designates my notes. / designates important.
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.
No idea is really bad, unless we are uncritical. What is really bad is to have no idea at all.
Trust, but keep your eyes open. Always follow your inspiration—with a grain of doubt.
(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.
Construct an equilateral triangle, being given a side.
Construct an equiangular triangle, being given a side.
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.