Errors in the Teaching/Learning of the Basic Concepts of Geometry Lorenzo J Blanco

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1 Errors in the Teching/Lerning of the Bsic Concepts of Geometry Lorenzo J Blnco 1. Activities in Techer Eduction The work tht we re presenting ws crried out with prospective primry techers (PPTs) studying in the Eduction Fculty of the University of Extremdur (Spin). The content of the work formed prt of the obligtory course Didctics of Geometry designed to be tken in the third yer of the officil Pln of Studies. The bsic objective of the course is tht the student should cquire the pedgogicl content knowledge (Blnco, 1994; Melldo, Blnco y Ruiz, 1998) 1 relted to the teching/lerning of Geometry in Primry Eduction. Our intention is tht the ctivities which we develop might generte simultneously mthemticl knowledge nd knowledge of the teching/lerning of Geometry. Also we tke it tht the curriculr proposls imply n epistemologicl chnge with respect to school-level mthemticl content nd to the clssroom ctivity which my result in the genertion of this knowledge. Preceding investigtions hve indicted to us tht our PPTs hve bsic errors concerning mthemticl content, nd in prticulr bout geometricl concepts. They lso hve deeply-rooted conceptions bout the teching/lerning of mthemtics deriving from their own experience s primry nd secondry pupils, nd which present contrdictions with the new school-level mthemticl culture. Our im therefore is not only to broden or correct their mthemticl knowledge reltive to the specific content of school-level mthemtics, but lso to put forwrd ctivities designed to encourge reflection on how mthemticl knowledge is generted nd how it is developed, tking into ccount the process of working towrds new mthemticl culture suggested in the current curriculr proposls nd in recent contributions bout the teching/lerning of Geometry. These ctivities should led them to reconsider their prior conceptions on mthemtics nd its teching/lerning. And consequently, this lerning environment must enble them to generte the metcognitive skills tht will llow them to nlyse nd reflect on their own lerning process s it is tking plce t tht moment. An importnt vrible in the process of lerning to tech is the cpcity to be ble to think bout one s own lerning process nd the wy in which it hs developed. The proposed tsks will enble epistemologicl chnge with respect to mthemticl knowledge: how this mthemticl knowledge is generted nd developed; nd how this knowledge is lernt. Encourge Strengthen Mthemticl knowledge Reflections on the lerning process on Working in groups, Conjecturing, Generlizing, Communicting,... Figure 1. Proposed tsk objectives 1

2 Our teching experience nd the conclusions of vrious studies suggest to us the dvisbility of posing chosen situtions from school-level mthemtics which the prospective techers might hve difficulties in resolving. This will mke it possible to nlyse nd evlute, nd consequently, to correct nd develop the PPTs mthemticl knowledge. 2. Errors concerning geometry concepts We re going to look t vrious ctivities which showed up mjor conceptul nd procedurl errors when the students techer resolved them. I consider tht the cuse hs to be sought in the teching process tht they themselves went through in primry school. Activities bout the ltitude of tringle It hs been found tht PPTs hve problems in performing ctivities relted to the concept of ltitude of tringle 2 (Gutiérrez y Jime, 1999; Azcárte, 1997) 3. This suggests situtions tht we my present s eductionl tsks to llow us to nlyse the difficulties nd errors presented by the teching/lerning of geometry in primry eduction. As I mentioned t the beginning of this rticle, my teching ctivity is with prospective techers of primry eduction, nd this is the context in which the resolution of the following ctivities is developed. Activity 1. Drw the orthocentre of n obtuse tringle. The ctivity described is set by wy of the following mthemticl tsk: Define ltitude of tringle Define the orthocentre of tringle Drw the orthocentre of the following tringle Figure 2. Activity 1. Drw the orthocentre of the tringle This mthemticl sitution is n ctivity which brings out mjor errors of concept nd procedure of the PPTs with respect to the specific concept of the ltitude of tringle, but lso with respect to the process of the teching/lerning of geometricl concepts. The nlysis of the students responses to this set ctivity presents n interesting contrdictory sitution. Thus most of the students write down correctly the definition of ltitude of tringle nd of orthocentre. They drw the ltitudes incorrectly, however, nd consequently lso the orthocentre of the tringle of the figure. They usully plce the orthocentre inside the tringle s the following figure shows. 2

