YOUNG CHILDREN’S UNDERSTANDING OF ANGLE IN A DYNAMIC GEOMETRY ENVIRONMENT
Harpreet Kaur Nathalie Sinclair Simon Fraser University Simon Fraser University h a56 sf .ca nathsinc sf .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.
ey ords Geometry and Geometrical and Spatial hin ing lementary School d cation Angle Gest res Introduction he concept of angle is m ltifaceted and can pose challenges to learners, even into secondary school Close, 1982, Mitchelmore White, 1995 . Despite these diffic lties, children sho sensitivity to the concept of angle from very early years Spel e, Gilmore, McCarthy, 2011 . Angles are normally introd ced to children ite late in formal school settings. For e ample, in ritish Col mbia, they are introd ced in grade 6 12 years old , even tho gh st dents are e pected to describe, compare, and constr ct 2 D shapes, incl ding triangles, s ares, rectangles and circles in grade 2. he strong capacity of yo ng children to attend to and identify angles in vario s physical conte ts motivated s to see hether a more dynamic conception of angle namely, angle as t rn might s pport their developing nderstanding at an earlier age. We have been investigating other geometry related concepts at this age too, sing DG s, incl ding shape identification, symmetry and parallel lines Sinclair, Moss, ones, 2010 Sinclair a r, 2011 . Previo s research reports on the effectiveness of rtle Geometry ogo for teaching the concept of angle Clements, attista, Sarama, S aminathan, 1996 Simmons Cope, 1990 . o ever, e believe that DG might be helpf l in thin ing of angles as t rns and rotation more effectively. n this paper, e report on an e ploratory st dy cond cted ith a split class of indergarten grade 1 children ages 5 6 or ing ith angle sing The Geometer’s Sketchpad. We foc s on the emergence of the concept of angle as t rn and disc ss the specific mediating role of the se of the soft are on this thin ing. Children’s Understanding of Angle n the research literat re, the concept of angle is sho n to have different perspectives, namely angle as a geometric shape, nion of t o rays ith a common end point static angle as movement angle as rotation dynamic angle as meas re and, amo nt of t rning Close, 1982 enderson aimina, 2005 . D e to different prevalent definitions of the term angle, teachers often face diffic lty in no ing hat definition of angle to se Close, 1982 . Mitchelmore and colleag es and Clements and colleag es have done ab ndant research in the area of angle concept over the past t enty years. M ch research has been cond cted on the development of the concept of angles, foc sing at the grades 3, 4 and higher levels. Mitchelmore and White 1995 s ggest that angles occ r in a ide variety of physical sit ations that are not easily correlated. Despite the e cellent no ledge of all sit ations, specific feat res of each sit ation strongly hinder recognition of the common feat res re ired for defining the angle concept Mitchelmore, 1998 . Mitchelmore and White 1995 proposed that children initially ac ire a body of disconnected angle no ledge sit ated in a large n mber of everyday e periences they then gro p sit ations to form angle conte ts s ch as t rns and corners and finally they form an abstract angle concept by recogni ing similarities across several angle conte ts. ater or s of Mitchelmore involved teaching e periments White Mitchelmore, 2003 in hich they divided angle sit ations into three cl sters 2 line angles corners of room, intersecting roads, pairs of
scissors , 1 line angles doors, indshield ipers , and 0 line angles the t rning of a door nob or a heel . he sit ation is more problematic for st dents here the t o arms of angle are not clearly visible. esearch sing Logo sho s that st dents tend to vis ali e the t rn of t rtle as t rn of their body b t ma ing these t rns involves riting n merical commands Clements, attista, Sarama, S aminathan, 1996 . he DG does not involve the riting of the commands and can th s be sed at an earlier age to develop more alitative nderstanding of angle. We believed that the DG approach o ld be helpf l in developing the dynamic as ell as static concept of angle.
