Richard M. Grassl

Team teaching and cooperative groups in abstract algebra: nurturing a new generation of confident mathematics teachers

Experiences in designing and teaching a reformed abstract algebra course are described. This effort was partially a result of a five year statewide National Science Foundation (NSF) grant entitled the Rocky Mountain Teacher Enhancement Collaborative. The major thrust of this grant was to implement reform in core mathematics courses that would effect a shift to a learning paradigm from the traditional instructional paradigm. The central teaching action involved in this reform of abstract algebra was the use of team teaching in a cooperative group setting. The liaison of a mathematician working collaboratively with a mathematics educator provided valuable lessons in understanding key components of effective teaching. A series of qualitative interviews were conducted along with several supporting surveys. The survey data along with various forms of student evaluations served as a basis for drawing conclusions for the study.

Impact of proof validation on proof writing in abstract algebra

Many undergraduate students have difficulty writing mathematical proofs even though this skill is important for the development of future teachers and those who may be involved in instruction or training as a graduate student or supervisor. In addition, research indicates that mathematics majors and secondary education mathematics majors possess little ability to determine whether arguments are valid proofs. Asking students to validate proofs using specific activities may not only help undergraduates develop the ability to validate proofs, but also may help increase their own proof writing abilities. The purpose of this study was to investigate the effect of proof validation on the ability to write proofs. Two sections of abstract algebra were the focus of a ‘guided research’ study–one section was taught using a weekly proof validation activity and the other was taught as a control group. Data analysis revealed that the validation group significantly outscored the control group on items involving proof writing on the final exam. Results of the proof validation activities are also presented.

PASCALIAN RECTANGLES MODULO m

Abstract We develop a recursive technique for optimizing the sums and as a by-product produce two new proofs of the Ramus formula Recently these special sums have surfaced in the study of fault tolerance in the hypercube computer.