I discuss my experience with teaching an advanced undergraduate Real Analysis class using both lecturing and the small-group guided discovery method. The article is structured as follows. The first section is about the organizational and administrative components of the class. In the second section I give examples of successes and difficulties that the students had with the mathematical content of the course. My evaluation is influenced by the main goal of such a Real Analysis class: students should be able to use, read, and write sound mathematical proofs that lead to the comprehension of the fundamental principles of Analysis. In the third section I present some of the students' comments regarding the method, and I summarize my own conclusions in the fourth section. In an Appendix, as a way to test the progress made by the students, I examine how the concept image of "real number" changed and evolved during the term. (Contains 4 footnotes.)