Sonia Palha
email: s.abrantes.garcez.palha@hva.nl

Why digital tools may (not) help by learning about graphs in dynamics events?

Keywords: visualization, intrinsic feedback, graphs, interactive tool, learning with technology

Understanding graphs representing dynamic events is a challenge for many students at all levels. And technological tools can provide support in overcoming some of these difficulties. In our research we developed a digital tool that enables students to create, modify and improve graphs from dynamic events using interactive animations and intrinsic feedback. In order to get insight about why the tool helped (or not), the students we conducted a qualitative study in which we interviewed nine students who used the tool. The results offer insight in students' learning and thinking about dynamic graphs and how digital feedback can afford that. These results are useful for researchers, developers and teachers.

Introduction

Dynamic graphs remain an essential subject at all levels of High School in mathematics, but it remains a challenging topic for students and even for teachers [START_REF] Carlson | Integrating a models and modeling perspective with existing research and practice. Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching[END_REF][START_REF] Moore | Students' images of problem contexts when solving applied problems[END_REF]. Reasons behind students' difficulties frequently involves problems in visualizing change and variation, limited understanding of functions and co-variational reasoning. Moreover the construction of a global graph from a realistic situation and stretch is similar to mathematical problems solving and requires mathematical thinking [START_REF] Moore | Students' images of problem contexts when solving applied problems[END_REF][START_REF] Thompson | Quantitative reasoning and mathematical modeling[END_REF]. Digital tools can support students to deal with these difficulties in several ways. Dynamic software as Geogebra and applets can support students in visualizing relations through enabling them to draw, move and modify graphs within different representations. And interactive applications that connects animations and graphs can be used to explore relationships between phenomena, and it's graphical representation. The learning potential of dynamic tools can be highly improved by including possibilities for students interaction with the tool like students' own productions and incorporated feedback-features [START_REF] Laurillard | Teaching as a design science: Building pedagogical patterns for learning and technology[END_REF]. However, this also put more demands on the tools' design especially when they build upon students free-hand productions. More knowledge and research on students learning with this type of tools are needed to inform tooldevelopers and teachers.

Interactive Virtual Math (IVM) is an example of such a tool. It generates and builds upon students free-hand graphs, and it has incorporated feedback. The tool was developed through developmental research [START_REF] Palha | Interactive Virtual Math: a tool to support self-construction graphs by dynamical relations[END_REF] with the aim to improve students' learning about graphs by dynamic events. The tool creates opportunities for students to experience the thinking and reasoning that is needed to generate, revise and modulate a graph from themselves. When entering the tool they get tasks that encourages them to imagine two variables changing simultaneously. The tool requests the students to produce the graphical representation and the verbal explanation for this relation, which requires students to represent their concept image graphically and verbally. With the help of the incorporated feedback, the student is challenged to think, reason and act upon his own construction. This is an innovative pedagogical feature of the tool that requires deep knowledge to be developed properly. The tool also includes the use of Virtual Reality (sound, movement, interaction), which is however very limited, but it is expected to improve the experience of the graphic situation.

Previous research [START_REF] Palha | Students learning with Interactive Virtual Math: an exploratory study in the classroom[END_REF] about IVM shows that students (age 13-17) find the tool useful because it assists them to improve graphs and/or to gain a more thorough understanding of the subject. The study was a teaching experiment involving three classes at secondary and one class at tertiary education in The Netherlands that used IVM during one lesson (45-50 minutes). Seventynine students reported through questionnaires about their experience with the tool and what supported them the most. However, this data didn't provide us much in depth knowledge about the way students improved their graphs and why. During the experiment we also interviewed nine students, and we collected their backlog files in the tool. This data has been analyzed recently, and it provides us with new insights on learning with the tool.

In this paper we report the results of these qualitative analyses. We use the framework for covariational reasoning of [START_REF] Carlson | The precalculus concept assessment: A tool for assessing reasoning abilities and understandings[END_REF] and the notion of intrinsic feedback [START_REF] Laurillard | Teaching as a design science: Building pedagogical patterns for learning and technology[END_REF] to interpret and evaluate the way students utilized the tool. The guiding research question is: how does the tool enable students to improve their graphical representation and/or understanding about dynamic events?

Theoretical background

Learning about dynamic graphs

An example of a dynamic event is the following situation: imagine a bottle filling with water. Sketch a graph of the water's height in the bottle [START_REF] Carlson | The precalculus concept assessment: A tool for assessing reasoning abilities and understandings[END_REF]. To solve the bottle-task, the students will need considering how the dependent variable (height) changes while imagining changes in the independent variable (volume). The coordination of such changes requires the ability to represent and discern relevant features in the shape of the graph. These mental actions are in the core of covariational reasoning and are clearly defined in the framework of [START_REF] Carlson | The precalculus concept assessment: A tool for assessing reasoning abilities and understandings[END_REF]. The authors define covariational reasoning as entailing five mental actions: (M1) coordinating the value of one quantity with changes in the other; (M2) coordinating the direction of the change; (M3) coordinating the amount of change of one quantity while imagining successive changes in the other quantity; (M4) coordinating the average rate of change of the function with uniform increments of change in the input variable; (M5) coordinating the instantaneous rate of change of the function with continuous changes in the independent variable for the entire domain of the function.

Several researches [START_REF] Thompson | Quantitative reasoning and mathematical modeling[END_REF][START_REF] Saldanha | Re-thinking co-variation from a quantitative perspective: Simultaneous continuous variation[END_REF][START_REF] Castillo-Garsow | Chunky and smooth images of change[END_REF] reported students' difficulties in engaging with the mental activities M4-M5. These students also have difficulty explaining why a curve is smooth and what is conveyed by an inflection point on a graph. According to [START_REF] Carlson | The precalculus concept assessment: A tool for assessing reasoning abilities and understandings[END_REF]. Students should have opportunities to experience the covariational nature of functions by dynamic events. [START_REF] Thompson | Quantitative reasoning and mathematical modeling[END_REF] too states it is critical that students first engage in the mental activity to visualize a situation and construct relevant quantitative relationships prior to determining formulas or graphs. [START_REF] Ellis | The influence of reasoning with emergent quantities on students' generalizations[END_REF] suggests that learners should be helped to focus on quantities and generalizations about relationships, connections between situations and dynamic phenomena.

Intrinsic feedback

Technological tools can be designed to help to create a learning environment that fits the characteristics mentioned in the previous section. The learning environment that we have in mind involves learning from experience and by reflection on ones' own productions. These are underlying ideas in theories that account learning as an active and social process. With the use of specific type of tasks, students can translate their concepts into practice. They can compare and evaluate how well they achieve some learning goal. And employ this to improve their initial concept and practice and develop knowledge [START_REF] Laurillard | Teaching as a design science: Building pedagogical patterns for learning and technology[END_REF].

Digital tools with capability to provide intrinsic feedback can assist students with learning because they: (i) enable the students to modulate their concept image and generate actions that brings them nearby a goal and; (ii) use the feedback of the tool to modulate their practice and revise their actions. According to the framework of Laurillard learning occurs when students engage in a successive cycle involving these actions. Actions and feedback drive the internal modulate-generate cycle that links the students' conception to their repertoire of actions as practice. Through reflective observation students can modulate the abstract concept in the light of concrete experience and generate new actions as active experimentation (p.168).

The prototypical tool IVM aims at creating a practice environment for the learning about graphs by dynamic events, in which student engage in the internal successive cycles described above. Moreover, the tool incorporates features designed to engage students in modulate-generate-revise actions that foster the development of covariational reasoning [START_REF] Carlson | The precalculus concept assessment: A tool for assessing reasoning abilities and understandings[END_REF]. With the tool there is a sequence of trials, mostly with a cyclic character that need to occur and be repeated to assure learning (see table 1 with the description of the tool).

Self-construction -task

Task Learner the tool sets a task to construct a graph from a non-trivial dynamic event the learner draws a graph by the dynamic event and explains his/her thoughts

Compare -task

Task Learner Task

The tool displays a second task with a dynamic event that corresponds to the more common mistake made by learners with regard to this type of task The learner uses the comparison to draw (if needed) an improved graph and/or adapt the explanation The tool offers a menu in which students can choose between seeing help or to submit their graph

Help 3D-animation (optional)

Learner Task

Learner

The learner chooses help 3D-animation The tool displays an animated version of the dynamical event in which the same amount of water is added each time student presses a button The learner notices that the height of the water increases in the jar and that, for the same amount of added water, the incremented heights are different and depending on the form of the jar; the learner reflects on these relations and becomes aware that the increment of the height is smaller when the jar is broader -in the middle and larger in the extremities. The learner uses the visualization to draw (if needed) an improved graph and/or adapts the explanation

Help interactive animation (optional)

Learner Task

Learner

The learner choose help interactive animation The tool displays an animation of the dynamical event and an empty cartesian graph in which the learner can move given dots to the estimated height every time the same amount of water is added; both actions, adding the water by pressing a button and moving the dot are performed by the learner. The learner moves vertically the first dot to the position he/she thinks that the height of the water will reach and then press the button. The water fills in the jar at a certain height. The learner compares the estimated and the reached height and uses the comparison to move the dot to a more precise location.

