FUZZY MEASURES FOR STUDENTS MATHEMATICAL MODELLING SKILLS

Transcription

1 Iteratoal Joural of uzzy Logc Systems (IJLS) Vol., No., Aprl 0 UZZY MEASURES OR STUDENTS MATHEMATICAL MODELLING SKILLS Mchael Gr. Voskoglou School of Techologcal Applcatos Graduate Techologcal Educatoal Isttute, Patras, Greece mvosk@hol.gr, voskoglou@tepat.gr ABSTRACT MM s oe of the cetral deas the owadays mathematcs educato. I a earler paper applyg deas from fuzzy logc we have developed a model formalzg the MM process ad we have used the total possblstc ucertaty as a measure of studets MM capactes. I the preset paper we develop two alteratve fuzzy measures for MM. The frst of them cocers a adaptato for use a fuzzy evromet of the well kow Shao s formula for measurg a system s probablstc ucertaty. The secod oe s based o the dea of the ceter of mass of the represeted a fuzzy set fgure, that s commoly used fuzzy logc approach to measure performace. The above (three total) fuzzy measures for MM are compared to each other ad a classroom expermet preseted our earler paper s recosdered here llustratg our results practce. KEYWORDS Mathematcal Modellg, uzzy Sets ad Logc, Possblty, Ucertaty, Ceter of Mass.. INTRODUCTION Before the 970 s Mathematcal Modellg (MM) used to be a tool hads of the scetsts workg maly Idustry, Costructos, Egeerg, Physcs, Ecoomcs, Operatos Research, ad other postve ad appled sceces. The frst who descrbed the process of MM such a way that could be used teachg mathematcs was Pollak ICME-3 (Karlsruhe, 976). Pollak represeted the teracto betwee mathematcs ad real world wth a scheme, whch s kow as the crcle of modellg [6]. Sce the much effort has bee placed by researchers ad mathematcs educators to develop detaled models for aalyzg the process of MM as a teachg method of mathematcs ([], []. [3], [9], etc). I all these models t s accepted geeral (wth mor varatos) that the ma stages of the MM process volve: Aalyss of the gve real world problem,.e. uderstadg the statemet ad recogzg lmtatos, restrctos ad requremets of the real system. Mathematzg,.e. formulato of the real stuato such a way that t wll be ready for mathematcal treatmet, ad costructo of the model. Soluto of the model, acheved by proper mathematcal mapulato. Valdato (cotrol) of the model, usually acheved by reproducg through t the behavour of the real system uder the codtos exstg before the soluto of the DOI : 0.5/jfls.0.0 3

2 Iteratoal Joural of uzzy Logc Systems (IJLS) Vol., No., Aprl 0 model (emprcal results, specal cases etc). Implemetato of the fal mathematcal results to the real system,.e. traslato of the mathematcal soluto obtaed terms of the correspodg real stuato order to reach the soluto of the gve real world problem. Durg the990 s we developed a stochastc model for the descrpto of the MM process across the above les by troducg a fte Markov cha o ts stages []. Applyg stadard results from the relevat theory we succeeded expressg mathematcally the gravty of each stage (where greater gravty meas more dffcultes for studets the correspodg stage) ad we obtaed a measure of studets modelg capactes. A mproved verso of ths model has bee preseted [5]. MM appears today as a dyamc tool for teachg mathematcs, because t helps studets to lear how to use mathematcs solvg real world or everyday lfe problems, thus gvg them the opportuty to realze ts usefuless practcal applcatos. or more detals about the MM process ad ts applcato as a method for teachg mathematcs see [4], ts refereces, etc. ally, cocerg the stages of the MM process preseted above, otce that the aalyss of the problem, although t deserves some atteto as beg a prerequste for the developmet of the MM process, s actually a troductory stage that could be cosdered as a sub stage of mathematzg. Next, we shall also cosder valdato ad mplemetato as a sgle (joed) state of the whole process. Ths hypothess, wthout chagg the substace of thgs at all, wll make techcally easer the developmet of the fuzzy framework for MM (as a process of three stages) that we are gog to preset below.. A UZZY MODEL OR THE MM PROCESS Models for the MM process lke those preseted the prevous secto (cludg our stochastc oe) are helpful uderstadg the modellers deal behavour, whch they proceed learly from real world problems through a mathematcal model to acceptable solutos ad report o them. However lfe the classroom s ot lke that. Recet research, ([4], [6], [8], etc), reports that studets school take dvdual modellg routes whe tacklg MM problems, assocated wth ther dvdual learg styles. Studets cogto utlzes geeral cocepts that are heretly graded ad therefore fuzzy. O the other had, from the teacher s pot of vew there usually exsts vagueess about the degree of success of studets each of the stages of the modellg process. All these gave us the mpulso to troduce prcples of fuzzy sets theory order to descrbe a more effectve way the process of MM classroom. Created by Zadeh [3], fuzzy logc has bee successfully developed by may researchers ad has bee prove to be extremely productve may applcatos (see, for example, [], [3]; Chapter 6, [0], [8], etc). There are also some terestg attempts to mplemet fuzzy logc deas the feld of educato ([7], [5], [9], [3], [6], [7], [8],[9], [30], [3] etc). I a earler artcle [7] we have developed a fuzzy model for the descrpto of the MM process. I the followg few paragraphs we cte parts of ths artcle. or specal facts o fuzzy sets ad ucertaty theory we refer freely to [3]. Let us cosder a group of studets,, durg the MM process the classroom. We deote by A, =,,3, the stages of aalyss./mathematzg, soluto ad valdato/mplemetato respectvely ad by a, b, c, d, ad e the lgustc labels of eglgble, low, termedate, hgh ad complete degree of studets success respectvely each of the A s. Set U={a, b, c, d, e} 4

