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First Seen: 04/26/2024

Last Indexed: 10/23/2024

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[ G.Polya ] Routine Problem May be called the probloem to solve the equation x^2-3x+2=0 if the solution of the gen eral quadratic equation was explained and illustrated before so that the student hasnothing to do but to substitute the numbers -3 and 2 for certian letters which appear in the general solution. iEven if the quadratic equatio ws not solved generally in letters but ha a doen ismilar quadtratic equations with numerical co efficients were solved just before, the problem should be called a "orutine problem" In general, a problem is a "routine problem" if it can be solved either by s ubstituting special data into a formerly solved general problem or by sollfiing step by step, without any trace of originality, some well orn conspicuous example. Settin ga routine problem, the teacher thrusts under the nose of the student an immediatel and decisive answer to the question Do you know a related problem? Thus the student needs nothing but a little care and patience in following a cut and dried precept, nad he has no opport unity to use his judggement or his inventive faculties. Routine problems, even many routine problems, may be necessary in teaching mathematics but to make the students do no other kind is inexcusable. Teaching the mechanical performance of routiine mathematical opera tions and nothing else is well undret he level of the cookbook because kitchen recipeis do leave som4ething to the imagination and jugdedmnt of the cook but mathem atical recipes do not. Rules of Discovery : The first rule fo 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. It may be good to remind somewhat ruldely that certian aspiracions are hopeles.s infaliable rules of dis covery leading to the solution of all possible mathematc al problesmw ould be more desirable thamn the philosop hers sonte,vainly sought by the alcemists. Such rules would work magic; but there is no such thing as magic. To find unaling rules applicable to all sorts of prob lem is anm old philosophical dream; but this dream wil never be mroea thna d ream. A reasonoable sourt of heuristic cannot aim at unfailing rules; but it may endeavior to study procedures, (mental oeprations, moves, steps) which are typically useful in solving problems. Such procedures are practiced in every sane person sufficiently interested in his problem. They are hinted by cretian stereoytped questions and suggest ions which intelligent people put themselves and in telligent teachers to their students. A collection of such questions and suggestions, stated with sufficient general ity and neatly ordered may be less desirable than the philosophers stone but can be provided. The list we study provides such a collection.