Home · Book Reports · 2018 · How to Solve It - A New Aspect of Mathematical Method

designates my notes. / designates important.

Thoughts

Overall I thought this was a great book. It is very straight and to the point, with plenty of examples. It repeats quite a bit, but, given he is a professor, he knows that is how you get something to stick.

I think it is plain enough to be taught to any high School class, at least an adequately educated one; I can’t speak for the state of affairs seen in today’s schools.

It applies to more than mathematics, though that is the angle it generally approaches from. Reverse planning, visualize the problem as solved and work backwards, is employed in military training and is an extremely valuable problem solving tool and used frequently within the book.

The format is more of a reference book, maybe even a dictionary in some aspects. There are no chapters but instead three parts.

The first part hammers into the reader a few critical points: find the unknowns, find the knowns, and draw a picture. These three pieces of advice are staggering in the effectiveness, yet seem so simple.

Next you would do well to look for similar problems whose solutions or techniques you might be able to apply to the current problem. There is, of course, no need to reinvent the wheel.

Alternatively is there something you can add or subtract from the problem to make it more approachable? Can you loosen a condition to make the problem easier and thus get a handle on?

There is also a good portion of part one dedicated to how one should approach not only learning, but teaching. A teacher should only guide the student, who should be allowed to struggle just enough so that they appreciate the results. If the teacher simply presents complete theorems and proofs, the student will not likely be motivated. By slogging thought the difficult process of discovery and understanding, getting down in the grass with the problem, the student will at once become more acquainted with problem solving and experience the suffering of failure and the elation of success. By giving away the secret, the teacher robs the student of self discovery.

Part two continues with techniques for how to approach problems, how to get a hold of them. To get acquainted with a problem, it is often useful to visualize it in as great a detail as you can. This dovetails, of course, with the advice of drawing a picture.

When drawing your pictures, try to draw generalities, lest you make assumptions based on your inaccurate representation.

The triangle having the angles 45°, 60°, 75° is the one which, in a precise sense of the word, is the most “remote” both from the isosceles, and from the right-angled shape. 5 You draw this, or a not very different triangle, if you wish to consider a “general” triangle.

Once you have a clear visualization in mind, and on paper, can you isolate parts of the problem? Can you see the problem from various, previously unseen, angles? Any of these new ideas should be followed as long as they seem fruitful.

Once you have an idea and you’ve formed a basic plan of attack, approach your target incrementally, combining a number of small steps into great steps. Check each step so that a simple error might not carry through and spoil your result.

Once you’ve solved your problem, after more or less iterative work, trying the various angles and ideas, you should reflect upon your solution and method. Look back and try to improve your solution; once you’ve arrived at a solution sometimes a more direct path becomes clear.

This reflective process is what will separate the great problem solvers from the average. It will give a more intimate understanding of each problem, which can subsequently act as a catalog of potential strategies in all future problems. This is where the subtlety of problems will become more apparent.

Part three is the most extensive part, containing many general strategies as well as the encyclopedia of concepts presented for easy reference.

The common advice of looking for analogies and simpler problems similar to your is again repeated. Adding auxiliary elements, such as lines to reveal triangles, or problem specific notation, can help to expose the ’trick’ in solving a problem. Care should be take, though, not to wantonly add auxiliary elements.

From a teaching perspective, you should not simply add these auxiliary elements, but motivate why you added them. Try to get the students to answer questions Socraticly about what would be helpful, wouldn’t it be nice if, etc.

Adding or removing data and conditions can allow you to create a simpler problem, or possibly trigger a memory of a similar problem.

A student’s experience will be incomplete until they, under their own motivation, invent their own problems. Transforming and augmenting problems is an exceptional way to cultivate this independent problem creation.

Again reverse planning is covered. It is often easier to start with the end and work your way, stepwise, to the beginning than to proceed from the data to the unknown.

Pappus: Assume what is required to be done as already done.