3 Iteratoal Joural of uzzy Logc Systems (IJLS) Vol., No., Aprl 0 We are gog to represet the A s as fuzzy sets U. or ths, f a, b, c, d ad e respectvely deote the umber of studets that have acheved eglgble, low, termedate, hgh ad complete degree of success at the state A =,,3, we defe the membershp fucto m A terms of the frequeces,.e. by for each x U. Thus we ca wrte m A (x)= x A = {(x, x ) : x U}, =,,3 I order to represet all possble studets profles (overall states) durg the MM process, we cosder a fuzzy relato, say R, U 3 of the form R={(s, m R (s)) : s=(x, y, z) U 3 } To determe properly the membershp fucto m R we gve the followg defto: A trple (x, y, z) s sad to be well ordered f x correspods to a degree of success equal or greater tha y, ad y correspods to a degree of success equal or greater tha z. or example, the profle (c, c, a) s well ordered, whle (b, a, c) s ot. We defe ow the membershp degree of s to be m R (s) = m (x). m (y). m (z) A A A 3 f s s a well ordered profle, ad zero otherwse. I fact, f for example (b, a, c) possessed a ozero membershp degree, gve that the degree of success at the stage of soluto s eglgble, how the proposed soluto could be valdated satsfactorly? I order to smplfy our otato we shall wrte m s stead of m R (s). The the possblty r s of the profle s s gve by ms r s = max{ m s } where max{m s } deotes the maxmal value of m s, for all s U 3. I other words r s s the relatve membershp degree of s wth respect to the other profles. I [7] t s further descrbed how the above model ca be used studyg - through the calculato of the pseudo-frequeces f(s) = m t s ( ) ad the correspodg possbltes f ( s) r(s)= - the combed results of the performace of two or more groups durg the max{ f ( s)} MM process of the same real stuato, or alteratvely the performace of the same group durg the MM process of dfferet stuatos. I order to llustrate the use of the above model practce we preseted [7] the followg CLASSROOM EXPERIMENT: k t = 5

4 Iteratoal Joural of uzzy Logc Systems (IJLS) Vol., No., Aprl 0 The subjects were 35 studets of the School of Techologcal Applcatos of the Graduate Techologcal Educatoal Isttute of Patras (Greece),.e. future egeers, ad the basc tool was a lst of 0 problems volvg mathematcal modellg gve to studets for soluto (see Appedx). Our characterzatos of studets performace at each stage of the MM process volved: Neglgble success, f they obtaed postve results for less tha problems. Low success, f they obtaed postve results for, 3, or 4 problems. Itermedate success, f they obtaed postve results for 5, 6, or 7 problems. Hgh success, f they obtaed postve results for 8, or 9 problems. Complete success, f they obtaed postve results for all problems. Examg studets papers we foud that 7, 8 ad 0 studets had acheved termedate, hgh ad complete success respectvely at stage of aalyss/mathematzg. Therefore we obtaed that a = b =0, c =7, d =8 ad e =0. Thus aalyss/mathematzg was represeted as a fuzzy set U the form: A = {(a,0),(b,0),(c, 7 35 ),(d, ), ( 35 8 e, 35 I the same way we represeted soluto ad valdato/mplemetato of the model as fuzzy sets U by ad 0 )}. A = {(a, 6 35 ),(b, 6 35 ),(c, 6 35 ),(d, 7 35 ),(e,0)} A 3 = {(a, 35 ),(b, 0 35 ),(c, 3 35 ),(d,0),(e,0)} respectvely. Usg the gve defto we calculated the membershp degrees of the 5 3 total (ordered samples of 3 objects take from 5) possble studets profles (see colum of m s () Table below). or example, for s=(c, b, a) oe fds that m s = m (c). m (b). m (a) = A A A = 0, It tured out that (c, c, c) was the profle of maxmal membershp degree 0,08. Therefore the possblty of each s U 3 s gve by ms r s = 0, 08. or example, the possblty of (c, b, a) s 0,09 0,08 0,353, whle the possblty of (c, c, c) s of course. A few days later we performed the same expermet wth a group of 30 studets of the School of Maagemet ad Ecoomcs. Workg as before we foud that ad A ={(a,0),(b, 6 30 ),(c, 5 30 ),(d, 9 30 ),(e,0)}, A ={(a, 6 30 ),(b, 8 30 ),(c, 6 30 ),(d, 0),(e,0)} 6

