Eirini Geraniou
email: e.geraniou@ucl.ac.uk

Cosette Crisan
email: c.crisan@ucl.ac.uk

University students' engagement with an asynchronous online course on digital technologies for mathematical learning

Keywords: Online learning, asynchronous communication, student engagement, digital technologies, mathematical learning

Promoting students' active and meaningful engagement in online learning environments, and especially in asynchronous courses where students are not interacting with the lecturer or their peers in real time, can be challenging. This paper presents the initial results from an exploratory study on how the tutors' design considerations of a ten-week online asynchronous master's course, together with their online pedagogic strategies have impacted on students' learning experiences and their engagement with its resources, such as content, peers and tutors. The data we focus on consists of students' online contributions and one-to-one interviews. Our research aims to gain an insight into students' views on how their engagement impacted on their online learning for this course, leading thus to identifying opportunities and barriers to take into account when designing for active learning in an online course.

Introduction

Active learning has a reputation of being effective in supporting students' retention of knowledge, addressing potential misconceptions, promoting engagement and encouraging positive attitudes towards learning [START_REF] Michael | Where's the evidence that active learning works[END_REF]. Considering these arguments, educators should be encouraging active learning in every mode of study, such as face-to-face traditional teaching, online synchronous, asynchronous or blended learning. Even though active learning strategies, such as Think-Pair-Share, Role Playing, Group Discussion, etc., are widely used and have been shown to be successful in face-to-face, as well as virtual yet synchronous teaching situations, achieving active learning in an online asynchronous learning environment presents a number of 'new' (as is not widely researched and addressed) challenges [START_REF] Riggs | Actively Engaging Students in Asynchronous Online Classes[END_REF]. For example, some authors (e.g. [START_REF] Picciano | Beyond student perceptions: Issues of interaction, presence and performance in an online course[END_REF] claim that online discussions and learning could distance students from convergent thinking, instructor directed inquiry and scientific thinking compared to face-to-face discussion. Others (e.g. [START_REF] Jiang | A study of factors influencing students' perceived learning in a webbased course environment[END_REF][START_REF] Dixson | Creating Effective Student Engagement in Online Courses: What Do Students Find Engaging[END_REF] found that students' online learning very much depends on the quantity and quality of their peers' postings in online discussions, which is certainly an unpredictable factor. [START_REF] Haythornthwaite | Building social networks via computer networks: creating and sustaining distributed learning communities[END_REF] and [START_REF] Rovai | A preliminary look at structural differences in sense of classroom community between higher education traditional and ALN courses[END_REF] found that the greater variability in the online social interactions between students in terms of regular presence, participation, etc. and their sense of belonging in a community and trusting their peers in their mutual learning journey can take longer than face-to-face learning in a classroom environment. And just as importantly, practical issues such as easy access to the online platform and the course resources and student support when technical difficulties arise, need to be offered (e.g. [START_REF] Hewitt | How habitual online practices affect the development of asynchronous discussion threads[END_REF].

There are great benefits of learning in an asynchronous environment. Students have more time to reflect on the resources, the tasks, their tutors' and peers' contributions, but they also have more time to invest in offering more thoughtful and insightful contributions [START_REF] Collison | Facilitating online learning: Effective strategies for moderators[END_REF], than perhaps in a face-to-face interaction. [START_REF] Riggs | Actively Engaging Students in Asynchronous Online Classes[END_REF] found that more students choose online asynchronous courses due to the flexible mode of study, i.e. managing their time to work on the given tasks as and when their own schedules permit, but also spending as much time as they want per task. For more introvert students and those who are not given the opportunity to share their thoughts in face-to-face learning environments due to time constraints in sessions, the authors concluded that online asynchronous courses can give them more time to compose their thoughts and contribute.

To address the above-mentioned challenges, educators have striven to design asynchronous online courses that promote active learning and meaningful engagement from students and offer support by adapting their own pedagogical strategies accordingly [START_REF] Riggs | Actively Engaging Students in Asynchronous Online Classes[END_REF] and developing new such pedagogical strategies. Considering our own experiences as tutors in teaching such a course for the past four years, we recognized our role shifting from that of a subject matter expert, a course developer and a tutor to that of: (i) a technology trainee, to ensure we were familiar ourselves with various digital tools, (ii) a trainer, to support our students in their interactions with these digital tools and (iii) a facilitator or moderator, to manage our students' online contributions and interactions with their peers in online discussion forums. In particular, our role in promoting students' active and meaningful engagement in online discussion forums has been one of our greatest challenges. To identify strategies that support and maintain such engagement throughout the duration of the course, while at the same time evaluating and revisiting the design of our course, we carried out an exploratory study.

In this paper, we present our online asynchronous master's course on digital technologies for mathematical learning, followed by a description of the methodology of our exploratory study that investigated the extent to which and how the asynchronous online experiences impacted on our taught postgraduate students' learning in this course. The initial analysis from this exploratory study data follows. Some conclusions are then drawn, while the key findings of our study are put forward as key recommendations for nurturing active learning opportunities in any online asynchronous course.

Our online asynchronous course

In 2014, we designed a brand new course Digital Technologies for mathematical learning, the first and only online course of the University College London (UCL)'s MA in Mathematics Education master programme, which uses the UCL Moodle platform for its delivery. There are two e-learning aspects of this course: (1) its online delivery and (2) the focus on digital technologies of the course itself, consisting of (i) familiarisation of the students with a wide range of digital tools and resources (graph plotters, dynamic geometry environments, statistical software, fully interactive online packages) and (ii) critical reflection on the implications of using such tools in the learning and teaching of mathematics mainly at secondary school level (11-18 years old students).

