Gal Nakash-Stern

Anat Cohen
email: anatco@tauex.tau.ac.il

Developing MAP for integrating mathematical applets in teaching sequence

Keywords: Mathematical applets, Map, Meta-data, teaching sequence

In this paper, we present the way in which the math teacher proceeds to select an applet, and especially its main considerations when preparing for the lesson or preparing the study unit. In addition, this study examines what the teacher's meta-data should be in order to make a knowledgeable select of applets and integrate them into the teaching sequence. This study aims to help the teachers cope with the heavy load of the Open Educational Resources (OER) and help them in their day-to-day management of the study units. Therefore, qualitative research was carried out to examine the teachers' choices and considerations regarding their goals and sequence of teaching. The present study offers a way to deal with this difficulty by means of a Map that will link the teacher's pedagogical-content considerations. This map may be used by the teacher as a tool for intelligent choice of applets.

Introduction

A well-informed integration of technology in teaching and learning at school positively affects students regarding cognitive features and effective ones (emotional and motivational), as well as features connected with the teachers' choices in the classroom and outside of it. Using technological tools for the purpose of teaching is very common nowadays [START_REF] Stehr | Selecting, Sequencing, and Connecting: Using Technology to Support Area Measurement through Tasks, Strategies, and Discussion[END_REF]. These tools are often used for the sake of innovation, and for introducing a fresh aspect to increase students' motivation [START_REF] Horzum | Interaction, structure, social presence, and satisfaction in online learning[END_REF]Smith, Shin, & Kim, 2017). This occasionally leads to the development of new technological tools that are not always best fitted to the students, and do not attach importance to mathematical content. Thus, if mathematical content is not taken into account, the choice of using certain technology for the purposes of teaching and studying does not grant any educational benefits, and may indeed miss the actual goal [START_REF] Dȩbowska | Report and recommendations on multimedia materials for teaching and learning electricity and magnetism[END_REF].

The recent development of various educational resources containing digital learning materials for the use of teachers is especially relevant to this study [START_REF] Cohen | The use of digital repositories for enhancing teacher pedagogical performance[END_REF]. These educational resources -most of which are free for use (such as the Israeli Ministry of Education's "Educational Catalogue") -offer various activities and material and in varying formats, from short texts, videos, and applets up to full lesson plans. Hence, teachers are given a chance to make use of existing learning materials in order to optimize and improve their work. Among these educational resources there is specifically a major supply of game-based applets, interactive web utilities [START_REF] Pejuan | An evaluation tool for physics applets[END_REF], aimed at various subjects in Mathematics and for a broad agerange. It is a well-known fact that using these applets has a positive impact on students [START_REF] Hershkowitz | Creative reasoning and shifts of knowledge in the mathematics classroom[END_REF], and that using the applets through a guided model can provide better outcomes in the acquisition of language and of mathematical terminology [START_REF] Dijanić | The impact of using GeoGebra interactive applets on conceptual and procedural knowledge[END_REF].

Despite the potential for teaching and learning encompassed within existing learning materials, and their many benefits, studies suggest that their usage, as well as their combinations to match the specific user's needs, are relatively low [START_REF] Hilton | Examining the reuse of open textbooks[END_REF][START_REF] Pirkkalainen | Removing the barriers to adoption of social OER environments[END_REF][START_REF] Bernstein | OER and the value of openness: Implications for the knowledge economy[END_REF]. One argument is that many educators are not aware of these growing educational resources, or that they are not entirely convinced of their effectiveness [START_REF] Cannell | Widening access and OER: Developing new practice[END_REF]. Furthermore, there is a concern that the abundance of displayed information might lead to confusion, discrepancies and wasting time when searching for information -due to the lack of a single unified system which would license and catalogue learning material [START_REF] Nash | Learning objects, learning object repositories, and learning theory: Preliminary best practices for online courses[END_REF]. It could also lead to difficulties in evaluating the quality and reliability of information and its management [START_REF] Clements | Open educational resources repositories literature review-Towards a comprehensive quality approaches framework[END_REF][START_REF] Gurung | Are OE Resources High Quality?[END_REF].

There are two significant challenges facing those teachers who are interested in using mathematical applets and making informed usage of them. The first is related to the stage of searching and selecting the materials that will best suit the goals of their specific teaching subject; and the second is regards to the optimum usage of those materials within the teaching process in class during a lesson, or during a few consecutive lessons.

Main research goals and questions

Three main goals for this research ( 1) Understanding what data is needed for teachers to integrate applets within their teaching sequence; (2) Identifying specific features within applets that will enable an informed integration within the teaching sequence; (3) Creating a Map based on metadata fields that teachers can choose an appropriate applet and integrate it within the classroom's teaching sequence. In order to achieve these goals two questions were asked: (1) What are the main considerations that teachers operate in order to integrate applets into teaching sequence? (2) What is the meta-data that can help teachers to make decisions about using those applets? Meta-data in educational aspects (pedagogic -content) and, Meta-data in technological aspects.

Methodology

The research was conducted in three main stages. The first stage included a review of various metadata models connected to the world of studies and education. A theoretical framework was built based on these models (e.g Dublin Core and LOM) including the meta-data of existing mathematical applets. The meta-data was divided into educational dimensions and technicaltechnological dimensions. The educational dimension contains the teachers' pedagogical requirements and contents, and the technological dimension describes the applets (Figure 1). The meta-data fields which describe the Mathematical applet in each dimension are grouped into different categories of metadata. The branching of these metadata categories shows a comprehensive view of all the characteristics that can be considered when examining applets.

The second stage included semi-structured interviews with 12 mathematics teachers who use applets in elementary schools. The interviews aimed to establish the theoretical framework, and to examine the teachers' perception of integrating applets within their teaching sequence in accordance with their teaching goals. The interviews were divided into four stages in accordance with the stages the teacher faced when implementing the applets: applet searching, locating, using, and feedback. The interviews were conducted as one-off sessions with each teacher. This qualitative stage draws its data from the natural system, in which the researcher and the interviewees are on the same communication level, and the research output is created from opportunities in their interaction.

The third stage dealt with analyzing the findings, and developing a tool in the form of a Map for applets integration. This stage of analysis and tool-building placed emphasis on the teachers' answers concerning their ways of choosing applets and integrating them into their math class teaching sequence. This research was based on Grounded Theory, in which data collection was carried out before the study is fully understood [START_REF] Shkedi | Words of meaning: Qualitative research-theory and practice[END_REF].

Preliminary findings

The findings' analysis was divided into two levels: addressing the teachers' considerations regarding the integration of applets within their teaching sequence, and addressing the meta-data they would find useful for their decision making process. Based on these, a comprehensive Map was constructed.

Teachers' considerations

A number of considerations arose regarding the teachers' select of an applet and planning of their teaching sequence -mainly pedagogical and technological considerations. From the teachers' point of view, the main considerations were pedagogical considerations which included mainly the teaching goals, the pedagogy, the mathematical content, the location and time frame. Moreover, it was found to be important to consider their teaching experience and professional knowledge, their teaching style and even their personality. The technological considerations include mainly authorizations, certification (Figure 2).

Meta-data

Beyond the considerations that the teachers must use when choosing an applet, there is also the matter of the specific data that is required from the applet or database they are examining. The teachers defined the meta-data from pedagogical aspects as well as technological ones. Pedagogical aspects included mainly the teaching sequence in relation to the objective of use, the way of studying, acquired skills, students' profiles in relation to difficulty levels, timeline, mediation levels, the student experience evoked by motivational factors and configuration. In accordance with the findings' analysis, a set of Metadata fields was created for the purpose of assisting the teachers in an optimal search for the applet most suitable for their teaching sequence. Figure 3. presents the main meta-data fields that can assist the teachers in their decision-making process while constructing their teaching sequence with the use of applets (Figure 3).

A map for applets integration in teaching sequence

The pedagogical and technological considerations as well as the set of metadata are presented by process map. This map shows the teacher's way of thinking in making the choice in four stages: the search stage, the detection stage, the use stage, and the feedback stage. At each stage the teachers referred to a number of different categories. In the search phase, the teachers referred mainly to the availability of information, pedagogical considerations and a structural model. At the point of identification, the teachers referred mainly to the data they had in the applet and to the superlative information that still did not exist, when they referred mainly to the information they knew and their experience. At the point of use, the teachers referred to their meta-goals, motivational factors, and time and place of use of the applet. In the fourth and final stage, the teachers referred to the effectiveness of feedback, the type of feedback required, feedback interactivity, and feedback as a learning too.