5 Iteratoal Joural of uzzy Logc Systems (IJLS) Vol., No., Aprl 0 A 3 ={(a, 30 ),(b, 9 30 ),(c, 9 30 ),(d,0),(e,0)}. The we calculated the membershp degrees of all possble profles of the studet group (see colum of m s () Table ). It tured out that (c, c, a) was the profle possessg the maxmal membershp degree 0,07 ad therefore the possblty of each s s gve by ms r s = 0, 07. Calculatg the possbltes of all profles for the two groups (see colums of r s () ad r s () of Table below) we obtaed a qualtatve vew of studets performace durg the MM process expressed mathematcal terms. ally the combed results of performace of the two groups were studed by calculatg the pseudo-frequeces f(s) ad the correspodg possbltes r(s) of all studet profles s (see Table ) Table : Studet profles wth o zero pseudo-frequeces Note: The outcomes of Table are wth accuracy up to the thrd decmal pot. 3. UZZY MEASURES O STUDENTS MM SKILLS A cetral object of the educatoal research takg place he area of MM s to recogze the attamet level of studets at defed stages of the MM process ad several efforts have bee made towards ths object ([0], [8], [], [5], etc). 7

6 Iteratoal Joural of uzzy Logc Systems (IJLS) Vol., No., Aprl 0 I [7] t s argued that the total possblstc ucertaty T(r) o the ordered possblty dstrbuto r of the studets profles ca be used as a measure of ther MM capactes. I fact, the amout of formato obtaed by a acto ca be measured by the reducto of ucertaty resultg from ths acto. Accordgly studets ucertaty durg the MM process s coected to ther capacty obtag relevat formato. The lower s T(r) - whch meas greater reducto of the system s tal ucertaty - the greater the ew formato obtaed,.e. the greater the studets effcecy solvg modellg problems. Wth the doma of possblty theory ucertaty cossts of strfe (or dscord), whch expresses coflcts amog the varous sets of alteratves, ad o-specfcty (or mprecso), whch dcates that some alteratves are left uspecfed,.e. t expresses coflcts amog the szes (cardaltes) of the varous sets of alteratves. Strfe s measured by the fucto ST(r) o the ordered possblty dstrbuto r: r = r. r m r m+ of the studet group (where m+ s the total umber of all possble studets profles), defed by ST(r) = log [ = I the same way, o-specfcty s measured by N(r) = log [ ( r r + ) log = j= r ( r r + ) log ]. Therefore, the sum T(r) = ST(r) + N(r) s a measure of the total possblstc ucertaty T(r) for ordered possblty dstrbutos ([4]; page 8). j ]. Gog back to the CLASSROOM EXPERIMENT preseted the prevous secto ad wth the help of Table oe fds that the ordered possblty dstrbuto for the frst studet group s: r =, r =0,97, r 3 =0,768, r 4 =0,5, r 5 =0,476, r 6 =0,45, r 7 =0,40, r 8 =0,378, r 9 =r 0 =0,34, r =0,39, r =0,37, r 3 =0,305, r 4 =0,93, r 5 =r 6 =0,56, r 7 =0,07, r 8 =0,95, r 9 =0,7, r 0 =r =r =0,59, r 3 =0,34, r 4 =r 5 =..=r 5 =0. Therefore, usg a calculator we foud that the total possblstc ucertaty of the frst group s T(r) 0,565+,405=,97. I the same way we foud for the secod group that T(r) = 0,45+,87 =,3. Thus, sce,3<, 97, the secod group demostrated a better performace geeral tha the frst oe. Ths happeed despte to the fact that the profle (c, c, c) wth maxmal possblty of appearace for the frst group s more satsfactory tha the correspodg profle (c, c, a) for the secod group. Aother well kow measure of a system s probablstc ucertaty ad the assocated formato was establshed by Shao 948. Whe expressed terms of the Dempster- Shafer mathematcal theory of evdece for use a fuzzy evromet, Shao s measure takes the form 8

