DETERMINATION OF MISCONCEPTIONS THAT ARE ENCOUNTERED BY TEACHER CANDIDATES AND SOLUTION PROPOSITIONS FOR RELIEVING OF THESE MISCONCEPTIONS

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1 DETERMINATION OF MISCONCEPTIONS THAT ARE ENCOUNTERED BY TEACHER CANDIDATES AND SOLUTION PROPOSITIONS FOR RELIEVING OF THESE MISCONCEPTIONS Dokuz Eylül Uiversity, Faculty Of Educatio Departmet Of Mathematics, Buca, IZMIR Cek KEŞAN, Deiz KAYA INTRODUCTION I order to thik, to iterpret ad judge correctly, the humas have to comprehed what they have leared. Cocepts are the abstract represetatives of the classificatios that are formed by objects, evets, ideas ad behaviors which have commo specificatios. (Fida, N., 1985). Cocepts reduce the complexity by simplifyig the eviromet that people live. They help us i defiig ad explaiig the objects i our eviromet. Learig cocepts caot be determied oly by classifyig objects or by tellig the ame or defiitio of a class of objects. Learig cocepts has three steps; iterpretig, traslatig ad trasitio. For a idividual to overcome these three steps, he has to have the ability to percept commo elemets objects, evets, ideas ad behaviors by abstractig them ad has to distiguish the commo ad ucommo sides of these. Glover, Roig ad Bruig (Glover,A.Roig,R.R.,Bruig, R. H., 1990) have divided the database eeded i problem solvig process ito three parts as subject kowledge, geeral ad specific strategic kowledge. The coceptual kowledge that is related to the subject of the problem costitutes the first type of kowledge. Rememberig a kowledge sigifies that it is kow. However, this remembrace ca both be by memorizig ad comprehedig. It is the comprehesio step, which is formed by the behaviors that distiguishes a perso who has compreheded, from a perso who has memorized. (Alka, H., Altu, M., 1998). As kowledge ad skill is ot trasferred geetically ad caot be cotrolled by istict, the behaviors that will be ecessary i life should be desigated to the perso. This ca be achieved by learig. The people agree that without learig life caot be successful. While teachig ew kowledge to idividuals, it should be made sure that the cocepts are formed correctly. I recet years, studets uderstadig of the scietific cocepts is oe of the subjects that researchers ad teachers give importace to. The reaso is that studets have difficulties i uderstadig scietific cocepts. Learig cocepts i a o-meaigful way leads to the formatio ad icreasig of miscoceptios. Miscoceptio is the perceptio of cocepts by studets i a differet way tha their scietifically accepted defiitios. The difficulties for studets that ca occur i learig cocepts ca also be related to; time, memory, strategies, cocetratio, culture, developmet ad isufficiecy of teachers. (Ülge,G.,1988). Miscoceptios are a big impedimet i meaigful learig. Especially, the permaet mistakes creates great difficulties for the math educatio to reach its goals if they are t avoided o time. Traditioal teachig techiques seem to be the major reaso i the occurrece of mistakes. (Lawso, A. E., Thompso, L. D & Ubuz, B., 1999 & Marek, E. A., Cowa, C. C., 1994). Studies have showed that, as miscoceptios are permaet ad cotiuous, ad at the same time they are ot sufficiet to make the studet develop the right cocepts, it is hard to relieve miscoceptios by traditioal teachig techiques. (Lawso, A. E., Thompso, L. D. 1988). The defiitio of miscoceptios ad distiguish these from isufficiet kowledge is a very importat area i educatioal research(teachig Physics, 1999). The reasos for miscoceptios ca be explaied as; lesso books, teacher factor ad ot havig iformatio of the studets past kowledge, ot preparig the appropriate eviromet, wrog usage of techological tools, ot makig the ecessary cocept exchagig i lessos. Seeig the missig part of the kowledge should be taught to the studets. Error-correctio operatio should take place. The questioaires which were made with differet teachers workig i private schools, govermet colleges, govermet high-schools showed that eve teachers have serious miscoceptios cocerig high-school mathematics. For this reaso, it is ecessary to determie teachers miscoceptios ad to search ways for relievig these. (Alka, H. ve arkadaşları, 1996 & Köroğlu, H. ve arkadaşları, 1996 & Alka, H. ve arkadaşları, 1995). 7

2 While their studies i educatioal faculties, by learig the ecessary kowledge, teories, basic priciples ad cocepts, teacher cadidates will be able make relatios with their kowledge ad practices i their workig areas. (YOK/Düya Bakası, 1997). This study is made uder the light of these thoughts ad is based o the results of the questios which were desiged for testig field kowledge. These questios were applied to the teacher cadidates which were attedig Applied Semiar classes held i the fial class of Uiversity of Dokuz Eylul (DEU) Educatioal Faculty of Buca (EFB) Mai Divisio of Secodary Sciece ad Mathematical Fields Educatio (DSSMFE) i the educatioal term of ad the attedees of certificatio program who were graduated DEU Faculty of Sciece ad Literature (FSL)i the educatioal term of **Dokuz Eylül Üiversitesi Eğitim Bilimler Estitüsü İlköğretim Matematik Eğitimi Tezli Yüksek Lisas Öğrecisi * Dokuz Eylül Üiversitesi Buca Eğitim Fakültesi İlköğretim Matematik A.B:D THE OBJECT AND LIMITATIONS OF THE RESEARCH The object of this research is, to evaluate the kowledge of the teacher cadidates which will graduate from Educatioal Faculty of Buca (DSSMFE) educatioal term ad certificatio program attedees which were graduated from Uiversity of Dokuz Eylul (DEU) Faculty of Literature Diviso of Mathematics ad to make cotributio to the studies that have bee made to determie the differet miscoceptios that these groups separately experieced, ad the miscoceptios that they commoly experieced. To achieve this object, aswers searched for the sub-problems below. 