Evaluating the Acquisition of the Van-Hiele Levels with the COG Defuzzification Technique L 1

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1 Research Article SCIFED Publishers Michael Gr. Voskoglou,, 08, : SciFed Journal of Artificial Intelligence Open Access Evaluating the Acquisition of the Van-Hiele Levels with the COG Defuzzification Technique * Michael Gr. Voskoglou * Graduate Technological Educational Institute of Western Greece in Patras, Greece Abstract The Centre of Gravity (COG) technique, one of the most popular defuzzification methods in fuzzy mathematics, suggests that a situation s fuzzy data can be represented by the coordinates of the COG of the area included between the graph of the membership function involved and the X-axis. In this paper a special form of the COG technique, developed in earlier works for use as an assessment method, is applied for assessing the acquisition of the van Hiele levels of geometric reasoning. Similar methods applied in the past on the van Hiele level theory are discussed and an example is presented illustrating our results in practice. The COG technique is also compared with a traditional assessment method of the bi-valued logic, the Grade Point Average (GPA) index Keywords Euclidean Geometry; Van Hiele (vh) Levels of Geometric Reasoning; Mathematics Education; Human Cognition; Mathematical Thinking; Fuzzy Sets; Fuzzy Logic (FL); Centre of Gravity (COG) Defuzzification Technique; Fuzzy Assessment Methods; Grade Point Average (GPA) Index Introduction It is generally accepted that the pedagogical value of the Euclidean geometry is great, mainly because it cultivates the student cognitive skills and connects directly mathematics to the real world. However, students face many difficulties for the learning of the Euclidean geometry, which fluctuate from the understanding of the space to the development of geometric reasoning and the ability of constructing the proofs and solutions of several geometric propositions and problems. The van Hiele (vh) theory of geometric reasoning [, ] suggests that students can progress through five levels of increasing structural complexity. A higher level contains all knowledge of any lower level and some additional knowledge which is not explicit at the lower levels, Therefore, each level appears as a meta-theory of the previous one [3]. The five vh levels are: L (Visualization): Students perceive the geometric figures as entities according to their appearance, without explicit regard to their properties. L (Analysis): Students establish the properties of geometric figures by means of an informal analysis of their component parts and begin to recognize them by their properties. L3 (Abstraction): Students relate the properties of figures, they distinguish between the necessity and sufficiency of a set of properties in determining a concept and they form abstract definitions. * Corresponding author: Michael Gr. Voskoglou, Graduate Technological Educational Institute of Western Greece in Patras, Greece. mvosk@ hol.gr; Tel: Received June 5, 08; Accepted July 3, 08; Published August 06, 08 Citation: Michael Gr. Voskoglou (08) Evaluating the Acquisition of the Van-Hiele Levels with the COG Defuzzification Technique. SF J Artificial Intel Copyright: 08 Michael Gr. Voskoglou. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. page of 8

2 Citation: Michael Gr. Voskoglou (08) Evaluating the Acquisition of the Van-Hiele Levels with the COG Defuzzification Technique. L4 (Deduction): Students reason formally within the context of a geometric system and they grasp the significance of deduction as means of developing geometric theory. L5 (Rigor): Students understand the foundations of geometry and can compare geometric systems based on different axioms. Obviously the level L 5 is very difficult, if not impossible, to appear in secondary classrooms, while level L4 also appears very rarely. Although van Hiele [] claimed that the above levels are discrete which means that the transition from a level to the next one does not happen gradually but all at once other researchers [4-8] suggest that the vh levels are continuously characterized by transitions between the adjacent levels. This means that, from the teacher s point of view, there exists fuzziness about the degree of student acquisition of each vh level. Therefore, principles of Fuzzy Logic (FL) could be used for the assessment of student geometric reasoning skills. Perdikaris [9] presented a fuzzy framework for the vh theory by introducing a finite absorbing Markov chain [0] on the fuzzy linguistic