Feeback at the end-reward feature

Learner Task Learner

The learner submits his/her solution The tool displays the figure of the jar corresponding to the learner drawings The learner compares the form of the jar with the initial form and reflects on the differences and similarities and on the relation between the graph and the shape of the jar. The learner can go again through the tool and use the visualization to draw an improved graph and/or adapt the explanation

Method

We interviewed nine students from 10th and 11th grade who utilized the tool in the classroom: two boys (Niels and Abel) and two girls (Miriam and Jenny) from school A, 11th grade, 16-17 yearsold; two girls (Olivia and Anne) from school B, 10th grade, 15-16 years-old and two boys (Gerry and Arion) and one girl (Manon) from school C, 10th grade 16 years old. In schools A and B students utilized a computer and construct the graph with the mouse. In the school C school students used a Iphone and draw with the finger. The interviews were semi-structured and focus on the way students used the tool and the reasons to modify or not the original graph. The interviews were performed by the author of this paper or by the teacher. They took about 5-10 minutes per student and were recorded in audio. Additional data included the results of a survey conducted during the lesson and the students productions registered in the tool self (backlog of the tool). The data analysis was qualitative and based on the covariational framework van Carlson et al. (2010) and the notion van learning cycles from [START_REF] Laurillard | Teaching as a design science: Building pedagogical patterns for learning and technology[END_REF].

Findings Students transformed an incorrect graph into a correct graph with the tool (n=5)

To draw a correct graph students need to coordinate the average rate of change of the height as function of the volume and imagining it changing instantaneously with continuous changes in the independent variable (levels MA4 and M5 of the covariation framework). Analysis of the backlog of the tool showed that five from the nine students generated a correct graph, but only after a second or third trial. In their first trial these students sometimes draw straight lines instead of curves or draw a concave up-down graph instead of a concave down-up or an inaccurate global shape. This suggests that the students could reason at the levels MA2 or MA3 of the covariation framework at the start of the assignment. In the interview students were asked to explain how they transformed their graph and why they transformed it. Table 2 provides an overview of the way students engaged in the learning cycle and generated, modulated and revised their initial graph with the tool.

Students How and why the graph was transformed and which tool-features enabled this

Jenny

Jenny modulated her concept image and generated a better-shape graph after two trials. The student used the feedback from the reward-feature to revise and modulate the steepness of the curves.

Niek

Niek modulated his concept image in two trials. The student used the feedback from the reward-feature to revise and modulate a concave up followed by a concave down graph into a correct graph.

Abel

Abel modulated his concept in two trials. The student used the feedback from the reward-feature to transform part of the graph (an increasing straight line) into a concave down curve.

Gerry

Gerry modulated his concept in three trials. The student used feedback from the reward-feature and help 3D-animation to revise and modulate three increasing straight lines into a concave down followed by a concave up curve and to improve the shape of the graph.

Manon Self-construction of the graphs (with no formula) enabled Manon to reflect before drawing the graph and therefore modulate her concept image; feedback from the reward-feature enabled her to generate and revise actions as she improved her graph. Both help-features were consulted but not used.

Table 2: the tool enables students to produce a correct graph with the tool (n=5)

All five students realized that their graph was not correct because of the feedback from the rewardfeature. One student, Niels explained:

"when I saw the feedback at the end (....) you can work towards a nice graph and you know how it really should look like, because you know that at a certain point will be slower than elsewhere (...) " when your vase is inside you know that then you have to go to the other side (...) so first slowly and then faster instead of first faster and then slower "

Manon found the tool very instructive especially because the self-construction-task that challenged her to generate an action from her practice repertoire and reflect on it:

"'I was already ready to put something in the Graphic Calculator or to do something like that. Gerry felt also helped by help 3D-animation and he explained why:

"you saw in the animation (3D) the round shape of the bowl and you saw better how the water was distributed. In the beginning that a lot more space had been taken and in the middle there is much wider and because of that the graphs were very different "

Further, all the students with exception of Abel considered that because of the tool they understand the subject better; Abel considered that he already understood the subject.

Students realized they had an incorrect graph and did not succeed in transforming their graph into a correct one with the tool (n=3)

Three of the nine students (Olivia, Anne and Arion) have done two, sometimes three trials with the tool but were unsuccessful to generate a correct graph. Analysis of the backlog of the tool showed that the students initially generated a graph with one or more straight lines, which corresponds to M2 in the covariational framework. Interviews with the students provided us with more insight about how students revised and modulated their graphs (Table 2).

Students How the graph was transformed and which tool-features enabled this

Olivia

The first trial is an increasing straight line, the second trial is a concave down line and third trial is a concave up. The comparison-task and the feedback from the reward-feature enabled student to modulate her concept in the sense that the lines should not be straight. Olivia generates and revise actions when she produces new graphs with curves. She revised and modulate her graph but she could not generate a correct graph

Anne

In the first and second trial the graph is an increasing straight line and the slope of the second trial is smaller.

The feedback from the reward-feature and help-3D animation enabled Anne to (partially) modulate her concept as she realized that the lines should not be straight. She revised and modulate her graph but she could not generate a correct graph.

Arion

In the first and second trials the graph has three increasing straight lines. The feedback from the rewardfeature enabled student to modulate his concept: that the graph should not have straight lines. However, for technical reasons the student could not work with the tool and use the feedback to improve his graph.

Table 3: the tool enables students to realize their mistake but not to improve it (n=3)

Olivia considered that the tool helped her to improve the graph and she explained why:

"it really helped that in the end I converted my drawing into that shape, as a sphere because I saw a bit connection" (…) "If there is no straight line then there must be a concave up line or a concave down line -I will try both.

Olivia didn't used the help-features because she doesn't think that it would be useful to achieve her goal of drawing a correct graph. The visualization of the context is not the problem according to the student but the fact that she can't draw the graphical representation:

"'I went very often trying the tool because I was curious (….) with the movies I do not know much what I had to show"

The other two students did not consider that the tool helped them to improve the graph. Anne because the straight line remained a straight line and Arion because the tool didn't work well and he could not keep trying (there were in some smartphone technical problems). For these students the intrinsic feedback provided by the tool enabled them to engage in the modulate-generate cycle and modulate their initial concept image towards a more sophisticated one (although not yet correct).

Students realized they had an incorrect graph but did not do nothing about it (n=1)

Miriam was the only student who produces an incorrect graph and did not invest effort to improve it with the tool. In her initial trial she constructs an incorrect graph with three straight increasing lines, which corresponds to the mental action M3 in the framework covariational reasoning. She didn't consult the help-features. In a second trial Miriam worked with a peer-student who produced a similar graph to her but with curves instead of straight lines. Noticing the difference enabled the student to revise and modulate her concept image. Additionally, her graph was discussed by the teacher in the classroom. This student only engaged in the learning cycle when the intrinsic feedback of the tool was combined with extrinsic feedback of a peer student and from the teacher.

Conclusions and discussion

Learning dynamic graphs at secondary school represent a challenge for many students. In this study we investigate how and why the IVM-tool enabled students to improve (or not) their graphical representation of a dynamic event. We have seen that all nine students failed to produce a correct graph in a first trial with the tool. Five of these students could improve their initial graph and produced a correct graph in their subsequent trial. The features of the tool that supported them the most were to see the bottle-form correspondent to their graph at the end (reward-feature) and the opportunity to self-construct the graph, because it challenged them to think and try to imagine how the graph it would be. One student referred also to the animation 3D, which have encouraged him to imagine better "how the water was distributed" in the bottle and with relation to the graph. Three other students have tried improving their initial graph with the tool and although they could not produce a correct graph the tool seemed to have helped them to improve their thinking, as they became aware that the graph should not be linear or contain linear parts. They conclude this because of the reward-feature and in combination with other features (comparison-task or with the animation help-3D). One student only improved her graph when this was combined with extrinsic feedback from peer-student and the teacher.

Reflecting upon these results, we realize that even though the tool showed potential to engage students in thinking about mathematical graphs this was insufficient for all the students to generate an appropriate representation. These students realized with the help of the tool what they have done wrong and the feedback maintained them on the task. But the students did not successfully use these and other features of the tool to move forwards. Why did the tool not help the students? Analysis with the covariational framework suggests the three students possessed a limited understanding of covariational reasoning. What can seem to be a reason for students 'difficulty in imagining the change of the height varying as function of the volume and [START_REF] Thompson | Quantitative reasoning and mathematical modeling[END_REF][START_REF] Carlson | The precalculus concept assessment: A tool for assessing reasoning abilities and understandings[END_REF]. The tool contains two animations that were explicitly designed for students to engage in covariational thinking: the animation 3D and the interactive animation. But as our results showed none of the students consulted it or they only did it superficially. We cannot therefore comprehend how these features could have supported the students. Another feature of the tool that we had expected to help the students was the reward-feature and, in a certain way it did. The feedback assisted students to realize they were not constructing the correct graph but did not provide other directions or explanations that help them forwards. This is an aspect that requires further thought and investigation. In addition, expanding the tool with more tasks and different contexts is needed in order to offer students enough opportunities for exploring and practice covariational thinking and reasoning.