3 Figure 3. Photocopy of the response of student to ctivity 1 (It is the point of intersection of the three ltitudes of tringle. The ltitude of tringle is the perpendiculr line which goes from the vertex of the tringle to the opposite side or its prolongtion). It is interesting tht the students re unwre of the contrdiction their response presents until we initite with them n nlysis of the process which they followed in resolving the ctivity. The interction tht we provoke with the students leds us to reject the hypothesis of confusion with some other concept such s tht of medin, or bisector of vertex, or perpendiculr bisector, or with the representtion of ny of them. And tht is why this sitution llows us to go deeper into the process of cquisition of geometricl concepts on the bsis of the students own process of lerning the concepts we re concerned with. A similr sitution to the bove occurs when we sk the students to drw the circumcentre of n obtuse tringle. Activity 2. Drw the ltitude of different tringles. The errors in representing the ltitudes of tringles re eqully mnifest when we set the following ctivity. In ech tringle drw the ltitude upon the side mrked with the letter Figure 4. Activity 2. The students mnifest mjor difficulties in drwing the ltitude of some of the tringles in the figure. Indeed, the errors in representtion nd nswers left blnk formed high percentge. 3

4 Recognition of specific prisms. Activity 3. In our clsses, we use bsic dictionry of geometricl concepts s resource for the students. From the definitions, we crry out different ctivities to estblish reltionships of similrity nd difference between concepts. This will help us to go deeper into these concepts, into their chrcteristics, nd to recognize different criteri of clssifiction nd inclusion. Well, these ctivities led us into prdoxicl situtions which hve elements in common with tht described before from the perspective of tringle geometry. Thus, for instnce, t one point in the course, we focus on the definitions of polyhedr, nd specificlly on the concept of prism. Now, t the beginning of the work on the concept of prism, once the definition hs been estblished nd memorized by the prospective techers, we sk them to identify different specific prisms mongst the polyhedr of the dictionry. Well, I hve to sy tht, in spite of knowing the definition of prism nd using the dictionry of geometricl concepts, they find it hrd to recognize further exmples of prisms other thn the right or oblique prisms or the tringulr or pentgonl prisms which re specificlly given in the dictionry. In most cses, they do not recognize the cube or rectngulr prism (clled orthohedr in spnish use) s prticulr cses of prisms. In other words, they hve difficulties in setting up reltionships of similrity between different geometricl definitions, nd therefore in being ble to understnd nd set up different clssifiction criteri. 3. Anlysis of these situtions. Definition nd representtion of geometricl concept The nlysis of these situtions shows us tht the students errors hve common elements tht re interesting to highlight. To understnd the sitution we re fced with, we hve to look t the nlysis of the concepts involved nd the different subconcepts tht mke them up, nd ssume tht the solving procedure followed by the students is closely relted to their own stge s primry school pupils. In other words, the errors tht the students mnifest re minly bsed on the teching/lerning process tht they went through in primry school. Let us go bck to ctivity 1, nd nlyse the procedure followed s function of the recognition nd use of the properties of the concepts involved (Figure 5). Problem posed: Drw the orthocentre of n obtuse tringle Orthocentre Intersection of the three ltitudes of tringle perpendiculr line segment vertex of the tringle side of tringle side opposite vertex perpendiculr to line segment from n externl point line segment drwn from one vertex of the tringle to the opposite side or its prolongtion Figure 5. Vribles of the ltitude concept 4

5 When we look further into the vribles of the ltitude concept, especilly into perpendiculrity onto the opposite side or onto its prolongtion, the students begin to recognize their error in the representtion of the orthocentre. The recognition of the error is n effective, interesting, nd motivting strting point from which to continue the ctivity nd propose specific new ctivities both designed to fill these gps in their mthemticl knowledge, nd on the teching of Geometry. Bering in mind the mp represented in the previous Figure, we shll proceed by recognizing ech of the concepts nd subconcepts indicted, giving emphsis to their pproprite representtion. This line is similr to tht put forwrd by Gutiérrez & Jime (1996) 4. In this regrd, those uthors note the following subconcepts of the concept of ltitude nd the ssocited ctivities: 1) The subconcept of perpendiculrity: Drw stright line perpendiculr to ech of these stright lines Figure 6. Activity 4. 2) The subconcept of perpendiculrity from point: Drw stright line from given point to given stright line segment or its prolongtion Figure 7. Activity 5. 5

