Deciding Which . . . How Techniques Are . . . Towards Selecting the . . . What is the Best Way to Towards Selecting the . . . Explicit Solution: . . . Distribute Efforts Among Need to Take . . . Students: Towards Case of Interval . . . Case of Fuzzy Uncertainty Quantitative Approach to Title Page Human Cognition ◭◭ ◮◮ ◭ ◮ Olga Kosheleva 1 and Vladik Kreinovich 2 1 Department of Mathematics Education Page 1 of 13 2 Department of Computer Science Go Back University of Texas at El Paso 500 W. University Full Screen El Paso, TX 79968, USA Emails: { olgak,vladik } @utep.edu Close Quit
Deciding Which . . . 1. Deciding Which Teaching Method Is Better: For- How Techniques Are . . . mulation of the Problem Towards Selecting the . . . • Pedagogy is a fast developing field. Towards Selecting the . . . Explicit Solution: . . . • New methods, new ideas and constantly being devel- Need to Take . . . oped and tested. Case of Interval . . . • New methods and new idea may be different in many Case of Fuzzy Uncertainty things: Title Page – they may differ in the way material is presented, ◭◭ ◮◮ – they may also differ in the way the teacher’s effort ◭ ◮ is distributed among individual students. Page 2 of 13 • To perform a meaningful testing, we need to agree on Go Back the criterion. Full Screen • Once we have selected a criterion, a natural question is: what is the optimal way to teaching the students. Close Quit
Deciding Which . . . 2. How Techniques Are Compared Now: A Brief De- How Techniques Are . . . scription Towards Selecting the . . . • The success of each individual student i can be natu- Towards Selecting the . . . rally gauged by this student’s grade x i . Explicit Solution: . . . Need to Take . . . • So, for two different techniques T and T ′ , we know the Case of Interval . . . corresponding grades x 1 , . . . , x n and x ′ 1 , . . . , x ′ n ′ . Case of Fuzzy Uncertainty • In pedagogical experiments, the decision is usually made Title Page based on the comparison of the average grades ◭◭ ◮◮ = x ′ 1 + . . . + x ′ = x 1 + . . . + x n def and E ′ def n ′ E . ◭ ◮ n ′ n Page 3 of 13 • Example: we had x 1 = 60, x 2 = 90, hence E = 75. 2 = 70, and E ′ = 70. In T ′ : Now, we have x ′ 1 = x ′ Go Back – the average grade is worse, but Full Screen – in contrast to T , no one failed. Close Quit
Deciding Which . . . 3. Towards Selecting the Optimal Teaching Strategy: How Techniques Are . . . Possible Objective Functions Towards Selecting the . . . • Fact: the traditional approach – of using the average Towards Selecting the . . . grade as a criterion – is not always adequate. Explicit Solution: . . . Need to Take . . . • Conclusion: other criteria f ( x 1 , . . . , x n ) are needed. Case of Interval . . . • Maximizing passing rate: f = # { i : x i ≥ x 0 } . Case of Fuzzy Uncertainty • No child left behind: f ( x 1 , . . . , x n ) = min( x 1 , . . . , x n ) . Title Page • Best school to get in: f ( x 1 , . . . , x n ) = max( x 1 , . . . , x n ). ◭◭ ◮◮ • Case of independence: decision theory leads to ◭ ◮ f = f 1 ( x 1 ) + . . . + f n ( x n ) for some functions f i ( x i ). Page 4 of 13 • Criteria combining mean E and variance V to take Go Back into account that a larger mean is not always better: Full Screen f ( x 1 , . . . , x n ) = f ( E, V ) . Close • Comment: it is reasonable to require that f ( E, V ) is Quit increasing in E and decreasing in V .
