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A mathematical exercise is a routine application of algebra or other mathematics to a stated challenge. Teachers assign mathematical exercises to develop the skills of their students. Early exercises deal with addition, subtraction, multiplication, and division of integers. Extensive courses of exercises in school extend such arithmetic to rational numbers. Various approaches to geometry have based exercises on relations of angles, segments, and triangles. The topic of trigonometry gains many of its exercises from the trigonometric identities. In college mathematics exercises often depend on functions of a real variable or application of theorems. The standard exercises of calculus involve finding derivatives and integrals of specified functions. Usually instructors prepare students with worked examples: the exercise is stated, then a model answer is provided. Often several worked examples are demonstrated before students are prepared to attempt exercises on their own. Some texts, such as those in Schaum's Outlines, focus on worked examples rather than theoretical treatment of a mathematical topic. ==Graduation== In primary school students start with single digit arithmetic exercises. Later most exercises involve at least two digits. A common exercise in elementary algebra calls for factorization of polynomials. Another exercise is completing the square in a trinomial. An artificially produced word problem is a genre of exercise intended to keep mathematics relevant. Stephen Leacock described this type:〔Stephen Leacock "A,B,C – The Human Element in Mathematics", pages 131 to 55 in The Mathematical Magpie (1962) by Clifton Fadiman (editor) Simon and Schuster〕 :The student of arithmetic who has mastered the first four rules of his art and successfully striven with sums and fractions finds himself confronted by an unbroken expanse of questions known as problems. These are short stories of adventure and industry with the end omitted and, though betraying a strong family resemblance, are not without a certain element of romance. A distinction between an exercise and a mathematical problem was made by Alan H. Schoenfeld:〔Alan H. Schoenfeld (1988) "Problem Solving",(see page 85), chapter 5 of ''Mathematics Education in Secondary Schools and Two-Year Colleges'' by Paul J. Campbell and Louis S. Grinstein, Garland Publishing, ISBN 0-8240-8522-1〕 :Students must master the relevant subject matter, and exercises are appropriate for that. But if rote exercises are the only kinds of problems that students see in their classes, we are doing the students a grave disservice. He advocated setting challenges: :By "real problems" ... I mean mathematical tasks that pose an honest challenge to the student and that the student needs to work at in order to obtain a solution. A similar sentiment was expressed by Marvin Bittinger when he prepared the second edition〔Marvin L Bittinger (1981) ''Fundamental Algebra and Trigonometry'', 2nd edition, Addison Wesley, ISBN 0-201-03839-0〕 of his textbook: :In response to comments from users, the authors have added exercises that require something of the student other than an understanding of the immediate objectives of the lesson at hand, yet are not necessarily highly challenging. Some comments in the preface of a calculus textbook〔L.J. Goldstein, D.C. Lay, D. I. Schneider (1993) ''Calculus and Its Applications'', 6th edition, Prentice Hall, ISBN 0-13-117169-0〕 show the central place of exercises in the book: :The exercises comprise about one-quarter of the text – the most important part of the text in our opinion. ... Supplementary exercises at the end of each chapter expand the other exercise sets and provide cumulative exercises that require skills from earlier chapters. This text includes "Functions and Graphs in Applications" (Ch 0.6) which is fourteen pages of preparation for word problems. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Exercise (mathematics)」の詳細全文を読む スポンサード リンク
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