According to the map (Figure 4), the teacher can filter out the considerations that lead her to decide whether to select an applet. The integration applet Map is presented in four main steps: filters, properties, receipt of products and choice of characteristics and in the end decision-making, from the search stage to reaching the desired applet. The filter stage includes all the pre-requisite knowledge the teacher needs when choosing an applet that includes her professional knowledge, her personality, her mathematical goal, and more. At this stage, the teacher also defines her target audience, the method of instruction and the timing of her teaching sequence. The properties stage includes the different characteristics of the applet and its target audience. The third stage, which is defined as the receipt of the products and the examination of characteristics, is from a stage in which all the knowledge that the teacher has is integrated into this stage and it examines the continuation of the process in order to select the applet. In the fourth and final stage, the teachers arrive at a crossroads where they have to make a choice: Should they choose the current applet, or should they go back to the second and third stages, and re-examine other applets.

Figure 4: A map for applets integration in teaching sequence

In conclusion, this research looked at the considerations and the metadata which would influence teachers when choosing applets and integrating them into the teaching sequence. The interviews indicated great significance attached to the teacher's personality, needs and authority. Another issue that emerged through the interviews was the students' needsthe teacher's understanding of what the student needs to get; how the students will connect to the applet; and how we get the students to understand mathematics by using the right applet. Another issue which is just as important as the teacher and student's needs, is the teachers' learning process during their search for the desired applet. During the interviews, the teachers explained how they approached searching and choosing the applet, as well as how they integrated it within their teaching sequence, and these led to conclusions regarding their method of learning.

Discussion and conclusions

In this research, the pedagogical consideration was the main one for choosing applets. According to the pedagogical consideration, teachers choose the applet most suited to their teaching sequence. Since the mathematical content is highly significant in building up the students' mathematical knowledge [START_REF] Kimeswenger | Identifying and assessing quality criteria for dynamic mathematics materials on platforms[END_REF], the teachers will first search for an applet appropriate for that content. Most of the teachers interviewed stated that without that element of information they would not be able to begin the search, hence -this is a primary filter when choosing a database or applet. Similarly, Namukasa, Gadanidis, Sarina, Scucuglia, and Aryee in their research (2016) found that among the most important considerations were the pedagogy and content that the teachers wanted to teach in their mathematics lessons. They also stated that mathematical terms or "mathematical thinking" would be the most critical when choosing a digital object. This issue was addressed as well by [START_REF] Kotsopoulos | A pedagogical framework for computational thinking[END_REF], who explained the importance of the students' computational mathematical thinking, which should take precedence.

The study found that technical components such as installation instructions and necessary hardware and software programs were not essential elements when choosing an applet. Most teachers, when asked about the technical aspect of choosing an applet, answered that they do not attribute much importance to those aspects when choosing to use an applet. The teachers stated that the only factor impacting on their choice would be the hardware or program's digital license -as defined by [START_REF] Green | Open licensing and open education licensing policy[END_REF] during his OER license research, where he examined how they influence studying. The teachers stated they are already used to the "wonders of technology" and that they know how to handle anything that comes their way. Much to our surprise, the technical aspect that we had emphasized at the beginning of the research did not end up being a hindrance or even a consideration when choosing an applet. This surprised us a great deal, since some populations find study-suitability very difficult during times of technological advancements [START_REF] Göthe | Age differences in dual-task performance after practice[END_REF].

During this research, we mapped out the considerations and information the teachers would need when choosing an applet or mathematical resources. Then, similar to [START_REF] Kimeswenger | Identifying and assessing quality criteria for dynamic mathematics materials on platforms[END_REF] we recognized the need for developing a tool, which will help teachers to navigate in the heavy load of resources and to assess learning materials. Thus, we built a Map for integrating applets in teaching sequence. The Map consists of the knowledge the teachers will need for choosing the applet and for integrating it into the teaching sequence. We believe that the Map we have created may assist in developing a technological tool with which teachers may integrate all the elements necessary for their teaching. Beyond the teacher's familiarization with the educational resources and applets, it is necessary to develop a tool in the form of Google, consisting of mathematical educational resources and applets, with the added option of feeding it criteria that would make it possible to obtain the desired product, and which could assist the teachers' choice of an applet, and shorten search duration.

Since this research did not deal with the construction of lesson units by teachers, it is vital for future research to deal with that subject in greater depth and regarding the following influencing factors: How long it takes the teachers to learn the usage of applets; how long it takes to build a lesson plan whilst relating to pedagogical components as well as to the teacher's teaching goals; the chronology of lesson building; and how the applets assist the teachers.

In an age of rapid and frequent changes, we must pay attention to the fact that teachers are required to implement new technologies on a daily basis as part of their lesson structure. Hence, the main importance of this research is in assisting teachers integrate mathematical applets within their teaching sequence in the classroom. Teachers nowadays are laden with a huge amount of applets and resources [START_REF] Peralta | Understanding learning resources metadata for primary and secondary education[END_REF] which is why we developed the presented Map. Teachers are expected to have an extremely high level of literacy due to current usage of many technological tools. Therefore, it is important for future research to address the way in which the teachers learn how to use technological tools, and to actually implement them in the classroom.

Figure 1 :

 1 Figure 1: The theoretical framework

Figure 2 :

 2 Figure 2: The teacher's considerations for integrating applets

Figure 3 :

 3 Figure 3: Pedagogical and technological metadata fields

Acknowledgment

This study is founded by the Education Ministry's chief scientist.


Dilek Tanışlı

Nilüfer Köse

Melih Turgut
email: mturgut@ogu.edu.tr

Professional development through a web-based portal: The progress of mathematics teachers teaching algebra based on hypothetical learning trajectories

Keywords: Mathematics teacher, Professional development, Hypothetical learning trajectories

The aim of this paper is to present preliminary findings of a large-scale project for designing a web-based portal to contribute to the professional development of mathematics teachers for teaching algebra in lower secondary education. Eleven teachers volunteered to participate in this research that aimed to design hypothetical learning trajectories (HLTs) for sixth-grade algebra context over a three-week period. The teachers designed their own HLTs through a web portal by receiving feedback from mathematics teacher educators online. Their processes of designing and applying lesson plans during the first and third week were analysed according to the components of HLT and the theoretical perspective of the quartet of subject matter knowledge. Data analysis revealed that the teachers managed to define more coherent learning goals and hypothesis for their students' learning as well as making connections between the concepts.

Introduction

In Turkey, as in many countries, studies on the mathematics teacher knowledge received particular attention from researchers [START_REF] Gökkurt | Examination of middle school mathematics teachers' pedagogical content knowledge: The sample of cone[END_REF][START_REF] Türnüklü | Mathematics teachers' perceptions of quadrilaterals and understanding the inclusion relations[END_REF]. Research on in-service teachers commonly indicates that a number of issues exist regarding teachers' subject matter knowledge of mathematics. Researchers have addressed major problems in the Turkish context: (i) a lack of certain professional development programs and their sustainability; and (ii) a gap between research and practice, which, in particular, means a lack of communication between mathematics teacher educators and in-service mathematics teachers [START_REF] Yıldırım | Teacher education research in Turkey: Trends, issues and priority areas[END_REF]. The latter is of crucial importance since professional development activities are generally organised independently by each public school whereby teachers prepare termly plans without any collaboration with mathematics teacher educators. This research study elaborates the design and use of a web-based portal to contribute to the professional development of mathematics teachers, in which they design their own teaching-learning algebra activities within a perspective of hypothetical learning trajectories (HLT) [START_REF] Simon | Reconstructing mathematics pedagogy from a constructivist perspective[END_REF] through close interaction with mathematics teacher educators.

We focus on the algebra context of Turkish (lower) secondary school curriculum and design a large-scale research project adopting a number of theoretical and conceptual elements, such as mathematics teaching cycle and HLT [START_REF] Simon | Reconstructing mathematics pedagogy from a constructivist perspective[END_REF] and the mathematics subject matter knowledge quartet, as described by [START_REF] Rowland | Elementary teachers' mathematics subject knowledge: The knowledge quartet and the case of Naomi[END_REF].

