To begin, we should ask: what is a problem? When you find yourself in one situation and you want to get to another but don’t know how to get there, you got a problem. The process that a problem solver needs to go through is firstly to define the problem and then explore ways that leads to a solution(s).
To understand problems some suggest that you first understand how you relate to them. This leads us into the theory about internal representation and external representation. When we refer to internal representation we are referring to our inner thoughts. To understand a problem the individual comes up with objects and relations that match up to the problem that he or she is dealing with externally.
An external representation is basically when you project the problem externally, like when you draw a diagram or when you write down an equation. You typically use external representation when the information is too much for you to remember or understand internally. With certain problems we don’t need any external representation like when we are dealing with an easy mathematical problem such as 2+3 in which most people will look at this problem and know the answer simply with internal representation.
External representations are of great aid though as they help you organize information that is difficult to achieve internally if there is too much information to memorize.
When facing a problem we must define certain points. The first is the original condition of the problem and the second is the goal. The goal is different from the solution. The solution is a tool or rather a road that leads to the end product, that is to say the goal.
We will discuss how to reach a solution later on in this post but what is important to consider for now is that there is not one way to find an answer, instead it’s a process that the problem solver needs to go through. You first ask yourself a question that you later understand has mislead you into a direction that doesn't lead to your goal. And, from here you ask a new question. You go about doing this until you have reached so many “wrong” directions that a more appropriate one becomes clear.
If you have trouble solving a problem it might be because you are working with a representation that is not fit either for you or for the particular problem in which you are working with. Try changing the way you work or your perspective to see if you find solutions.
Methods for Finding Solutions
When we have learned how to define problems we can begin searching for them. It is often in this stage most people feel lost because you don’t know where to start. To creatively find your own way of applying methods to problems, you first need to learn some basic methods that are used in the “discovery process”. So, without further ado here are basic methods in which every problem solver should accustom herself/himself with.
For small problems it is useful to use the systematic trial-and-error method. I say systematic because this is the efficient form out of two forms, the other one being called blind. Trial-and-error is used when you have no clue where to begin your search and you randomly start searching until you have reached your goal.
To take a classic example, consider that you have a bag of 40 balls: 1 of them is red while the rest are blue. You are supposed to pull out the red ball without looking inside the bag. If you do this using the blind trial-and-error, you will go about picking a ball from the bag that is blue and then place it back into the bag before you continue your search. I think you agree with me when I say that this is not an effective way of finding the red ball. Instead we approach the problem systematically. Once we get a blue ball we place it somewhere else to increase our probability of finding the red ball as we continue our search.
In this case the goal is quite clear, it's the red ball and with this method we learn to be organized and consider our "mistakes" before we proceed with our search. When diminishing the clutter as we reach them, we increase our chances of finding what it is that we are looking for.
You use the means-ends analysis when you have a bit more knowledge on where you are heading than in the cases in which trial-and-error is applied. With means-ends you clearly define what the goal is and then segment it into smaller parts. Sub-solutions for the parts can then lead to solutions on how to solve the entire goal as a whole.
What is great with the means-ends analysis is that it can help you reach other aspects of the problem that you didn’t recognize when you looked at it from a contextual point of view.
When you’re dealing with difficult problems that you feel completely lost within it might be because you’re new to the thinking that goes into that specific problem. Most problems can be sorted into categories in which the process of solving them differs, like for instance if you are solving interest rate problems and triangle problems. The way in which you approach them follows a certain pattern. In short, you need to acquire certain knowledge for each problem-category.
One typical way of doing this is by using auxiliary problems. With this technique you begin by finding an easier problem that is in some way related to your initial one. Since auxiliary problems are easier, the problem solver is able to unravel the auxiliary problem with comparatively little effort. By solving the auxiliary problem you will gain knowledge on how the problem's pattern is structured, which will help you find solutions for the initial, more complex problem.
When we have reached a solution for our goal it's common to take a deep breath of relief and check the process as completed. But by leaving the process like this we miss out on some valuable knowledge, because instead of leaving the problem that is now solved, we should question what it is we have learned by going through the problem solving process again. You need to go back and investigate your steps and your thinking throughout the process.
• What was specifically difficult and how did you overcome this?
• How would you approach this kind of problem if you would encounter it again?
• What kind of methods would you use?
• How can this problem be used as auxiliary if you are facing more advanced problems that are related to this one?
Make sure to not procrastinate this last step of analysis as it is a very effective way of remembering what you have just accomplished.
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