I base this question off where I got my motivation for math and science. Throughout several attempts in my junior years, I was able to design a perpetual motion machine, design a free energy device, prove Einstein $E=mc^2$ wrong. Obviously I was way, way off, and I eventually found that out by myself. However, the amount of interest in math and science and the amount I learned was greatly benefited. So I ask the question: should students be told they're wrong about an idea they have?
The example I will give is a student "proving" $1=2$ by method of division by 0, such as the one described here. I classify this in 4 ways:
They should be told they're right. This makes the student feel smart and makes them have a higher interest in the subject. Eventually they will find out they are wrong. (This obviously being a lie.)
They are told that they came up with good thoughts, not confirming or denying they are right. They may not receive as much interest, but they are still in the process of confirming if they are right and wrong, and if curious enough and believe they are right, perform their own study.
They are told they are wrong, but are not told why. Which still may present some curiosity, maybe enough to find it themselves. In this way they are not risking the trust between the teacher and student.
They are told they are wrong, and explained why. This way they are taught the process themselves, but may not receive as much interest from it.
What's the most effective way for their education?