This study extends previous research about learning dynamic graphs and covariational reasoning. It adds knowledge about students thinking in contexts of change and their difficulties in constructing graphs from themselves. The results provide directions to improve the instructional environment of IVM that can be useful for teachers and researchers interested in using the tool. Further research is needed to gain more insight in the help-features and to improve the intrinsic feedback. We also need to investigate the learning with the tool in larger settings: with more students, teachers and whole classrooms. The prototypical version of the tool presented at CERME 11 is available at https://app.dwo.nl/dwo/apps/player.html#570660.

Table 1 : description of the tool Interactive Virtual Math (IVM)

 1


Hana Ruchniewicz
email: hana.ruchniewicz@uni-due.de

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

Digital media support functional thinking: How a digital selfassessment tool can help learners to grasp the concept of function

Keywords: Digital media, technology, functions, formative self-assessment, metacognition

Developing functional thinking is a central objective in mathematics education. Students especially need to internalize three aspects of functions: mapping, covariation, and object besides learning to translate flexibly between various representations. However, many learners struggle to recognize and interpret functional relationships. To help these students, a formative self-assessment approach with digital media offers new possibilities. Technology can support understanding not only due to the use of multiple, dynamic, interactive, or linked representations, but offer a constraint-support structure to guide students' actions. We present two case studies in form of task-based interviews from the third cycle of a design-based research study that aims at the development and evaluation of the "self-assessment for functional thinking electronic" (SAFE) tool. The cases show how certain technological features of the SAFE tool support learners' self-assessment and functional thinking.

Why support functional thinking?

Many real-world phenomena, such as measuring the temperature at a weather station over a period of time, can be comprehended functionally. Because of this variety of applications and its importance to grasp further mathematical concepts, the discussion of functional relationships is central in many mathematics curricula. For example, it is one of five key content domains in lower as well as upper secondary mathematics education in Germany. Although the function concept is fundamental in the learning of mathematics, many students have difficulties in its comprehension. Thus, they need support to acquire 'functional thinking'. For this reason, we use a formative student self-assessment approach as it has proven to enhance student learning [START_REF] Stacey | Technology and Assessment in Mathematics[END_REF].

Theoretical Background Formative self-assessment and metacognitive activities

According to [START_REF] Wiliam | Integrating assessment with learning: what will it take to make it work[END_REF], formative assessment (FA) can be conceptualized in five key strategies. These regard mainly the teacher to be responsible for FA. Especially in computer-based assessment, features such as automatic feedback, a limitation of students' responses by e.g. multiple choice formats, or a focus on measuring achievements [START_REF] Stacey | Technology and Assessment in Mathematics[END_REF], leave little room for students to recognize, reflect upon and react to their work. This is why, the FA framework by Wiliam and Thompson was refined in the EU-project FaSMEd to allow formative self-assessment to be understood as a process performed, observed and reflected on by students. It is conceptualized in these five key strategies (KS): 1) understanding learning intentions and criteria for success, 2) eliciting evidence of understanding, 3) providing self-feedback, 4) activating peers as instructional resources for one's learning, and 5) regulating one's learning process. However, strategy four is only relevant when observing formative self-assessment in classroom situations [START_REF] Ruchniewicz | Can I sketch a graph based on a given situation? -Developing a digital tool for formative self-assessment[END_REF]. For all these strategies, but especially for the fifth, learners need to use metacognitive activities. They describe the procedural component of one's metacognition. This means all actions of regulation of one's cognitive activities in learning processes. They include: 1) planning problem-solving steps with appropriate mathematical tools, 2) monitoring in form of controlling tool-use and comparing what is achieved to set goals, and 3) reflection on given problems or understanding of mathematical concepts [START_REF] Cohors-Fresenborg | The role of metacogntitive monitoring in explaining differences in mathematics achievement[END_REF]. To enable students to self-assess their functional thinking abilities, this mathematical content needs careful consideration.

Functional thinking

Functional thinking is a didactical concept, that describes all mental images learners need to build and use when dealing with functions, their representations, and their applications in modelling and problem solving to gain a comprehensive understanding of the function concept. "Functional thinking is a way of thinking, that is typical for working with functions" (translated from Vollrath, 1989, p. 6). Three aspects, noticed in German didactics today as Grundvorstellungen (GVs), characterize this "typical" and allow for various views on functions [START_REF] Vollrath | Funktionales Denken [Functional thinking[END_REF]: 1) Mapping: a function assigns exactly one value of a dependent quantity to the value of an independent one. As such a unique mapping, it is viewed in a local and static way. 2) Covaritation: in a dynamic view, a function describes how two quantities change in relation to one another. While the independent quantity runs through a set of values from a domain, it causes the values of the dependent quantity to change accordingly. 3) Object: a function viewed globally is a mathematical object, that has its own specific properties (e.g. characteristic graph, symmetry) and can be operated upon.

Although the three GVs of functions apply for all forms of semiotic representation, they can appear variously according to a function's visualization. Usually, students interact with functions represented verbally, numerically, symbolically, or graphically. The mapping aspect, for instance, is highlighted when calculating a y-value by inserting the x-value into an equation. By contrast, the covariation aspect is of interest when looking at the change of a graph's values for an interval of xvalues. Each representation emphasizes different properties of the function [START_REF] Duval | A cognitive analysis of problems of comprehension in a learning of mathematics[END_REF]. Thus, it is central for functional thinking that students learn to work with and translate between various representations. This leads them to a comprehensive understanding of the function concept [START_REF] Duval | A cognitive analysis of problems of comprehension in a learning of mathematics[END_REF].

Due to the concept's complexity, such a comprehension and, thus, the acquiring of functional thinking is challenging. Numerous misconceptions are described in regards to learners struggling to conceptualize functions. For example, [START_REF] Clement | Misconceptions in graphing[END_REF] states that many students falsely treat graphs as literal pictures of the underlying situations (graph-as-a-picture mistake). Others overgeneralize function types or properties, such as using linear functions in inappropriate situations (illusion of linearity). Further, students might swap the x-and y-coordinates or disregard the uniqueness of a function (e.g. [START_REF] Hadjidemetriou | Children's graphical conceptions[END_REF][START_REF] Leinhardt | Functions, graphs, and graphing: Tasks, learning, and teaching[END_REF]. To react to such misconceptions and support students' self-assessment, digital media offer new opportunities.

Digital media

Digital media have the potential to enhance formative self-assessment by altering the assessment process due to changes in, e.g. the nature of tasks, types of feedback, or even assessed skills. What is more, they present new chances for learning by e.g. providing dynamic or interactive representations of mathematical objects [START_REF] Drijvers | Uses of technology in lower secondary mathematics education: A concise topical survey[END_REF]. Various studies examine if the use of technology improves achievements. In an ICME survey concerning lower secondary learners, Drijvers et al. (2016, p. 5) summarize: "[t]he overall image is that the use of technology in mathematics education can have a significant positive effect, but with small effect size." The authors conclude that existing studies give hints for whether achievements can improve, but do not explain why [START_REF] Drijvers | Uses of technology in lower secondary mathematics education: A concise topical survey[END_REF]). To answer why digital media can support functional thinking, it is necessary to consider their potential and appraise it against possible risks. Following, we list some of the most important arguments for teaching, learning and assessing functional thinking with digital media: 1) Fast availability of representations: Quickly available visualizations of functions leave time for examining functional relationships, generating examples, or checking one's hypotheses. Yet, the large amount of representations and speed of their availability might lead to a complexity that hinders students to reflect on their actions (Barzel, Hußmann, & Leuders, 2005). Cavanagh and Mitchelmore (2000, p. 118), for instance, identified the tendency "to accept whatever was displayed in the initial window without question" as one of three typical mistakes of 10th and 11th graders asked to interpret linear and quadratic graphs on graphic calculator screens. Students did not reflect upon the visual image on screen or relate it to an inserted algebraic equation [START_REF] Cavanagh | Graphics calculators in mathematics learning: studies of student and teacher understanding[END_REF].

2) Multiple representations: The aspect of fast availability plays a key role when it comes to screening various representations of the same function at once. Each one stresses different aspects of a function [START_REF] Duval | A cognitive analysis of problems of comprehension in a learning of mathematics[END_REF]. Simultaneous visualizations can support the construction of mappings between those aspects and the translation between representations, which helps learners to solve problems quicker and conceptualize functions (van Someren, Boshuizen, de Jong, & Reinmann, 1998).