Reverse planning and augmenting problems can be made easier if the problem is first decomposed. By seeing the individual pieces, which may be seen as sub-problems, that might be easier to solve separately and then reassemble into a final solution much progress can be made.

Similar to augmenting or decomposing you should try to remove the technical terms from a problem by ‘going back to definitions’ and using foundational concepts.

A simple step-by-step process for approaching your problem; translate from words to math, set up the problem by following these steps:

1. Analysis
2. Practical problems need to limit data, you can only collect so much
3. What is the unknown? Did I use all the data?
4. What are the conditions?
5. Find connections
6. Do I know any similar problems to compare with this problem?
7. Can I loosen or remove conditions, changing the problem into one I know?

As you are setting up or working out a problem, remember that quick tests can be the extremes of a problem, but success isn’t guaranteed. Use extrema for guides, not as hard and fast rules.

Finally, the book concludes with several great pieces of advice:

With regard to signs:

Trust, but keep your eyes open. Always follow your inspiration—with a grain of doubt.

When you’re stuck, sleep on it:

‘Take counsel of your pillow’ is an old piece of advice.

The end of the book contains 20 problems plus hints and their solutions. I found these very interesting and spent several days playing with them. Some were, at least for me, straightforward, while wit others I was left scratching my head for hours.

No idea is really bad, unless we are uncritical. What is really bad is to have no idea at all.

Further Reading

• Mathematics and Plausible Reasoning continue the line of thinking begun in How to Solve It.

Exceptional Excerpts

Teaching to solve problems is education of the will.

No idea is really bad, unless we are uncritical. What is really bad is to have no idea at all.

Pappus: Assume what is required to be done as already done.

The triangle having the angles 45°, 60°, 75° is the one which, in a precise sense of the word, is the most “remote” both from the isosceles, and from the right-angled shape. 5 You draw this, or a not very different triangle, if you wish to consider a “general” triangle.

The first rule of discovery is to have brains and good luck. The second rule of discovery is to sit tight and wait till you get a bright idea.

Trust, but keep your eyes open. Always follow your inspiration—with a grain of doubt.

Table of Contents

• Part 1: In the Classroom
• Part 2: How to Solve it A Dialogue
• Part 3: Short Dictionary of Heuristic
• Part 4: Problems, Hints, Solutions

• Pages numbers from the pdf.

· Part 1: In the Classroom

page 30:

• It is foolish to answer a question that you do not understand.

page 32:

• We know, of course, that it is hard to have a good idea if we have little knowledge of the subject, and impossible to have it if we have no knowledge. Good ideas are based on past experience and formerly acquired knowledge. Mere remembering is not enough for a good idea, but we cannot have any good idea without recollecting some pertinent facts; materials alone are not enough for constructing a house but we cannot construct a house without collecting the necessary materials.

· Part 2: How to Solve it A Dialogue

· Part 3: Short Dictionary of Heuristic

page 64:

• simplex sigillum veri (simplicity is the seal of truth).

page 65:

• Auxiliary elements.

page 66:

• We should not introduce auxiliary elements wantonly.

page 69:

• Teachers and authors of textbooks should not forget that the intelligent student and THE INTELLIGENT READER are not satisfied by verifying that the steps of a reasoning are correct but also want to know the motive and the purpose of the various steps. The introduction of an auxiliary element is a conspicuous step. If a tricky auxiliary line appears abruptly in the figure, without any motivation, and solves the problem surprisingly, intelligent students and readers are disappointed; they feel that they are cheated. Mathematics is interesting in so far as it occupies our reasoning and inventive powers. But there is nothing to learn about reasoning and invention if the motive and purpose of the most conspicuous step remain incomprehensible.

page 83:

• The mathematical experience of the student is incomplete if he never had an opportunity to solve a problem invented by himself. The teacher may show the derivation of new problems from one just solved and, doing so, provoke the curiosity of the students. The teacher may also leave some part of the invention to the students. For instance, he may tell about the expanding sphere we just discussed (under 5) and ask: “What would you try to calculate? Which value of the radius is particularly interesting?”