Theoretical Perspective n previo s research, e have fo nd Sfard s 2008 commognition approach is s itable for analysing the geometric learning of st dents interacting ith DG s see Sinclair, Moss, ones, 2010 Sinclair a r, 2011 . For Sfard, thin ing is a type of disc rsive activity. Sfard s approach is based on a participationist vision of learning, in hich learning mathematics involves initiation into the ell defined disco rse of the mathematical comm nity. he mathematical disco rse has fo r characteristic feat res ord se vocab lary , vis al mediators the vis al means ith hich the comm nication is mediated , ro tines the meta-discursive rules that navigate the flo of comm nication and narratives any te t that can be accepted as tr e s ch as a ioms, definitions and theorems in mathematics . earning geometry can th s be defined as the process thro gh hich a learner changes her ays of comm nicating thro gh these fo r characteristic feat res. We have previo sly presented a developmental tra ectory related to identifying shapes in terms of different levels of disco rse and no e are trying to do the same thing ith angles, b t e ill loo first at ho the different components of the disco rse change as the st dents or ithin the DG . We are partic larly interested in investigating ho the st dents might move bet een different ord ses and to e amine the informal lang age they se to tal abo t angles. Similarly, given the importance of gest res in comm nication of abstract ideas Coo Goldin Meado , 2006 , and their potential to comm nicate temporal conceptions of mathematics e , 2003 Sinclair Gol abaghi, 2010 , e chose to e tend Sfard s approach to incorporate gest ral forms of vis al mediators. Given the fact that e are or ing ith very yo ng children ho have had little e pos re to a mathematical disco rse aro nd angle, e ill be interested in seeing hether they ma e se of ind of mismatch gest res that Goldin Meado 2004 describes as indicating a readiness to learn. ita 2000 foc ses on the cognitive f nctions of gest res, hich play an important role in comm nication. e points o t that the prod ction of a gest re helps spea ers organi e rich spatio motoric information, here spatio motoric thin ing organi es information differently than analytic thin ing hich is sed for speech . We th s e pect that children ill se gest res to convey spatio motoric information, even tho gh they might not be able to convey the analytic thin ing sed in speech. Moreover, children s gest re might be non red ndant ith their speech. r goal in loo ing at the gest res ill be to see ho they comm nicate different ideas abo t angles partic larly the mobile ones associate ith angle as t rn. We are less interested in classifying the st dents gest res in terms of Mc eill s 1992 categories than in nderstanding the embodied, concept al basis of the gest res.
Exploring the Concept of Angle Participants and Tasks We or ed ith indergarten grade1 children aged 5 6 from a school in a r ral lo S S to n in the northern part of ritish Col mbia. here are 20 children ith diverse ethnic bac gro nds and ith a ide range of academic abilities. We designed lessons related to angle along ith the classroom teacher, ho has a Master s degree in mathematics ed cation and has been developing her practice of sing DG s for a co ple of years. he teacher and st dents or ed ith angles in different ays sing S etchpad for si lessons in a hole class setting ith an W nteractive Whiteboard . ach lesson lasted appro imately 30 min tes and as cond cted in a gro p ith the children seated on a carpet in front of a screen. essons ere videotaped and transcribed.
Dynamic Angle Sketches n this st dy, e sed t o different s etches to e plore the concept of angle ith the children. We began ith a simple angle diagram Fig.1 . n the s etch, dragging the verte of an arm of the angle changes the angle. he p rpose of sing this s etch as to enable children to foc s on the standard form of angle as a geometric shape and to b ild an nderstanding of angle thro gh its behavio ral properties. he research s ggests that children have diffic lty seeing a static angle as a t rn. he sit ation is more problematic hen the t o arms of the angle are not clearly visible. he second s etch sed is a driving angle model, hich sho s both a static as ell as dynamic sense of angle Fig.2a, 2b . t incl des a car that can move for ard as ell as t rn aro nd a point. he t rning is controlled by a little dial hich has t o arms and a centre . o n mbers are sed. here are fo r action b ttons rn, Drive For ard, rase races and eset that control the movement of the car. St dents can reg late motion and t rns to create different shapes li e random paths, s ares, 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 amo nt of t rn. he p rpose of this s etch as to move to a more dynamic presentation of angle related to a real life conte t, here the foc s of the children o ld be more on the contin o s behavio r of the t rning heels and o ld enable them to see the process of t rning along ith the