3) Dynamic representations: In static media, variations of a mathematical object need to be observed, interpreted, and projected upon its representation by the user. Dynamic representations let students experience these changes directly [START_REF] Kaput | Technology and Mathematics Education[END_REF]. Especially the covariation GV of functional thinking can, thus, be supported by digital media as the investigation of changes in function values is easier. 4) Interactivity and linked representations: Digital media permit "not simply to display representations but especially to allow for actions on those representations" (Ferrara, Pratt, & Robutti, 2006, p. 242). Moreover, it is possible to link representations so that the variation of one is automatically reflected in another. Such links enable learners to investigate functional relationships as they provide immediate feedback and encourage them to change between representations. Although interactive and linked representations offer great potential to support functional thinking, the technological speed entails the risk of students being overwhelmed by quick changes in visualizations and driven to act upon them without reflection [START_REF] Zbiek | Research on technology in mathematics education: A perspective of constructs[END_REF]. 5) Effecting student actions: Student actions are effected by a digital tool's design. If learners are asked to draw a function graph, the technology might supply them with a suitable coordinate system. Kaput (1992, p. 526) refers to a tool's "constraint-support structure" stating that "whether a feature is regarded as one or the other does not depend inherently on the material itself, but on the relation between the user's intentions and those of the designer of the material and the contexts for its use."

In our study, these arguments for using digital media to support functional thinking guide the design of a digital self-assessment tool and offer hypotheses for the analysis of students' learning processes.

Methodology

We use design-based research that aims at developing and evaluating the SAFE tool [START_REF] Ruchniewicz | Can I sketch a graph based on a given situation? -Developing a digital tool for formative self-assessment[END_REF]Ruchniewicz & Barzel, in print). Several versions of the tool are designed, investigated, and re-designed. This cyclic process is guided by the following research question: How do certain technological features of the SAFE tool support students' self-assessment and functional thinking?

Design of the SAFE tool

The SAFE tool's aim is for students to be self-assessors and, thus, no direct feedback is generated. For all tasks, users compare their answer to a sample solution and evaluate on their own whether it is correct or which steps to take next in their learning. The design means to create a balance of providing enough information and autonomy for learners. Therefore, the SAFE tool intends to assess and repeat basic competencies after they have been taught. The learning conjecture is the ability to sketch graphs based on given situations as this reflects a key aspect of functional thinking. The SAFE tool runs as an iPad application and consists of five parts: Test, Check, Info, Practice, and Expand. These are joined in a hyperlink structure and labelled with various symbols for easy user orientation (fig. 1).

Figure 1: Hyperlink structure of the SAFE tool

Students start by solving the Test task, which presents the story of a bike ride and asks them to draw a time-speed graph. They can label the axes using drop-down menus and sketch the graph directly on the screen using their fingers. Afterwards, they move to the task's sample solution. It consists of a simulation of the situation that is linked to one possible solution graph. The simulation can be started and stopped by the user and a qualitative speedometer can be viewed as well (fig. 3a). The learner moves to the Check, that provides five statements regarding important aspects about the graph in question alongside common mistakes that could arise. For each statement, the student decides whether it is true or false for his/her own solution. For this diagnostic step, the Check includes a pictorial visualization of the situation together with a static representation of the sample graph and the learner's own solution in the same coordinate system (fig. 3b). If an error is identified, the student can choose to read an Info, which entails a general explanation to repeat basic ideas about the function concept and specifies them in the time-speed context of the Test task, or work on a specified Practice task. If the initial graph is correct or a student checks all statements, he/she is presented with two more Practice tasks as well as an Expand task with a more complex context.

Data collection and analysis

We use case studies in form of task-based interviews with individual students. The interviews are videotaped, transcribed, and analyzed qualitatively. Three subject groups are included: grade 7-9 students (aged 12-15) as they are novices regarding the function concept; grade 10 students (aged 15-16) as they have to repeat curricula contents of years 5-10 for a state-wide assessment; and 2 nd semester university students in their bachelor in mathematics education (aged 19-22) as they are experienced, but often need to repeat basic competencies in the transition from school to university. This paper presents two cases from the third design cycle, which give a first glance of the latest SAFE tool's potential as data collection is not completed. So far 3 interviews with year 9 and 5 with university students are recorded. One university student solves the Test and self-assessment without difficulties. One younger student shows poor knowledge of the content and does not recognize graphs as representations of functions. She is unable to identify her mistake on her own. Six learners show similar self-assessment processes. As the diagnostic step of assessing their solution to the Test occurs when viewing either the Test solution or Check, two of the (university) students were chosen as cases to represent both of these ways of self-assessment. As our focus is assessment, the learner's work with other tool parts (Info, Practice, Expand) is not included, as it portrays their next steps of learning.

For data analysis, we use qualitative content analysis with deductive categories. Additional inductive categories will be developed once data collection is completed [START_REF] Mayring | Qualitative Content Analysis[END_REF]. In relation to our theoretical basis, three main categories with different subcategories were identified to guide analysis:

Results

The Test task, that students use for the diagnostic step of their self-assessment, asks them to draw a time-speed graph for this situation: "Niklas gets on his bike and starts a ride from his home. Then he rides along the street with constant speed before it carves up a hill. On top of the hill, he pauses for a few minutes to enjoy the view. After that he rides down and stops at the bottom of the hill." Paul and Ayse solve the task and describe their reasoning in the interviews (fig. 2). Thus, they elicit evidence of their own understanding (KS2). Afterwards, they both move to the sample solution (fig. 3a). Paul starts the simulation, stops it when the bike starts to ride uphill and says: "The first thing I saw was that it is assumed that not -(points at the origin in the sample graph) so that he does not start directly with constant speed (points along the first increasing graph segment) but that starting from home is also its own time period. That means that you also have to consider that he has to start first and that he is not at any speed at first." Thus, the linked simulation encourages him to reflect his own solution. He identifies a mistake in his answer by realizing that he previously did not model the first part of the situation, namely starting the bike ride from home. Afterwards, he resumes the simulation and stops it again when the bike stops on top of the hill: "Okay, and then he becomes again, when he rides up the hill, that he does not reach the speed with which he rides uphill at once, but that the speed decreases in shifts." Here, Paul reflects part of his answer (riding uphill) by comparing it to the sample graph. He recognizes that the speed and, thus, the graph cannot decrease suddenly, but that it slows down with time. Even if Paul does not realize that his graph disregards the uniqueness of the underlying functional relationship (vertical lines in his solution, fig. 2a), the simulation does inspire him to observe the covariation of the quantities time and speed more closely.

Ayse watches the entire simulation at once and directly moves on to the Check. Her screen shows the checklist and a multiple visualization of the bike's path together with her own and the sample solution's graph in one coordinate system (fig. 3b). Ayse reads checkpoint 2: "I realized correctly when the graph is increasing, decreasing, or remaining constant." Relating to this, she assesses her solution: "Basically yes, but the duration! So with the variable with the time, so I should have much longer (points to the part of her graph that models the bike stopping on top of the hill) along zeroshould have gone along the x-axis when he stops for a few minutes." Thus, she denies the statement of the checkpoint for her answer and crosses it off (fig. 3b). The checkpoint together with the multiple representation of the functional relationship prompt Ayse to reflect on her solution and discover an error regarding her graph's slope: while she depicted the bike stopping on top of the hill only as one point of her graph reaching the x-axis, the tool reveals that it should be a constant graph segment with the value of zero. Her reflection shows that Ayse did regard the variation and value of the dependent quantity (speed) while drawing her graph, but she missed to consider the change of the independent quantity (time) as well. The SAFE tool helps her to focus more on the covariation of both quantities.

Conclusion

The two cases of Paul and Ayse show that the SAFE tool can support students' self-assessment and functional thinking. This becomes apparent when considering how the analysis' categories of selfassessment, functional thinking, and technological features interact with each other in the interviews.

Paul's case reveals that an interactive representation of a simulation linked to a graph (TF1) stimulates him to use the metacognitive activity of reflection (MA3). Further, he starts and stops the simulation several times in order to evaluate different aspects of the graph, which shows that his reflection is supported by this tool functionality (TF3). He evaluates his own graph by comparing it to the sample solution and identifies his own mistakes while regulating this process (KS5). Paul realizes that he missed to model part of the situation (start of bike ride) and translate it into its graphical representation (CR). For another part of the situation (riding uphill), he focusses on the covariation of both quantities. While he initially assumes a more prototypical graph with many constant segments (fig. 2a,M), he addresses a non-linear decrease of the graph's slope to model the bike slowing down when looking at the simulation (GV2). In his reflection, he shows an understanding of criteria for success, namely translating all parts of the situation and correctly sketching the graph's slope (KS1). Additionally, he describes how to change his initial work to correct it, thus, giving himself feedback (KS3).