labels (characterizations) of no, low, intermediate, high and complete acquisition respectively of each vh level. He also used [] the fuzzy possibilities of student profiles and the generalized possibility theory of the classical Shannon s entropy for measuring a system s uncertainty in order to compare the intelligence of student groups by means of the vh level theory. However, this method needs laborious calculations in its final application. The novelty of this paper is that an alternative method is utilized for the assessment of student acquisition of the vh levels, the Center of Gravity (COG) defuzzification technique. The COG method, in contrast to the measurement of the uncertainty which evaluates the mean performance, focuses on the student quality performance and it is very simple in its final application, while its outcomes can be easily interpreted. The rest of the paper is formulated as follows: In Section the utilization of the COG technique as an assessment method is sketched, while in Section 3 an example is presented illustrating its applicability in the vh theory. In Section 4 the COG technique is compared to the Grade Point Average (GPA) index, a classical assessment method of the bi-valued logic. The last Section 5 is devoted to the final conclusions and to some hints for future research. For general facts on fuzzy sets and logic we refer to the book [].. The COG Defuzzification Technique as an Assessment Method It is well known that the solution of a problem in terms of FL involves in general the following steps: Selection of the universal set U of the discourse. Fuzzification of the problem s data by defining the proper membership functions. Evaluation of the fuzzy data by applying rules and principles of FL to obtain a unique fuzzy set, which determines the required solution. Defuzzification of the final outcomes in order to apply the solution found in terms of FL to the original, real world problem. One of the most popular in fuzzy mathematics defuzzification methods is the Centre of Gravity (COG) technique [3]. In applying the COG technique, assume that A= {( X, M(X)) : X U} is the fuzzy set determining the problem s solution. In order to construct the graph of the membership function y = m(x) one has to relate to each X U an interval of values from a prefixed numerical distribution, which actually means that the universal set U is replaced with a set of real intervals. There is a commonly used in FL approach to represent the system s fuzzy data by the coordinates ( Xc, Y c) of the COG, say F c, of the area F (e.g. see [3]). The COG coordinates are calculated by the following well-known [4] from Mechanics formulas: x c F F =, y c = () dxdy dxdy F xdxdy F ydxdy page of 8

3 Citation: Michael Gr. Voskoglou (08) Evaluating the Acquisition of the Van-Hiele Levels with the COG Defuzzification Technique. Subbotin et al. [5], based on a Voskoglou s [6] fuzzy model for the process of learning a subject matter in the classroom, adapted the COG technique for use as an assessment method of student learning skills. Since then, Subbotin and Voskoglou, working either jointly, or individually improved and used the COG technique as an assessment method in several other human activities (e.g. see [7-0]). This method is sketched below for the needs of the present work: Let G be a group of individuals (or of any other objects; e.g. CBR systems []. We choose as set of the discourse the set U = {A, B, C, D, F} of the fuzzy linguistic labels (grades) of excellent (A), very good (B), good (C), fair (D) and unsatisfactory (F) performance respectively of the group s members. When a numerical value, say y, is assigned to a group s member (e.g. a mark in case of a student), then its performance is characterized by F, if Y [0, ], by D, if Y [, ], by C, if Y [, 3], by B if Y [3, 4] and by A if Y [4,5] respectively. Consequently, we have that Y = m( x) = m( F) for all x in [0,), Y = m( x) = m( D) for all x in [,), Y3 = mx ( ) = mc ( ) for all x in [, 3), Y4 = m( x) = m( B) for all x in [3, 4) and Y5 = m( x) = m( A) for all x in [4,5]. Therefore, the graph of the membership function y = m(x) takes the form of Figure, where the area of the level s section F contained between the graph and the X-axis is equal to the sum of the areas of the rectangles S i, i=,, 3, 4, 5. Then, by calculating the double integrals in formulas () (e.g. see Section 3 of [9]), it is straightforward to check that ( ) ( y ) + 3y + 5y3 + 7y4 + 9 y5,y c = y + y + y3 + y4 + y5 () With,, 3, 4, 5 x = F x = D x = C x = B x = A j= ( ) m xi and yi =, i =,,3, 4,5. 