text:
Melih Turgut
email: mturgut@ogu.edu.tr

Students' reasoning on linear transformations in a DGS: A semiotic perspective

Keywords: Semiotic mediation, linear algebra, DGS

The aim of this paper is to analyze students' reasoning on linear transformations while using a Dynamic Geometry System (DGS) from a semiotic mediation perspective. Considering design heuristics of Realistic Mathematics Education and the semiotic potential of certain tools and functions of DGS, I have developed a hypothetical learning trajectory and have designed a task for inventing fundamental properties of linear transformations. The task was field-tested in a case study with pair of undergraduate linear algebra students. An analysis of the task-based interviews, with a semiotic mediation lens, shows that the students managed to (re-)invent the fundamental properties of linear transformations.

Introduction

One major issue in the teaching and learning of linear algebra is providing students with readymade mathematics using different representations and different contexts [START_REF] Dorier | The role of formalism in the teaching of vector spaces[END_REF] without considering the students' intellectual needs for learning [START_REF] Harel | Two Dual Assertions: The First on Learning and the Second on Teaching (or Vice Versa)[END_REF]. An example might be to introduce the notion of linear transformations with two fundamental properties as in numerous textbooks, where such introduction to the topic could trigger epistemological issues for students' conceptualization of non-linear transformations [START_REF] Dreyfus | Cabri based linear algebra: Transformations[END_REF]. In this paper, I acknowledge a contrary introduction to the topic and consider a research question: Is it possible for students to (re-)invent fundamental properties of linear transformations? To answer this question, I consider a dynamic geometry system (DGS), which invite students into a progressive process of epistemic exploring, conjecturing and generalizing [START_REF] Leung | Discernment of invariants in dynamic geometry environments[END_REF]. Consequently, I focus on specific tools and functions of GeoGebra, such as dragging and grid functions, ApplyMatrix command and slider tool of the DGS as a tool of semiotic mediation for students' reinvention of proposed mathematics.

Theoretical Perspectives

In this work, I consider two theoretical insights: (i) Realistic Mathematics Education and (ii) Theory of Semiotic Mediation, for designing instructional activity and analyzing the teaching-learning process.

Realistic Mathematics Education (RME)

RME is a domain-specific instructional theory developed by Dutch researchers (Van den [START_REF] Van Den Heuvel-Panhuizen | Realistic Mathematics Education[END_REF]. The word realistic, here, does not directly refer to real-world task situations, but to paradigmatic situations that invite the development of meaningful mathematics. The problem situations do not necessarily come from real life directly, they can be related to an imaginary world or to real mathematics that students experience as meaningful: task situations have to be experientially real [START_REF] Gravemeijer | How emergent models may foster the constitution of formal mathematics[END_REF] to students. In parallel to such views, RME offers three interacting design heuristics for curriculum developers and educational designers [START_REF] Freudenthal | Didactic Phenomenology of Mathematical Structures[END_REF][START_REF] Gravemeijer | How emergent models may foster the constitution of formal mathematics[END_REF][START_REF] Van Den Heuvel-Panhuizen | Realistic Mathematics Education[END_REF]: guided reinvention, didactic phenomenology, and emergent modelling. Guided reinvention means providing students with an environment for their exploration, elaboration and inventing of mathematics. Didactic phenomenology refers to finding certain experientially real phenomena, which might form an environment where students create mathematics. The objective of emergent modelling is to enable students to shift from informal task situations to formal mathematics through support, enabling them to create their own informal mathematics.

Theory of Semiotic Mediation (TSM)

TSM was presented by Bartolini [START_REF] Bartolini Bussi | Semiotic mediation in the mathematics classroom: Artifacts and signs after a Vygotskian perspective[END_REF] with the following main idea: to construct mathematical meanings, the teacher intentionally uses artefacts as a tool of semiotic mediation, which are used in carefully-designed tasks. The aim of the TSM is to transform students' personal meanings to mathematical meanings. The teacher exploits the semiotic potential of the artefact, in which he or she uses an epistemological and didactical analysis to picture out possible learning steps from personal meanings to shared conventional mathematical meanings.

Here, taking into account the didactic goals, the teacher considers what students know, what their experience with the artefact is and how they will accomplish the task by using the artefact. As a next step, the teacher designs a didactic cycle for classroom interventions.

Students' interaction with the artefact produces a complex semiotic process. Artefact signs (aS) appear when students who use the artefact relate in some way to the activity; specifically, to the use of artefact. Mathematical signs (mS) appear, when the students make a definition, conjecture, generalization or proof corresponding to didactic goals. Pivot signs (pS) have an interpretative link between personal meanings and mathematical signs and can appear in the accomplishment of the task. In the application of the didactic cycles, the teacher's role is orchestrating students' learning.

Methodology

This paper, in which I focus and present the results of a single task, is part of an extensive Design-Based Research (project) [START_REF] Bakker | An Introduction to Design-Based Research with an Example From Statistics Education[END_REF]. Due to page limitation, I will present a case limited to a pair of students (A male, B female), who were sophomore level undergraduate linear algebra students, aged twenty. The students had experience solving linear equations, matrix algebra, (geometric) vector spaces and subspaces, and they had learned that every linear transformation can be represented through matrices. They also had experience in the use of GeoGebra's main functions, specifically forming a slider and a matrix, and applying the ApplyMatrix construction tool from previous task sequences, where they constructed meaning of a transformation and linear transformation. However, the students did not know the fundamental properties of linear transformations. Task-based interviews were video-recorded and lasted around half an hour. The data was analyzed through a semiotic lens using categories of signs [START_REF] Bartolini Bussi | Semiotic mediation in the mathematics classroom: Artifacts and signs after a Vygotskian perspective[END_REF]: aS, mS and pS.

Mathematical context, semiotic potential of DGS and task design

A linear transformation is a specific transformation between V and W can be represented as W V T  : for vector spaces V and W , where T satisfies: (i)

) ( ) ( ) ( v u v u T T T    for all vectors V  v u,
, and (ii)

) ( ) ( u u kT k T  for all V  u and all scalars R  k (Lay, 2006). Here, I took 2 R   W V
because of DGS availability (for example as in GeoGebra) and considered the semiotic potential of the following tools and functions of DGS for students' (re-)invention of the fundamental properties above: (i) the dragging function allows the user to manipulate figures and explore independency-dependency of drawings and constructed objects, (ii) the grid function activates specific lines for integer values on the x and y axes and this function enables the user to observe variations of the coordinates of the objects in different windows, (iii) the slider tool offers a means to define a parameter and this may evoke meaning for dynamic (co)variation [START_REF] Turgut | Students' thinking modes and the emergence of signs in learning linear algebra[END_REF], (iv) the ApplyMatrix tool works through an input line that enables the user to apply certain matrix transformations to geometric figures. I postulate that students' dragging sliders connected to a matrix and applying matrix transformations to arbitrary vectors could provide an understanding for a meaning: matrix (and therefore linear) transformations preserve vector addition and scalar multiplication.

The synergy between the definitions of guided reinvention and didactic phenomenology heuristics and the notion of semiotic potential in TSM implies the construction of a possible learning route, in other words, a Hypothetical Learning Trajectory (HLT) [START_REF] Simon | Reconstructing mathematics pedagogy from a constructivist perspective[END_REF] which has to be elaborated on by the designer before the experiment by following four points [START_REF] Bakker | An Introduction to Design-Based Research with an Example From Statistics Education[END_REF]: (i) learning goals, (ii) students' pre-knowledge, (iii) assumptions for students' learning, and (iv) the teacher's role (also in our case, the role of artefacts). Therefore, in Table 1, I express (i), (ii) and (iii) points of a HLT for invention of fundamental properties of linear transformations in a DGS. As aforementioned before, students worked on GeoGebra interface in the previous didactic cycles, which were about transformation of geometric vectors, figures, and constructing meaning for linear transformation. Consequently, the tools and functions of GeoGebra and proposed concepts were experientially real for them. Following Table 1 and considering guided reinvention heuristic, the task was formulated as follows (a possible interface for the task steps is presented in Figure 1), and also for students' making their own models (cf. emergent modeling).