7 Iteratoal Joural of uzzy Logc Systems (IJLS) Vol., No., Aprl 0 H= - m s l m s, l s= where s the total umber of elemets of the correspodg fuzzy set ([4]; p.0). The above measuremet s kow as the Shao etropy or the Shao- Weer dversty dex. I the above formula the sum s dvded by l order to ormalze H, so that ts maxmal value s regardless the value of. It should be metoed here that the probablty of a studet s profle s defed by ms. m p s = 3 s U I adoptg H as a measure of a group s performace o MM t becomes evdet that the lower s ts value (.e. the hgher s the reducto of the correspodg ucertaty), the better the group s performace. A advatage of adoptg H as a measure stead of T(r) s that H s calculated drectly from the membershp degrees of all profles s wthout beg ecessary to calculate ther probabltes p s. I cotrast the calculato of T(r) presupposes the calculato of the possbltes r s of all profles frst. However, we must meto that accordg to Shackle [7] the huma reasog ca be formalzed more adequately by possblty rather, tha by probablty theory. Cocerg our CLASSROOM EXPERIMENT, usg Table oe fds that H 0, 48 for the frst group ad H 0,386 for the secod group, whch shows aga that the geeral performace of the secod group was better tha that of the frst oe. I [3] we have formalzed the process of learg a subject matter by the dvduals (ad especally the process of learg mathematcs by studets) usg a fuzzy logc approach smlar to that descrbed the prevous secto for the process of MM. Later [6] we have expaded ths argumet by usg the total possblstc ucertaty of a studet group as a measure of ts learg sklls. Meawhle, Subbot et al. [9], based o our fuzzy model for the learg process [3], they developed a dfferet approach to a comprehesve assessmet of studets learg sklls. Recetly, together wth Prof. Subbot, we have appled ths approach for measurg the effcecy of a Case-Based Reasog system [0] ad as a assessmet tool of a studet group s Aalogcal Reasog abltes [9]. Here we shall apply the above approach for developg a alteratve fuzzy measure for studets MM capactes. or ths, gve a fuzzy subset A = {(x, m(x)): x U} of the uversal set U wth membershp fucto m: U [0, ] we correspod to each x U a terval of values from a prefxed umercal dstrbuto (whch actually meas that we replace U wth a set of real tervals) ad we costruct the graph of the membershp fucto y=m(x). There s a commoly used fuzzy logc approach to measure performace wth the par of umbers (x c,y c ) as the coordates of the ceter of mass c of the represeted fgure (see for example, [5], [] ad []), whch we ca calculate usg the followg well-kow formulas: () xc =, yc = dxdy xdxdy s ydxdy dxdy. 9

8 Iteratoal Joural of uzzy Logc Systems (IJLS) Vol., No., Aprl 0 It s ot a problem to calculate such umbers usg the formulas above; however t could take some sgfcat amout of tme. However, as ay assessmet, our approach s very approxmate. So t would be much more useful practce to smplfy the stuato by substtutg the trapezods of our graph by rectagles. I ths way our graph s approxmated wth a bar graph, lke gure below. It s easy to see that the case whe our fgure cossts of rectagles, the formulas () ca be reduced to the followg formulas: Ideed, thscase () ( ) y y = = xc =, y c =. y y = = x c =, yc =, dxdy dxdy xdxdy ydxdy dxdy s the total mass of the system whch s equal to y. = xdxdy s the momet about the y-axs ad t s equal to y xdxdy = dy xdx = y = = 0 = = = = 0 0 = = xdx = ( ) y. ydxdy s the momet about the x-axs ad t s equal to y y ydxdy = ydy dx = ydy = y. y 4 y y y 3 y 5 c (x c, y c ) 0 a b c 3 d 4 e 5 gure : Bar graphcal data represetato 0