5 Figure m x ( j ) Figure : The Graph of the COG Method y m(b) s 4 m(d) 5 then mx ( ) = (00%) i= s m(f) s m(c) s 3 m(a) O F D C B A x Note that the membership function y = m(x) can be defined, as it happens with fuzzy sets in general, in any compatible to the common logic way, according to the user s choice. In this work we define y = m(x) in terms of the frequencies. For example, if n is the total number of the members of G and n A is the number of the members of G whose performance was characterized by the label A, we set na yt =, etc n i Using elementary algebraic inequalities one finds that the minimal value of the y-coordinate corresponds to 5 the COG F M =, 0 ([9], Section 3). Also, using x c and y c given in () it is straightforward to observe that the s 5 etc. page 3 of 8

4 Citation: Michael Gr. Voskoglou (08) Evaluating the Acquisition of the Van-Hiele Levels with the COG Defuzzification Technique. group s ideal performance ( y5 = and yi = 0 for i 5) corresponds to the GOG F 9,, while its worst performance corresponds to the GOG F W,. There fore the COG lies in the area of the triangle FMFF W of Figure. Figure : The Area where the COG Lies Making elementary geometric observations on Figure it is straightforward to obtain the following assessment criterion: Between two groups, the group with the greater value of x c performs better. If two groups have the same xc.5, then the group with the higher y c performs better. If two groups have the sam x c <.5, then the group with the lower y c performs better. As it becomes evident from the above criterion, a group s performance depends mainly on the value of the x-coordinate of the COG, which is calculated by the first of formulas (). In this formula, greater coefficients (weights) are assigned to the higher grades. Therefore, the COG method focuses on the group s quality performance. 9 Since in case of the ideal performance the first of formulas () gives that x c =, values of x c greater than half of the above value, i.e., greater than 9 =.5, could be considered as demonstrating a more than satisfactory group s performance. 4 Remark: Since in the above adaptation of the COG defuzzification technique as an assessment method the graph of the membership function takes the form of Figure, where the area of the level s section contained between the graph and the X-axis is equal to the sum of the areas of five rectangles, this method was called Rectangular Fuzzy Assessment Model (RFAM). Subbotin in an effort to confront better the ambiguous assessment scores being at the boundaries between two successive assessment grades (e.g. something like 84-85% being at the boundaries between A and B) developed several variations of the RFAM by changing the shape of the corresponding membership function s graph; e.g. see [, 3] for the Trapezoidal Fuzzy Assessment Model, [4] for the Generalized RFAM, etc. However, all these variations have been finally proved to be equivalent to the original RFAM [5]. 3. An Application of the COG Technique to the Van Hiele Level Theory Gutierrez et al. [7] presented a paradigm for evaluating the acquisition of the vh levels in three dimensional Geometry by three different groups, say G, G and G 3, of 0, and 9 students respectively. Here, in order to illustrate the applicability of the COG technique as an assessment method in the vh level theory we shall use the data of this page 4 of 8

5 Citation: Michael Gr. Voskoglou (08) Evaluating the Acquisition of the Van-Hiele Levels with the COG Defuzzification Technique. paradigm, which are depicted in Table. Table : Data of the Paradigm (Degree of Acquisition) Group vh level F D C B A G L G L G L G L G L G L G3 L 0 4 G3 L G3 L From the first row of Table one finds that y5 = and y = y= y3 = y4 = 0 for LinG. Therefore, the first of equations 9 () gives that x c = = 4.5. Similarly, from the fourth row of Table, one finds that y = y = 0, y 8 3 =, y4 = and y5 = Therefore xc = for LinG. In the same way one finds that the values of x c are.5 for L in G 3, 3.7, 3.07,.39 for L in G, G, G 3 respectively and.75,.4, 0.5 for L 3 in G, G, G 3 respectively. Therefore, according to the first case of the assessment criterion of Section, G demonstrated the best performance in the first three vh levels, followed by G and G 3. Also, on comparing the above values with.5, one concludes that G demonstrated a more than satisfactory performance in these three levels, while G demonstrated a more than satisfactory performance in the first two levels and G 3 demonstrated a more than satisfactory performance only in the first level. Finally, since the mean value = , (demonstrated a more than satisfactory overall performance with respect to the first three vh levels. In the same way, since the corresponding mean values for G and G3are.93 and.46 respectively, G demonstrated also a more than satisfactory overall performance, while the performance of G3 was not satisfactory. 