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: THE AXIS OF REFLECTIVE SYMMETRY AS REPRESENTATION IN MATHEMATICS LEARNING
Athanasios Gagatsis*, Areti Panaoura**, Iliada Elia*, Nicolaos Stamboulidis***, Panayiotis Spyrou***
* Department of Education, University of Cyprus, Cyprus
** Department of Education, Frederick University, Cyprus
*** Department of Mathematics, University of Athens, Greece
The present study explores students constructed definitions for the concept of the axis of reflection for a function in relation to their abilities in dealing with tasks involving different forms of representations and problem solving tasks. A major concern is also to examine interrelations between these three ways of thinking dealing with the concept of the axis of reflection. Findings revealed students difficulties in giving a proper definition for the concept and resolving tasks involving conversions between diverse modes of representations (algebraic, graphical, tabular). Several inconsistencies among students constructed definitions, their competence to use different representations of the axis of reflection and their problem solving ability were also uncovered, indicating lack of flexibility between different ways of approaching the concept.
INTRODUCTION
Mathematics, as a human creation dealing with objects and entities relies heavily on visualization in its different forms and at different levels, far beyond the obviously visual field of geometry and spatial visualization. Nowadays, visualization is being recognized as a key component of reasoning, problem solving and proving (Arcavi, 2003). Although, the mental processes on mathematics learning have received extensive research in the field of mathematics education (Presmeg, 1992), there are still many issues concerning visualization and the use of different representations in mathematics education, which require further investigation.
The term representation is defined as the concept that includes the represented world to elements of the representing world and a process that uses the information in the representing world (Markmann, 1999). In the field of mathematics teaching and learning visual representations play an important role as a means in communicating mathematical ideas (Elia, Gagatsis, & Deliyianni, 2005). Besides recognizing the same concept in multiple systems of representations, the ability to manipulate the concept within these representations as well as the ability to translate the concept from one system of representation to another are necessary for the acquisition of the concept.
The concept of function is of fundamental importance in the learning of mathematics (Eisenberg, 1992; Dubinsky & Harel, 1992). The present study concentrated on the axis of reflection for a function and the understanding on the concept in relation to the definition and the transformation from the geometric representation to the algebraic one and vice versa. The question is whether the understanding of the definition plays a central role in transferring the knowledge from the algebraic representation to graphic and vice versa. Symmetry is an important concept in mathematics learning. Principles and Standards for School Mathematics (NCTM, 2000) sets symmetry as one of the significant geometry concepts. Symmetry is repeatedly stressed grade after grade in school curriculum. Besides, it has been applied to other mathematics strands as well as many other areas including physics.
Mathematics instructors traditionally have focused their instruction on the use of algebraic representations with the intention to avoid confusion between mathematical objects and their representations. This is because they think that the algebraic difficulties in the construction of concepts are linked to the restriction of representations when teaching.
The concept of function in mathematics education research
The notion of function is a most important one for mathematics (Mesa, 2004) and it is of fundamental importance in the learning of mathematics. A substantial number of research studies have examined the role of different representations on the understanding and interpretation of functions (Gagatsis & Shiakalli, 2004). The concept of function admits a variety of representations and consequently has the capability of being taught using diverse representations, each of which offers information about particular aspects of the concept without being able to describe it completely.
Akkoc and Tall (2002) investigated the representational complexity involved in different representations (set diagrams, ordered pairs, graphs, formulae) in respect to the concept of function. Much of the literature regarding the teaching and learning of functions refers to the learners image of function (Sfard, 1994; Thompson, 2000). Vinner and Dreyfus (1989) found that a learner might know the formal definition of function yet not fully be able to apply it. Hence from an instructional point of view, before a teacher can expect learners to use and apply functions accurately, he/she must assist in accurately developing their concept image to encompass the definition.