9 Iteratoal Joural of uzzy Logc Systems (IJLS) Vol., No., Aprl 0 rom the above proof, where, =,,,, deote the rectagles of the bar graph of gure, t becomes evdet that the trasto from () to () s obtaed uder the assumpto that the tervals legth s ad the tervals start from 0. I fact, let us go back to the fuzzy model for the MM process preseted the prevous secto. The, each of the stages of mathematzg, soluto ad valdato ca be graphcally represeted as gure, where the lgustc labels a, b, c, d, e of eglgble, low, termedate, hgh ad complete degree of success are takg values the tervals [0,), [,), [,3), [3,4) ad [4,5] respectvely. Ths meas practce that a studet earg, for example, the grade, a partcular stage of the MM process s characterzed by the teacher as achevg low success, earg the grade 3,7 s characterzed as achevg hgh success, etc. Now formulas () wll be trasformed to the followg formulas: x y + 3y + 5y + 7y + 9y c = y + y + y3 + y4 + y , y + y + y3 + y4 + y 5 yc =. y + y + y3 + y4 + y5 Sce we ca assume that y + y + y + y + y =, we ca wrte (3) xc = ( y + 3y + 5y3 + 7 y4 + 9 y5 ), yc = ( y + y + y + y + y ) where y, 5, s the rato of the cases the group havg the labels a, b, c, d, ad e to the umbers of all cases the group (.e. wth the termology used the model sketched the prevous secto we ca wrte y = x ). But, 0 (y -y ) =y +y -y y, therefore y +y y y wth the equalty holdg f, ad oly f, y =y. I the same way oe fds that y +y 3 y y 3, etc. Hece t s easy to check that (y +y +y 3 +y 4 +y 5 ) 5(y +y +y 3 +y 4 +y 5 ) wth the equalty holdg f, ad oly f, y =y =y 3 =y 4 =y 5. I our case y +y +y 3 +y 4 +y 5 =, therefore 5(y +y +y 3 +y 4 +y 5 ) wth the equalty holdg f, ad oly f, y =y =y 3 =y 4 =y 5 =. The the frst o formulas (3) gves that x c = 5. urther, 5 combg the equalty 5(y +y +y 3 +y 4 +y 5 ) wth the secod of formulas (3) oe fds that 0y c, or y c. Therefore the uque mmum for y c correspods to the ceter of mass 0 m ( 5, ). 0

10 Iteratoal Joural of uzzy Logc Systems (IJLS) Vol., No., Aprl 0 The deal case s whe y =y =y 3 =y 4 =0 ad y 5 =. The from formulas (3) we get that x c = 9 ad y c =. Therefore the ceter of mass ths case s the pot ( 9, ). O the other had the worst case s whe y = ad y =y 3 =y 4 = y 5 =0. The for formulas (3) we fd that the ceter of mass s the pot w (, ). I ths way the area for c could be approxmately represeted as the tragle of the gure below. The from elemetary geometrc cosderatos t drectly follows that for two groups wth the same x c,5 the group havg the ceter of mass whch s stuated closer to s the group wth the hgher y c ; ad for two groups wth the same x c <.5 the group havg the ceter of mass whch s stuated farther to w s the group wth the lower y c. gure : Graphcal represetato of the area of the ceter of mass Based o the above cosderatos t s logcal to formulate our crtero for comparg the groups performaces the followg form: Amog two or more groups the group wth the bggest x c performs better; (4) If two or more groups have the same x c.5, the the group wth the hgher y c performs better. If two or more groups have the same x c <.5, the the group wth the lower y c performs better. I the CLASSROOM EXPERIMENT preseted the prevous secto the stages of aalyss/mathematzg, soluto ad valdato/mplemetato of the model for the frst studet group ca be represeted the followg form: A = {(a,0),(b,0),(c, 7 35 ),(d, ), ( 35 8 e, 35 0 )}. ad A = {(a, 6 35 ),(b, 6 35 ),(c, 6 35 ),(d, 7 35 ),(e,0)}, A 3 = {(a, 35 ),(b, 0 35 ),(c, 3 35 ),(d,0),(e,0)}. Smlarly for the secod group we ca wrte: A = {(a,0),(b, 6 30 ),(c, 5 30 ),(d, 9 30 ),(e,0)},