4. Comparison of the COG Technique with the GPA Index The assessment methods which are commonly used in practice are based on principles of the bi-valued logic. The calculation of the mean value of the scores assigned to each one of its members is the classical method for assessing the mean performance of a group of objects with respect to an action. On the other hand, a very popular in the USA and other Western countries assessment method is the calculation of the Grade Point Average (GPA) index [6]. This index is a weighted average in which greater coefficients (weights) are assigned to the higher scores. Therefore, it is connected to the quality group s performance. 0nF + nd + nc + 3nB + 4nA The GPA, is calculated by the formula GPA = (3) n, where n is the total number page 5 of 8

6 Citation: Michael Gr. Voskoglou (08) Evaluating the Acquisition of the Van-Hiele Levels with the COG Defuzzification Technique. of the group s members and n A, n B, n c, n D and n F denote the numbers of the group s members that demonstrated excellent (A), very good (B), good (C), fair (D) and unsatisfactory (F) performance respectively [6]. In case of the worst performance ( n F = n ) formula (3) gives that GPA = 0, while in case of the ideal performance ( n A = n ) it gives that GPA = 4. Therefore we have in general that 0 GPA 4. Consequently, values of GPA greater than could be considered as indicating a more than satisfactory performance. Formula (3) can be also written in terms of the frequencies defined in Section in the form GPA= y + y3 + 3y4 + 4 y5 (4). Therefore, in order to find the numerical relation between GPA and the x-coordinate of the COG technique, we write the first of formulas () of Section as x c= ( y + y3 + 3y4 + 8y5) + ( y+ y + y3 + y4 + y5) = ( GPA + ) or finally in the form xc = GPA (5). Consider now two groups, say G and G with GPA values GPA and GPA and values of the x-coordinate of their COG x c and x c respectively. Then, if GPA > GPA, equation (5) gives that xc > xc. Also, if GPA>, then by (5) we get that x c >.5. Therefore, by applying the first case of the assessment criterion of Section, one concludes that in this case the GPA index and the COG technique provide always the same assessment outcomes. However, if GPA = GPA, equation (5) gives that xc = xc. Therefore, in this case one of the last two cases of the assessment criterion of Section is applicable, which means that the GPA index and the COG technique provide different assessment outcomes on comparing the performance of the two groups. The following simple example [4], paragraph (vii) of Section 4) shows that in case of the same GPA values the application of the GPA index could not lead to logically based conclusions. Therefore, in such situations, the assessment criterion of Section 3 becomes useful due to its logical nature. Example: The student grades of two Classes with 60 students in each Class are presented in Table. Table : Student Grades The GPA index for the two classes is equal to *0 + 4*50 3*0 + = 4* which means that the two Classes demonstrate the same performance in terms of the GPA index. 5 c c i i= 5 6 Also, equation (5) gives that x = x = > But y = + = for the first and yi = + = for the second Class. Therefore, according to the second case of the assessment criterion of i= Grades Class I Class II C 0 0 B 0 0 A Section, the first Class demonstrates a better performance in terms of the COG technique. Now which one of the above two conclusions is closer to the reality? For answering this question, let us consider the quality of knowledge, i.e., the ratio of the students received B or better to the total number of students, which is equal to 5 6 for the first and for the second Class. Therefore, from the common point of view, the situation in Class II is better. However, many educators could prefer the situation in Class I having a greater number of excellent students. Conclusively, in no case it is logical to accept that the two Classes demonstrated the same performance, as the calculation of the GPA index suggests. page 6 of 8

