The first thing my dad taught me to do homework on my own was how to think on paper. I would, for example, have a math problem to solve, and my dad would show me how to break it down and visualize it using ink. Where words fail, drawings come in handy.
Even though I eventually learned to visualize problems in my head, I still rely a lot on paper to clear my mind and lay down resolution steps.
Let's take programming, for example. There are many visual standards to define requirements and design software programs, such as UML or Entity/Relationship diagrams. Sometimes, however, it's best to just let your mind roam on paper and draw whatever comes to you without restriction: mind-maps, Venn diagrams, flowcharts, to-do lists... or all of them together.
Back in high school, my biology teacher wanted us to come up with new diagrams to represent data. It was one of my favorite assignments because it allowed us to literally play with numbers, express our creativity, and merge art and science. I remember representing the evolution of the cranial volume throughout hominization using three-quarter circles of increasing diameters and different angles. Applied to more abstract problems, this is how visual thinking helps us open up our imagination to think outside the box.
I find it more productive to spend some time paper thinking before actually getting into the details of the craft, or as Lincoln puts it: "Give me six hours to chop down a tree and I will spend the first four sharpening the ax." Seizing and breaking down problems before chopping down atomic tasks is the best way to get things done without losing sight of the overall system.