Theoretical Framework and Description of the Project

The notion of a learning trajectory corresponds to possible (anticipated) learning paths that students could follow in a proposed content designed under certain objectives. Since learning, in itself, is a complex process and depends on phenomenological experiences, [START_REF] Simon | Reconstructing mathematics pedagogy from a constructivist perspective[END_REF] adds the word hypothetical to learning trajectory to emphasise a teacher's prediction about students learning and elaborates the notion of HLT under three components; 'the learning goal that defines the direction, the learning activities, and the hypothetical learning process' (p. 136). In fact, [START_REF] Simon | Reconstructing mathematics pedagogy from a constructivist perspective[END_REF] considers the HLTs as a part of mathematics teaching cycle, which includes certain key components such as a teacher's knowledge and an assessment of student knowledge (Figure 1). In Figure 1, particular emphasis is given to the teacher's knowledge as a starting point. Therefore, in this research, we consider and refer to a quartet of subject matter knowledge of mathematics [START_REF] Rowland | Elementary teachers' mathematics subject knowledge: The knowledge quartet and the case of Naomi[END_REF]) that has four dimensions; foundation, transformation, connection, and contingency. The first component, foundation refers to a repertoire of the teacher's academic knowledge for teaching-learning mathematics including his/her beliefs regarding why mathematics is important and why it should be taught. Here, transformation refers to the transformation of theoretical knowledge into practice by designing and planning pedagogical tasks in terms of choosing appropriate examples and activities for the construction of mathematical meanings. The notion of connection refers to the coherence of designed parts of a lesson or series of lessons through deliberatively chosen activities and domain-specific tasks. Such pedagogical task sequences enable students to make a connection between different concepts as well as to the interplay between different representations. The final component contingency refers to 'classroom events that are almost impossible to plan for' (Rowland et al., 2005, p. 263).

Related literature confirms that [START_REF] Simon | Reconstructing mathematics pedagogy from a constructivist perspective[END_REF] mathematics teaching cycle within HLTs could be a heuristic tool for teachers' professional development (e.g. Wilson, Mojika, & Confrey, 2013). Consequently, we exploit mathematics teaching cycles as twofold: as a perspective for designing interfaces of the web portal (which will be explained in the next subsection); and as an analysis tool to explore teachers' progress as a part of their professional development. In order to elaborate development of teachers' knowledge in depth, we also refer to the subject matter knowledge quartet as an additional analysis tool while looking deep into the teachers' lesson plans.

The Project and Research Question

Since the Internet is now widely used in different platforms as well as in educational contexts, we designed a sustainable and specific web portal to ensure mathematics teachers' professional development regarding algebra content of the (lower) secondary school mathematics curriculum. We specifically focused on algebra content as the researchers in the project team have background knowledge in epistemological issues and misconceptions regarding learning algebra. In order to create such a portal, at first, a domain http://megedep.anadolu.edu.tr was taken, where 'megedep' referred to a shortcut of the project's title in Turkish. The interface includes some practical information in addition to theoretical foundations that are necessary for participating teachers in the project. More precisely, the interface includes a video describing the aim of the project, a manual for using the website, an introductory description for constructivism and the notion of HLT, as well as exemplary lesson designs under the notion of HLT. After registering to the website, the teachers could connect to a user interface including the components of the mathematics teaching cycle, which is sketched in Figure 3a. In the portal, we followed [START_REF] Khan | Managing e-Learning: Design, Delivery, Implementation and Evaluation[END_REF] for designing an e-learning platform where there exists an effective feedback system. When teachers enter the learning goal, possible activities and their hypotheses for students learning, the mathematics (teacher) educators (MTEs) (in this paper, the authors) of the project receive alerts by email. Through entering their own interface, MTEs review the teachers' input and give feedback to the teachers to improve the anticipated teaching and learning processes. Thereafter, the teachers make revisions, and if they complete necessary improvements, then they could proceed to design a more fine-grain teaching plan. After that, the MTEs again review their plans and provide feedback. The progress continues in a cycle like this. However, it ends when the teachers record their own teaching episode and upload videos to the web portal after having written their responses to some reflective questions (described in the methods section) regarding the process. Thereafter, the MTEs watch the teaching episodes and review the teachers' responses to reflective questions by taking notes and then communicate their ideas to the teachers through the portal for the teachers to prepare and develop the next episodes.

In this paper we provide a brief analysis of the teachers' initial lesson designs and the final (third) lesson designs created within the web portal to explore their professional development progress. We address the research question: How does mathematics teachers' professional development progress when they use megedep portal to design HLTs and lesson plans for teaching algebra?

Methods

Theoretical insights of the mathematics teaching cycle [START_REF] Simon | Reconstructing mathematics pedagogy from a constructivist perspective[END_REF] and quartet of subject matter knowledge [START_REF] Rowland | Elementary teachers' mathematics subject knowledge: The knowledge quartet and the case of Naomi[END_REF] refer to a qualitative paradigm. In this paper, in order to explore progress of teachers' professional development in depth, we consider the same perspective and adopt a case study design [START_REF] Bogdan | Qualitative Research for Education: An Introduction to Theory and Methods[END_REF]. Eleven (all six grade teachers) secondary school mathematics teachers (T1-T11) with different teaching experiences (ranging 1-15 years) volunteered to participate in the project. All of the participants work in the same city located in central Turkey and have experience of the objectives (algebra) of the national curriculum, having all graduated from faculties of education from different universities. In addition, while T1, T5, T6, T8 and T10 have only a bachelor degree, T7 is studying for a master's degree and also T2, T3, T4 and T11 are studying for doctorates. One participant, T9, already holds a master's degree. All of the participants learnt of the notion of HLT for the first time as they become involved in the project.

During the first project meeting (lasting around three hours), all of the participants were introduced to the philosophy behind constructivism, the use of the megedep web portal, and the notion HLT with a number of exemplary cases.

The Data and Analysis

The data comes from two sources: the first is based on the MTEs' evaluations of the teachers' designs of HLTs. In other words, the teachers designed their initial HLTs and MTEs wrote feedback to improve each part. After the teachers' improvements on the HLTs, the designs were reviewed once more. The first part of the data comes from the MTE final reviews of the HLTs. The second part of the data comes from the teachers' responses to reflective questions that were proposed after they completed teaching episodes. Three reflective questions were asked before the teachers applied their HLTs whereas four questions being proposed after they applied their lesson plans.

Table 1 shows the reflective questions that were proposed to the teachers through the megedep interface.

Before application of the lesson plans

After application of the lesson plans -What were the key elements in your lesson plan? -How and why did you determine the activities in your plan? -How did you determine your hypotheses for students learning? -Which learning goals were obtained and which were not? Explain in detail.

-Which parts of your plan worked well, or which did not? Explain in detail.

-Which hypotheses of student learning appeared in the classroom? Did all? Were there any contingency cases? If so, what did you do? -In the teaching process, which concepts were conceptualised and which were not? Explain in detail.

Table 1: Reflective questions proposed to the teachers through the megedep interface

The teachers prepared three teaching episodes corresponding to three weeks. In other words, they prepared an HLT for each week considering the objectives of the curriculum. MTEs evaluated the first and third week's HLTs, in order to explore the teachers' progression. Similarly, the MTEs also collected all of the responses given to the reflective questions. A thematic analysis [START_REF] Miles | Qualitative Data Analysis: An Expanded Sourcebook[END_REF] was employed to elaborate emerging main themes within a perspective of components of HLT and the quartet of subject matter knowledge.

Findings

Since our focus is on designing HLTs and lesson plans as components of professional development, we will present our findings in two subsections; an analysis of the teachers' designs of HLTs and an analysis of the teachers' designs and applications of lesson plans.

Analysis of the Teachers' Design of HLTs

The analysis of the teachers' first HLT designs revealed: four themes appeared for determining learning goals; three themes for determining activities; and four themes for determining the hypotheses for students learning. Table 2 describes these themes and their respective frequencies.