Step 1: Open GeoGebra and activate grid function. Next, form two sliders a and b and, using a and b, form an arbitrary 2 2 matrix. . Move the sliders and drag u and v in itself. Discuss with your pair and explain your observations. Step 4: Form a new slider k . Now, obtain matrix transformation of u k and also compute u k . Drag the vector u and explain your observations, and make conjectures. What happens when you move the sliders? . For a while, the students discussed the steps of the task to determine which matrix application is the first, the second or the third, which seemed rather confusing for them. After they had completed the three steps, while dragging the sliders, they were surprised because a number of vectors and some transformation vectors overlapped. At this moment a few aS appeared (see 18-20): 18 A: … [pointing on the grid (see Figure 2a)] look, how this happened, these are overlapping… 19 B:

No. I think, it is because of matrix, look, [dragging sliders and pointing matrix entries with pencil (see Figure 2b)] it is changing. Next, the teacher intervened to make the students focus on the transformations of the vectors, because they had spent a lot of time dragging sliders, changing matrix entries (i.e., trying a unit or zero matrix and so on) to figure out why some overlapped (see Figure 3a). Then students rechecked the steps and wrote up the findings in their own way. Some pS appeared here, reflecting the students' new meanings through the semiotic potential of the artefact (37-38), and also appeared on the students' productions (Figure 3b).

T:

… what about the transformations of vectors? What did you observe? … 37 B:

… I think we will find a relationship between these [pointing on the notes (see Figure 3b)]. Here, we have the sum vector's transformation and sum of each vector's transformation. 38 A:

However, this could be dependent on the choice of matrix? What will happen for the matrices where their determinants are zero? … As a next step, aS and pS interlaced with the students' personal meanings associated with matrix transformations. They re-analyzed their findings, and finally, mS corresponding to students' reinvention of the fundamental properties appeared (93-97). … [First, she is writing her conclusion, but not mathematically (see Figure 4a), then she is trying to write it mathematically with her partner's help (see Figure 4b)] … Next, while trying to express the situation mathematically, which was under the teacher's orchestration, the students reinvented the first fundamental property (see Figure 5a). However, the teacher was orienting students to prove their result considering their pre-knowledge on representing linear transformations with matrices. Student B immediately related the situation with her preknowledge and proved her conclusion (119 and Figure 5b). 129 B: … Oh yes, I think this is obvious; this is also a result of property of matrix algebra. For a matrix, k can be multiplied with each entry of a matrix or it can be expressed a factor [writing

) ( u u k f k  
]. Therefore, their [meaning u k and u k ] transformations are the same.

Conclusions

In this paper, I consider the research question, 'Is it possible for students to (re-)invent fundamental properties of linear transformations?' Students work on the task formed through the design heuristics of RME and the semiotic potential of some tools and functions of GeoGebra provide an affirmative answer, but with some doubts and limitations. For instance, the students spent much time determining vectors when they overlapped. This issue to be considered is the students' frequent analysis of matrix entries, where they think that an overlapping situation depended on this. I think that such frequent analysis of matrix entries stems from previous experience, where the students were continually trying to find matrices of linear transformations. Interestingly, in the students' analyses for characterizing the matrix, different semiotic resources beyond aS, pS and mS appeared; for instance, gestures and mimics attached to students' analyses process. A multimodal perspective [START_REF] Arzarello | Semiosis as a multimodal process[END_REF] could provide a detailed view for our case. However, in the present case within a TSM perspective, I observe a semiotic chain [START_REF] Bartolini Bussi | Semiotic mediation in the mathematics classroom: Artifacts and signs after a Vygotskian perspective[END_REF], which shows the connection between semiotic resources of students' learning, for inventing fundamental properties of linear transformations as follows (Figure 6).

  Step 2: Form two arbitrary vectors v u, and construct v u  through an Input line. Step 3: Apply matrix transformation to v u, and v u  . Name these vectors, respectively: u , v and w respectively, and then calculate

Figure 1 :.

 1 Figure 1: An expected DGS interface for the task

  Figure 2: a, b, c Emergence of aS during analyzing the overlapping situation

d

  Figure 3: a, b Students' productions as pSInterestingly, once more, they focused on the entries of the matrix, because in the previous step they had employed a unit or a zero matrix, and they began to check other possibilities for the cause of the overlapping situation. Consequently B figured 'they always overlap'. Here, aS 'overlaps' in the previous analyses, and can now be considered a pS (see 63, 86), because it is mediating the transformation of personal meanings to mathematical meanings.63 B:It is clear that they always overlap … Why is this happening? 64 A: Exactly… but why? …

  . What was the meaning of w? It was a transformation of v u  ? … [moving sliders and thinking]… are overlapping, this means, we have obtained the same vector. … Does transformation of the sum vector [meaning v u  ] equals the sum of the separate transformations? 96 A: Absolutely, right. … 97 B:

Figure 4

 4 Figure 4: a, b Students' conclusions for the first fundamental property as mS

Figure 5

 5 Figure 5: a, b Emergence of mS in relation to the task's goal As a final step, the teacher asked the students to consider Step 4. As expected they placed a slider for k, and applied a matrix transformation to u k and also computed vector u k . As soon as one student saw that the transformation of u k and u k overlapped and were exactly the same, by the help of the first property she invented, B expressed her views. The final mS emerged in the discourse (129).

Figure 6 :

 6 Figure 6: A semiotic chain for inventing fundamental properties

Table 1 : HLT for the inventing of fundamental rules for linear transformations

 1

	Associated Concepts	Expected Steps in the DGS	Exemplary Task	Epistemic Artefacts in DGS	Expected Mathematical Meanings
	-Geometric vectors -Addition of vectors -Multiplication with scalars -Matrix transformations -Fundamental properties of transformations linear	-Exploring the the objects effects of sliders on (arbitrary) linear transformations of arbitrary vectors -Comparing the initial and final versions of vectors while moving sliders or dragging	-Form sliders the objects -Construct 2 ´2 matrix -Form arbitrary vectors -Use Apply sliders and drag -Move the command Matrix	-Slider tool tool construction -Dragging -Grid function -Apply Matrix	-Comprehending that the situation is independent from matrix entries or vectors -Formulating the first rule situation,

Proceedings of CERME10

Acknowledgment

This paper is part of an on-going post-doctoral project, which is supported by the Scientific and Technological Research Council of Turkey (TUBITAK), under 2219-International Post-Doctoral Research Fellowship Programme. I would like to thank my supervisor Paul Drijvers for his kind contributions to the paper and also thank TWG16 reviewers for making constructive comments.


domain: math shs

text:
Sonia Palha

Stephan Koopman
email: s.a.h.koopman@hva.nl

Interactive Virtual Math: A tool to support self-construction graphs by dynamic events

Keywords: Visualization, virtual reality, interactive tool, mathematical modeling, reasoning

An essential condition to use mathematics to solve problems is the ability to recognize, imagine and represent relations between quantities. In particular, covariational reasoning has been shown to be very challenging for students at all levels. The aim of the project Interactive Virtual Math (IVM) is to develop a visualization tool that supports students' learning of covariation graphs. In this paper we present the initial development of the tool and we discuss its main features based on the results of one preliminary study and one exploratory study. The results suggest that the tool has potential to help students to engage in covariational reasoning by affording construction and explanation of different representations and comparison, relation and generalization of these ones. The results also point to the importance of developing tools that elicit and build upon students' self-productions.

Introduction

Students' difficulties with constructing graphs that model dynamic events are well documented in literature (e.g. [START_REF] Thompson | Quantitative reasoning and mathematical modeling[END_REF][START_REF] Carlson | Applying covariational reasoning while modeling dynamic events: A framework and a study: A Framework and a Study[END_REF][START_REF] Carlson | The precalculus concept assessment: a tool for assessing students' reasoning abilities and understandings[END_REF]. When modeling a dynamic situation into a graph (e.g. the speed variating with time or the height of water in a bottle variating with volume), it has to be conceptualized as a covariation relation, that is a relationship between two variables that vary simultaneously [START_REF] Thompson | Quantitative reasoning and mathematical modeling[END_REF][START_REF] Carlson | Applying covariational reasoning while modeling dynamic events: A framework and a study: A Framework and a Study[END_REF]. However, students have a tendency to view functions in terms of symbolic manipulations and procedures rather than as relationships of dependency between two variables. These students might encounter difficulties imagining how the output values of a function are changing while imagining changes in function input values. And therefore they might fail in successfully construct a graph of a function modeling a dynamic situation.

