I guess this is the order mathematics was taught in the authors university, in my experience with taking mathematics courses as part of a engineering degree, way too often there is no attempt at a "pre-rigorous" stage at all, as applications and intuitions are seen with contempt by the more formally-minded teachers. Vladimir Arnold's essay on this topic is a classic that I reread every few months for inspiration:
Richard Courant, the classic book of which is referenced in the article, remarks in the preface of one of his other books how the logical order of presentation of mathematical theories is much different from the order that works best pedagogically. I wish more educators would understand this seemingly simple observation.
As a teacher, the reason I teach with formality before allowing students to be informal, is that formality is training wheels.
Initially, the students do not know how to write a coherent proof. That is why we give them formal tools such as induction, and require them to write clearly, detailing their steps. When we do not do this, we see catastrophic lack of logical and coherent thought. Sometimes, this is the result of them not being able to _write_ their thoughts coherently, but more often, it is a results of their thoughts not _being_ coherent in the first place. Formality is used, partly, as forced coherency in writing, in an attempt to induce coherency in thought.
Once the students are able to write coherent proofs, sure, let them ride fast, let them "freestyle" proofs, with the confidence that what they're writing is what they intend to write, and with the skill to communicate concisely, clearly and unambiguously with informal language. If they can do it, more power to them. Often times, however, they need a lot of time with the training wheels, before they can ride fast and not fall disastrously.
Indeed, by the time that you get to an introductory analysis course in the US (a 300-level undergraduate course!), students generally are confused about what the givens are in a proof, and what exactly they are trying to prove, let alone getting from point a to point b. Most have not had to "prove" anything since 9th grade geometry, and are used to plug-and-chug solutions backed by geometric concepts stripped of formality. Still, I think that few students would get to this point without having those concepts, and if I get confused on a problem my first step is usually still to "draw a picture"! The two go hand in hand in my opinion, but due to the demand from the engineering departments for teaching mathematical solutions (I don't know the full history of how the curriculum developed but I suspect this and its associated budgetary reasons are a large part), half of the learning process is severely delayed.
I think you are talking about a different thing than I do. I have nothing against demanding some rigour in reasoning very early on in mathematics courses, what I am concerned about is the purely formal presentation of mathematical concepts, especially those that were first discovered by studying physical or geometrical situation and which naturally arise in such contexts. For example, is the formal epsilon-delta definition of a limit the best way to begin a calculus course? Yes, this is logically one of the basic building blocks of the theory, but what value does it have pedagogically for a beginning student? It is known that learning happens to a large degree via associating new concepts with ones already known, what can such a definition be associated with? This is carried out to extremes some times, I've seen introductory calculus books replacing sentences in natural language with formal-logic apparatus like quantifiers etc. and advertising it as some great pedagogical improvement.
Those are in my opinion excellent examples of courses developing intuition without sacrificing rigour:
I also love the books by Richard Courant for how they manage to explain the concepts in crystal-clear English writing, present both the applications and the theory, building up the rigour gradually.
You're confusing rigor with mathematical formality. Computing with equations that were taught to you formally, and solving mathematical problems rigorously are two different things. One of the easiest ways to tell the difference between the two modes of teaching is that in the former, you're taught a method and asked to apply. In the latter, you're frequently told the answer (though not always), and asked to find the method.
http://pauli.uni-muenster.de/~munsteg/arnold.html
Richard Courant, the classic book of which is referenced in the article, remarks in the preface of one of his other books how the logical order of presentation of mathematical theories is much different from the order that works best pedagogically. I wish more educators would understand this seemingly simple observation.