final prod ct position after a specific angle t rn . Classroom Discussion he classroom teacher tried to s pport the disco rse of angle as t rn as she or ed ith the s etches. n hat follo s, e report on the children s or ith the first s etch and then their developing sense of angle as they or ed ith the second one. Introducing the angle as a geometric shape. he teacher began by sho ing the children the s etch in Fig re 1 and as ing them hat they sa . nitially, the st dents attended to the feat res of the fig re li e points, circle etc. ne st dent ttered the ord angle, altho gh, hen prompted, didn t elaborate. o st dents, Colin and asmine, compared the s etch ith triangle. ristian as the first to impose motion on the diagram Teacher: Kristian, what do you see? (Pointing towards figure1 on the screen) Kristian: I see the point is going back and that point is going up…Inaudible… ristian foc ssed here on the t o arms of the angle, not mentioning e plicitly the point at hich they meet. he teacher as ed abo t other similar e amples and the children described hat they sa in terms of concrete ob ects li e a s ing, slide, the letter , a ho se, the bottom of a hill, and a nose in their responses. h s, initially their disco rse as dominated by everyday lang age and they made little or no se of mathematical ords and their comparisons involved very loose vis al resemblance. Follo ing initial disc ssion, the teacher dragged the verte of one arm of the angle. Teacher: What happens if I do this? (dragging one vertex of one arm of the angle) Will: It turns wider
Teacher: It does turn wider. What about if I do this? (dragging the vertex further) Morris: It widens even more now. Teacher: It widens even more. What am I doing if I am moving this line? What kind of movement is that? Jasmine, Will, Kristian: Square, it’s a clock Teacher: It’s a clock movement? Students: Yeah Teacher: What am I doing to the line? Students: Moving, clock. When the teacher drags the verte of one arm of the angle to ma e it ider, Will and Morris describe the change in terms of idening Will also ses the ord t rn to describe the motion. Several st dents describe the movement of the arm of the angle as the movement of the cloc . his is the first instantiation of angle as t rn, hich the st dents associated ith a cloc . ote that they do not e plicitly se the ord angle, b t they tal abo t the same diagram they sa before, no sing an entirely ne comparison, hich becomes shared idely. he teacher finishes the lesson by tal ing abo t the space bet een the t o arms and sho ing them ho to mar an angle bet een t o arms in Sketchpad, hich can be done sing the Mar er tool. Introducing driving angle model. n the s bse ent lesson, the teacher presented the driving angle model to the st dents and as ed them hat they sa . nitially children foc sed more on feat res li e color of lines, circle and points. he teacher prompted them to find the similarity bet een the t o s etches Fig res 1 and 2a . Most of the st dents foc sed on describing the absence of the car in first s etch Fig.1 and the presence of the circle in the second s etch Fig. 2a . D ring comparison, only the second s etch as displayed on the W . Will: And there is a blue one on this one Teacher: There is a blue one on this one. That’s the same? Will: Yeah, Surrounded by a circle (draws circle in the air to show the surroundedness. Repeats gestures a few times) Will recalled the angle diagram of first s etch and noticed its presence in both the s etches, b t his description bl e one on this one sho ed a dominance of everyday lang age. Despite the se of ord angle by the teacher hile mar ing the angles at the end of previo s lesson, none of the st dents sed the ord in their comparison of the t o s etches. D ring the classroom disc ssion, the children asserted that in order to drive the car, the heels sho ld be t rned. Teacher: ay, have another estion for yo g ys. What does the car do Will: t drives on the gro nd. gesturing with both arms, turning both his arms together from left to right Teacher: t does drive on the gro nd. So e re going to pretend o r screen is the gro nd. o is it going to drive on the gro nd What ays can it drive asmine
Jasmine: o have to t rn the heel. With both arms turning as if turning a steering wheel Will and asmine both sed gest res to describe the t rning of the car. Many children in the class imitated their gest res. asmine sed the gest re of t rning the steering heel as a vis al mediator hile associating the motion of car ith the t rning of heels. he teacher then pressed the rn b tton and the car t rned by the same amo nt indicated by the angle dial, also leaving a trace of the t rn behind. Teacher: ay am going to move something for yo , here. ay f tell yo that this is ind of li e the car s steering heel, hat do yo thin that car is going to t rn li e no What do yo thin is going to happen hin abo t it. Making the turning angle smaller in the size , What is it going to do no Students: Turn, turn Teacher: Turn…is it going to turn the same?
