Multidimensional Scaling for Selecting Small Groups in College Courses
Pardoe, Iain, The American Statistician
I have used group work in my undergraduate statistics course for a number of years, and have found it to be a useful method for improving student learning by raising student interest and increasing class participation. As well as working together during class time, students work extensively together in groups outside of class on homework assignments and projects. In managing small groups in the classroom--in my case groups of 3-5 students in a class of approximately 60--I have experimented with various ad-hoc methods for selecting the groups, each of which have had their drawbacks. For example, randomly selecting groups has led to frequent student complaints that they have difficulties meeting as a group outside of class time due to incompatible schedules. On the other hand, allowing students to self-select groups has tended to produce groups of friends in which there is very little diversity (gender, age, and ethnicity, as well as academic ability). In an attempt to balance the conflicting goals of selecting groups whose members have mostly similar schedules while at the same time maintaining group diversity. I have developed a method for using multidimensional scaling (MDS) to accomplish this task.
The use of small groups in college courses stems from the concept of cooperative learning, whereby groups of, say, three to five students work together as a team to solve a problem or complete an assignment [see Garfield (1993); Giraud (1997); Keeler and Steinhorst (1995); and Magel (1998) for examples in the field of statistics]. The National Council of Teachers of Mathematics (2000) and the National Research Council (1989) advocated cooperative learning in elementary and secondary education, while Johnson, Johnson, and Smith (1991) and Garfield (1993) extolled the virtues of cooperative learning in the college classroom. Johnson et al. (1991) showed that when students work together, they often accomplish more, and at a higher level, than they could individually. Garfield (1993, par. 8) cited published research that suggests that "the use of small group learning activities leads to better group productivity, improved attitudes, and sometimes, increased achievement."
With reference to the question of how to select cooperative learning groups, Garfield (1993, par. 18) noted that "the instructor may allow students to self-select groups or groups may be formed by the instructor to be either homogeneous or heterogeneous on particular characteristics (e.g., grouping together all students who received A's on the last quiz, or mixing students with different majors)." The remainder of this article describes how to use MDS to select groups to be homogeneous on student schedules. The method also enables inclusion of further criteria for group selection, such as making sure that each group has at least one member with a particular skill. The method is described in sufficient detail that it can be applied to any course in which cooperative learning groups are used, and can easily be adapted to work with characteristics other than student schedules.
The next section briefly describes MDS and how it can be applied to the problem of grouping students with similar schedules. The following section presents results from my undergraduate course in the winter quarter of 2004 and an evaluation of how well the method worked. Presenting and evaluating the results also provides valuable opportunities for covering a number of statistical concepts in class, including surveys, association measures, multidimensional scaling, and statistical graphics. The final section contains a discussion.
2. MULTIDIMENSIONAL SCALING FOR SELECTING GROUPS
MDS is a series of methods for displaying a set of objects in low-dimensional space (often 2D) that reflects similarities between the objects (see Kruskal and Wish 1978 for an overview). MDS results can be used to create a 2D map where the physical distances between the objects on the map are meant to correspond closely with the measured object similarities. …