Idea Seeds #01 - Problem Solving, 101

Learning to Learn

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Life's little mysteries

Idea Seeds

Ernő Rubik, the inventor of the well known puzzle, the Rubik Cube, said and I quote: “The problems of puzzles are very near the problems of life, our whole life is solving puzzles.”

If this is true, why then do we not spend more time learning how to solve puzzles and thus be better able to solve the problems of life? Lets plant some Idea Seeds in the hope that they will stimulate debate, in particular, between family members, their friends and the young about how best to tackle solving some of the problems of life. Idea-seeds pertinent to issues such as:

  • problem solving
  • choosing a career
  • what it means to be educated
  • learning to learn

will be presented as starting points to seed these discussions.

Like the evolution of allowing real seeds to naturally produce stronger healthier plants so too families and friends can encourage the young to discuss the idea-seeds and then if found appropriate to adopt them for themselves and keep them growing. If they are found to be inappropriate then to help them to modify or add to them until each idea-seed meets the needs of that person's own individuality. Any idea-seed found inappropriate should be rejected but not before a suitable alternative idea-seed is found to replace it.


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Real life problem to solve

Here is a topic close to my father's heart:

"Try the following: Imagine that you work for an engineering company that gives one bursary each year to a suitable applicant to study engineering at a university. Two applicants have applied and you see that they have both written the same regular South African Matriculation Examinations and have each done all their subjects at Higher Grade. Applicant X has achieved distinctions in English and History and C’s for Mathematics and Science. Applicant Y has achieved C’s for English and History and distinctions in Mathematics and Science. If this is the only information that you are given to base your decision on, who would you recommend for the bursary. Please decide now who you would give the bursary to and importantly, the reasons for your choice before reading any further?

If all we have to compare X and Y on are their results then, in my view, Applicant X with A’s in English and History is the better ‘educated’ of the two at this stage in their development. I consider English and History to be far more important in a person’s ‘World View’ than Mathematics and Science. The latter are both, at matriculation level in South Africa, just swot subjects. You either know the method or the facts or the process to get to the answer or you don’t. If you do, then an A is not difficult to achieve. Creative skills are not needed. On the other hand for English and History, you have to be able to ‘analyze’ each question to establish exactly what it is the examiner is asking for and then to ‘synthesize’ an answer in a creative way from what you know and have learnt that covers everything asked for in the question before being considered for an A.

A’s are not easy to get in these subjects; they have to be earned and provide the only real proof, from matriculation results, that the person has the ability to ‘analyze’ and ‘synthesize’. The latter is a skill that is not easily taught and essential, as far as I am concerned, for getting high marks in examinations but more importantly solving the problems of life and making wise decisions while doing so. (I have used English in this example but an A in any other home language done at an equivalent level would provide the same proof of the ability to ‘analyze’ and ‘synthesize’. However, there are other very sensitive issues that need to be considered when home language and the language of instruction in tertiary level courses are not the same."

2-match puzzle
2-match puzzle
3-match puzzle
3-match puzzle

Puzzles and solutions

Here is the first of six puzzles that my father has used with success when teaching both individuals and large classes to solve problems. Each puzzle demonstrates one or more of the ‘basic processes’ he believes one should go through each time one sets about solving a new problem.

Find some matches and lay them out as shown in the diagram below.

Your task is to reposition two matches to leave four squares. Reposition means exactly that; select and move two matches to another position where they still form part of a square. The squares must be the same size and every match must be part of a square. In the diagram below two matches have been repositioned still showing five squares but now with an added appendage. Appendages are not allowed because they aren’t part of a square.

When possible he does this puzzle on an overhead projector and allow students to work on the problem for a minute or two or until someone says that they know how to do it. At that point he tells them that his interest in not on who can or cannot do the problem but on whether they knew what ‘processes’ they had had used to get to their solution. Please try and solve the problem now and while doing so think about the ‘processes’ you are using to get to a solution, before reading further.