The main aim of this 10-week course is to encourage students to reflect critically on the potential and limitations of digital technologies for the learning and teaching of mathematics by providing opportunities for students to apply knowledge of relevant research and theory in practice. Besides getting familiarised with digital tools and reflecting upon their use as 'learners' and 'users' themselves, students are also encouraged to design activities to use with at least three different digital tools and then evaluate them by trialling them with learners. In this process they carry out mini-research studies and reflect upon the use of digital technologies for mathematical learning. The course is delivered in 10 weeks and each week the tutors upload on the UCL Moodle platform: (1) the learning objectives and a description of the week's content, (2) the key readings: one essential and two indicative ones and (3) the offline and online weekly tasks, which are tightly structured towards the course aims and the learning objectives. Instead of direct teaching, learning is designed to take place as the result of doing short, manageable offline and online tasks. The offline tasks include (1) familiarising with a piece of software and going through scaffolded activities using specific software, and (2) designing and trialling mathematics activities involving the specific software that bridge learners' interactions with digital media and the mathematical concepts. The online tasks include: (1) engaging with the ideas in the key readings and writing a response about the points they agreed or disagreed with or by contributing to online discussion forums with written observations on views and perspectives of fellow students; (2) reflecting on the application of the ideas encountered in the key readings in specific learning contexts such as the offline tasks they designed and trialled. Every week, the students are required to carry out the reading and the weekly tasks, then post their reflections online, and respond to their peers' posts. There are three themes covered in our course: (a) Visualising, (b) Generalising and Expressing and (c) Modelling. In these three themes, the students interact with many digital tools (DT), such as Excel, GeoGebra, Autograph, the eXpresser microworld, Logo, Scratch and a variety of modelling applications and online digital platforms. Research (e.g., [START_REF] Dixson | Creating Effective Student Engagement in Online Courses: What Do Students Find Engaging[END_REF] indicated that it can be helpful for instructors of online asynchronous courses to create an architecture for their own engagement, too. As tutors, we made sure that we had social presence demonstrated on a weekly basis through: uploading materials, posting messages, guiding student learning, providing feedback according to set times, sending reminders, contributing to some online discussions.

Methodology

For the four years we have run this course, we have conducted end of theme, as well as end of course evaluations and while the vast majority of our students showed 'online' presence by posting their work and occasionally posting questions to tutors regarding practical issues about the work, we have recognised that they were not engaging in online collaborations with their peers as much as we wanted them to, for a full learning experience. In our efforts to encourage our students' active and meaningful engagement in online collaborations, informed by findings of research (e.g. [START_REF] Dixson | Creating Effective Student Engagement in Online Courses: What Do Students Find Engaging[END_REF], we trialled some actions, which included: tasks that formed the basis of an online group discussion; collaborative tasks with tutor-nominated groups; pick-a-paper tasks, i.e. 'choose one paper and its commentary from a member of your small group and post your comparative remarks and reflections'. While these strategies worked in terms of increasing the likely hood of students contributing online, we also became interested in why some students were more engaged than others, why they displayed a higher degree of learner autonomy, evidenced in their inquiring further, seeking assistance and generating new threads of discussion, than others. As such, we became interested in enquiring into our students' perceptions of the extent and the way their asynchronous experiences contributed to their active and meaningful engagement on this course.

To address this research question, we decided to carry out an exploratory study focusing on data collected from our latest cohort of students. This data comprises 20-min interviews with a sample of 10 of the 17 students enrolled on this course and their online contributions throughout the delivery of the course (January-March 2018). Of the 10 students who volunteered for the interviews, three were English native speakers, while seven were international students, for whom English was an additional language. The interview questions focused on students' mode of study and students' experiences of online asynchronous learning, in particular their experiences of participating in online discussion forums. Our intention is to use the outcomes of this exploratory study to inform our thinking and identify issues to research further, but also rethink our practices in terms of communicating with our students and providing effective instructional strategies for improved student communication and collaboration. This will lead to the re-design of our course in its next presentation, focusing on encouraging active learning and engagement with the online course resources.

The interview data was analysed in conjunction with the students' online contributions and written assignments, following the grounded theory approach [START_REF] Strauss | Basics of qualitative research; grounded theory procedures and techniques[END_REF]) and using NVIVO to organize, manage and analyse this data. Our students' social processes regarding their mode of study and online contributions were analysed by going through the interview transcripts and coding their comments using labels (i.e. short phrases or sentences) that described what they said. This process was repeated a number of times until there were no more issues identified in the data, no improvements to the current codes were needed and no more codes were produced. At the end of this iterative process, these labels had become analytical representations of the data and turned to the identified categories we share later in the Results section. By doing a constant comparative analysis of the emerging categories, we were able to identify two emerging themes and finalised the categories. Next, we present the emerged themes and categories in the way we linked and integrated them aiming at capturing all the variations in our data and allowing a theory to emerge, that of students' motivation and challenges they faced regarding online contributions in an online asynchronous course.

Results

The two main emerging themes grounded in students' responses are: Motivation to contribute in online asynchronous learning encouraged and sustained through Learning from others; Peer Assistance; Peer students' online study patterns; Interest in and enjoyment of topics, and Challenges to contribute in online asynchronous learning such as Non-participation; Visibility of learning; Style and Etiquette of communication; Feeling at ease with your peers. For each of these two emergent themes, students had commented upon the tutors' influence in their learning.

Motivation to contribute in online asynchronous learning: Learning from others Those students who engaged with the online activity felt that they benefitted as a result. One international student, Jennie, who enrolled on this course following her undergraduate studies, admitted feeling comfortable to participate in the online discussion forum as she learned a lot from reading the contributions of what she thought of as 'her more experienced and older than her' peers. She admitted looking specifically for their contributions and trusting their views.

Yvette felt that reading research papers, reading other students' work and having to give feedback and comment on others' contributions helped her with her own learning: gaining a deeper insight into the key readings, thus learning about how to engage with research and how to use it to reflect on mathematical learning when digital technologies were used. Similarly, John reckoned he learned from his peers: "I felt like you were in a classroom with other students in a way. Obviously with it being asynchronous it's a lot slower than that, but you kind of need it to be asynchronous so you can actually go and read what people have written. And then it was easy to find again what you've written and how other people have responded", while Jack said that seeing other students' contributions triggered him to respond, hence explicitly engaging with the knowledge base of this course.

Mary and Anne mentioned the opportunities to learn from others, which were facilitated by the tutors' collated feedback on all students' contributions per theme; they would revisit this feedback often, which helped them be more aware of the expectations of this course and improve their future work.

Motivation to contribute in online asynchronous learning: Peer assistance All 10 students commented on how useful the UCL Moodle Discussion Forum was for sharing ideas, for reading other students' contributions and reflecting upon their own thinking and understanding of the course material, tasks and resources. "Sometimes if I had no idea how to start my work and I saw how my classmates did it, then I had some directions" (Janet). Having an online record of the peers' contributions enabled the students to revisit aspects of the course at any point throughout the course. One student, Frances, extended this argument by comparing the online contributions to those of a face-to-face learning environment, where "you can only hear a few people's ideas" and mentioned how useful it was to have the time to read and reflect upon them before responding and/or asking questions, which wouldn't have been the case in a time constrained face-to-face interaction: "You have all these online resources to access and explore, and you can ask directly the creators".

Even though there was some peer support offered in Discussion Forums, many students told us that they needed "immediate feedback". For example, Mary argued for the importance of having a tutor 'close by' when learning about and interacting with a new DT. She felt that she needed to ask questions that needed immediate answers in order for her to make progress. Similarly, Patricia and Helen argued about their preference for face-to-face sessions when familiarizing themselves with a new DT, as in that scenario the help was immediate and support was always there (we offered two optional face-to-face sessions as part of this course).