Determining Learning Goals

Determining Activities Determining Hypotheses -Defining learning goals coherent with curriculum objectives (T1) -Not sequencing learning goals (T2, T3, T4, T5, T6, T7, T8) -Referring to objectives as learning goals (T9, T10, T11) -Defining learning goals, not in relation to objectives (T8, T7, T6, T2) -Determination of coherent activities (T7, T8, T6, T11, T9, T3, T2) -Activities not based on the curriculum objectives (T8, T6, T7, T10) -Writing only the title of activities (not based on any objective) (T5, T1, T4) -Writing hypotheses based on activities (T2, T3, T11) -Formulating a partial hypothesis not focusing on entire student learning (T4, T9, T1, T5, T10, T8) -Hypotheses not based on the activities (T7) -Not a clear formulation (T6)

Table 2: Themes regarding the teachers' design of the first HLT

Table 1 reveals that only one teacher defined learning goals according to the objectives of the curriculum, whereas seven teachers could not define sequencing learning goals, i.e. they were not able to express learning goals in order. Three teachers directly referred to the curriculum objectives as learning goals, while four of them defined the learning goals independent of the curriculum objectives. With respect to a determination of the activities, seven teachers managed to define coherent activities, i.e. activities that are coherent with curriculum objectives and also with the learning goals that they defined. Four teachers expressed activities not based on the objectives and three teachers only wrote the title of activities without giving any details. Three teachers expressed their hypotheses that are coherent with the learning objectives and activities that had been written before. However, it should be noted that these three teachers thought that students would follow the same learning trajectory. Six teachers formulated their hypotheses partially, without focusing on the students' entire learning of the proposed concepts, while one teacher proposed a hypothesis independent of the activities and learning goals. Moreover, one teacher expressed his possible teaching steps, including no trace of the hypotheses of students learning.

After all the teachers had used the megedep portal for three weeks, we analysed their resulting HLTs. Tables 3 shows that there were two emerging themes for determination of the learning goals; one theme for the determination of the activities and two themes for the determination of the hypotheses.

Determination of Learning Goals

Determining Activities Determining Hypotheses -Defining learning goals based on the curriculum objectives (T6, T1, T3, T4, T7, T11, T5, T2) -Defining learning goals part by part, not based on all curriculum objectives (T10, T8, T9) -Determination of coherent activities (T6, T1, T3, T10, T4, T7, T11, T5, T2, T8, T9) -Writing hypotheses based on activities (T3, T11, T5, T2) -Formulating a partial hypothesis not focusing on entire student learning (T6, T1, T10, T4, T7, T8, T9) Table 3: Themes regarding the teachers' design of the third HLT Eight teachers managed to define learning goals based on the curriculum objectives. However, there were still three teachers who defined learning goals part-by-part, not based on the curriculum objectives. All of the teachers were able to define activities coherent with the learning goals and curriculum objectives and also explained them in detail. With respect to a determination of the hypothesis, four teachers expressed the hypotheses of students' learning that were coherent with the learning goals and activities, while seven teachers formulated hypotheses part-by-part, not focusing on the conceptual learning.

Analysis of the Teachers' Designs and Applications of the Lesson Plans

The teachers' responses to the reflective questions posed after the first and the third weeks' lesson plans were analysed according to the quartet of subject matter knowledge. When the authors compared the personal analyses on the reflective questions, it was agreed to consider the same themes for the first and the third week's results. As a result, four themes emerged for the foundation component, three themes emerged for transformation component, five themes emerged for connection component, and three themes emerged for the contingency component. Table 4 summarises the emerging themes with the arrows indicating some themes that have specific meanings. For instance, a→b means that a is the number of teachers in the first week, while b is the number of teachers in the third week. 4 indicates that there is an increase in the number of the teachers, who recognised students' misconceptions, proposed effective questions and referred to multiple ways of assessing student knowledge by the end of the third week's application. In parallel with this finding, the teachers' focus on computational applications decreased. No change occurred in the use of multiple representations while selecting and/or preparing carefully-designed tasks, and the teachers' emphasis on explaining mathematical concepts increased. Similarly, at the end of the third week, the teachers' interplay between mathematical concepts and interplay between mathematical problem-solving techniques increased. With regard to the contingency component, after the first week's application, two of the teachers believed that all of the goals had been achieved by the students. However, at the end of the third week, there is an indication that most of the teachers' beliefs were changed. Moreover, at the end of the third week, the teachers managed to explain how and why certain parts of the plans and proposed activities did not work.

Conclusions and Further Steps

In this paper, we present preliminary findings of an on-going large-scale project aiming to contribute to the professional development of mathematics teachers through a web-based portal. The findings of the study suggested that at the end of three weeks' usage of the megedep portal, the participant teachers managed to define more coherent learning goals, classroom activities and hypotheses for students learning regarding (lower) secondary school algebra. Most progress is observed in the teachers' interplay between the mathematical concepts by referring to different mathematical problem-solving techniques. These conclusions were possible for the teachers' own readings of the existing literature on learning algebra and the MTEs feedback that was given at each step for designing HLTs through the megedep portal. However, expressing hypotheses for students learning as well as determining learning goals based on curriculum objectives are not trivial tasks for the teachers. We aim to acknowledge this fact by showing or preparing more examples from the literature (e.g. [START_REF] Simon | Reconstructing mathematics pedagogy from a constructivist perspective[END_REF][START_REF] Simon | Explicating the role of mathematical tasks in conceptual learning: An elaboration of the hypothetical learning trajectory[END_REF] in order to provide teachers with a detailed understanding to write hypotheses for their own classroom context. In summary, the main contribution of this paper is to show how a web-based portal could mediate the professional development of mathematics teachers, where the key component in such progress is peers' review and feedback in lower algebra context provided by the megedep portal. The use of acknowledged digital technologies here constructs a link between a collaborative design for teaching-learning algebra and classroom practice, which can be considered as novel progress for their professional development. The main ease here is the collaboration, which is independent of time and place.

The next steps of the project can be summarised by conducting the same process with seventh grade teachers, but over seven weeks and this time focusing on the students learning and the use of pretest and post-test tasks. The results of these steps will be the focus of future research papers.

Figure 1 :

 1 Figure 1: Mathematics Teaching Cycle (adapted from Simon, 1995, p. 136)

  Figure 2 displays the (homepage) interface of the website.

Figure 2 :

 2 Figure 2: Megedep's (Web portal) interface

Figure 3

 3 Figure 3: a. The user interface (translated), b. The user interface to enter components of HLTWhen a teacher selects 'Process', Figure3bis shown, where the teacher enters the class level, the topic and the related learning objectives determined by the Turkish Ministry of National Education (MoNE, 2018) of Turkey. In the portal, we followed[START_REF] Khan | Managing e-Learning: Design, Delivery, Implementation and Evaluation[END_REF] for designing an e-learning platform where there exists an effective feedback system. When teachers enter the learning goal, possible activities and their hypotheses for students learning, the mathematics (teacher) educators

Table 4 : Overview of the teachers' design and application of the lesson plans

 4

Table

Acknowledgement

This research is supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK) under SOBAG 1001 programme. We would like to thank TWG15 reviewers for their careful reading of the paper and substantial suggestions that improved the paper.


Daniel Thurm
email: daniel.thurm@uni-due.de

Bärbel Barzel
email: baerbel.barzel@uni-due.de

Self-efficacythe final obstacle on our way to teaching mathematics with technology?

Keywords: Self-efficacy, technology, teacher-education, graphing calculators, professional development

Despite the potential of digital technology to enhance the learning of mathematics the use of graphing tools with or without computer algebra systems (CAS) remains low. For this reason, there is a need to identify and understand factors that lead to greater technology uptake. In this quantitative study with 198 upper secondary in-service teachers from Germany the relation between self-efficacy beliefs and frequency of technology use in the mathematics classroom is examined. Results show that higher self-efficacy beliefs are strongly associated with a more frequent use of technology. Based on the results, recommendations for teacher education are given.