Deciding Which . . . 4. Towards Selecting the Optimal Teaching Strategy: How Techniques Are . . . Formulation of the Problem Towards Selecting the . . . • Let e i ( x i ) denote the amount of effort (time, etc.) that Towards Selecting the . . . is need for i -th student to achieve the grade x i . Explicit Solution: . . . Need to Take . . . • Clearly, the better grade we want to achieve, the more Case of Interval . . . effort we need. Case of Fuzzy Uncertainty • So, each function e i ( x i ) is strictly increasing. Title Page • Let e denote the available amount of effort. ◭◭ ◮◮ • In these terms, the problem of selecting the optimal ◭ ◮ teaching strategy takes the following form: Page 5 of 13 Maximize f ( x 1 , . . . , x n ) Go Back under the constraint Full Screen e 1 ( x 1 ) + . . . + e n ( x n ) ≤ e. Close Quit
Deciding Which . . . 5. Explicit Solution: Case of Independent Students How Techniques Are . . . • Maximize: f 1 ( x 1 ) + . . . + f n ( x n ) under the constraint Towards Selecting the . . . Towards Selecting the . . . e 1 ( x 1 ) + . . . + e n ( x n ) ≤ e. Explicit Solution: . . . Need to Take . . . • Observation: the more efforts, the better results, so we Case of Interval . . . can assume e 1 ( x 1 ) + . . . + e n ( x n ) = e. Case of Fuzzy Uncertainty • Lagrange multiplier: maximize Title Page n n � � J = f i ( x i ) + λ · e i ( x i ) . ◭◭ ◮◮ i =1 i =1 ◭ ◮ • Equation ∂J Page 6 of 13 = 0 leads to f ′ i ( x i ) + λ · e ′ i ( x i ) = 0 . ∂x i Go Back • Thus, once we know λ , we can find all x i . Full Screen � n • λ can be found from the condition e i ( x i ( λ )) = e . Close i =1 Quit
Deciding Which . . . 6. Explicit Solution: “No Child Left Behind” Case How Techniques Are . . . • In the No Child Left Behind case, we maximize the Towards Selecting the . . . lowest grade. Towards Selecting the . . . Explicit Solution: . . . • There is no sense to use the effort to get one of the Need to Take . . . student grades better than the lowest grade. Case of Interval . . . • It is more beneficial to use the same efforts to increase Case of Fuzzy Uncertainty the grades of all the students at the same time. Title Page • In this case, the common grade x c that we can achieve ◭◭ ◮◮ can be determined from the equation ◭ ◮ e 1 ( x c ) + . . . + e n ( x c ) = e. Page 7 of 13 • Students may already have knowledge x (1) ≤ x (2) ≤ . . . Go Back • In this case, we find the largest k for which Full Screen e 1 ( x (0 k ) + . . . + e k ( x (0) k ) ≤ e and then x ∈ [ x (0) k , x (0) k +1 ) s.t. Close e 1 ( x ) + . . . + e k − 1 ( x ) + e k ( x ) = e. Quit
Deciding Which . . . 7. Explicit Solution: “Best School to Get In” Case How Techniques Are . . . • Best School to Get In means maximizing the largest Towards Selecting the . . . possible grade x i . Towards Selecting the . . . Explicit Solution: . . . • The optimal use of effort is, of course, to concentrate Need to Take . . . on a single individual and ignore the rest. Case of Interval . . . • Which individual to target depends on how much gain Case of Fuzzy Uncertainty we will get: Title Page – first, for each i , we find x i for which e i ( x i ) = e , and ◭◭ ◮◮ then ◭ ◮ – we choose the student with the largest value of x i as a recipient of all the efforts. Page 8 of 13 Go Back Full Screen Close Quit
Deciding Which . . . 8. Need to Take Uncertainty Into Account How Techniques Are . . . • We assumed that: Towards Selecting the . . . Towards Selecting the . . . – we know exactly the benefits f ( x 1 , . . . , x n ) of achiev- Explicit Solution: . . . ing knowledge levels x i ; Need to Take . . . – we know exactly how much effort e i ( x i ) is needed Case of Interval . . . for each level x i , and Case of Fuzzy Uncertainty – we know exactly the level of knowledge x i of each Title Page student. ◭◭ ◮◮ • In practice, we have uncertainty : ◭ ◮ – we only know the average benefit u ( x ) of grade x Page 9 of 13 to a student; Go Back – we only know the average effort e ( x ) needed to bring a student to the level x ; and Full Screen – the grade � x i is only an approximate indication of Close the student’s level of knowledge. Quit
Deciding Which . . . 9. Average Benefit Function How Techniques Are . . . • Objective function: f ( x 1 , . . . , x n ) = u ( x 1 )+ . . . + u ( x n ) . Towards Selecting the . . . Towards Selecting the . . . • Usually, the benefit function is reasonably smooth. Explicit Solution: . . . • In this case, if (hopefully) all grades are close, we can Need to Take . . . keep only quadratic terms in the Taylor expansion: Case of Interval . . . u ( x ) = u 0 + u 1 · x + u 2 · x 2 . Case of Fuzzy Uncertainty Title Page • So, the objective function takes the form ◭◭ ◮◮ n n � � x 2 f ( x 1 , . . . , x n ) = n · u 0 + u 1 · x i + u 2 · i . ◭ ◮ i =1 i =1 Page 10 of 13 n n � � • Fact: E = 1 x i and M = 1 Go Back x 2 i = V + E 2 . n · n · Full Screen i =1 i =1 • Conclusion: f depends only on the mean E and on the Close variance V . Quit
Deciding Which . . . 10. Case of Interval Uncertainty How Techniques Are . . . • Situation: we only know intervals [ x i , x i ] of possible Towards Selecting the . . . values of x i . Towards Selecting the . . . Explicit Solution: . . . • Fact: the benefit function u ( x ) is increasing (the more Need to Take . . . knowledge the better). Case of Interval . . . • Conclusion: Case of Fuzzy Uncertainty – the benefit is the largest when x i = x i , and Title Page – the benefit is the smallest when x i = x i . ◭◭ ◮◮ � n � � � n ◭ ◮ • Resulting formula: [ f, f ] = u ( x i ) , u ( x i ) . i =1 i =1 Page 11 of 13 • Reminder: for quadratic u ( x ) and exactly known x i , Go Back we only need to know E and M . Full Screen • New result: under interval uncertainty, we need all n Close intervals. Quit
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