This paper presents a case study of following visual attention during collaborative geometry problem solving. We first analyse the emergence and spread of the incorrect idea of curved lines as the optimal shortest solution. Then, we examine the different visual representations made for solution ideas and how these are observed by the collaborating peers.Peer reviewe

Hannula, Markku S.

Toivanen, Miika

English

Helsingin yliopiston digitaalinen arkisto

Hannula & ToivanenHannula & Toivanenmaking and observing visual representations during problem solving: An eye tracking studyMarkku S. Hannula1, 2 & Miika Toivanen1, 31University of Helsinki, Finland; 2Volda University College, Norway; and 3SeeTrue Technologies, FinlandThis paper presents a case study of following visual attention during collaborative geometry problem solving. We first analyse the emergence and spread of the incorrect idea of curved lines as the optimal shortest solution. Then, we examine the different visual representations made for solution ideas and how these are observed by the collaborating peers.IntroductionThis paper will present a 20 minute episode of collaborative problem solving in a Finnish grade nine mathematics classroom. The episode is rich in students using different representations for the ideas they are creating, making this an interesting case study on multimodality. We have been using mobile eye-tracking devices, which provides us an insight not only for what representations students are using when they work with ideas on different media, but also to what visual presentations the students are attending to during collaboration.Eye-tracking is a method to study student cognition as it occurs. Hartmann and Fischer (2016) compare it to mind-reading: the target of a fixation (maintaining of the visual gaze on a single location) tells what we think about, and the fixation duration corresponds to processing time. Glöckner and Herbold (2011) summarize research evidence to suggest that gazes related to more automatic processes would have shorter fixations (below 250 ms) and more elaborate information processing generally requires long fixations of more than 500 milliseconds. However, there is also much evidence to the effect that sometimes the gaze-mind connection breaks and the fixation and thoughts are not aligned (e.g.  Schindler & Lilienthal, 2017).Our research will focus on examining student eye movements in the context of collaborative problem solving in geometry. This study will address three under-examined research areas. First, while eye-tracking research is strong in the context of language processing (Rayner, 1998), the method has been far less used in the area of mathematics (Hartmann & Fischer, 2016). Second, eye-tracking research has mainly been conducted in laboratory situations and studies conducted in real classrooms have only recently started to appear (e.g. McIntyre, Mainhard, & Klassen, 2017). And third, research on multiple persons interacting (e.g. Rogers, Speelman, Guidetti, & Longmuir, 2018) is so far extremely limited. Although methodologically challenging, moving into these poorly explored areas is important for mathematics education.Information in a learning context is often visual. Gestures and diagrams are part of multimodal communication (Radford, 2008; Arzarello, Paola, Robutti & Sabena 2009). The student cannot look at all possible sources of visual information at the same time, but must choose what to attend to. We need to study visual attention in the context of mathematical behaviour in ecologically valid ways. In this paper we will present results from the MathTrack project that uses mobile gaze-trackers to study students’ collaborative problem solving.MethodsParticipantsThe data was collected during a grade ninth mathematics lessons in Finnish lower secondary school. The teacher and a collaborative group of four target students volunteered for wearing gaze-tracking glasses throughout the lesson. ApparatusWe recorded the actions and conversations of the problem-solving session using audio recording and three stationary video cameras in the classroom. The gaze-tracking device consisted of two eye cameras, a scene camera, and simple electronics attached to 3D-printed frames (see, Toivanen, Lukander, & Puolamäki, 2017). The accuracy of the device was approximately 1.5 degrees of visual angle. We used software that computed the gaze target coordinates on the scene video. The camera frame rate depends on lightning conditions, and maximum rate in optimal conditions is 30 frames/second. Data was recorded on laptop computers that were carried in backpacks allowing subjects to move freely in the classroom.Procedure for data collection and analysisThe research group visited two mathematics lessons. The purpose of the first lessons was to help students to get used to the equipment and to test the equipment in the data collection environment. The students solved a non-routine mathematics problem collaboratively in groups of four students. The actual data collection and the collaborative geometry problem solving took place during the second lesson. The goal of the problem task was to find the optimal way to connect four imaginary cities located at the vertices of a square with cable. The problem solving session lasted almost an hour. The students worked both with paper and pencil as well as with GeoGebra software. In addition, three video cameras recorded the teacher and target students’ activity and we collected students’ written work. We also recorded screen capture videos form student computers and conducted post lesson interviews with the teacher and focus students, but these have not been used in the current analysis.The students were seated in pairs sitting opposite to each other. The students in the same pair could easily see each other’s computer screens while seated, but not the screen of the other pair. However, all students could see at least two other students’ notebooks.We had scripted most of the problem solving session for the