Likewise, Ayse's case shows that the SAFE tool encourages a reflection of her solution (MA3) and points her attention towards the covariation of both variables (GV2). Her reflection is initiated by the Check's multiple representation of the situation together with her and a sample graph (TF2) as well as a provided checkpoint (TF3). Ayse grasps that change in both variables needs consideration when sketching graphs (KS1). Finally, she formulates a self-feedback (KS3) and regulates her assessment process by reflecting her work, identifying a mistake and crossing off the checkpoint (KS5).

Both students use four key strategies of formative self-assessment and the metacognitive activity of reflection when working on technological features, such as multiple, interactive, and linked visualizations. This enhances their functional thinking by shifting the learners' focus towards the Grundvorstellung of covariation, paying attention to the graph's slope, and translating the entire situation into a graphical representation of the underlying functional relationship. Future investigations of students' work with the SAFE tool will reveal whether these findings can be generalized to inform the design of other digital media to support learners' conceptions of functions.

Figure 2

 2 Figure 2: a) Paul's and b) Ayse's Test solution and reasoning

Figure 3

 3 Figure 3: a) Linked simulation and graph as the Test's sample solution, b) Ayse's Check screen


Axel Schulz
email: axel.schulz@uni-bielefeld.de

Daniel Walter
email: daniel.walter@uni-muenster.de

'Practicing place value': How children interpret and use virtual representations and features

Keywords: Place value, digital media, representational change, primary mathematics education

The tablet-app 'Stellenwerte üben' (Practicing place value) 1 was developed to provide a software for practicing place value concept. There are two goals associated with this project: 1) Developing the app and 2) Investigating how children interpret the given non-symbolic representations and how they use the implemented features. This article presents the underlying theoretical background, a short description of selected features, and a sketch of initial empirical findings on the interpretation and use by primary school children in their second and third school years. First findings suggest that previous experience with concrete physical material overlaps and influences the intended use of the implemented virtual features.

Place value understanding

The development of the understanding of place value and therefore of the composition of multidigit numbers is one of the most important goals in mathematics education in primary school [START_REF] Wittmann | Teaching aids in primary mathematics: Less is more[END_REF]. In addition to that, lack of understanding of the concept of place value is a main indicator for learning difficulties in arithmetic (Scherer et al., 2016, p. 636). Understanding of place value can be assumed when someone 1) knows that the place value system is based on continually grouping by tens (powers of ten), 2) knows about the meaning of place value: The quantity of the value is determined by its position represented by the digitsthe values of the positions increase in powers of ten from right to left, 3) knows about the meaning of face value: The digits have to be multiplied by the power of ten assigned to its position, 4) knows that the quantity of the whole numeral is the sum of the values represented by the individual digits and 5) knows the rules and structure of number word formation corresponding to the written number (e. g. [START_REF] Fuson | Children's conceptual structures for multidigit numbers and methods of multidigit addition and subtraction[END_REF].

In the literature, we can find numerous appropriate activities to foster the development of students' place value understanding and to strengthen it in practice phases. Some of these activities are the following: Bundle and unbundle unstructured sets of objects according to the power ten (van de [START_REF] Van De Walle | Elementary and middle school mathematics: teaching developmentally[END_REF][START_REF] Hiebert | Links between teaching and learning place value with understanding in first grade[END_REF], arrange the bundles according to the place value to introduce and practice the place value system and name the bundles to compile and practice the rules of number word formation (van de [START_REF] Van De Walle | Elementary and middle school mathematics: teaching developmentally[END_REF][START_REF] Gaidoschik | Zehner und Einer: Die ersten Schritte. Anregungen für die Erarbeitung von Stellenwertverständnis im Zahlenraum bis 99[END_REF]. In addition to these activities there are several manipulatives and representations that seem to be appropriate to work with: Unstructured objects such as linking cubes and tiles, pre-structured objects, for example baseten blocks (Dienes-Blocks), 'place value mats' for sorting or trading base-ten material (e. g. [START_REF] Van De Walle | Elementary and middle school mathematics: teaching developmentally[END_REF], pp. 165-169, Gerster & Walter, 1973) and place value charts with tiles or digits for a standardized and conventional representation.

(Virtual) representations and their potentials

Theoretical analyses and empirical findings suggest that there are several features of virtual representations that have the potential to support mathematical learning processes. Some examples for these features are the following: Multitouch-Technology(e.g. [START_REF] Sinclair | Developing number sense with TouchCounts[END_REF], support in structuring an unsorted set of objects [START_REF] Walter | How Children Using Counting Strategies Represent Quantities on the Virtual and Physical 'Twenty Frame[END_REF], reducing an extraneous cognitive load by the means of computational offloading [START_REF] Rogers | HCI Theory: Classical, Modern, and Contemporary[END_REF] and informative feedback (e.g. [START_REF] Harrass | Computereinsatz im Arithmetikunterricht der Grundschule -Theoretische Grundlegung und empirische Forschung zum Üben mit Lernsoftware[END_REF]. For the present study, we focus on the following two features (see below): Fitting between virtual representations and mathematical ideas (e.g. [START_REF] Peltenburg | Mathematical power of special educational needs pupils: An ICT-based dynamic assessment format to reveal weak pupils' learning potential[END_REF] and synchronously linked representations (e.g. Ainsworth, 2006). However, it must be pointed out that the existence of the features mentioned above does not automatically lead to an intended use (e. g. [START_REF] Walter | How Children Using Counting Strategies Represent Quantities on the Virtual and Physical 'Twenty Frame[END_REF].

Fitting between representations and mathematical ideas

Mathematical objects can not be empirically observed (like for example astronomical or biological objects) and therefore mathematical knowledge can not be developed through this observation. "The only way to have access to […] [mathematical objects] and deal with them is using signs and […] representations" (Duval, 2006, p. 107). These representations can be symbolic (e.g. written or spoken numbers "30", "three tens") or non-symbolic such as iconic or concrete representations (e.g. three ten-rods of the base-ten blocks; thirty apples in a picture), or something in between (e.g. three tiles in the tens place in a place value chart). In addition to that, these representations can be 'material/external' or 'mental'for example three dots in a place value chart or thinking of three dots in a place value chart (Duval, 2006, p. 105). In this context, there are (among other things) two aspects that have to be taken into account: 1) The intended way from external representations to mathematical ideas 'is not straight, easy or clear' [START_REF] Söbbeke | Language use, mathematical visualizations, and children with language impairments[END_REF](Söbbeke, , p. 1499) and these representations are not self-explanatory (Dreher & Kuntze, 2015, p. 91). 2) Mathematical ideas are abstract concepts and the representations must not be confused with the mathematical idea or object (Duval, 2006, p. 107).

Nevertheless, the structure of a given non-symbolic representation (no matter if these are concrete, virtual or mental) should 'fit' with the structure of a given mathematical idea (e.g. [START_REF] Sarama | Physical and Virtual Manipulatives: What is "Concrete[END_REF]. On the one hand this fitting should be provided by an 'intended structure' built in by the manufacturer (Söbbeke, 2005, p. 135;Sarama & Clements, 2016, p. 82), on the other hand this fitting is the result of an epistemological process (Steinbring, 2015, p. 288). These aspects and conditions are taken into account through developing the tablet-app and analyzing its interpretation and use (see section "Design research and study design" below).

Synchronously linked representations

"Comprehension in mathematics assumes the coordination of at least two registers of […] representations" (Duval, 2006, p. 115). On the one hand this coordination is a learning goal in mathematics education. On the other hand, it is a challenge because representations can be misinterpreted and the representations of different registers do not have to be congruent and, therefore, a translation between registers is not simple, obvious or self-evident (Duval, 2006, pp. 115-120). An appropriate method to foster the ability to coordinate and translate between different registers and to realize what is mathematically relevant is to analyze representation variations in at least two registers (Duval, 2006, p. 125). In contrast to non-virtual representations, virtual representations can automatically be linked so that changes to one representative have a direct effect on another representation. This feature has the potential to support children in overcoming difficulties in switching between different representations (e.g. [START_REF] Ainsworth | The functions of multiple representations[END_REF][START_REF] Goodwin | A Framework for Examining Technologies and Early Mathematics Learning[END_REF].

Design research and study design

There are two goals associated with the presented project: 1) Developing the app considering the described theoretical background and 2) Investigating how children interpret the given nonsymbolic representations and how they use the implemented features. To give an insight into the process of this design research project, we give a short description of the implemented features, a short description of the study design and in the next section a sketch of initial empirical findings on the interpretation and use by primary school children.

Implemented features

There are numerous apps focusing on place value (e. g. [START_REF] Burris | Virtual place value[END_REF][START_REF] Ladel | Development of a Flexible Understanding of Place Value[END_REF]; Utah State University, n. d.). Though, none of these focus on the practicing phase of the learning, which provide self-generating tasks and informative feedback after entering a result. In addition to that, the implemented features of the app 'Stellenwerte üben' as described below can also be found in these apps. However, there is no app that combines all of these features.