Students: Yeah. he teacher told the st dents that the angle dial as li e the car s steering heel and she changed the si e of the angle in the dial. hen she prompted the st dents to predict the o tcome on pressing the rn b tton several st dents predicted that car o ld t rn altho gh they co ld not predict the amo nt of t rn. his sho s that st dents did not associate the t rn of the car ith the angle in the dial they had no ro tine for assessing the magnit de of the t rn. he teacher invited the children to compare the si e of the ang lar t rns. She changed the si e of the angle and as ed the children to decide hether this t rn as smaller or larger than the previo s amo nt of t rn. Teacher: he car is stopped b t if yo loo , that s li e right here. he car is t rned t rnn ... and stops here. And then moved this steering heel p here ith the angle Fig. 3a and e press t rn again and it only ent from here to here Fig.3b . Are these the same
Figure 3a Figure 3b Figure 3c: Morris’s gestures Figure 3d
Students: No, no, no…that one is too bigger (pointing towards the screen) Teacher: Which one is bigger? Morris: They don’t match. There is no match. One is bigger (Lying on the carpet, gesturing with his fingers as if he is comparing the sizes (Fig. 3c). And then shaking his head no). Morris compared the t o different amo nts of angle t rns by comparing the si e of the traces. Morris lin ed the difference in the si e of t o fingers Fig. 3c and 3d ith the difference in t o amo nts of t rn si es of traces . e sed the gest res ith t o inde fingers to e plain his reasoning. hese are ne vis al mediators that are sed to e press the idea of angle. hey also point to a ne ro tine for comparing t rn. is gest re here is non red ndant see ita, 2000 ith his speech since he e presses meanings for angle t rn that are not evident in the speech. ndeed, the children se very fe verbal e pressions in these lessons. We note that the initial se of concrete vis al mediators in first lesson as replaced by the se of embodied mediators in the form of gest res. Understanding of benchmark angles. he teacher introd ced some benchmar angles in the ne t lesson by sing the first s etch ith ords li e right t rn. She sho ed the 90° angle in the s etch Fig.4a . he children compared the right angle ith the side of a ho se and corner of a bo . he teacher sho ed a hole movement of the one arms of angle from 0° to 180° by dragging the verte of one arm Fig. 4b .
Figure 4a: Sketch showing 900 Figure 4b: Movement of one arm from 00 to 1800 he children compared 180° to a line and a road. Morris made a gest re ith his hand moving along a straight path, hile describing a car moving along a straight road Fig re 5a th s pic ing p the conte t of the previo s day s or . he teacher as ed the st dents ho they co ld ma e the car t rn by 1800.
Teacher: How can you make that car go that far…go in a straight line…Remember we did the 180 that was a straight line on the board. You said that it looked like a road. What can you do to the steering wheel? Students: Press it Teacher: No Will: Move it Teacher: How can you move it, Will? Show us. Could you show us that up there, Will? Give it a try. Can you show how to make it go 180? Show us how to go halfway around? (Will comes on the board and try to move the vertex (point) on angle dial to adjust angle, but fails to move the point (Fig.5b)). You have to hold it. Tell me when to stop? (Will hold the vertex while dragging and adjusts angle to approx. 1800 (Fig.5c)) one of the children, other than Will, ere able to recogni e that they needed to change the angle in the dial in order to ma e the car to t rn by 180°. Will sho ed an nderstanding of the association of the t rn of car ith the angle in the dial. his shift in the Will s nderstanding might be the res lt of seeing the teacher repeatedly changing the angle in the dial d ring the lesson and of seeing her systematic dragging sho n in Fig. 4b. At the teachers re est, asmine made the car t rn half ay by pressing the rn b tton Fig. 5d . When the teacher as ed abo t the direction of the car after pressing the rn b tton again, both Chloe and asmine predicted that the car o ld complete the f ll t rn.