1- What are lack of kowledge ad miscoceptios of the graduates of EFB DSSMFE Mai Divisio of Mathematics ad FSL Divisio of Mathematics related to sequece cocept? - What are lack of kowledge ad miscoceptios of the graduates of EFB DSSMFE Mai Divisio of Mathematics ad FSL Divisio of Mathematics related to permutatio ad combiatio which are oe of the basic subjects of secodary level possibility cocept? 3- What are lack of kowledge ad miscoceptios related to logarithm? 4- What are lack of kowledge ad miscoceptios related to complex umbers? 5- What are lack of kowledge ad miscoceptios related to trigoometry, geometry ad applicatio of these to secod degree equatios. 6- What are lack of kowledge ad miscoceptios related to relatio ad fuctio? 7- What are lack of kowledge ad miscoceptios related to modular arithmetic ad sets? The data of the research is limited to the field kowledge of the teacher cadidates which will graduate from Educatioal Faculty of Buca (DSSMFE) educatioal term ad certificatio program attedees which were graduated from Uiversity of Dokuz Eylul (DEU) Faculty of Literature Diviso of Mathematics that has bee tried to be determied by a test. I the applied test a target ad a behavior has bee determied for each questio s right alterative ad misleadigs. Statistics I this research, iside the teacher cadidates which will graduate from Educatioal Faculty of Buca (DSSMFE) educatioal term ad certificatio program attedees which were graduated from Uiversity of Dokuz Eylul (DEU) Faculty of Literature Diviso of Mathematics, it is decided that a group of 104 idividuals has the quality to represet the other mathematics teachers. Limitatio Limitatios are give as below. a) The data of the research is limited to test aswers give by the teacher cadidates which will graduate from Educatioal Faculty of Buca (DSSMFE) educatioal term ad certificatio program attedees which were graduated from Uiversity of Dokuz Eylul (DEU) Faculty of Sciece ad Literature Diviso of Mathematics. b) Distributio of the test questios with respect to the subjects has bee based upo quizzes ad the test has bee limited to 5 questios. c) The validity of the evaluatio tool is limited to the time sectio. METHOD The research is directed towards revealig miscoceptios of the teacher cadidates that are graduated from Educatioal Faculties ad teacher cadidates who had atteded certificatio program ad graduated from Faculty of Sciece ad Literature, ad to preset suggestios that could prevet these mistakes. 8

3 Populatio The samplig of the research is formed by the teacher cadidates which will graduate from Educatioal Faculty of Buca (DSSMFE) educatioal term ad certificatio program attedees which were graduated from Uiversity of Dokuz Eylul (DEU) Faculty of Sciece ad Literature. I the samplig there are 49 teacher cadidates of Educatioal Faculty of Buca (DSSMFE) Divisio of Mathematics ad 55 certificatio program attedees graduated from Faculty of Sciece ad Literature. Data Collectio Tool ad Aalysis I the research, data collectio tool cosists of a test that is applied to 104 teacher cadidates. Five alteratives have bee determied for each questio, i each alterative differet behavior ad mistakes have bee searched. It is give as a fore coditio that four wrog aswers will disvalue oe right aswer. This coditio preveted the oes who aswered this test to aswer radomly because of iefficiet kowledge. I the applied test there were 5 questios related to field kowledge. I the evaluatio of the data, SPSS 8.0 pack software was used. Average, stadard deviatio, frequecy ad percetage display results were evaluated. The Object of Askig the Questios of the Data Collectio Tool ad the Target ad Behaviors That are Wated to be Measured. 1) I alterative A, this kowledge was searched. Does he/she kow that ifiity is t icluded i R? If he/she is t aware that ifiity is t icluded i R, he/she will thik that the result that was obtaied for belogs to the sequece. For alteratives B ad C, this kowledge was searched. If a N içi, 0 < a < b coditio is provided i the divisio sequece, the divisio sequece is b limited, a ad b.are coverget, the divisio sequece is also coverget. Otherwise it is ot proved. For alterative D, are they aware that limit may ot belog to the sequece? For alterative E, there should t be ay N that causes idefiicy for the rule that will be attaied to the sequece. ) For A ad B, to cofirm Each coverget sequece is limited but opposite is ot right. the alteratives C,D ad E were asked to measure the basic cocepts that determies the relatio betwee coverget sequeces ad mootoy. 