MEANINGS ABOUT DYNAMIC ASPECTS OF ANGLE WHILE CHANGING PERSPECTIVES IN A SIMULATED 3D SPACE Xxxx and xxxx We report findings from design-based research aiming at shedding light on the meanings about angle in 3D space generated by 12 year olds while changing virtual perspectives of their angular constructions with a specially designed Turtle Geometry with dynamic manipulation microworld. The results show that the various perspectives taken of the constructed graphical objects as well as the fact that this could be achieved through various means mediated the notion of angle in quite distinct ways with interesting influences on meaning construction. THEORETICAL BACKGROUND Even though angle (along with length and distance) is one of the most important mathematical tools for describing and analysing physical space and present in a wide variety of physical situations, these are not easily correlated or connected in relation to angle concepts by children at the end of primary (Freudenthal, 1983, Mitchelmore & White, 2000). According to Henderson and Taimina (2005) angle can be defined from at least three different perspectives: (a) angle as a geometric shape, i.e. formed between two geometrical objects embodying directionality which can be either segments or 2d geometrical figures (b) angle as a dynamic notion, indicating a change of one direction both as a turn and as the result of a turn; and (c) angle as a measure represented by a number. In typical school education angle is basically approached as a static geometric shape while the notion of angle as turn is usually underrepresented although it is considered the most natural, the most instinctive aspect of angle (Freudenthal, 1983). Even in cases where angle is approached as turn this is done only through static 2d representations, which, no matter how cleverly designed, may delay the development of dynamic aspects of the concept and their integration with the static ones (Clements et al, 1996). Digital media seem to provide the potential to re-address the use of dynamic perspectives for students to form meanings about angle. Its contribution to the teaching and learning of geometry in general is perceived to be strongly linked with interactivity, multiple interlinked representations including symbolic ones, dynamic manipulations and dynamic visualisations (Laborde et al., 2006). Here we report how students’ intuitions and ideas concerning angle as a spatial visualisation concept were challenged as they worked with a set of activities we designed adopting a constructionist theoretical perspective (Kafai & Resnick, 1996, xxxx, 2007). The students used a digital medium called MaLT which integrates a 3d Logo Based Turtle Geometry with a) dynamic manipulation of procedure parameter values and the resulting constructed figures and 1- 1 2011. In xxxx (Eds.). Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education, Vol. 1, pp. XXX-YYY. Ankara, Turkey: PME.
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b) dynamic manipulation of the students’ viewpoint inside the simulated 3d space containing the constructed objects. Turtle geometry is based on a different geometrical system to those usually associated with the learning of geometry and it has been characterised as differential by Papert (1980) and as intrinsic by Abelson & diSessa (1981). It’s considered as differential since a given geometrical state of the turtle is fully defined by its relation to the turtle’s immediately previous state. In a similar vein it is characterised as intrinsic in the sense that there is no need to refer to places outside the turtle’s immediate vicinity when deciding on an input to a procedure to change turtle’s state. Researchers seem to conclude that carefully designed Logo-based microworlds are an effective medium in offering rich mathematical experiences and encouraging the construction of meaning in relation to the notion of angle as turn in 2d through the turtle metaphor (Clements & Sarama, 1997, xxxx, 1997). However extending such microworlds to 3d space raises new issues related to the way the turtle metaphor may be put to use and the way deeply rooted intuitions about experiencing space and locomotion can be exploited so as to make sense of angle (xxxx & xxxx, 2010). In viewing the images in the computer screen as ‘signifiers’ mediated by the system in which they are created and acted upon (Morgan et al., 2009), we wanted to investigate how observing an object from various perspectives would interact with the actual process of constructing meanings about angle as a determinant of its spatial properties (xxxx, 1997). TOOLS METHOD AND TASKS MaLT is a constructionist microworld environment that extends ‘Turtleworlds’ to 3d geometrical space. ‘Turtleworlds’ blends Logo based Turtle Geometry with tools to dynamically manipulate procedure variables and observe the resulting ‘continuous’ change to the respective figural constructions (xxxx et al, 1997). In MaLT, we used a well established method to extend Turtle Geometry to 3d by adding two kinds of turn commands (Reggini, 1985): ‘UPPITCH/DOWNPITCH n degrees’ (‘up/dp n’) which pitches the turtle’s nose up and down on a plane perpendicular to the one defined by right-left turns and ‘LEFTROLL/RIGHTROLL n degrees’ (‘lr/rr n’) which moves the turtle around its own axis. A second feature of MaLT is that we kept the ‘Turtleworlds’ feature of variation tools. These tools recognise the procedure responsible for any figural construction and afford dynamic manipulation of variable values resulting in DGS-style change in the figures. A third feature also affords dynamic manipulation but this time what is changed is the users’ viewpoint of the Turtle Geometry space a) by a toggle fashion by using buttons to pick among 3 default views (front, side, top-down) (Fig. 3) and b) by dragging a specially designed vector tool, which we called ‘the active vector’, where the user can define the camera’s direction or position. The work reported in this paper is part of a design experiment in the sense that Cobb et al. (2003) have described it. The research took place in a classroom with 12 year olds in a public school in Greece. The class had totally 16 teaching sessions with the experimenting teacher over two months. The activity sequence was divided in two 1- 2 PME 35 - 2011
