Teaching Problem Solving : Mastering bit by bit Vs Try it all by yourself? Background:

I'm a teacher who trains hundreds of high school students in Math, Physics &amp; Chemistry to crack probably toughest competitive exam in India.

I deal with the students who are hardworking and scored at least 70%+ in their 10th Grade. They are definitely in a position to figure out the variables and required formula and solve it, if the problem is direct. For rest of the post when I say problem solving, I'm referring to the problems of Math &amp; Physics which are not as tough as 12th grade Olympiad problems but definitely a lot tougher than typical textbook level problems. You can see some of such problems here. Each paper shown in the link, needs to be solved in 3 hours and more than 70% must be secured. 

Typical way of training: [Method1] Try it all by yourself
Students are taught (it is ensured students understood the) concept. Solve 3-4 variety mid-level example problems, to illustrate application of concept to the problem and reinforce their understanding. Move on to the next concept. Do the same with all the concepts and finish the chapter. Then, students are given about 50-60 unsolved problems, no two of same model. Students spend lot of time solving these problems, they discuss with teachers and they discuss among themselves. 

Problem with above method: The success percentage is very less. Its as low as 2-3%. I mean, only 2-3% of the students get through all the difficulties and finish those problems. Most of the other students find those problems too difficult to solve and are either losing motivation to solve such problems or think that they are not intelligent enough to solve the problems.

Fixing the problem: [Method2] Master bit by bit
I've read few books like, Why Don't Students Like School, How to Solve It: A New Aspect of Mathematical Method and analyzed myself (how do I solve the problems?) and understood that we try to recollect the patterns of problems. By practicing problems we are learning the patterns of problems and when solving an unsolved problem trying to figure out our put the patterns that we already learned. So, I've broken down larger problems into smaller problems. I've arranged these problems in the increasing level of difficulty. For the patterns to be remembered properly, I've increased no. of simple problems. I'm ensuring that students are mastering problems concept by concept, level by level.

Result is, now my students are not demotivated, they feel problem solving is simple and they are doing problems well. 

Problem with my method:
I realized that my students are entering a dangerous zone. They are not willing to solve difficult problems and they are unable to solve unknown model problems. Did I overtrain them? I consulted my college professors regarding this and they were furious with my method. They said, that I am spoon feeding the students. They said that I must follow the method 1.  That is what is going to help them in future in their higher studies and research.



What exactly is happening? Why is method 2 failing? Why will method 1 work? Is it just about the psychological difference in feelings that method 1 guys feel more prepared for solving any kind of problem? Whats optimal way? Where can I learn more?
There are two theoretical constructions that may be of use to you:

Scaffolding


  "...what the child is able to do in collaboration today he will be able to do independently tomorrow" -Vygotsky


You are right to graduate the level of difficulty of problems the students encounter. Intuitively, a student has before her a level of task which, although perhaps within her reach, nonetheless is difficult and, as such, allows students to develop their skills by doing tasks at this level. 

Lev Vygotsky, a soviet psychologist who stressed the role of play in learning (a perspective which is particularly germane to problem solving skills, which rely on an organic schematisation of knowledge and approaches- relational understanding), codified this intuition in the formal notion of The Zone Of Proximal Development. In the ZPD, we have the tools to play with the problems we face, and will organically develop the skills to solve similar problems- and, as with our intuitive version, this is where the best learning takes place.

The crucial thing about the ZPD, however, as distinct from our intuitive construction, is that tasks in the learner's ZPD are tasks that the student requires assistance to solve. In this handy diagram from wikipedia, the ZPD is represented by the middle circle.


A psychologist named Jerome Bruner went on to analyse the kinds of "assistance" or scaffolding needed in the ZPD- support frameworks which aid the student in tackling the task on her own (not, it must be noted, merely doing half the work- cf. BenCole's remark). 

Quoting from Wiki's Promoting better learning: scaffolding:


  
  Effective learning environments use instructional scaffolding to aid the student in his/her construction of new knowledge. 
  Avoid telling the learner exactly how to accomplish the task; do not solve the problem for the learner. This may help the learner
  immediately, but it hinders the learning process. 
  It is important to promote better learning by helping the learner achieve his/her learning goal through the use of instructional
  scaffolding. The use of scaffolding helps the learner to actively
  build and construct new knowledge.
  


To elucidate the whole Scaffolding theory in the abstract would take time, so I'll leave concrete recommendations for later and link the relevant wikipedia page, if you're curious.

Metacognition


  "Students can learn to think better if schools concentrate on teaching them how to do so"- Presseisen


Thinking about thinking has almost become a cliche in modern british education (and, I suspect, further afield!), but it is with good reason. The process of deliberating on strategies and approaches to to problems, including attitudes and emotional responses, called metacognition is invaluable both to problem solving and to learning in general.

It is what separates a boy who makes a mental list of all of his 'tools' and selects the best for the job, and the boy with the hammer ready to hand, to whom everything looks like a nail. It is what separates the girl for whom failure is a miasma to sulk in, and the girl who sees her own misery and changes tack. Incorporating metacognitive elements into pedagogy essential to develop students independent enough to solve problems.

Again, the theory in the abstract is enormous, so here's another wiki page:Metacognition. 

Now for some concrete recommendations (these are just what springs to mind right now, I may add to it later):


Incorporate metacognitive maxims into your pedagogy "You're stuck! Excellent- that means you're learning" "If you don't know what to do- make a list" "I like to take a while to plan my mode of attack!" and so on...
Model your internal dialogue in approaching a problem- do this often, and heavily exaggerate every aspect of it 
The most effective model for scaffolding is interpersonal- find some TAs!
If not TAs- getting them to work in groups or pairs can mimic this
Getting students to make a list of the tools and equations at their disposal at the start of a given problem
Forcing students to draw a neat, clear diagram
Resources, such as mini whiteboards (students hold up their answers at the end of an allotted time) allow you to tailor your problems to their ZPDs
Make problems manifest in different ways so they can incorporate their intuitions, approaches and playfulness from other spheres
A flow chart or worksheet for approaches to general problems incorporating the above as well as socratic questions (make toolkit, draw picture, is there a general equation? what can we sub into it? etc) can be a real boon
Most importantly, be creative (or at least appear to be so), and your students will show their own creativity- Vygotsky says learning is about play (and, as I've said, this model is certainly apposite to problem solving), you must put them in a playful place any way you can!