3.3.1 Maximally Diverse Groups Many group assignment algorithms have focused on creating maximally diverse groups where the sum of pairwise differences of some characteristics is maximized. The difference between two students and can be defined as
The goal of creating maximally diverse groups is then formulated as adding up the differences for all groups and all student pairs within each groups, that is,
The members of the groups are of being represented as . This is the same formulation used by other authors [ 22].
3.3.2 Evenly Skilled Groups One should not assume that groups will work well just because they are formed properly according to some criteria. In some group projects, it makes a lot of sense to encourage the teams explicitly to teach each other relevant skills and knowledge that some but not all group members have, consistent with reciprocal teaching [ 14]. Originally, reciprocal teaching was used to improve reading comprehension where students would take turns in leading the discussion. In this way, the students took on the role of the teacher. In a situation where students have a slightly different skill set, they should help the other students to learn it. However, this may not come naturally without the explicit encouragement of the instructor who is not present in most of the group meetings. Thus, students should be assigned to teams such that all the relevant skills are covered. This not only improves the chance of doing a good project, but it also enables reciprocal teaching.
The even skilled criterion is especially interesting for skilled, yet diverse student populations. They can be found interdisciplinary programs such as cognitive science or human-computer interaction. Other typical situations matching this student profile are business administration (MBA) courses and programs helping professionals to find new careers. The latter has become especially important in the current economic situation where many jobs disappear and people have to learn a new profession. It surely would be useful to take advantage of these students' existing skills in the classroom.
Table 1 shows a small example with three groups, two skills and , and four students per group. Each skill is rated on a scale from 1 to 5 with 5 being best. The last line displays the maximal rating for each skill and group. For instance, skill gets a rating of 3 for groups 1 and 2, suggesting that these groups will have problems because nobody is very good at skill . On the other hand, group 3 has a rating of 5 for skill which means that this group will do fine for skill . Furthermore, if the students indeed teach each other as they were asked to by the instructor at the beginning of the course, the first student in group 3 with the high rating will teach the others. Thus, the other students in that group will improve for this skill probably more than the members in groups 1 and 2. Therefore, the worst value for skill across all three groups is and for it is . These two values are computed by (3). Since we want to have high values for all skills, we maximize the worst skill, that is, needs to maximized. As explained below, a slightly different function shown in (4) is maximized to improve the performance of the search algorithm.
Therefore, we define a new optimization criterion, evenly skilled groups, by maximizing the minimal skill for each group. Assume that we want student groups to have some minimal expertise in designing a system architecture, in user experience design and writing reports. The group skill for each of these individual characteristics is the maximum over all the members' skills. Since we want to maximize the worst skill of the group, we maximize over the minimal group skill. This avoids the problem of having imbalances that can result from a global sum of differences only as discussed in Section 2. Let be the worst value for skill across all groups, that is,
Then, we ought to maximize over the worst skill, that is, maximize . However, this is problematic because this function is not smooth, giving the search very little information about the quality of the individual groups. For instance, if the skills are rated on an -level Likert scale, can assume different values. Considering that tends be small, say five or seven, this is not enough information for the search algorithm to differentiate between solutions of somewhat different quality. Therefore, an additional factor is added such that the sum of the values is also considered. This results in up to more values than without this factor. Although this reintroduces the danger of having imbalanced groups to a minor degree, it contains much more information than the max factor alone. Thus, the actual optimization criterion is
3.4.1 Preferring Friends and Avoiding Foes Evidence suggests that groups perform better if the students can be in a group with students they like or prefer to work with for some other reason [ 9], [ 17]. Assume that we let the students prefer to work with some students and avoid some of the other students. Here, the former will be called friends and the latter foes. represents the preference of student to be in the same group as student , where . Of course, is not necessarily equal to :
A variation of this approach is to change the definition of such that being not in the same group is a stronger preference than being in the same group. This can be simply accomplished by replacing with a smaller value, say , in the definition of in (6).
Assuming that the friends-and-foes preferences should not influence the assignment too much, has to have a relatively small value. Based on the observation that relatively few of these preferences were stated by the students, was set to resulting in for most cases which is much less than . The value of is the worst skill across all groups which tends to be 3 or more (see Tables 3, 4, 5, and 6). This is less arbitrary than it may appear at first. First, the s are normally set once for a certain context. Second, the results are quite stable even if the values are changed. What matters is that the relative sizes of the s are chosen reasonably.
3.4.2 Distributing Subsets of Students Sometimes, it can be useful to request that a subset of students be evenly distributed over the groups. For instance, in a mixed course, we have a few students that are participating in the course from remote locations via a synchronous connection. One way of dealing with group projects is to have roughly one remote student per project. This can be easily accomplished using the “foes” idea from the previous section.
3.4.3 Assigning Students to Specific Groups Assigning certain students to certain types of groups can be useful. For instance, students can be assigned to larger groups if the students are a bit less experienced or new to the specific academic program. In this case, we simply add a penalty if this condition is violated. Assume that we want to make sure that the students in set are assigned to groups with at least members. Function returns if a student is assigned to a too small group and 0 otherwise:
The three functions , , and represent a sample of optimization criteria that come from actual classroom requirements. Of course, this sample is not exhaustive, though it shows how new requirements can be added.
3.4.4 More Preferences These examples show that some preferences occurring in educational contexts can be expressed in a relatively simple way without having to change the algorithm itself. However, can all interesting preferences be expressed in the given framework, and how does an instructor with a mathematically average background express such preferences?
Any preference needs to be translated into a function of the form . The larger , the better the student assignment is for this preference. It is advisable to scale so that it has values in a known range, because this makes it easier to weigh it relative to other preferences using . The optimization criteria used or suggested by other group formation approaches are quite easy to formulate in the framework used here. These criteria include heterogeneous groups [ 13], [ 21], [ 16], homogeneous groups [ 17], mixtures of homogeneous and heterogeneous groups [ 25], student location and availability [ 4], meeting capability and gender balance [ 38], and compatibility of students [ 29].
However, we should not expect a regular instructor to have to develop the mathematical formulation of the preferences. They need a proper user interface that allows them to formulate the preferences in pedagogical and organizational terms, not mathematical formulas. This is another reason why it is important that all optimization criteria have a clear interpretation in the domain in which the criteria are originally stated. If the only explanation for a group formation approach is an interesting algorithm, we may lack the possibility of providing a useful and usable interface to the end user.
The author is with the Department of Information Design and Corporate Communication, Bentley University, 175 Forest Street, Waltham,
MA 02452-4705. E-mail: email@example.com.
Manuscript received 23 Mar. 2009; revised 18 June 2009; accepted 23 Mar. 2010; published online 15 July 2010.
For information on obtaining reprints of this article, please send e-mail to: firstname.lastname@example.org, and reference IEEECS Log Number TLT-2009-03-0039.
Digital Object Identifier no. 10.1109/TLT.2010.17.
Roland Hübscher received the PhD degree in computer science from the University of Colorado at Boulder. He is currently an associate professor at Bentley University, where he teaches in the Human Factors in Information Design program. His research focuses on intelligent user interfaces and algorithms to support students and teachers.