The Road Less Travelled and Dead Ends
You cannot be afraid to fail. If I am working on a problem and I try a solution, then there is a possibility that it may not work. However, I can take the attempt at a solution and learn from it. Sometimes students get the idea that math is clean and easy. If you don't get the solution right away, then you must not be very good. In reality, very few problems worth solving have a straightforward solution. It may take several false starts and "failures." This is all part of the process. And the sooner that students understand that, the more likely they will be to persevere.
This is reinforced in the "real-world." I have noticed an anomaly with the data that I am studying. I keep coming back to the issue and I still haven't resolved it. I have to accept that I may not solve it before I leave, but perhaps someone else can resolve it after I leave. There of course are historical parallels in the world of mathematics, Fermat's Last Theorem probably being the most famous.
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