6 3) The subconcept of opposite vertex: Identify the vertex of tringle tht is opposite certin side. Indicte with B the vertex opposite the side b of ech tringle. Then drw line segment tht goes perpendiculrly from the vertex opposite b to this side b or to its prolongtion b b b b b Figure 8. Activity 6. 4) The concept of ltitude of tringle: Drw the ltitude of tringle onto certin side. It is t this point tht we cn return to proposing ctivity 2 (Figure 4), which will be resolved by the students tking into ccount these vribles tht we hve been discussing. Let us return now to the students responses. It is interesting to look gin t the contrdiction tht occurs when the students write down the correct definitions of ltitude nd orthocentre, but drw grphicl representtion tht does not correspond to wht they hd written (Figure 3). Even more importnt, however, is the filure to recognize their error until we initite the nlysis of the vribles of the concept ltitude. This sitution llows us to spek of the difference between definition nd representtion of concept, nd consequently we cn look more deeply into the mentl imge tht the students hve of the concepts involved. In this cse, it is the specific mentl imge tht they hve ssocited with the concept of ltitude of tringle. Thus, on reclling their time s primry eduction pupils, the students recognize n imge ssocited with the height of n cute tringle stnding on horizontl bse nd rrnged so tht the representtion of the ltitude lies within the tringle. h h h is the ltur" (ltitude or height) of the tringle Figure 9. Trditionl representtion of the ltitude of tringle in textbooks 6

7 The buse of this representtion lso helps to crete the imge of the ltitude of the tringle s perpendiculr line segment which is unique for ech tringle. (The expression h is the ltitude of the tringle mkes this explicit). This ide hs its field of vlidity in the everydy use of the word ltur (= height). This imge explins why some students resolve ctivity 7 in the sense indicted in the figure itself. Activity 7. Drw the ltitude of the following tringle h Figure 10. Activity 7. Drw the ltitude of the tringle. And, in generl, it prtly explins the difficulties the students hve in resolving ctivity 2. Likewise, the imge of the orthocentre ppers linked to the representtion of the orthocentre of this cute tringle, so tht the orthocentre is locted inside it. In other words, they cpture the imge of the orthocentre in prticulr exmple. The orthocentre of cute tringles is lwys in the interior of the tringle Figure 11. Orthocentre of cute tringles As consequence of the sitution tht the students re experiencing, we cn mke three importnt observtions: First, the mentl imge of the orthocentre in the interior of the tringle predominted over the recognition nd use of the vribles of the concept, s expressed in the definition. Second, the mentl imge is lsting, despite the contrdiction tht rises between the definition nd the representtion. And third, the students begin to become wre of their contrdiction when we initite the nlysis of the vribles of the concept, nd not before. 7

8 The predominnce of the metl imge ssocited with concept is the reson why, when we propose ctivity 8, they identify it just with rhombus, excluding from our responses the concept of squre. Identify the figure. Activity 8 Figure 12. Confusión betwen squre nd rhombus concepts It is cler tht in this exmple, the student hs been crried long by the mentl imge ssocited with the rhombus, insted of by the nlysis of the properties of the figure, which in this cse hs ll four sides equl nd the four right ngles. Similrly, the difficulties in nlysing the vribles of concept, nd the imge tht they hve ssocited with prticulr cses of tht concept, constitute the cuse of the students difficulties in finding similrities nd differences or reltionships of inclusion between mthemticl concepts in generl, nd geometricl concepts in prticulr. And this leds us on to the third ctivity in which, to resolve it, we sked our students to mke squre, in which there pper the nmes of prisms, rectngulr prism (clled orthohedr in Spnish use), nd cube, nd underneth, the subconcepts nd reltionships which mke up the definition of ech of these nmes. At this point, it is convenient to recll tht to cquire concept mens building conceptul scheme of tht concept. Therefore, memorizing the definition of concept is no gurntee of understnding its mening. In relity, understnding mens hving conceptul scheme so tht certin menings re ssocited to the word tht designtes the concept: mentl imges, properties, procedures, experiences (Azcárte, 1997, 29). 8