Research has revealed that traditional approaches have not been successful in overcoming the above described difficulties. Technological tools can however afford alternative approaches to the subject. Also, most of the research that provides insight in students difficulties with understanding graphical situations is done in clinical environments. We need to develop a better understanding of students learning in classroom settings.

In our research we developed a tool that intends to provide an alternative way to approach the learning of graphs by dynamic events and an opportunity for examining its learning in the classroom. The tool Interactive Virtual Math (IVM), which can be found at https://virtualmath.hva.nl, is designed to support 14-17 years old students at secondary school to understand the graphical representation of relations between variables in dynamic situations. IVM supports this process by addressing the visualization of these relationships. The aims of this paper are to introduce a prototype of the tool, its main features and design and, to discuss its added value for students' learning based on the results of one preliminary study and one exploratory study.

Theoretical framework

Covariational reasoning

An example of a mathematical task that requires understanding of covariational reasoning is Task A from Figure 1. The task is about a dynamic situation involving the height of water in a bowl and the volume and, it was taken from [START_REF] Carlson | The precalculus concept assessment: a tool for assessing students' reasoning abilities and understandings[END_REF], who used it to diagnose students' understanding of graphs of this type of events. To solve task A, students will need to consider 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 interpret relevant features in the shape of the graph [START_REF] Carlson | The precalculus concept assessment: a tool for assessing students' reasoning abilities and understandings[END_REF]. [START_REF] Carlson | Applying covariational reasoning while modeling dynamic events: A framework and a study: A Framework and a Study[END_REF][START_REF] Carlson | The precalculus concept assessment: a tool for assessing students' reasoning abilities and understandings[END_REF], developed a framework that allows to investigate students' covariational reasoning abilities when responding to dynamic function tasks. The framework describes covariational reasoning as entailing five mental actions, which are successively more complex: (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. We used this framework to evaluate the quality of students' graphs and explanations in our study.

Guiding principles and main features of the tool

There are many technological tools available for learning graphs from dynamic events, but very few request students' own productions. They are often simulation-tools, which involve whole figures or part of figures that have to be moved, changed or dragged. When students are asked to construct a graph with these kind of tools, construction actually means using representations that are already given or can be synthesized by putting parts together. In this case there is not a true visualization of students' concept image [START_REF] Vinner | Concept definition, concept image and the notion of function[END_REF], since part of the representation is already given. A distinguishing feature of the IVM is that it builds solely on students' graphical productions.

The tool Interactive Virtual Math allows students to draw, analyze and compare graphs for themselves and improve the graphs if they conclude this improvement is needed. At CERME 10 we presented a second prototype version of the tool in which the students work on an assignment involving a single graphic situation: the dynamic event described in task A (Figure 1). In later versions we expect it to be possible to use more contexts and varied assignments so that all students can practice at their own level. In Table 1 we present a short description of the main features of the tool: Self-construction, Contrast, Help 1 and Help 2, Reward and flow. These features are based on general learning principles that include building on students' previous knowledge, interaction and feedback. We expect that the use of the tool will challenge students to create their own graphs and explanations, to make assumptions, conjectures and to reflect upon these (feature Flow).

The tool was also built according to topic specific learning principles. [START_REF] Thompson | Quantitative reasoning and mathematical modeling[END_REF] states that it is critical for students to first engage in mental activity to visualize a situation and construct relevant quantitative relationships prior to determining formulas or graphs. Therefore, the graphs in the tool must be drawn by the student themselves and the tool elicit students to imagine relationships from scratch, without presenting any (partial) graphical representation that has not been drawn by the student themselves (feature Self-construction).

A second guiding idea behind the tool-design is the focus on visualizing quantities. Results from [START_REF] Ellis | The influence of reasoning with emergent quantities on students' generalizations[END_REF] indicate that instruction encouraging a focus on quantities can support generalizations about relationships, connections between situations, and dynamic phenomena. To help students to focus on the relation between the height of the water and the volume we provide two kinds of help with the tool: the features Help and Help 2. In Help 1 the student visualizes the increasing height of the water in the bowl and he can start and stop the water falling in bowl. In Help 2 students must assume the height of the water in the bowl and represent it in the graph with dots. We expect that the students, while guessing where to put the dot for the height, will notice that the difference in height between consecutive dots (values of the height) decreases in certain situations and increases in others.

Another guiding principle was to provide constructive feedback to the students' final graph and to give them a way to evaluate their production. The students get to see, after submitting their graph, the corresponding bowl-figure to the graph they draw (feature Reward).

Finally, the tool also includes the use of Virtual Reality (VR), which is still limited to Help 1. Here the use of VR (sound, movement, interaction) is expected to improve the experience of the graphic situation.

Table 1: main features of Interactive Virtual Math

Feature Description

Self-construction

The student is given two assignments. The first assignment is task A from Fig. 1 and the second assignment is a variation of the same task with a cylinder instead of a bowl. In both assignments they are requested to draw a graph that describes the relationship between two variables in the corresponding dynamic situation. The student constructs the graph with a finger, a digital pen or a mouse.

Contrast

The student compares her/his own graph and explanation of the two situations, referred to as a and b. The student can then submit the graphs or improve them.

Help 1

The student visualizes the increasing height of the water in the bowl. He listens to the water he moves the platform with the ball and he can start and stop the water falling.

Using a mobile device and a cardboard, Help 1 can be experienced as Virtual Reality

Help 2

The student connects the graphical representation to the context representation. A Cartesian coordinate system in the plane and the bowl appear next to each other. The student must construct a dot graph that represents the height of the water in the Cartesian graph. He does this by dragging and dropping dots into the graph.

Reward

The student gets the corresponding form of the bowl.

Thematic Working Group 16

Methodology Preliminary study

Previous to the development of the first version of the IVM tool, we conducted a preliminary study to explore students' knowledge, skills and difficulties with constructing covariation graphs. The study (February-March 2016) involved N=98 students from 4 classes age 15-17 years old and we used three versions of the same task with different questioning (Figure 1). The students in each of the four classes were divided into three groups and each group was presented with one of the three versions.

Analyses of students' written answers showed that the majority of the students (64%) failed to successfully solve task A (see also Table 2). Nineteen of them presented an increasing but incorrect graph, suggesting that they understand that the water increases or that the height increases with the amount of water but they don't have a consistent concept image of this process. Most of these students (13 out 19) produced one straight line (9 students) or a combination of two/three straight lines (4 students). These findings point that the majority of students that solved the self-construction tasks (tasks A and C) could not construct for themselves an acceptable representation. These results motivated the importance of engaging students in self-construction assignments and the development of the IVM-tool. No answer 2 (6%) 2 (5%) 1 (4%)

Exploratory study about the first version of the tool

The first version of the tool was developed in February -April 2016 by a team composed by one researcher-math educator (first author), a high school teacher (second author) and ICT -designers. We decided to use task A (Fig. 1) that we considered suitable to explore students' understanding of covariation and within a broad age group. To explore its learning potential and usability we investigated through a small qualitative study the learning of four students age 14-15 years old (two boys and two girls) with different school performance for mathematics. Kevin 1 has high grades for mathematics, Lisa and Anton have average grades and Wilma has low grades. We observed and interviewed the students while working with the tool. The aims of the exploratory study were: (i) to understand how the students construct a graphical representation with IVM; (ii) to identify features of the tool that support or constrain students' successful construction; (iii) to get a better understanding about how the guiding principles work and can be used to develop later versions of the tool. The collected data consisted of video records and students' written work and it was collected at two different moments in April 2016. In both situations the students were asked to go first through the whole application on their own. Lisa was the first student to be interviewed; she used the application on a computer. The other three students Kevin, Wilma and Anton were interviewed together at their school. Wilma and Anton use a tablet and Kevin a mobile device. The data was first organized chronologically with relation to each student's attempt to construct the graph and use of the tool. Secondly, a global description of how each student attempted to construct and transform the graph was made and how they used the main features of the tool. We used the covariational framework [START_REF] Carlson | Applying covariational reasoning while modeling dynamic events: A framework and a study: A Framework and a Study[END_REF] to get insight in students' covariational reasoning abilities. A summary of the results are presented in Table 3. These results and the data were shared and discussed with the ICTteam and used to evaluate the tool and to make decisions for the development of a next version.