Bundle and unbundle:

The module 'Bundle' requests the bundling of exactly ten objects (ten ones or ten tens) to create a bundle of the next higher value (ten ones  one ten; ten tens  one hundred). To this end, one has to 'activate' exactly ten objects (e.g. by using the lasso-function) and to push the bundle-button. Then the ten objects assemble automatically into the next higher bundle and this slides to the respective column of the place value sorting map (see Fig. 1). An essential aspect of this assembling is the transparent value preserving: Even after bundling ten objects, these objects are still apparent (a ten is made of ten ones, a hundred is made of ten tens). In reverse, unbundling is possible, for example, by sliding one hundred in the tens (or ones) column. After doing this the hundreds automatically split into ten tens (or hundred ones). This feature is implemented in all modules with virtual representation of the base-ten material. Sarama and Clements point out that these virtual bundling activities are "more in line with the mental actions that we want students to learn" (Sarama & Clements, 2016, p. 85, emphasis in original) than activities with concrete-physical base-ten blocks. Sorting bundles: The module 'Sort' requests the sorting of an unorganized set of ones, tens and hundreds in the place value sorting map. By sliding the objects in the respective columns, the students practice the assignment of a given bundle to the appropriate place value column. In this context, the sorting is aligned to the visible quantity of a given bundle. This visual assignment is supposed to strengthen the integration of the 'quantity' value aspect and the 'column' value aspect of place value understanding (Sayers & Barber, 2014, pp. 24-25).

It has to be taken into account, that these practicing activities (virtual bundling and sorting) do not only by themselves lead to a sustainable place value understanding, but there have to be several prior activities and accompanying instruction (e. g. van de [START_REF] Van De Walle | Elementary and middle school mathematics: teaching developmentally[END_REF]). In addition to this, it has to be taken into account that the described activities are based on representations that are visually clearly distinguishable by their quantity. These activities can be interpreted as didactic stages on the way to an abstract understanding of place value. A rigid adherence to these stages and an unreflected use of this representation could lead to common errors (as described by Ladel & Kortenkamp, 2016, p. 297).

The research interests arising from these considerations are the following: Synchronized linked representations: Changes in the iconic register (e.g. unbundle a ten rod) lead to changes in the nonverbal-symbolic register at the bottom of the surface (0+10+2 changes into 0+0+12, see Fig. 3).

Study design

One focus of our research is to analyze students' methods of usage and interpretation while working with the tablet-app 'Stellenwerte üben' (see the research questions developed above). To this aim individual clinical interviews with n=29 German second and third graders have been conducted. All children knew the physical base-ten material from classroom. Main tasks have been, among other things: 1) the creation of non-symbolic representations: "Please show me the number two-hundred-fourty-five using the tablet-app." (RQ1), 2) the explanation of a created non-symbolic representation: "How do you know this is the number you are looking for?" (RQ1 and 3), 3) an interpretation of 2 ten-rods in the tens place: "Another child has told me this is a representation of twenty tens. What do you think?" (RQ2), and 4) taking away two tens from two bundled hundredsquares "Can you please remove two tens." (RQ1). All interviews have been transcribed (obtaining both speech and actions) and analyzed using a qualitative content analysis [START_REF] Mayring | Qualitative Inhaltsanalyse. Grundlagen und Techniken[END_REF].

Initial empirical findings

It was examined whether and how children make use of digital media's features implemented in the developed tablet-app. Furthermore, we wanted to evaluate how children interpret the given nonsymbolic representation. These findings might lead to implications for further development, research and for lessons using the described app.

Using the fitting between representations and mathematical ideas

As expected, we found different types of usage. On the one hand, there are children who use the implemented features (here bundling and unbundling by sliding and lassoing) independently and are aware of what they are doing (for an example see the following transcript).

Interviewer: What would you have to do to remove twenty?

Student: I know, I know. Like this. (Unbundles a hundreds-square by sliding it to the tenscolumn. Swipes away two ten-rods.)

Interviewer: Ok, and what is your result?

Student: Hundred-and-eighty.

On the other hand, there are children who imitate the action with concrete base-ten blocks (namely trading). These children erase a given hundred-square, create ten ten-rods instead and take away two of these ten-rods. This finding suggests that previous experience with concrete physical material overlaps and influences the intended use of the implemented feature. This assumed interdependence between virtual and concrete actions is worth a closer lock in further investigation.

Interpretation of the non-symbolic representation

The non-symbolic representation of the tablet-app was developed to integrate both the quantity value aspect and the column value aspect of place value understanding (Sayers & Barber, 2014, pp. 24-25). To examine children's interpretation of this representation (two ten-rods in the tenscolumn) we asked explicitly for this interpretation ("Another child told us… What do you think?", see above). By doing this we found two different types of explanation: 1) Children who are aware of the fact that there are exactly two tens. Some of these children referred explicitly to the intended integration of quantity value and column value. 2) Children who mix up the naming of the given representation ("There are twenty tens"). Through further analyses and interviews we want to find out if these answers have their roots in an inaccurate use of mathematical vocabulary or if they are indicators for a conceptual misinterpretation of the intended structure.

Using the synchronously linked representations

As expected, our findings indicate that not all children consider the linked representations independently in their methods of use. Most children only focus on and refer to the non-symbolic representation. There are at least two possible explanations for this finding: 1) in the surface's graphical proportioning there is much more room for the non-symbolic representation than for the symbolic representation and 2) the main tasks requested actions with the non-symbolic representation. Nonetheless, it can be stated that there are both low-and high-achieving students who do independently use the linked representations, and all students use this feature when asked for an explanation. In these cases, the linked representations are used to support children's argumentation and as self-monitoring.

Interviewer: Please show me the number two-hundred-fourty-five.

Student:

(Creates two hundreds in the hundreds-column, four tens in the tens-column, five ones in the ones-column.)

Interviewer: Ok, and how did you know, how many hundreds, how many tens and how many ones you needed?

Student: One square is hundred ones (points at the hundred-square) and there are two, thus, two-hundred. You can read this here, too (points at the symbolic representation).

Conclusion

The described study investigates children's usage and individual interpretation concerning a virtual practicing material. The tablet-app was developed to foster and strengthen children's place value understanding and it provides several features which are supposed to support mathematical learning processes. The implemented features are based on mathematical didactic theories and findings. In the following, we summarize the main results and draw some conclusions for teaching and further research.

The described findings validate the previous assumptions [START_REF] Walter | How Children Using Counting Strategies Represent Quantities on the Virtual and Physical 'Twenty Frame[END_REF] that the existence of the described mathematical didactic features of virtual representations does not automatically lead to an intended use. Especially instead of using the synchronously linked representations children mainly focused on the non-symbolic representation, which is quite explainable. Nevertheless, we found that some children (both low-and high-achieving) used the linked representations independently and others used them for argumentation when asked for an explanation. This knowledge gives teachers the opportunity to discuss the connection between the different representations in the context of 'math conferences' with all students. In these settings students may become aware of the possibility to change between and use different representations, which they do not yet do independently.

Another feature which was not used as intended by some children is the fitting between the virtual actions and the intended mental actions (Sarama & Clements, 2016, p. 85). Some students traded a hundred for ten tens and vice versa, instead of bundling or unbundling. These different types of usage (trading vs bundling and unbundling) could be used for constructive discussions in class: Do these different actions lead to the same results? What aspects of these actions are the same, what are different? This discussion about actions, representations and the underlying mathematical ideas might lead to a better understanding of some basic aspects of the place value concept.

As mentioned above, some of the reported findings lead to further research interests like the investigation of an interdependence between using concrete and virtual representations. Also of interest is the question if the different interpretations of the non-symbolic representation ground merely on inaccurate use of mathematical vocabulary or if they are indicators for a conceptual misinterpretation of the intended structure. Since the tablet-app was developed for practice, further investigations also might focus more on autonomous usage (without an interviewer) and ways of practicing.

Figure 1 :

 1 Figure 1: Bundling ten ten-rods



  Research question 1 (RQ1): In which ways do students interpret and use the intended fitting between representations and mathematical ideas?  Research question 2 (RQ2): How do students interpret the non-symbolic representation particularly with regard to the hundred-squares, ten-rods and unit-squares in the place value sorting mat?

Figure 3 :

 3 Figure 3: Synchronously linked representations in 'Stellenwerte üben'  This leads to research question 3 (RQ3): Do children notice the synchronously linked representations and in what ways do they use this feature?

Figure 2 :

 2 Figure 2: Sorting objects

The German version of the tablet-app is provided on Google Play. An English version is in progress.


Georgios Thoma
email: g13thoma@gmail.com

Irene Biza
email: i.biza@uea.ac.uk

Problem-solving techniques in the context of an educational video game: the Mudwall puzzle in Zoombinis

Keywords: Problem-solving, video games, techniques, global-local problem

This paper aims to highlight the problem-solving nature of video games by examining problemsolving techniques in the context of the educational game Zoombinis. Using screen-capturing software, the gameplay of an 8-year-old participant was recorded and analysed in relation to his strategies towards solving the implicit Mudwall problem in the game, from a global-local perspective. Findings indicate the development of techniques -'stepping-stone' and 'probableexclusion'-that through Polya's problem-solving steps, the game's design and participant's experimentation, can be used to solve the Mudwall puzzle, illustrating such game's potentiality as a problem-solving environment.