11 Iteratoal Joural of uzzy Logc Systems (IJLS) Vol., No., Aprl 0 ad A = {(a, 6 30 ),(b, 8 30 ),(c, 6 30 ),(d, 0),(e,0)}, A 3 = {(a, 30 ),(b, 9 30 ),(c, 9 30 ),(d,0),(e,0)}. Therefore, for the stage of aalyss/mathematzg we fd that x c = 7 8 ( )=3,3 ad x c = ( ) =, By the crtero (4), the frst group demostrates a better performace. or the stage of soluto we fd that X c = ( ),86 ad x c = ( ), By the crtero (4), the frst group demostrates aga better performace. ally, for the thrd stage of valdato/mplemetato we have X c3 = 0 3 ( ),59 ad x c3 = 9 9 ( ) =, So ths step, the performaces of both groups are close, but the frst group performs slghtly better. Based o our calculatos we ca coclude that the frst group demostrated better at all three stages. We ca also compare each group s performace at each stage. Both groups performed better at the frst stage ad worse at the thrd stage. Ths drectly reflects the ascedg complcato of the tasks at the secod stage ad especally at the thrd stage. 4. DISCUSSION AND CONCLUSIONS MM s oe of the cetral deas the owadays mathematcs educato. I ths paper we have developed a fuzzy framework for the represetato of MM as a process cosstg of three stages: Aalyss/mathematzato, soluto ad valdato/ mplemetato. Applyg fuzzy logc formalzg the MM process helps obtag quattatve formato for ths process (comparg studets performaces, etc), as well as a qualtatve vew of the degree of success ts successve stages through the calculato of the possbltes of all studets profles. I a earler paper we troduced the total possblsc ucertaty T(r) o the ordered possblty dstrbuto r of the studets profles as a measure of studets MM capactes. I the preset paper we troduced two alteratve fuzzy measures. The frst oe s the well kow Shao- Weer dversty dex H, properly adapted for use a fuzzy evromet. I the secod oe we measure the dvduals performace MM by graphcally represetg the formato as a two dmesoal fgure ad work wth the coordates of the ceter of mass c of ths fgure. We emphasze the fact that the above approaches (three total) are treatg dfferetly the dea of a group s performace. I fact, the frst two cases (measures T(r) ad H) the studet group s ucertaty durg the MM process s coected to ts capacty obtag the relevat formato. Uder ths sese, the lower s the system s fal ucertaty (whch meas greater reducto of the tally exstg ucertaty), the better s ts performace. O the other had, the thrd case the weghted average plays the ma role,.e. the result of the performace close to the deal performace have much more weght tha the oe close to the lower ed. I other words, whle the frst two cases are lookg to the average performace, the thrd oe s mostly lookg at the qualty of the performace. Therefore some dffereces could appear boudary cases. Ths explas why, the classroom expermet preseted ths paper, accordg to the 3

12 Iteratoal Joural of uzzy Logc Systems (IJLS) Vol., No., Aprl 0 frst two approaches the frst group was foud to have a better performace tha the secod oe, whle just the opposte happeed accordg to the thrd approach. I cocludg, t s argued that the kowledge of all the above approaches helps fdg the deal profle of performace accordg to the user s persoal crtera of goals ad therefore to fally choosg the approprate approach for measurg the results of hs/her expermets. Iearler papers we have also developed a stochastc model for the same purposes by troducg a fte Markov cha o the stages of the MM process. Nevertheless, ths model s helpful oly uderstadg the deal behavour whch modellers proceed learly from real-world problems through a mathematcal model to acceptable solutos ad report o them. However t has bee observed that studets take dvdual modellg routes whe tacklg MM problems. Therefore a qualtatve approach of all possble studets profles durg the MM process becomes ecessary for ts deeper study, whch s obtaed by calculatg ther possbltes through the use of our fuzzy model. O the other had the characterzato of the studets performace terms of a set of lgustc labels whch are fuzzy themselves s a dsadvatage of the fuzzy model, because ths characterzato depeds o the researcher s persoal crtera. Therefore a combed use of the fuzzy ad stochastc models seems to be the best soluto achevg a worthy of credt mathematcal aalyss of the MM process. ACKNOWLEDGMENT The author wshes to thak hs colleague ad collaborator Prof. Igor Ya. Subbot (Natoal Uversty, LA, Calfora, USA) for hs valuable suggestos that played a mportat role wrtg ths paper. REERENCES [] Berry J. & Daves A. (996), Wrtte Reports, Mathematcs Learg ad Assessmet: Sharg Iovatve Practces. I: C. R. Haes & S. Duthorr (Eds.), Lodo, Arold, O? [] Blomhψj, M. & Jese, T.H. (003), Developg mathematcal modelg competece: Coceptual clarfcato ad educatoal plag, Teachg Mathematcs ad ts Applcatos,, [3] Blum, W. & Leβ, D. (007), How do studets ad teachers deal wth modellg problems? I C.R. Haes et al. (Eds.): Mathematcal Modellg: Educato, Egeerg ad Ecoomcs, (ICTMA ), -3, Chchester: Horwood Publshg. [4] Borroeo err, R. (007), Modellg problems from a cogtve perspectve. I C.R. Haes et al. (Eds.): Mathematcal Modellg: Educato, Egeerg ad Ecoomcs, (ICTMA ), 60-70, Chchester: Horwood Publshg. [5] Caversa. L., uzzy Computg: Basc Cocepts. [6] Doer, H. M. (007), What kowledge do teachers eed for teachg mathematcs through applcatos ad modelg? I W. Blum et al. (Eds.), Modellg ad Applcatos Mathematcs Educato, 69-78, NY: Sprger. [7] Esp, E. A. & Olveras, C. M. L. (997), Itroducto to the Use of the uzzy Logc the Assessmet of Mathematcs Teachers, Proceedgs st Medterraea Coferece o Mathematcs Educato, 07-3, Cyprus. [8] Galbrath, P. L. & Stllma, G. (00), Assumptos ad cotext: Pursug ther role modelg actvty. I J.. Matos et al. (Eds.): Modellg ad Mathematcs Educato: Applcatos Scece ad Techology (ICTMA 9), , Chchester: Horwood Publshg. [9] Greefrath, G. (007), Modellere lere mt offee realtatsahe Aufgahe, Koh: Auls Verlag [0] Haes C. & Crouch R. (00), Recogzg costructs wth mathematcal modelg, Teachg Mathematcs ad ts Applcatos, 0(3), [] Hellma M., uzzy Logc Itroducto, [] Jamshd, M., Vadee, N., & Ross, T. (993), uzzy logc ad Cotrol, Pretce-Hall. [3] Klr, G. J. & olger, T. A. (988), uzzy Sets, Ucertaty ad Iformato, Pretce-Hall, Lodo. 4

