the 5Es. The connection with the different constructs is made explicit within the teacher materials, with a range of activities such as Scratch-based tasks that require pupils to explore within the Scratch environment and then subsequently explain and exchange their programs with the rest of the class as well as 'unplugged' tasks (away from the computer) which require pupils to envisage outcomes of particular programs and bridgE to their existing mathematical knowledge to calculate the correct inputs.

Findings Measurable constructs for the fidelity of ScratchMaths

Fidelity of implementation within an education context has been defined as "the determination of how well an intervention is implemented in comparison with the original program design during an efficacy and/or effectiveness study" [START_REF] O'donnell | Defining, Conceptualizing, and Measuring Fidelity of Implementation and Its Relationship to Outcomes in K-12 Curriculum Intervention Research[END_REF]. Using the criteria of program differentiation, the SM team evolved some critical components for implementation with fidelity of the SM intervention, leading to the derivation of quantifiable measures for the identification of high-, middle-and low-fidelity implementation. There is a tension between a high-fidelity implementation of the original designed intervention and the potential for the intervention to be adaptable and flexible enough to fit within a range of contexts, which could lead to higher rates of adoption and sustainability [START_REF] O'donnell | Defining, Conceptualizing, and Measuring Fidelity of Implementation and Its Relationship to Outcomes in K-12 Curriculum Intervention Research[END_REF]. Therefore, finding an appropriate measure of fidelity that identifies the critical components of an intervention that must remain unchanged but allows for appropriation to local context was important.

The fidelity of implementation was defined in terms of five school-level measures as: days of professional development attended; availability of computer technology; curriculum coverage; the amount of time spent teaching SM and the sequence of progression followed through the materials. The criteria are given in Table 1.

Fidelity criteria High Medium Low

Days of professional development Y5 and Y6 teachers each attended at least two days of PD.

Y5 and Y6 teachers each attended at least one day of PD.

Y5 and/or Y6 teacher had limited PD from a more experienced person away from the organised sessions.

Technology

Computers running Scratch 2.0 online or offline.

Minimum 2:1 pupil to computer ratio.

No classification

Computers running Scratch 2.0 online or offline.

Minimum 3:1 pupil to computer ratio.

SM curriculum coverage

Pupils taught at least some of the core activities across 5 different modules.

Pupils taught at least some of the core activities across 4 different modules.

Pupils taught at least some of the core activities across 3 or fewer different modules.

Curriculum time

Time spent teaching SM is at least 20 hours in Year 5 and at least 12 hours in Year 6.

Time spent teaching SM is at least 12 hours in each year.

Time spent teaching SM is fewer than 12 hours per year.

Progression

The order of modules and order of activities are mostly followed in general.

The order of modules and order of investigations are mostly followed in general.

The order of modules is mostly followed in general.

Table 1: The ScratchMaths fidelity criteria

All schools participating in the trial received online surveys to complete at the end of the 2015-16 and 2016-17 school years to collect information about SM implementation. 28 of the 55 'treatment schools' provided data to both surveys and are included in our analysis presented below. Whilst a more complete set of research data would have increased the validity and reliability of our findings, we could only work with the data that we received. Indeed, it is a finding that, despite their initial enthusiasm to participate in the project, with institutional participation agreed by the headteacher, approximately half of the schools did not respond to project communications. Hence the survey results of these 28 schools, alongside data triangulation by follow-up communications and school visits, were then used to classify the schools according to their fidelity.

School level fidelity

The spread of fidelity scores for the 28 schools is shown in Figure 1. Probing the survey and interview data further revealed a number of findings.

Professional development:

The school-level fidelity measure reveals high commitment to the PD. However, probing the data further revealed that 43% of schools were hampered by internal staff changes and teacher movement in and out of the schools (teacher 'churn'). So, in fact the teachers who were trained did not necessarily match with the actual classes being taught SM.

Technology access: Only one school was unable to provide the pupil:computer ratio of at least 2:1, which resonates with recent OECD data on the high levels of computer access in UK schools (OECD, 2015).

Coverage: Three quarters of the schools reported that they had struggled to cover the SM curriculum. This was particularly evident in Y6 where the demands of the high-stakes National tests are great, with the result that a huge proportion of class time was spent 'teaching to the test' and associated revision activities.

Curriculum time:

The allocation of curriculum time was challenging for about half of the schools, again more evident in Y6 due to pressures of the National test.

Curriculum progression: All of the schools were high fidelity in that they followed the order of the teaching modules and the activities within them. (In lower fidelity schools, teachers reported that they varied the sequencing or skipped activities). This analysis identified 15 schools that were high fidelity for all five criteria. From this group, an opportunity sample of 2 schools was selected for follow-up visits and semi-structured interviews with teachers and headteachers during the 2017-18 school year, focusing on three general questions to elicit respondent's experiences and perceptions of the SM curriculum: How and why they got involved in the SM project? How they implemented SM, and what were their impressions of SM? Both schools were in the same geographical (rural) area about one mile apart. The resulting data was used firstly, to triangulate the fidelity judgement for each school and secondly, it was coded in relation to the 5 Es as a means to establish a deeper understanding of the quality of the implementation. Arriving at a school level judgement for a project that had spanned two years proved to be complex due to the differing experiences, perceptions and memories of the people involved.

The demographic information for the two schools is given in Table 2 Although similar in overall size, the pupil demographic and attainment measures were notably different; one might have expected School A to achieve rather lower progress given the social background of the pupils. This difference is highlighted in School B's lower 'progress score', a comparative measure of pupil progress in similar schools. Both schools taught the SM curriculum in the teaching time allocated for the teaching of computing and none of the teachers in either school also taught mathematics to the same pupils. We now highlight aspects of the quality of delivery that were discernible from the research data, to provide deeper insight in relation to schoollevel implementations.

