Students develop these skills only if genuine mathematical problems solving is taking place. The solved examples in the textbooks give the idea that problem solving is a linear process with no false start or illogical attempts. Also the way the solution is presented does not show how much time and efforts are needed to come up with a useful solution. Even the problems suggested at the end of a chapter are usually of the same type as those already solved. Students may start to practice on less demanding problems. However as they become more confident, problems should be varied and more demanding. Students need to understand that even when no solution to the problem is obtained, learning is taking place. It is the time and efforts spent on finding that contribute to the learning process. All the thinking taking place and the organization of one’s thoughts during the problem solving process contribute to the learning process.
ACKNOWLEDGEMENT
The author would like to thank all colleagues at UGRU with whom he had fruitful discussions about this work.
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APPENDIX A: SOLUTION TO PROBLEM 6.
We first rewrite the equations obtained.
S1 t1 = 1400 (1)
1400 + S2 t1 = X (2)
S1 t2 = X + 600 (3)
S2 t2 = 2X - 600
The question in problem 6 is to find X and not all the unknowns. We use equations (1) and (2) to write
S1 = 1400 / t1 (5) and S2 = (X - 1400) / t1 (6)
We next substitute S1 and S2 in equations (3) and (4) by their expressions in (5) and (6) to write
( 1400 / t1 ) t2 = X + 600 (7)
( (X - 1400) / t1 ) t2 = 2X - 600 (8)
Let T = t2/t1 and write equation (7) and (8) as follows
1400 T = X + 600 (9)
X*T - 1400T = 2X - 600 (10)
We now end up with two equations with two unknowns that can easily be solved to give X = 3600 meters.
1,
2,
3,
4,
5,
6,
7,