What are strategies or tips, which research mathematicians have discovered through their work and experience, that would help undergraduates learn how to discover counterexamples or find an object on their own? Here, I define an object to be a bijection or function or or matrix or ... that satisfies the demanded properties. Although an undergraduate, I am posting here because I am interested in opinions and perspectives from research mathematicians. My question is motivated by the following:
Lecturers and textbooks almost never illustrate or teach how to construct counterexamples and requested objects. They start immediately with a claim, without any explanation or even summary of the claim's origins, and then check the definition(s). Yet I find the hard part to be the rough work and thought processes that produced the counterexample.
Exams and problems have always required me to discover counterexamples or objects. So my question seems to constitute fundamental skills that are significantly assessed and graded, but they are rarely taught, if at all
I realise that counterexamples and examples may require years to conceive and that experimentation is crucial. So the rough work and thought processes might be muddled. Nonetheless, what are some steps towards increased productivity? For example, exam conditions encumber "playing around" for a counterexample or object.