School A was confirmed as a high-fidelity school as it became clear that the two teachers involved (Peter and Carol) had worked in a highly collaborative way and had both embraced the SM curriculum, dedicating their planning and preparation time to becoming familiar with all of the pupil activities in advance of the lessons, as highlighted by the following response:

Peter: You couldn't just pick it up and follow it, and then expect the children to understand. And, then we thought, 'Well, if we have got to spend this amount of time getting our heads around it, it is a big ask for the children to do it in the same time'.

Concerning the 5Es, Carol referred to the ways in which she encouraged children to Explain,

Carol:

Once they got it, we kind of said, 'Well, try to show the person next to you, if they haven't,' and then sometimes, if they showed them one little bit, they were alright then…

Peter highlighted how the pupils Exchanged their ideas, Peter: And, I make a point at the end of sessions now, I leave 15 minutes before the end so that anyone who has done what I have asked them to do, essentially gets to share their work. And, they absolutely love it, whether they have not quite got it, or they have done more… … I put it on the big screen for them.

In addition, School A had made the SM materials locally accessible and created files of exemplar student work, which had supported alignment between the two teachers' pedagogic approaches.

The overall fidelity criteria for school B was also confirmed to be high as the two teachers described how one of them (David) had taught the Y5 curriculum to both classes and the second teacher (Tim) had done the same in Y6. During interview, Tim described his pedagogic approach thus, Tim:

We sort of gave them the option to explore the bits and for them to … trying to link it to the real world, to explain the steps behind it… why they have done what they have done, why they think it is may be more efficient or whatever. And as I say, I think we did the envisioning with a lot of the debugging ones [activities].

David commented on his use of the unplugged activities, highlighting how he encouraged pupils to Envisage, Explain and Exchange their ideas, David: I would put up, for example, one of the algorithms and I say, "Well, what would this algorithm do if we clicked on it," and they would talk it through with a partner and then… I could pop one up on the board and say, "Right, what would this do?" And, "If we changed this bit, how would that change the outcomes?" and things like that.

Final comments

It is hard to assess in such a large-scale study how far the implementation of SM was faithful to the designers' goals [START_REF] Fullan | The meaning of educational change[END_REF]. Many of these goals were generic, and we make no claim to originality on this score. For example, teacher 'churn' was both ubiquitous and destructive -quite frequently SM had not engaged fully with the PD. However, where we saw specific variation in fidelity, it tended to be related to the particularities of attempting a novel and demanding curriculum based on a somewhat daunting field of knowledgeprogrammingwhere teachers had little or no background. In such circumstances, the curtailment of teaching time due to the pressures of national testing was a strong impediment.

Clearly, some factors supported SM implementation: local access to the resources, exemplar student work and pre-written computer models. The excellent collaborative practice of School A that enabled them to overcome the challenges is noteworthy and echoed in other high-fidelity schools.

We noted little teacher adaptation at this first stage of implementation, but with more fluency and familiarity with the SM approach, this is likely to grow, leading, we conjecture to less emphasis on 'curriculum coverage' and more time to explore the mathematical ideas.

Figure 1 :

 1 Figure 1: Spread of fidelity measures (n=28 schools)

  .

	School	Number	of	Scaled score in	School progress	%	of	pupils
		pupils on roll		mathematics	in maths score 2	eligible for free
				(100 = National		school		meals
				expected score)		(Measure of socio-
						economic status)
	School A	420		102	Average		28.9%
					(-0.9)
	School B	424		101	Below average		3.6%
					(-3.1)

Table 2 : Overview of the schools 3

 2

https://www.ucl.ac.uk/ioe/research/projects/scratchmaths

A government-defined measure of school progress with respect to pupils' mathematical outcomes at age 11. https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/560969/Primary_scho ol_accountability_summary.pdf.pdf

2017 data taken from https://www.compare-school-performance.service.gov.uk

Acknowledgment

This research was funded by the Education Endowment Foundation.


Dimitris Diamantidis

Chronis Kynigos
email: kynigos@ppp.uoa.gr

Ioannis Papadopoulos

The co-design of a c-book by students and teachers as a process of meaning generation: The case of co-variation

Keywords: c-book, Documentational Approach, design of digital tools, co-variation

In this study we focus on the role of students as co-designers of digital resources for learning mathematics. Students with the use of c-book technology construct in collaboration with their teachers units of narrative blended with artifacts and constructions designed with digital tools for mathematics education, acting as designers. We see the design phase as a learning process for students, searching for new mathematical meanings that emerge around the concept of covariation. For our analysis we use the lens of Documentational Approach as a means of a deeper understanding of the learning process.

Introduction

Proposing answers on how to create didactic objects that may provoke reflective mathematical discourse in the classroom, fostering students to make significant mathematical advances and transforming grounded conceptual understandings about certain mathematical concepts constitute a generic aim of instructional design research [START_REF] Thompson | Didactic objects and didactic models in radical constructivism[END_REF]. The design of didactic objects entails making assumptions on students' and teachers' way of participating in this discourse. However, there is a productive tension between teachers' intentions and actions when they design tasks and the students' activity when they are involved in them [START_REF] Johnson | Mathematical tasks and the student: navigating "tensions of intentions" between designers, teachers, and students[END_REF], that could be studied in more detail. Research into the design and use of mathematical tasks accommodating students' intentions, actions and interpretations to the same extent as those of teachers' is fragmented. Focusing on the use of digital media in mathematics education, the students' agency during the learning process and its confluence on instructional design could become more visible; students along with mathematics teachers may share the role of co-designers of digital resources that can be used as didactic objects. In this report we build on previous research about students' activity with a 'c-book' technology, a new digital environment where storytelling is blended with mathematical tasks [START_REF] Papadopoulos | Meanings around Angle with Digital Media Designed to Support Creative Mathematical Thinking[END_REF]. But, now we make a step further, using c-book technology to put students in the role of the co-designer (the 'narrator') in the sense of building on the notion of design as a learning process and as a means to intervene in educational practice (Kynigos, 2015a). We focus on the question: How does the students' role as co-designer affect the mathematical meanings they make through the design process? In our research, we employed the Documentational Approach (DA) a framework hitherto used to understand and support teachers as they iteratively design and use resources in the classroom [START_REF] Gueudet | Towards new documentation systems for mathematics teachers?[END_REF], in order to understand classroom practice and student meaning making.