Introduction

Research in the last decades has shown that digital technology like function plotters or computer algebra systems can facilitate the learning of mathematics in many different ways [START_REF] Drijvers | Uses of technology in lower secondary mathematics education: A concise topical survey[END_REF]. For this reason, there is an ubiquitous call for the integration of these technologies into mathematics teaching. However current use of technology in the mathematics classroom still remains low: "Nowadays there exist countless ideas, classroom suggestions, lesson plans and research reports for the use of new technologies in mathematics classrooms. But the situation concerning the real use of DT [digital technology] in mathematics classrooms has not succeeded in the way many had expected in the last decades." (Weigand, 2014, p. 5; see also Heid, Thomas, & Zbiek, 2013, p. 599) This indicates that the implementation of technology into the mathematics classroom is not a straightforward task and draws attention to the need to understand the factors influencing technology uptake. This is particularly important for the design of professional development (PD) programs which aim at supporting teachers' when integrating digital technology. Research shows that beliefs and knowledge as well as external factors like school culture and resources play an important role (Heid, Thomas, & Zbiek, 2013, p. 630). However, less attention has been given to self-efficacy beliefs which capture a persons' perceived ability to organize and execute courses of action (Bandura, 1997, p. 3). Therefore, this paper explores to what extend self-efficacy beliefs in the domain of teaching mathematics with technology influence the uptake of technology.

Theoretical background

Digital technology in the mathematics classroom comprises a plethora of different technologies. These range from general technology that can be used across different subjects (e.g. word processing software like MS-Word) to subject-specific technology like digital learning environments, function plotters, geometry packages and computer algebra systems (CAS) that are specifically used in mathematics education. In this paper, unless stated otherwise, the terms "technology" is used to refer to these specific tools, that are also known under the term "Mathematics Analysis Software" (MAS; [START_REF] Pierce | Perceptions that may affect teachers' intention to use technology in secondary mathematics classes[END_REF]. In the last 30 years, a great number of research projects has shown that MAS can support student learning in many different ways. For example, these tools can facilitate constructivist teaching approaches like discovery learning by giving pupils the opportunity to explore mathematical links on their own. In addition, digital tools can enhance conceptual understanding of mathematics by providing easy access to multiple and linked representations [START_REF] Drijvers | Uses of technology in lower secondary mathematics education: A concise topical survey[END_REF]. For this reason, the use of technology in the mathematics classrooms is advocated for by researchers as well as teachers.

Despite this call for a deep integration of technology into mathematics teaching there is a "widely perceived quantitative gap and qualitative gap between the reality of teachers' use of ICT and the potential for ICT suggested by research and policy" (Bretscher, 2014, p. 43). Reasons for this are multifaceted and many different factors have been proposed in the last decade which can affect the uptake of technology. These factors include external factors like time constraints and resources (Thomas & Palmer, 2014, p. 72;Heid, Thomas, & Zbiek, 2013, p. 630). However, research indicates that the most important factor is the teacher who is ultimately responsible for how and when technology is used: "Clearly, schools can go only so far to encourage ICT use; actual take-up depends largely on teachers' personal feelings, skills and attitudes to IT in general" (Mumtaz, 2000, p. 337). Therefore, internal factors like teacher knowledge and teacher beliefs have become the focus of attention. For example, the concept of TPACK [START_REF] Mishra | TPACK: A framework for teacher knowledge[END_REF] acknowledges, that a special type of knowledge is needed by teachers to teach mathematics with technology. Also, pedagogical beliefs, for example about constructivist or traditional philosophies of teaching mathematics, and beliefs referring to the teaching and learning mathematics with technology have been studied, showing that these beliefs can play an important role.

However, it seems that research has given little attention to teacher self-efficacy-beliefs when teaching mathematics with technology. Self-efficacy beliefs are defined as "a judgement of one's ability to organize and execute the courses of action required to produce given attainments" (Bandura, 1997, p. 3). This means, self-efficacy beliefs don't capture the actual ability of a person but rather describe how someone perceives it. Research about the integration of general technology like word processing or presentation software across different subjects has found that self-efficacy beliefs play a crucial role for the uptake of these technologies [START_REF] Ertmer | Teacher technology change: How knowledge, confidence, beliefs, and culture intersect[END_REF][START_REF] Wang | Increasing preservice teachers' self-efficacy beliefs for technology integration[END_REF]. As Ertmer & Ottenbreit-Leftwich note: "Although knowledge of technology is necessary, it is not enough if teachers do not also feel confident1 using that knowledge to facilitate student learning" (Ertmer & Ottenbreit-Leftwich, 2010, p. 261). Therefore, it is particular surprising that self-efficacy beliefs have not been studied in much detail in the context of teaching with MAS. Apart from the works of Cavanagh & Mitchelmore (2003, p. 16), Doerr & Zangor (2000, p.159) and [START_REF] Hong | Factors influencing teacher integration of graphic calculators in teaching[END_REF] where the role of self-efficacy beliefs becomes apparent subliminally at some points, we could only identify the work of [START_REF] Thomas | Teaching with digital technology: Obstacles and opportunities[END_REF] explicitly taking self-efficacy beliefs into account. They show that 42% of the surveyed teachers do mention lack of self-efficacy as an obstacle preventing teachers to use calculators more in their teaching. In addition, teachers with higher self-efficacy beliefs showed a more positive attitude to technology and to the use of technology in the learning of mathematics. Hence, they include confidence as a key factor in their model for technology integration. In summary, there seems to be some indication that teacher self-efficacy beliefs play an important role for the integration of technology into the mathematics classroom, but empirical evidence that teachers' self-efficacy beliefs about teaching with MAS is indeed linked to the classroom usage of technology is lacking.

Research question & methodology

The study aims to explore the link between self-efficacy and use of MAS. To achieve this goal, it is necessary to measure teachers' self-efficacy beliefs as well as teachers' technology use. Since the frequency of technology use can be more validly be self-reported by teachers compared to the quality of their teaching [START_REF] Mayer | Measuring instructional practice: Can policymakers trust survey data? Educational Evaluation and Policy Analysis[END_REF] we focus on self-reported frequency of technology use and addresses the following questions: Do teachers with higher self-efficacy beliefs use MAS more often than teachers with lower self-efficacy beliefs? Which areas of technology use show particular strong links to self-efficacy beliefs?

To capture the frequency of technology use we used a likert-scale questionnaire [START_REF] Thurm | Teacher Beliefs and Practice When Teaching with Technology: A Latent Profile Analysis[END_REF] assessing the use of technology in the following areas: (f1) use of technology for discovery learning, (f2) use of technology for linking multiple representations, (f3) use of technology when practicing, (f4) use of technology to support individual learning. In addition, a category capturing how often the use of technology was subject to critical reflection was included. Items in this category asked for example how often limitations of technology were discussed or how often there was a critical reflection about when to use pen & paper skills and when to use technology to solve a given task.

To measure teachers' self-efficacy beliefs, we developed a questionnaire that assessed self-efficacy in two domains (see below). For each of these domains a set of items was generated and refined until validity of the items was agreed on by teachers and experts. Response category was chosen according to Banduras "Guide for constructing self-efficacy scales" [START_REF] Bandura | Self-Efficacy. The exercise of control[END_REF] which recommends that teachers rate the strength of their belief in their ability on a scale from 0-100. In the following we give a brief description of each domain:

(s1) Self-efficacy related to task design & task selection when teaching with technology (4 Items)

The design of tasks is one of the most important factors when teaching with technology: "The inclusion of technology requires an understanding of the kinds of tasks that may utilize the resources provided by the technology to support students' high-level thinking" (Sherman, 2014, p. 225; see also [START_REF] Drijvers | Digital technology in mathematics education: Why it works (or doesn't)[END_REF]. If teachers don't feel able to select and design appropriate tasks they are most likely to use their old tasks that are usually not tuned to the use of technology. Two sample items of the scale were: I can design task for the use with MAS. I can distinguish between good and bad tasks for the use with MAS.

(s2) Self-efficacy related to lesson design & lesson implementation (4 Items) These tasks however will not unfold their potential on their own. They need to be embedded in carefully designed lessons. Hence teachers must feel able to design and carry out appropriate lessons: "[...] the teacher has to orchestrate learning, for example by synthesizing the results of technology-rich activities, highlighting fruitful tool techniques, and relating the experiences within the technological environment to paper-and-pencil skills or to other mathematic activities." (Drijvers, 2015, p.148) Two sample items of the scale were: I can design and implement lessons that support discovery learning using MAS. I am able to design lessons that make versatile use of MAS.