3) It is asked to cofirm that a commo multiplier ca be egative i a geometric sequece ad the sequece will be coverget for r < 1 4) It is asked with the idea if a teacher cadidate could distiguish the theory: For a serial to be coverget, it is ecessary but ot eough to make the geeral term s limit go to zero. 5) It is searched if the teacher cadidates kow that ; R is the Cauchy sequece of every coverget sequece 6) It is searched how the sub-sequece of a sequece is obtaied, give that there is sub-sequece. 7) It is searched that, if a teacher cadidate will express a behavior of solvig the problem of a total of a serial that ca state covergece by movig from the S total of pieces. 8) It is asked to measure if it is kow that i a biomial expasio, the coefficiets of the terms that are i the equal distace from the startig poit ad the ed, also if they use the kowledge that the possibility of the complemetary of a evet is obtaied by subtractig the evet s possibility from 1. 9) To solve a questio, problem, a research it is ecessary to uderstad the questio completely. To provide this, the teacher cadidate should catch the lightig words i the questio. Is he/she is able to reach aother type of questio by leavig the type of questio that he/she decided before? It is asked to search how much he/she reflects the mistakes to the combiatio. 10) It is asked to measure if they are aware of a serious relatio betwee the set ad the possibility. 11) It is asked to measure if the teacher cadidates could exactly percept the differece betwee the arragig ad groupig. 1) It is asked to measure, if the teacher cadidates keep i mid that a umber which is possible to take the logarithm of, caot have a egative matis (decimal fractio). 13) It is asked to measure, their capacity to solve the logarithmic equality system ad if they kow that the base of the logarithm should be differet tha 1. 14) It is asked to measure, that the absolute of the differece betwee the two complex umbers is equal to the distace betwee the two poits that are equivalet to these complex umbers, ad if they are aware that whe oe is variable the there is a geometrical area problem or ot. 9

4 15) It is asked to measure, that the product of two complex umbers argumet is equal to its argumets total separately, ad if they could use their kowledge of secod degree equality together with the kowledge of complex umber. 16) It is asked to measure, if they kow that istead of Cartesia coordiates the polar coordiates ca be used to desigate the poit o a plae ad if they are aware of the dese relatio betwee trigoometry ad complex umbers. 17) It is asked to measure, if they kow that the relatio betwee the roots of a secod grade equality ca also be used i trigoometry. 18) ad 19) It is asked to measure if they are of the dese relatio betwee geometry ad trigoometry. 0) It is asked to measure the kowledge level of teacher cadidates that it is possible to obtai special results by the help of total, differece ad radius formulas i trigoometry, ad whe oe of the trigoometric rates is kow the the other ca be calculated. 1) It is asked to measure the how successfully they ca use these kowledge o a example: Why equality relatio? Why equality classificatio? What are the coditios of beig a equality relatio? ) It is asked to measure, if they ca produce alterative solutio steps for the ability of obtaiig a fuctio s opposite. 3) It is asked to see, how these kowledge are applied o a example: What is mode? Whe solvig a problem i modular arithmetic, how do you search its relatio with the umber cocept? 4) ad 5) It is asked to fid out, how much they are aware of the basic specificatios of a set. How frequetly they ca use these kowledge o the examples? EXAM QUESTIONS 1-Which oe of the below is wrog? A) 5 + sequece does t have the lowest elemet i) 15,53 ii) B) The divisio of two limited sequeces is also limited i) 30,53 ii)30,83 C) The divisio of two diverget sequeces ca be coverget I)0,53 ii)4-83 D)The limit of a positive termed sequece may ot be positive I) 0,53 ii) E).expressio caot be geeral term of a sequece I)15,53 ii) Which oe of the below expressios is wrog? A) A coverget sequece may ot be limited i)44,8-ii)61-76 B) A limited sequece may ot be coverget i)14,8-ii)3-76 C) A coverget sequece may ot be mootoous i)8,8-ii)08-76 D) A mootoous sequece may ot be coverget i)00,8-ii)00-76 E) I a mootoous icreasig upper limited sequece, the limit is the LUB i)14,8-ii) What is the limit of the S sequece with a geeral term of :S= 1-(0,4)+(0,4)-(0,4)3+...+(-1)-1.(0,4)-1? A)7/3 i)00,40-ii)08,45 B) 0,64 i)05,40-ii)08,45 C) 5/7 i)40,40-ii)3,45 D) 0 i)30,40-ii)3,45 E) 1 i)5,40-ii)0,45 4 Which oe of the below is diverget? + 3 A) i)38,53-ii)7-85 B) 1 =1 5 =13 i)11,53-ii)17-85 C) i)03,53-ii)04-85 D) i)35,53-ii)36-85 =1 ( + 1) = E) i)11,53-ii)14-85 =1 ( 1)( + 1) 5 Which oe of the below is ot a basic ( CAUCHY ) progressio? 1 A) i)0,0-ii)00,43 B) i)00,0-ii)16,43 C) ( + 1) 3 i)10,0-ii)16,43 D) cos + i)10,0-ii)9,43 3 E) ( + 1) i) 60,0-37, Which oe of the below is (a ) for the (a ) progressio that has a geeral term of; a = A) i)18,34-ii)13,56 B) 1 1 i)35,34-ii) 51,56 C) i)05,34-ii)03,56 D) 1 i)1,34-ii)10,56 ( ) 30