Theoretical Framework

The focus of this research is on the design of a digital medium as a learning process; a group of teachers and students jointly design such a resource for learning mathematics through the pages of a digital narrative. Assuming that the element of storytelling may provoke students' initiative, the design process could be a field of dense interaction between students and teachers. It could yield a series of digital artefacts (i.e. narrative meshed with constructionist digital 'widgets') as foregoing versions of the end product, along with discussion among the members of the group that refers mostly to argumentation about them, reconstruction and exchange of different versions of artifacts. From the constructionist point of view, this communication between teacher and students results to a learning collective where meaning making process occurs naturally [START_REF] Papert | Situating Comstructionism[END_REF]. In this sense, the digital medium to be designed is more than a product for the designers' group; it may be an object-to-think-with [START_REF] Brennan | Objects to think with[END_REF][START_REF] Kynigos | Designing Constructionist E-Books: New Mediations for Creative Mathematical Thinking?[END_REF]. Especially for students, the design process of this kind of medium could foster the generation of new meanings around mathematics.

To investigate this aspect in detail we have chosen the approach of DA [START_REF] Gueudet | Towards new documentation systems for mathematics teachers?[END_REF][START_REF] Guin | The complex process of converting tools into mathematical instruments: The case of calculators[END_REF] that describes the creation of new resources. Taking into account that DA has been used to study teacher's role as a designer of digital resources, we adopted it to study the students as they co-design resources with the teacher being part of the mathematics course. According to Documentational Genesis (DG) [START_REF] Gueudet | Towards new documentation systems for mathematics teachers?[END_REF][START_REF] Guin | The complex process of converting tools into mathematical instruments: The case of calculators[END_REF], a 'document' is the set of the existing resources the designer uses together with the scheme of use for these resources in a certain class of situation; i.e. when a mathematics teacher constructs a certain task for teaching functions with a digital medium (this is the class of situation), he/she attaches a scheme of use for a set of resources (the curriculum, another related task, a discussion with other teachers etc), shaping a document. In parallel, this task can be a new resource for him/her or others. In this study, we have students collaborating with teachers in the design of new resources, shaping new documents about mathematics blended with the element of narrative, which might affect the set of the existing resources. Teachers, at the same time have their own teaching agenda. Thus, in terms of instructional design, the documents that teacher and students jointly create can be seen as didactic objects, as well. Thus, we found DA useful to help us enhance our understanding of a process through which new mathematical meanings for students may emerge.

In this paper, we present instances of students as they generate meanings around the concept of covariation. Following [START_REF] Thompson | Variation, co-variation, and functions: Foundational ways of thinking mathematically[END_REF] approach of co-variational thinking we build on the hypothesis that co-variation emerges when students conceptualize a situation envisioning two quantities varying simultaneously and dynamically. Quantity refers to students' conceptualization of an object's attribute that can be measured. A quantity is varying in the sense that students envision the object having momentary states, thus its attribute has momentary values [START_REF] Thompson | Didactic objects and didactic models in radical constructivism[END_REF].

The Digital Medium

The c-book technology C-book technology affords the design and use of modules named c-book units. Each c-book unit consists of a narrative blended with diverse "widgets"; hyperlinks, videos and mostly instances of digital tools like GeoGebra, DME (Digital Mathematics Environment) and MaLT2 [START_REF] Kynigos | Designing Constructionist E-Books: New Mediations for Creative Mathematical Thinking?[END_REF]. DME provides also an authoring tool for the teacher to design new modules, while MaLT2 is a web-based Turtle Geometry environment which affords the design through Logo programming along with dynamic manipulation of 3D geometrical objects using sliders. GeoGebra and MaLT2 allow customization and personal construction of tools [START_REF] Healy | Charting the microworld territory over time: design and construction in mathematics education[END_REF] in the form of microworlds, fostering the meaning making process. The end users (mostly students) of a c-book unit can explore the narrative, experiment with the "widgets between the lines" and be involved in mathematical tasks. C-book environment also includes the "Workspace", a tool that affords asynchronous discussions between the designers or the users of a c-book unit organized in 'trees' (Fig. 1).

The Design of the Study

Jewelry c-book unit was jointly designed from scratch by a group of students and teachers, for the purposes of this study. The storyline in this c-book unit is about a dinky merchant named Sonier back in 1820 who got an inheritance after his uncle's death. However, the size of the uncle's assets was not clear, so Sonier got in a quest to detect the inherited goods. His adventure -full of math related riddles and playing cards -involved a trip from Berlin to his uncle's house in Vienna. His uncle's passion for gambling -in contrast to his love for mathematics-was well-known. Widgets were used by the designers' group to make up challenging tasks as parts of a riddle. A riddle was formed either by one task alone, or by a sequence of tasks, and the answer of each task was prerequisite for solving the next one and moving forward with the story. The plot was narrated through comic frames and text giving hints that might help the readers to solve the riddles (Fig. 1).