The reflective symmetry in mathematics
Rosen (1995) indicates that symmetry is immunity to a possible change (p.2). This change is considered as a transformation of a geometrical object that preserves the geometrical properties of the object. Leikin, Berman and Zaslavsky (1997) extending the previous definition assume that symmetry involves three elements: a geometrical object, its properties and a transformation. A reflection is like a mirror image. The line of reflection acts as the mirror and is halfway between the point and its image. If the point is its own reflection, then it is a point on the line of reflection. Symmetry is more a geometrical than algebraic concept, but the subject of symmetry does come up in a couple of algebraic contexts. For instance, when you are graphing quadratics, you may be asked for the parabolas axis of symmetry. This is usually just the vertical line x=h, where h is the xcoordinate of the vertex (h,k). That is a parabolas axis of symmetry is usually just the vertical line through its vertex.
Until the beginning of secondary school, symmetry is only the reflection through a line which is the usual conception too. Then pupils learn the reflection through a point, and then they learn translation and rotation (Bulf, 2007). According to the NCTM (2000) Standards in grades 912 all students should use Cartesian coordinates and other coordinate systems, such as navigational, polar or spherical systems, to analyze geometric situations, they should understand and represent translations, reflections, rotations, and dilations of objects in the plane by using sketches, coordinates, vectors, function notation and matrices and they should use various representations to help understand the effects of simple transformations and their compositions.
Hoyles and Healy (1997) suggest that success in reflective symmetry is accomplished by giving students the opportunity to view things from a different perspective. They analyze the processes through which students come to negotiate mathematical meanings for reflective symmetry and they describe a microworld, Turtle Mirrors, designed to provide tools to help students focus simultaneously on actions, visual relationships and symbolic representations. At the same framework, Healy (2004) investigated how learning systems in which learners knowledge about reflection become connected to the institutional knowledge about geometrical transformations that they are intended to learn in school mathematics.
The study of Son (2006) examined how preservice teachers understand reflective symmetry and what types of pedagogical strategies they use to help a student who has misunderstanding of reflective symmetry. It was found that a large portion of them have limited understanding of reflective symmetry. It was also revealed that preservice teachers have tendency to rely on procedural aspects of reflective symmetry when helping a student understand reflective symmetry correctly although they recognized a students misconceptions in terms of conceptual aspects.
Although there are a lot of studies dealing with students conceptions of reflective symmetry and their difficulties in understanding the concept there is more to be uncovered and explored with the combination of the main concern of these studies: concept definition and different representations of function (algebraic, graphical, tabular) with an axis of reflection. Limited attention has been given to interrelations among students concept image of reflective symmetry and the use of different representations of the mathematical concept. The present study aimed to explore students definition of the axis of reflection for a function and their abilities to identify the concept in different modes of representation and transfer from one mode of representation to another. Secondly it aimed to investigate the relationships between students definitions of the axis of reflection, their ability to handle different representations of the concept and their problem solving performance.
METHOD
Participants
Data were collected from 139 students attending the second class of the lyceum (1617 years old) from two high schools.
Research Instrument and Variables
A test which consisted of nine tasks was developed and administered to the students. Below we give a brief description of the test (see Appendix) and the corresponding codification of the variables used for the analysis of the data.
The first task asked students to explain When is a line an axis of reflection for a function . It actually asked them to provide a definition. A correct definition was coded as D1 , an ambiguous definition was coded as D2 and an incorrect definition or the lack of any definition was coded as D3 . We wanted to investigate the degree of understanding the concept by connecting different information they were taught and their ability to express it verbally. At the second task they were asked to choose the correct sentence for the function y=f(x) with domain R and the axis yy as an axis of reflection. This task would be an indication whether they had understood practically the definition. At the third task students had to indicate the coordinates of the symmetrical point A (2,5) for different axes (xx, yy, the line y=x, the line x=1 and the line y=1). The variables that were used were S1, S2, S3, S4 and S5 with values 0 for wrong answers and 1 for the correct ones. The purpose of the fourth task was to investigate students ability to recognize a function with an axis of reflection at a graphic representation. A figure with a part of a function with the axis yy as an axis of reflection was presented and students had to recognize and choose the correct answer for the rest part of the function. The variables were Cga, Cgb, Cgc, Cgd. At a similar context at the sixth task students had to recognize the axis of reflection at four different functions when those are presented graphically (Rga, Rgb, Rgc and Rgd).
