Konyalioglu, Serpil. Konyalioglu, A.Cihan. Ipek, A.Sabri. Isik, Ahmet

Transcription

1 The Role of Visalization Approach on Stdent s Conceptal Learning Konyaliogl, Serpil Department of Secondary Science and Mathematics Edcation, K.K. Edcation Faclty, Atatürk University, Erzrm-Trkey; serpilkonyali2002@yahoo.com Konyaliogl, A.Cihan Department of Secondary Science and Mathematics Edcation, K.K. Edcation Faclty, Atatürk University, Erzrm-Trkey. ackonyali@atani.ed.tr Ipek, A.Sabri Department of Primary Mathematics Edcation, K.K. Edcation Faclty, Atatürk University, Erzrm-Trkey. Isik, Ahmet Department of Secondary Science and Mathematics Edcation, K.K. Edcation Faclty, Atatürk University, Erzrm-Trkey. The aim of this stdy is to investigate the role of visalization approach on stdents conceptal nderstanding. The reslts of this stdy, while there is no statistical difference between the control and experiment grops in terms of procedral learning, experimental grop stdents were more sccesfl in conceptal learning statistically. Key words: visalization, conceptal knowledge, procedral knowledge. INTRODUCTION Researches in mathematics edcation has changed especially over the last for decades. Mathematical knowledge is among the foremost sbjects in the change process. Abot learning psychology, Skemp (1971) searched firstly mathematics knowledge mentioned two kinds of knowledge. The first one is to recognize a set of symbols, which is mechanical knowledge that does not inclde conceptal nderstanding, bt incldes the ability to make procedres. The second one is the knowledge that can symbolize mathematical concepts; relate each other, and the knowledge that based pon abilities of making procedres with mathematical concepts (Baki 1998). Baykl (1999) defines that procedral knowledge is symbols, rles and knowledge sed in solving mathematical problems and on the other hand, Baykl (1999) states that conceptal knowledge is described as mathematical concepts and relationship to each other.

2 Althogh many researches recently have done in mathematics edcation in showing that there are an important difference between conceptal knowledge and procedral knowledge (Ma 1999), conceptal and procedral knowledge complete and dependent on each other even if these types of knowledge seem to be independent from each other (Baki 1998). This knowledge distinction has discssed mathematics edcators and has been accepted as general, there is no consenss these type of knowledge and their relation. They freqently try to make a distinction between conceptal knowledge and procedral knowledge, thogh the difference between these knowledge is not clear (Isleyen &Isik 2003). There is a relation between these knowledge. Bt, note that, in mathematics edcation, fnctional and permanent learning can be possible only by balancing conceptal and procedral knowledge (Noss & Baki 1998). Since traditional mathematics teaching mainly cltivates skills, neglecting conceptal nderstanding of the nderlying domain (Kadijevic 1999). The stdents learning difficlties in acqiring the concepts of mathematics is abstract natre of mathematics. Since mathematical concepts are abstract, stdents learns mathematics by memorizing. One of the most important problems associated with the teaching mathematics is risen from the stdents nderstanding difficlties in establishing the relationship between their knowledge and intition abot concrete strctres and abstract natre of mathematics. It is not easy to find concrete examples in mathematical concepts. There is a special importance of geometrical strctres called as semi-concrete on teaching mathematics. An important component of forming concrete or at least semi-concrete of or mental representation of a concept is an external or physical reference (Konyaliogl et al. 2003). It is sitable sage of semi-concrete strctre pointed ot as geometric system in teaching of the abstract concept in mathematics. Graph, diagram, pictres and geometrical shape or models are a tool for visalization of the abstract concept in mathematics. By means of these, hman reason sets p a relation between physical or external world and the abstract concepts (Konyaliogl 2003). It can be considered concepts sch as geometric strctres and mathematical-physical models for meaningfl teaching mathematics. Also, mathematical concepts are abstract that one needs highly cognitive achievements to assimilate them (Baki 2000). By sing visalization approach many mathematical concepts can become concrete and clear for stdents to nderstand. The term visalization is sed in different meaning between mathematics edcators. It is sed the paper as it was defined by Zazkis, Dbinsky