9 4. Teching geometricl concepts The recognition by the prospective techers of the contrdiction nd their expecttions s future techers of Mthemtics motivte them to nlyse the cuses of the sitution. And this llows us to initite the nlysis of the methodologicl process with respect to the teching of Geometry lived through by the prospective techers during their stge s primry level pupils. We could represent this schemticlly in the following wy: i. Definition - recognition of the specific figure - memoriztion drills ii. Exmples of figures - description of their mthemticl chrcteristics - definition - drills of recognition nd recognition of the specific figure Figure 13. Scheme shown in Gutiérrez & Jime (1996, 145). Respect to the concepts involved, the prospective techers recognize tht their teching ws bsed on: i. Stndrd exmples tht displyed the vrious concepts. Thus, the tringles were usully presented in the sme position; the ltitudes of the tringles were drwn normlly on equilterl nd cute tringles stnding on horizontl bse. ii. The gretest emphsis ws given to the definition, obviting nlysis of the properties, nd with no importnce given to the fct tht the visuliztion produces more lsting nd influentil effect thn spoken or written text. iii. The ctivities reflected sttic nd repetitive process on templte exmples chosen lmost exclusively from the textbook. iv. Lck of specific experimenttion with other possible situtions imed t going deeper into the comprehension of the concepts nd id in trnsferring the knowledge to other problems. v. A pucity of mterils nd resources. In most cses the textbook ws prcticlly the only resource. 9

10 In my opinion, this model of teching s reclled by the prospective techers lies t the origin of the difficulties they hve in working with different geometricl concepts. Fortuntely it is worth reclling tht this model does not correspond with the current proposls on teching geometry. These proposls point to the need for the students to shre in the construction of the mthemticl concepts ctively nd cretively, with the ide tht this will encourge communiction, the elbortion of conjectures, problem solving, etc. Define nd drw the orthocentre of n obtuse tringle Uncovers Errors of concept nd procedure Correct enuncition of the definition Unsuitble mentl imge Deficient recognition nd use of the properties Predominnce of the mentl imge over the properties. Anlysis of the methodologicl process lived through by the PPT s during their own primry eduction dys. (i) Definition - recognition of the specific figure - memoriztion drills (ii) Exmples of figures - description of their mthemticl chrcteristics - definition - drills of recognition nd recognition of the specific figure The definition is emphsized Visulistion produces lsting mentl effect. Stndrd exmples Sttic process Lck of specific experimenttion Pucity of mteril resources Abusive dependence on the textbook Figure 14. Prospective techers during their stge s primry level pupils. 10

11 Note: 1. In Spnish, n ltitude of tringle is simply clled the ltur, n everydy word which lso mens height. References Azc rte, C. (1997) Si el eje de ordends es verticl, Àqu podemos decir de ls lturs de un tri ngulo?. Sum, Blnco, L.J. (1994). Initil Trining nd Teching Prctice. Methodologicl Issues in lerning to tech. First Itlin-Spnish Reserch Symposium in Mthemtics Eduction. Diprtmento di Mtem tic de l Universit de Moden (Itli) Guti rrez, A. y Jime, A. (1996). Uso de definiciones e im genes de conceptos geom tricos por los estudintes de Mgisterio. En Gim nez, J.; Llinres, S.; y S nchez, M.V. (eds.): El proceso de llegr ser un profesor de primri. Cuestiones desde l educci n mtem tic Guti rrez, A. nd Jime, A. (1999). Preservice Primry Techers Understnding of the Concept of Altitude of Tringle. Journl of Mthemtics Techers Eduction, Vol. 2, n¼ Melldo, V.; Blnco, L.J. y Ruiz, C. (1998). A frmework for lerning to tech sciences in initil primry techer eduction. Journl of Science Techer Eduction 9(3)