Figure 5a Figure 5b & 5c: Will adjusting dial to 1800 Figure 5d Teacher: Why do you think it is going to go all the way around? Chloe: Because when you press it, it will go…and stop Teacher: Okay, can you think why it is going to do that? Chloe: Because it goes that way and stops right there. (Pointing towards the screen and reflecting, as if imagining the turn (Fig. 6a))
Figure 6a: Chloe while responding Figures 6b & 6c: Jasmine gesturing half and full turn From Chloe s e planation incl ding her gest res , it seems that she imagines the car t rning and e plains her reasoning in terms of the stopping point of the heel. Teacher: It is going to close. Why do you think it is going to close? Jasmine: Because it is like this [gestures shown in Fig. (6b, 6c)] Teacher: Yeah, because it is half way now…oh…that’s interesting
While Chloe and asmine both vis alised the final position of the car s ccessf lly, Chloe sed ords to describe the t rn of the car, hereas asmine sed the gest res to e plain the reasoning in terms of half t rning and f ll t rning position. oth seemed to nderstand that t o half t rns res lt in a f ll t rn. nce again, their nderstandings ere comm nicated by the non red ndant gest res, in hich arms are sed as sides of angle, hich provided ne vis al mediators for their reasoning. Discussion and Conclusion At the beginning of the episodes, the children had virt ally no mathematical disco rse aro nd angle. y or ing on sit ations involving t rn, hich e tho ght o ld provide a strong embodied connection for the children, the goal of these lessons as to see hether st dents co ld develop a more sophisticated disco rse aro nd angle as t rn and not necessarily involving any n merical antification of angle. he dragging of the one arm of the angle foc sed the children s attention on the t rning behavior of the segment in the diagram. his seemed to act as a vis al mediator to hich the children associated the movement of a cloc . he cloc movement metaphor enabled the children to see the angle diagram as t rning of arms, and hence, initiated the disco rse angle as t rn. When the children ere as ed to or ith the Drive model, they initially sed informal ays to describe the t rning of the car by gest ring the t rning of a steering heel. his steering heel metaphor differs in important ays from the cloc metaphor. While the latter foc ses on the changing position of the arms and th s on the changing si e of the angle bet een the arms , the former foc ses on the angle as a movement from one arm to another and th s, on the si e of a given angle . nly one st dent as able to associate the t rn of car ith the angle dial e plicitly. Will s recognition of the need to ad st of angle dial to 1800, in order to move the car half ay, sho s that the diffic lty in vis ali ing the 0 arm angle as reported by White and Mitchelmore 2003 co ld be eased by the trace feat re of Sketchpad, enabling st dents to see the process of t rning as ell as the prod ct. Some children ere s ccessf l in comparing t o different amo nts of t rns. Also, some children ere able to reason abo t half and f ll t rns in fact, they co ld e plain that a f ll t rn o ld re ire the repetition of t o half t rns. hey co ld not tal abo t angle and t rns sing mathematical vocab lary. At the beginning, the st dents described the angle diagram in terms of its parts, sing concrete, everyday lang age they event ally came to associate the notion of angle and t rn to both a cloc and a steering heel both of hich capt re something abo t the idea of angle as t rn. Several st dents also sed gest res to e press the notion of angle, as can be seen in Fig res 3 and 6. Finally, the st dents co ld also tal abo t angle in terms of vario s si es especially, the half angle . We arg e that the gest res sed by the st dents became part of their vis al mediators for the concept of angle. hese gest res ere at least in part evo ed by the diagrams in Sketchpad. For e ample, the t o inde fingers sho n in Fig re 3 arise from the angle dial sed by the teacher. he children s se of t rning gest res in their responses to the car s etch sho s that they created gest res as mediators for the p rpose of comm nication. Also, asmine s gest re Fig re 6 relating half t rn to f ll t rn is a clear e ample of the effect to dynamic vis ali ation offered by S etchpad. he traces of half t rn in Sketchpad acted as a vis al mediator and then asmine sed her gest res of half t rn straight opened arms to ma e a f ll t rn covering the hole movement of arms to close hands as a means of comm nication. his interplay bet een gest re and diagram resonates ith Ch telet s 1993 theory of mathematical inventiveness, hich de Freitas and Sinclair 2012 disc ss in the conte t of