The group of designers consisted of three Grade-9 students in a public Experimental School in Athens (where teachers follow the same curriculum with other schools, adopting innovative teaching methods and/or conducting teaching experiments in collaboration with the University) and two teachers of mathematics. One of the teachers was the school teacher of the students and he additionally shared the role of the "participant observer". The communication between students and teachers while making the c-book unit, took place exclusively in a shared digital Workspace. The whole process lasted almost two months. Students and teachers were exchanging ideas and widget drafts in the Workspace in order to discuss and decide their inclusion to the c-book unit. Whilst the students were akin to the use of digital media, they did not have the competence to make changes in an already designed widget, so when they wanted to suggest a modification of an existing widget, they used to "draw a picture" to inform their teachers. The 68 posts uploaded in the Workspace and the produced c-book unit constituted our data. The analysis of the collected data took place on the level of identifying and analyzing critical episodes defined as selected segments of (discourse) activity with a single theme as a focus in the discussion among the design group members in the Workspace [START_REF] Kynigos | Teachers as designers of digital educational resources for creative mathematical thinking[END_REF]. The focus in these episodes lies on instances that show how the storytelling affected the emergence of new mathematical meanings around co-variation.

Results

The theme of "Jewelry" was used by the teachers as a sparker for the content of the c-book unit from the very first moment. As a result, one student posted in the Workspace a short story that triggered the whole group to make a decision about the main idea of the plot: The heir of a probably great but mysterious fortune. From the beginning of the discussion among students and teachers, the question "what kind of task should be constructed to be included in the c-book unit?" became the epicentre. Teachers (T1 & T2) proposed a couple of tasks, and posted them on the Workspace. The main issue for the reader was to fix a buggy-by-design diamond constructed in MaLT2 (Figure 2). The teachers' aim was to underpin learning through tinkering. Students (S1 & S2) expressed a discontent, saying that this could be boring. Instead, they made a suggestion: 1 S1: I think this could be a little boring. It seems better if we had riddles, 2 not pure math problems; like an orientation game we recently played.

S2:

Yes! And the riddles could be connected to each other; you must solve one 4 riddle before moving to the next, and go on with the story.

This was the designers' first goal, to construct a 'chain of riddles' connected to each other in a way that the narrative would be garbled if one or more riddles remained unsolved. The teachers posted suggestions on possible twists of the narrative, including clues about riddles, as an effort to find out what a proper riddle might be for students, while students made their own suggestions. Soon, it was obvious that riddles' selection would leave its mark on the narrative or vice versa:

S1:

The uncle could have hidden a different diamond in each room at his house. 6

Sonier should replicate the correct diamond using MaLT2. The challenging 7 thing is that Sonier should find the correct room number, by solving an 8 equation, or by calculating a value of a variable x (Figure 2). 5 T2: Why is it challenging? 10 S1:

It is more difficult to make these calculations, than having a number 11 ready to be used.

In the previous extract, S1 uses the word "variable" referring to a numeric expression (Fig. 2), maybe just because the value of x is not obvious or easy to calculate. Students made a suggestion about a riddle, where the challenge was to unlock a room, by calculating x's value which was the door's 'key'. In the next couple of days students reflected on their design, inferred with frustration that if someone solves the door' riddle once, then he/she had no interest in reading the c-book unit again, since he/she knows the answer:

12 S2:

The value of x must be different every time someone opens the c-book unit. 13 T1:

We could use a slider in GeoGebra. 14 S1: T1, in which way? S2's idea reminds me of a book I read. The story had many different endings. We could also do the same; the flow of 16 the narrative can vary in relation to the value of the variable x. This might be considered the ultimate aim of the designers; A c-book unit that would be interesting to be read repeatedly. Soon, they realized that it was not feasible to have a different answer for the riddle every time someone opens the c-book unit. Hence, there was a shift in the group's effort. They turned to the connection between the readers' choices and the alternate versions of the narrative. Teachers made up a microworld that afforded the variation of a line's position on the Cartesian plane, by the use of two sliders in GeoGebra, as a sparker for students to make a riddle. Therefore, the use of the slider came up again, as an option (Workspace):

17 T1: See the graph in GeoGebra. What changes when you move the sliders? 18 S1:

The position of the line.

S2:

We can relate the diamond's shape that Sonier asks for, with the line.

T1:

To what characteristic of the line? 21 S2:

The sliders change the angle between the line and x-axis. We could imply 22 through the narrative that this angle should be equal to diamond's angle.

Ιt seems that the designers came up with the idea of using the notion of angle as an element that would give a potential relation between two riddles. Thus, students made up a story in GeoGebra as the first riddle. There was a map of Vienna on a Cartesian plane with a line on it. The position of the line was determined by two sliders. The reader should put the line in a position of his/her choice, in order to make a path. Then, using the size of the angle between the line and x axis in degrees, as a hint, the reader should fix the buggy diamond in MaLT2 (Fig. 2) that had been proposed by the teachers at the beginning; this was the second riddle. The twist of the narrative was depended on the diamond that was constructed, since its shape was crucial for the storyline. It seems that the designers used the answer of the first riddle as an output magnitude necessary for solving the second one. When the content (microworlds and narrative) was ready to be used, students suggested making a trial on how it works focusing on the use of constructions:

S1:

We should play a couple of times, maybe more, to see how it works. 24 S2:

Yes, we should draw a line that changes smoothly and normally in order to coordinate its angle with the angle of the diamond.

So, the students tested the two microworlds many times, to observe if the variation of the line's position in GeoGebra had an elegant effect on the diamond's shape in MaLT2, through the common angle measure. Finally, they decided to use the same name 'phi' as the name of the variable for both the diamond's angle in MaLT2 and the angle in GeoGebra, too (the angles mentioned in line 25).