teacher. During the lesson, the teacher first gave general instructions for the lesson and students got their GeoGebra software started (3 min 45 s), then posed the problem on the board and students worked on the task individually (4 min 33 s). Next, students worked with a partner (4 min 10 s), and then the four students worked together (26 min 42 s; fast solvers were provided an extension task). Finally, the solutions were collected on the board and discussed (4 min 54 s). Furthermore, the students downloaded a GeoGebra app to further examine the correct shape (5 min 57 s). In this article, we focus on the phase during which the four students were working together.When students were working individually, in pairs, or in groups of four, the teacher’s activity consisted of roaming in the classroom and stopping to scaffold one group at a time. The teacher was instructed to provide encouragement and to ask questions that required students to explicate their thinking but to not provide hints on how to solve the problem.The episode was transcribed using eye-tracking video to provide rich information about visual attention to gestures and images. Where necessary and possible, multiple videos were examined to better discern details.Results and discussionDuring individual and pair work, the students as a group had produced all the common easy solutions (Figure 1) and each student had found the solution “X”, which is best of these solutions. 						Figure 1. The common easy solutions, which we will call “C”, “X”, “H”, and “Z”.I begin the description where the students are challenged to move beyond these basic solutions. The teacher comes to check how the students are doing, and while she discusses with students, they start interacting as a group. In what follows, T is for teacher, S1 to S4 the four students. S1 and S3 are seated as a pair, and S2 and S4 as another pair. Teacher	How are you doing here?S3	We thought that this [pointing at X on her screen]. We were thinking, if one can get through here [tracing the diagonal from one vertex to the centre and then along the other diagonal to the other vertex].Teacher	Yes, it’s possible.S3	Then this is [our favourite]S4	I feel there is something, just “Ha!” [Expressing surprise, smiling, gestures a rapid short hand movement.] 	[Teacher and S2 smile.]Teacher	You can still continue thinking. Some good elements from different solutions. Brainstorm still a little further. [Moves away.]S3	[To S4.] So that means that there is still some better solution.S4	So there is. S1	Circle.S2	Some triangle.For this first segment, I want to point out four things. First, the eye-tracking data allowed us to see what the student was actually pointing at and what her question was about. The same level of detail will be informative in some of the events that will come later. Second, we see that the students know that it is common that a problem has an unexpected solution and they joke about it. As the teacher encourages them to explore further, they know that there must be another, better solution to be found. S1 suggests a circle. This idea catches fire in the group and very much influences the process that follows. I will next report how this idea develops and after that I will discuss later why this might have been such a tempting idea.Teacher 	[Comes to S2 and S4.] What do you have here? … Which, in your mind, would be the most convincing? S4	That. [Points an X on her screen.] Would it be circle? [Makes a small unclear gesture in the air.]Teacher	You can test it. You can get the length of the circle, when you take the circle tool. [Points at the tool on GeoGebra menu bar.]S3	But how…?S1 	I said circle before (----). [Pointing at S4, smiling.]Teacher 	[Smiles to S1.] Yes. You try it, too.S1 	I don’t want to (---). S2	It does not have to be a whole circle, it can be only. [Pointing at X on own screen, tracing an incomplete circle to connect all four vertices.] S1	[Begins to draw.]S3	How can the circle even (--) [Draws a circle in her notebook.] …S2	(To S3, smiling.) Do you know the thing, circle and inside it a square, and then circle and square and circle and square [gesturing with hands, making first a square with both hands, and then ‘collapsing’, them into a smaller circle and repeating this a few times.]S3	Yes, but. Aha! [Surprised.] But, but, it becomes longer, still. Or does it?The teacher makes the decision not to call the circle either a good or a poor suggestion. Instead, she invites S4 to test it. As S1 claims ownership for the idea, she suggests him to test it, too. S2 realizes immediately, that the connection between two vertices could be left out. Only S3 opposes the idea, but also joins drawing it.The teacher then announces to the class that it is time to move on to work in groups of four. She soon returns to challenge this group to reconsider the circle as a solution. Meanwhile S4 used GeoGebra to compute the length of the circle as the solution connecting the four vertices. She got a solution that it would be shorter than X, possibly due to not having the figures in the same scale. However, the solution is not questioned, even though S1 is puzzled by the apparent fact that the solution he suggested is so good.S4	No, it IS the fastest! [Cheering.] S1, you are the best! [Gestures V for victory with both hands.] Eight point four. Okay, it is the fastest. S1 you are a god.S1	Weird. [Shakes his head a little.]S2	And, if you remove that part, because that is not needed. [Pointing at S4 screen, circle between two vertices.] …Teacher	I would ask about this circular solution, what are the benefits in this in comparison to, for example this, there they are straight. [Points at the solution C on S2 screen.]I would suggest that this phase of events is influenced by reading incorrectly the social cues given by the teacher. For S3, the social pressure might be at play. However, S2 points out to an image in her mind, probably something like Figure 2. A circle has beauty and power in mathematics. Although the discussion ends with a conclusion