Problem Solving as a Process (steps) part 2

As a process, according to George Polya (1957), problem solving originally has 4 steps. The 1^st step is understanding the problem. In this step, students have to carefully read the problem, capable to point out the principal parts of the problem, the unknown, the data, and the condition. George Polya subdivided this step into two stages: 1) getting acquainted and 2) working for better understanding. The 2^nd step is devising a plan where students consider some possible actions or strategies such as drawing a graph, finding a pattern, or making a list. Furthermore, the next step is carrying out the plan in which students implement a particular plan to solve the problem, if necessary, create a new plan.

Finally, students reflect and look back at what they have done, what worked, and what didn't. This is also important for students since by looking back at the completed solution, by reconsidering and reexamining the result and the path that led to it, students could consolidate and develop their ability to solve problems. All these 4 processes should be seen as a dynamic, non-linear and flexible approach. By using these steps, students will deal more effectively and successfully with most types of mathematical problems.

Some experts modified Polya’s steps into five steps or more. Charles Craig (2007), a member of EARAT Manuals and a lecture in University of Georgia, proposes 7-step method to solve math problem. He developed these steps to solve mathematics economy problems since EARAT Manuals deals more with business world. But, these steps also can be applied on any mathematics problems using different strategies. The steps are:

1.    Read Through the whole problem carefully.

Students need to read the whole problem not partially and try to understand the problem.

2.    List the important information that are given in the question.

In order to help them understand the problem, students should make a list of the important information. Listing can be done in many ways such as write it on a paper formally or just underline it. Listing all the important information also help students remembering the problem.

3.    Decide what needs to be found.

Students have to be sure and clear about what the question wants them to answer and also what needs to be found first to help them find the solution. This helps students avoiding unnecessary actions and also influences the way they answer the problem or give a conclusion.

4.    Decide what method can be used to solve the problem.

Students consider some possible actions or strategies to solve the problem such as drawing a graph, finding a pattern, or making a list and decide the most suitable method.

5.    Do the calculations using the steps decided on.

Students solve the problem based on strategies they have decide on the 4^th step.

6.    Write your answer

Writing out the answer is not easy. Students have to be aware about what the problem asked and how they state the conclusion.

7.    Check to see that your steps and answer seems reasonable and it provides the answer to the problem.

Students check every step they have done. Reexamining the problem and the calculations are very important to anticipate any mistake they made during the process.

In the Polya’s 1^st step, students have to read carefully the problem. So, it is obvious that Charles’s first step is belong to polya’s 1^st step. In the 2^nd Charles step, students need to list every important information given on the problem. This activity similar to what Polya said that students have to be able to point out the principal parts of the problem. Therefore, 2^nd Charles step also belongs to Polya’s 1^st step.

3^rd Charles step encourages students to find out what the question asked and what needs to be found before finding the solution and in the 4^th step, students consider and decide the most suitable strategies to solve the problem. All these activities are part of devising plan on Polya 2^nd step.

Students do the calculation or the plan in the 5^th and write the final answer in the 6^th Charles step in which Polya united it in his 3^rd step. All the activities are the same with Polya 3^rd step. Finally, the 7^th step of Charles is part of Polya’s 4^th step where students look back and check every step to fix any mistake they made.

Charles divides the first Polya’s step into two points, read and list, since this 7-step is mostly used to solve mathematics economy problems which have more important information such as figures, charts, or diagrams. Furthermore, he also separate Polya’s 2^nd step into two points, decide what needs to be found and decide what method can be used for the same reason as the 1^st. Furthermore, step 6 becomes different points from the 5^th step because he argued that many mathematics problems, especially mathematics economy problems, are not easily concluded with simple sentences and many students have difficulties in it.  

There is an example included in Charles paper. The problem is:

“An apprentice earns $25.85 per hour and $36.35 for overtime. If he works 40 hours at the regular rate and then works 8 hours of overtime in one week, how much does he make in total?”

How to solve it:

Step 1: Read the problem carefully

An apprentice earns $25.85 per hour and $36.35 for overtime. If he works 40 hours at the regular rate and then works 8 hours of overtime in one week, how much does he make in total?

Step 2: List the important information

 An apprentice earns $25.85 per hour and $36.35 for overtime. If he works 40 hours at the regular rate and then works 8 hours of overtime in one week, how much does he make in total?

Step 3: Decide what needs to be found

The words “how does he make…” tell us to figure out what he should be paid.

Step 4: Decide what method can be used

The words “per hour” tell us to multiply and the words “and … in total” tell us to add. Multiply to find the two amounts and then add them to find the total earnings.

Step 5: Do the calculations

$25.85 x 40 = $1034.00

$36.35 x 8 = $290.80

Then add those two numbers

          $1034.00

            $290.80
          $1324.80 in Total

Step 6: Write your answer

The apprentice makes a total of $1324.80

Step 7: Check to see that your steps and answer seems reasonable.

References

George Polya.1957.How to Solve It, A New Aspect of Mathematical Method.USA.

Charles Craig.2007.Basic Steps in Problem Solving.EARAT Manuals.

Charles Craig.2011.Mathematics Skills: Manipulation of Variables.EARAT Manuals.