Notes and Quotes from the text

1. Solving problems is a practical skill. We acquire any practical skill by imitation and practice.. Trying to swim, you imitate what other people do with their hands and feet to keep their heads above water, and finally, you learn to swim by practicing swimming. Trying to solve problems, you have to observe and to imitate what other people do when solving problems and, finally, you learn to solve problems by doing them.

2. Trying to find the solution, we may repeatedly change our point of view, our way of looking at the problem. We have to shift our position again and again.

3. It is foolish to answer a question that you do not understand. It is sad to work for an end that you do not desire. The student should not only understand the problem, he should also desire its solution.

4. It is important, however, that the suggestions from which we start should be simple ,natural, and general, and that the list should be short. The goal is to develop ability not just a special technique. The goal is assimilation and the development of mental habit.

5. The ingenuity of the problem-solver shows itself in the originality of the solution.

6. It would be a mistake to think that solving problems is a purely intellectual affair; determination and emotions play an important role.

7. Solving problems is a fundamental human activity. In fact, the greater part of our conscious thinking is concerned with problems. When we do not indulge in mere musing or daydreaming, our thoughts are directed toward some end; we seek means ,we seek to solve a problem.

8. Going around an obstacle is what we do in solving any kind of problem; the experiment has a sort of symbolic value.

9. There is a certain difficulty in turning round, in going away from the goal, in proceeding without looking continually at the aim, in not following the direct path to the desired end.

Summary

Your problem may be modest; but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery. Such experiences at a susceptible age may create a taste for mental work and leave their imprint on mind and character for a lifetime.

The opportunity may be lost even if the student has some natural talent for mathematics because he, as everybody else, must discover his talents and tastes; he cannot know that he likes raspberry pie if he has never tasted raspberry pie. He may manage to find out, however, that a mathematics problem may be as much fun as a crossword puzzle, or that vigorous mental work may be an exercise as desirable as a fast game of tennis.

Such interest (in the ways and means of invention and discovery) may be more widespread than one would assume without reflection. The space devoted by popular magazines and newspapers to crossword puzzles and other riddles seems to show that people spend some time in solving unpractical problems.

The student should acquire as much experience of independent work as possible. The teacher should help the student discreetly, unobtrusively, naturally.

Four Phases of Problem Solving:

Something very undesirable and unfortunate may result if the student leaves out any of the four phases without having a good idea. The worst may happen if the student embarks upon computations or constructions without having understood the problem.

Many mistakes can be avoided if the student checks each step. Some of the best effects may be lost if the student fails to reexamine and reconsider the completed solution.

Phase 1. Understand the problem. See what is clearly required. Student should be able to state the problem fluently, its principal parts, the unknowns the data, the condition. If external representation is needed.

Question List;

A. Most important: What is the unknown?

B. What don't we know?

C. What is required?

D. What do you want to find?

E. What are you supposed to seek?

F. What is the goal?

Proverbs

1. Who understands ill, answers ill.

2. Think on the end before you begin.

3. A fool looks to the beginning, a wise man regards the end.

4. A wise man begins in the end, a fool ends in the beginning.

5. Where there is a will there is a way.

Phase 2. Devise a plan. See how the various items are connected, how the unknown is linked to data. We have a plan when we know, or at least know in outline, which calculations, computations, or constructions we have to perform in order to obtain the unknown. The main achievement in the solution of a problem is to conceive the idea of a plan. Good ideas are based on past experience and formerly acquired knowledge. If this does not work we must look around for some other appropriate point of contact, and explore the various aspects of our problem; we have to vary, to transform, to modify the problem. Variation of the problem may lead to some appropriate auxiliary problem.

Question List:

A. Do you know a related problem?

B. Look at the Unknown. Try to think of a familiar problem having the same or similar unknown.

C. Could you restate the problem?

D. If at first you cannot solve the problem, try to solve first some related problem.

E. Did you use all the data?

F. Did you use the whole condition?

Proverbs

1. Diligence is the mother of good luck

2. Perseverance kills the game.

3. An oak is not felled at one stroke.

4. If at first you don't succeed, try, try again.

5. Try all the keys in the bunch.

6. Arrows are made of all sorts of wood.

7. As the wind blows you must set your sail.

8. Cut your coat according to the cloth.

9. We must do as we may if we can't do as we would.

10. A wise man changes his mind, a fool never does.

11. Have two strings for your bow.

12. Do and undo, the day is long enough.

13. The end of fishing is not angling but catching.

14. A wise man will make more opportunities than he finds.

15. A wise man will make tools of what comes to hand.

16. A wise man turn chance into good fortune.

17. Have an eye to the main chance.

Phase 3. Carry out the Plan. Patience, step by step procedure, with careful checking.

Question List:

A. Can you see clearly that the step is correct?

B. Can you prove that each step is correct?

Proverbs

1. Look before you leap.

2. Try before you trust.

3. A wise delay makes the road safe.

4. If you will sail without danger you must never put to sea.

5. Do the likeliest and hope the best.

6. Use the means and God will give the blessing.

7. We soon believe what we desire.

8. Step after step the ladder is ascended.

9. Little by little as the cat ate the fickle.

10. Do it by degrees.

11. What a food does at last, a wise man does at first.

Phase 4. Look Back. Even fairly good students, when they have obtained the solution of the problem and written down neatly the argument, shut their books and look for something else. Doing so, they miss and important and instructive phase of the work. By looking back at the completed solution, by reconsidering and reexamining the result and the path that led to it, they could consolidate their knowledge and develop their ability to solve problems. The students will find looking back at the solution really interesting if they have made an honest effort, and have consciousness of having done well.

Question List:

A. Can you check the result?

B. Can you check the argument?

C. Can you derive the result differently?

D. Can you see it at a glance?

E. Can you use the result, or the method, for some other problem?

Proverbs

1. He thinks well that thinks not again.

2. It is safe riding at two anchors.

Miscellaneous Heuristics and Proverbs:

1. Too many details or too minute particulars are a burden on the mind. i.e. don't lose sight of the forest for the trees.

2. After having decomposed the problem, we try to recombine its elements in some new manner. Difficult problems demand hidden, exceptional, original combinations, and the ingenuity of the problem-solver shows itself in the originality of the combination.

3. Allow the "subconscious" to work or "Take counsel of your pillow." or "if today will not, tomorrow may."

4. The end suggests the means.

5. Your five best friends are What, Why, Where, When, and How. You ask What, you ask Why, you ask Where, When, and How - and ask nobody else when you need advice.

6. Do not believe anything but doubt only what is worth doubting.

7. Look around when you have got your first mushroom or made your first discovery; they grow in clusters.

8. Working backward is very difficult but often provides a solution.

(P. 226) "There is a certain difficulty in turning round, in going away from the goal, in proceeding without looking continually at the aim, in not following the direct path to the desired end."