I hope you have solved the puzzle and are aware of how you arrived at your solution. Compare the ‘processes’ you used with his. "My first step in solving a new problem is to be sure that I have understood the problem. To do this I walk away from the problem. I switch off the overhead projector and walk away from it to demonstrate this. If you keep staring at the matches your mind is distracted and trying to solve the problem with “if I move this here then….” when you should be concentrating on asking yourself if you can ‘verbalize accurately what you have been asked to do’, and whether you have accurately ‘identified everything that has been given’. When I ask students what has been given, the overwhelming response is always five squares which is not really helpful. I then tell them their reply should have been something like this:

“I have been given sixteen matches arranged in a pattern that shows five squares. I have to rearrange the pattern by repositioning two matches to leave four squares all of the same size and where every match is part of a square. The latter means no appendages are allowed.” The minute they heard me say “sixteen matches” there was an increase in chatter as they realized the problem was now half solved. All that was needed was to look for was a pattern where no match shared a side with another square. In other words each square had to be ‘stand alone’. With a bit of ‘logic’ the problem is quickly solved. No matches can be repositioned from squares 1, 3, and 4 and by repositioning one match from 2 and one from 5 a fourth stand alone square can be made".


The solution
The solution

From here to there

I hope too that you will have worked out how the five squares were formed using only sixteen matches. Four matches share a common side to produce the fifth square. Make sure you can identify them in the five-square layout. The problem is easy to change. Layout the four-square solution and redefine the task as follows. “Reposition two matches to make five squares. Reposition means exactly that; select and move two matches to another position where they still form part of a square. The squares must be the same size and every match must be part of a square. The latter means no appendages allowed.

The main ‘idea-seed’ in this article is: ‘When given a problem reflect back to yourself, or whoever has posed the problem, everything you have been given and exactly what it is you think you have been asked to do.’ If you are reflecting back to an actual person and that person agrees then you know you are starting with the right information and by doing so you will avoid all those “Oh I thought that is what you meant” or “I didn’t see it that way” situations that cause so many hassles at a later stage. When things get serious and real money is added into the mix make sure you do this in writing so you can defend yourself legally if things go awry.’

When you watch any sporting professional getting ready to perform some task you will see that they all go through some much practiced ‘ritual’ to get their minds and bodies properly focused before performing the task. A kicker getting ready to kick a goal or penalty; a golfer getting ready to putt; a tennis player getting ready to serve; are examples, from many. There is real merit in following their example so, if you haven’t already done so, start developing your own ‘problem solving ritual’ and your efforts will be well rewarded.


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Problem solving riual

Now that he has outlined the first step in his ‘problem solving ritual’ − he reflects back to himself, or to whoever has posed the problem, what I understand has been given and what it is I am expected to do.

The five-square match puzzle demonstrates very clearly the benefits of developing a ‘problem solving ritual’. After going through the questions: “What has been given?” and “What has to be done?” and finding that sixteen matches have been given to make four squares with one-match-per-side it becomes obvious the pattern sought must have four stand-alone squares. To find the two matches that have to be moved from where they are situated in the five-square pattern is then a simple task.

Wikipedia's definition of problem solving

The term problem-solving is used in many disciplines, sometimes with different perspectives, and often with different terminologies. For instance, it is a mental process in psychology and a computerized process in computer science. Problems can also be classified into two different types (ill-defined and well-defined) from which appropriate solutions are to be made. Ill-defined problems are those that do not have clear goals, solution paths, or expected solution. Well-defined problems have specific goals, clearly defined solution paths, and clear expected solutions. These problems also allow for more initial planning than ill-defined problems.[1] Being able to solve problems sometimes involves dealing with pragmatics (logic) and semantics (interpretation of the problem). The ability to understand what the goal of the problem is and what rules could be applied represent the key to solving the problem. Sometimes the problem requires some abstract thinking and coming up with a creative solution.

Mindtools on problem soving

Defining the Problem

The key to a good problem definition is ensuring that you deal with the real problem – not its symptoms. For example, if performance in your department is substandard, you might think the problem is with the individuals submitting work. However, if you look a bit deeper, the real issue might be a lack of training, or an unreasonable workload.

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