7 Citation: Michael Gr. Voskoglou (08) Evaluating the Acquisition of the Van-Hiele Levels with the COG Defuzzification Technique. 5. Conclusion and Hints for Future Research In this work we applied the COG defuzzification technique as an assessment method for the vh level theory. The COG technique provides the same assessment outcomes with the GPA index, unless if the groups under assessment have the same value of the GPA index. In the last case the application of the GPA method could not lead to logically based conclusions. Therefore, in this case the COG technique is useful due to its logical nature. Other fuzzy assessment methods have been also used in earlier author s works like the measurement of a system s uncertainty [7] and the application of the fuzzy numbers [8]. These methods, in contrast to the COG technique and the GPA index which focus on the corresponding group s quality performance, measure the group s mean performance. Our plans for future research include the effort of comparing the outcomes of all these methods in order to obtain the analogous conclusions. References. Van Hiele PM, Van Hiele-Geldov D (958) Report on Methods of Initiation into Geometry. In H. Freudenthal (Ed.), Groningen, The Netherlands: JB. Wolters Van Hiele PM (986) Structure and Insight, Academic Press, New York. 3. Freudenthal H (973) Mathematics as an Educational Task. 4. Burger WP, Shaughnessy JM (986) Characterization of the vam Hiele Levels of Development in Geometry. Journal for Research in Mathematics Education 7: Fuys D, Geddes D, Tischler R (988) The van Hiele Model of Thinking in Geometry among Adolescents. Journal for Research in Mathematics Education Monograph Wilson M (990) Measuring a van Hiele Geometric Sequence: A Reanalysis. J Res Math Edu : Gutierrez A, Jaine A, Fortuny JK (99) An Alternative Paradigm to Evaluate the Acquisition of the van Hiele Levels. J Res Math Edu : Perdikaris SC (0) Using Fuzzy Sets to Determine the Continuity of the van Hiele Levels. J Math Sci and Math Edu 6: Perdikaris SC (996) A Systems Framework for Fuzzy Sets in the van Hiele Theory of Geometric Reasoning. International J Math Edu Sci and Tech 7: Kemeny JG, Snell JL (976) Finite Markov Chains. Springer.. Perdikaris SC (0) Using Possibilities to Compare the Intelligence of Student Groups in the van Hiele Level Theory. Journal of Mathematical Sciences and Mathematics Education 7: Klir GJ, Folger TA (988) Fuzzy Sets, Uncertainty and Information, Prentice-Hall. 3. Van Broekhoven E, De Baets B (006) Fast and accurate centre of gravity defuzzification of fuzzy system outputs defined on trapezoidal fuzzy partitions. Fuzzy Sets and Systems. 57: Center of mass: Definition. Wikipedia, Igor Ya. Subbotin, Badkoobehi H, Bilotckii NN (004) Application of fuzzy logic to learning assessment. Didactics of Mathematics, Problems and Investigations. : Voskoglou MGr (999) An Application of Fuzzy Sets to the Process of Learning. Heuristics and Didactics of Exact Sciences. 0: 9-3. page 7 of 8

8 Citation: Michael Gr. Voskoglou (08) Evaluating the Acquisition of the Van-Hiele Levels with the COG Defuzzification Technique. 7. Igor Ya. Subbotin, Mossavar-Rahmani F, Bilotckii NN (006) Fuzzy logic and iterative assessment. Didactics of Mathematics: Problems and Investigations 5: Voskoglou MGr, Igor Ya. Subbotin (03) Dealing with the Fuzziness of Human Reasoning. International Journal of Applications of Fuzzy Sets and Artificial Intelligence 3: Voskoglou MGr (0) A Study on Fuzzy Systems. Amer J Comp App Math : Voskoglou MGr (04) Assessing the Players Performance in the Game of Bridge. Amer J App Math Stat : Voskoglou MGr (03) Case-Based Reasoning in Computers and Human Cognition: A Mathematical Framework. International Journal of Machine Intelligence and Sensory Signal Processing : 3-.. Subbotin, IYa (04) Trapezoidal Fuzzy Logic Model for Learning Assessment, arxiv (math.gm). 3. Voskoglou M (04) A New Fuzzy Method for Assessing Human Skills, Progress in Nonlinear Dynamics and Chaos : Subbotin IYa, Voskoglou MGr (06) An Application of the Generalized Rectangular Model to Critical Thinking Assessment. Amer J Edu Res 4: Voskoglou MGr (06) Comparison of the COG Defuzzification Techniques and its Variations to the GPA Index. American Amer J Comp App Math 6: Swinburne University of Technology,Grade Point Average Assessment Voskoglou MGr (0) Stochastic and Fuzzy Models in Mathematics Education, Artificial Intelligence and Management, Lambert Academic Publishing, Saarbrucken, Germany. 8. Voskoglou MGr (05) Defuzzification of Fuzzy Numbers for Student Assessment. American Journal of Applied Mathematics and Statistic 3: Citation: Michael Gr. Voskoglou (08) Evaluating the Acquisition of the Van-Hiele Levels with the COG Defuzzification Technique. page 8 of 8