YOUNG CHILDREN’S UNDERSTANDING OF ANGLE IN A DYNAMIC GEOMETRY ENVIRONMENT Harpreet Kaur Nathalie Sinclair Simon Fraser University Simon Fraser University ha56sf.ca nathsincsf.ca This paper examines the effect of the use of dynamic geometry environments on children’s thinking about angle. Using a driving angle model in Sketchpad, kindergarten children were able to develop an understanding of angle as “turn,” that is, of angle as describing an amount of turn. Gestures and motion played an important role in their developing conceptions. eyords Geometry and Geometrical and Spatial hining lementary School dcation Angle Gestres Introduction he concept of angle is mltifaceted and can pose challenges to learners, even into secondary school Close, 1982, Mitchelmore White, 1995. Despite these difficlties, children sho sensitivity to the concept of angle from very early years Spele, Gilmore, McCarthy, 2011. Angles are normally introdced to children ite late in formal school settings. For eample, in ritish Colmbia, they are introdced in grade 6 12 years old, even thogh stdents are epected to describe, compare, and constrct 2D shapes, inclding triangles, sares, rectangles and circles in grade 2. he strong capacity of yong children to attend to and identify angles in varios physical contets motivated s to see hether a more dynamic conception of anglenamely, angleastrnmight spport their developing nderstanding at an earlier age. We have been investigating other geometryrelated concepts at this age too, sing DGs, inclding shape identification, symmetry and parallel lines Sinclair, Moss, ones, 2010 Sinclair ar, 2011. Previos research reports on the effectiveness of rtle Geometry ogo for teaching the concept of angle Clements, attista, Sarama, Saminathan, 1996 Simmons Cope, 1990. oever, e believe that DG might be helpfl in thining of angles as trns and rotation more effectively. n this paper, e report on an eploratory stdy condcted ith a split class of indergartengrade 1 children ages 56 oring ith angle sing The Geometer’s Sketchpad. We focs on the emergence of the concept of angleastrn and discss the specific mediating role of the se of the softare on this thining. Children’s Understanding of Angle n the research literatre, the concept of angle is shon to have different perspectives, namely angle as a geometric shape, nion of to rays ith a common end point static angle as movement angle as rotation dynamic angle as measre and, amont of trning Close, 1982 enderson aimina, 2005. De to different prevalent definitions of the term angle, teachers often face difficlty in noing hat definition of angle to se Close, 1982. Mitchelmore and colleages and Clements and colleages have done abndant research in the area of angle concept over the past tenty years. Mch research has been condcted on the development of the concept of angles, focsing at the grades 3, 4 and higher levels. Mitchelmore and White 1995 sggest that angles occr in a ide variety of physical sitations that are not easily correlated. Despite the ecellent noledge of all sitations, specific featres of each sitation strongly hinder recognition of the common featres reired for defining the angle concept Mitchelmore, 1998. Mitchelmore and White 1995 proposed that children initially acire a body of disconnected angle noledge sitated in a large nmber of everyday eperiences they then grop sitations to form angle contets sch as trns and corners and finally they form an abstract angle concept by recogniing similarities across several angle contets. ater ors of Mitchelmore involved teaching eperiments White Mitchelmore, 2003 in hich they divided angle sitations into three clsters2 line angles corners of room, intersecting roads, pairs of scissors, 1line angles doors, indshield ipers, and 0line angles the trning of a doornob or a heel. he sitation is more problematic for stdents here the to arms of angle are not clearly visible. esearch sing Logo shos that stdents tend to visalie the trn of trtle as trn of their body bt maing these trns involves riting nmerical commands Clements, attista, Sarama, Saminathan, 1996. he DG does not involve the riting of the commands and can ths be sed at an earlier age to develop more alitative nderstanding of angle. We believed that the DG approach old be helpfl in developing the dynamic as ell as static concept of angle. Theoretical Perspective n previos research, e have fond Sfards 2008 commognition approach is sitable for analysing the geometric learning of stdents interacting ith DGs see Sinclair, Moss, ones, 2010 Sinclair ar, 2011. For Sfard, thining is a type of discrsive activity. Sfards approach is based on a participationist vision of learning, in hich learning mathematics involves initiation into the elldefined discorse of the mathematical commnity. he mathematical discorse