Four principles How to Solve It suggests the following steps when solving a mathematical problem: First, you have to understand the problem. pp .6-8 After understanding, make a plan. pp .8-12 Carry out the plan. pp .12-14 Look back on your work. pp .14-15 How could it be better? If this technique fails, Pólya advises: p114 "If you can't solve a problem, then there is an easier problem you can solve: find it." Or: "If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?" "Understand the problem" is often neglected as being obvious and is not even mentioned in many mathematics classes. Yet students are often stymied in their efforts to solve it, simply because they don't understand it fully, or even in part. In order to remedy this oversight, Pólya taught teachers how to prompt each student with appropriate questions, p33 depending on the situation, such as: What are you asked to find or show? p214 Can you restate the problem in your own words? Can you think of a picture or a diagram that might help you understand the problem? Is there enough information to enable you to find a solution? Do you understand all the words used in stating the problem? Do you need to ask a question to get the answer? Guess and check p99 Make an orderly list p2 Eliminate possibilities p94 Use symmetry p199 Consider special cases p190 Use direct reasoning Solve an equation p172 Pólya advises teachers that asking students to immerse themselves in routine operations only, instead of enhancing their imaginative / judicious side is inexcusable. Look for a pattern p108 Draw a picture pp103-108 Solve a simpler problem p114 Pólya notes that 'human superiority consists in going around an obstacle that cannot be overcome directly' Use a model p105, p29-32 , for example, Pólya discusses the problem of water flowing into a cone as an example of what is required to visualize the problem, using a figure. Work backward p105, p225 Use a formula pp141-148. Pólya describes the method of analysis Be creative p172 (Pólya advises that this requires that the student have the patience to wait until the bright idea appears (subconsciously).) Applying these rules to devise a plan takes your own skill and judgement. pp148-149. In the dictionary entry 'Pedantry & mastery' Pólya cautions pedants to 'always use your own brains first' Polya lays a big emphasis on the teachers' behavior. A teacher should support students with devising their own plan with a question method that goes from the most general questions to more particular questions, with the goal that the last step to having a plan is made by the student. He maintains that just showing students a plan, no matter how good it is, does not help them. This step is usually easier than devising the plan. p .35 In general, all you need is care and patience, given that you have the necessary skills. Persist with the plan that you have chosen. If it continues not to work, discard it and choose another. Don't be misled; this is how mathematics is done, even by professionals. Pólya mentions that much can be gained by taking the time to reflect and look back at what you have done, what worked and what did not, and with thinking about other problems where this could be useful. p .36 pp.14-19 Doing this will enable you to predict what strategy to use to solve future problems, if these relate to the original problem.
Four principles How to Solve It suggests the following steps when solving a mathematical problem: First, you have to understand the problem. pp . 6–8 After understanding, make a plan. pp . 8–12 Carry out the plan. pp . 12–14 Look back on your work. pp . 14–15 How could it be better? If this technique fails, Pólya advises: p. 114 "If you can't solve a problem, then there is an easier problem you can solve: find it." Or: "If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?" "Understand the problem" is often neglected as being obvious and is not even mentioned in many mathematics classes. Yet students are often stymied in their efforts to solve it, simply because they don't understand it fully, or even in part. In order to remedy this oversight, Pólya taught teachers how to prompt each student with appropriate questions, p. 33 depending on the situation, such as: What are you asked to find or show? p. 214 Can you restate the problem in your own words? Can you think of a picture or a diagram that might help you understand the problem? Is there enough information to enable you to find a solution? Do you understand all the words used in stating the problem? Do you need to ask a question to get the answer? Guess and check p. 99 Make an orderly list p. 2 Eliminate possibilities p. 94 Use symmetry p. 199 Consider special cases p. 190 Use direct reasoning Solve an equation p. 172 Pólya advises teachers that asking students to immerse themselves in routine operations only, instead of enhancing their imaginative / judicious side is inexcusable. Look for a pattern p. 108 Draw a picture pp. 103–108 Solve a simpler problem p. 114 Pólya notes that 'human superiority consists in going around an obstacle that cannot be overcome directly' Use a model p. 105, pp. 29–32 , for example, Pólya discusses the problem of water flowing into a cone as an example of what is required to visualize the problem, using a figure. Work backward p. 105, p. 225 Use a formula pp. 141–148. Pólya describes the method of analysis Be creative p. 172 (Pólya advises that this requires that the student have the patience to wait until the bright idea appears (subconsciously).) Applying these rules to devise a plan takes your own skill and judgement. pp. 148–149. In the dictionary entry 'Pedantry & mastery' Pólya cautions pedants to 'always use your own brains first' Polya lays a big emphasis on the teachers' behavior. A teacher should support students with devising their own plan with a question method that goes from the most general questions to more particular questions, with the goal that the last step to having a plan is made by the student. He maintains that just showing students a plan, no matter how good it is, does not help them. This step is usually easier than devising the plan. p . 35 In general, all you need is care and patience, given that you have the necessary skills. Persist with the plan that you have chosen. If it continues not to work, discard it and choose another. Don't be misled; this is how mathematics is done, even by professionals. Pólya mentions that much can be gained by taking the time to reflect and look back at what you have done, what worked and what did not, and with thinking about other problems where this could be useful. p . 36 pp. 14–19 Doing this will enable you to predict what strategy to use to solve future problems, if these relate to the original problem.
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.6-8
. 6–8
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.8-12
. 8–12
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.12-14
. 12–14
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.14-15
. 14–15
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p114
p. 114
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p33
p. 33
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p214
p. 214
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p99
p. 99
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p2 Eliminate possibilities p94 Use symmetry p199
p. 2 Eliminate possibilities p. 94 Use symmetry p. 199
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p190
p. 190
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p172
p. 172
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p108
p. 108
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pp103-108
pp. 103–108
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p114
p. 114
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p105, p29-32
p. 105, pp. 29–32
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p105, p225
p. 105, p. 225
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pp141-148.
pp. 141–148.
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p172
p. 172
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pp148-149.
pp. 148–149.
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.35
. 35
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.36 pp.14-19
. 36 pp. 14–19