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Moreover the connections between the different modes of representations influence. mathematical learning and strengthen students ability in using mathematical. concepts of functions Romberg Fennema Carpenter 1993 p iii Knuth 2000. indicated that students have difficulties in making the connections between different. representations of functions formulas graphs diagrams and word descriptions in. interpreting graphs and manipulating symbols related to functions. The theoretical perspective used in the present study is related to a dimension of the. framework developed by Moschkovich Schoenfeld Arcavi 1993 According to. this dimension there are two fundamentally different perspectives from which a. function is viewed The process perspective and the object perspective have been. described From the process perspective a function is perceived of as linking x and y. values For each value of x the function has a corresponding y value Students who. view functions under this perspective could substitute a value for x into an equation. and calculate the resulting value for y or could find pairs of values for x and y to draw. a graph In contrast from the object perspective a function or relation and any of its. representations are thought of as entities for example algebraically as members of. parameterized classes or in the plane as graphs that are thought of as being picked. up whole and rotated or translated Moschkovich et al 1993 Students who view. functions under this perspective could recognize that equations of lines with the form. y 3x b are parallel or could draw these lines without calculations if they have. already drawn one line or they can fill a table of values for two functions e g f x. 2x g x 2x 2 using the relationship between them e g g x f x 2 Knuth. 2000 The algebraic approach is relatively more effective in making salient the. nature of the function as a process while the geometric approach is relatively more. effective in making salient the nature of function as an object Yerushalmy. Schwartz 1993, Sfard 1992 has argued that the ability of seeing a function as an object is. indispensable for deep understanding of mathematics Furthermore developing. competency with functions means moving towards the object perspective and. graphical representation Moschkovich et al 1993 Students being able to profitably. employ object perspective can achieve a deep and coherent understanding in. functions We believe that this point to point approach that is the algebraic approach. gives students only a mere and local image of the concept of function On the. contrary the geometric approach gives students a global approach of the concept of. functions so that students who can manipulate and use it will perform better in. solving complex geometric problems, The purpose in this study is to contribute to the mathematics educational research. community s understanding of the algebraic and geometric approach students. develop and use in solving function tasks and to examine which approach is more. correlated with students ability in solving complex geometric problems. 3 386 PME28 2004, The analysis was based on data collected from 95 sophomore pre service teachers. enrolled in a basic algebra course during the spring semester in 2003 The subjects. were for the most part students of high academic performance admitted to the. University of Cyprus on the basis of competitive examination scores Nevertheless. there are big differences among them concerning their mathematical abilities. The instruments used in this study were two tests The first consisted of four tasks. implementing simple tasks with functions In each task there were two linear or. quadratic functions Both functions were in algebraic form and one of them was also. in graphical representation There was always a relation between the two functions. e g f x x g x x 2 Students were asked to interpret graphically the second. function The second test consisted of two problems The first problem consisted of. textual information about a tank containing an initial amount of petrol and a tank car. filling the tank with petrol Students were asked to use the information to draw the. graphs of the two linear functions and to find when the amounts of petrol in the tank. and in the car would be equal The second problem consisted of a function in a. general form f x ax bx c Numbers a b and c were real numbers and the f x was. equal to 4 when x 2 and f x was equal to 6 when x 7 Students were asked to find. how many real solutions the equation ax bx c had and explain their answer The. tests were administered to students by researchers in a 60 minute session during. algebra course, The results concerning students answers to the above tasks and problems were. codified in three ways a A was used to represent algebraic approach function. as a process to tasks and problems b G stands for students who adopted a. geometric approach function as an entity c The symbol W was used for. coding wrong answers A solution was coded as algebraic if students did not use. the information provided by the graph of the first function and they proceeded. constructing the graph of the second function by finding pairs of values for x and y. On the contrary a solution was coded as graphical if students observed and used. the relation between the two functions e g g x f x 2 in constructing the graph. of g x This paper is focused on the first two types of responses Moreover. implicative analysis Gras Peter Briand Philipp 1997 was used in order to. identify the relations among the possible responses of students in the tasks and. problems Therefore twelve different variables representing the algebraic and. geometric approaches emerged More specifically the following symbols were used. to represent the solutions involved in the study a Symbols T1A T2A T3A. and T4A represent a correct algebraic approach to the tasks and P1A to the first. problem second problem could not be solved algebraically b Similar symbols. T1G T2G T3G and T4G represent a correct geometric approach to the tasks. and P1G and P2G correct graphical solutions to the problems respectively. For the analysis and processing of the data collected implicative statistical analysis. was conducted using the statistical software CHIC A similarity diagram and a. PME28 2004 3 387, hierarchical tree were therefore produced The similarity diagram represents groups.
