A little while back I discussed some of the problem solving strategies that my son and I have used to help him tackle his Maths homework. These “Rules of Maths”, as we call them, can be generalized to grown-up sized problems. Here are some of our latest discoveries:
Rule 7: Make it Real
If Jonathan is given the sum, “18 ÷ 2 =”, he might not be able to find the solution. On the other hand, if I were to give him 18 beads and ask him to share them between the two of us, you can bet he’d get the right answer fairly quickly. It is often easier to deal with concrete problems than it is to think about theoretical or abstract ones.
Of course, you can be flexible about how “real” you make things. For example, if you’re counting red cars you don’t have to go out and find real cars that happen to be red – toy cars do just as well, as do beads or bits of paper or crayons. Kids have great imagination!
Speaking of imagination, physicist Richard Feynman tells a story in about his use of “Make it Real” in the book, “Surely You’re Joking, Mr. Feynman!” (the story is called “A Different Box of Tools”). He recalls how he “gave a lot of trouble” to mathematics students during his time at Princeton be telling them that their subject was “trivial”. He would get the mathematicians to explain a complex and strange scenario to him, and then make a statement about that scenario. Feynman would immediately tell the mathematicians whether their statement was true or false. He achieved this apparently impossible feat of mental gymnastics by building a mental picture of the problem in his head… involving hairy green billiard balls or whatever… and then base his answer on whether or not the statement was true of his model.
Rule 6: Break the Problem into Smaller Pieces
Sometimes a problem seems overwhelming at first, but can be tackled successfully if it is broken down into bite-sized pieces. For example, adding two 3-digit numbers can be broken down into 3 simpler stages: adding the units, adding the tens, adding the hundreds.
Most adult-sized problems are similar. Moving house, for example, is a big challenge, but is much easier to handle if once broken down into steps like calculating your budget, deciding on a location, finding a suitable property and so on.
Rule 7: Try a Simpler Version
If the problem you’re working in is just too complex for you, why not try working on a simpler version of the problem? For example, figuring out the area of all the rooms in a house is quite an undertaking for Jonathan, but one room he can cope with.
In a recent article on his blog the brilliant Eric Lippert explains how this technique works for computer programmers. He suggests the following benefits of this technique:
- The solution to the simpler problem might be adaptable to solve the harder problem. Now you have a plan: solve the easier problem, look for an adaption.
- Conversely, the inability to adapt the simpler problem to the harder problem might shed light on precisely what makes the harder problem harder. Come up with a plan to tackle the specific thing that makes the problem harder.
- Solving simpler problems that you know you can do is good practice and builds confidence. And you’re more likely to be successful if you start with code that already correctly solves a problem.
- If you fail to solve the harder problem you can always submit the solution to the simpler problem for partial credit.
Rule 8: Try Turning the Problem Around
For example, adding 27 to 2 is harder than adding 2 to 27, and 12 of 2 is harder than 2 lots of 12.
All sorts of problems are easier to solve if you start with a goal and work backwards. For example, journey planning is sometimes easier if you start with the destination and ideal arrival time and work backwards to your starting point.
A word of warning: not all problems work the same backwards as they do forwards (think subtraction, for example). Nevertheless, lots of problems do work well that way, and many are easier to solve in reverse.
(Of course, starting in the middle can work, too – as long as you can keep track of where you are and where you’re going)
Rule 9: Make Some Sort of Start
If all else fails, just start somewhere and see where it takes you.
For big puzzles, I often start with the easy bit. The benefits are:
- It gets some of the problem out of the way, so you have less to worry about
- You tend to learn things on the way that help you tackle the next bit.
If all else fails, just write out the question (Rule 4). At least you’ll get some clarity on what’s expected of you.
Rule 10: Don’t Be Afraid of Big Numbers
Big numbers are intimidating for Jonathan, but he’s learning that he can tackle them in the same way that he tackles smaller numbers.
As adults we’re sometimes faced with big problems that seem intimidating at first. However, it is often the case that the techniques that we use to solve smaller problems work perfectly well on their scaled-up cousins. Don’t be intimidated! Just take things one step at a time.
Image courtesy kalyana sundaram via Stock.Xchng.