It is Advantageous to Make a Syllabus as Precise as Possible: Decision-Theoretic Analysis

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1 Joural of Iovatve Techology ad Educato, Vol. 4, 2017, o. 1, 1-5 HIKARI Ltd, It s Advatageous to Make a Syllabus as Precse as Possble: Decso-Theoretc Aalyss Fracsco Zapata 1, Olga Kosheleva 2 ad Vladk Kreovch 1 1 Departmet of Computer Scece 2 Departmet of Teacher Educato Uversty of Texas at El Paso 500 W. Uversty El Paso, TX 79968, USA Copyrght c 2016 Fracsco Zapata, Olga Kosheleva ad Vladk Kreovch. Ths artcle s dstrbuted uder the Creatve Commos Attrbuto Lcese, whch permts urestrcted use, dstrbuto, ad reproducto ay medum, provded the orgal work s properly cted. Abstract Should a syllabus be precse? Shall we dcate exactly how may pots we should assg for each test ad for each assgmet? O the oe had, may studets lke such certaty. O the other had, structors would lke to have some flexblty: f a assgmet turs out to be more complex tha expected, we should be able to crease the umber of pots for ths assgmet, ad, vce versa, t t turs out to be smpler tha expected, we should be able to decrease the umber of pots. I ths paper, we aalyze ths problem from a decso-theoretc vewpot. Our cocluso s that whle a lttle flexblty s OK, geeral, t s beefcal to make a syllabus as precse as possble. Mathematcs Subject Classfcato: 62C05, 91B06, 97A99 Keywords: syllabus, decso makg uder ucertaty, Hurwcz optmsm-pessmsm crtero

2 2 Fracsco Zapata, Olga Kosheleva ad Vladk Kreovch 1 Should a Syllabus Be Precse? Formulato of the problem. Shall we dcate exactly how may pots we should assg for each test ad for each assgmet? O the oe had, may studets lke such certaty. O the other had, structors would lke to have some flexblty: If a assgmet turs out to be more complex tha expected, we should be able to crease ts weght. Vce versa, t t turs out to be smpler tha expected, we should be able to decrease the umber of pots. What we do ths paper. I ths paper, we aalyze ths problem from a decso-theoretc vewpot. Our cocluso s that whle a lttle flexblty s OK, geeral, t s beefcal to make a syllabus precse. 2 Decso Makg: A Bref Remder Decso makg: geeral case. Accordg to decso theory (see, e.g., [1, 4, 5, 6, 7]) decsos of a ratoal aget ca be equvaletly descrbed as maxmzg a approprate objectve fucto u(a). Ths objectve fucto s kow as the utlty fucto. Decso makg uder ucertaty. I some cases, we do ot kow the exact cosequeces of each possble acto. I ths case, for each acto a, stead of the exact value u(a) of the correspodg utlty, we oly kow the terval of possble values: [u(a), u(a)]. I such stuatos, a ratoal aget should select a acto a that maxmzes the expresso u(a) def = α u(a) + (1 α) u(a). Ths optmsm-pessmsm crtero was frst formulated by a Nobelst Leo Hurwcz [2, 3, 5, 6]: The optmsm value α = 1 meas that a perso oly takes to accout best-case cosequeces. The pessmsm value α = 0 meas that a perso oly takes to accout worst-case cosequeces. A realstc approach s to take α (0, 1). I partcular, there are reasoable argumets favor of selectg α = 0.5.

3 It s advatageous to make a syllabus as precse as possble 3 3 Aalyss of the Stuato I geeral, the overall grade g for the class s a weghted average of grades g o dfferet assgmets: g = w 1 g w g, wth w = 1. The grade g o each assgmet depeds o the studet s efforts g = f(e ). Let us assume that a studet has a certa overall amout of effort E dedcated to ths class; the: amog all possble combatos e wth e = E, the studet selects the oe that maxmzes hs/her utlty. 4 Case of a Precse Syllabus I a precse syllabus, the weghts w are explctly stated. I ths case, the studet maxmzes w f(e ). For equal weghts, Lagrage multpler approach leads to ( ) w f(e ) + λ e E m. Dfferetatg wth respect to e ad equatg dervatve to 0, we get w f (e ) = λ. I partcular, whe assgmets are of equal complexty ad w = cost, we get e = cost. Thus, a precse syllabus ecourages studets to lear all the topcs. Ths s exactly what we structors wat. 5 Case of a Imprecse Syllabus Imprecse syllabus: geeral case. Let us ow cosder the extreme case of a mprecse syllabus, whe o formato s provded about w. I ths case, the best-case ga s u = max g = max f(e ). Ths ga correspods to the case whe:

4 4 Fracsco Zapata, Olga Kosheleva ad Vladk Kreovch the assgmet wth the hghest grade gets weght 1, ad other assgmets get weght 0. The worst-case ga s u = m g = m f(e ). Ths ga correspods to the case whe: the assgmet wth the lowest grade gets weght 1, ad other assgmets get weght 0. Thus, a studet maxmzes the expresso u = α max f(e ) + (1 α) m f(e ). What f a studet dlgetly studes. If a studet dlgetly studes each topc, we have e = E ( ) E, ad u = f. What f a studet gambles. O the other had, f the studet gambles ad places all hs/her efforts to oe topc, the I ths case, u = α f(e). max g = f(e) ad m g = 0. So what wll a studet do? So, f α f(e) > f gamble stead of studyg each topc. No matter what α > 0 s, for suffcet large, we have ( ) E f f(0) = 0. ( E ), the studet wll Thus, for large, the above equalty wll be satsfed. So, a mprecse syllabus ecourages gamblg approach stead of a dlget thorough study. 6 Cocluso: It Is Advatageous To Make Syllab Precse A precse syllabus ecourages a studet to study all the topcs ths s what we structors would lke to see. O the other had, a mprecse syllabus ecourages gamblg approach stead of a dlget thorough study.

5 It s advatageous to make a syllabus as precse as possble 5 Thus, t s advatageous to make a syllabus as precse as possble. Ackowledgemets. Ths work was supported part by the Natoal Scece Foudato grats HRD ad HRD (Cyber-ShARE Ceter of Excellece) ad DUE , ad by a award from Prudetal Foudato. The authors are thakful to all the partcpats of the 19th UTEP/NMSU Workshop o Mathematcs, Computer Scece, ad Computatoal Scece (El Paso, Texas, November 5, 2016) for valuable dscussos. Refereces [1] P. C. Fshbur, Utlty Theory for Decso Makg, Joh Wley & Sos Ic., New York, [2] L. Hurwcz, Optmalty Crtera for Decso Makg uder Igorace, Cowles Commsso Dscusso Paper, Statstcs, o. 370, [3] V. Kreovch, Decso makg uder terval ucertaty (ad beyod), Chapter Huma-Cetrc Decso-Makg Models for Socal Sceces, P. Guo ad W. Pedrycz (eds.), Sprger Verlag, 2014, [4] D. Luce, Ivdual Choce Behavor: A Theoretcal Aalyss, Dover, New York, [5] R. D. Luce ad H. Raffa, Games ad Decsos: Itroducto ad Crtcal Survey, New York, Dover, [6] H. T. Nguye, O. Kosheleva ad V. Kreovch, Decso makg beyod Arrow s mpossblty theorem, wth the aalyss of effects of colluso ad mutual attracto, Iteratoal Joural of Itellget Systems, 24 (2009), o. 1, [7] H. Raffa, Decso Aalyss, Addso-Wesley, Readg, Massachusetts, Receved: November 22, 2016; Publshed: Jauary 3, 2017