that the straight lines are shorter, the students present some quite interesting arguments.S3	Don’t have to stop, just goes on wheeeee. [Rolls head in small circles.] S4	[The verbal explanation was incomprehensible, and could not be translated. However, her gestures focus on turning around the vertex.]S3	…If you change the lines of the square into a circle, the circle will be bigger. [S4 agrees.]Here, we see two kinds of arguments. The first kind refers to physical movement, and there, the curved path feels better. The other argument refers to the circle being the optimal solution for the ratio between area and perimeter. Figure 2. Squares inscribed in circles inscribed in squares.The students finally agreed that the curved lines are just “extra length”, and continued generating new ideas. I will present those that remained in the discussion and developed further. However, we see the curved lines to follow in their ideation, see X’ and H’ in Figure 3.Figure 3. The evolving solutions, which we will call X’, H’, Y’, and Y.The following stage in student progress is best described as a diagram (Figure 4). The students generated several ideas that were visually presented either as gestures, or images, either on paper or on computer. These ideas were effectively communicated among the students in the group and the arrows present the flow of information from one student to another, when a student was looking at another student’s gestures or drawings. In this diagram, you can see, for example, how H’ was drawn by S4, and then observed by S2 and S3, who both produced their own versions of the solution somewhat later However, the students struggled to produce the solution with GeoGebra. In fact, this led to S2 and S4 making little progress, as they struggled with the tools to produce a solution of the right shape. On the other hand, S3 decided to approximate H’ using lines, leading first, accidentally, to small Y, which she deleted, and then to the solution Y’, which was developed further into the optimal solution Y.	Student 1	S3	S4	S2Time	HG	PP	GG	HG	PP	GG	HG	PP	GG	HG	PP	GG22								H’				23											H’	24					H’						X’	25				X’								26												X’27									X’			28							H’					29						Y						30		H’		>								31						Y’						32									H’			33			X’									H’34												35												36									H’			37						Y					H’	38												39											Y	40												YFigure 4. Diagram of visual representations and their observations. The arrow presents the direction of information flowing from visual representation to observer. HG = Hand Gesture, PP = Pen on Paper, GG = GeoGebra. The symbol “>” in the graph refers to a student gesturing a shape like that.ConclusionsThis case study illustrates how an erroneous idea may emerge and survive in collaborative problem solving. The original the idea of circle as the solution was not grounded on anything more than a false intuition. However, it got rooted in the group’s discourse, probably because students were reading the social cues falsely. Although the students discussed and logically refuted the idea, new solution attempts with curved lines appeared and spread in the group.However, the erroneous solutions (X’ and H’) both have some resemblance to the optimal shortest path (Y), providing opportunities to find the optimal solution. Moreover, the use of technology both assisted and hindered the finding of optimal solution. Student S2 and S4 had their focus on how to draw curved lines, which effectively prevented them from generating new ideas. The student S3, on the other hand, used line segments to make and approximation of the curved line, which opened her the opportunity to find the optimal solution.AcknowledgmentThis research was funded by the Academy of Finland grant no. 297856 (MathTrack -project).ReferencesArzarello, F., Paola, D. Robutti, O., & Sabena, C. (2009). Gestures as semiotic resources in the mathematics classroom. Educational Studies in Mathematics, 70(2), 97–109.Glöckner, A., & Herbold, A. K. (2011). An eye‐tracking study on information processing in risky decisions: Evidence for compensatory strategies based on automatic processes. Journal of Behavioral Decision Making, 24(1), 71-98.Hartmann, M. & Fischer, M. H. Exploring the numerical mind by eye-tracking: a special issue. (2016) Psychological Research 80(3), 325–333. https://doi.org/10.1007/s00426-016-0759-0McIntyre, N. A., Mainhard, M. T., & Klassen, R. M. (2017). Are you looking to teach? Cultural and dynamic insights into expert teacher gaze. Learning and Instruction, 49, 41–53.Radford, L. 2008. Why do gestures matter? Sensuous cognition and the palpability of mathematical meanings. Educational Studies in Mathematics 70(2), 111–126. Rayner, K. (1998). Eye movements in reading and information processing: 20 years of research. Psychological Bulletin, 124(3), 372.Rogers, S. L., Speelman, C. P., Guidetti, O., & Longmuir, M. (2018). Using dual eye tracking to uncover personal gaze patterns during social interaction. Scientific reports, 8(1), 4271.Schindler, M., Lilienthal, A. (2017) Eye-tracking and its domain-specific interpretation: a stimulated recall study on eye movements in geometrical tasks. In: B. Kaur, W. K. Ho, T. L. Toh, & B. H. Choy (Eds.), Proceedings of the 41st Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 153-160). Singapore, Singapore: PME.Toivanen, M., Lukander, K., & Puolamäki, K. (2017). Probabilistic approach to robust wearable gaze tracking. Journal of Eye Movement Research, 10(4). DOI: 10.16910/jemr.10.4.2 	1 - 1	2019. In NNN (Eds.). Proceedings of the 43rd Conference of the International Group for the Psychology of Mathematics Education, Vol. 1, pp. XX-YY. Pretoria, South Africa: PME.1 - 8	PME 43 – 2019	PME 43 – 2019	1 - 7

Making and observing visual representations during problem solving : An eye tracking study

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Making and observing visual representations during problem solving : An eye tracking study

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