Discussion

For the analysis of the Jewelry's design, under the lens of DA we have distinguished four phases. In the first phase, although they shared the same class of situation, i.e. to design a c-book-unit, students and teachers seemed to have their own separate agendas. Teachers suggested an instance with a buggy-by-design figure of diamond in MaLT2 which mostly promoted the added value of learning mathematics through tinkering, while students overrated the amusement of the reader. Furthermore, students at first almost rejected the microworld as 'pure mathematics' (line 2), suggesting to use riddles 'like in a game they recently played' instead of mathematical problems. This reaction may be in line with the hypothesis that students' set of resources was mostly related to storytelling, than to the design of mathematical tasks. This became more evident in the second phase, where the designers tried to make up riddles linked to each other, which probably is a good description of the new class of situation for the group. Despite the consensus on that, there was no common understanding of what kind of riddles should be used. Students made up an intriguing story about hidden diamonds in locked rooms, trying in the same time to involve a variable x as the 'key number' to unlock the door. We assume that they made an effort to use resources with mathematical content, such as a narrative based on the use of a variable, representing it with x (line 8). However, this effort was fruitless, since they did not manage to operate the variation of x. In the sense of co-variational thinking, the dysfunctional use of x seems to be congruent to Thompson's view (2002); although a person operates successfully on algebraic representations, it is hard to talk about the represented relationship. In the third phase, the aim of the design was more apparent. The riddles should be constructed in a way to provoke the reader to read the c-book-unit many times, which as a class of situation was more specific. To construct this kind of riddles, teachers proposed an instance in GeoGebra where a line changed position through the use of sliders (line 13). The core idea was to relate each time the 'key number' of the locked door to a different position of the line; Unfortunately, this did not work. It seems that the relation between sliders' manipulation and the line's changing position was not a usable case of co-variation for the students, since the varying attributes of the line might not be clear in the momentary states of its variation [START_REF] Thompson | Didactic objects and didactic models in radical constructivism[END_REF].

Using this kind of instance as a design unit might not be feasible for the students according to their scheme of use for their set of resources. However, students made a contribution which determined the final product, the c-book-unit as a 'document'. They proposed that the c-book-unit should have alternate endings, in order to be provoking. This idea, came up from a resource related with storytelling, (i.e. a book read by S1, line 14), and was soon adapted by the rest of the designers as the main goal of their efforts, during the fourth phase. Therefore, the role of instances-riddles was to give to the narrative the characteristic of different twists of the plot. To achieve this goal students and teachers used the instance in GeoGebra, along with buggy diamond in MaLT2, trying to 'link' them. Thus, it seems that they conceptualized this linking as a situation of two objects (the diamond and the line) varying simultaneously and dynamically. This became clearer, when students tried to coordinate the manipulation of the line in GeoGebra with the diamond in MaLT2 (lines 23-25), so that a move of the sliders in GeoGebra should correspond to an 'elegant' and 'normal' change of shape of the diamond. This action seems to be in line with [START_REF] Thompson | Didactic objects and didactic models in radical constructivism[END_REF] view of co-variation that points out the importance of coordinating actions among two attributes with an 'operative image' of their co-variation so that one can act out how the two are related. In phase four, students' goal was to conceptualize and define a 'co-variational relationship' between two attributes; the position of the line in GeoGebra and the diamond's shape in MaLT2. It seems that students, by making use of the two sliders that represented angles' measures (one related to the line and another related to the diamond), they envisioned these two attributes' momentary states varying simultaneously (lines 17-22). Thus, they decided to use a common variable 'phi' related with both angles as an operative image of the co-variation between line's position and diamond's shape, linking the two riddles. According to [START_REF] Johnson | When a critical aspect is a conception: Using multiple theories to design dynamic computer environments and task to promote secondary students' discernment of co-variation[END_REF] students are facing fewer difficulties to discern co-variation among quantities (in the sense of Thompson) that can be measured with linear units. Building on this position, we assume that, in our case, students employed such a quantity 'phi' (that represented an angle's measure) to define and use a 'co-variational relationship' among two objects/attributes (the position of the line and the shape of the diamond) that could not directly be measured.

Conclusions

In this study, the focus was on the role of students as co-designers of a digital medium, a c-bookunit, i.e. a digital narrative blended with tasks and constructions with mathematical content. We used the DA theory to understand in detail the way new meaning around co-variation during the design process emerged. During the four phases of the design process the students' set of resources and their scheme of use seemed to be in flux. In the beginning, the use of resources had mostly to do with storytelling. Τhis changed progressively, and in the last phase a combination of narrative with mathematical resources was used. It was in this phase when, according to our analysis, students made meanings around the concept of co-variation, while 'building' a relationship for the needs of the storyline. It is remarkable that, after building this relationship, students tried to attune the mathematical content to the narrative, referring to the line in GeoGebra, as a path on a map. It seemed that initially, the differentiation between students' and teachers' resources and their schemes of use, worked as a barrier to the design process. However, from the analysis of the 'document' is evident that when resources were related to both narration and available mathematical tools, this led to the emergence of new meanings. From this point of view, it seems that when students act as designers using this kind of media, they take the role of both the storyteller and learner. Given the lack of research in this field, we believe that such enquiries should come up, assuming that these two roles -of storyteller and learner-should not be seen as separated, since the interaction between different kinds of resources was crucial for the design as a learning process.

Figure 1 :

 1 Figure 1: A c-book unit page on left hand side. The Workspace on right hand side.

Figure 2 :

 2 Figure 2: The GeoGebra widget-riddle on left hand side. The 'diamond'-riddle in MaLT2 on right hand side. Designers used these artifacts in the c-book unit to form a narrative with alternate endings. On top left hand side: the 'x' of line 8.


Anders Støle Fidje
email: anders.s.fidje@uia.no

Ingvald Erfjord
email: ingvald.erfjord@uia.no

Investigating similar triangles using student-produced videos

Keywords: Student-produced videos, inquiry-based teaching, mathematical discussions

The task of eliciting student talk in mathematics teaching can be daunting for teachers. An approach to this is using student work actively in teaching. In this paper, the teacher moves in a lower secondary full class discussion regarding similar triangles is investigated. The results show that the teacher utilised different teacher moves to steer the discussion towards both didactical and mathematical goals.