5 E) 1 i)30,34-ii)3,56 7-What is the total for x=1+x+3x +4x ? A) 9/4 i)79,7-ii) 5,41 B) /3 i)00,7-ii)00,41 C) 3/ i)07,7-ii)6,41 D) 4/9 i)07,7-ii)14,41 E)1/3 i)07,7-ii)08,41 8- I the expasio of (x+y)1 expressio the differet coefficiets formed are writte o differet cards ad put iside a he a card is radomly picked, what is the possibility of umber o the card to be higher that 1? A) /7 i)00,47-ii)04,45 B) 3/4 i)00,47-ii)00,45 C)9/13 i)70,47-ii)7,45 D) 5/7 i)30,47-ii)0,45 E)7/8 i)00,47-ii)04, teams are playig i a league, i each match oe is elimiated ad always wiig teams are playig. I this league, y matches ca be played at most? 3 A) 16 i)1,51-ii)06,56 B) i)05,51-ii)09,56 C) 96 i)00,51-ii)06,56 D) 6 i)00,51-ii)00,56 E) 31 i)84,51-ii)77, I a class of 40 studets, the umber of studets who passed Deutsch is 6, the umber of studets who passed matics is ad the umber of studets who passed both is 18. Whe you radomly call a studet from this class, if it is kow that he/she is failed mathematics, what is the possibility for him/her to be also failed from Deutsch? A) 7/10 i)13,44-ii)00,36 B) /5 i)00,44-ii)10,36 C) 5/9 i)69,44-ii)55,36 D) 4/9 i)18,44-ii)35,36 E) 1/3 i)00,44-ii)00, girl ad 4 boy studets had goe to a theatre. If the girls caot sit side by side, how may differet sittig combiatios ca be? A) 880 i)10,38-ii)18,9 B) 1440 i)1,38-ii)37,9 C)430 i)53,38-ii)31,9 D) 160 i)06,38-ii)00,9 E) 5040 i)11,38-ii)1,9 1- If log3=0,48 the what is log 3 0, 03? A),48 i)04,51-ii)00,35 B),493 i)00,51-ii)05,35 C) 1,506 i)48,51-ii)47,35 D),506 i)04,51-ii)15,35 E) 1,493 i)44,51-ii)31,35 13 If logx + logy = logx. logy ad log x y = the what is (x,y)? A) (1,1) i)10,96-ii)09,78 B) (, ) i)0,96-ii)00,78 C) ( 3 10,10) i)00,96-ii)06,78 D) (10 10,1000) i)80,96-ii)67,78 E) (10,100) i)06,96-ii)16, What ca be the highest base argumet of z umbers which proves the equality of z 4 i =? π 5π π π 5π A) i)06,3-ii)00,14 B) i)06,3-ii)1,14 C) i)56,3-ii)6,14 D) i)18,3-ii)6,14 E) i)06,3-ii)00, What are the total degree of base argumets of the equivalecy of z (6 i) z + 5 5i = 0? A) 45 0 i)00,1-ii)33,05 B) 10 0 i)16,1-ii)00,05 C) 10 0 i)00,1-ii)00,05 D) 40 0 i)00,1-ii)00,05 E) i)84,1-ii)67, What is the distace betwee the complex umbers give with their polar coordiates of z1 = (,780) ad z=(5,180)? A) 3 i)00,6-ii)00,09 B) 19 i)9,6-ii)60,09 C) 0 i)00,6-ii)00,09 D) i)00,6-ii)00,09 E)3 3 i)07,6-ii)40, I a ABC triagle, if ma<90 0 ad tab ad tac are the roots of x+3x-1=0 equality, the what is the measure of A? A) 45 0 i)100,0-ii)41,30 B) 15 0 i)00,0-ii)00,30 C) 30 0 i)00,0-ii)05,30 D) 60 0 i)00,0-ii)36,30 E) 75 0 i)00,0-ii)18, I a ABC triagle, if AC = BD = DC = L dir taα=m the what is taβ? A) m i)50,04-ii)00,11 B) m 1 C) 3 m i)00,04-ii)33,11 i)00,04-ii)50,11 31

6 D) E) m 1 3 m 3 i)50,04-ii)16,11 i)00,04-ii)00,11 α 19- If ta = 3 the what is Cosα? A) i)86,30-ii)90,36 B) 1 3 i)00,30-ii)00,36 C) i)06,30-ii)00,36 D) i)06,30-ii)05, E) i)00,30-ii)05,36 0- I a O cetered circle with 6 cm. radius, [PT is taget to the circle ad PT = 8.cm. If [ PA] [ AT] the what is the value of taα? A) 1 i)00,18-ii)11,16 B) 3 i)00,18-ii),16 C) 4 3 i)11,18-ii)00,16 D) 3 4 i)11,18-ii)11,16 E) 3 i)77,18-ii)55,16 1-Accordig to β= (x,y):4, divides x-y. relatio defied i the set of A= 0,1,,3,4,5,6,7,8,9,10,11,1, which oe of the below is ot i the equivalecy class of 0? A) 0 i)00,33-ii)03,55 B) 4 i)06,33-ii)03,55 C) 6 i)94,33-ii)90,55 D) 8 i)00,33-ii)00,55 E) 1 i)00,33-ii)03,55 -f:1-1 ad is a oto fuctio. What is f -1 (x) for the value of f(x) that provides the coditio of [ f ( x) ] 1 f ( x) + = 0? x x A) x x i)00,45-ii)00,7 B) + 1 i)77,45-ii)60,7 C) x 1 i)09,45-ii)0,7 D) x i)09,45-ii)0,7 x + x x + 1 x x E) i)04,45-ii)00,7 x How may whole umbers are there, that provides the equivalecy of x 3 4[ mod( x ) ]? A) 1 i)76,7-ii)48,4 B) i)00,7-ii)13,4 C) 3 i)15,7-ii)7,4 D) 4 i)07,7-ii)08,4 E) 5 i)00,7-ii)04,4 4- If s(a)=5, s(b)=6, s(c)= 9 ad if the highest value of s(a B C) is a, the lowest value is b ad the highest value of a b s(a B C) is c, the lowest value is d, the what is the rate of? c + d A) i)04,43-ii)0,6 B) i)04,43-ii)0,6 C) i)04,43-ii)08,6 D) i)86,43-ii)88,6 E) i)00,43-ii)00, For A,B,C sets, which oe of the propositios below is precisely right? A) ( A B = φ ve B C = φ) A C = φ dir i)07,80-ii)16,65 B) A B = A C B = C dir i)07,80-ii)08,65 C) A B = A C B = C dir i)33,80-ii)3,65 D) A B C C ( A B)dir i)49,80-ii)44,65 E) ( A B) = B dir A i)0,80-ii)00,65 Note: The i) aa,bb-ii)aa,bb symbols give i tables above sigifies these; The umbers comig after symbol i) are the percetages that are related to the aswers give by the fial class teacher cadidates at Educatioal Faculty of Buca, Divisio of Mathematics Teachig. 3