Introduction

Compared to other media, video games offer interactivity to the player. As this unique attribute is constantly explored and worked upon by game designers, educational researchers examine the role video games can play to learning. Playing video games, for example, can create highly motivational and engaging environments for players [START_REF] Van Eck | SAPS and digital games: Improving mathematics transfer and attitudes in schools[END_REF]. [START_REF] Devlin | Mathematics education for a new era: Video games as a medium for learning[END_REF] and [START_REF] Monaghan | Games: Artefacts in Gameplay[END_REF] view video games as ideal means of presenting and learning mathematics, which, through their design, can potentially represent mathematics by overcoming the symbolisms used in paper. Literature has focused on ways video games could influence mathematical learning, either be played during students' spare time [START_REF] Lowrie | Problem solving in technology rich contexts: Mathematics sense making in outof-school environments[END_REF] or be included in traditional classrooms [START_REF] Moore-Russo | A study of how angry birds has been used in mathematics education[END_REF] or other classroom settings [START_REF] Calder | Using Mathematical Apps with Reluctant Learners[END_REF]. However, even when researchers are optimistic about the future, they are still reserved on what has been realized so far by video games [START_REF] Ke | A case study of computer gaming for math: Engaged learning from gameplay?[END_REF]. Video games have been studied in many mathematics education research studies mainly as motivational and engaging educational environments [START_REF] Kebritchi | The effects of modern mathematics computer games on mathematics achievement and class motivation[END_REF]Chang et al., 2015). Nevertheless, few studies focus on how mathematical content is presented in these games. The study [START_REF] Thoma | A case study of young students' formation of problem solving methods and development of techniques in the context of Zoombinis, an educational video game[END_REF], part of which is presented in this paper, examined the educational game Zoombinis as a mathematical problem-solving environment for young children. Zoombinis has been viewed as engaging, challenging and motivational by previous studies [START_REF] Yelland | Playing with ideas and games in early mathematics[END_REF][START_REF] Moss | Playful thoughts: A study of the effects of "Logical Journey of the Zoombinis" on elementary students' mathematical attitudes and reasoning skills[END_REF]. The aim of this study was to examine how the problems are presented in Zoombinis, how young players interact with the game, and how they form techniques to deal with said problems. In this paper, we focus on the techniques developed by one participant in a specific puzzle of the game, called Mudball Wall (Mudwall thereafter). After presenting the theoretical framework of the study, a description of the game-and then the Mudwall puzzle-will be offered showcasing the key features that make it a problem-solving environment. Next, the analysis is presented using excerpts of the participant's playthrough highlighting his techniques, followed by a discussion with concluding remarks. [START_REF] Downs | Local and Global Perspectives in Problem Solving[END_REF] propose two perspectives on mathematical problem: the local and global. The authors argue that every problem's structure implies a global system of principles and assumptions that are related to this and other similar problems. However, the structure of a certain problem remains highly local and specific. Problem-solving approaches and arguments can develop at a local level, that in turn might be generalised from a global perspective, namely a metalevel of similar problems. According to [START_REF] Downs | Local and Global Perspectives in Problem Solving[END_REF]) the "switch" (p.2271) between the two perspectives might happen in two ways: by viewing problem-solving as an amalgamation of both global and local perspectives, or by examining strategies emerging from a local perspective, that become important in the global. One construction that could emerge through the interrelationship of the local-global could be a technique. In this paper, the term technique is used based on the definition offered in Mamona- [START_REF] Mamona-Downs | Realization of techniques in problem solving: the construction of bijections for enumeration tasks[END_REF]:

Theoretical framework

A technique, as does a method, conveys a mathematical argument that can be extended over a variety of tasks. Our convention is that a technique splits the argument into pre-determined stages, each of which addresses a specifically formulated aim. For a method such explicit structuring is not stressed. (p. 236) While techniques can be linked with a specific mathematical theory, some techniques can be formed as "products of problem-solving activities" and "successfully completed solutions" (p. 237). After formed, techniques can be used as problem-solving tools. This way one technique could deal with many problems, as well as objects, that share similar attributes, very often without the problem-solver being aware of this technique. Additionally, Mamona- [START_REF] Mamona-Downs | Realization of techniques in problem solving: the construction of bijections for enumeration tasks[END_REF] view each solution as a construction, which can guide a problem-solver towards understanding the technique as a response to past problem-solving experiences. Finally, techniques could be viewed as existing between heuristics and algorithms (Mamona- [START_REF] Mamona-Downs | Επίλυση προβλήματος στα μαθηματικά [Problem solving in mathematics[END_REF].

Video games as problem solving environments

Video games generally have an overarching theme, story or goal that create a setting for the playerproblem solver. This is what we consider as the global perspective of the problem (global problem thereafter). The global problem is not always explicit, but rather it is experienced through solving problems from a local perspective that players face at each puzzle (local problem thereafter). Most of these local problems in games hint or build towards a solution of a global problem, either explicitly, shared with the players, or hidden from them until they make the connection themselves. We see video game problem-solving endeavour, having similarities to Polya's basic steps of problem solving (1973, pp. xvi-xvii): trying to understand the problem; making connections between available data to devise a plan; carrying out said plan; and reflecting on actions. Since video games are interactive media and usually offer real-time feedback the player may be guided by the game's design towards the steps mentioned. Additionally, variations of the same problem can guide the development of certain techniques. However, for the above to happen a player must have experience and a clear understanding of the games rules. Thus, the understanding for this paper is that techniques can be formed only once an initial exploration phase has past, when the player has some awareness of the actions they are allowed to perform and has formed an idea or expectations on what results said actions might yield. Therefore, experimentation is important in these early stages and is usually the first step of a developing technique. In its mathematical essence regardless the problem, experimentation has the following elements: "Locate the variables, the key-elements of the problem. Keep all except one constant, and start experimenting with this one to understand its role towards the solution of/solving the problem. Afterwards, keep this one constant and experiment using another one. Do the same for each element of the problem until you can understand the role each one plays towards solving the problem." (translated from Greek, Mamona- Downs and Papadopoulos, 2017, p. 81) Based on the above, a game can be defined as a collection of global problems, that through experimentation and consecutive playthroughs of local problems, can lead players into reflecting on their actions, and developing techniques that lead them to a solution.

The participant and data collected

The participants of the study were four children aged 6 to 8 years old. A requirement for the study was participants had never played the game before. This way, the exploration and the formation of problem-solving techniques could be captured from their first stage of development. Data were collected by the first author through video captured from participants' gameplay, recording their onscreen actions, exclamations, inquiries, ideas as well as conversations between them and the researcher who is the first author of the paper. To enrich audio data, participants were advised to express their thoughts and actions verbally as has been done in similar studies [START_REF] Ke | A case study of computer gaming for math: Engaged learning from gameplay?[END_REF]. Additionally, suggestive and encouraging prompts were used by the researcher in an effort to produce more data, while creating a comfortable and game-friendly environment for the participants. The 8-year-old participant presented in this paper, with pseudonym Damian, played the game in four hourly sessions, in different days in a week.

For each participant, the screen recordings were categorized to short segments of playthroughs, namely successful or unsuccessful attempts of the same game puzzle. Then, the data were analysed according to participants' actions to go through the puzzle (experimentation); approaches to understand what works or not (local problem); being able to see the underpinning rules (global problem); being able to transfer what they have learned in one playthrough to the next one (shift from the local to the global); development of techniques. The total amount of time allocated in each stage and the gameplay session were also taken into account.

Zoombinis 1 : the game and the global-local problems

Zoombinis are creatures that try to find a new home. Guided by the player in groups of 16, they visit and must go through a total of 12 puzzles before reaching their goal. Each Zoombini has four attributes: hair, eyes, nose and feet, and these are often the variables used in each game puzzles (three Zoombinis can be seen in Figure 2). Each puzzle has a penalty system that stops Zoombinis from progressing, thus forcing the player to backtrack, collect any Zoombinis left behind, and try the puzzle again. This way, players are encouraged to work towards more efficient and effective solutions, improving them through continuous playthroughs of local problems. Based on the above, we define local problems of a puzzle the individual attempts of this puzzle. The puzzle has an underpinning rule that is controlled by variables that are not known to the player, which we call global problem. However, in each of player's attempts the values of these variables change randomly as we explain in the context of Mudwall in the next section. That means the player begins a new attempt with new values of the variables, which justify a new local problem. However, the global problem remains the same and the solver has to unpack it, often working unknowingly towards its solution. While a local problem might be solved by chance, this solution will not work the next time the player visits the puzzle, due to the randomization mentioned before. In order to be effective problem solvers, players need to find techniques for the global problem that corresponds to this puzzle that will allow them to solve every local problem they will encounter at this puzzle. Considering the specific penalty of each puzzle, the global problem is the search for an optimal solution with minimum and ideally zero Zoombinis left behind after the problem is solved.