At the fifth task students were asked to write the algebraic expression (Ex) of any function which has an axis of reflection. At the seventh task they had to recognize from four functions presented algebraically those with axis of reflection the axis yy (Raa, Rab, Rac and Rad). At the eighth task the tables of the values of four functions were presented and students had to recognize those which corresponded at functions with an axis of reflection (Rta, Rtb, Rtc and Rtd). Finally at the ninth task a problem was presented. Students had to find out the values for a,b and c for the parabola y = ax EMBED Equation.3 +bx+c, with a `" 0 if it cuts the axis xx at the point (3,0) the axis yy at the point (0,3) and its acme (which is on the axis of reflection) has a x intercept x=1. The variable which was used was Pr with the following values: 1 for a correct solution, 0.75 if they used correctly the two point of intercepts without giving a correct solution, 0.5 if they used correctly two of the three data, 0.25 if they used correctly one of the three data and 0 for a wrong solution.
Statistical Analysis
Primarily, the success percentages were tallied for the tasks of the test by using SPSS. For the analysis of the collected data, the similarity statistical method was carried out using a computer software called C.H.I.C. (Classification Hirarchique, Implicative et Cohsitive), Version 3.5. (Bodin, Coutourier, & Gras, 2000).
RESULTS
Success Percentages
Firstly the percentages of success on the tasks of the test are presented (see Table 1).
Table 1: Success Percentages of the Tasks of the Test
TasksVariablesPercentages (%)
Task 1D1 (correct)6 D2 (ambiguous)34D3 (wrong)60Task 2Da18Db40Dc (correct)36Dd15Task 3S150S249S315S424S511Task 4Cga 40Cgb22Cgc (correct)45Cgd3Task 5Ex30Task 6Rga43Rgb (correct)75Rgc4Rgd (correct)58Task 7Raa (correct)29Rab25Rac35Rad (correct)16Task 8Rta39Rtb (correct)49Rtc (correct)24Rtd (correct)38Task 9Pr1.4
Only 6% of the students presented a correct definition indicating their difficulties in understanding the concept and in expressing verbally the definition. At the second task with the practical definition of the concept , only 36% answered correctly indicating that they did not understand the relation between the algebraic representations and the axis yy as an axis of reflection for the function (the sum is above 100% because many students had chosen more than one answers).
At the third task the highest percentages of correct answers were in the case of the point of reflection regarding the axis xx (S1) and yy (S2). The highest percentage at the forth task (45%) was about the correct answer. Nevertheless 40% of the pupils have chosen the answer a where the rest of the graph was symmetrical regarding the axis of reflection xx, indicating that they had not realized the differences concerning the different axis of reflection or they confused the concepts of function and relation. The students responses at the sixth task, of the recognition of the axis of reflection indicated that 75% had chosen correctly the second choice and 58% the forth one. Nevertheless 43% of students had chosen the first graph which had not an axis of reflection but a point of reflection which was the center of the coordinates. At the same time the percentage of students who had chosen correctly both the b and d graphs were 40.3% and only 14.3% had chosen the first graph as well. In general from the abovementioned results we can say that students succeeded in tasks where they had to recognize the vertical axis of symmetry for functions which are represented graphically. On the contrary results indicated that students did not recognize the functions with axis of reflection the yy represented in algebraic forms. The percentage of students who had chosen correctly the first choice (parabola) was 29% and the forth choice was only 16%. At the eight task, although many students had chosen correctly the b, c or d answers, 39% of them had chosen the first table. The students performance at the last task, regarding the problem solving procedure was disappointing. Only 14.4% succeeded in using the data of the point of the parabola algebraically and only 0.72% (1 student) used the data for the axis of reflection.
