Over the years, I’ve developed a number of questions/suggestions I often use in assisting students in the development of new problem-solving strategies. The goal is to guide students find strategies for approaching novel problems (the good kind, where you aren’t just adjusting for different values, but are really trying to figure out something ‘new’).
Write it down
Nothing is more frustrating than to boldly go where you have been before. And even dead ends often prove to have interesting branches–so long as the dead ends are not covered over by the mists of time. Keeping a log or graphic allows you to fruitfully revisit old endpoints while avoiding going down the exact same trail.
A problem’s representation alone can allow you to wrap your head around a tough problem, perceive new relationships or discover an analogical way to think about it that makes you powerful by providing you with new tools. One neat example is Feynman diagrams, which (I have read 🙂 ) sufficiently simplified some aspects of the strange world of quantum physics that they actually allowed new ideas to be had.
Each of us knows an awful lot about the world. In many cases, understanding in domain 1 can be useful in attacking mysteries of domain 2. Some of the most famous intellectual discoveries in the sciences have arisen from this mode of thinking. Darwin compared the ‘red in tooth and claw’ hand of nature (‘Natural’ selection) to artificial selection he saw going on everywhere around him (horses, pigeons, cattle). Einstein’s ‘gedanken’ (thought experiments) often involved likening abstract ideas from physics to more mundane natural situations. Any textbook on electricity will be rife with comparisons to the flow of water.
The risk here is that the source analogy will always be distinct from the thing analogized; for an analogy to be fruitful, it must contain core areas of similarity with the target. Further, the one using it must be able to recognize the inevitable breakdown of the analogy–places where truths about thing 1 mislead if assumed to apply to thing 2.
‘One of these things is not like the other’ (from Sesame Street!)
How we represent a problem to ourselves can leads us to the solution… or block it from us. Some of the prompts I use:
“Write down every way you can describe the two cases/hypotheses/possibilities as different”
At the outset, we cannot guess which avenues are going to be fruitful, so a ‘complete’ listing of all the available avenues cannot be a bad idea. There is a huge difference between a ‘hunch’, which is based on a familiarity with prior, informative cases, and a wild stab in the dark, which is ‘this is what popped into my head first for no reason’
There is a fine line between fishing and just standing on the shore like an idiot.
“Take each of the above and write down several different phrasings/points of view”
Sometimes, there is a critical difference between approach the same data as “You are taller than I am” vs. “I am shorter than you are”. While this can obviously be reduced to the trivial (adding “you have more centimeters of height than I do” likely adds nothing), ‘trying out’ different points of view on the same concept or data can trigger a new insight
Is there a ‘low road’ from where you are now?
Not all challenges are created equal. A reverse example might be “Describe your friend”. Faces are hard to describe. Gender generally isn’t. That’s why police reports tend to describe someone as “sex, height, clothing, skin color.” Someone’s personality or home address might be really helpful, but they’re tough stuff to figure out. In Sudoku, there is often (always? I’m not a big player) a square where there are very limited possibilities, and many that at the start admit of too many options for guessing to be fruitful.
This is one of the distinctions about newbies vs. ‘experts’ in solving any particular puzzle: identifying the low-hanging fruit and going for it before attacking the tougher problems. This isn’t just a matter of convenience–initial successes may limit secondary possibilities, rendering the problem soluble.
Just walk away
I have read many, many times about studies that have demonstrated the utility of leaving your brain alone to do its work. The thinking involves the idea of ‘walling in’ your ideas, and the more you think, the more deeply you cut the mental channels you’ve already traveled. Dreaming, (day- or otherwise!), relaxing your attention (long walks, bike rides, taking of hot showers…) relaxes the inhibitors of lateral connection-making in the brain, allowing you to ‘discover’ linkages that might illuminate your path forward.
Bottom line: regardless of how you phrase it, I’m pretty sure each of us has our own anecdotal cases of figuring out where the keys had been put, remembering a forgotten word, etc. that appeared ‘out of the blue’ after we’d stopped pounding away at the problem with focused attention.