Concept
A mathematical exercise is a routine application of algebra or other mathematics to a stated challenge. Mathematics 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. 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 quadratic polynomial. An artificially produced word problem is a genre of exercise intended to keep mathematics relevant. Stephen Leacock described this type: 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: Students must master the relevant subject matter, and exercises are appropriate for that.