Problem Solving In Mathematics

Learning problem solving is never a spectator game. The learners have to be actively involved if any meaningful learning has to take place. Different teachers use different strategies and techniques. In teaching contents, the teacher has no option but to master strategies and skills that will inspire learners to become motivated and actually enjoy learning. (Leamson, R, 2000). Also learners must be taught intensively and extensively the strategies. The learners  “… must make what they learn part of themselves.” (Chickening, A.W. & Gamson, Z.F,  1987). The teacher should strive to trigger intrinsic motivation in the learners, as this is likely to make them succeed.

Poor masterly of strategies deny students the power to be flexible as they lack approach options to choose from in solving problems. When learners know a variety of techniques, they do not give up at the first failure, they tend to apply different approaches until they get it correct (persistence). When they get it correct, they get motivated and learn to be flexible and persistent instead of giving up.

            Most students at the grade between 7-12 have a dislike for mathematics. Poor masterly of techniques has been held culprit for this. In the attempt to enable the learners between 7-12 grades develop persistence and flexibility in mathematics, a number of problem solving strategies and techniques can be instrumental and they include:

            1). Work Backward Strategy

Problem: Cleo got his salary on Wednesday, on Thursday he spent $1.50 at the hotel. On Friday, Ben paid Cleo the $1.00 he owed him. If Cleo now has $2.00, how much is his salary?

Understanding the problem

How much money did Cleo have on Friday? ($2.00)

How much money did Cleo spent in the hotel? ($1.50)

How much money was given on top of his salary? ($ 1.00)

Planning a solution

Had Cleo got Ben’s $1.00 on Thursday night? (No)

How much money did Cleo have at the end of Thursday?($2.00-$1.00=$1.00)

How much money did Cleo have before he spent $ 1.50 on Tuesday? (2.50)

Finding the answer

Start with $ 2.00

Subtract $ 1.00

Add $ 1.50

End with $ 2.50

 Extension of problem

This strategy can be applied in solving all problems that deal with spending. For instance, John spend ¼ of his gas on day 1, and 2/3 of the remaining on the second day, if the remaining was one liter, what capacity was his gas before use?

2) Make a table strategy

            Cleo and Tom began reading a novel the same day. If Cleo reads 5 pages each day and Tom 3 pages each day, what page will Tom be when Cleo will be reading page 20?

Fig.1

Understanding the problem

How many pages does Cleo read each day? (5) Tom? (3). Did they start reading their books on the same day? (Yes)

Planning a solution

How many pages had each read at the end of the day 1? Cleo (5) Tom (3)

Find the number of pages read for the first 3 days. 5, 10, 15

Finding the solution

Fig 1 shows that, Tom will be reading page 12 when Cleo is reading page 20

Problem extension

Cleo digs 10 hectors a day, Tom digs 8 while Ben digs 6, what hector will Tom and Ben be digging when Cleo will be digging his 50^th hector? Using the table the student will be able to work it.

• Bean Toss

Materials: the student will need 10 beans, a cup, pencil and a plain paper.

 Purpose: Not many students are abstract thinkers. The beans painted on one side with a color like white and black on the other side to represent positive and negative is a concrete reference for the concept of integers. This will represent a positive and a negative side in an activity such as (+2) – (-1) = +3

Activities and procedure: the students’ pair up and decide on which color is negative and which represents positive. Each student tosses the beans and records the outcome, for instance, (+3) + (-2) = +1. As the game continues, the students internalize the rules of working with signed numbers  and the rule can be extended to  division and multiplication

Extension: they learn the real life application. For example I received $ 6. I owe Tom $ 4. What does my account read? (+6) + (-4) = +2

4) Guess and check technique

This strategy arrives at a verifiable answer through guessing possible answers and checking to see which answer fits the problem. Students should predetermine a likely starting point and work in the right direction to solve the problem. The best student achieves this by eliminating as many numbers as possible with every guess.

Example: Caleb has 40 balls. If he had 10 more white types than the black type, how many of each ball did he have?

Understanding the problem.

There are more 10 white balls than black balls so white ball +10. Black  balls  –10.Half of 40 = 20 so if there were 20 black the white would be 30 (20+10) hence not correct. So the number of black is less than 20. Half the difference between the 2 total s and subtract from previous guess different is 10 half = 5 hence the black balls are 20-5 = 15.White ball = 25 so that 15+25 = 40. The second guess is correct

5). Solve a simpler problem

Some problems are too complex to solve in one step. The learner should divide it into cases and solving each separately.

Example; how many palindromes are there between 0-1000? The student can solve this by starting at how many of the numbers 1-9 are palindromes? All the nine are palindromes. How many of the number 10-99 are palindromes?

11

22

33         ý = 9

..

• From 100-999
• 202 …..909

111 212 …. 919

121 222 …. 929

….   ….         …

191 292 …. 999

Working out

9 columns´10 palindromes=90

90 palindromes from 100-999

The answer is (90+9+9)=108

 The above techniques and others must be accompanied with creative problem variation, such as changing context/setting. It is only after students have mastered various techniques to solve similar or varied problems that they can develop flexibility and persistence. So the teacher should competently teach the application of various strategies. The learners should know how to select appropriate techniques for each problem and how to justify their solutions using different approaches. When students develop flexibility and persistence, they learn to view the difficulty of complex mathematical investigations as a challenge rather than a bother. When they solve a problem successfully, they experience a feeling of accomplishment. This motivates them to attempt harder problems.

Remember the extension of problems help in generalization of problems and makes the learner to be creative, make value judgment and incorporate other branches of mathematics.

              In conclusion, techniques for solving different problems coupled with plenty of examples, motivating exercises that build skills and confidence, visually appealing graphics presented in fun, and highly engaging manner should all be used to help learners develop flexibility and persistence in solving problems.

References

Charles, R.L., Mason, R.. P., Nofsinger, J.M, & White, C. A.  (1985).

          problem-solving experiences in mathematics. Addison: Wesley publishing

          company.

Leamson, R. (2000). Algebra in simplest terms. From

           www.learner.org/resources/series66htm

           as retrieved on  Nov 1 2007. 19:43:52. GMT

Polya, G. (1973). How to solve it. Princeton: Princeton university press.