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phases with a strand of two tasks each. In task 1 (phase 1) the students were asked to navigate the turtle in such a way so as to simulate the take-off and the landing of an aircraft. In task 2 (phase 1) the students were asked to construct rectangles and to position them in at least two different planes. In the second phase students experimented with two half-baked microwords (xxxx, 2007). In each case they were given a ‘buggy’ procedure and were asked to experiment, figure out what was wrong or superfluous in the code and correct it. In particular, in task 3 (phase 2) students were asked to use the variation tool to control and experiment with the three variables of the procedure ‘movedoor’ (Fig. 1) that corresponded to different turtle turns so as to create the simulation of door opening and closing.The procedure was designed to have on purpose more than the variables needed. Variables a and c need to have a constant value of 90 while variable b needs to remain a variable. The students were not told of this of course. They first had to decide which was the role of each variable and which values could be given to them. Then they had to make changes to ‘movedoor’ so as to develop a procedure that only creates the simulation of a door opening and closing with the least possible variables. In task 4 (phase 2) the students were asked to use the variation tool to control the four variables in procedure ‘revolving’ (Fig. 1) which corresponded to turtle turns so as to create the simulation of a revolving door with four rectangular flaps. The procedure was again designed to have more than the variables needed. Students had firstly to work out the role of each variable and which values could be given to them. As before, they had to make changes to ‘revolving’ so as to develop a procedure that creates the simulation of a revolving door with the least possible variables. Finally the students were asked to extend the procedure of the revolving door in order to create a simulation of the fan of a watermill. to movedoor :a :b :c uppitch(:a) leftroll(:b) repeat 2 [forward(3) right(:c) forward(2) right(:c)] end
to revolving :a :b :c :d up(:a) lr(:b) repeat 4 [repeat 2 [fd(7) rt(:c) fd(4) rt(:c)] lr(:d)] end
Figure 1: The Logo code of the two half-baked microworlds In order to describe pupils’ learning trajectories as they happened in real time we adopted a participant observation methodology while the main corpus of data included video-recorded observational data, researchers’ observational notes as well as the sorting and archiving of pupil’s work on and off computer. Data were categorized in clusters of specific critical episodes that do not represent some quantifiable entity but are chosen to represent clearly the kind of activity that was going on in a specific time in the classroom. The results presented here are based on the work of one focus group.
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CONSTRUCTING MEANINGS ABOUT ANGLES WHILE CHANGING PERSPECTIVES The analysis that follows shows that the various perspectives taken of the constructed graphical objects as well as the fact that this could be achieved through various means mediated the notion of angle in quite distinct ways with respective consequences on meaning construction. Perspective taking through the use of microworld’s cameras The students seemed to have initially focused on angle as a directed turn trying to syntonize their embodied motional experiences with turtle’s motion as its is evident both from the view of the 3d space which they preferred as well as from the kind of turn commands they used (Fig. 2). Flying the turtle along the z axis by maintaining a frontal view the orientation of the vehicle of motion coincided both with the orientation of students’ bodies in the lived in 3d space and with the standard way of referring to the orientation of information on the computer screen. Students’ comments corroborate this result. When asked why they preferred this kind of ‘flight’ they replied: ‘If we wanted to turn turtle right or left, we could see from our hands. If we wanted to turn it right, let’s say, we would think where our hand is and we would send it to the right’. Moreover in this view the turn commands given to the 3d turtle was in accordance with our earthly experiences where the directions of ‘up’ and ‘down’ are fixed as a result of the gravitational effect. This was rather critical in particular as far as the use of the commands uppitch/downpitch are concerned, as other relative researches have underlined students’ difficulty in applying the above mentioned set of commands in an intrinsic way (xxxx et al., 2009) and not in relation to the standard up/down directions that are automatically applied to 3d space, Uppitch(45) Forward(2) Downpitch(45) Forward(2) Downpitch(45) Forwardd(2) Uppitch(45)