Results and discussion

As we can see in Table 3, all four students improved their graphs on basis of the tool. Kevin produced in the first trial an incorrect graph with three straight lines and he improved it in second trial after comparing the form of the bowl he got in the Reward with the bowl in the bowl-assignment. Wilma produced in the bowl-assignment, in the first trial two incorrect graphs: a straight line and afterwards a raising curve. She 'improved' the graph after seeing the cylinder-assignment (Contrast). Through consulting Help 1 and Help 2 she constructed in a second trial a final acceptable graph. Anton produced in the bowl-assignment several incorrect graphs. His final graph in the first trial is a curve raising slowly. He consulted Help 1 several times and, based on that, he produced a graph with three straight lines and adapted the length of the line segments. Anton's improvement did not lead to a final acceptable solution and the student remained in doubt whether the pieces of the graph should be curved or not. All students have produced an acceptable graph at first trial (straight line)

Contrast

First, all students draw a straight line at assignment one but improve their drawing after constructing the graph of assignment two. Based on the analyses of students reasoning while constructing and explaining their graphs, we identified a number of aspects through which students could be brought to a better understanding of graphical situations, while working with the tool. One aspect is students engagement in covariational reasoning and their progression through the mental actions [START_REF] Carlson | Applying covariational reasoning while modeling dynamic events: A framework and a study: A Framework and a Study[END_REF]. For instance, Wilma identifies and represents the two quantities changing together (M1). She draws initially a straight line which suggests that she attends only to the direction in which the height changed while imagining increases in the amount of water (M2). After consulting help 1 she changes her straight line into a rising curve and then into a curve-down followed by a curve-up graph and she is able to explain how changes in the amount of water were related to changes in the height of the water at various locations in the bottle (M3).

Another aspect is students' involvement in actions that underpin mathematical reasoning such as the construction and explanation of different representations and, comparing, relating and generalizing these ones. Examples that we observed include students comparing their own graph and bowl filling up with water, which was the case of Wilma when she used Help 1 or Anton switching from Help 1 to his own graph several times; students evaluating the relation between the reward and initial graph.

Visualizing the bowl of the reward made Kevin to think about the relation between the form of the bowl and the form of the graph. He used the reward to improve the smoothness of the graph curve; students contrast the relation between graphical situations of assignment one and two. For instance, Anton switches between one and two and adapt the graph one after seeing assignment two.

As Table 3 shows, different students used different features to improve their graph, which suggests that tool with possibilities to choose to view additional help or not and to be able to switch between the graphical situations, allows for diversity. Furthermore, all students had difficulty with constructing a graph, even with the tool support. This result suggests that self-construction tasks are needed to reveal these difficulties, which can remain unnoticed when using simulation-tools or tools in which the representations are already given.

A final aspect concerns the usability of the tool. Students valued the opportunity of choice and the interactivity of Help 2 (one can drag and decide where to put the point). And, one student (Lisa) who view Help 1 in VR with the cardboard valued this experience as a more enriching one.

There are also some critical issues with regard to the methodology of the study and the tool design. The small amount of students involved in the use of the IVM tool allowed for a fairly detailed study of their interaction with the tool. But, we should carefully interpret our findings since they regard only 4 students. We need to experiment more with the tool in classrooms, in combination with other tasks and forms of interaction and teacher support to better understand its potential and to what extend these findings can be generalized. With regard to the tool design, a number of aspects should be improved in follow up versions. One challenge concerns the self construction-and reward-features.

It is left to the tool to decide what is an acceptable representation and how accurate it can be. We programmed the tool to accept any sketch of concave up followed by a concave down graphs starting at the origin. And, for the graph to be considered accurate, the line must be smoothly drawn. Sometimes the tool rejects answers that are accepted by the researchers and teachers. Another concern is the amount of variables involved in the assignments (height, accumulated volume, time, volume per unit of time, shape of the bottle). It is reasonable that the students should focus on one or two variables but not so many that are changing simultaneously. At the CERME conference we also received useful suggestions to improve the tool. For instance the time-counter in Help 1 can be replaced by a volume-counter and, students could fill the bowl by adding themselves cups of water.

This could help students to focus on the relation between height and volume rather than height and time. Another suggestion was allowing students to change the shape of the bottle as this might afford students' awareness of the phenomenon.

Concluding, this paper reports on the experiences of students learning graphical representations by dynamic events with the aid of a new learning technology (IVM); a topic which many students struggle to understand. We have learned that the prototype-tool has potential to engage students in covariational reasoning and we identified a number of aspects that could bring the students, while working with the tool, to a better understanding of graphical situations. Namely, the tool affords construction and explanation of different representations and, comparison, relation and generalization of these ones. The results also point to the importance of elicit and build upon students selfproductions.

Figure 1 :

 1 Figure 1: tasks used in preliminary study

Table 2 : results of preliminary study

 2

		Task A (self-construction graph) Task B (multiple choice )	Task C (self-construction bowl)
	Acceptable 12 (36%)	25 (66%)	3 (11%)
	Incorrect	19 (58%)	11 (29%)	22 (79%)

Table 3 : students' use of the features of the tool during the exploratory study

 3

	Features	Kevin	Wilma	Anton	Lisa
	Construction	Acceptable final graph	Acceptable final graph	Incorrect final graph	Acceptable final
	(round bowl)	after two trials	after two trials	after two trials	graph after two trials
	Construction
	(cylinder bowl)

Proceedings of CERME10

The real names of the students were modified Thematic Working Group 16 Proceedings of CERME10


text:
Øistein Gjøvik
email: oistein.gjovik@ntnu.no

Iveta Kohanová
email: iveta.kohanova@ntnu.no

Preliminary testing of apps in mathematics, based on international collaborative work

Keywords: Mobile learning, game based apps, mathematics learning, collaborative work

In this paper, we report on a project about developing mobile applications for learning mathematics through game playing. Several different types of applications were developed in a collaboration between universities in Norway and Slovakia, and between teacher education and information science. We give some preliminary results on how two of these applications were received and used by Slovakian pupils.

Introduction

Mobile devices, such as smartphones, tablets and laptops, have become an integral part of our lives. Teachers and pupils use them daily for communication, searching for information or for entertainment. Pupils today, born from 1990 to 2010 and recognized as generation Z, are the most technologically advanced generation, often known as digital natives. They were born into the era of the Internet and Facebook; they always want to stay connected with their friends and to use highspeed digital devices [START_REF] Baker | Digital Information Strategies. From Applications and Content to Libraries and People[END_REF]. Hence, the wide spread of mobile devices causes a natural social pressure and challenge for educators to include these devices into education, to support learning. Computers, laptops, and netbooks have all been added to classroom settings with the hopes of revolutionizing education, promising vast improvements to pupil outcomes. These technologies, largely, have left education unchanged and in a continual state of need for improvement [START_REF] Mcquiggan | Mobile learning: A Hand-book for Developers, Educators, and Learners[END_REF]. Mobile learning offers a novel approach to reach current pupils. By the term mobile learning we follow McQuiggan, Kosturko, McQuiggan and Sabourin (2015, p. 31).

It is anywhere, anytime learning enabled by instant, on-demand access to a personalized world filled with the tools and resources we prefer for creating our own knowledge, satisfying our curiosities, collaborating with others, and cultivating experiences otherwise unattainable. Mobile learning implies adapting and building upon the latest advances in mobile technology, redefining the responsibilities of teachers and students, and blurring the lines between formal and informal learning.

Mobile learning offers flexibility in when learning takes place, personalization of content, and gives pupils experience with contemporary technology and relevant skills for the future. So unsurprisingly, mobile learning has been considered as the future of learning or as an integral part of any other form of educational process in the future [START_REF] Trifonova | Mobile Learning -Review of the Literature[END_REF].

