How to Divide Optimal Division into . . . Students into Groups - PowerPoint PPT Presentation

Formulation of the . . . How to Describe the . . . Possible Objective . . . Optimal Division into . . . How to Divide Optimal Division into . . . Students into Groups Combined Optimality . . . A More Nuanced Model so as to Optimize Learning: Optimal Division into . . . Case of Uncertainty Towards a Solution Home Page to a Pedagogy-Related Title Page Optimization Problem ◭◭ ◮◮ ◭ ◮ Olga Kosheleva 1 and Vladik Kreinovich 2 Page 1 of 18 Go Back Departments of 1 Teacher Education and 2 Computer Science University of Texas at El Paso, El Paso, TX 79968, USA Full Screen olgak@utep.edu, vladik@utep.edu Close Quit

Formulation of the . . . How to Describe the . . . 1. Formulation of the Problem Possible Objective . . . • Students benefit from feedback. Optimal Division into . . . Optimal Division into . . . • In large classes, instructor feedback is limited. Combined Optimality . . . • It is desirable to supplement it with feedback from A More Nuanced Model other students. Optimal Division into . . . • For that, we divide students into small groups. Case of Uncertainty Home Page • The efficiency of the result depends on how we divide Title Page students into groups. ◭◭ ◮◮ • If we simply allow students to group themselves to- gether, often, weak students team together. ◭ ◮ Page 2 of 18 • Weak students are equally lost, so having them solve a problem together does not help. Go Back • It is desirable to find the optimal way to divide students Full Screen into groups. This is the problem that we study. Close Quit

Formulation of the . . . How to Describe the . . . 2. Need for an Approximate Description Possible Objective . . . • A realistic description of student interaction requires a Optimal Division into . . . multi-D learning profile of each student: Optimal Division into . . . Combined Optimality . . . – how much the students knows of each part of the A More Nuanced Model material, Optimal Division into . . . – what is the student’s learning style, etc. Case of Uncertainty • Such a description is difficult to formulate and even Home Page more difficult to optimize. Title Page • Because of this difficulty, in this paper, we consider a ◭◭ ◮◮ simplified description of student interaction. ◭ ◮ • Already for this simplified description, the correspond- Page 3 of 18 ing optimization problem is non-trivial. Go Back • However, we succeed in solving it under reasonable as- Full Screen sumptions. Close Quit

Formulation of the . . . How to Describe the . . . 3. How to Describe the Current State of Learning Possible Objective . . . • We assume that a student’s degree of knowledge can Optimal Division into . . . be described by a single number. Optimal Division into . . . Combined Optimality . . . • Let d i be the degree of knowledge of the i -th student S i . A More Nuanced Model • We consider subdivisions into groups G k of equal size. Optimal Division into . . . • If two students with degrees d i < d j work together, Case of Uncertainty then the knowledge of the i -th student increases. Home Page • The more S j knows that S i doesn’t, the more S i learns. Title Page • In the linear approximation, the increase in S i ’s knowl- ◭◭ ◮◮ edge is thus proportional to d j − d i : ◭ ◮ d ′ i = d i + α · ( d j − d i ) . Page 4 of 18 • In a group, each student learns from all the students Go Back with higher degree of knowledge: � Full Screen d ′ i = d i + α · ( d j − d i ) . Close j ∈ G k ,d j >d i Quit

Formulation of the . . . How to Describe the . . . 4. Discussion: Group Subdivision Should Be Dy- Possible Objective . . . namic Optimal Division into . . . • Students’ knowledge changes with time. Optimal Division into . . . Combined Optimality . . . • As a result, optimal groupings change. A More Nuanced Model • So, we should continuously monitor the students’ knowl- Optimal Division into . . . edge and correspondingly re-arrange groups. Case of Uncertainty Home Page • Ideally, we should also take into account that there is a cost of group-changing: Title Page – before the student start gaining from mutual feed- ◭◭ ◮◮ back, ◭ ◮ – they spend some effort adjusting to their new groups. Page 5 of 18 Go Back Full Screen Close Quit