Both scales were administered in 2014 to 198 upper secondary school teachers within a larger research study [START_REF] Thurm | How to professionalize teachers to use technology in a meaningful way-Design research of a CPD program[END_REF] in North Rhine-Westphalia, Germany. In this state the use of MAS is compulsory in the final examination in upper secondary since the schoolyear 2014/2015. The average age of the teachers was 43 years, which is comparable to the average age of 45 years for all teachers in North Rhine-Westphalia. Teaching experience with mathematics was distributed as follows: 0-5 years (27%), 6-11 years (19%), 12-17 (16 %), 18-23 (11%), more than 23 years (25%). Due to the fact that MAS had been made compulsory only shortly before the study took place, there was a large number of teachers without any previous experience in teaching with MAS. In total, 52% had no previous experience teaching mathematics with technology and only 12% had more than 5 years' experience in teaching mathematics with technology.

Results

In a first step we used a confirmatory factor analysis to scrutinize the quality of the questionnaires on an empirical base. The goodness of fit was judged using the chi-square-fit index, the root mean square error of approximation (RMSEA), the standardized root mean square residual (SRMR) and the comparative fit index (CFI). The result for the self-efficacy questionnaire showed a good model fit with RMSEA = 0.041 SRMR = 0.022, CFI = 0.994, and χ2 /df = 1.327. The scales (s1) and (s2) also showed a good reliability with Cronbach' s alpha equal to 0.92 and 0.9 respectively. The goodness of fit and the reliability of the questionnaire assessing frequency of technology use were also high (see [START_REF] Thurm | Teacher Beliefs and Practice When Teaching with Technology: A Latent Profile Analysis[END_REF].

In first step we checked to what extend experience in teaching mathematics and teaching with MAS correlated with self-efficacy and frequency of technology use. As can be seen (Table 1) there is a small correlation between self-efficacy and the number of years teachers previously used technology in their classroom. Subsequently, we calculated the correlation between all scales on a latent level (Table 1). It is obvious that there are significant correlations between all subscales indicating that higher self-efficacy beliefs correspond to more frequent use of technology. Selfefficacy in the area of task design (s1) is most strongly linked to discovery learning (f1) and practice (f3), self-efficacy in the area of lesson design and implementation (s2) is most strongly linked to discovery learning (f1) and individual learning (f4). In addition, it is remarkably that the scale (s2) consistently shows higher association with frequency of technology use than the scale (s1). Lowest correlation between self-efficacy and frequency of technology use is observed for the areas of multiple representation (f2) and reflection (f5).

Self

Discussion

The study aimed to provide empirical evidence for the link between self-efficacy and using MAS.

The results clearly indicate a strong relationship showing that higher self-efficacy beliefs are associated with more frequent use of technology. In addition, self-efficacy correlates only to a limited extend with teaching experience with technology, which indicates that teachers need additional support for developing their self-efficacy beliefs. In total, the results support the model of [START_REF] Thomas | Teaching with digital technology: Obstacles and opportunities[END_REF] who include self-efficacy as a key variable in their model of technology integration. Furthermore, the importance of self-efficacy becomes even more pronounced when noticing that the size of the correlation between self-efficacy and frequency of technology use is higher compared to the correlation of technology-related beliefs and frequency of technology use found in [START_REF] Thurm | Teacher Beliefs and Practice When Teaching with Technology: A Latent Profile Analysis[END_REF]. This indicates that self-efficacy beliefs might be even more influential for technology uptake than beliefs about the value of technology in the learning of mathematics.

But why do self-efficacy beliefs seem to play such an important role? One explanation might be, that teaching with technology poses a plethora of new challenges because it requires many changes in the teaching routines and attitudes. For example, traditional task that requires routine pen & paper skills easily become obsolete in the presence of a CAS. In addition, teaching with MAS can lead to more individualized learning, with greater uncertainty for the teacher who cannot longer simply follow a strict teaching plan. For example, Dunham & Dick (1994, p. 443) stress, that teachers teaching with technology need to be much more used to deal with unplanned situations, show more flexibility and may experience a perceived loss of control. Especially for teachers that are used to a traditional style of teaching this can pose a major challenge: "[...] the use of technology often means more individualized, student-centered classrooms in which teachers are no longer the sole source and authority of knowledge. This could be very disturbing to many teachers who are used to lecturing and other teacher-centered approaches because it require them to abandon their routines and learn new ways of teaching." (Zhao & Cziko, 2001, S.18) From the results several recommendations for teacher education and professional development can be inferred. Firstly, programs must place a stronger focus on developing teacher's self-efficacy beliefs. Currently, knowledge, pedagogical beliefs and beliefs regarding the teaching of learning with technology seem to be the focus of most technology related professional development programs (e.g. Chamblee et al. 2008). Even though these factors might also increase self-efficacy, the most effective way of increasing self-efficacy is to enable successful mastery experiences for teachers. For example, Tschannen-Moran & McMaster (2009) compared different approaches for fostering self-efficacy beliefs in PD-programs and concluded that authentic mastery experience embedded in the teachers regular teaching context was the most powerful way to foster self-efficacy beliefs. In particular, they stress the role of individual support for teachers: "Without coaching to assist teachers in the implementation of the new skill, a significant proportion of teachers were left feeling more inadequate than they had before" (Tschannen-Moran & McMaster, 2009, p. 241). However, programs that individually assist teachers to put theory into practice in this way appear to be quite rare in the context of teaching with technology (Grugeon et al., 2009, p.343). This might be due to the fact that individual coaching is quite resource intensive and hard to scale up.

For this reason, it might be fruitful to develop strategies for increasing self-efficacy even when individual coaching is not possible due to a lack of resources. For example, teachers might prepare a class for their fellow PD-participants and carry out part of the class during the PD-meetings. With this respect, approaches like micro-teaching or approximations-of practice (Grossmann, Hammermess & McDonald, 2009) could be a fruitful approach. Another way to increase selfefficacy could be the use of vicarious experiences which refers to hearing about or observing other teacher's successful technology integration [START_REF] Wang | Increasing preservice teachers' self-efficacy beliefs for technology integration[END_REF].

Finally, the results from this study also give some indication about content selection for PDprograms. The result that self-efficacy beliefs are stronger correlated with frequency of technology use for supporting discovery learning than with frequency of technology use to support multiple representation might reflect that discovery learning requires much stronger self-efficacy beliefs due to its complexity. In contrast, the use of multiple representations is perceived as less complex and less challenging. PD-programs could therefore choose the support of multiple representations as starting point for novice teachers and advance to discovery learning in later stages.

In summary, this study shows that teachers' self-efficacy beliefs are an important factor when teaching mathematics with technology. PD-programs must provide means to strengthen teacher's self-efficacy beliefs if we want to take teachers needs seriously and change the degree of technology integration. Otherwise the outcome of PD-programs might be limited to knowledgeable teachers that are convinced about the benefits of technology, but lack the self-efficacy to put their knowledge and beliefs into practice. We should not miss the final step that might be self-efficacy beliefs.

Table 1 : Result of the correlation analysis (* p<0.1, ** p<0.01, *** p<0.001, TE=Teaching experience)

 1

	-efficacy

The term "confidence" is often used interchangeably with the terms "self-efficacy" or "self-efficacy beliefs". However Bandura notes that "Confidence is a catchword in sports rather than a construct embedded in a theoretical system"(Bandura, 1997, p. 382). Therefore, we will use the term "self-efficacy beliefs" throughout this paper.


Jan Franz Wörler
email: woerler@mathematik.uni-wuerzburg.de

How to distinguish simulations? Development of a classification scheme for digital simulations for teaching and learning mathematics

Keywords: Simulation, mathematical models, educational experiments, representations

Computer simulations are used to teach and learn mathematics, and didactic research also uses the term. However, there are sometimes very different meanings of the term 'simulation'. As a possible result didactic research on the use of simulations can only be compared to a limited extent. The present article therefore clarifies the concept of simulation. Based on this, a scheme for the classification of digital simulations, independent of the concrete content, is designed, which allows to distinguish simulations a priori and objectively according to theoretically derived didactic criteria and thus to order the wide field of "simulations". Finally, it is briefly outlined how the scheme can help teachers to select suitable simulation implementations for teaching and learning purposes.

