The process of problem solving                          1/1
INTRODUCTION
The
goal
The goal is to make it possible for you to solve problems that use mathematics in a reliable, confident manner. The problem solving process that you will come to understand is applicable to all subjects that use mathematics for problem solving.
Though our goal is to learn to solve problems which require use of mathematics, it is necessary to understand the properties and solution methods of more general problems. We will start this by exploring some simple routine problems that we encounter in daily life. Doing so will bring out the essential features of problems and their solutions.
Some
perspective.

Problem
Solving
as a
natural
process.
Easily the most important thing to understand about problem solving is that problem solving is a natural, indeed an innate, process. We are born problem solvers.

To appreciate your built-in problem solving abilities you need only to look at the many problems you routinely solve daily. Such problems as brushing your teeth, riding a bicycle, driving a car, going to a store are examples of routine daily problems which serve as informative examples of the problem solving process. Observing and thinking about solving such problems brings out quite basic problem solving principles.

First you will observe that the goal, the problem to be solved, is known. It is the conditions that need to be satisfied, the constraints, that make attaining the goal a problem. These constraints also bring life to the solution process.

Definition
of a
problem
These observations make it possible to have an operational definition of the word problem.
A problem is a request for a result subject to conditions that must simultaneously be satisfied.
To understand the meaning of this definition and the manner in which it specifies how to solve problems, we will explore the rather routine problem of leaving your home and going to the mall.
Respond to
what is
asked for.
Having chosen your trip to the mall as our problem, we first consider the resources available to accomplish the goal. There are choices. You could realize this goal by driving a car, riding a bicycle, walking or a variety of other means. Specifically, we must respond to the request, "how to get to the mall?"
Request
Response
Let's make the likely choice - driving. This response immediately generates a request; "Where are the keys?" With the keys in hand you go to the car, unlock the door, place the key in the ignition, start the car, put it in gear (forward or reverse according to existing constraints) and so forth.

What we see here is an ordered process of Request-Response. Each response is a consequence of the previous request and a precursor to the next request. (We put the key in before turning it, for example.

Request
Response
Result
You complete the trip to the mall by responding to requests as they arise: stopping at red lights, making left turns or right turns as needed, accommodating to an unexpected detour, deciding to pick up your dry cleaning and so forth.

Thus solving the problem of driving to the mall consists of responding to requests in a logically sound sequence to generate the result.

SureMath Problem Solving There are usually a number of different solution paths for a given problem, generated by choosing differently at places where options present themselves. The diagram at the left illustrates this idea. At the start of the trip-to-the-mall problem there was a choice of type of transportation. This is illustrated by the different logical paths arising out of the starting point. Conditions may arise that encourage change to a different solution path as depicted in the diagram.

It is seen that solving a problem is a natural process carried out in sensible steps. The steps are dictated by the current state of the solution. Essentially, problems solve themselves. You need to supply knowledge (location of the mall) and skills (how to drive a car).

Knowledge




Resources




Tools




Skills
The trip-to-the-mall problem illustrates that you do not need to figure-out how to solve a problem before you start. How to solve a problm is well defined by the problem. The solution steps themselves will lead you to the result. The difficulty arises in knowing how to respond to a request. This requires knowledge. You may not possess or be able to acquire the knowledge needed to respond to a request. If we limit the problems of interest to types which are within your current knowledge field, the next thing needed is appropriate resources. One resource immediately available is your brain. You can think out the response to a request. Another resource that is widely available is pencil and paper. This is a powerful supplement to the brain. Other resources include dictionaries, encyclopedia, textbooks, reference books, calculators, computers, the world wide web, teachers, friends, parents and many more.

Problems of any kind require various tools to implement the steps of the solution. The basic tools for mathematical problem solving are the tools available in algebra. These consist of such things as translating words to equations, manipulating equations, algebraic substitution, factoring, expanding, graphing functions and so forth. The mathematical tools available are virtually endless and the degree to which you might need more advanced tools, such as those available in the calculus, will depend on your professional objectives. However, algebra is the cornerstone of mathematical problem solving.

Finally, skills are needed in order to use the tools efficiently. Skills are developed through practice

To
summarize


Request
Response
Result
This introduction serves to establish the main idea involved in problem solving. Problem solving consists of a natural step by step process .Each step is a consequence of the previous step and a precursor to the next step.

In the next chapter we will see this idea at work.

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