In June 2016, gaming apps were the most popular apps based on availability, as about 23 % of all apps available in the Apple App Store fit in this category. The second most popular category was Thematic Working Group 16 Proceedings of CERME10 Business (10.22 %), closely followed by the Education category (9.21 %) ('Most popular ', 2016). [START_REF] Shuler | iLearn II: An Analysis of the Education Category on Apple's App Store[END_REF] has analyzed the Education category from Apple App store. In 2011, more than three quarters (77 %) of the top selling apps targeted preschool or elementary aged children. Early learning was by far the most popular subject/skill-set (47 %), followed by mathematics (13 %). [START_REF] Drigas | A Review of Mobile Learning Applications for Mathematics[END_REF] have analyzed the most representative studies of recent years (2002 -2013), involving online and mobile applications and tools for mathematics as well as their effect in the educational process. The results of the studies revealed that online and mobile learning applications motivated pupils, making mathematics instruction more enjoyable and interactive than ordinary teaching practices. The analyzed applications were targeted towards one specific area of mathematics, like graphs and functions, arithmetic, algebra, geometry, problem solving or mathematical programming and they were available only in English or Spanish. In light of this, we see it as an important contribution to ongoing research into mathematics education to engage in projects that examine the process of developing applications for mobile technologies as well as studying the effects they could have on learning. Also, providing tools readily available for school teachers was an important factor for running the Apps in Math project, as detailed in the next section.

Design and implementation of the Apps in Math applications

The main goal of the Apps in Math project (AiM) was to develop 25 applications in 15 months for supporting teaching and learning mathematics in lower and upper secondary schools in Slovakia and Norway. In Norway, pupils have relatively good access to technology, compared with European countries. Almost 90 % of pupils use Internet in schools but the most common use is probably the computer and not mobile platforms. After school hours, as much as 94 % of all children aged 9-16 have access to a mobile phone, and 83 % have a smartphone. [START_REF] Medietilsynet | år) bruk av opplevelser og medier[END_REF] Several schools have a policy of buying one laptop for each child in school. Most publishing houses have their own apps and games connecting to their textbooks, and there are usually many choices teachers can do regarding software for their pupils. Much is not translated into Norwegian, but this is generally not seen as a big difficulty.

In Slovakia not all pupils have their own smartphone or tablet; the further east one goes, the less pupils have their own mobile device [START_REF] Michálková | Využitie aplikácií pre mobilné zariadenia na hodinách matematiky na strednej škole[END_REF]. In the primary and secondary schoolsthe typicality is to have three computer rooms per school, in which Informatics is mainly taught, so there is rarely room for mathematics lessons in these specialized classrooms. Pupils usually do not have their own PC. During 2013-2015, thanks to national project supported by EU funding, 22 000 tablets were given to Slovak schools, which usually meant set of 30 tablets per school. Pupils in one school are sharing those tablets; teachers bring them for lesson, at the end of the lesson pupils have to return them, because they will be used in other classrooms. In Google Play or App Store there are very few mathematical apps in Slovak language that are intended to be used in mathematics classes at lower or upper secondary schools. So there is a need for applications, which teachers could use in math classes and for different levels of schooling.

The applications (modules) developed within Apps in Math project focus on various mathematical topics that are part of Slovak or Norwegian curriculum for pupils aged 9-19. The development of modules went in coherence with Design based research [START_REF] Wang | Design-based research and technology: Enhanced learning environments[END_REF] and its iterative cycles. The mathematics teacher educators from Trondheim and Bratislava have cooperated with academics and bachelor students of applied informatics at the Comenius University in Bratislava.

Slovak bachelor students in Applied Informatics have programmed the modules based on the specifications from mathematics teacher educators and master and PhD students, as part of their bachelor thesis in informatics. The modules were tested extensively within the local participating groups in Slovakia and in Norway, as well as with pupils in Slovak and Norwegian schools. Reflective analysis of problems and obstacles was done and changes were implemented after each testing. All modules are part of one framework application called Apps in Math and they are divided into five main categories: Numbers, Functions, Geometry, Chance and Logic. Apps in Math is available for Android and iOS1 platform and in Slovak, Norwegian and English language. [START_REF] Ebner | Mobile Applications for Math Education -How should they be done? Mobile learning and mathematics: Foundations, Design, and Case Studies[END_REF] has divided applications into four categories: stand-alone learning apps, game-based learning apps, collaborative apps and learning analytics apps. Apps in Math has the characteristics of being gamebased learning application. [START_REF] Diah | Discover Mathematics on Mobile Devices using Gaming Approach[END_REF] have introduced the framework for mobile educational games consisting of four important segments: Learning Theories, Mobile Learning Approach, Games Development Approach and Learning and Education Medium. Most of the modules in Apps in Math apply the constructivism as the learning theory and for the mobile learning approach the games use activity-based themes for informal and lifelong learning.

Case studies

This section describes two case studies (Study 1 and Study 2) that were conducted to evaluate the effectiveness of mobile learning with Apps in Math application in real-world settings, with lower and upper secondary school pupils. We have chosen the SAMR-model for a quick categorization of the modules, where digital technologies can be placed on a scale from just replacing already existing practicing to facilitate types of tasks that could not have been done without the digital tools [START_REF] Hudson | Best Practices for Evaluating Digital Curricula. Dreambox Learning Inc[END_REF]. Limited resources and limited time made it necessary to choose for evaluation those modules that were closest to being finished. The module Lucky Hockey is based on the classic learning game Green Globs [START_REF] Dugdale | Green Globs: A Microcomputer Application for Graphing of Equations[END_REF], and several versions of this game has been implemented over the years. The pupils who play the game are going to shoot a hockey puck across an ice hockey arena in order to collect as many coins as possible. The coins are shattered around the play field, sometimes in a random manner, sometimes to provoke a particular shot. The pupil shoots by entering a function expression, using the touch screen controls to alter the parameters of the function (Figure 2). By playing this game pupils should understand what impact the parameters of the function have on a graph. Using the SAMR-model we can say that Lucky Hockey acts as a direct tool substitute, but that the functional improvement allows for a more dynamic and dual view of the representations of a linear graph and the corresponding expression. Hence we can say this app is an augmentation of traditional instruction. The module House of cards focus on arithmetic and geometric sequences in two separated submodules called Arithmetricks and Geometricks. By playing this game pupils should discover relations between the terms of the sequence and be able to write down basic formulae related to these relations. The number sequences are displayed on playing cards. Both submodules have a Learn mode, in which basic principles of the sequence are explained. The pupil has to determine the number, which is added/multiplied to/with each of the following sequence terms (Figure 3). The pupil has to answer five tasks correct within the time limit. After 3 incorrect attempts the correct answer is shown. In the next three levels the pupil should select the card, which belongs to the empty red spot in the given sequence within time limit (Figure 4). In the first level first 3 terms are given and the pupil should select the missing card for 4 th and 5 th term. Again, using the SAMR-model on Arithmetricks and Geometricks, we note that the effectiveness and readiness of the app makes work with sequences easier than in traditional teaching, or teaching done with real cards. Hence this app too provides an augmentation over traditional instruction. The target group for the Lucky Hockey game study was Slovak pupils between the age 14 and 15 (grade 9), who had had no experience in linear functions yet. The goal of Study 1 was to determine which aspects of the linear function concept students seem to approach more effectively through the use of the Lucky Hockey game. Time limited gaming (25 minutes) was meant as an adidactical situation (Brousseau, 1997). The adidacticity was promoted by giving the students full responsibility for the technology-supported exploration of mathematical tasks by retroacting only with the milieu and not the teacher (Sollerval, de la Iglesia, 2015). All together 54 pupils from 2 different schools in Slovakia participated in Study 1 in November and December 2015.

The target group for the House of Cards game study was Slovak pupils between the age 16 and 17 (grade 11), who had not learned about sequences yet and had no previous knowledge about arithmetic and geometric sequences. The goal of Study 2 was to determine which aspects of the arithmetic/geometric sequence concept students seem to approach more effectively through the use of the House of Cards game. All together 49 pupils from schools in Bratislava participated in Study 2 in March 2016. They first played the Arithmetricks game (starting with Learn mode and consequently going through all three levels) for 25 minutes. The next lesson (in the same day) they played the Geometricks game with the same conditions. During both Studies 1 and 2 all pupils used an iPad. No pre-test was conducted since pupils did not have any knowledge on these topics. The post-tests were used to determine the level of acquired knowledge. All pupils of Study 1 and 2 completed the post-tests as part of the evaluation, right after playing the game. The phase of institutionalization took place a few months after Studies 1 and 2, due to prescribed curriculum.