Introduction

The DIM-project (Digital Interactive Mathematics Teaching) is a developmental research project exploring the use of digital tools in three lower secondary mathematics classrooms in Norway. One such innovation is the use of student-produced videos (later termed SPVs) as a medium for presenting strategies related to inquiry-based tasks. This idea emerged from the teachers as they wanted to explore different uses of videos in teaching, both teacher-and student-produced. The three-part lesson sequence used in the analysed lesson was developed by both teachers and researchers in a workshop. The aim for both teachers and researchers was to develop an approach for eliciting student talk in full-class mathematics discussions.

The transcripts originated from a follow-up discussion related to similar triangles and the use of this idea to measure the heights of objects, the task and lesson sequence is presented in detail in the methodology section. While planning the discussion, the teacher viewed SPVs to identify aspects of interest in the students' work. The teacher also used videos while enacting the discussion. As such, the paper addresses the topic of using student thinking in mathematics teaching. The paper aims to characterise how the teacher used the SPVs in teaching to promote students' active engagement in a full-class discussion. Our analysis is based solely on video recordings of the teaching, as the authors sought to investigate how looking through the lens of teacher moves can inform how the teacher is positioning the students in the discussion. [START_REF] Scherrer | Investigating the relationship between "effective" teachers and theoretical notions of effective teaching: An analysis of whole-group discussions[END_REF] Research has demonstrated that effective instructional practices demand students' mathematical talk [START_REF] Walshaw | The teacher's role in classroom discourse: A review of recent research into mathematics classrooms[END_REF]. To achieve this, teachers need approaches to elicit students' thinking and promote mathematical discussion. An area of interest for the teachers and researchers in the DIM-project is to foster what Goos defines as a community of inquiry; a classroom "…where students learn to talk and work mathematically by participating in mathematical discussions and solving new or unfamiliar problems" (Goos, 2004, p. 259). This moves beyond being mathematically correct as the students are "…expected to propose and defend mathematical ideas and conjectures and to respond thoughtfully to the mathematical arguments of their peers" (Goos, 2004, p. 259). An approach to studying such discussions is analysing the moves used by the teacher to facilitate students' active engagement. Krussel, Edwards and Springer define a teacher move as "a deliberate action taken by a teacher to participate in or influence the discourse in the mathematics classroom" (Krussel, Edwards, & Springer, 2004, p. 309). In the following paragraphs, we introduce frameworks of teacher moves and actions used as guidance for the research presented in this paper. [START_REF] Scherrer | Investigating the relationship between "effective" teachers and theoretical notions of effective teaching: An analysis of whole-group discussions[END_REF] classifies twelve types of moves to investigate teaching practices in relation to theoretical notions of effective teaching. Scoring guidelines are used to code and evaluate the effectiveness of teachers' discussions, where each move is scored according to the surrounding moves. In this framework, the teacher moves that are prompted by student thinking generaly receive positive scores. Scherrer refers to five such teacher moves, categorised as rejoinders, used to elicit and use student thinking rather than evaluate it. The moves categorised as rejoinders are uptake, collect, connect, lot and repeat. Uptake is an open-ended question that invites student thinking while using a student response to extend, clarify, or elaborate the discussion. Collect is a move aiming to provide different student responses to the same question. Connect is a question related to different responses and how they may be connected. Lot are occurrences where the teacher acknowledges a student's response and indicates that the topic will be revisited in the future. Repeat is a move related to echoing a student's response. [START_REF] Mueller | Teachers promoting student mathematical reasoning[END_REF] perceive teacher interventions as crucial to introduce the students to the mathematical discourse. The authors state that "[t]houghtful interventions, implemented according to students' developing ideas, allow students to take ownership of their learning and solutions" (Mueller et al., 2014, p. 2). In order to investigate teacher moves, they present three categories which encompass student thinking; making ideas public, eliciting student ideas, and encouraging justification and explanations. In this paper, we use these categories as an initial coding scheme for the interactions. We expand the categories to incorporate the elements needed to investigate the use of SPVs to promote a community of mathematical inquiry.

Key concepts

We view the investigated episodes to involve a didactical performance in accordance with how the term is outlined in the framework of instrumental orchestration [START_REF] Drijvers | The teacher and the tool: instrumental orchestrations in the technology-rich mathematics classroom[END_REF]. This performance is the ad-hoc decisions made by the teacher within the orchestration. Teacher moves can be planned, but they also involve some aspect of improvisation from the teacher, as the teacher must manage the feedback provided by the students. The teacher's moves are related to specific didactical intentions. Such intentions are typically implicit rather than explicit. As such, the didactical intentions referenced in this paper are deduced by the authors based on observed patterns in the data. The teacher moves are integral in this process as they are used by the teacher to benefit the didactical intentions. In the transcripts analysed for this paper, the teacher is asking the students to explain why a solution "makes sense". Thus, the perceived didactical intention is "a discussion using a SPV to validate a strategy for solving similar triangles problems".

Methodology

The lesson analysed for this paper was number two of in all four lessons the teacher enacted using SPVs in plenary discussions over a six-month period. Four lessons were video recorded from three different teachers' classrooms totalling twelve lessons of about 45 minutes. All these lessons utilised a fixed three-part sequence. Part one is a group investigation of an inquiry-based task in school (see an example of such a task below). Part two is to present the work of the group in a video, this serves as homework, and every student produces one video and sends it to the teacher. Part three is a full class discussion related to the topic investigated in part one and two. The discussion is based on the presentations in the SPVs, and the teacher watches all the videos while planning the full class discussion. The transcripts used in this paper is from one such discussion. One of the aims of this paper is to pilot an analytical approach to investigate the discourse in these discussions. As such, one lesson was selected at random and analysed for this paper. The process of selecting a lesson was done by double-clicking a random folder.

23 tenth grade students and one teacher attended the 43-minute lesson analysed for this paper. The mathematical topic was similar triangles, and the lesson was a follow-up discussion at the end of this topic. One of the teachers in the DIM-project designed the task below after a brainstorming session at a workshop where both teachers and researchers attended.