Motivation to contribute in online asynchronous learning: Peer students' online study patterns

The variance in the students' online presence depended on many factors, such as their interest in the topic, their other commitments (work, studying, deadlines to assignments, etc.), but also their preferred style of studying. All the 10 students agreed they liked the flexible mode of study this course offered, as most of them found it very convenient to choose their own time to study.

Six of the 10 students (Janet, Anne, Patricia, Jack, Frances, Mary) admitted that time management was a challenge in this course. In some cases, it was as result of their own struggle at times to selfmanage their work and not because of the structure of the course. But in other cases, it was due to 'wait-time' for others to contribute online to Discussion Forums. Anne agreed that timing is very important even if the course is asynchronous and flexible in that sense. She would never read any postings that were uploaded too late in the week, as she would have already finished her work by then. Similarly, Patricia argued: "I can't be bothered to respond to that now, because I'm on to the next one. It's not fresh in my mind anymore". Frances, too mentioned how important it was for peers to post on time otherwise "You are delaying your partner's work too".

Motivation to contribute in online asynchronous learning: Interest in and enjoyment of topics

& tasks Looking at the patterns of all students' online participation on a weekly basis throughout the course, we noticed that there were weeks when their online presence was much higher than in others. The 10 students we interviewed argued that this was due to their own interests in certain topics covered during certain weeks. For example, Janet said that being interested in a topic encouraged her to work on the task earlier in the week than when she didn't like the topic or she was not very familiar with it, e.g. programming, and she would engage with others' online contributions too.

All 10 students valued the task which required them to design and trial an activity with learners, then reflect on the mathematical learning. John and Jack argued how great it was to reflect on pedagogical issues regarding students' learning mathematics with DT. They all commented about learning not only new DT, but also learning how to utilize them in the classroom and use them effectively with learners. Jennie felt that this course was indeed a great preparation for her future career as a mathematics teacher. She felt she got more time to think about the relationship between the DT and mathematics and how it could be used in her own way in mathematics classrooms.

Challenges to contribute in online asynchronous learning: Peer students' non-participation

Lack of sustained participation from students seemed to be a key factor for disengagement from online discussion forums. As the course progressed, Jack noticed that not many of his peers participated nor did they respond to his comments. He felt de-motivated and as a result lost his keenness for ongoing online participation. Similarly, due to the lack of peer participation, Anne found it difficult to contribute online, as there were no contributions from others that she could comment on. Even if there were contributions, if too few, there were only so many opportunities to comment on them.

Challenges to contribute in online asynchronous learning: Visibility of learning Amongst the reasons mentioned by our students as holding them from contributing online included being afraid of showing their insecurity and lack of confidence in their own expertise with DT or subject matter of the course. Mary confessed: "I know they won't say are you stupid, how can't you not know this? But I still feel people get the impression that you don't know anything and they would hesitate comparing or checking your contributions".

Visibility of online contributions was even more of a challenge for the non-native English students, who were also worried about the clarity of expressing their thinking in written format. Six out of seven of our international students admitted that they were reluctant to submit comments online. Mary admitted that she would probably hesitate to make a contribution in a face-to-face session for that same reason, but online her contributions remain visible to others for the duration of the course. This was also a concern for Vicky, who commented that she preferred the online mode of communication to the verbal one as it gave her time to think through what she wanted to say, with the advantage that she would 'correct her English and grammar before posting anything'.

Challenges to contribute in online asynchronous learning: Style and Etiquette of communication

Helen mentioned the challenge of communicating in the Discussion Forums; when asking a question, she would need to spend time thinking and articulating the question accurately and in a concise format, whereas a face-to-face scenario is more like having a chat and the question might be formed by itself. The formality of the Moodle discussion forum was mentioned by Anne too. She was put off by the formality of such communications and the fact that tutors would see such contributions and that their posts would also be assessed. Her perception of the role of online contributions led to her being reluctant to upload anything that she hadn't thought through very carefully. Being a native English speaker did not make it easy to adhere to the more formal style of communication; Patricia too admitted feeling occasionally reluctant to post any comments as she had to make them formal.

However, all 10 students admitted that this formality required in all of their online contributions did lead to clarity and coherency of arguments that they were trying to put forward. Having tried to set up a group chat on Facebook Messenger (the seven international students) Jennie found the communications were very informal, with far too many details which she didn't need to read, and with ideas that lacked clarity, although she, like all the others liked the instant feedback.

Challenges to contribute in online asynchronous learning: Feeling at ease with your peers Another reason for being disengaged from the Moodle Discussion Forum was the lack of familiarity with their peers, having not met them in person before the start of the course, which led to a lack of a sense of being part of a community. For example, for Helen it was difficult to post on the forum discussion and comment on work from students she never met face-to-face. Anne thought that her peers who she was not acquainted to were too 'polite' in their comments to her work, so as not to hurt her feelings. This familiarity and friendship with the people on Messenger allowed them to share comments freely, not thinking that they would be upset or offended. Another disadvantage of posting feedback to a peer in a forum is that the way the feedback is articulated might be considered bad criticism and upset their peer, whereas in a face-to-face situation, they could see that the feedback is expressed in a positive manner and their peers can recognize that a student is trying to help and give constructive feedback. For example, in Messenger you can put a smiley face too, and so no matter how the feedback was articulated (as a criticism, for example) the receiver would not see this as a negative comment.

Some conclusions and our key findings

Some of our findings are similar to the ones by authors we referred to earlier in the introduction. All students were very appreciative of the flexible mode of study of our online asynchronous course and the time given to reflect upon peers' contributions, which are similar results to [START_REF] Riggs | Actively Engaging Students in Asynchronous Online Classes[END_REF] work. Our students' learning did depend on the quantity and quality of peers' postings [START_REF] Dixson | Creating Effective Student Engagement in Online Courses: What Do Students Find Engaging[END_REF] and the tutors' feedback and their trust in their peers [START_REF] Haythornthwaite | Building social networks via computer networks: creating and sustaining distributed learning communities[END_REF].

However, our study brought to the fore two important aspects not mentioned in the literature we reviewed. We learned from our data analysis that online communication was perceived by our students as a double-edge sword; on one 'edge' they all liked the flexible mode of study, managing their own time, taking time to think through their responses or others' contributions, but the formality of their contributions deterred them from engaging in conversations with peers more fully. We also learned that we needed to orchestrate better the timing of online contributions in an asynchronous course: while having the flexibility to contribute at their own convenient times, deadlines are even more important to be set and met, in order to ensure an active and meaningful engagement and exchange of views. We thus conclude this paper by putting forward these two key findings as recommendations to designers and tutors of online asynchronous courses, with the aim of nurturing active learning opportunities.