13 Iteratoal Joural of uzzy Logc Systems (IJLS) Vol., No., Aprl 0 [4] Klr, J. G. (995), Prcples of Ucertaty: What are they? Why do we mea them?, uzzy Sets ad Systems, 74, 5-3. [5] Perdkars, S. (0), Usg uzzy Sets to Determe the Cotuty of the va Hele Levels, Joural of Mathematcal Sceces & Mathematcs Educato, 6(), [6] Pollak H. O. (979), The teracto betwee Mathematcs ad other school subjects, New Treds Mathematcs Teachg, Volume IV, Pars: UNESKO. [7] Stllma, G. A. & Galbrath, P. (998), Applyg mathematcs wth real world coectos: Metacogtve characterstcs of secodary studets, Educatoal Studes Mathematcs, 96, [8] Subbot, I. Ya., Badkoobeh, H. & Blotsk, N. (004), Applcato of uzzy Logc to Learg Assessmet, Ddactcs of Mathematcs: Problems ad Ivestgatos. Volume, [9] Subbot I. Ya. & Voskoglou, M. Gr. (0), Applcatos of uzzy Logc to Case-Based Reasog, Iteratoal Joural of Applcatos of uzzy Sets,, 7-8. [0] Va Broekhove, E. & De Baets, B. (006), ast ad accurate ceter of gravty defuzzfcato of fuzzy system outputs defed o trapezodal fuzzy parttos, uzzy Sets ad Systems, 57, Issue 7, [] Voskoglou, M. G. (995), Measurg mathematcal model buldg abltes, Iteratoal Joural of Mathematcal Educato. Scece ad Techology, Vol. 6, [] Voskoglou, M. G. (999), The Process of Learg Mathematcs: A uzzy Set Approach, Heurstcs ad Ddactcs of Exact Sceces, 0, 9 3. [3] Voskoglou, M. G. (006), The use of mathematcal modellg as a tool for learg mathematcs, Quader d Rcerca Ddattca (Sceze Mathemathe), Uversty of Palermo, 6, [4] Voskoglou, M. G. (007) A stochastc model for the modellg process, I Mathematcal Modellg: Educato, Egeerg ad Ecoomcs, C. Chaes, P. Galbrath, W. Blum & s. Kha (Eds), Horwood Publ.. Chchester, [5] Voskoglou, M. G. (009), Trasto Across Levels the Process of Learg, Joural of Mathematcal Modellg ad Applcato (Uversty of Blumeau, Brazl), Volume, [6] Voskoglou, M. G. (00), A fuzzy system s framework for solvg real world problems, WSEAS Trasactos o Systems, Vol. 9, Issue 6, [7] Voskoglou, M. Gr. (0), Stochastc ad fuzzy models Mathematcs Educato, Artfcal Itellgece ad Maagemet, Lambert Academc Publshg, Saarbrucke, Germay ( look at ). [8] Voskoglou, M. Gr. (0), A uzzy Model for Aalogcal Problem Solvg, Iteratoal Joural of uzzy Logc Systems, (), -0. [9] Voskoglou, M. Gr. (0), uzzy Logc ad Ucertaty Problem Solvg, Joural of Mathematcal Sceces & Mathematcs Educato, Vol. 7, No., [30] Voskoglou, M. Gr. & Subbot, I. Ya. (0), uzzy Models for Aalogcal Reasog, Iteratoal Joural of Applcatos of uzzy Sets, Vol., -38. [30] Zadeh, L. A. (965), uzzy Sets, Iformato ad Cotrol, 8, APPENDIX Lst of the problems used the classroom expermet Problem : We wat to costruct a chael to ru water by foldg the two edges of a orthogoal metallc leaf havg sdes of legth 0cm ad 3 cm, such a way that they wll be perpedcular to the other parts of the leaf. Assumg that the flow of the water s costat, how we ca ru the maxmum possble quatty of the water? (Remark: The correct soluto s obtaed by foldg the edges of the loger sde of the leaf) Problem : A car dealer has a mea aual demad of 50 cars, whle he receves 30 ew cars per moth. The aual cost of storg a car s 00 euros ad each tme he makes a ew order he pays a extra amout of 00 euros for geeral expeses (trasportato, surace etc). The frst cars of a ew order arrve at the tme whe the last car of the prevous order has bee sold. How may cars must he order order to acheve the mmum total cost? 5

