Two variables are defined. Let Y equal the number of persons in the group, and X the number of personal relationships between the members. Then using the formula X = (Y2 - Y)/2 (that is, X equals Y squared minus Y all divided by 2), as Bossard1 has, we find that the larger the group, the more disproportionate the increase in personal relationships.
Size of group: 2 3 4 5 8 12 15 35
Relationships: 1 3 6 10 28 66 105 595
Note how radically the number of relationships increases with the addition of one or two people. What does this do to the individual in terms of communication, understanding, and ability to participate without pressure or frustrations?
Hundreds or thousands may be spectators. Working, interacting groups seem to do best when composed of five to eight members. If the group is larger, some become performers and others spectators. At age six, spontaneous groups seldom exceed three or four children. Sizes now accepted for school classes are much too large for good cooperative work.2
If the mathematical formula is a hang-up, try drawing the relationships on a page of paper to convince yourself of the truth of this work. See the example at the end of this blog.
1 James H. S. Bossard. The Sociology of Child Development. New York: Harper and Brother, 1948. p. 146.
2 This whole section is adapted from an article by Mary Margaret Scobey, entitled "Developing and Using Classroom Groups," 1960. See http://ascd.com/ASCD/pdf/journals/ed_lead/el_196312_scobey.pdf.
3 Cole, Neil. Cultivating a Life For God: Multiplying Disciples Through Life Transformation Groups. Carol Stream: ChurchSmart Resources, 1999.