Acknowledgment

This work was supported by UCL Grand Challenge of Transformative Technology, which funded our project: Supporting active and meaningful engagement in an asynchronous online course.


Lisa Göbel
email: lisa.goebel@uni-due.de

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

Dynamic vs. static! Different visualisations to conceptualize parameters of quadratic functions

Keywords: Visualisations, parameter, technology

Background

The technology has gotten faster over the last years; students became more proficient in the use of smartphones etc. But is the power of speed with a quick visualisation when dragging a slider bar or grabbing a graph really beneficial while learning and understanding the concept of parameters? The study presented here uses a guided discovery approach (adapted from [START_REF] Mosston | Teaching: From Command to Discovery[END_REF] to investigate different visualisations to conceptualize parameters of quadratic functions as it is the first non-linear function taught and can serve as a prototype for the conceptualization of parameters of higher polynomial functions. To specify the intended learning process in the study, the core elements for a conceptual understanding are presented: For this study in the frame of quadratic functions the two mathematical objects variables and functions and their "Grundvorstellungen" (as defined in vom Hofe & Blum, 2016) are important for the conceptualization. For functions it is important to develop the ideas of a function as mapping, covariation and object [START_REF] Vom Hofe | Grundvorstellungen" as a Category of Subject-Matter Didactics[END_REF]. All these three aspects are crucial, when learning how a graph is changing when one of the parameter changes. Parameters can be classified as a specific type of variables. While many authors distinguish the Grundvorstellungen of variables as unknown, generalized number and changing variable (e.g. [START_REF] Malle | Didaktische Probleme der elementaren Algebra[END_REF], [START_REF] Drijvers | Learning algebra in a computer algebra environment: design research on the understanding of the concept of parameter[END_REF] applies these specifications on parameters. The role of parameter is not fixed to one aspect, but can be placeholder, unknown, changing quantity or generalizer. All roles are crucial to understanding the concept and the conceptualization can be supported through technology. Another important aspect to understand the concept of a function is to be aware and to be able to change between different representations of functions (algebraicsymbolical, numeric-tabular, graphic-visual, situative-linguistical, e.g. [START_REF] Duval | A Cognitive Analysis of Problems of Comprehension in a Learning of Mathematics[END_REF]. [START_REF] Duval | A Cognitive Analysis of Problems of Comprehension in a Learning of Mathematics[END_REF] describes it as the critical threshold for learning. This can be supported by technological tools [START_REF] Drijvers | Uses of technology in lower secondary mathematics education: A concise topical survey[END_REF]. It is still an open question whether the conceptualization differs when different visualisations are used when learning. These points lead to our research questions: How can technology-assisted guided discovery support the conceptualization of parameters in the field of quadratic functions?

Methodology

In a control-group design intervention study with one control-and three experimental groups a total of 14 Year 9 classes worked on a series of tasks designed using a guided discovery approach, which was implemented through hints on the tasksheet how to investigate the influence (for information on the tasks see [START_REF] Göbel | Power of Speed" or "Discovery by Slowness": Technology-assisted Guided Discovery to Investigate the Role of Parameters in Quadratic Functions[END_REF]. The different control-and experimental groups differed only in the kind of visualisation used. The control group had no technological visualisation while the three experimental groups used different types of digital environments (standard function plotter tool, pre-programmed file with the possibility to drag the graph, pre-programmed file with sliders for the parameters) to investigate the influence of parameters in the vertex form of quadratic functions . Students designed a summary sheet of their findings during the intervention, which was collected. For an overview on the intervention and the differences between the four groups see [START_REF] Göbel | Power of Speed" or "Discovery by Slowness": Technology-assisted Guided Discovery to Investigate the Role of Parameters in Quadratic Functions[END_REF]. A paper-pencil test was given to all students to collect baseline data for the comparison between classes. As well as the summary sheets collected a focus group in 13 classes was filmed. The summary sheets were analysed using a coding manual developed by qualitative content analysis. The videos transcripts were analysed with regards to the role of parameters, as well as using technology to explore and validate students' own hypotheses.

Results

Overall it can be said that the dynamic visualisations seem to support the students learning better than the static visualisations. Dynamic groups design significantly more overall appropriate summary sheets. Students in the dynamic visualisations groups, so students using sliders (SL) or drag mode (DM), found out more details about the influence of the parameters than students in the control group or students using a standard function plotter (FP). While in the control group 70,4% of students identified the influence of parameter c adequately (FP 46,3%, DM 76,5%, SL 73,8%), this number drops to 52,1% while identifying the influence of parameter b (FP 40,3%, DM 68,2%, SL 72,3%). Analysis of the summary sheets show that students in the dynamic groups seem to view parameter more as a changing quantity compared to students in the static groups who view parameter as placeholder. The video analysis shows that students use the technology to explore on their own, explain something to each other or to check their hypotheses even though they worked on their own in a guided discovery. Technology-assisted guided discovery can support the learning, but the pre-structuring and dynamic manipulation of the visualisation is crucial for benefiting the learning.


Markku S Hannula
email: markku.hannula@helsinki.fi

Miika Toivanen
email: miika.toivanen@helsinki.fi

The effect of digital tools on

Keywords: Attention, computers, cooperative learning, eye movements

Introduction

Paper and pen is a different medium than computer and screen and dynamic geometry environment (DGE) is very different from paper when solving a geometry problem. It is well documented that technology in general [START_REF] Chauhan | A meta-analysis of the impact of technology on learning effectiveness of elementary students[END_REF] and DGE in specific [START_REF] Chan | Dynamic geometry software improves mathematical achievement: Systematic review and meta-analysis[END_REF] have positive effects on mathematics achievement. There is much research on the specific affordances of DGE in learning and problem solving. For example, [START_REF] Christou | Problem solving and problem posing in a dynamic geometry environment[END_REF] argue that DGE as a mediation tool encourages students to use modeling, conjecturing, experimenting, and generalizing in problem solving. [START_REF] Healy | Software tools for geometrical problem solving: Potentials and pitfalls[END_REF] claim that DGE can scaffold the solution process and help students move from argumentation to logical deduction, while for less successful students the DGE may prevent them from expressing their mathematical ideas.