7 The umbers comig after symbol ii) are the percetages that are related to the aswers give by the teacher cadidates attedig Faculty of Sciece ad Literature, Divisio of Mathematics. aa symbol is the markig percetage of the related alteratives by teacher cadidates that aswered the related questio, bb symbol is the percetage of studets that aswered the related questio. The expressios writte i bold sigifies the correct aswers. FINDINGS For teacher cadidates attedig Educatioal Faculty of Buca, the rate of aswerig the questio that measures the cocepts related with sequeces was lower comparig to the graduates of the Faculty of Sciece ad Literature. For this reaso, it is determied that there is a real lack of kowledge cosiderig sequece subject. Despite this, it was oticed that the miscoceptios were same cosiderig the defied values of the Sequece ad that its value was ot at real umbers. They could t comprehed that two diverget sequeces divisio ca be coverget. The reaso for this is thought that the teacher cadidates could t achieve trasitio, which is oe of the steps of comprehesio. To be able to determie that the positive termed sequece s limit may ot be positive, it is eeded to be able to iterpret that zero is ot positive. For the graduates of Faculty of Sciece ad Literature, we feel that there is a serious lack of kowledge trasitio ad miscoceptio i determiig the rightess of two diverget sequeces divisio may be positive. Here, the trasformatio, which is oe of the steps of comprehedig i limitatio subject, could t be achieved. The teacher cadidates of DSSMFE i the Educatioal Faculty of Buca by ot agreeig that a limited sequece may ot be coverget ad that limit is LUB i a mootoous icreasig upper limited sequece, showed that due to the lack of iterpretatio they have mistakes i the subjects of limitatio ad covergece, mootoous icreasig ad upper limited. As the umber of teacher cadidates who teded that a coverget sequece may ot be mootoous was higher, it is clear that there is a miscoceptio related to the subject of mootoy. The two of groups has aswered the questio related to calculatig the limit of a sequece with its geeral term give almost i the same rate. Educatioal Faculty of Buca had a higher percetage of correct aswers tha the teacher cadidates of Faculty of Sciece ad Literature, had serious mistakes i the subject of partial total i determiig geeral term. I determiig diverget sequece the teacher cadidates of Faculty of Sciece ad Literature aswered the questios at a higher rate tha the teacher cadidates of Educatioal Faculty of Buca. Both of the two groups had right aswers i a lower rate. They had mistakes about covergece criteria. The umber of the oes who could t achieve trasitio, which is a step of comprehesio i the subject of covergece divergece, was ot low. I determiig the specificatios of Cauchy sequece the groups of Faculty of Sciece ad Literature ad Educatioal Faculty of Buca had difficulties almost at the same rate. I both of the groups, there were oes who did t use the expressio; Every coverget sequece is a Cauchy sequece, ad fell ito cotradictive side. They had mistakes i calculatig the geeral term of sequece, which is give as partial total, ad reachig to sub-sequece. Cosiderig this, the teacher cadidates of Educatioal Faculty of Buca had mistake more tha the teacher cadidates of Faculty of Sciece ad Literature. I calculatig a certai value of a serial expasio, the teacher cadidates of Educatioal Faculty of Buca has aswered i a lower rate tha the teacher cadidates of Faculty of Sciece ad Literature. However, the teacher cadidates of Educatioal Faculty of Buca had a higher rate i fidig the correct aswer. They could t achieve trasitio operatio, which is a step of comprehesio. I Biomial expasio, the terms after certai terms may have same coefficiets. The percetage of givig the right aswer was at the same rate. However, by stickig the cotradictio above or ot readig the questio carefully, the rate of persos givig wrog aswer was high. 33

8 Eve the rate of aswerig possibility questio was low i both groups, it was lower for the graduates of Faculty of Sciece ad Literature. The percetage of correct aswerig for the possibility questio which icluded sets was higher for Educatioal Faculty of Buca tha Faculty of Sciece ad Literature. However, we caot avoid the percetage of teacher cadidates who mixed with the coditio ad had mistake. The percetage of correct aswerig for the questio related to arragig was low for both two groups. A lack of iterpretatio was observed i subject of arragig. The percetage of aswerig the logarithm questio related to Matis ad Logarithm was higher for the group of Educatioal Faculty of Buca tha the group of Faculty of Sciece ad Literature. The percetage of correct aswerig was also higher for the group of Educatioal Faculty of Buca. They fell ito cotradictive side i the same ad uavoidable rate. The mistakes related to the operatios i logarithm was ot high. I both of two groups there was lack of kowledge related to the absolute value ad argumet of complex umber. But the percetage of correct aswerig was higher for the group of Faculty of Sciece ad Literature tha the Educatioal Faculty of Buca. The percetage of the oes who fell ito cotradictio because of wrogly determiig the complex umbers that supplied equality, caot be avoided. There is lack of kowledge i fidig the roots of equivalecies with complex varieties. However, the percetage of correct aswers was high. There is a very high lack of kowledge i polar displayig of the complex umbers. However, the percetage of aswerig this questio correctly was high. The group of Educatioal Faculty of Buca has give more correct aswers tha the group of Faculty of Sciece ad Literature. Cosiderig a triagle, there is lack of kowledge i usig trigoometric relatios for solvig equivalecies. The teacher