3 and Datermann(1996), that is, as an act in which an individal establishes a strong connection between an internal constrct and something to which access is gained throgh the sense. Sch a connection can be made in two direction. An act of visalization may consist of any mental constrction of object or processes that an individal associates with objects or events perceived by an external sorce. Alternatively, it may consist of the constrction, on some external medim sch as paper of objects or events. Conseqently, the act of visalization is translation from external to mental. Visalization can be alternative method and powerfl resorce for stdents doing mathematics, a resorce that can pon the way to different ways of thinking abot mathemathics than the lingistic and logico-propositional thinking of traditional and the symbol maniplation of traditional algebra (Konyaliogl et al. 2003). Use of the visalization approach provides stdents to look at mathematics corse, which was seen as a cmlation of abstract strctres and concepts from a different perspective. METHOD The stdents sed in this stdy were 60 sophomores enrolled in linear algebra corse designed for the profesional teaching of mathematics program. All stdents have had the same formal edcation in mathematics. They took calcls and set theory corse in the first year. Those corses did not inclde linear algebra content.the stdents were divided randomly into two grops consist of 30 stdents. All stdents were given basis knowledges deal with vector concept reqired for linear algebra.they were taght vector space concept by one instrctor for three one-hor lectres per week. In the process, vector space concept to experimental grop was presented in the two hors geometrically and one hor algebraically. Control grop was also presented in the two hors algebraically and one hor geometrically. At the end of for weeks, the two grops were given same test. The qestion in the test were chosen to be simple problems on the vector space concept, which can be solved directly by applying the vector space definition. Problem 1: Which of the following sets is a sbspace of vector space V? Give yor answer with explanition. a) W = { (x,y) R 2 : y = x + 1 } b) W = { (x,y) R 2 : y = x 2 }

4 Problem 2: W 1 and W 2 are non-trivial sbspaces of a vector space V. Is W 1 W 2 a sbspace of vector space V? Problem 3: Let V be a vector space and ω a fixed vector in V. If W is a set of all scalar mltiplications of ω, is W a sbspace? The stdents were asked to answers these qestions both algebraically and geometrically. It has been explanied V is eqal to R 2 or R 3 for geometric descriptions. Geometric description is the stdents answers involved only written geometric descriptions. Model answers for the three problems set in this stdy corsewerw given in Appendix. The responses which the stdents handed to the qestions have been sbmitted at following tables. On the end of for weekly corse process, two grops were asked to answer three qestions given above. Problem 2 and problem 3 reqire conceptal knowledge. The reslts analysed by SPSS packet program. The reslts are presented by percentages, freqencies and t-test is carried ot. Significance level was taken as p= FINDINGS AND DISCUSSION It is allowed to their correct and incorrect answers withot interesting in algebraic and geometric descriptions of the stdents at following table. Frthermore, it is clarified that the stdents cold not reply to the qestion on others colmn.. Table: The stdents general responses Problem Grop Correct Answer Incorrect Answer No response p-vale 1-a A %86,6 %6,7 %6,7 B %80,0 %13,3 %6,7 0,497 1-b A %80,0 %13,3 %6,7 B %73,3 %20,0 %6,7 0,549 2 A %63,3 %30,0 %6,7 B %26,7 %60,0 %13,3 0,004 3 A %86,6 %3,4 %10,0 B %53,3 %20,0 %26,7 0,004

5 Test reslt show that the stdents who were exposed to experimental grop stdents are,on average, more sccessfl than control grop stdents at the 0.05 significance level. As seen from Table, it was fond that the stdents in experimental grop were more sccesfl than the stdents in control grop withot regarding conceptal and procedral learning. Most of the stdents in both grop answered problem 1-a and problem 1-b correctly by sing only sbspace description. This high percentage of correct answers may be de to solving similar exercices in the teaching process in the classroom. Althogh a different settings of problem 1-a and problem 1-b were discssed dring the instrction, other problems were not discssed at any time dring the instrction. Stdents in experimental grop were fond to be more sccesfl than the stdents in control grop in solving the problem 2 and problem 3 which reqired more conceptal nderstanding. When two grops were compared, stdents responses in experimental grop inclded less nmber of incorrect answers than control grop stdents responses. Althogh there is not a meaningfl difference between procedral knowledge of stdents from both grops, there is a meaningfl difference between conceptal knowledge. In addition, stdents responses to problem 2 and problem 3 inclded less nmber of correct answers than the responses given to the other problems. Althogh problems 2 and problem 3 reqired more conceptal nderstanding dealing with sbspace concept, the sccess for these problems can be explained that stdents in experimental grop than stdents in control grop comprehend well intittionally. The distinction between concept definition and concept image spports or conclsion that differences between the achievements of two grops can be attribted to the different instrctional approaches sed. Althogh these concepts are given nder eqal circmtances to two grops, stdents performance in solving these problems are fond to be different. An explanation to this cold be that thogh these two grops received the same concept definition of vector space, they were exposed to different experiences which reslted in forming different concept image. Stdents from experimental grop had accomplished these problems becase they had well concept image deal with vector space. CONCLUSION It can be considered that geometrical strctres spported algebraically on the teaching of sbspaces increase in meaningfl learning of the stdents and help stdent s conceptal learning. In visalization approach stdents can perceive relations between abstract concepts and semi-concrete strctre and make sense of abstract concepts in mathematics, and ths this