One way to examine how the learners' experiences are different in DGE and paper is eye-tracking. Human gaze consists of approximately three to four fixations (maintaining of the visual gaze on a single location) in a second [START_REF] Rayner | Eye movements in reading and information processing: 20 years of research[END_REF]. [START_REF] Hartmann | Exploring the numerical mind by eye-tracking: A special issue[END_REF] compare eye-tracking information to mind-reading: the target of a fixation usually tells what we think about and the fixation duration corresponds with processing time. Fixation duration is an established indicator for perceptual or cognitive processing difficulty, also in the context of mathematics [START_REF] Rayner | Eye movements in reading and information processing: 20 years of research[END_REF]. [START_REF] Glöckner | An eye-tracking study on information processing in risky decisions: Evidence for compensatory strategies based on automatic processes[END_REF] summarize research evidence to suggest that gazes related to more automatic processes would have shorter fixations (below 250 ms) and more elaborated information processing generally requires long fixations of more than 500 milliseconds.

Visual attention has been studied mostly in the context of reading and research on mathematics is much less frequent [START_REF] Hartmann | Exploring the numerical mind by eye-tracking: A special issue[END_REF][START_REF] Rayner | Eye movements in reading and information processing: 20 years of research[END_REF]. In a systematic review on reading on paper and digitally, [START_REF] Singer | Reading on paper and digitally: What the past decades of empirical research reveal[END_REF] found out that reading comprehension is influenced by the text presentation. However, in the context of mathematics, we did not find any comparative eye-tracking studies between digital and non-digital learning environments.

Existing eye-tracking research on mathematics shows that the method is relevant. [START_REF] Andrá | Reading mathematics representations: An eye-tracking study[END_REF] investigated how students read mathematical texts. In their study, fixation durations were typically in the range from 190 to 250 milliseconds. Fixations were longer for formulas than for graphs or text, but graphs attracted more fixations, leading to longer time spent looking at the graphs. In [START_REF] Lin | Tracking eye movements when solving geometry problems with handwriting devices[END_REF] study students were solving problems on tablets, and they suggest fixation counts, time on target, and run counts as relevant measures, because these differentiate successful from unsuccessful solvers and correlate with perceived difficulty.

We find the dominant methodologies for eye-tracking problematic. So far, most studies have been conducted in laboratory situations. We believe that problem solving needs to be studied in ecologically more valid contexts. We are interested in problem solving in contexts, where multiple modalities are present [START_REF] Arzarello | Gestures as semiotic resources in the mathematics classroom[END_REF] and multiple goals need to be addressed [START_REF] Hannula | Motivation in mathematics: Goals reflected in emotions[END_REF]. Most importantly, we are interested in problem solving in collaborative situations. Our earlier studies show that mobile eye tracking provides interesting data on attentional behavior in real classroom situations (e.g. Garcia [START_REF] Moreno-Esteva | Using gaze tracking technology to study student visual attention during teacher's presentation on board[END_REF][START_REF] Haataja | Teacher's gaze behavior when scaffolding peer interaction and mathematical thinking during collaborative problem-solving activity[END_REF]). While we have not found studies examining the visual attention in digital and non-digital mathematics learning environments, it seems likely that a digital tool would have an effect on visual attention even in the context involving collaboration. We formulate our research question as follows: Does the choice of learning environment (paper vs. GeoGebra) have an overall effect on student attentional processes as indicated by fixation durations when students are solving a geometry problem.

Method Participants

We examined fixation durations for one teacher and her eight students. The teacher taught the same problem solving session twice in two different Finnish grade nine classrooms. The first lesson was recorded in May 2017 and then the students solved the task using paper and pencil. We call this Paper lesson, even if at the end of the lesson the students continued examining the same problem using computers. The second lesson was recorded in May 2018 and then the students solved the task using GeoGebra software. This lesson we call the GeoGebra lesson, even if the students used also pen and paper to some extent. The ethics review has approved our research procedures.

The mathematics teacher Joanne was an experienced teacher. The students were four girls (using paper) and three girls and one boy (using GeoGebra). The students were selected among volunteers. Data from the first lesson has previously been analyzed from the perspective of student and teacher eye-contact [START_REF] Haataja | Teacher's gaze behavior when scaffolding peer interaction and mathematical thinking during collaborative problem-solving activity[END_REF]Haataja, Salonen, Laine, Toivanen, & Hannula, forthcoming). The focus in this paper is to examine whether use of computers has an influence on students' fixation durations as an indication of an effect on their visual attention.

In both of the lessons, the teacher first introduced the lesson structure and when students got their respective tools (i.e. paper and rulers or laptop and GeoGebra) ready, the teacher posed the geometry problem to the class. Students first worked individually, then with a pair, then four together, and finally the students' solutions were collected on the board and discussed. During the first lesson, the teacher also posed an extension problem and during both lessons they continued to examine the problem with a GeoGebra application after discussion. However, that end part of both lessons is beyond the current paper's analysis. The relevant lesson phases are summarized in Table 1.

Lesson phase

Paper and pen (s) GeoGebra (s) The teacher was instructed to provide encouragement and to ask questions that require students to explicate their thinking but to not provide hints for how to solve the problem. When students were working individually, in pairs, or in groups of four, the teacher's activity consisted of roaming in the classroom and stopping for scaffolding one group at a time.

For the analysis, we included only those lesson phases that took place during both lessons. Moreover, we grouped together four separate instances when the teacher was giving instructions to the class or posing the problem. This way, we have identified six lesson phases that captured altogether 41 minutes and 16 seconds of the paper lesson and 34 minutes and 15 seconds of the GeoGebra lesson. Our analysis will focus on these parts of the lesson.

Apparatus

We recorded the actions and conversations of the problem-solving session using audio recording and three stationary video cameras in the classroom. In this paper, we analyze data form five gazetracking glasses that recorded eye movements of the teacher and the target students. The gazetracking device consisted of two eye cameras, a scene camera, and simple electronics attached to 3D-printed frames (Figure 1). The devices and software were self made (see, [START_REF] Toivanen | Probabilistic approach to robust wearable gaze tracking[END_REF]. The camera frame rate depended on lightning conditions, and maximum rate in optimal conditions was 30 frames/second. Data was recorded on laptop computers that were carried in backpacks allowing subjects to move freely in the classroom.

Procedure

The recorded data was first analyzed to identify all fixations for all subjects. The fixation durations were estimated from the eye image difference between consecutive cropped eye image frames and setting a threshold for the average pixel-wise difference. This results in more accurate measure, compared to using gaze coordinates which might fluctuate due to algorithmic miscomputations. The information of fixation onsets and durations was then combined with information about timing of lesson phases from a video following teacher actions in the class. After this preliminary organization of data in Excel, the data was imported to SPSS 24 for statistical analyses.

First we analyzed the descriptive statistics of fixation durations. As the distributions were nonnormal, we used non-parametric tests in our consequent analyses. When comparing fixation durations of two samples, we decided to use the Kolmogorov-Smirnov Z-test rather than Mann-Whitney's U-test, because it has more power to detect changes in the shape of the distributions. We analyzed the variation of gaze durations across the different lesson phases separately for students and the teacher for both the paper lesson and the GeoGebra lesson.

