Should a Syllabus Be . . . Decision Making: A . . . Decision Making . . . Analysis of the Situation It Is Advantageous Case of a Precise Syllabus to Make a Syllabus Case of an Imprecise . . . Case of an Imprecise . . . As Precise As Possible: Conclusion: It Is . . . Acknowledgments Decision-Theoretic Analysis Home Page Title Page Francisco Zapata 1 , Olga Kosheleva 2 , and Vladik Kreinovich 1 ◭◭ ◮◮ ◭ ◮ 1 Department of Computer Science 2 Department of Teacher Education Page 1 of 10 University of Texas at El Paso 500 W. University Go Back El Paso, Texas 79968, USA Full Screen fazg74@gmail.com, olgak@utep.edu, vladik@utep.edu Close Quit
Should a Syllabus Be . . . Decision Making: A . . . 1. Should a Syllabus Be Precise? Decision Making . . . • Shall we indicate exactly how many points we should Analysis of the Situation assign for each test and for each assignment? Case of a Precise Syllabus Case of an Imprecise . . . • On the one hand, many students like such certainty. Case of an Imprecise . . . • On the other hand, instructors would like to have some Conclusion: It Is . . . flexibility. Acknowledgments Home Page • If an assignment turns out to be more complex than expected, we should be able to increase its weight. Title Page • Vice versa, it it turns out to be simpler than expected, ◭◭ ◮◮ we should be able to decrease the number of points. ◭ ◮ • In this talk, we analyze this problem from a decision- Page 2 of 10 theoretic viewpoint. Go Back • Our conclusion is that while a little flexibility is OK, Full Screen in general, it is beneficial to make a syllabus precise. Close Quit
Should a Syllabus Be . . . Decision Making: A . . . 2. Decision Making: A Brief Reminder Decision Making . . . • According to decision theory:’ Analysis of the Situation Case of a Precise Syllabus – decisions of a rational agent Case of an Imprecise . . . – can be equivalently described as maximizing an ap- Case of an Imprecise . . . propriate objective function u ( a ). Conclusion: It Is . . . • This objective function is known as the utility function . Acknowledgments Home Page • In some cases, we do not know the exact consequences of each possible action. Title Page ◭◭ ◮◮ • In this case, for each action a : ◭ ◮ – instead of the exact value u ( a ) of the corresponding utility, Page 3 of 10 – we only know the interval of possible values: Go Back [ u ( a ) , u ( a )] . Full Screen Close Quit
Should a Syllabus Be . . . Decision Making: A . . . 3. Decision Making under Interval Uncertainty Decision Making . . . • In such situations, a rational agent should select an Analysis of the Situation action a that maximizes the expression Case of a Precise Syllabus Case of an Imprecise . . . def u ( a ) = α · u ( a ) + (1 − α ) · u ( a ) . Case of an Imprecise . . . Conclusion: It Is . . . • This optimism-pessimism criterion was first formulated Acknowledgments by a Nobelist Leo Hurwicz. Home Page • The optimism value α = 1 means that a person only Title Page takes into account best-case consequences. ◭◭ ◮◮ • The pessimism value α = 0 means that a person only ◭ ◮ takes into account worst-case consequences. Page 4 of 10 • A realistic approach is to take α ∈ (0 , 1) . Go Back • In particular, there are reasonable arguments in favor of selecting α = 0 . 5. Full Screen Close Quit
Should a Syllabus Be . . . Decision Making: A . . . 4. Analysis of the Situation Decision Making . . . • In general, the overall grade g for the class is a weighted Analysis of the Situation average of grades g i on different assignments: Case of a Precise Syllabus Case of an Imprecise . . . n � g = w 1 · g 1 + . . . + w n · g n , with w i = 1 . Case of an Imprecise . . . i =1 Conclusion: It Is . . . Acknowledgments • The grade g i on each assignment depends on the stu- Home Page dent’s efforts g i = f ( e i ). Title Page • Let us assume that a student has a certain overall amount of effort E dedicated to this class; then: ◭◭ ◮◮ ◭ ◮ – among all possible combinations e i with � e i = E , i =1 Page 5 of 10 – the student selects the one that maximizes his/her Go Back utility. Full Screen Close Quit