Figure 2: ‘Flying’ the turtle along the Z axis and the respective Logo code Progressively however, the students were not so concerned about body-syntonicity and they used various views of the simulated 3d space when they wanted to focus on the graphical results of specific turtle’s turns. For instance during task 4 students had extra difficulties in finding out the role of the :d variable, which determined the measure of turtle’s turning and respective position in the 3d space before drawing each successive door of the revolving door model. In the following episode students conjectured about the number of the visible rectangles (doors) if the value given to d is 720. However they did not find the front default view convenient and after testing 1- 4 PME 35 - 2011
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all the available default views they choose to continue working with the top-down view active, where the doors/rectangles created by the turtle was more clearly visible. S1: Lets see how many doors there are if the value is Only one? This perspective is not convenient, I will change it (He activates successively all the 3 default views and opts for the top –down one). S2 Yes, exactly like in the case of 360. It turns two rounds. S1: Yes, it collects all of them in one. When we move it, the doors are changing position. They are sticking together or they are unsticking. S2: We can’t say that. With d we determine their place. Look, If it is 90o they are turning and they are forming a cross, they form right angles, yes right angles, with 360o or 720o they are placed together in the same line.
Front view
Side view
Top-down view
Figure 3: Episode 1 and the 3 default views of the simulated space It should be noticed that with the default views students came in contact with simplified 2d views of the simulated 3d space which possibly helped them focus on specific aspects of their construction as a result of turning commands. In the above episode trying to explain screen phenomenology in relation to the measure of a leftroll turn command students opted for the top – down view where they could easily observe turtle’s rolling and the number of drawn rectangles. In this view the dihedral angle between the parallelograms is rather more easily discernable as it looks more like the 2d geometrical figures that they are accustomed to. Thus students can in a way coordinate turtle’s turning – one round for 360o and two rounds for 720o- with the static geometrical figure of 360 and 720 angle where the position of the two rays that form the angle coincide. Changing perspective of the constructed 3d object through the use of the variation tool. During the 3rd and 4th task the simulated 3d object’s position and orientation could be dynamically arranged through the combined use of Logo programming and the variation tool. The sequential change of the values of variable a and b in both respective procedures created a film-like succession of the different instances of the 3d model that gave the impression of rotation. For instance the dynamic manipulation of the values of variable a helped students view their constructions from different perspectives while not changing perspective as observers of the 3d space. In the following abstract students are trying to change the orientation of the revolving door model in relation to the ground plane so as to create the fan of a watermill. As a result PME 35 - 2011
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of their previous experimentation they can easily discern that it is the same construction with a different orientation in 3d space (we should lie it down), while their construction in a front 3d view should look like viewing it from the top. They also instantly recognise that they should change the values of variable a and they start experimenting with it through the variation tool. S2:Now we should lie it down. (First print screen on the right) S1: It’s the same as before. S2:Yes, it’s like viewing it from the top. (Second print screen on the right) S1: Move variable a…yes 360, it makes a whole circle and lies down S2: or 720, 1080 etc’
Figure 4: Episode 2 Changing the values of variable a sequentially, they observed the turtle and the whole 3d construction rolling around the x axis and having the desired orientation in 3d space for multiples of 360 values. Dragging the variation tool didn’t provide thus just a kinaesthetic sense of dynamic manipulation and animation of mathematical objects changing only their visual characteristics. It provided an action/notation context that fostered experimentation and rendered the various turn commands descriptors of evolving geometrical objects’ place and orientation in 3d space. Imaginary perspective taking through the turtle metaphor In the end of Task 2 there was some free time available and students spontaneously decided to try to construct a closed figure building upon their experimentation during Task 1 where they simulated the flight of an aircraft (Fig 5). Each take-off and landing of the turtle was used as the building block of a ‘peculiar’ figure that came as result of four repeats of the initial turtle’s journey while turning turtle 90 degrees before each re-execution. It is also interesting (Fig. 5) that students adopted a more analytic strategy, visualising the whole turtle’s journey and explaining it to each other before entering commands. Moreover they adjusted the view of the 3d space with the active variation tool so as to have a clear 3d view of the simulated space and they kept it fixed throughout their construction. It seems that as students got progressively more accustomed to the turtle’s motion and the software’s representational infrastructure the crucial point was not so much body syntonicity with the turtle but the coordinatation of two different view points: the view- point of the turtle which must be moved in an appropriate way so as to draw a figure and the view point of an external observer who looks at the figural results of turtle’s movement.
