Like many professors, I am strongly concerned with the transition mathematics majors must make from technique-oriented learning to proof-oriented learning. In the spring of 2006, I developed and taught a course entitled "Introduction to proof-oriented mathematics". The main idea of the course is to take a "problems-first" approach to teaching proofs. Instead of the standard "hear lecture, do problems, get feedback" order of affairs, the problems are introduced first, and the students get almost immediate feedback, so that they can appreciate the proofs done in the following lecture. As the syllabus elaborates,
We break each topic into two-day units. Before the first day, you will be given some problems to work on. You will then hand your progress on these problems in the form of a "working paper" at the beginning of the first day. Progress may range from exploring some examples and breaking the problem down into smaller questions, to giving full solutions. Then on the first day of the unit we will work on groups on these problems (and beyond), handing in further progress when appropriate. The second day will be a lecture, based on the material covered by the problems from the first day. We will formalize problem-solving techniques and give proofs of facts which apply generally. The proofs will hopefully be easier to follow, since we will have encountered the main ideas on the first day.The problems were from the text Mathematical Thinking: Problem-solving and Proofs by D'Angleo and West. This book has an unusual abundance of good problems. Even the section on basic logic (contrapositives, and the like) had fun problems such as "Say whether the following assertion is likely true: `All of my five-legged dogs can fly.'" The book also introduces a healthy variety of topics. One challenge for the students is that the book uses concise mathematical language throughout (in contrast to say The Heart of Mathematics, which I like to use for freshman seminars and "math for poets" classes). But because we could read the book together during our group-work, coping with the level of language of the book became a useful exercise in itself. Students have appreciated this, saying "My current math texts seem so easy that I can read them for fun." In general the students seemed to gain a great deal of confidence from taking the class, saying "I feel bad at this class, but my other math classes seem easy now." From what I could tell it strongly encouraged their mathematical development. This course is probably not ideal for all students and professors (no course is), but it seemed to work well in this instance.
In many senses, I have been developing this course for years. The greatest influence on this development was my four years at the PROMYS program in mathematics for gifted high-school students, which employs a similar philosophy. I took such an example-first philosphy as the primary focus in developing a class on experimentation and proof in mathematics, based on the mathematics laboratory class at Mt. Holyoke college. But the current class is more old-fashioned, not relying on using a computer to generate conjectures. Elements of this philosophy are also present in smaller doses in some of my other classes such as linear algebra, where I introduce iterated function systems as a way in which affine-linear transformations can encode remarkably intricate geometry, which provides strong motivation to analyze and prove facts about such transformations.