A preliminary study was conducted in September 2015 with 77 pupils of different age (7 -16), in order to introduce them the early versions of five different games, including the Lucky Hockey game. At this stage, the game was more or less fully working, apart from minor graphical issues. Part of the group (about 20 pupils) tested the Lucky Hockey game. During the testing pupils thought (while playing the Learn mode -Figure 1), that the expression is always y = 0x + b, because they were able to hit the goalie only by changing parameter b. This was an obstacle in Level 1, so we had to refine Learn mode and control the possible movements of a shooting player. Most of the pupils liked the game and did grasp the notion of linear function. In the preliminary study we also asked all the pupils about their interest in using smartphones or tablets to learn mathematics in school. Figure 5 shows their answers. 92.3 % of pupils, who answered positively on this question, also said that they would like to play tested games at home. Out of them 46.7 % in the situation when they are bored, 28.3 % for practicing mathematics and 25 % when doing homework.

Results

Figure 6 shows that pupils performed quite well in the post-test of Study 1. The average score was 5.11 and median score of 5. Pupils could obtain a maximum of 7 points, which were obtained by 13 pupils (24 %). Half of the pupils (50 %) scored 4-6.5 points, but there was also one pupil whose score was 0. The results indicate that most pupils learned the slope and intersection-aspects of the function concept on an acceptable level. The lowest score performance they had occurred in the last task, in which they were asked to explain what impact the parameters a and b in the expression y = ax + b had on the corresponding linear graph. Only 46 % of pupils explained it correctly. Nevertheless, they performed better in tasks in which they were supposed to draw a line in correspondence with a given equation (76 %) or select the correct line/equation out of four possibilities that is corresponding to a given equation/line (86 %). Figure 7 shows the results of pupils in the post-test of Study 2. It is clear that these pupils performed better in the Geometricks post-test. Here, 50 % of the pupils obtained 8-10 points, while 10 was the maximum. 18 pupils (36.7 %) obtained maximum score, and one pupil obtained the minimum score of 2. The second lowest score was 5, also obtained only by one pupil. The lowest performance was in the last task in which they had to write down the formula for how to find the 10 th term, if they knew the quotient (𝑞 = 1 3

) and the 1 st term was given as 𝑎 1 . Only 51 % of the pupils wrote the correct formula and explained their answer correctly. The most frequent error was made by 7 pupils (8.2 %), claiming that 𝑎 10 = 𝑎 1 . 1 3 10 . In all the other tasks pupils were able to determine the unknown term, if they knew specified values of the 1 st term and the quotient, or specified values of two consecutive or two nonconsecutive terms, with successfulness of 89 % -100 %. The scores in the Arithmetricks post-test were slightly lower, with an average 6.82 and 6 as a median score. 8 pupils (16.3 %) obtained the maximum score and two pupils obtained the minimum score of 3. Distribution of scores within the box chart shows that approximately one quarter of the pupils obtained the same score, 6 points.

Discussion and conclusive remarks

We note from the results that most of the pupils did learn the important principles of linear functions or sequences at an acceptable level. However, only about 50 % of the pupils were able to answer the last questions correctly. This might have improved if the pupils were to play the games additional times. This hypothesis also arises from differences between how the pupils scored in Arithmetricks and Geometricks post-tests. The pupils did play the Geometricks game after playing the Arithmetricks game and since the principle is not very different, it could cause that they performed better in the Geometricks post-test. As mentioned above, the phase of institutionalization took place 6 months after the Studies 1-2. While pupils who participated in Study 1 did not remember much about the linear function, it was different with pupils of Study 2. Pupils recalled the main principles of arithmetic/geometric sequence and told the teacher that it was not needed to explain it again: "It's like in that game we have played." From observation of the teacher we also note that traditional teaching of sequences went this time easier, probably also due to the mobile learning. The low performance on the last task of the Lucky Hockey post-test could be caused by the nature of the task. In all the previous questions we used numerals instead of parameters a and b, whereas on the last question some generalization and explanation were expected. While some pupils may have misunderstood the meaning of the parameters, some didn't give any reply at all or they only explained the role of one parameter. If the phase of institutionalization in form of, say, a discussion among pupils and a teacher took place right after the gaming activity, pupils' understanding of the parameters' role might have been better. The interest of pupils to learn mathematics with a mobile device was visible during testing both in Slovakia and in Norway. According to the results of the questionnaire it seems that most of the Slovak pupils would like to include mobile learning in their schooling. Testing of the other various applications from the project, not mentioned in this paper, also confirms that Slovak pupils and teachers consider mobile learning as a motivational way of learning and teaching mathematics [START_REF] Michálková | Využitie aplikácií pre mobilné zariadenia na hodinách matematiky na strednej škole[END_REF][START_REF] Kapitulčinová | Využitie aplikácií pre mobilné zariadenia na hodinách matematiky na základnej škole[END_REF]. Mobile phone games in classroom is a novel idea and it might still cause the engagement of being a contemporary, "fresh" way of learning mathematics, which could be the reason of pupils' and teachers' enthusiasm.

The results of Study 1 and 2 suggest that mobile learning can be both motivational for pupils when learning mathematics, and helpful when acquiring new knowledge effectively. Gamification of education has also reached mathematics instruction but resources and research are just beginning to surface. Ideas from the project are being further developed at both participating institutions. Current issues can include utilizing the small touch screen sensibly and also collecting data from how and when pupils use the applications. The mobile phone is a tool we can expect to see more in mathematics education as learning becomes further individualized and online. One lesson learned from this project is the difficulty of communicating mathematical ideas from the idea stage to the actual implementation. This became quite apparent when collaborating with different countries, different levels of study and different study branches. Another lesson from the project is that it turned out to be much easier to develop ideas with a narrow mathematical theme, than to make applications that facilitates exploration and discovery.

Figure 1 :Figure 2 :

 12 Figure 1: Learn mode of Lucky Hockey

Figure

  Figure 3: Learn mode of Arithmetricks Figure 4: Third level of Arithmetricks

Figure 5 :

 5 Figure 5: Interest of pupils to learn mathematics with mobile devices

Figure 6 :

 6 Figure 6: Box and Whisker Chart of Lucky Hockey Post-test Results

Figure 7 :

 7 Figure 7: Box and Whisker Charts of Arithmetricks and Geometricks Post-tests Results

Proceedings of CERME10

http://www.project-aim.eu/eng/download Thematic Working Group 16 Proceedings of CERME10


text:
Krista Francis
email: kfrancis@ucalgary.ca

Brent Davis
email: brent.davis@ucalgary.ca

Number, arithmetic, multiplicative thinking and coding

Keywords: Coding, robotics, arithmetic, number concepts, elementary education

With frequent predictions of upcoming technological and economic difficulties triggered by an impending shortage of information and communications technologies (ICT) professionals, the calls are growing stronger to include coding as a core element of school curriculum. These calls are bolstered by the suggestion that coding supports the development of thinking skillswhich echoes a longstanding argument for teaching mathematics. Motivated by the parallel, we attempted to investigate some of the common ground between learning to code and the development of core mathematical concepts. We photographed and video recorded children, aged 9-10, as they learned to build and program Lego Mindstorms TM EV3 robots over four days. Our findings suggest that programming supports children's understandings of decimal numbers and their transitions from additive to multiplicative thinking.

Introduction

In recent years there has been a growing recognition that information and communications technologies (ICT) are a major contributor to innovation and economic growth. For instance, the Organization for Economic Cooperation and Development (OECD, 2016) considers computer programming a necessity for a highly skilled labour force. Shortages are already felt across the world and demand for highly skilled ICT professionals is expected to rise. In our home country of Canada, for instance, there are predicted shortages of more than 150,000 skilled ICT workers in the next few years. This shortage is impacting IT innovations and revenues (see [START_REF] Arellano | Why Canada has an 182,000 IT talent shortage while lots of tech professionals are out of work[END_REF][START_REF] Clendenin | Canada's IT labour shortage: Challenges and opportunities[END_REF].

