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1) Give students a fairly involved problem that they will struggle to solve without knowledge of a particular concept (like integration, for instance), and have them attempt to solve it. Some may even succeed, but either way they will have a deep insight into the concept about to be presented.
2) Then place them in groups, and have them try to solve the problem as groups. Encourage them to be creative and to transform the problem in to equivelent ones, or approximate ones, or smaller ones, etc. Have them go at it as if they were trying to solve a puzzle. Many more will likely solve the problem, without the aid of the concept being taught. Keep track of the approaches the students take that are most promosing, or interesting. Have them discuss/present the most interesting approaches.
3) Develop the concept from its founding principles. This is the usual exposition. Then try and relate the theoretical foundation to the approches students took on the problem. The teacher will likely trip up in several spots trying to do this. That's OK, and actually prefered; it shows that even proffesionals have to struggle with math sometimes, and shows how they struggle with it. Point out that there "maybe some connection here or there." But then go back and summarize the core theoretical concept concisely. Don't show them the solution to the orignal problem, yet.
4) Place them back into groups, and have them have one last go at the orignal problem, now armed with the concept just presented. For those who did just understand the concept may marvel at the "magic" of using the new concept in making the problem easier.
5) Present theoretical solution to the problem. Take questions for clarification, etc.
6) Quiz/Test the individuals on many similar problems in different contexts to the one solved. If a review of the concept is needed, then repeat steps 1-6 and a different appropriately chosen problem.
7) Quiz/Test individual on extremely diverse sets of problems requiring the same concept. If a review of the concept is needed, then repeat steps 1-7 and a different appropriately chosen problem.
To have time for this approach (which I estimate to be 3x-4x normal exposition), give students a short time to prepare for a pretest of review information, and only allow those who "pass" to take the class.
Sorry, but without the foundations, going further is often futile.