Teachers’ Practices and Beliefs Regarding Teaching Tuning in Elementary and Middle School Group String Classes

The purpose of this study was to examine teachers’ practices and beliefs related to the teaching of stringed instrument tuning in elementary and middle school group classes. The aspects examined included the following: (a) teachers’ beliefs about teaching tuning in their string classes, (b) activities teachers used when teaching tuning in string classes, (c) methods for assessing students’ tuning skills, (d) the grade level that tuning instruction begins, (e) the amount of instructional time used for teaching tuning, (f) teachers’ goals for students’ tuning independence, and (g) teacher, program, or school characteristics that affect how and when tuning is taught. Participants ( N = 139) reported that the average time required to develop tuning independence is 4.5 years. The amount of instructional time available and the level of students’ aural skills were the greatest perceived obstacles to developing tuning independence. Significant differences were found in tuning activities, beliefs about students’ tuning abilities, and assessment procedures and were based on participants’ age, teaching experience, and grade levels taught. The findings indicate a need for further development of tuning pedagogy and greater use of formal assessment to determine if students have developed the prerequisite skills for tuning.

LOOP GROUPS, STRING CLASSES AND EQUIVARIANT COHOMOLOGY

We give a classifying theory for LG-bundles, where LG is the loop group of a compact Lie group G, and present a calculation for the string class of the universal LG-bundle. We show that this class is in fact an equivariant cohomology class and give an equivariant differential form representing it. We then use the caloron correspondence to define (higher) characteristic classes for LG-bundles and to prove a result for characteristic classes for based loop groups for the free loop group. These classes have a natural interpretation in equivariant cohomology and we give equivariant differential form representatives for the universal case in all odd dimensions.

On the vanishing problem of string classes

The ordinary string class is an obstruction to lift the structure group LSpin(n) of a loop group bundle LQ → LM to the universal central extension of LSpin(n) by the circle. The vanishing problem of the ordinary string class and generalized string classes are considered from the viewpoint of the ring structure of the cohomology H*(M; R).