6 approach facilitates stdent s nderstanding abstract concepts. Inclding visalization into the teaching process increases the stdents conceptal learning. It cold be sggested that the teaching method applied in this stdy cold be extended to teaching the other abstract concepts in mathematics. Visalization mst be sed both a tool and an aim in mathematics edcation. REFERENCES Baki, A., (1998): Balance of conceptal and procedral knowledge in mathematics edcation. In: Mathematics Symposim for the 40th Anniversary of Atatürk Univesity, Erzrm, Trkey, Special Nmber, pp Baykl, Y. (1999): Primary mathematics edcation. Ankara, Trkey, Ani Printing Press Bernardz.N.& Kieran.C.(1996): Approaches to Algebra:Persctives for Research and Teaching, Klwer Academic Pbl. Vinner.S. (1983): Concept definition, Concept Image and Notation of Fnction, International Jornal of Mathematics Edcation, Science and Technology, 14, pp Isleyen, T. & Isik, A.(2003). Conceptal learning in mathematics edcation. Jornal of the Korea Society of Mathematical Edcation Series D: Research in Mathematical Edcation, 7(2), pp Konyaliogl, A.C., Ipek, A.S. & Isik, A. (2003): On the teaching linear algebra at the University level: The role of visalization in the teaching vector spaces. Jornal of the Korea Society of Mathematical Edcation Series D: Research in Mathematical Edcation, 7(1), pp Konyaliogl, A.C. (2003): Investigation of Effectiveness of Visalization Approach on Understanding of Concepts in Vector Spaces at the University Level, Atatürk University, Gradate School of Natral and Applied Sciences, Department of Mathematics Edcation, Unpblished Doctoral Dissertion, Erzrm, Trkey. Kadijevic, D.(1999): Conceptal task in mathematics edcation. The Teaching of Mathematics, 2(1), pp Ma, L. (1999): Knowing and teaching elementary mathematics. Teachers nderstanding of fndamental mathematics in China and the United States. Mahwah, NJ: Erlbam. Noss, R.& Baki, A. (1996): Liberating school mayhematics from procedral view. Jornal of Hacettepe Edcation (Ankara) 12, pp Skemp, R.R. (1971): The psychology of learning mathematics. Middlesex, UK: Pengim Boks Ltd. Zazkis, R., Dbinsky, E., & Datermann, J.(1996): Coordinating visal and anayltic strategies: a stdy of stdents nderstanding. Jornal for Research in Mathematics Edcations, 27(4),

7 APPENDIX: Model answers for the three problems set. Problem 1: It has been explanied V is eqal to R 2 or R 3 for geometric descriptions. Geometric description (GD) is the stdents answers involved only written geometric descriptions, sch as: The set of W in problem 1-a is not a sbspace of R 2 has been geometrically shown in Figre-1. +v W W v c W (c>1) Figre 1 Any element of W set in problem 1-a is an ordered pair of the form (x, x+1). The set of W is not a sbspace of R 2 since it cannot obtain an ordered pair of the form (0,0) for no vale of x variable. + v W v Figre 2 The set W which is the set of ordered pair of the form (x, x 2 ) in Problem 1-b as a different in Problem 1-a contains the point (0,0) for x=0. Becase the set W is a sbspace of R 2, thogh it

8 contains the point (0,0) is necessary, it is not adeqate. The sm of any two elements in W is not in W. So W cannot a sbspace of R 2. Problem 2: W 1 and W 2 are two sbspaces of V. By chosening V=R 3, it can be considered as W 1 is the set of the points on the axis-x and W 2 is the set of the points on the plane-xy. Namely, W 1 = {(x, y, z) R 3 : x=0, y=0, z R} W 2 = {(x, y, z) R 3 : x, y R, z=0} That W 1 is a sbspace of V=R 3 is shown in Figre 3 and that W 2 is a sbspace of V=R 3 is shown Figre 4. v +v Figre 3 c +v v c Figre 4 The addition of two elements in W 1 and the mltiplication of an arbitrary element by scalar c are in the set. It can be shown in Figre 3. So, W 1 is a sbspace of V=R 3. The same sitation is valid for W 2 (Figre 4). Becase W 1 W 2 V=R 3, W 1 W 2 is formed from points on the axis-z and the plane-xy ( Figre 5).

9 v +v W 1 W 2 Figre 5 c Consider any two elements in W 1 W 2., v W 1 W 2 for W 1 and w W 2. As shown in Figre 5, + v W 1 W 2 for, v W 1 W 2. So V is not a sbspace of R 3. Problem 3: Let W be eqal to R 2. If w is any vector in V=R 2, then it can be shown in Figre 7. W w c 1 w (c 1 >1) w c 3 w (-1<c 3 <0) c 2 w (c 2 <1) Figre 6 Figre 7 W= {c i w: w= (x,y), x,y,c i R, i=1,2,3, } The all scalar mltiplication of W changes its orientation and magnitde, bt not change its direction. The scalar mltiplication of w by c i R is visalized in Figre 7. The explanation of Problem 3 is similar to other problems.