of variables which are based on the similarity of students responses to these. variables The hierarchical tree shows the implications A B between sets of. variables This means that success in A implies success in B Gras et al 1997. The main purpose of the present study was to examine the mode of approach students. use in solving simple tasks in functions and to test which approach is more correlated. with solving complex geometric problems Most of the students correctly solved the. tasks involved linear functions T1 and T2 Their achievement radically reduced in. tasks involved quadratic functions T3 and T4 and especially in solving complex. geometric problems only 27 4 and 11 6 of the 95 subjects correctly provided. appropriate solutions Table 1 More than 60 of the students that provided a. correct solution chose an algebraic approach even in situations in which a geometric. approach seemed easier and more efficient than the algebraic Furthermore in the. second problem most of the students failed to recognize or suggest a graphical. solution as an option at all,Tasks and Problems,T1 T2 T3 T4 P1 P2. Geometric approach,24 2 23 2 19 21 16 11 6,with correct answer. Algebraic approach,59 56 8 22 1 24 3 11 4 0,with correct answer. Incorrect Answer 16 8 20 58 9 54 7 72 6 88 4,Table 1 Students responses to tasks and problems. Numbers represent percentages,T1 T4 Tasks in functions P1 P2 Geometric problems.
Students correct responses to the tasks and problems can be classified according to. the approach they used and they are presented in the similarity diagram in Figure 1. More specifically two clusters i e groups of variables can be distinctively. identified The first cluster consists of the variables T1A T2A T4A and. T3A which represent the use of the algebraic approach process perspective The. second cluster consists of the variables T1G T2G P1G P2G P1A. T3G and T4G and refers to the use of the geometric approach object. perspective and solving geometric problems The emergence of the two clusters is in. line with the assumption of the study and reveals that students tend to solve tasks and. 3 388 PME28 2004, problems in functions using the same approach even though in tasks that a different. approach is more suitable, It can also be observed from the similarity diagram that the second cluster includes. the variables correspond to the solution of the complex geometric problems with the. variables representing the geometric approach,1st cluster 2nd cluster. Figure 1 Similarity diagram of the variables, Note Similarities presented with bold lines are important at significant level 99. More specifically students geometric approach to simple tasks in functions is. closely related with their effectiveness in solving complex geometric problems This. close connection may indicate that students who can use effectively graphical. representations are able to observe the connections and relations in geometric. problems and are more capable in problem solving It is also important to. acknowledge that almost all of the similarities in the second cluster are statistically. significant at level 99 and this refers to the geometric approach and the complex. geometric problem solving, Significant implicative relations between the variables can be observed in the.
hierarchical tree which is illustrated in Figure 2 First three groups of implicative. relationships can be identified The first group and the third group of implicative. relations refer to variables concerning the use of the geometric approach object. perspective and variables concerning solution to the geometric problems The second. group provides support to the existence of a link among variables concerning the use. of algebraic solution process perspective This finding is in line with the findings. emerged from the similarity diagram The formation of these groups of links. PME28 2004 3 389, indicates once again the consistency that characterizes students provided solutions. towards the function tasks and problems Second the implicative relationship P2G. P1G T2G and T1G indicates that students who solved the second problem by. applying the correct graphical solution have implied the application of the object. perspective graphical representation for the other problem and the four tasks An. explanation is that students who have a solid and coherent understanding of. functions can identify relations and links in complex geometric problems and thus. can make the necessary connections between pairs of equations and their graphs and. easily apply geometric approach in solving simple tasks in functions. 1st group 2nd group 3rd group, Figure 2 Hierarchical tree illustrating implicative relations among the. Note The implicative relationships in bold colour are significant at a level of 99. DISCUSSION, It is important to know whether pre service teachers are flexible in using algebraic. and geometric representations in function problems Although problems used in this. study are some of those taught at school subjects had difficulties especially when. they needed to implement geometric approach Many students have not mastered. even the fundamentals of the geometric approach in the domain of functions. Students understanding is limited to the use of algebraic representations and. approach while the use of graphical representations is fundamental in solving. geometric problems,3 390 PME28 2004, The most important finding of the present study is that for the group of pre service. teachers two distinct sub groups are formatted with consistency the algebraic and the. geometric approach group The majority of students work with functions is restricted. to the domain of algebraic approach and this process is followed with consistency in. all tasks As a consequence few students develop ability to flexibly employ and. select graphical representations thus geometric approach The present study is in line. with the results of previous studies indicating that students can not use effectively the. geometric approach which engenders within the object perspective Knuth 2000. The fact that most of the students chose an algebraic approach process perspective. and also demonstrated consistency in their selection of the algebraic approach even in. tasks and problems in which the geometric approach object perspective seemed. more efficient or that they failed to suggest a graphical approach at all is particularly. distressful considering that the students participated in the study thought to be. representative of our best students, Moreover an important finding is the relation between the graphical approach and.
geometric problem solving This finding is consistent with the results of previous. studies Knuth 2000 Moschkovich et al 1993 indicating that geometric approach. enables students to manipulate functions as an entity and thus students are capable to. find the connections and relations between the different representations involved in. problems The data presented here suggest that students who have a coherent. ALGEBRAIC AND GEOMETRIC APPROACH IN FUNCTION PROBLEM SOLVING Nikos Mousoulides and Athanasios Gagatsis Department of Education University of Cyprus This study explores students algebraic and geometric approach in solving tasks in functions and the relation of these approaches with complex geometric problem solving Data were obtained from 95 sophomore pre service teachers enrolled in a

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