The task investigated and solved in the SPVs was (the authors' translation from Norwegian):

Look at the drawing (Figure 1). There is a mark on the wall (green dot). Nils has placed a mirror in front of him and placed himself so that he sees the mark in the mirror.

He can measure the following lengths; from the ground to his eyes, from himself to the mirror, and from the mirror to the wall. Can he, by using these measurements, find the final measure (from the ground to the green dot)? Data was collected using a video camera. The SPVs and the google presentation used in the lesson were shared with the first author before the lesson. The data was transcribed and translated by the first author.

The data was analysed using the approach presented in Powell, Francisco, and Maher (2003, p. 413). The authors perceive the process of analysis in seven non-linear phases (comments related to our process in brackets); Viewing attentively the video data, describing the video data (data reduction), identifying critical events (identifying instructional episodes where student thinking is central), transcribing (selected episodes of interest for the topic of this paper), coding (using the codes presented in this paper and constructing new codes if needed), constructing storyline (looking at the context of the code), and composing a narrative.

The transcripts presented in this paper are from one specific episode within the lesson. The notion of an episode is to be understood as a sequence within a lesson related to a specific goal, with a specific didactical intent. There were six episodes in the lesson. The time-span of the episodes were from three to ten minutes. The episodes with titles indicating the content were:

1. Discussion related to misconceptions and issues identified in the SPVs 2. Introduction to figures often used in similar triangle problems 3. Discussion related to validating a strategy for solving similar triangles problems 4. Discussion related to validating a similar strategy for solving similar triangles problems 5. Discussion related to a different strategy for solving similar triangles 6. Discussion related to solving similar triangle problems using GeoGebra

The third episode was selected to be analysed for this paper based on; a SPV being shown in the episode, the narrative being constructed by both teacher and students, and being an episode of length fitting for the page constraints of this paper. The episode is mainly a conversation between the teacher and one student (Harald). The transcript also presents the narrative from the video (SPV) as it is central to the dialogue which emerges. Additional information is in square brackets. (..) is used to indicate a short pause in the narrative.

Results and analysis

At the start of the lesson, the teacher elaborated on two issues emerging from the videos. The first issue was the definition of similar triangles, as some students stressed that all the angles and the shape had to be the same, not recognising that they are the same criteria. The second issue was a lack of explanation of why the two triangles used to solve the task were similar. This issue can be identified in the transcript below, from Svein's video. SPV (Svein): Okay, if we have (..) for example (..) two triangles and all the angles are the same, and they have the same shape. They do not necessarily need to have the same size, but. (..) We can see that all the angles are the same in both figures. Therefore, these are similar. Okay, we measured from the feet to the eyes.

Teacher: And the measurements are here [pointing to the measurements on the screen].

SPV (Svein): It was 165 cm, and from the feet to the mirror, which is from here to here [points with the mouse on the video], it was 178 cm. And (..) we want to find the ratio between these, so we divide (..) 165 by 178 and get 0,926966 and so on. And then we measure from the mirror, which was here (..) to the church [points with the mouse on the video], because we wanted to find the height of the church. Eh (..) and we got 1448 cm. Eh (..) then we calculated 1448 times 0,9269 and so on. And we got 1342,247 cm. And that is 13,42 m. So, from the church (..) up to here [points to the top of the church in their figure] it is about 13,42 m.

Teacher: And we will stop Svein there. Eh (..) we will look further later with. That is good.

If you look at the drawing here, (..) the height of Svein or Harald, I do not know who is measuring, it is 165 from the ground to the eyes. And from the ground to the mirror it is 178 cm, and 1448 from the mirror to the church. And then he calculates that the church wall is 1342 cm or 13,42 m. My question to you; does this make sense?

The transcript above illustrates how the teacher guides the students towards his didactical intent. It appears as if the purpose for showing the video was an introduction to the task and one approach to solve it. The teacher interacts with the video to highlight the measurements used in the calculation, adding emphasis to the presentation in the video. We consider this a parallel to a pushback rejoinder move, to add emphasis to important aspects of the presentation through intervention. Pushback is usually done through a question, as in "could you point to the measurement?". However, as the presentation is through a pre-produced video, the teacher elects to point to the measurements himself. It appears as if the objective is to make students reflect on how to validate that the strategy works; if this calculation makes sense. This is to be explored through a discussion, hence the didactical intent appears to be a discussion using a SPV to validate a strategy for solving similar triangles problems.

The teacher sets the scene to further the student's presentation, all the while emphasising that this is the student's work. Furthermore, it is not clear from the context of the question what kind of validation the teacher wants, as such; the request is an open-ended Uptake. As the question is also directly related to student thinking (presented in the video), the teacher is requesting the students to make their ideas public. In the following transcript, the teacher further establishes what such a validation entail.

Student:

Yes.

Teacher: Why does it make sense?

Student: Because the church is about 13 meters.

Teacher: You know that, but I did not know that. [START_REF] Cazden | Classroom Discourse: The Language of Teaching and Learning[END_REF]. The teacher's statements exemplify the complex relationship between the planned teacher moves and the didactical performance of the teacher, used to bring the discussion towards a validation. At first, there is a rebuttal; the student knows that the church is about 13 meters, but this is not the kind of validation the teacher expects. This move must be adhoc as the teacher did not know the student's validation strategy before the question was asked.

As the reinitiation of the original uptake is left unanswered, the teacher provides a starting point towards a validation, by changing the ratio and finding an alternative solution. The level of planning in this "thinking aloud" move is unclear, but one would assume that this is a preplanned move being enacted. In this process, the teacher is limiting the cognitive demands from the initial "find a strategy to validate this calculation" to "find a strategy to determine which calculation is correct". The initial request to make ideas public is limited to following through with the specific strategy initiated by the teacher, by encouraging an explanation from one of the students with ownership of the work. The following extract shows the reply from Harald and the teacher's response.

