Déjà vu: a study of study group formation

By Chris Brooks and Maggie Penn

Introduction

There are twenty students in a fifteen weeklong computational economics class. They are given one homework assignment each week, which they are allowed to collaborate on. The assignments are mathematical in nature, but also need to be written up nicely. The amount of math and writing needed varies from assignment to assignment. Since assignments are due the very next day, students must form into study groups before they know the exact nature of each assignment. Some students are better at math, others are better at writing, and some are good or bad at both. For the first assignment the students are randomly paired up. They get to know the type of their partner, but no one else. For all subsequent assignments students get to form into groups of their own choosing. However students only get to propose to be in one other group, and the group may reject them. As the semester progresses, they learn the types of the other students who they have worked with. Students also learn from other students over the course of the semester, and can thus boost up their own type by working with smarties.

The Model

· Each student can be described by a two-dimensional vector, (m, w), where m represents the student’s ability to do math problems and w represents the student’s writing ability. Before the first homework assignment the students’ types are randomly drawn from a uniform [0, 1] X [0,1] distribution.

· Before each assignment is assigned, study groups are formed. For the first assignment, 10 groups are randomly assigned. For all subsequent assignments, the students form their own groups as follows: first the students are randomly ordered. Then the first student in line makes an offer to another student to form a group. The other student either accepts or rejects the offer. Then the second student in line makes an offer to either another student or a group, asking to join. Again, the offer is either accepted or rejected. The process is repeated until everyone has made an offer. Each student can make only one offer.

· Students will offer to be with the individual or group who he believes will help him get the highest assignment grade. A student will accept the offer of another student if he believes the expected value of his homework grade does not go down as a result of accepting the student. A group will accept a student into the group by voting unanimously on the prospective student, where everyone votes according to his or her beliefs. Beliefs will be discussed later.

· For group i, let m_i^max and w_i^max denote the maximum values that m and w take over all members of the group. Let m_i^AVG and w_i^AVG denote the average math and writing values over all of the individuals in group i. After each assignment a student in group i with type (m, w) changes types if m < m_i^max or w < w_i^max or both. His new math type will be 0.5 * (m + m_i^max ) if, for example, m < m_i^max .

· Each student holds beliefs about every other student’s type. If a student has never been in a group with another student, his belief is that the other student has type (0.5, 0.5). If a student has been in a group with another student, then he believes that student to have whatever that student’s updated type was after they turned in their last assignment together.

· Each homework assignment requires different levels of math and writing ability, and these levels are unknown to the students before they form into groups. Let a_k and b_k denote the importance of math and writing, respectively, on homework assignment k. The weight a_k is distributed uniformly over [0, 1] and b_k equals 1 - a_k .

· The homework grade is a function of the math and writing grades of the people in the group. Group i’s grade on assignment k is represented by the function f(m, w), where

f(m, w) = .5[ a_k (m_i^max + m_i^AVG ) + b_k (w_i^max + w_i^AVG )]

Results

• Under majority rule, students separate quickly into one large group and several small ones (regardless of the starting size of groups). The largest group has 7-9 students generally.

• Students in small groups seem to be there by choice; they have high starting values and don’t get an opportunity to learn about other students. Generally both values are above 0.5, and one value is significantly higher than 0.5.

• Under unanimity rule groups remain small and there is more movement of students. We found no group with more than two students in it.

• While there appears to be no trend in grades over time, the average group grade is above 0.5.

• People that consistently do not belong to any group generally have high or low types. The more average students tend to group together.

• Extensions could include:
• Changing how individuals update their beliefs about types
• Increasing the dimensionality of the type space
• Looking at different voting rules
• Nonstationary student populations
• Changing how people “learn”