It's not as helpful, but I like to teach my high school students: Say you're planning to rob a jewelry store. You can rob it constructively or by contradiction.

It's not as helpful, but I like to teach my high school students: Say you're planning to rob a jewelry store. You can rob it constructively or by contradiction. You could case the joint, figure out the guards and the employees, their schedules, their uniforms, the jewels on display, the layout of the place, and so on. You can make a plan, an appropriate scenario, disguises, make a copy of keys, some costume jewelry, some fake uniforms. Really know the place inside and out, Oceans-11 the whole thing. And if you do it right, you can make it out with the jewels before anyone even knows they were stolen. Or you can get some masks and some baseball bats, steal a semi truck, and drive it through the front window of the jewelry store. If the jewel cases aren't open from the impact, or if someone gets in your way, well, refer back to the baseball bats. Both accomplish what you set out to do, but the first is clearly more elegant and often takes more work and study, requiring/granting a greater understanding of the system. And future thieves would learn a lot more studying Danny Ocean than they would Johnny Smash-and-Grab.