Introduction

There is a multitude of computer programs, apps and files for learning and teaching mathematics, which are called "simulations" by their authors. These are, for example, Excel data sheets for the replication of cube throwing [START_REF] Böer | Erst per Hand, dann am Rechner. Simulationen stochastischer Prozesse mit Excel[END_REF] or algorithms in computer algebra systems for calculating hiking trails within the framework of mathematical modelling [START_REF] Bracke | Mit Hilfe der Mathematik zum Gipfel: Bergsteigen als Thema der mathematischen Modellierung[END_REF]. In the online portal GeoGebra Materials (www.geogebra.org/materials), on which authors can publish GeoGebra files, over 1000 hits are listed under the search term simulation for the year 2017. It can be assumed that at least some of these documents were designed for teaching-learning contexts and are actually used for learning mathematics. MathEduc (www.zentralblatt-math.org/matheduc/), a database for publications with a mathematics didactic background, currently lists over 2000 publications that contain the term simulation.

These are just a few examples of "simulations". However, their consideration already suggests the interpretation that research, theory and practice for learning and teaching mathematics are currently concerned with simulations: simulations are used to teach mathematical content or procedures at present. The above examples also show how differently the term simulation is used: the examples refer to a broad variety of mathematical contents, use different representations, are based on different types of mathematical models, allow partly very extensive manipulations and partly none, are used to pursue different objectives or to fulfill different functions. This leads us to the thesis that the term simulation is used in connection with the teaching of mathematics, but that its meaning remains in general individual and vague.

If, however, didactic research aims to collect empirical research data on the use of computer simulations for teaching and learning mathematics and if these data and results are to be comparable, then a common conceptual basis for simulations is indispensable. This article presents ideas for the design of such a basis by theoretically deriving a multidimensional classification scheme for digital simulations. For this purpose, essential characteristics of simulations are first deduced, which then depending on their traits are combined into a system to distinguish between simulations. The article therefore investigates the following research questions:

(Q1) Which central characteristics and features do digital simulations for learning mathematics have? Which traits of these characteristics can be derived? (Q2) How can these characteristics and traits be used to assign computer simulations a priori and objectively into classes and to distinguish them from each other?

Methodological approach

In order to clarify research question Q1, a hermeneutical approach is pursued: it derives essential characteristics of simulations from existing theories (e.g. systems theory) and application sciences by means of a literature research and thus searches for the theoretical meaning of the term simulation. The focus will be on those aspects that are important from a didactic point of view. The approach is alternately confronted with a comparison with such properties of programs, apps and files which exist in reality (on the Internet, in textbooks, as learning apps or the like) and which are called "simulations" by their authors. They show concrete characteristics of the properties. In this way, the question is examined in the alternation of deduction from theory and induction from teaching-learning practice and thus does not lose its relation to the practice of teaching mathematics.

Theoretical foundations

When it comes to simulations, a distinction was usually made between physical simulation with concrete models and simulation with abstract or mathematical models [START_REF] Krüger | Simulation: Grundlagen, Techniken, Anwendungen[END_REF]. In the latter case, simulation is a calculation carried out on a computer, which Krüger (1974, p. 27) calls "calculation experiment" and which Hockney and Eastwood (1988, p. xix) call "computer experiment". On the one hand, this distinction is very rough. On the other hand, the two categories become blurred as soon as physical experiments are virtually mapped on the computer, as explained in detail in [START_REF] Wörler | Computersimulationen zum Lernen von Mathematik[END_REF]. Therefore, they are not useful for classification.

For this reason, we assume a more general approach: we interpret simulationin close agreement with [START_REF] Krüger | Simulation: Grundlagen, Techniken, Anwendungen[END_REF], [START_REF] Kobayashi | Modeling and analysis: An introduction to system performance evaluation methodology[END_REF] and [START_REF] Greefrath | Simulieren: Mit Modellen experimentieren[END_REF] as the conduct of experiments with models of systems. It is therefore necessary to clarify the terms experiment and model in order to understand the concept of simulation.

Models

According to Henn (2011, p. 419), "models are simplifying presentations, which consider only certain, somehow objectifiable parts of reality". This characterization corresponds well with the examples of simulations mentioned above, because they refer to a real situation or a real object, which is reproduced virtually on the computer. In addition, simulations are also conceivable that take up intra-mathematical phenomena: experimental work with geometric figures or with integral and derivative can serve as examples for simulations that have no explicit reference to reality. Therefore, the term model has to be understood more broadly with regard to simulations. For example, Velten (2009, p. 3) states: models are "simplified descriptions of a system". A system in turn "is an object or a collection of objects whose properties we want to study" (Velten, 2009, p. 8). It consists, according to [START_REF] Bossel | Simulation dynamischer Systeme: Grundwissen, Methoden, Programme[END_REF], of individual elements connected by relations. A model therefore contains a mapping of the relevant elements and relations of the system. This is illustrated in Figure 1.

Experiments and variations

Experiments are particularly widespread in the social and natural sciences, where they are a central method of gaining scientific knowledge. They can be used to create or to test hypotheses and usually they use the observation of the effects of targeted manipulations of a system. For physics education, for example, Kircher, Girwidz and Häußler (2009, p. 244) describe how to proceed in experiments: "Under fixed and controllable framework conditions, observations and measurements are carried out on physical processes and objects; variables are systematically changed and data are collected". Accordingly, the systematic variation of independent variables within defined framework conditions is the central component of conducting experimentsand thus also of simulations.

The important role of systematic variation in the learning and teaching of mathematics has already been emphasized: [START_REF] Schoenfeld | Mathematical Problem Solving[END_REF] sees variation as a heuristic principle in problem solving. [START_REF] Schupp | Thema mit Variationen: Aufgabenvariation im Mathematikunterricht[END_REF] adds that variation as a student activity can help to convey an authentic picture of mathematics and enable a deeper understanding. [START_REF] Philipp | Experimentelles Denken: Theoretische und empirische Konkretisierung einer mathematischen Kompetenz[END_REF] interprets intra-mathematical experimentationin the sense of hypothesis formation and hypothesis testing -as explorative problem solving. She considers this type of problem solving to be a process characteristic of mathematics, but also sees it as a fundamental competence with four partial competences: generating examples, structuring, formulating hypotheses, checking hypotheses.

These activities are addressed in the same way when experimenting with models of objects or systems in simulations. To vary allows the creation of different system states (called "data" by Kircher et. al., 2009, p. 244, andcalled "examples" by Philipp, 2013, p. 156) that can be analysed to learn something about the given system or to understand it. This applies to students as well as to scientists, which is why experimentation is also a central activity inquiry-based approaches (e.g. [START_REF] Artigue | Conceptualizing inquiry-based education in mathematics[END_REF]. Therefore, in the following we will distinguish simulations according to what types of variation they allow.

Types of variation

According to the above considerations, models can be traced back to their elements and relations. Furthermore, experiments are characterized by variation within controlled framework conditions. Under these assumptions, six different types of variations in simulations can be derived theoretically [START_REF] Wörler | Computersimulationen zum Lernen von Mathematik[END_REF]. Only the following three should be mentioned here, because the other three have mainly theoretical-epistemological value: these types are the possibility of variation of ... 1) the number of model elements, 2) the properties of the model elements, 3) the relations between model elements, They are hereinafter referred to as Possibilities of Interaction between the experimenter (e.g. the learner) and the underlying (mathematical) simulation model. Their existence can be checked objectively and a priori for each concrete implementation of a simulation. In this way, simulations can be distinguished according to the type and the number of implemented Possibilities of Interaction. We use this approach to clarify the research question Q2 below.

Specifics of digital simulations

With regard to the research question Q1 we conclude: Experiments involve the activity of varying. They also require the observation of the effects of the manipulation on the modelled system and its behavior. If experiments are transferred to the computer as digital experiments, these two aspects have to be translated, too.