has for characteristic featres ord se vocablary, visal mediators the visal means ith hich the commnication is mediated, rotines the meta-discursive rules that navigate the flo of commnication and narratives any tet that can be accepted as tre sch as aioms, definitions and theorems in mathematics. earning geometry can ths be defined as the process throgh hich a learner changes her ays of commnicating throgh these for characteristic featres. We have previosly presented a developmental traectory related to identifying shapes in terms of different levels of discorse and no e are trying to do the same thing ith angles, bt e ill loo first at ho the different components of the discorse change as the stdents or ithin the DG. We are particlarly interested in investigating ho the stdents might move beteen different ord ses and to eamine the informal langage they se to tal abot angles. Similarly, given the importance of gestres in commnication of abstract ideas Coo Goldin Meado, 2006, and their potential to commnicate temporal conceptions of mathematics e, 2003 Sinclair Gol abaghi, 2010, e chose to etend Sfards approach to incorporate gestral forms of visal mediators. Given the fact that e are oring ith very yong children ho have had little eposre to a mathematical discorse arond angle, e ill be interested in seeing hether they mae se of ind of mismatch gestres that GoldinMeado 2004 describes as indicating a readinesstolearn. ita 2000 focses on the cognitive fnctions of gestres, hich play an important role in commnication. e points ot that the prodction of a gestre helps speaers organie rich spatiomotoric information, here spatiomotoric thining organies information differently than analytic thining hich is sed for speech. We ths epect that children ill se gestres to convey spatiomotoric information, even thogh they might not be able to convey the analytic thining sed in speech. Moreover, childrens gestre might be nonredndant ith their speech. r goal in looing at the gestres ill be to see ho they commnicate different ideas abot angles particlarly the mobile ones associate ith angleastrn. We are less interested in classifying the stdents gestres in terms of Mceills 1992 categories than in nderstanding the embodied, conceptal basis of the gestres. Exploring the Concept of Angle Participants and Tasks We ored ith indergartengrade1 children aged 56 from a school in a rral lo SS ton in the northern part of ritish Colmbia. here are 20 children ith diverse ethnic bacgronds and ith a ide range of academic abilities. We designed lessons related to angle along ith the classroom teacher, ho has a Masters degree in mathematics edcation and has been developing her practice of sing DGs for a cople of years. he teacher and stdents ored ith angles in different ays sing Setchpad for si lessons in a hole class setting ith an W nteractive Whiteboard. ach lesson lasted approimately 30 mintes and as condcted in a grop ith the children seated on a carpet in front of a screen. essons ere videotaped and transcribed. Dynamic Angle Sketches n this stdy, e sed to different setches to eplore the concept of angle ith the children. We began ith a simple angle diagram Fig.1. n the setch, dragging the verte of an arm of the angle changes the angle. he prpose of sing this setch as to enable children to focs on the standard form of angle as a geometric shape and to bild an nderstanding of angle throgh its behavioral properties. he research sggests that children have difficlty seeing a static angle as a trn. he sitation is more problematic hen the to arms of the angle are not clearly visible. he second setch sed is a driving angle model, hich shos both a static as ell as dynamic sense of angle Fig.2a, 2b. t incldes a car that can move forard as ell as trn arond a point. he trning is controlled by a little dial hich has to arms and a centre. o nmbers are sed. here are for action bttons rn, Drive Forard, rase races and eset that control the movement of the car. Stdents can reglate motion and trns to create different shapes lie random paths, sares, rectangles, and so on. Figure 1: Angle as a Shape Figures 2a and 2b: Driving Angle Model he traces offer a visible, geometric record of the amont of trn. he prpose of this setch as to move to a more dynamic presentation of angle related to a real life contet, here the focs of the children old be more on the continos behavior of the trning heels and old enable them to see the process of trning along ith the final prodct position after a specific angle trn. Classroom Discussion he classroom teacher tried to spport the discorse of angleastrn as she ored ith the setches. n hat follos, e report on the childrens or ith the first setch and then their developing sense of angle as they ored ith the second one. Introducing the angle as a geometric shape.