Harald:

Because, if you divide 165 by 178 you get 0,9 and so on, (..) and if you multiply that by 178 you get the height of the person. But if you had divided the opposite way, and multiplied 178 by 1,078, you would have 180 and so on. That is not the height we measured. So, we have the ratio between the bottom and the height.

Then we can multiply with the other height to find the height [of the church].

Teacher: I will repeat what I perceive he said. He divided this by this [points to the ratio calculation] and got 0,9. And then he is saying that if you use that number, the bottom [178], and multiply by 0,9 you get 1,65. And then I understand you said that if you then take the other bottom and multiply by 0,9 you get this [points to the church height]. Did I understand you correctly?

Harald: Yes.

In this exchange, the teacher obtains what appears to be a satisfactory validation in his view and confirms this through his comments. The teacher's moves are interesting, as the aim of the initial and follow-up moves was to get the validation. However, the rephrasing from the teacher does not stress the validation presented by the student. The student explained that the ratio calculated by the teacher would provide the wrong height for the person. As such, the ratio used by the group must be the right calculation in this situation, as it provides the correct height of the person. The teacher's rephrasing is mostly related to the overall strategy used and does not highlight the validation, why it provides the correct solution. This may be regarded as a missed opportunity to zoom out of the computational perspective and connect the task to the general proportional relationship, which underpins similar triangles problems. While the teacher does acknowledge the backwards calculation to get the height of the person, it is not clear from the extract that this is the student's validation of the strategy.

Teacher: So, we must think about which numbers to multiply. Not just choose freely. But, there is also something logically correct with these numbers. Can you look at the numbers, why is it logically correct? You can keep going Harald.

Harald: Because the height is less than the bottom. The student's affirmation regarding the teachers rephrasing is followed by two statements, or "words of warning", to conclude Harald's validation. The statement "…think about which numbers to multiply" is the key to the deduced goal of the episode, to validate why the initial strategy worked. The student's idea of backwards calculation showed this, but as the teacher did not highlight this during his rephrasing, it might get lost on the students. Rather than generalising or expanding the idea, the teacher moves on to another strategy, by relating the question directly towards the measurements and visual information. It appears as if this is what the teacher was working towards the entire episode. Evidence of this is the emphasis on measurements at the start of the episode, and the need to introduce two approaches to validate the strategy provided in the SPV.

Concluding remarks

To encourage students' active engagement in mathematical discourse is instrumental in establishing a community of mathematical inquiry. In the transcripts provided, the process of negotiating what a validation entail is directly related to student work. By addressing the students and narrowing the initial uptake to steer the discussion towards the goal, the teacher appears to get a satisfactory validation. Still, the analysis illustrates the need to dig deeper than codes, exemplified by the teacher's rephrasing of Harald's idea of backwards calculation. The teacher validates the student's response. However, he de-emphasises what we perceive to be an important contribution to the discussion. Furthermore, the teacher keeps a computational perspective throughout the episode, not zooming out to connect the discussion (and initial task) to the concept of similar triangles in general. To account for such instances, we emphasise the need to analyse transcripts qualitatively as oppose to quantitatively to understand the potential of the moves throughout the discussion fully.

The three-part sequence presented in this paper might unlock possibilities for more active engagement from the students. More work is needed to understand the potential of this model. In future work, we aim to incorporate the teacher's perspective and to investigate the design process of the individual teachers in the main study. We made a conscious decision to only use the video recordings as data for the analysis in this paper as one of the aims was to investigate the potential of using the lens of teacher moves to analyse mathematical discussions. The analysis showed that the teacher utilised teacher moves to steer the discussion towards the didactical intent, and the goal of the episode. In the future, we hope to contribute to the understanding of both the intention and impact of teacher moves in a classroom discussion.

In the context of the Norwegian school system, digital tools are on the rise. Such tools are integral in the framework presented in this paper. Throughout the discussion, the teacher utilised the interactive blackboard in varying ways. To play videos, pause at desirable moments and show actual student work are integral elements in the design of the lesson. These elements enable the teacher to use students' own voices extensively throughout. However, the implementation of such ideas does require a lot from the teacher. In the analysis presented, we realise the importance of both being alert and open towards student ideas and being conscious about the goals of the episodes. The episode investigated is illustrative of this dialectic relationship. Although it would appear as if the teacher had a specific validation strategy in mind when the episode starts, he does not avert from Harald's strategy of backwards calculation. Not only does he confirm that the student's idea is usable to validate the strategy, he also repeats the idea while acknowledging that this is the student's work and not his own.

Figure 1 :

 1 Figure 1: Drawing provided to the students ("Kan måles" translates to "Can be measured") Record a video where you show your measurements and explain how you can find the height of the building. Also, show generally how you can find the sides of similar figures if you know measurements of some of the sides. Maximum 5 minutes.

  But why does it make sense? It could be about 14 m. Why does it make sense? [awaits response]. If they had done it like this, divided 178 by 165, and found another solution. They chose to do 165 divided by 178, but if they had changed these numbers they would have gotten 1,079. And if they had taken the length to the church[14,48] and multiplied by 1,079 the answer would be (..) eh (..) they would have [looks in his notes]. I have not calculated it, but they would have about 1500. That is another answer. Why is one of them correct? You need to know the height [of the church] to say that it is 13 m. Harald, why does what you did make sense?In this section, the teacher's idea of a validation becomes clearer. It would appear as if, in his view, the validation should result from the information at hand, not from prior knowledge. This might explain the decision to repeat the measurements from the students' work in the prior section, and not to go into detail about the calculations they used. Such negotiation of what the argumentation should entail is important in understanding what mathematical discourse is

  Teacher: Yes, do you see that the height is a bit smaller than the bottom here? [points to the smaller triangle]. Then it makes sense that this [points to the church's height] is a bit smaller than here (..) to the church [from the mirror to the church]. So it makes sense. Okey, now there is some red text [in the teacher's notes]. You get two minutes to write. What was Svein and Harald's strategy?