cadidates of Educatioal Faculty of Buca has aswered 100% correctly but teacher cadidates of Faculty of Sciece ad Literature has aswered correctly at a lower rate. The rate of the oes who aswered the questio related to legth i a triagle was almost oe, so there is a high lack of kowledge. The rate of aswerig this questio correctly was 0% for the group of Educatioal Faculty of Buca. For the group of Faculty of Sciece ad Literature, it is 50%. Here, we ca see that there are serious miscoceptios related to geometry; trasitio, trasformatio or iterpretatio was ot made i these questios. The rate of aswerig the questio related to trigoometric idetity was low ad both of the two groups has give correct aswers at a higher rate. About the questio that demaded trigoometry ad circle kowledge, a lack of kowledge was observed i both of the two groups. The teacher cadidates of Educatioal Faculty of Buca (EFB) have give correct aswers at higher rate tha the teacher cadidates of Faculty of Sciece ad Literature (FSL). The EFB cadidates had a higher rate of aswerig the questio that demaded the equivalecy grade of a set accordig to relatio, but the rate for the oes who aswered correctly was high i both of the two groups. As the rate for aswerig the questio related to determie reverse fuctio was low, there is a lack of kowledge cosiderig this subject. The correct aswerig rate is high, but it is higher for the the group of Educatioal Faculty of Buca. Cosiderig the subject related to modular arithmetic, the group of Educatioal Faculty of Buca has more lack of kowledge tha the group of Faculty of Sciece ad Literature. However, the group of Educatioal Faculty of Buca has give more correct aswers. Miscoceptio is higher for FSL. Gorup of EFB has more lack of kowledge cosiderig the subject related to elemet umber of a set. However, both of the groups have aswered at the same ad a higher rate. The questio determiig the propositios related to the operatios i a set was aswered by both of the two groups at a higher rate, but the group of Educatioal Faculty of Buca had a higher rate of correct aswers. The correct aswers rate was low for the two groups almost at the same amout ad there is miscoceptio about this subject. Whe determiig the uity of two sets, eve oe set is evidet; it may ot be expressed i a uique type. The mistakig rate is high at this poit ad there is miscoceptio. DISCUSSION The teacher cadidates which will graduate from Educatioal Faculty of Buca (DSSMFE) educatioal term ad certificatio program attedees which were graduated from Uiversity of Dokuz Eylul (DEU) Faculty of Sciece ad Literature were tested i this research. 34

9 The research is orieted to determie the geeral kowledge levels related to the steps of iterpretatio, trasformatio ad trasitio of high school mathematics which these teacher cadidates will service i the future. It is evidet that the evaluatio tool, which was applied, was ot eough to measure all the dimesios of a subject. For each subject, the kowledge levels ad lack of kowledge of the teacher cadidates should be researched ad the ways of relievig these should be determied. Furthermore, these subjects should be researched ad the solutio steps should be determied for the teacher cadidates of Educatioal Faculty ad Faculty of Sciece ad Literature which will soo gai to the army of educatio. By the fidigs ad results obtaied from this study which the umeratios ad limitatios are determied of, we thik that it will make it ecessary to reach the decisios below. The available educatioal model is still based o straight expressio method; the priciple of explaiig the lesso books ad otes i a straight way. This meas that, educatio which is made by a passive method, caot achieve its duty (Gürol, M. 1997). It is very importat to make the teacher cadidates comprehed the basic poits while dealig with the sequeces, which are oe of the basic subjects of aalysis. It is ecessary to give a better defiitio of sequece ad teach the cocepts like limit of sequece, limitatios i sequeces, covergece, divergece, the lowest elemet, the highest elemet, i a careful way. Especially, i Educatioal Faculties ad Sciece ad Literature Faculties these cocepts should be hadled very carefully. I explaatio of the subjects, active educatio should be preferred. Geeral mathematics is the begiig poit of advaced mathematics ad the basic for other advaced subjects. The last twety years is the witess for the publicatios related to the uderstadig of geeral mathematics by teacher cadidates (Moagha, J.,1986). I sequeces, the relatio betwee covergece ad limitatio, mootoy ad covergece, mootoy ad limitatio ad the theories related to determiatio of limit by utilizig these properties should be examied very well. Especially i aalysis lessos, these cocepts should be discussed i a better way. What are the ecessary coditios of determiig the limit of a sequece ad how ca we determie the limit of a sequece? These poits should well be illumiated i lessos, homework should be give to the teacher cadidates if ecessary. These difficulties should be relieved by applyig differet teachig methods. The properties of the sequece give as a Cauchy sequece should be examied more carefully i aalysis lessos at Educatioal