14 Iteratoal Joural of uzzy Logc Systems (IJLS) Vol., No., Aprl 0 Problem 3: A mportato compay codes the messages for the arrvals of ts orders terms of characters cosstg of a combato of the bary elemets 0 ad. If t s kow that the arrval of a certa order wll take place from st utl the 6 th of March, fd the mmal umber of the bary elemets of each character requred for codg ths message. Problem 4: Let us correspod to each letter the umber showg ts order to the alphabet (A=, B=, C=3 etc). Let us correspod also to each word cosstg of 4 letters a X matrx the obvous way; e.g. the matrx 9 5 correspods to the word SOME. Usg the matrx 3 5 E= 8 5 as a ecodg matrx how you could sed the message LATE the form of a 7 camouflaged matrx to a recever kowg the above process ad how he (she) could decode your message? Problem 5: The demad fucto P(Q d )=5-Q d represets the dfferet prces that cosumers wllg to pay for dfferet quattes Q d of a good. O the other had the supply fucto P(Q s )=Q s + represets the prces at whch dfferet quattes Q s of the same good wll be suppled. If the market s equlbrum occurs at (Q 0, P 0 ) producers who would supply at lower prce tha P 0 beeft. d the total ga to producers. Problem 6: A ballot box cotas 8 balls umbered from to 8. Oe makes 3 successve drawgs of a lottery, puttg back the correspodg ball to the box before the ext lottery. d the probablty of gettg all the balls that he draws out of the box dfferet. Problem 7: A box cotas 3 whte, 4 blue ad 6 black balls. If we put out balls, what s the probablty of choosg balls of the same colour? Problem 8: The populato of a coutry s creased proportoally. If the populato s doubled 50 years, how may years t wll be trpled? Problem 9: A we producer has a stock of we greater tha 500 ad less tha 750 klos. He has calculated that, f he had the double quatty of we ad trasferred t to bottles of, 5, or 40 klos, t would be left over 6 klos each tme. d the quatty of stock. Problem 0: Amog all cyldrcal towers havg a total surface of 80π m, whch oe has the maxmal volume? (Remark: Some studets dd t clude to the total surface the oe base (groud-floor) ad they foud aother soluto, whle some others dd t clude both bases (roof ad groud-floor) ad they foud o soluto, sce we caot costruct cylder wth maxmal volume from ts surroudg surface.) Author Mchael Gr. Voskoglou (B.Sc., M.Sc., M.Phl., Ph.D. Mathematcs) s curretly Professor of Mathematcal Sceces at the Graduate Techologcal Educatoal Isttute of Patras, Greece. He s the author of 8 books (7 Greek ad Eglsh laguage) ad of about 40 papers publshed reputed jourals ad proceedgs of teratoal cofereces of coutres 5 cotets, wth may refereces from other researchers. He s a revewer of the AMS ad member of the Edtoral Board or referee several mathematcal jourals. Hs research terests clude algebra, Markov chas, fuzzy logc ad mathematcs educato. 6