Formulation of the . . . How to Describe the . . . 5. Possible Objective Functions Possible Objective . . . n � = 1 Optimal Division into . . . def • First, we will consider the average grade a n · d i . Optimal Division into . . . i =1 Combined Optimality . . . • Another reasonable criterion is minimizing the number A More Nuanced Model of failed students. Optimal Division into . . . • In this case, most attention is paid to students at the Case of Uncertainty largest risk of failing, i.e., with the smallest d i . Home Page • From this viewpoint, we should maximize the worst Title Page def = i =1 ,...,n d i . min grade w ◭◭ ◮◮ • Many high schools brag about the number of their ◭ ◮ graduates who get into Ivy League colleges. Page 6 of 18 • From this viewpoint, most attention is paid to the best Go Back students, so we should maximize the best grade Full Screen def b = max i =1 ,...,n d i . Close Quit

Formulation of the . . . How to Describe the . . . 6. Optimal Division into Pairs: Our Theorems Possible Objective . . . • To maximize the average grade a : Optimal Division into . . . Optimal Division into . . . – we sort the students by their knowledge, so that Combined Optimality . . . d 1 ≤ d 2 ≤ . . . ≤ d n , A More Nuanced Model – in each pair, we match one student from the lower Optimal Division into . . . half with one student from the upper half. Case of Uncertainty Home Page • To maximize the worst grade w : Title Page – we sort the students by their knowledge; – we pair the worst-performing student (corr. to d 1 ) ◭◭ ◮◮ with the best-performing student (corr. to d n ); ◭ ◮ – if there are other students with d i = d 1 , we match Page 7 of 18 them with d n − 1 , d n − 2 , etc.; Go Back – other students can be paired arbitrarily. Full Screen • In this model, subdivision does not change the best grade b (this is true for groups of all sizes g .) Close Quit

Formulation of the . . . How to Describe the . . . 7. Optimal Division into Groups of Given Size g Possible Objective . . . • To maximize the average grade a , we: Optimal Division into . . . Optimal Division into . . . – sort the students by their knowledge, and, based Combined Optimality . . . on this sorting, divide the students into g sets: A More Nuanced Model L 0 = { d 1 , d 2 , . . . , d n/g } , . . . , L g − 1 = { d ( g − 1) · ( n/g )+1 , . . . , d n } ; Optimal Division into . . . – in each group, we pick one student from each of g Case of Uncertainty sets L 0 , L 1 , . . . , L g − 1 . Home Page • If there is only one worst-performing student, then, to Title Page maximize the worst grade w , we: ◭◭ ◮◮ – sort the students by their knowledge d 1 ≤ d 2 ≤ . . . ; ◭ ◮ – combine the worst-performing student (corr. to d 1 ) Page 8 of 18 with best ones (corr. to d n , . . . , d n − ( g − 2) ); Go Back – group other students arbitrarily. Full Screen • If we have s equally low-performing students d 1 = d 2 = . . . = d s , we match each with high performers. Close Quit

Formulation of the . . . How to Describe the . . . 8. Combined Optimality Criteria Possible Objective . . . • If we have several optimal group subdivisions, we can Optimal Division into . . . use this non-uniqueness to optimize another criterion. Optimal Division into . . . Combined Optimality . . . • Example: A More Nuanced Model – first, we optimize the average grade; Optimal Division into . . . – among all optimal subdivisions, we select the ones Case of Uncertainty with the largest worst grade; Home Page – if there are still several subdivisions, we select the Title Page ones with the largest second worst grade, etc. ◭◭ ◮◮ – etc. ◭ ◮ • Optimal subdivision into pairs: Page 9 of 18 – sort the students by their knowledge, d 1 ≤ d 2 ≤ . . . Go Back – match d 1 with d n , d 2 with d n − 1 , . . . , d k with d n +1 − k , Full Screen . . . Close Quit

Formulation of the . . . How to Describe the . . . 9. Combined Optimality Criteria (cont-d) Possible Objective . . . • Optimality criterion (reminder): Optimal Division into . . . Optimal Division into . . . – first, we optimize the average grade; Combined Optimality . . . – among all optimal subdivisions, we select the ones A More Nuanced Model with the largest worst grade; Optimal Division into . . . – if there are still several subdivisions, we select the Case of Uncertainty ones with the largest second worst grade, etc. Home Page – etc. Title Page • Optimal subdivision into groups of size g : ◭◭ ◮◮ – sort the students by their knowledge, and divide ◭ ◮ into g sets L 0 , . . . , L g − 1 ; Page 10 of 18 – match the smallest value d 1 ∈ L 0 with the largest Go Back values from each set L 1 , . . . , L g − 1 , – match the second smallest value d 2 ∈ L 0 with the Full Screen second largest values from L 1 , . . . , L g − 1 , etc. Close Quit