We predicted that the gaze behavior might be different across different lesson phases and, therefore, we analyzed the data separately for different lesson phases. Our expectation was that the fixation durations would on average be similar across those lesson phases where the use of tool would not be central (i.e. teacher instruction, students to the board, and discussion) while the possible effect of computer as a tool could be seen during individual work, pair work, and group work.

To check for sufficient similarity of visual behavior of compared individuals in the two conditions, we made pairwise Kolmogorov-Smirnov Z-tests for fixation distributions between individuals. For this analysis we selected the lesson phases where we expected the effect of computers to be minimal, i.e. teacher instruction, students to the board, and discussion. We used the information of pairwise differences to identify individuals whose fixation distribution was not comparable with other subjects' distribution. We then removed from future analysis those students whose fixation distribution deviated from other students the most. Finally, we made a comparison between student and teacher fixation distributions for the two lessons (paper and GeoGebra) for the different lesson phases using Kolmogorov-Smirnov Z-test.

Results

The total number of fixations during the analyzed lesson phases for all participants was 41,119. The shortest possible fixation to observe was 80 ms (two frames). The observed fixation durations varied from 80 ms to 15066 ms (Mdn = 234 ms, M = 430 ms, SD = 654.00). The durations were non-normally distributed, with skewness of 6.52 (SE = 0.012) and kurtosis of 69.62 (SE = 0.024).

We then analyzed the similarity of fixation durations for both sessions for the teacher and all students for the lesson phases that were less tool dependent (i.e. teacher instructions, students at the board, and discussion). The Kolmogorov-Smirnov Z-tests showed that the teacher fixations over both sessions were similar across the two sessions and different from all but one students' fixations.

All but one compared student pairs had statistically significant differences for at least one of the lesson phases. However, it was possible to identify two students, who stood out more strongly from the group as having a different distribution of fixation durations. We removed these students from the analysis. After this removal, we had three female students from the paper lesson and two female and one male students from the computer lesson.

We then made Kolmogorov-Smirnov tests for each lesson phase to compare fixation durations between the two conditions (working with paper or working with GeoGebra) (Table 2). The results show that during pair work and group work the distributions differ statistically very significantly.

The medians indicate that students using GeoGebra had shorter fixations during pair work than students using pen and paper, while the difference was small and opposite during group work. Differences for the other lesson phases were statistically significant (p < .01) for pair work and group work. To further explore the difference between paper and GeoGebra for visual attention, we looked at the distributions of different fixation lengths. We used the whole session for these students to identify the cut points for deciles. These cut points were then used to divide the distribution of fixation durations for the two lesson phases for paper and GeoGebra condition. We see that the GeoGebra condition had more deviation from the expected distribution of ten percent in each category. For pair work (Figure 2), the students using GeoGebra had more fixations in the time range 100 ms to 200 ms and fewer fixations of longer duration in comparison to paper condition. For group work (Figure 3), students using GeoGebra had fewer gazes in the time range 200 ms to 300 ms and more very long fixations. While the two distributions in the GeoGebra condition are somewhat different, there seems to be a trend for somewhat more short fixations at the cost of average duration fixations.

Discussion

The results show an effect in student fixation durations for the choice between computer and paper as a media to solve a geometry problem. Use of GeoGebra is related with slight shift in fixation durations towards short fixations during collaborative phases of the problem solving. During this phase, students are comparing their individual solutions and discussing for alternative options. With respect to visual attention, this phase should include attention to other students and their drawings.

Shorter fixations usually indicate lower cognitive load. In this context, it might mean that students who worked interactively with GeoGebra, executed more often simple search tasks, such as searching for cursor on screen, which have a low cognitive demand. Alternatively, it might be related to easier extraction of information from neat GeoGebra drawings in comparison to peers' hand written solutions. Thirdly, the computer screen has a lot of visual attracters (e.g. menus) that may lead to short fixations on distracting elements. We provide one more possible explanation. In our earlier studies we have noticed that during interpersonal interaction in class, short fixations at the other person's face are frequent. It may be that the students working with GeoGebra have more short fixations on peers' faces, either because they have longer discussions or as it is more difficult to share screens than notebooks, they have less time for looking at solutions. These possible explanations can't be answered before we have analyzed student behavior during these events and annotated the targets of the fixations.

These results need to be taken with caution. There is significant variation in fixation durations between individuals and even within an individual across time. Although we tried to control for this variation and removed students with obviously deviating fixation duration distributions, the remaining students are by no means identical in their gaze behavior. Future studies should address lager variation of individual and situational differences. A larger sample of students would allow us

Group work

Paper GeoGebra to identify clusters of students whose visual attention follows a similar pattern and then compare these groups with different attentional profiles across digital and non-digital contexts. There is also need to examine qualitatively the nature of visual attention where a difference has been found. For example, what are the targets of longest fixations when solving the problem with GeoGebra and not? We also need additional data with different types of tasks. Perhaps some of the differences identified here are specific to the task used? Overall, this is just a beginning of a long journey.

Figure 1 :

 1 Figure 1: The gaze-tracking equipment.

Figure 2 :Figure 3 :

 23 Figure 2: The distribution of fixation durations (ms) for the selected students for pair work

Table 1 : Lesson phases and their durations. The numbered phases were analyzed in this study

 1

	Teacher gives instructions regarding the lesson structure (1)	44	38
	Students fetch papers and rulers	139
	Teacher gives instructions for GeoGebra		92
	Teacher poses the problem (1)	273	111
	Individual work (2)	373	387
	Teacher gives instructions for pair work (1)	45	22
	Pair work (3)	205	513
	Teacher gives instructions for group work (1)	30	40
	Group work (four) (4)	1210	570
	Teacher poses task extension	362
	Students come to the board (5)	190	130
	Whole class discussion (6)	106	244

Table 2 : Z-test results for tool effect on fixation durations.

 2

	Lesson phase	Tool	n	Mdn (ms)	Z	p
		Paper	933	269
	Teacher instruction				.75	.627
		GeoGebra	1826	269
		Paper	1775	269
	Individual work				1.38	.045
		GeoGebra	1918	267
		Paper	2692	264
	Pair work				3.10	.000
		GeoGebra	1151	204

Acknowledgment

This research was funded by the Academy of Finland grant no. 297856 (MathTrack -project).


