By Paul Zeitz, PhD, University of San Francisco
Problem-solving requires a change in your attitude—almost a new mindset. Discover how thinking creatively can help you solve math problems (and daily life problems) with new ease.
Chainsaw the Giraffe: Creatively Solving Problems
There is an inspiring article that highlights thinking outside of the box when it comes to solving problems. The article compares a gifted class of middle school students with a remedial sort of vocational class, where it was measuring the creativity of the students. The experiment asked the students in each class one question: How do you weigh a giraffe?
The students in the gifted class were not so much gifted as successful, and they were used to succeeding and pleasing their teacher. They panicked because they didn’t know how to answer this question. This was way before the Internet, and the students couldn’t go online and look it up.
Meanwhile, in the vocational class, almost immediately some kid just blurted out and said, “Hey, I know what to do. Just take a chainsaw, and chainsaw that giraffe into chunks. Then weigh the chunks.”
Chainsawing the giraffe is an attitude that a good problem solver should have because you want to be fun, and you also want to be a little bit bad. Breaking rules is a good thing when we’re not talking about actual cruelty to animals here. We’re just talking about thinking outside the box or breaking mathematical rules.
To break the mathematical rules, we need a healthy dose of the three C’s.
This is a transcript from the video series Art and Craft of Mathematical Problem Solving. Watch it now, on Wondrium.
Solving Problems with the Three C’s
Concentration, creativity, and confidence are psychological attributes that are important for just about everything, but they’re vital for solving problems. How do we enhance them? All three of them are linked, but confidence is the least important of the three because it’s truly derived from the other two.
As your concentration ability increases, and if your creativity gets stronger, then you’ll naturally become more confident.
Learn more about thinking like a problem solver
To master concentration, you must set aside a quiet time and place for your work. You need to relax, to develop good work habits, and you need to find problems to concentrate on that are interesting to you, approachable by you, and addictive. Pretty much, that will do the trick. To build up your concentration, you want to build up from level one, which is a minute or so of concentration, at least to level three, getting up to an hour.
Collect a stock of back-burner problems. Start cultivating problems that you cannot solve. Make sure they’re interesting and then you’ll think about them. If you can find a problem that’s exciting to you, annoys you, that sort of gnaws at you, then you’ll think about it. Interesting problems will force you to become a better concentrator.To master concentration, you must set aside a quiet time and place for your work. Click To Tweet
Solving Problems with Confidence and Creativity
Concentration leads to confidence, which frees you to explore and facilitates investigation and creativity.
To build both your confidence and creativity, you need to be disciplined about using those interesting problems. You need to set up a problem-solving routine, some workplace, a lucky pen—and then you should keep to your routine to get your mind in a relaxed state.
Then, occasionally, deliberately break your routine. If you like to work in the morning, work late at night. If you like quiet, go to a noisy café. If you like to work in a restaurant, go sit in a library, etc.
You should also, as a strategy, specifically think about peripheral vision. The peripheral vision strategy is to realize that many problems cannot be solved with direct focus. It’s just like your eyes. Your fovea has very good focus, but it has less sensitivity than the perimeter of your eyes, the periphery of your vision.Peripheral vision strategy—many problems cannot be solved with direct focus. Click To Tweet
Many problems need to percolate in your unconscious in this way. You need to cultivate a good supply of back burners, and just get in the habit of not solving problems. The more you do this, the more you’ll get into a state of investigative and purposeful contemplation, the more powerful your mind will get.
Learn more about the psychological aspects of problem solving
Don’t be Afraid to Use Tools
Let’s look at a tool made famous by Carl Gauss. He was a prodigy, and as a teenager, he solved a problem that had been unsolved since Hellenistic times. He found a way to construct a regular heptadecagon, 17-gon, using a compass and straightedge. The rest of his career was not much different. What Gauss could do in an afternoon was equivalent to what an ordinary mathematician could do in a lifetime.
When he was 10, he was faced with the problem: How do you find the sum of the numbers 1 + 2 + 3 up to 100? How do you compute this in 1787 when there are no calculators? What little Gauss did was to pair the terms, the beginning term and the end term, (1 + 100); and then the second term and the next to last term, (2 + 99); and then (3 + 98); (4 + 97); and so on down to (50 + 51). Each of those pairs adds up to 101, and there are 50 such pairs. Thus, the sum is 5050, and that’s pretty clever. This is called Gaussian pairing and is an example of a powerful and useful tool.
Learn more about the power of specific tools, or “tricks”, to make a mathematical expression simpler
Wishful thinking is one of your first strategies for this because pretending to solve a problem, even an easier one, keeps you happy. It allows you to keep thinking about solving problems. Even delusion helps—deluding yourself into thinking that you’ve solved a problem allows you to solve it later because you can relax and be happy. Making yourself happy and confident, even if it’s through such a transparent thing as delusion, is fine.
A corollary of wishful thinking is a sensible idea, which we can simply call the “make it easier” strategy. The idea is completely common sense. If your problem is too hard, simply make it easier by removing the hard part. Either make the size smaller or remove an element that makes it hard. For example, if it involves square roots, remove them temporarily.
What you should keep in mind is strategy and tactic is what makes someone a good problem solver, not the tools. Now, if you’ve never seen the Gaussian pairing tool, which Gauss used to sum the numbers from 1-100, you are undoubtedly impressed. Gaussian pairing is quite clever, but they are tools and tools are just tricks. These are things that can be acquired. What you should keep in mind is that strategy and tactic are what makes someone a good problem solver, not the tools.
You should use these new ideas. Any time you see a new, interesting idea, learn it, use it, and make it yours. Ideas are collective human property. They are not private property. Don’t forget that what you’re doing is chainsawing the giraffe. It’s okay to mess around and break some rules.
Learn more about three strategies for achieving a problem-solving breakthrough
Solving Problems with Creative Thinking
Let’s look at a quickie, a problem that requires “think outside the box” thinking.
If you consider the problem of nine dots in a grid and ask how do you join them all by drawing no more than four absolutely straight lines? If you think outside the box, as demonstrated in the diagram, it’s pretty obvious what to do.
As long as you go outside the box, you’re able to get all 9 dots. It’s a fun and challenging problem if you’ve never seen it before.Thinking outside the box helps you become a good problem solver. Click To Tweet
People may be endowed unequally with confidence, creativity, and power of concentration, but all of these are trainable skills. It’s possible to practice them and improve them, but to do so, you will need to see lots of creativity in action and you need lots of open-ended opportunities to experiment.
Learn more about the hidden world of problem solvers
Common Questions About Problem Solving
The seven steps of problem-solving are: Identify a problem, define goals, brainstorm, consider alternatives, agree on the solution, execute the solution, and evaluate the outcome.
Good problem solvers generally have less drama in their lives and react less emotionally to difficulties, thus they can follow the steps of problem-solving more coherently.
Become a better problem solver by stimulating the brain with mathematical problems and games, removing drama from your life, and following the steps of problem-solving while allowing for random clues to appear.
Solving problems gets better with experience and team leaders must lead by offering solutions and roads to solutions that arise from experience.