TITLE: Teaching undergraduate students to write proofs
QUESTION [43 upvotes]: In my experience, there are roughly two approaches to teaching (North American) undergraduates to write proofs:

Students see proofs in lecture and in the textbooks, and proofs are explained when necessary, for example, the first time the instructor shows a proof by induction to a group of freshman, some additional explanation of this proof method might be given.  Also, students are given regular problem sets consisting of genuine mathematical questions - of course not research-level questions, but good honest questions nonetheless - and they get feedback on their proofs.  This starts from day one.  The general theme here is that all the math these students do is proof-based, and all the proofs they do are for the sake of math, in contrast to:
Students spend the majority of their first two years doing computations.  Towards the end of this period they take a course whose primary goal is to teach proofs, and so they study proofs for the sake of learning how to do proofs, understanding the math that the proofs are about is a secondary goal.  They are taught truth tables, logical connectives, quantifiers, basic set theory (as in unions and complements), proofs by contraposition, contradiction, induction.  The remaining two years consist of real math, as in approach 1.

I won't hide the fact that I'm biased to approach 1.  For instance, I believe that rather than specifically teaching students about complements and unions, and giving them quizzes on this stuff, it's more effective to expose it to them early and often, and either expect them to pick it up on their own or at least expect them to seek explanation from peers or teachers without anyone telling them it's time to learn about unions and complements.  That said, I am genuinely open to hearing techniques along the lines of approach 2 that are effective.  So my question is:
What techniques aimed specifically at teaching proof writing have you found in your experience to be effective? 
EDIT: In addition to describing a particular technique, please explain in what sense you believe it to be effective, and what experiences of yours actually demonstrate this effectiveness.

Thierry Zell makes a great point, that approach 1 tends to happen when your curriculum separates math students from non-math students, and approach 2 tends to happen math, engineering, and science students are mixed together for the first two years to learn basic computational math.  This brings up a strongly related question to my original question:
Can it be effective to have math majors spend some amount of time taking computational, proof-free math courses along with non-math majors?  If so, in what sense can it be effective and what experiences of yours demonstrate this effectiveness?
(Question originally asked by Amit Kumar Gupta)

REPLY [9 votes]: I taught the `transitions' course at a large state university a number
of years ago, with reasonable success. 
The clientele of this (purely elective) course was mainly B students 
in calculus who would likely have done poorly in real analysis
or abstract algebra, and would have had difficulty completing a math major.
To maximize the impact on students' ability to understand and produce 
proofs, several things were important:
a) The text was Velleman's How to Prove It: A structured approach, 
which is readable by average students, clearly delineates the structure and
construction of typical proofs, and is full of problems which are 
elementary but not boring.  (For a regular beginning analysis class
I just ask the students to read this book---esp. chapter 3 as mentioned
by Jon Bannon---and I discuss the basics of this material 
for a few lectures.)
b) The format of most class sessions was discussion not lecture. To have these students passively listen, like in their
previous courses which they demonstrably failed to master, would be
useless.  Discussion was structured like in a humanities or language
course, led by the instructor with specific goals in mind and calling
on individual students to involve everyone and make sure they get it.
The 22 students were informed that it was essential that they come to 
class prepared, having read the day's material and having worked the
relevant problems, laid out in each week's syllabus.
c) Why we insist on ``proof beyond unreasonable doubt'' was explained,
referring to the great discoveries of 19th and early 20th century 
analysis (especially regarding infinite sets and fractals)
that demand the enormously skeptical approach to establishing truth
which now dominates much of modern mathematics.
Many of the students were weak at the start and apparently benefited
from all this. For one, this course was a big step in eventually
changing his career from fisherman to gaining a Masters and working in a scientific software company. Another later did A work in a senior-level ODE course I taught.
But I did not conduct a randomized controlled study.