Faculties ad Sciece ad Literature Faculties ad coditios have to be determied very well. The theory of Every coverget sequece i R is a Cauchy sequece, every Cauchy sequece is coverget eeded to be very well examied. Cosiderig the determiatio of geeral term i sequeces ad how the sub- sequeces are obtaied, differet exercises should be made at aalysis lessos at EFB ad FSL by cosolidatig with examples. This subject is eeded to be isisted i lessos. Computer supported teachig, which icludes ad helps to uderstad these subjects, ca be made. I computer supported teachig, the teacher cadidates ca overcome mistakes by obtaiig the errors faster. I a computer supported teachig eviromet, the studets use the softwares iteractively, solve the problems step by step, by receivig errors lear their mistakes. I this maer, computer plays a bridge role which brigs studets kowledge ad talets ito foregroud (Baki, A., Budak, İ., 1999). The basic properties i the subject of determiig series total ad sequece cocept, which costitutes the base for itegral subject, eeded to be compreheded well. Otherwise, it is possible to ecouter importat problems i makig studets comprehed itegral ad other related subjects. Uderstadig possibility cocept etirely ca oly take place i the adolescece period (Piaget, J., Ihelder, B., 1975). I our coutry, possibility subject is both icluded i 8 th Class ad 10 th Class programs (Natioal Miistry of Educatio, 1988). Despite possibility subject is so importat, as i most of the foreig coutries, also i our coutry for various reasos these cocepts are t effectively educated. (Aksu,M., 1990 & Bar-O, E., Or-Bach, R., 1988 & Carpeter,T. P., Corbitt, M. K., Keper, H. S., Liquıst, M. M.,Reys, E. R., 1981 & Fischei, E., Scharch, D., 1997). Havig miscoceptios related to possibility subject, affects studets success (Fischei,E., Scharch, D.,1997). 35

10 Oe of the reasos for this, is that most of the mathematics teachers does t have the ecessary talet ad kowledge for teachig possibility subject effectively (Tobuk, Z., 1994). Beig oe of the basic subjects of possibility cocept, the Biomial expasio by isistig o its properties, we should make both two groups of teacher cadidates comprehed that i the expasio the terms, which are i same distace, have equal coefficiets. It is delighted if they ca apply combiatio i daily life. But this is ot eough. I calculatig coditioal possibility, it should be isisted that coditios are eeded to be determied well. The properties of the sets give i questios should be poited out well. Distiguishig poits should be brought ito foregroud. Details about the subject of permutatio s properties ad its meaig should be put forward. The subject should be explaied by may differet examples. I logarithm subject, properties of characteristic ad matis are geerally beig forgotte. These properties should be isisted more. A good level of success is observed o aalyzig logarithmic equivalecies. However, Educatioal Faculties are more successful. Also, Sciece ad Literature Faculties should attach importace to this subject. Aalogies that are formed by studets ot oly support studets to thik, but also help the teachers to see the right or wrog cocepts related to that cocept, which studets have i their mid (Wog, E. D., 1993). To uderstad a ew evet, aalogy supplies the prior segmeted kowledge to be used completely. Aalogies ot oly play a productive role i studets self learig ad educatio, but also have importat duties i solutio of wrog cocepts ad miscoceptios (Şahi, F., Gürdal,A.,Berkem,M.L.,000). We thik that, it is ecessary to make the studets comprehed the base argumet i complex umbers, iterpretatio of what ca it be at most or least. I complex umbered equivalecies, it is ecessary to be isisted o determiig the roots ad calculatig the base argumet. It is ecessary to attach importace to aalogy. As with other subjects, aalogy should also be practiced whe studyig other subjects. It should be isisted o givig differet expressios of complex umbers ad operatios that are ecessary for determiig the distace betwee two complex umbers. It is observed that this subject should be isisted carefully especially i Sciece ad Literature Faculties. The studets eed to have strategic kowledge o the subject. This kowledge should be trasferred from teacher to the studet before. Usig sius ad cosie terms whe writig all expressios related to taget ad cotaget, ca be give as a example for a this type of kowledge (Wilso,J.W., Feradez,M.L., Hadaway,N.,1999). For learig root totals ad root products of a secod-degree equivalecy ad solvig problems by combiig trigoometric rates with these ad likely properties, exercises should be made by both of the two groups. Models, which are a tool of material educatio, play a key role i iterpretig sciece ad help i uderstadig complex cocepts. Also supports studets to produce ew ad creative thoughts (Gilbert, K. J., Boulter, C., 1998). Accordig to Va Hiele the most importat reaso for the mistakes that studets make (Va Hiele, P. M., Va Hiele-Geldof, D., 1958) is the visio which is oe levels of geometric thikig levels. To make them able to use distace ad agle relatios, geometry lessos which iclude plae geometry, should also be give. It should be isisted more