Canada is hardly unique on this count, as evidenced by major pushes around the world to include coding as a core part of school curriculum. In response, some educators and educational systems are shifting from teaching "how to use" software programs toward "how to code." Estonia and England, for example, have implemented a national curriculum that makes computer programming mandatory for all school-aged students across all grades, and other nations appear to be moving in this direction. For instance, it is currently a topic of political debate in Australia, where the opposition party is calling to have computer programming taught in every primary and secondary school in the country [START_REF] Roumeliotis | Back to school: Canada lagging in push to teach kids computer coding[END_REF][START_REF] Sterling | An education for the 21st century means teaching coding in schools. The Conversation[END_REF].

In North America, national-level discussions and calls have yet to gather the same sort of momentum, but more and more initiatives are emerging at the local level. For example, the Chicago school district is adopting computer science as a core subject in all public high schoolsprompted in large part by support from Google and Microsoft and through initiatives such as Code.org and Hour of Code, which are dedicated to expanding access to computer science for all U.S. students. Despite the absence of a national strategy in the U.S., messages on the importance of learning to code are frequent, with some emanating even from the President's office. In fact, coding skills have been associated not only with empowering individuals and meeting employment needs, but with many aspects of the country's future and security [START_REF] Pearce | Why You Should Learn To Code (And How To Do It![END_REF].

Trends toward including coding in school curriculum were preceded by a broadly effective worldwide push to get computers in schools. In 2011, most students (71%) in OECD countries reported having access to computers and the Internet at school. However, most students reported using the computers at school for email, browsing the Internet, word processing or doing individual homework. For the most part, such activities require low-level cognitive thinking and do not challenge students to develop more than basic user skills. Learning how to program a computer, it is typically argued, involves higher-level cognitive processes and provides opportunities for developing higher-level ICT skills.

These sorts of arguments for teaching computer coding parallel long-standing rationales for teaching mathematics. Similarly, many of the structures and strategies within coding bear strong resemblances to elements of mathematical concepts [START_REF] Papert | Mindstorms: Children, computers, and powerful ideas[END_REF]. We discuss a few of these resemblances in this paper, focusing on arithmetic.

Conceptual metaphors are one of the ways we understand mathematics [START_REF] Núñez | Mathematical Idea Analysis: What Embodied Cognitive Science Can Say about the Human Nature of Mathematics[END_REF]. With regard to the concept of number, [START_REF] Lakoff | Where mathematics comes from: how the embodied mind brings mathematics into being[END_REF] describe "four fundamental metaphors of arithmetic": arithmetic as object collection, arithmetic as object construction, the measuring stick metaphor, and arithmetic as object along a path. The metaphor of arithmetic as an object collection is based on a one-one correspondence of numbers to physical objects. With this metaphor a greater size corresponds to a bigger number. For instance, 5 is greater than 2 because it forms a bigger collection. The metaphor of arithmetic as object construction is based on fitting objects/parts and arithmetic operations. For instance, 5 is greater than 2 because an object comprising 5 units is larger than one comprising two. The measuring stick metaphor maps numbers onto distances, whereby 5 is greater than 2 because it is longer. The metaphor of arithmetic as an object along the path is based on arithmetic as motion, by which 5 is greater than 2 because it entails moving further from a common starting point (i.e., zero). Programing robots provides opportunities for illustrating and experiencing these arithmetic metaphors.

Context

In this interpretive study we asked what mathematics children learn by building and programming Lego Mindstorms TM EV3. Interpretive research is about what meaning individuals construct in their lived experiences [START_REF] Bhattacharya | Interpretive research[END_REF]. We co-designed learning tasks with a graduate engineering student and co-taught the tasks with the classroom teachers over a course of four sequential days in three-hour daily sessions. The study's participants were 22 children, Grades 4-5 (aged 9-10), at Pakan School at Whitefish Lake 128 First Nation in rural Northern Alberta. Once the children knew the basic coding blocks for moving the robot, they were given [START_REF] Papert | Mindstorms: Children, computers, and powerful ideas[END_REF] task of programming a robot to follow a trace out of a triangle, square, pentagon or hexagon. On the third day, they were given the final challenge of building a robot that could find and douse a fire in any of four rooms in a building. Data included video-recordings, GoPro digital images, field notes, and artifacts including saved computer programs.

We video-recorded the four sessions to obtain rich contextual detail of children's mathematical interactions when programming the robots. Using interpretive video analysis [START_REF] Knoblauch | Videography. Focused ethnography and video analysis[END_REF] we selected videos and GoPro digital images that exemplified instances of children's mathematical thinking. Video data enables repeated viewing, slow motion, fast motion and frame-by-frame analysis. The selected videos formed the basis for emergent understandings of the children's experiences. The analysis developed through an iterative process of rereading the literature, reviewing the video and GoPro data, and rewriting. As is evident in our analysis, below, video data was vital. In particular, it permitted us to slow down the process and identify the integrated/nested processes of learning that occurred. The three instances that we use to focus our discussion were:

(1) a trio of girls learning to program their robot for the final challenge to move a certain distance into the hallway to illustrate a developing understanding of number, (2) a boy tapping the vertices and sides of a triangle to count the number of programming steps necessary for the robot to move around the triangle as an example of additive thinking, and (3) a boy learning how the number of sides and angles of a polygon connects to the number of repeats in a loop, which illustrates a developing shift from additive to multiplicative thinking.

Findings

In the numberline video (see https://vimeo.com/144996708video), Krista was helping the pink team program their robot to move into the building. This action required manipulating one block of EV3 code to move the wheels a specified number of rotations. The team members started out with a guess of 0.4 rotations to move the robot into the first corridor of the building. After testing how far the robot moved and observing that the robot needed to move a considerably greater distance, Krista prompted the girls by asking what they should try next. Celina suggested they try 0.5. The small incremental change was still not enough, so Krista suggested they try 2. Two rotations moved the robot too far.

Krista: What is between 0.5 and 2? Celina: 5.

Suspecting that Celina's response indicated that she and her teammates were unable to summon an appropriate interpretation of decimal numbers, Krista drew a simple number line on the whiteboard. Krista: What is between 0.5 and 2? Celina: Oh! 1.8.

The number Celina chose was close to the number of rotations actually required, which indicated she understood the meaning of 1.8. In the exchange above, we take Celina's immediate and satisfactory response to the repeated question as evidence that Krista was justified in her suspicion that the learners were lacking an appropriate interpretation for understanding decimal numbersor, at least, were unable to extend whatever interpretations that had available to a situation in which distance was measured in wheel rotations. Coding the robot to move compelled the learners to elaborate their understandings. Invoking the number line appeared to provide an appropriate metaphor for helping Celina understand.

In the following sequence of images and descriptions, we summarize how the task of coding the robot to move into a room calls for all four of [START_REF] Lakoff | Where mathematics comes from: how the embodied mind brings mathematics into being[END_REF] representations of arithmetic. To begin, the metaphor of arithmetic as an object collection is used in most counting situations, whenever the forms being counting are perceived as discrete objects. It is by far the most common interpretation of number through the task of assembling a robot, by simple virtue of the fact that the robots begin as large collections of separate items. Less obviously, it is also called for in coding moments as programmers translate complicated actions into discrete steps or instructions. And more obscurely too, such conceptual moves as the discretizing of wheel turns, so that they can be counted and thus used as a tool in programming, might be argued to rely on this metaphor. Figure 1 (left) presents an instance of this metaphor, showing that 2 turns is less (i.e., forms a smaller set than) 5 turns. shows how the metaphor of arithmetic as an object construction might be encountered when programming a robot to move. Celina wanted a larger wheel rotation than 0.4, so she added an incremental amount of 0.1 wheel rotations to make 0.5 wheel rotations. Contrasted to the previous metaphor, in this instance, wheel turns are not perceived as discrete objects, but as parseable continuities. Those parsed elements can then be assembled into an appropriate object to move the robot a precise distance.

The measuring stick metaphor also featured prominently in the children's programming, and was particularly prominent in the frequent need to interpret wheel turns in terms of actual distances (e.g., when the phrase "1 wheel turn" was deployed not as a description of movement but was a reference to a distance of roughly 12 cm). Figure 2 (left) in reference to the instance in which the room of the hall was shorter than approximately 1.8 wheel turns. In this instance, programming the code block requires understanding measurement. In this case, starting place becomes a critical element is that, for example, occurs when the robot enters the room and recurs in the opposite direction when the robot leaves.