Crosstabs analysis allowed us to investigate students performance to different tasks more analytically in relation to the definitions they proposed. Results indicated that 37.5% of the students who proposed a correct definition had chosen the correct definition Dc, as well. At the same time, 27.7% of the students who gave an ambiguous definition (D2) had chosen the Dc and 40.5% of the students who gave an incorrect definition (D3) had chosen the Dc. All the students (100%) who gave a correct definition solved correctly the task S1, 87.5% of them the task S2, 25% the task S3, 50% the task S4 and 25% the task S5. Even though the percentages are greater than the respective for the total of students (Table 1) they behaved similarly, indicating that the first two items were easier. The purpose of the forth task was to investigate students ability to recognize a function with an axis of reflection at a graphic representation. The percentage of correct solution (Cgc) of the students who proposed a correct definition was 50%. High percentages of students who gave correct definitions succeeded at recognizing the correct answers at the sixth task. Specifically 74.8% of them had chosen Rgb and 75% of them the Rgd. At the seventh task they had to recognize from four functions presented algebraically those with axis of reflection the axis yy. Only 50% of the students who gave a correct definition had chosen Raa and 12.5% of them the Rad. Finally, at the eight task the tables of the values of four functions were presented and students had to recognize those which corresponded at functions with an axis of reflection (Rtb, Rtc and Rtd). Only 50% of the students who gave a correct definition had chosen Rtb, 25 % of them the Rtc and 12.5% of the students the Rtd.
Results Based on the Similarity Diagram
The similarity analysis (Lerman, 1981) is a classification method which aims to identify in a set V of variables, thicker and thicker partitions of V, established in an ascending manner. When fitted together these partitions are presented in a hierarchically constructed diagram (tree) using a similarity statistical criterion among the variables. The similarity is defined by the crosscomparison between a group V of the variables and a group E of the individuals (or objects). This kind of analysis allows the researcher to study and interpret in terms of typology and decreasing similarity, clusters of variables which are established at particular levels of the diagram and can be opposed to others, in the same levels.
Figure 1 illustrates the similarity diagram of students responses to the tasks of the test. Five clusters of tasks were identified in the similarity diagram of Figure 1. The first cluster (G1) involves the finding of symmetrical points according to an axis of reflection (S1, S2, S3, S4, S5) with the problem solving performance (Pr). This group of tasks is related to the group of ambiguous definition (D2, Dd) and finally with the group Ex and Rad. Important is the relation with the correct definition D1. This cluster is considered to be conceptual because it consisted of two correct definitions (D1 and Dc) with the correct finding of symmetrical points at graphs and the problem solving ability, indicating that they can handle correctly the algebraic expressions of the concept. In contrast, the fifth cluster (G5) involves the incorrect identification of a function with an axis of reflection presented graphically (Rgc) and algebraically (Rac) and the incorrect choices for the construction of a part of a graph for a function with an axis of reflection yy. In the same context is the third cluster (G3) of tasks which involves incorrect choices in graphical representation and in tables (Rga, Cgb, Rta) with the presentation of an incorrect definition (D3).
The second (G2) and the forth (G4) clusters presented a similar mixed performance. Specifically at the second cluster the incorrect definition Da is related with the correct choices Cgc, Rtd and Rgb. This is an indication that there is a group of students who are able to recognize functions with an axis of reflection, presented graphically or as a table without being able to define correctly by using a verbal expression for the concept of the axis of reflection. Finally the forth cluster involves an incorrect choice of a definition (Db) with the correct recognition of a choice at a table representation (Rtc), the incorrect Rab (algebraic representation) and the correct Rgb.
Figure1: Similarity diagram of the tasks of the test according to students responses
To sum up, the above connections in the similarity diagram and more specifically the task separation according to the mode of representation indicate that different types of function representation with an axis of reflection were approached in a distinct and inconsistent way. Students who define correctly (or even ambiguously) the concept of the axis of reflection or choose a correct (or ambiguous) definition, succeed at the same time in solving the problem correctly, they find out correctly the points which are symmetrical to given points in regard to different axis of reflection and they recognize correctly the algebraic expression of function with axis of reflection. The conceptual understanding of the concept is related with its algebraic representation. At the same time the understanding of the concept by using other forms of representations (tabular, graphically) seems to be depended on surface features of the representation.
DISCUSSION
Findings revealed students difficulties in giving a proper definition for the concept of axis of reflection and resolve problems of functions involving conversions between diverse modes of representation. This result is consistent with the finding of Elia et al. (2007) that revealed pupils difficulties in tasks and problems of functions that required the connection and relations between different representations.