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Last names of authors in order as on the paper S1: Let’s make a square. So not right 45 but right 90, so as to go this way and then again 90 S2: Yes, and again 90 and we will come back. (So far they were talking to each other and now they return to the microworld inserting the commands)
Uppitch(45) Forward(2) Downpitch(45) Forward(2) Downpitch(45) Forward(2) Uppitch(45) Right(90)
Figure 5: Episode 3 and the respective Logo and graphical result Constructing the simulation of a 3d object while viewing the simulated space in perspective was not only more realistic or more familiar but allowed students to reconceptualise 3d objects in terms of Logo commands while paying attention not only to turtle’s immediately previous state but to the whole 3d space. In the above episode students executed four times the command right (90) so as to come back, to use students’ words. Changing turtle’s perspective in 3d space while having a fixed 3d view of the simulated space students seem to have intuitively articulated the total turn trip theorem (Papert, 1980): walking all the way round a polygon and returning to its initial position with the same orientation the turtle makes a full turn. CONCLUSIONS The above analysis addressed some of the meanings generated by the 12 year olds as they interacted with a microworld that afforded opportunities to act on multiple interlinked representations of angular concepts in 3d space (Morgan et al., 2009). The students used the simplified 2d views of the simulated 3d space through the manipulation of cameras to coordinate the figural results of the turtle’s turning in 3d space. They addressed angle as a directed turn in using the variation tool to experiment with figural changes resulting from changes in variable values in the context of noticing and understanding 3d objects’ spatial and geometrical properties. They seemed to see sense in moving from a turtle metaphor to that of a vehicle of motion metaphor (e.g. flying the turtle) in order to coordinate the intrinsic perspective of a moving entity with the perspective of an external observer of the 3d figural constructions as a whole and thus to use angle as a spatial visualisation concept. The students’ mathematical expressions may have been more or less divergent from institutionalised mathematics. However their constructionist activity seemed to offer them rich mathematical and phenomenological experiences which we suggest could scaffold meaning construction later on when formal teaching of 3d space takes place (Papert, 1980, Freudenthal, 1983). This research addressed meaning generation in the context of using dynamic and symbolic representations of 3d space in constructionist tasks. However, further research is needed in order to investigate the way angular concepts can be integrated with spatial navigation and orientation as well as with the use of metaphor in virtual 3d environments. For instance the didactical notion of building upon intuitive embodied metaphors of locomotion (Clements & Sarama, PME 35 - 2011
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1997) is highly questionable in 3d space as it is not obvious how those metaphors can be associated with the different kinds of turn in 3d space, taking into account that our earthly experiences are confined to a two dimensional (ground) plane. References Abelson H. and DiSessa A. (1981). Turtle Geometry: The Computer as a Medium for Exploring Mathematics. Cambridge M.A.: MIT Press. Clements D. and Sarama, J. (1997). ‘Children’s mathematical reasoning with the turtle programming Metaphor’, in English, L (ed.), Mathematical Reasoning, Analogies, Metaphors and Images Lawrence Erlbaum Publishers, Mahwah, pp. 313–338. Clements, D., Battista, M. T., Sarama, J., & Swaminathan, S. (1996). Development of turn and turn measurement concepts in a computer-based instructional unit. Educational Studies in Mathematics, 30, 313-337. Cobb, P., Confrey, J., DiSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational Researcher, Vol. 32-1, 9-13. Freudenthal, H. (1983) Didactical Phenomenology of Mathematical Structures. Reidel Publishing Company, Dordrecht Harel, I. & Papert, S. (1991). Constructionism: Reseasch Reports and Essays. Ablex. Publishing Corporation. Norwood, New Jersey. Henderson & Taimina (2005) Experiencing geometry. Euclidean and non Euclidean with History, Cornell University, Ithaca, New York xxxx. xxxx xxxx xxxx xxxx Mitchelmore, M. C. & White, P. (2000). Development of angle concepts by progressive abstraction and generalisation. Educational Studies in Mathematics, 41: 209–238. Morgan, C., Mariotti, M., A. and Mafei, L. (2009). Representation in Computational Environments: Epistemological and Social Distance. In International Journal of Computers for Mathematical Learning, 14 (3), 241-263 Reggini, H. (1985) Ideas y Formas, Buenos Aires, Ediciones Galapago
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