To re-emphasize, we observed each of Lakoff and Núñez' four metaphors of arithmetic to be present in programming the robot to move a required distance in the room. The ability to identify to the particular metaphor(s) that a situation is calling for is a critically important teaching competence, as Krista demonstrated in the interaction with Celina. Re-interpreting that brief episode, Krista recognized that Celina was not interpreting number as a distance (i.e., she was not using a measuring stick metaphor), and thus reminded her of that metaphor by offering the image of a number line. No explanation other than an image of number that fitted the application at hand was required.

Arithmetic Topic 2 -Moving from "additive" thinking to "multiplicative" thinking.

The need for appropriate metaphors and images of number isn't sufficient for making sense of that entire episode, however, closer analysis reveals a further issue with the children's arithmetic, namely the tendency to default to additive actions rather when multiplicative actions would have been more suitable. That episode began with the group's realization that an entry of "0.4" moved the robot only a small portion of a desired distance. Asked what else they might try, they increased the distance only incrementally by 0.1 (to 0.5) rather than the necessary factor of (roughly) three.

This same tendency to default to additive actions when multiplicative action would have been more productive was witnessed many times across many groups over the four-day project. The additive thinking video (see https://vimeo.com/144820583) provides a window into any instance of the same phenomenon. In this case, Gene, who was on the floor in orange, is figuring out how many blocks of code were needed for the program. As he counted "one, two, three, four, five, six," he tapped each vertex and side of the yellow triangle, finally announcing that six steps are needed. Gene's step-by-step of the same two steps (straight, turn, straight, turn, straight, turn sequence) is an example of additive thinkingthat is, of construing the situation in terms of a sequence of increments rather than a repetition.

Phrased in terms of coding, Gene opted to repeat the same line of instructions six times rather than employing a loop that ran six times. This happened in spite the fact that he and his group mates had learned how to use loops the day before when they programmed their robot to dance.

In fact, only one of the 8 groups in the class used a loop for the polygon tasksuggesting that the move from additive/increment-based thinking to multiplicative/loop-based thinking is more conceptually demanding than is often assumed. The additive to multiplicative thinking video (see https://vimeo.com/144826969) further illustrates this point, as the classroom teacher along with Krista attempted to help Liam program with loops. Liam, on the left, identified that a pentagon has three sides. When asked to count the sides, he walked around the pentagon counting aloud and announced "5 times." Krista explains that 5 times is the number of times to repeat the two block codes (go straight and turn) in the loop. In response, Liam exclaimed excitedly, "Yes!"

In the same clip there are two boys who were fine-tuning their robot's program to follow a triangle. Their robot never stopped, which indicates that they are using an infinite loopsuggesting that they are making use of a concept of "repeating," but likely not a concept of multiplication. After three attempts at tracing out a triangle, they still hadn't crafted a program that would stop their robots. [START_REF] Davis | The math teachers know: Profound understanding of emergent mathematics[END_REF] have identified a number of common instantiations for multiplication that are encountered in elementary school classrooms, including grouping, hopping, repeated sums, stretching and compressing, array-and area-making, and making combinations. Looping, it seems, is another, distinct instantiation of multiplication that is particularly powerful in the activity of programmingin a manner, we suspect, that might be used reflexively to support mathematics learning. Figure 3 below, illustrates two programs for following a triangle. Additive thinking is found with the sequential accumulation of six programming blocks: move forward, turn, move forward, turn, move forward turn. Multiplicative thinking requires recognizing that the triangle can be traced by repeating the move forward and turn blocks three times in a loop. In the exchanges above, Liam appeared to be developing fluency with multiplicative thinking. Across the participants there was a pervasive tendency to program robots to trace out polygons as a sequence of same-steps rather than as a repetition of a set of steps (i.e., as enabled with a loop). This tendency was not easily interrupted through instruction, which provides evidence of the complexity of thinking multiplicatively. Even at the end of the four days, during the final challenge, only two of the teams had managed to appreciate the power of loops sufficiently to incorporate them into their programs. Not surprisingly, theirs were also the robots that performed the best. In one of these cases, the code for the winning robot (see https://vimeo.com/145404678) involved a loop determining if a fire is present, announcing "Yes" or "No" as appropriate, and activating an arm motion to dump retardant if "Yes."

Part of the reason that we dwell on this point is that the operation of multiplication is, arguably, the most important concept in grade-school mathematics. Multiplicative thinking is the cornerstone of proportional thinking, which is foundational to advanced mathematics for reasons that include the access it affords to an extended range of numbers (for example, larger whole numbers, decimals, common fractions, ratio and percent), its role in recognizing and solving a range of problems involving direct and indirect proportions, and the power it offers with its prominent place in schoolbased concepts and processes (Education and Training, 2013). In brief, multiplicative thinking is a key in the transition from early ideas to later ideas (see, e.g., ACME, 2011, p. 20).

Closing remarks

Our preliminary findings suggest programming robots can support learning mathematics. In the episodes reported, the tasks of programming robots required more than parsing complicated actions into singular direction; they entailed flexible engagement, [START_REF] Lakoff | Where mathematics comes from: how the embodied mind brings mathematics into being[END_REF] conceptual metaphors and mathematical models.

Computer programming aligns closely with concepts and structures in mathematics and we suspect that it might provide other powerful instantiations for mathematical concepts that have not yet been noticed. That suggestion is perhaps not surprising, given the mathematical roots of computer programming. However, to our reading, it is not an aspect of programming that has garnered much consideration in either mathematics education or the technology education literature. Considering that mathematics literacy and competency with coding are of growing relevance, engagement with emergent technologies can complement and co-amplify mathematics learning, and contribute to evolving understandings of what "basic" mathematics might be for our era.

With regard to important complementarities between learning mathematics and learning to code, the Lego Mindstorms TM EV3 robots and the associated programming language provide a powerful instance of multiple solutions. They afford tremendous flexibility for accomplishing a range of tasks, from the trivial to the complex. None of the coding tasks set for the children in our study had pregiven or optimal "solutions." Despite thator perhaps because of thatthe children were able to engage in manners that they could recognize as successful, even when "complete" solutions were not reached. With incremental tasks and iterative refinements, children were able to learn more sophisticated and efficient methods for programming the robot. It is not difficult to imagine a mathematics class with similar standards of success.

That said, it is not a coincidence that the winning robot had the most efficient and sophisticated program of the group. Some answers are better than others, and those answers appear to reflect powerful mathematical thinking. Our future longitudinal research will investigate how children's understandings of mathematical concepts and programming robotics develop over several years.

We believe that the results of this study underscore the importance of developing and implementing a computer programming curriculum in schools. Coding is an emergent literacy that can amplify other critical literacies, while affording access to a diverse range of cultural capitals. The reasons to teach coding go beyond the technical and economic; for us, they are fundamentally ethical.

Figure 1 :

 1 Figure 1: Arithmetic as object collection. Number of wheel rotations | Arithmetic as Object constructioncombining portions of wheel rotations into single objects Figure 1 (right)shows how the metaphor of arithmetic as an object construction might be encountered when programming a robot to move. Celina wanted a larger wheel rotation than 0.4, so she added an incremental amount of 0.1 wheel rotations to make 0.5 wheel rotations. Contrasted to the previous metaphor, in this instance, wheel turns are not perceived as discrete objects, but as parseable continuities. Those parsed elements can then be assembled into an appropriate object to move the robot a precise distance.

Figure 2 :Figure 2

 22 Figure 2: Measuring Stick: The length of hall | Arithmetic as an object along the path. The robot travels further with 2 than 0.5 Figure 2 (right) shows how programing the robot to move draws upon the metaphor of arithmetic as on object along the path.In this case, starting place becomes a critical element is that, for example, occurs when the robot enters the room and recurs in the opposite direction when the robot leaves.

Figure 3 :

 3 Figure 3: An additive and a multiplicative program for moving a robot in a triangle

Proceedings of CERME10

Acknowledgments

We thank participating teachers and students. We are also grateful to Michael Poscente for designing and coaching the Lego TM robotics task. This work was funded by the Imperial Oil Science Technology Engineering and Mathematics Education Initiative.