The close similarity relation between students correct definitions and recognition of the axis of reflection in graphs indicates that students were more likely to apply their definition in the graphical mode of representation rather than in the other modes of representation. This behavior may be interpreted as a product of the didactic approaches. General lack of similarity connections among tasks involving different representations indicated students incompetence in flexibly handling different modes of representation, which is a main feature of the compartmentatization phenomenon (Elia et al., 2007). This inconsistent behavior can be seen as an indication of students conception that different representation of an axis of reflection which means different representations of a function are distinct and autonomous mathematical objects and not just different ways of expressing the meaning of the particular notion. At the teaching of the concept important is the understanding of the definition and its algebraic expression in order to support the conceptual understanding.
Even though few students could give a correct definition of the axis of reflection for a function they were not essentially in a position to identify the axis of reflections for functions, which were represented in different representational forms. The proportion of pupils who gave an accurate definition for the concept but did not provide a correct solution to other tasks also evidenced this. Similarity connections between students incorrect definitions and success in dealing with recognition tasks indicated that similar inconsistencies also hold for students who have inaccurate conceptions.
The findings of this study allowed us to describe students ability in giving a definition of the reflective symmetry for a function, recognizing the axis of reflection for functions in different forms of representation and investigate their interrelations. On the basis of students behavior, the aforementioned types of mathematical understanding seemed to give different information for the acquisition of the concept and were not necessarily consistent with each other. The above inference has direct implications for future research as regards the teaching practice of function and the axis of reflection for a function.
References
Akkoc, H. & Tall, D. (2002). The simplicity, complexity and complication of the function concept. In A. Cockburn & E. Nardi (Eds). Proceedings of the 26th conference of the international group for the psychology in mathematics education, V. 2, 2532.
Arcavi, A. (2003). The role of visual representations in the learning of mathematics. Educational Studies in Mathematics, 52, 215241.
Bodin, A., Coutourier, R., & Gras, R. (2000). CHIC : Classification Hierarchique Implicative et CohesitiveVersion sous Windows CHIC 1.2. Rennes: Association pour le Recherche en Didactique des Mathematiques.
Bulf, C. (2007). The use of everyday objects and situations in teaching mathematics: the symmetry case in French teaching geometry. Proceedings of the CERME 5. Larnaka.
Dubinsky, E., & Harel, G. (1992). The nature of the process conception of function. In E. Dubinsky & G. Harel (Eds.), The Concept of Function. Aspects of Epistemology and Pedagogy (pp. 85106). United States: The Mathematical Association of America.
Eisenberg, T. (1992). On the development of a sense for functions. In E. Dubinsky & G. Harel (Eds.), The concept of function. Aspects of epistemology and pedagogy (pp. 153174). The Mathematical Association of America.
Elia, I., Panaoura, A., Eracleous, A. & Gagatsis, A. (2007). Relations between secondary pupils conceptions about functions and problem solving in different representations. International Journal of Science and Mathematics Education, 5, 533556.
Elia, I., Gagatsis, A. & Deliyianni, E., (2005). A review of the effects of different modes of representations in mathematical problem solving. In: A. Gagatsis, F. Spagnolo, Gr. Makrides and V. Farmaki (Eds) Proceedings of the 4th Mediterranean Conference on Mathematics Education, Vol.1, pp. 271286. (Palermo, Italy: University of Palermo, Cyprus Mathematical Society).
Gagatsis, A., & Shiakalli, M. (2004). Ability to translate from one representation of the concept of function to another and mathematical problem solving. Educational Psychology, 24(5), 645657.
Healy, H. (2004). The role of tool and teacher mediations in the construction of meanings for reflection. In M.van den. Heuvel Panhuizen (Eds) Proceedings of the 28th conference of the international group for the psychology of mathematics education. V. 3 (pp. 3340).
Hoyles, C. & Healy, L. (1997). Unfolding Meanings for Reflective symmetry. International Journal of Computers for Mathematical Learning, 2 (1), 2759.
Leikin, R., Berman, A. & Zaslavsky, O. (1997). Defining and Understanding Symmetry. In E. Pehkonen (Ed.), Proceedings of the 21st International Conference for the Psychology of Mathematics Education: Vol. 3J^uvno# #%+++f,g,,(ps`.01266::<<<<긴괮긤}njI
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