mathematics ed cation in this case, the dynamism of the diagram seems to evo e ite directly the moving of the children s fingers and hands. At this early stage, e have foc sed on ord se and vis al mediators. here as little sense of any ro tines sed for identifying angles or eval ating their si e. F rther, given the emergent sense of angle and t rn, there eren t any endorsed narratives. o ever, the fre ent se of non red ndant gest res by Morris, Will, Chloe and asmine d ring the lessons indicates, as ell as the relatively infre ent se of verbal e pression, s ggests that the children had a certain readiness to learn Goldin Meado , 2004 abo t angle as t rn. he ne t phase of o r research, for hich e are no in the process of gathering data, involves e ploring the ay in hich the children s emergent sense of angle can be developed into ro tines for identifying angles and tal ing abo t their si e. We have some initial evidence that the children are able
to identify t o angles as being the same even hen their arms have different lengths beca se they se a ro tine of comparing t rn. his o ld be a very important res lt given the e isting research that h h h h h h h h
Acknowledgments We ish to than ileen ennison for her participation in this pro ect, both practically and concept ally. We d also li e to than Shiva Gol abaghi for val able comments on previo s drafts. References Ch telet, G. 1993 . Les enjeux du mobile. Paris Se il nglish translation by . Shore M. agha, Figuring space: Philosophy, mathematics and physics. Dordrecht l er, 2000 . Clements, D. ., attista, M. ., Sarama, ., S aminathan, S. 1996 . Development of t rn and t rn meas rement concepts in a comp ter based instr ctional nit. Educational Studies in Mathematics, 30 4 , 313 337. Close, G. S. 1982 . Children’s understanding of angle at the primary/secondary transfer stage. ondon Polytechnic of the So th an . Coo , S., Goldin Meado , S. 2006 . he role of gest re in learning Do st dents se their hands to change their minds Journal of Cognition and Development, 7 2 , 211 232. De Freitas, ., Sinclair, . 2012 . Diagram, gest re, agency theori ing embodiment in the mathematics classroom. Educational Studies in Mathematics 8 December 2011 , 1 20. Goldin Meado , S. 2004 . Gest re s role in the learning process. Theory into Practice, 43 4 , 314 321. enderson, D. W., aimina, D. 2005 . Experiencing geometry. Euclidean and non-Euclidean with history. thaca, Cornell University. ita. 2000 . o representational gest res help spea ing. n D. Mc eill d. , Language and gesture pp. 162 185 . Cambridge Cambridge University Press. Mc eill, D. 1992 . Hand and mind: What gestures reveal about thought. Chicago University of Chicago Press. Mitchelmore, M. C. 1998 . o ng st dents concepts of t rning and angle. Cognition and Instruction, 16 3 , 265 284. Mitchelmore, M. C., White, P. 1995 . Development of the angle concept by abstraction from situated knowledge. Paper presented at the ann al meeting of the American d cational esearch Association, San Francisco, CA. e , . 2003 . Do real n mbers really move ang age, tho ght, and gest re he embodied cognitive fo ndations of mathematics. n . ersh d. , 18 unconventional essays on the nature of mathematics pp. 160 181 . e or Springer. Sfard, A. 2008 . Thinking as communicating: Human development, the growth of discourses and mathematizing. Cambridge, U Cambridge University Press. Simmons, M., Cope, P. 1990 . Fragile no ledge of angle in t rtle geometry. Educational Studies in Mathematics, 21, 375 382. Sinclair, ., Gol abaghi, S. 2010 . Dra ing space Mathematicians inetic conceptions of eigenvectors. Education Studies in Mathematics, 74 3 , 223 240. Sinclair, ., a r, . 2011 . o ng children s nderstanding of reflectional symmetry in a dynamic geometry environment. n . Ub d. , Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education ol. 4, pp. 193 200 . An ara, r ey PM . Sinclair, ., Moss, ., ones, . 2010 . Developing geometric disco rse sing DGS in 3. n M. M. F. Pinto, . F. a asa i ds. , Proceedings of the 34th Conference of the International Group for the Psychology of Mathematics Education ol. 4, pp. 185 192 . elo ori onte, ra il PM . Spel e, . S., Gilmore, C. ., McCarthy, S. 2011 . indergarten children s sensitivity to geometry in maps. Developmental Science, 14 4 , 809 821. Stavy, ., irosh, D. 2000 . How students (mis-)understand science and mathematics: Intuitive rules. e or eachers College Press. White, P., Mitchelmore, M. 2003 . Teaching angles by abstraction from physical activities with concrete materials. Paper presented at the 27th nternational Gro p for the Psychology of Mathematics d cation Conference eld ointly ith the 25th PM A Conference ol. 4, pp. 403 410 , onol l , .