This article describes an innovative method for teaching of mathematics which was utilized to teach abstract algebra to a class of mathematics education majors at a small liberal arts college. A variation of R.L. Moore's Discovery Method was employed in conjunction with substantial use of mathematical software. Although student reactions were initially negative, students grew to accept and even embrace the instructional methods. Anonymous teaching evaluations administered at the end of the semester were favorable. As measured by examination performance, student achievement in this experimental class section was high. This teaching experiment was extremely labor-intensive from the instructor's perspective. Otherwise this teaching experiment was a positive experience for both the instructor and the students.