page 85:

• Let us sum up. Euclid’s manner of exposition, progressing relentlessly from the data to the unknown and from the hypothesis to the conclusion, is perfect for checking the argument in detail but far from being perfect for making understandable the main line of the argument. It is highly desirable that the students should examine their own arguments in the Euclidean manner, proceeding from the data to the unknown, and checking each step although nothing of this kind should be too rigidly enforced. It is not so desirable that the teacher should present many proofs in the pure Euclidean manner, although the Euclidean presentation may be very useful after a discussion in which, as is recommended by the present book, the students guided by the teacher discover the main idea of the solution as independently as possible.

page 88:

• It appears that starting the reasoning from the unknown is usually preferable (see PAPPUS and WORKING BACKWARDS ). Yet the alternative start, from the data, also has chances of success, must often be tried, and deserves illustration.

page 92:

• There is a formal classification in which the most usual and useful combinations are neatly placed. In constructing a new problem from the proposed problem, we may:

(1) keep the unknown and change the rest (the data and the condition); or 
(2) keep the data and change the rest (the unknown and the condition); or 
(3) change both the unknown and the data. 

• We are going to examine these cases. [The cases (1) and (2) overlap. In fact, it is possible to keep both the unknown and the data, and transform the problem by changing the form of the condition alone. For instance, the two following problems, although visibly equivalent, are not exactly the same:

Construct an equilateral triangle, being given a side. 
Construct an equiangular triangle, being given a side. 

page 101:

• Remove technical terms by ‘going back to the definition’. Ex: don’t think of a parabola, focus, and directrix, think in terms of points and distances.

page 106:

• Teaching to solve problems is education of the will.

page 110:

• No idea is really bad, unless we are uncritical. What is really bad is to have no idea at all.

page 114:

• Pappus: Assume what is required to be done as already done.

page 116:

• The triangle having the angles 45°, 60°, 75° is the one which, in a precise sense of the word, is the most “remote” both from the isosceles, and from the right-angled shape. 5 You draw this, or a not very different triangle, if you wish to consider a “general” triangle.

page 124:

• In the physical sciences, there is no higher authority than observation and induction but in mathematics there is such an authority: rigorous proof.

page 129:

• Latin saying is: “respice finem.” That is, look at the end. Remember your aim. Do not forget your goal. Think of what you are desiring to obtain. Do not lose sight of what is required. Keep in mind what you are working for.

page 169:

• The first rule of discovery is to have brains and good luck. The second rule of discovery is to sit tight and wait till you get a bright idea.

page 179:

• Trust, but keep your eyes open. Always follow your inspiration—with a grain of doubt.

page 182:

page 191:

• Repeatedly, I tried to recollect that name but all in vain. The next morning, as soon as I thought of the annoyance of the evening before, the name occurred to me without any effort. The reader, very likely, remembers some similar experience of his own. And, if he is a passionate problem-solver, he has probably had some similar experience with problems. It often happens that you have no success at all with a problem; you work very hard yet without finding anything. Butwhen you come back to the problem after a night’s rest, or a few days’ interruption, a bright idea appears and you solve the problem easily.

• “Take counsel of your pillow” is an old piece of advice.

page 211:

• Proverbs:
  • Who understands ill, answers ill
  • A fool looks to the beginning, a wise man regards the end
  • A wise man begins in the end, a fool ends in the beginning

page 212:

• Proverbs:
  • Diligence is the mother of good luck
  • A wise man changes his mind, a fool never does
  • A wise man will make more opportunities than he finds
  • A wise man will make tools of what comes to hand
  • A wise man turns chance into good fortune

page 213:

• Step after step the ladder is ascended

· Part 4: Problems, Hints, Solutions

• 20 problems, hints, and solutions.