The Mudwall puzzle

In the Mudwall puzzle, the Zoombinis are faced with a tiled wall that obstruct their journey. The player tries to get the Zoombinis to go over the wall using a mudball launcher machine. This machine fires mudballs onto the tiled wall and the player needs to hit the marked tiles that have dots in them (a double-dotted marked tile can be seen in Figure 2). The number of dots on each marked tile corresponds to how many Zoombinis will go over the wall successfully. The machine has buttons of five colours (blue, red, yellow, purple and green) and five shapes (square, triangle, star, circle and rhombus). In each playthrough the matrix (named A for the purposes of this paper) is a permutation of five shapes on one axis, and a permutation of five colours on the other axis, which is not known to the player. Each cell of this matrix corresponds to a unique combination of shape and colour. Additionally, there is a hidden permutation of the two axes as well, as the colours or the shapes can either be on the horizontal axis or the vertical one. These random permutations allow for: possible different matrices. Furthermore, on each such matrix, marked tiles are distributed randomly. The number of marked tiles can be from one to eight, based on the number of Zoombinis the participant has at the beginning of this puzzle. We consider the local problem of the Mudwall as: "Are you able to find the coordinates (i.e. shape and colour combination) of all the marked tiles, on this specific wall?". The player does not have an unlimited amount of mud due to the penalty system of the game. Essentially, the player's aim is to use the least possible amount of mudballs to the player to locate the coordinates of the marked tiles. We consider the underpinning global problem: "Can you find the coordinates of the marked tiles, in each possible wall (out of the 28800 that might appear), by firing the least amount of mudballs every time?". In the following section, we describe Damian's playthroughs of the Mudwall puzzle and illustrate the development of his problem-solving techniques.

Analysis of Damian's Mudwall puzzle playthroughs

Damian visited the Mudwall puzzle six times, spending 2 to 4 minutes in each playthrough, for a total of 15 minutes. It is important to highlight that due to the linear progression of the game described above, Damian was engaged with other puzzles in Zoombinis between playthroughs of the Mudwall. Thus, a considerable amount of time has passed between each of the events described below, ranging from ten-minute gaps up to days. In his playthroughs he was mostly quiet and engaged, and only spoke about any of his strategies when he was sure of their results. In the following pages, matrices (Figures 1 and3) created by the first author as part of the analysis demonstrate Damian's sequence of actions. Additionally, a screenshot (Figure 2) of the overall playthrough is offered where a step-by-step summary is not necessary.

At his first encounter with the Mudwall puzzle Damian is mostly exploring and experimenting, trying to understand how to interact with the puzzle. He located and used the available buttons of shapes and colours but he was not able to identify the goal of the problem and subsequently solve it. This experimentation phase was expected, as the Zoombinis video game constantly enforces an exploratory attitude through gameplay and narration, where players have to identify the variables, the goal and finally the solution of the problems they come across. Before ending his first playthrough, without having successfully launched any of the Zoombinis over the wall, he noticed that he run out of mud, which marks his first realisation of the penalty system of this puzzle. On his second playthrough, Damian shot four times consecutively the same yellow-star mudball (Figure 1, #1 to #4). During his shots, his cursor movements indicated how he was trying to hit adjacent tiles to yellow-star, by point-and-clicking on the tiles themselves, then firing the mudball. After his fourth shot, he stops trying to direct his mudballs this way, and explores different combinations of shape and colour. In his last two shots (Figure 1, #10) a pattern can be seen, the process of keeping one variable unchanged. This will be explored by Damian on his next playthrough. Finally, he managed to hit one marked tile A 4,3 (Figure 1, #9). After conversing with the researcher, he became aware that this is his aim on the Mudwall puzzle, to aim for the marked tiles.

In the third playthrough (Figure 2), Damian focused on keeping the colour constant and changing the shape. Once the column with the yellow shapes was completed, he tried to complete the blue column until his mud finished. During this playthrough, Damian identified the relationship of colour and shape to columns and rows of the tilled wall. In his fourth playthrough (Figure 3) Damian found the coordinates of all the marked tiles, getting all his Zoombinis over the wall. This is his first totally successful playthrough, and from this point onwards he was able to develop techniques which he then uses on his last recorded playthroughs.

We present this playthrough in detail, aiming to illustrate the techniques Damian developed. His first shot (#1) is a yellow-circle and the second (#2) a red-square (Figure 3). He changed both variables keeping neither colour nor shape constant. The current information does not allow him to relate the shapes and colours to the rows and columns of the matrix. With his third shot (#3) of a red-triangle, Damian keeps one variable constant (colour) and changes the second (shape). This creates a connection between the two mudballs: both share one variable, in this case the red colour (Figure 3, #2 and #3). These two mudballs now reveal that the matrix has colours in the columns and therefore shapes in the rows as illustrated below (Figure 3, #3-known data). We can see the approach of keeping one variable unchanged as systematic experimentation, which we name the stepping-stone technique.

Our interpretation is that Damian used the stepping-stone technique with the aim of landing on a marked tile. However, as the colours are in the columns he did not manage to land on the marked tile A 5,1 and lands instead on tile A 2,3 . By observing that the colours are arrayed on the columns, he can deduce that shapes are on rows. With the new information gained, Damian now focuses to hit either the tile A 3,1 or A 4,5 which are marked. He knows that the first row will be circles, the second triangles and the fifth squares. Thus, he eliminates all shapes except rhombus and star, as well as the colours red and yellow, as they belong on the third and fourth column respectively (Figure 3, #3known data). Finally, he chooses the colour blue, and the shape of a rhombus, both choices for which he has no information yet. His fourth mudball is blue-rhombus that lands on A 4,5 with success. We see this elimination of available options as another type of systematic experimentation, which we call probable-exclusion technique.

For his fifth (#5) mudball, Damian uses again the probable-exclusion technique and fires a pink-star (Figure 3). This time his shot lands on A 3,2 . He finally finds both tiles A 3,1 and A 5,1 using both the probable-exclusion and stepping-stone techniques.

On his fifth and sixth playthroughs Damian employs the strategies shown in his fourth playthrough.

He begins by keeping one of the two variables constant in his first two mudballs. From that point he uses interchangeably the stepping-stone and the probable-exclusion techniques, aiming for marked tiles whenever he gains some new information. Both his last playthroughs were successful as he managed to get all his Zoombinis through with mud to spare.

Discussion and conclusion

The aim of this paper was to examine the development of problem solving techniques in the context of the educational video game Zoombinis. We analysed Damian's experimentation approaches in relation to solving the local and global problem of the Mudwall puzzle, a problem he had no experience with. During his first local problem, he explored Mudwall by locating the variables, both colour and shape; as well as the penalty system of the mud decreasing after each shot. In his second local problem, he explored the machine's controls showing awareness of its function and managed to hit a marked tile thus experiencing a successful result, indicating his global problem goal. On his third local problem, he worked with only two colours, minimizing the variety of his choices, but establishing the connection between colour-shape to row-column. Finally, in his fourth playthrough he used the two techniques of stepping-stone and probable-exclusion, that combined lead him to his first successful solution in terms of the game. It is at this stage that Damian can be considered as having simultaneously solved the global problem of finding all the marked tiles without his mud running out. Thus, we identified strategies that go beyond random trial and improvement.

However, not every participant of the original study was as successful as Damian with the Mudwall puzzle. For the limited time of less than five hours of gameplay, all participants showed examples of experimentation, formed techniques and adapted them, some successfully reaching a global solution in the various in-game problems. It was also interesting how similar patterns emerged from the different participants' data, leading to the suggestion that due to carefully crafted game's design, players can be smoothly guided to a specific form of problem solving approaches.

This paper examined the video game Zoombinis as an environment used to learn problem-solving techniques. In their study Mamona- [START_REF] Mamona-Downs | Realization of techniques in problem solving: the construction of bijections for enumeration tasks[END_REF] highlight some issues regarding teaching techniques, such as students not wanting to use a technique after they had been taught it, and the difficulty to identify the underlying technique through the mathematical tasks which were provided for that reason. In my view, in the context of the video game, the first issue potentially could be resolved as students are the ones discovering and adapting the technique themselves.

Problem-solvers implicitly guided by a global problem, they have the agency to interact and experiment in a story-driven and challenging environment and thus find the need for a more efficient solution. Along the way, techniques and other constructions formed from this process, would be something players developed on their own and something that is efficient at providing a solution.

Figure 1 :

 1 Figure 1: Damian's second playthrough, where he hit his first marked tile #9, the yellow-rhombus marked with a '+' sign in its center

Figure 2 :

 2 Figure 2: Damian's third playthrough, one Zoombini has made it over the wall successfully

Figure 3 :

 3 Figure 3: Damian's fourth playthrough and his first fully successful one

https://external-wiki.terc.edu/display/ZOOM/The+Game