Exercise (mathematics)


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.

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:
A distinction between an exercise and a mathematical problem was made by Alan H. Schoenfeld:
He advocated setting challenges:
A similar sentiment was expressed by Marvin Bittinger when he prepared the second edition of his textbook:
The zone of proximal development for each student, or cohort of students, sets exercises at a level of difficulty that challenges but does not frustrate them.
Some comments in the preface of a calculus textbook show the central place of exercises in the book:
This text includes "Functions and Graphs in Applications" which is fourteen pages of preparation for word problems.
Authors of a book on finite fields chose their exercises freely:
J. C. Maxwell explained how exercise facilitates access to the language of mathematics:

Proprietary sets

The individual instructors at various colleges use exercises as part of their mathematics courses. Investigating problem solving in universities, Schoenfeld noted:
Such exercise collections may be proprietary to the instructor and his institution. As an example of the value of exercise sets, consider the accomplishment of Toru Kumon and his Kumon method. In his program, a student does not proceed before mastery of each level of exercise. At the Russian School of Mathematics, students begin multi-step problems as early as the first grade, learning to build on previous results to progress towards the solution.
In the 1960s, collections of mathematical exercises were translated from Russian and published by W. H. Freeman and Company: The USSR Olympiad Problem Book, Problems in Higher Algebra, and Problems in Differential Equations.

History

In China, from ancient times counting rods were used to represent numbers, and arithmetic was accomplished with rod calculus and later the suanpan. The Book on Numbers and Computation and the Nine Chapters on the Mathematical Art include exercises that are exemplars of linear algebra.
In about 980 Al-Sijzi wrote his Ways of Making Easy the Derivation of Geometrical Figures, which was translated and published by Jan Hogendijk in 1996.
An Arabic language collection of exercises was given a Spanish translation as Compendio de Algebra de Abenbéder and reviewed in Nature.
In Europe before 1900, the science of graphical perspective framed geometrical exercises. For example, in 1719 Brook Taylor wrote in New Principles of Linear Perspective
Taylor continued
The use of writing slates in schools provided an early format for exercises. Growth of exercise programs followed introduction of written examinations and study based on pen and paper.
Felix Klein described preparation for the entrance examination of École Polytechnique as
Sylvestre Lacroix was a gifted teacher and expositor. His book on descriptive geometry uses sections labelled "Probleme" to exercise the reader’s understanding. In 1816 he wrote Essays on Teaching in General, and on Mathematics Teaching in Particular which emphasized the need to exercise and test:
Andrew Warwick has drawn attention to the historical question of exercises:
In reporting Mathematical tripos examinations instituted at Cambridge University, he notes
Explaining the relationship of examination and exercise, he writes
Explaining how the reform took root, Warwick wrote:
Warwick reports that in Germany, Franz Ernst Neumann about the same time "developed a common system of graded exercises that introduced student to a hierarchy of essential mathematical skills and techniques, and...began to construct his own problem sets through which his students could learn their craft."
In Russia, Stephen Timoshenko reformed instruction around exercises. In 1913 he was teaching strength of materials at the Petersburg State University of Means of Communication. As he wrote in 1968,