| The process of problem solving 2/1 |
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| Simple Examples |
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Problem solving is a natural process. Understanding this process can be developed by considering the definition of a problem. |
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Definition |
A problem is a request for a result subject to conditions that must be simultaneously satisfied.
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Word
Problems
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Though the definition applies to problems in general, the material presented here will be limited to the simpler subset of problems for which mathematical methods are appropriate. These are generally referred to as word problems, story problems or puzzles. |
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Examining simple problems provides an excellent way of understanding the process of solving problems. The problem below starts you on the journey to learning to solve problems reliably.
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Figure 1 |
Comparison
with the
arithmetic
solution
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The answer to this simple problem can, quite properly, be obtained by writing (or thinking) 6 + 13 = 19. However, in writing this your mind has gone through the mathematical process shown in Figure 1. |
Relation
to the
definition
of a
problem
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The problem is explicit as to the result to be obtained, namely, the total number of apples. Knowledge provides the process by which this will be obtained. The total is obtained by adding the components.
The conditions to be simultaneously satisfied are that there are two contributors to the total and that these have the particular values given in the problem. Satisfying the conditions makes it possible to generate the result. |
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Developing reliable problem solving skills requires studying how problems are solved. The modification of the previous problem shown in Figure 2 continues the process of studying how problem are solved. It is another simple problem chosen so as to show the problem solving process using the simplest of mathematical machinery. |
Figure 2 |
Algebra
vs
arithmetic
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If we were not trying to discover how to solve problems we would simply write or think 3(8) + 8 = 32. However, in order to learn to solve problems reliably, we need to look carefully at how problems are solved, giving particular attention to the thinking that goes into obtaining the answer. In the present case simply writing 3(8) + 8 = 32 hides the thought processes. These processes are displayed explicitly in Figure 2. The problem in Figure 3 is a bit more challenging.
Note the ease with which the following solution is started. Even students in elementary grades can understand this starting point.
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Figure 3 |
Start
simple
Less
is
more
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Starting a problem solution is the simplest, least time-consuming part of solving a problem. There are quite a few words and pieces of data in the problem statement. There is no way of knowing how all this stuff fits into the solution. For this reason we ignore it. What we are looking for is what the problem asks for. We scan till we find the How in the sentence "How much more money did Harry have than Jon had before Harry gave Jon $6.00?"
Words such as how, when and where alert us to the fact that we have found what is asked for. What is asked for follows the how type of word. Often there is a question mark at the end. What is asked for precedes the question mark.
Difficulty frequently arises by trying to include too much in the starting statement. Strive for doing as little as you possibly can in responding to what is asked for.
Once the starting equation has been written it requests additional information. The information needed for a response to this request is frequently in the problem statement. Look at a problem statement as a source of information like a telephone book, dictionary or reference book. There is considerable irrelevant information in this problem statement. The equations never ask for any of that information so it is automatically ignored. Irrelevant material is frequently present in so called real-life or context-rich problems. Request-Response-Result approach automatically filters out such noise in artificial problems.
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