An education major wanted to test the efficacy of teaching methods
for the division of fractions.  Two new methods along with the
standard method were studied.  Five teachers were trained in all
methods and taught a total of twelve classes.  Assignments were
made in the following manner:

Teacher 1: Method A, Method A, Method B
Teacher 2: Method A, Method C
Teacher 3: Method A, Method B, Method C
Teacher 4: Method B, Method B
Teacher 5: Method A, Method B

The response was a test score for the students in the classes

Discuss what design approaches you would have used for this experiment;
you can be flexible in the number of teachers you use, but keep the
number of classes at 12. What are some of the shortcomings of the 
current approach?  What difficulties will they cause in the analysis?

Since teachers had been trained in all methods, students agreed that
a RCBD (with teacher as block) would have been appropriate here.  In
the current design, teacher and treatment are not orthogonal and there's
some unnecessary partial confounding (e.g., Teacher 4 and Method B).
The missing cells cause all least squares means for Method in the 
interaction model to be non-estimable (see exercises).  The Type 
III contrasts are unusual as well.

Students thought a RCBD with only 4 teachers would have worked well 
(provided we were constrained to study 12 classes).  Many noted that 
the class effect was probably important but was not accounted for 
with a RCBD. Several students have suggested BIBD's. If we want to
maintain the same marginals for each teacher, we could use a
CRD for Teachers 1 and 3 and a BIBD for Teachers 2, 4 and 5.
Some students thought the classes might be
ordered and created a replicated latin square with 6 teachers.
Other students used 6 teachers and a BIBD with k=2.