Aoife Harrison
email: aharrison@mountanville.ie

Aibhín Bray
email: brayai@tcd.ie

Brendan Tangney
email: tangney@tcd.ie

An Exploration of the effect of Bray's Activity Design Heuristics on Students' Learning of Transformation Geometry

Keywords: transformation geometry, contextual mathematics, technology-mediated, active learning

Introduction

The Irish Mathematics Curriculum, Project Maths, which was introduced into all post-primary schools on a phased basis from 2010, aligns with the core principles that underpin the Programme for International Student Assessment (PISA) mathematical framework [START_REF] Merriman | Project Maths and PISA: Comparing the coverage of PISA mathematics items by the Project Maths and pre-Project Maths curricula[END_REF]. However, there is evidence to suggest that teachers have been slow to move to the more constructivist teaching and assessment style required by the new curriculum [START_REF] Jeffes | Research into the impact of Project Maths on student achievement, learning and motivation: final report[END_REF][START_REF] Merriman | Project Maths and PISA: Comparing the coverage of PISA mathematics items by the Project Maths and pre-Project Maths curricula[END_REF]. In particular, in comparison to other participating countries, students in Ireland performed relatively poorly on the transformation geometry items in the PISA tests in 2003 and 2012.

One approach that has been proposed to address this issue is the use of Bray's Heuristics [START_REF] Bray | Collaborative, Contextual, and Technology-Mediated Mathematics Learning Activities: Design Heuristics and Effects on Student Engagement[END_REF]; a set of guidelines for designing technology-mediated, cross-curricular, collaborative and contextual mathematics learning activities. The theoretical framework that underpins Bray's Design Heuristics combines Realistic Mathematics Education (RME) and a particular constructivist model of 21 st Century teaching and learning -Bridge21 (www.bridge21.ie). Participation in activities of this kind have been shown to have a positive effect on students' conceptual understanding and engagement with mathematics [START_REF] Bray | Collaborative, Contextual, and Technology-Mediated Mathematics Learning Activities: Design Heuristics and Effects on Student Engagement[END_REF][START_REF] Tangney | Constructionist STEM Activities Using the Bridge21 Model[END_REF]. The research presented in this paper extends previous work by focusing on student attainment in exam-type questions, asking: How might participation in a transformational geometry activity created using Bray's Design Heuristics, help participants to achieve higher levels of attainment in the topic?

The Activity

Following a number of classroom-based, technology-mediated, exploratory activities designed to provide students with a foundation in basic transformational geometry concepts, participants were presented with a laminated orienteering map of the school campus overlaid with a square grid. An orienteering circuit with 5 bases, each labelled as a set of coordinates from A to E, was set up on the school grounds, with QR codes fixed at each base. Scanning each QR code generated the geometrical transformation required to locate the next base.

The class group was divided into five teams of 4/5 students. Each team was given a card with the coordinates of their team's starting point (e.g. A (-3,5)). Teams were timed from the moment they began to discuss the location of their starting base on the map. The teams dispersed in different directions to locate their first QR code. In order to scan the codes and receive the geometrical transformation, students used a built-in feature of the Snapchat app, which they had on their mobile phones. The geometrical calculations, needed to navigate to the next location, involved Axial Symmetry, Central Symmetry and Translations and were carried out by the students using white board markers on the laminated maps of their school grounds.

Methodology

The results presented in this paper focus on the quantitative aspect of the overarching exploratory study, exploring changes in attainment in written exams pre-and post-intervention. Participants in this research were 63 students, aged 14-15. The treatment group consisted of one class (n = 22) of students and the control group was made up of two classes (n = 41) graded at the same level of mathematical ability. The first author was mathematics teacher to the students in the treatment group, and the control group classes were taught by two other mathematics teachers within the school. The transformation geometry module was presented as outlined above to the treatment group while the control group were taught in a traditional, classroom-based fashion. The module took place in 10 x 40-minute single class periods over two weeks, within the normal school timetable. In an attempt to ascertain the impact of the learning experience on the students' ability to apply their knowledge of transformation geometry, the treatment and control cohorts sat the same written exam at the end of the module. This exam consisted of five exam questions, focused on transformation geometry, drawn from past papers of the state examination that Irish students sit at the end of grade 10.

Results

Average scores of the treatment group were 3% lower than the control groups in a pre-intervention, mid-year exam whereas in the post-intervention exam their results were 9% higher than those of the control group (Treatment: M = 71%, SD = 18; Control: M = 62%, SD = 15). Results of a one-tailed independent samples t-test indicated a statistically significant difference in performance between the participants in the treatment group and the control group (t(61) = 2.021, p = .029, one-tailed).

Conclusion

This paper provides a small sample of the results of a more significant exploratory study into the possible effects of participation in collaborative, technology-mediated, active learning experiences. The quantitative results described here are limited but have furthered the development of the hypothesis that this approach to activity design has the potential to positively impact on student learning in mathematics in general, and in transformation geometry in particular.


Eivind Hillesund
email: eivind.hillesund@uia.no

Didactical resource purposes as an aspect of students' decision making regarding resources used to learn mathematics

Keywords: Resources, documentational approach to didactics, university mathematics

Introduction

In my ongoing PhD project, I investigate first year engineering students' use of resources to learn mathematics. The project focuses on resource use from a student-centered perspective. Rather than focusing on any particular resources, I look at the use of resources as whole. In my data collection, I aimed to uncover all the resources the students used to learn mathematics, some of their strategies for how to use them, how the strategies evolved over time and what factors influenced which resources they used and how at any given time. I later added students' decision making to my focus based on its prevalence in the collected data. For the poster, I restrict myself one of the themes I uncovered in my thematic analysis of interview data, which I have named didactical resource purposes (DRPs).

Based on a literature review of leading journals in mathematics education, going back to 2010, there are very few studies about resources at university level, with a broad focus and a student-centered perspective. The few that exist (i.e. Gueudet & Pepin, 2016), still cover different aspects of the resource use than I do. Hence, my data is more relevant to didactical resource purposes.

Theoretical framework

I use the documentational approach to didactics (DAD) by Gueudet and Trouche (2009) as my theoretical framework. It is focused on how teachers use resources, both for teaching and professional development. Trouche and Pepin (2014) argue that the documentational approach may also be used to look at students' learning and suggest university mathematics as a good place to experiment with the approach. I hope my research can help expand upon the documentational approach to include perspectives on how students use resources to learn mathematics. I use an inductive approach with a deductive component. After identifying themes, I consider whether it can be related to an aspect of the documentational approach. I derived the term 'didactical resource purpose', from DAD's view that a resource has a material component, a mathematical component and a didactical component. The didactical component of a resource is tied to the organization of activity. DRPs can be considered phases of learning that structure students' use of resources.