o trigoometric properties ad idetities. The taget ad vertical properties i circle should be determied well ad the sythetic examiatio of the circle should be made i a better way. While workig o these subjects, there should be a tool laboratory for mathematics ad as well as objects of plae geometry, objects of space geometry should be kept i there. As Corell also stated, Math educatio should be etertaiig ad iterestig. Whe studets are pleased i a math lesso which is equipped with projects, cocepts, shows ad likely activities, learig ad motivatio icrease i educatio(corell,c.,000). Both of the two groups have successfully achieved determiig equivalecy classes of a set accordig to a give relatio. The operatios for determiig fuctio ad reverse fuctio should be hadled more detailed by Sciece ad Literature Faculties. More studets should comprehed the operatios i modular arithmetic. More 36

11 exercises should be made o this subject. The lack of prelimiary ad talet about sets subject effects possibility cocept egatively (Bar-O, E., Or-Bach, R., 1988). A good level of success was observed o sets subject. However, some basic properties related to the operatios i sets was t determied completely by two groups. For this reaso, operatios i sets should be examied more carefully i lessos of Abstract Mathematics ad Aalysis. Lesso books are aother factor that causes miscoceptios. Lesso books should be ivestigated by this poit of view. It is agreed that Lesso Books ad Assistat Lesso Books have a importat part i educatio. This situatio bears the obligatio for arragig publicatios i a suitable way. Otherwise, oe of the compoets of educatio remais missig (Alka, H.ve arkadaşları, 1996). May differet researches show that Elemetary ad Secodary Level studets also have miscoceptios. For this reaso, miscoceptios of the teachers workig i Secodary Educatio Istitutios should be relieved by ier service courses, semiars ad publicatios. We thik that it is ecessary to coduct these kids of studies i Educatioal Faculties for teacher cadidates to be relieved from these miscoceptios. LITERATURE Aksu, M. Problem Areas Related to Statistics i Traiig Teachers of Mathematics i Turkey. A. Hawkis (Ed.), Traiig Teachers to Teach Statistics. Iteratioal Statistical Istitutio. Voorburg, ss:17-137, (1990). Alka, H., Altu, M. Matematik Öğretimi, Aadolu Üiv. Yayıları, No: 107, Eskişehir, (1998). Alka, H., Sezer, M., Özçelik, A.Z., Köroğlu, H. Matematik Öğretimide Yei Bir Model Yaklaşımı, II. Ulusal Eğitim Sempozyumu, Marmara Üiv. Eğt. Fak., 19-0 Eylül, Alka, H., Özçelik, A.Z., Köroğlu, H. Ülkemizde Uygulaa Matematik Öğretimide Görüle Yalışlıklar ve Temel Nedeleri, II. Ulusal Fe Bilimleri Eğitimi Sempozyumu, ODTÜ Eğt. Fak., Eylül, Alka, H., Sezer, M., Özçelik, A.Z., Köroğlu, H. Matematik Öğretimide Ölçme ve Değerledirmei Etkisi, II. Ulusal Eğitim Sempozyumu, Marmara Üiversitesi, Atatürk Eğitim Fakültesi, İstabul, Baki, A., Budak, İ. Excel yardımıyla yei bölüebilirlik kuralları taımlama, EBİD 99 Sempozyumu Kitapçığı, s.76-78, (1999). Bar-O, E. Ve Or-Bach, R. Programmig Mathematics: A New Approach i Itroducig Probability to Less Able Pupils, Joural of Mathematics Educatio i Sciece ad Techology, 19(): 81-97,(1988). Carpeter, T.P., Corbitt, M.K., Keper, H.S., Liquist, M.M. ve Reys, E.R. What are the Chaces of Your Studets Kowig Probability?, Mathematics Teacher, 73: , (1981). Corell, C. Matematikte Nefret Ediyorum, Yaşadıkça Eğitim, 65 Ocak-Mart: 15-, (Çev: Eyüboğlu, N.), (000). Gilbert, K. J., Boulter, C. Models i explaatios. Part 1: Horses for courses?, It. J. Sci. Educ. 0 (1), 83-97, (1998 a). Glover, A.,Roig, R. R., ve Bruig, R. H. Cogitive Psychology for Teachers, The Uited States Of America, Macmilla Publishig Compay, (1990). Gürol, M. Tekik Öğretme ve Adaylarıı, Tekik Öğretme Eğitimide Bilgisayar Kullaımıa ilişki Görüşleri, Eğitim ve Bilim Dergisi, Akara, (1997). Fida, N. Okulda Öğreme ve Öğretme, Aklım yayıları, sayfa: 189, Akara, (1985). Fischei, E. Ve Scharch, D. The Evolutio with Age of Probabilistic, Ituitively Based Miscoceptios, Educatioal Studies i Mathematics, 9: , (1997). Köroğlu, H., Albayrakoğlu, Kayser, S. Matematik Öğretimide Temel Kavramları Verilmeside Karşılaşıla Güçlükler ve Giderilme Yolları, II. Ulusal Eğitim Sempozyumu, Marmara Üiv. Eğt. Fak., 19-0 Eylül, Lawso, A.E. ad Thompso, L.D. Formal reasig ability ad miscoceptios cocerig geetics ad atural selectio, Joural of Research i Sciece Teachig, Vol. 5: , (1988). Marek, E.A. ad Cowa, C.C. ad Cavallo, A.M.L. Studets miscoceptios about difusio: How ca they be elimiated, The America Biology Teacher, Vol, 56: 74-77, (1994). Milli Eğitim Bakalığı, XII. Milli eğitim şurası kararları, Tebliğler Dergisi, 51, (74), (1988). Moagha, J. Adolecet s Uderstadig of Limits ad ifiity, Upublished Ph.D. Thesis, Uiversity of Warwick, (1986). Piaget, J. Ve Ihelder, B. The Origi of the Idea of Chace i Chidre. New York, W.W. Norto Şirketi, (1975). Şahi, F., Gürdal, A., Berkem, M. L., Fizyolojik Kavramları Alamlı Öğreilmesi ile İlgili Bir Araştırma, IV. Fe Bilimleri Eğitimi Kogresi 000, Hacettepe Üiversitesi, Eylül, (000). Teachig Physics, Volume 34, umber 5,, pp:94, September, (1999). 37

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