This means that at least some of the Possibilities of Interaction (see above) have to be technically implemented and offered to the experimenter. Some control elements like checkboxes, sliders, input fields or selection menus must be available in every digital simulation to set and change parameters; [START_REF] Moreno | Interactive Multimodal Learning Environments Special Issue on Interactive Learning Environments[END_REF] call this type of interaction "manipulating" (p. 311). They distinguish manipulating from "controlling" (p. 311), which includes control elements for the speed or sequence of content (such as pause/play key) but does not influence properties of the modelled system. It occurs especially with moving images or animations. It is, however, consequent also to consider controlling in the classification scheme for simulations: On the one hand, both types of interaction occur in simulations side by side, although only manipulating allows experiments. On the other hand, the connection to preliminary work on the potential of moving images for learning mathematics [START_REF] Kautschitsch | Der Videofilm: Geeignetes Mittel zur Visualisierung und Entwicklung mathematischer Begriffe und Modelle[END_REF][START_REF] Samson | Video clips and cell phones: VITALmaths in the palm of your hand[END_REF] succeeds.

Digital experimentation also requires the possibility to observe and evaluate the modelled system and its states. Therefore, a computer output device is indispensable. This can be a conventional PC or smartphone screen. Alternatives could also be virtual reality glasses or simply a PC printer. On such output devices the result of a digital simulation (as final state after a manipulation) or the respective simulation model itself (with intermediate states during manipulation) has to be represented, for example as table, graph or diagram, but also as dynamic visualization of processes or experimental environments. Each digital simulation is therefore necessarily connected with an external representation of these model states.

Based on Bruner's modes of representation (enactive, iconic, symbolic) from 1964, [START_REF] Ladel | Multiple externe Repräsentatio putereinsatz: Zur Bedeutung für das Mathematiklernen im Anfangsunterricht[END_REF] proposes a scheme for the analysis of mathematical learning software on the basis of the forms of external representation used. She derives eight computer-based forms of external representation and specifically distinguishes between static and dynamic representations. [START_REF] Ladel | Multiple externe Repräsentatio putereinsatz: Zur Bedeutung für das Mathematiklernen im Anfangsunterricht[END_REF] also shows that how computer programs can be classified with regard to these forms.

Result: classification scheme

In order to answer the research question Q2, the above explanations were combined into a twodimensional classification scheme: The first dimension is formed by the Level of Manipulation (LoM). This number indicates how many Possibilities of Interaction are implemented in a concrete digital simulation, i.e. whether and to what extent a concrete digital simulation allows variations by the experimenter.

For clarification, imagine a dice throwing simulation with two sliders: One controls the number of the dice, one the number of the throws. There are then two ways to interact with the underlying simulation model, which gives the simulation a LoM of 2 (labeled as: LoM 2).

In the broadest sense, the LoM indicates the degrees of freedom that the implementation gives the experimenter for interaction with the underlying simulation model (in the sense of manipulating, according to [START_REF] Moreno | Interactive Multimodal Learning Environments Special Issue on Interactive Learning Environments[END_REF]. In this way, different implementations of a simulation can be analyzed with regard to their LoM and arranged on an ordinal scale.

If we accept LoM 0 (zero) on this scale, animations (generally: moving images) can also be located in the range between LoM 0 (no controlling or manipulation implemented) and LoM 1 (manipulation implemented). We thus take into account the type of interaction in the model that Moreno and Mayer (2007) call "controlling" (p. 311). The distinction between animation and simulation, which is not clear to outsiders in many cases, is transferred into the concrete specification of LoMs: If any controlling option is implemented, the LoM is increased by 0.5.

The second dimension of the classification scheme presented here can be defined according to the scheme of Ladel ( 2009 The result is a 5 × n-classification scheme with the dimensions external representation (with the four traits ISR, IDR, MSR, MDR) and plurality of variations (indicated in LoM), see Figure 2.

Each digital simulation can be assigned to the scheme: Let's assume that the dice throwing simulation from above represents only the final results of the experiment as bar chart. That would be an isolated static representation (ISR). The simulation would therefore be classified in the cell ISR × LoM 2 of the schema. If, in contrast, the dice throws are represented sequentially so that the result of the experiment is assembled step by step and the bars in the chart "grow", the representation is dynamic (IDR), so that the classification IDR × LOM 2 results. And if, in addition, a further controller is integrated which allows the speed of the process to be regulated, the LoM would be increased by 0.5 (classification: IDR × LoM 2.5); in this case, the position between two cells indicates that a dynamic sequence can be controlled that does not affect the final result.

Discussion and outlook

The classification scheme currently presented indicates the possibilities of experimentation in the form of LoM. With this construct, simulations can be ordered locally, i.e. two implementations of the same phenomenon or mathematical model can be compared with regard to their cognitive requirements: The higher the LoM is, the higher the cognitive requirements are.

Limitations of the scheme

Global ordering is currently not possible with the scheme, however, as the following example shows: A simulation is given in which one unbound variable b can be manipulated (LoM 1) and which displays the function graph (IDR). While the effect of the manipulation is easy to interpret for the graph of the function f(x) = x + b, it is much more difficult for f(x) = x 2 + b⋅x or even f(x) = x b + 2. Thus, the scheme currently lacks an analysis of the complexity or difficulty of the content, which is reflected in the complexity of the underlying mathematical model. It should take into account the prior knowledge of the experimenter (or learner). Therefore, it is planned to extend the scheme by a didactic analysis of the concrete simulation that examines the mathematical requirements for each possibility of interaction. In this way, the model could also enable global ordering of simulations.

Learning and teaching mathematics

Since extra-mathematical references are often taken up in simulations in order to learn mathematics, content references to related sciences are indispensable in mathematical simulations. Therefore, the scheme is not limited to learning and teaching mathematics. It can be useful whenever mathematical representations are used to experiment with models as a learner activity. That is the case, for example, in teaching and learning of modelling or problem solving in mathematics, but also in biology, physics or technology. This makes the classification scheme useful for Science, Technology, Engineering and Mathematics (STEM) education, too. Here, mathematical competences (e.g. Process Standard "Representation", National Council of Teachers of Mathematics, 2000, pp. 67-71) are also addressed via subject-specific content (physics, biology and so on).

Implications

A first explorative application of the category system to randomly selected "simulations" from GeoGebra Materials provides that in most cases the user can vary one to four aspects of the application (LoM 1-4; hardly LoM > 5). Often dynamic representations occur and also different forms of representation are combined to multiple dynamic representations (MDR).

One of the aims of the research work on the scheme is also to use the scheme to assess whether a concrete simulation is good or less suitable for a (concrete) learner. According to this, there should be good and less suitable classes in the scheme, depending on the individual prerequisites and goals. First cautious pointers to such classes are: for a pure visualization of dynamic processes, an implementation with 0 < LoM < 1 is sufficient; this includes video clips as well as animations. If simulations are to be used for explorative work in the context of teaching and learning, they have to allow some variations. Thus, only implementations with LoM ≥ 1 are suitable. With them, the modeled system can be understood by linking variation and effect, which can serve in problem solving, mathematical modelling or generating and testing hypotheses. It can be assumed that simulations with a high LoM generate a high cognitive load (according to [START_REF] Chandler | Cognitive Load Theory and the Format of Instruction[END_REF] for the experimenters (i.e. learners). The performance of the users would thus determine how high the LoM should be at most when selecting a simulation. The respective external representations offered in a simulation, although they bring benefits for the learning process, can also create further obstacles in the learning process (e.g. [START_REF] Ainsworth | Analysing the Costs and Benefit of Multi-Representational Learning environments[END_REF][START_REF] Ainsworth | DeFT: A conceptual framework for considering learning with multiple representations[END_REF][START_REF] Duval | A cognitive analysis of problems of comprehension in a learning of mathematics[END_REF]. Building on this, at least one implication is conceivable: in order to keep the simulations understandable, rather small LoMs should be used for MDR, whereas higher LoMs would be suitable for ISR and IDR (as indicated in Figure 2). By means of such considerations, the presented scheme can help teachers to find suitable simulations for a certain group of learners.

A detailed analysis of the classification scheme as well as the empirical and theoretical validation of the conclusions is still pending.

Figure 1 :

 1 Figure 1: Components of a simulation

  ): The eight types of external representations she proposes can be aggregated into the disjoint categories isolated static representation (ISR), isolated dynamic rep. (IDR), multiple static rep. (MSR) and multiple dynamic rep. (MDR). Every concrete implementation of a simulation can be analyzed in terms of how the changes in the model states triggered by the variation are represented on the output device.

Figure 2 :

 2 Figure 2: Two-dimensional classification scheme for digital simulations