Methodology

The study involves nine students from introductory mathematics courses for engineers at three different universities in Norway (here called Alpha, Beta and Charlie). The universities were chosen on the basis of courses that I considered having a different focus (Alpha on material resources, Beta on digital resources and Charlie on social resources). Beyond that, I used convenience sampling. All volunteers from each course were accepted (4 from Alpha, 2 from Beta and 3 from Charlie). They were interviewed three times, near the beginning middle and end of the semester. The interviews were semi-structured, starting with open questions (i.e. "can you say a bit about how you use resources to learn mathematics") and moving on to more specific questions (i.e. how many do you usually study with when you study with fellow students). I analyzed the interviews using a thematic analysis approach (Braun & Clarke, 2006), including open coding; test of intercoder reliability; creating themes; reviewing themes; and then naming and defining them. While codes were numerous and often quite specific, themes were fewer and more general. When identifying themes, I took care to count the number of statements from each student that related to the theme and keep track of the extent to which these statements were in response to open or narrow questions. I also compiled a list of related quotes. Both numbers and quotes related to each DRP feature in the poster.

Results

The didactical resource purposes I identified in my analysis can be considered phases of the students' learning process. Each DRP related to statements from at least seven out of the nine students:

 Introduction. Some resources were commonly used for students' first encounter with mathematical topics, including lectures, textbook and lecture videos.  Practice. After introduction, students work mathematically, mostly using resources emphasized in the courses. The textbooks and learning platforms were used to find exercises and calculator, pencil and paper and fellow students were employed to solve them. At University Beta, the program MyMathLabs was used to find and solve exercises.  Evaluation. Students often checked whether their answers were correct or their work processes were constructive. Resources included answers in the textbook, Wolfram Alpha, GeoGebra, and discussion with fellow students, the lecturer or a parent.  Explanation. After introduction, during practice or after evaluation, students would look for elaborations on the topic at hand to solidify their understanding or 'fill gaps'. This process involved a more active search for information than introduction did. Gueudet, G., & Trouche, L. (2009). Towards new documentation systems for mathematics teachers?. Educational Studies in Mathematics, 71(3), 199-218. Trouche, L., & Pepin, B. (2014). From instrumental to documentational approach: towards a holistic perspective of teachers' resource systems in higher education. Research in Mathematics Education, 16(2), 156-160.

  Resources for this included textbook, fellow students, student assistants, lecturer, parent, Google, lecture videos, YouTube, Khan Academy, course pages and worked examples in MyMathLabs. PROCEEDINGS of INDRUM 2016 First conference of the International Network for Didactic Research in University Mathematics (pp. 444-453). Montpellier, France: Université de Montpellier.

	Literature
	Braun, V., & Clarke, V. (2006). Using thematic analysis in psychology. Qualitative research in
	psychology, 3(2), 77-101.
	Gueudet, G., & Pepin, B. (2016). Students' work in mathematics and resources mediation at entry
	to university. In E. Nardi, C. Winsløw, & T. Hausberger (Eds.),


Ingi Heinesen Højsted
email: ingi@edu.au.dk

Guidelines for design of didactic sequences utilizing dynamic geometry affordances related to mathematical reasoning competency

Keywords: Dynamic geometry environments, reasoning competency, research-based guidelines

HAL is

Identifying guidelines for fruitful teaching with technology in mathematics education is an important research objective. However, technology, such as dynamic geometry environments (DGE), can be used for different purposes. Therefore, to make sense of guidelines for teaching with DGE, it is necessary to clarify the mathematical aim of the teaching. To this end, the notion of mathematical competencies (Niss & Jensen, 2011), which includes the reasoning competency (RC), can be used. Research on DGE affordances has found potentials regarding development of students' mathematical reasoning (e.g. [START_REF] Leung | Discernment and reasoning in dynamic geometry environments[END_REF], which is promising, because students' inadequate abilities in reasoning is well documented (e.g. [START_REF] Hoyles | Curriculum change and geometrical reasoning[END_REF]. However, students' accessibility to DGE, does not guarantee greater learning outcome. The manner in which the DGE is used is essential [START_REF] Jones | Research on the use of dynamic geometry software: Implications for the classroom[END_REF]. Therefore, to support the utilization of the potentials of DGE regarding RC, it is important to investigate: which research-based guidelines may be formulated for using DGE to support students' development of RC?

An extensive review was conducted using the hermeneutic circle approach [START_REF] Boell | A hermeneutic approach for conducting literature reviews and literature searches[END_REF], to map out potentials of DGE described in the literature. Initial searches were made with search words such as "dynamic geometry", "geometry software", "geometry technology", "interactive geometry" and "proof", "reasoning", "conjecture" in MathEduc and ERIC databases, as well as reading CERME technology TWG proceedings. After reading literature acquired from the primary search, interesting references were followed and if suitable, added to review. In addition, after reading and gaining some insight into the area of interest, adjusted search words were used in new searches. 136 publications were included. The definition of the RC played a decisive role in the review process. It influenced the choice of search words and was the optic used to decide which DGE potentials described in the literature were deemed relevant for this study.

In synthesizing the review findings, three categories of guidelines were identified: students' cognition, task design and teaching practices. The guidelines emerge from the utilization of four types of DGE affordances: feedback, dragging, measuring and trace. Theoretical constructs from two predominant theoretical frameworks, the instrumental approach [START_REF] Artigue | Learning mathematics in a CAS environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work[END_REF] and the Theory of Semiotic Mediation [START_REF] Bartolini Bussi | Semiotic mediation in the mathematics classroom: Artifacts and signs after a Vygotskian perspective[END_REF], were found to be suitable in the design of guidelines. Specifically, the notion of semiotic potential is useful in the consideration of what dimensions guidelines might entail. Furthermore, a description can be made of the instrumented techniques to be appropriated along with the utilization schemes needed. The role of the teacher can be described in supporting the mediated signs of the students towards mathematical meanings in order to support their development of RC. The guidelines are considered a priori in an ongoing project, in which the next step is to implement them in practice and refine them through a designbased research approach. To illustrate, table 1 below shows a segment from the a priori guidelines.

Table 1 : Example from a priori guidelines

 1

	Instrumented	Utilization	Tasks	Teacher´s role	Reasoning
	techniques	schemes			competency
	Constructing	Understanding	Construction tasks	Instruction, discussion and	Prerequisite
	direct	the difference	with direct	feedback: focus on	for working on
	invariants,	between direct	invariants, where	mathematical meanings -	conjecturing
	which induce	and indirect	dragging free points	Address that direct invariants	tasks.
	indirect	invariants, and	in the construction	can induce indirect invariants	Some initial
	invariants	the connection	unveils (surprising)	because of Euclidian	conjecturing
	because of	between them.	indirect invariants.	geometry governing the	tasks.
	Euclidean			environment.
	theory.