Should a Syllabus Be . . . Decision Making: A . . . 5. Case of a Precise Syllabus Decision Making . . . • In a precise syllabus, the weights w i are explicitly Analysis of the Situation stated. Case of a Precise Syllabus n Case of an Imprecise . . . • In this case, the student maximizes � w i · f ( e i ) . Case of an Imprecise . . . i =1 • For equal weights, Lagrange multiplier approach leads Conclusion: It Is . . . to Acknowledgments � n � n Home Page � � w i · f ( e i ) + λ · e i − E → min . Title Page i =1 i =1 • Differentiating with respect to e i and equating deriva- ◭◭ ◮◮ tive to 0, we get w i · f ′ ( e i ) = − λ . ◭ ◮ • In particular, when assignments are of equal complex- Page 6 of 10 ity and w i = const, we get e i = const . Go Back • Thus, a precise syllabus encourages students to learn Full Screen all the topics. Close • And this is exactly what we instructors want. Quit
Should a Syllabus Be . . . Decision Making: A . . . 6. Case of an Imprecise Syllabus Decision Making . . . • Let us now consider the extreme case of an imprecise Analysis of the Situation syllabus, when no information is provided about w i . Case of a Precise Syllabus Case of an Imprecise . . . • In this case, the best-case gain is Case of an Imprecise . . . u = max g i = max f ( e i ) . Conclusion: It Is . . . i i Acknowledgments • This gain corresponds to the case when: Home Page – the assignment with the highest grade gets Title Page weight 1, and ◭◭ ◮◮ – other assignments get weight 0. ◭ ◮ • The worst-case gain is u = min g i = min f ( e i ). i i Page 7 of 10 • This gain corresponds to the case when: Go Back – the assignment with the lowest grade gets weight 1, and Full Screen – other assignments get weight 0. Close Quit
Should a Syllabus Be . . . Decision Making: A . . . 7. Case of an Imprecise Syllabus (cont-d) Decision Making . . . • Thus, a student maximizes Analysis of the Situation Case of a Precise Syllabus u = α · max f ( e i ) + (1 − α ) · min f ( e i ) . Case of an Imprecise . . . i i Case of an Imprecise . . . • If a student diligently studies each topic, we have Conclusion: It Is . . . � E � e i = E n , and u = f Acknowledgments . n Home Page • On the other hand, if the student gambles and places Title Page all his/her efforts into one topic, then ◭◭ ◮◮ max g i = f ( E ) and min g i = 0 . ◭ ◮ i i Page 8 of 10 • In this case, u = α · f ( E ) . Go Back � E � • So, if α · f ( E ) > f , the student will gamble in- Full Screen n stead of studying each topic. Close Quit
Should a Syllabus Be . . . Decision Making: A . . . 8. Conclusion: It Is Advantageous To Make Syl- Decision Making . . . labi Precise Analysis of the Situation � E � Case of a Precise Syllabus • If α · f ( E ) > f , the student will gamble instead n Case of an Imprecise . . . of studying each topic. Case of an Imprecise . . . • No matter what α > 0 is, for sufficient large n , we have Conclusion: It Is . . . � E � Acknowledgments → f (0) = 0 . f Home Page n Title Page • Thus, for large n , the above inequality will be satisfied. ◭◭ ◮◮ • So, an imprecise syllabus encourages gambling ap- ◭ ◮ proach instead of a diligent thorough study. Page 9 of 10 • Thus, it is advantageous to make a syllabus as precise Go Back as possible. Full Screen Close Quit
9. Acknowledgments Should a Syllabus Be . . . Decision Making: A . . . Decision Making . . . This work was supported in part: Analysis of the Situation Case of a Precise Syllabus • by the National Science Foundation grants: Case of an Imprecise . . . Case of an Imprecise . . . – HRD-0734825 and HRD-1242122 Conclusion: It Is . . . (Cyber-ShARE Center of Excellence) and Acknowledgments – DUE-0926721, and Home Page • by an award from Prudential Foundation. Title Page ◭◭ ◮◮ ◭ ◮ Page 10 of 10 Go Back Full Screen Close Quit
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