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the subject, and did not object to the heavy increase in their work load. The main difficulty was with the teachers – or more precisely, with the examiners, who were accustomed to basing their exams on the book. Putting practical problems on the exams complicated their job. They were persons along in years...the only hope was to bring younger people into teaching.
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287:
consequence of self-evident axioms, is not always, to our minds, self-evident; but the mathematician, who by long practice has acquired a familiarity with many of these forms, and has become expert in the processes which lead from one to another, can often transform a perplexing expression into another which explains its meaning in more intelligible language.
533:...by the 1830s it was the problems on examination papers, rather than exercises in textbooks, that defined the standard to which ambitious students aspired... not only expected to find their way through the merest sketch of an example, but were taught to regard such exercises as useful preparation for tackling difficult problems in examinations.
483:...a course of "mathematiques especiales". This is an extraordinarily strong concentration of mathematical education – up to 16 hours a week – in which elementary analytic geometry and mechanics, and recently infinitesimal calculus also, are thoroughly studied and are made into a securely mastered tool by means of many exercises.
426:, first published in 1751, 70 per cent of which was devoted to exercises as opposed to about 1 per cent by Recorde. The inclusion of exercises was one of the most significant subsequent developments in arithmetical textbooks, and paralleled the development of education as teachers began to make use of
299:
Upper division offerings for mathematics majors, where for the most part students worked on collections of problems that had been compiled by their individual instructors. In such courses emphasis was on learning by doing, without an attempt to teach specific heuristics: the students worked lots of
566:
exercises were not given at the
Institute, and on examinations the students were asked only theoretical questions from the adopted textbook. I had to put an end to this kind of teaching as soon as possible. The students clearly understood the situation, realized the need for better assimilation of
541:
It was widely believed in
Cambridge that the best way of teaching mathematics, including the new analytical methods, was through practical examples and problems, and, by the mid-1830s, some of the first generation of young college fellows to have been taught higher analysis this way were beginning
507:
The inclusion of illustrative exercises and problems at the end of chapters in textbooks of mathematical physics is now so commonplace as to seem unexceptional, but it is important to appreciate that this pedagogical device is of relatively recent origin and was introduced in a specific historical
286:
As mathematicians we perform certain mental operations on the symbols of number or quantity, and, proceeding step by step from more simple to more complex operations, we are enabled to express the same thing in many different forms. The equivalence of these different forms, though a necessary
498:
The examiner, obliged, in the short-term, to multiply his questions enough to cover the subjects that he asks, to the greater part of the material taught, cannot be less thorough, since if, to abbreviate, he puts applications aside, he will not gain anything for the pupils’ faculties this
524:
Such cumulative, competitive learning was also accomplished more effectively by private tutors using individual tuition, specially prepared manuscripts, and graded examples and problems, than it was by college lecturers teaching large classes at the pace of the
247:
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
197:
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
457:...for the true and best way of learning any Art, is not to see a great many Examples done by another Person; but to possess ones self first of the Principles of it, and then to make them familiar, by exercising ones self in the Practice.
231:
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
421:
Firstly, it was almost all exposition with very few exercises — The later came into prominence in the eighteenth and nineteenth centuries. As a comparison we might look at another best seller, namely
Walkingame’s
214:
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
133:: 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
550:
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
267:, we have included worked-out examples at appropriate points in the text and have included lists of exercises for Chapters 1 — 9. These exercises range from routine problems to alternative
449:
will find much more pleasure in observing how extensive these
Principles are, by applying them to particular Cases which he himself shall devise, while he exercises himself in this Art,...
559:
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430:
as sources of exercises. Recorde was writing mainly for those who were teaching themselves, scholars who would have no one to check their answers to the exercises.
223:
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.
490:
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
465:
in schools provided an early format for exercises. Growth of exercise programs followed introduction of written examinations and study based on pen and paper.
1120:
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John
Denniss & Fenny Smith, "Robert Recorde and his remarkable Arithmetic", pages 25 to 38 in Gareth Roberts & Fenny Smith (editors) (2012)
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368:
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320:, students begin multi-step problems as early as the first grade, learning to build on previous results to progress towards the solution.
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Examples in
Mathematics, Mechanics, Navigation and Nautical Astronomy, Heat and Steam, Electricity, for the use of Junior Officers Afloat
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for each student, or cohort of students, sets exercises at a level of difficulty that challenges but does not frustrate them.
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problems because (according to the implicit instructional model behind such courses) that’s how one gets good at mathematics.
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This text includes "Functions and Graphs in
Applications" (Ch 0.6) which is fourteen pages of preparation for word problems.
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to the instructor and his institution. As an example of the value of exercise sets, consider the accomplishment of
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The student of arithmetic who has mastered the first four rules of his art and successfully striven with sums and
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328:
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The individual instructors at various colleges use exercises as part of their mathematics courses. Investigating
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A similar sentiment was expressed by Marvin
Bittinger when he prepared the second edition of his textbook:
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Some comments in the preface of a calculus textbook show the central place of exercises in the book:
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assign mathematical exercises to develop the skills of their students. Early exercises deal with
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316:. In his program, a student does not proceed before mastery of each level of exercise. At the
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arithmetic exercises. Later most exercises involve at least two digits. A common exercise in
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reformed instruction around exercises. In 1913 he was teaching strength of materials at the
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of key theorems, but containing also material going beyond what is covered in the text.
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137:, focus on worked examples rather than theoretical treatment of a mathematical topic.
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Essais sur l’enseignement en general, et sur celui des mathematiques en particulier
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Stephen
Leacock "A,B,C – The Human Element in Mathematics", pages 131 to 55 in
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Andrew
Warwick has drawn attention to the historical question of exercises:
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both to undertake their own research and to be appointed Tripos examiners.
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In the 1960s, collections of mathematical exercises were translated from
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Essays on Teaching in General, and on Mathematics Teaching in Particular
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were used to represent numbers, and arithmetic was accomplished with
67:
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Explaining the relationship of examination and exercise, he writes
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Masters of Theory: Cambridge and the Rise of Mathematical Physics
554:
through which his students could learn their craft." In Russia,
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by Paul J. Campbell and Louis S. Grinstein, Garland Publishing,
626:
Mathematics Education in Secondary Schools and Two-Year Colleges
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is a genre of exercise intended to keep mathematics relevant.
1032:
862:
Robert Recorde: The Life and Times of a Tudor Mathematician
826:
The Ways of Making Easy the Derivation of Geometric Figures
741:, translated by John Maykovich, revised by Irving Sussman,
397:
collection of exercises was given a Spanish translation as
384:
Ways of Making Easy the Derivation of Geometrical Figures
263:
In order to enhance the attractiveness of this book as a
27:
Educational puzzle to be solved by symbol manipulation
624:(1988) "Problem Solving",(see page 85), chapter 5 of
560:
Petersburg State University of Means of Communication
694:
Introduction to Finite Fields and their Applications
537:
Explaining how the reform took root, Warwick wrote:
102:. In college mathematics exercises often depend on
1134:
1101:
438:framed geometrical exercises. For example, in 1719
668:L.J. Goldstein, D.C. Lay, D. I. Schneider (1993)
278:explained how exercise facilitates access to the
494:which emphasized the need to exercise and test:
1019:Pacific Institute for the Mathematical Sciences
419:
925:Development of Mathematics in the 19th Century
777:, translator and editor J.L. Brenner (1963,6)
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157:In primary school students start with single
8:
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129:Usually instructors prepare students with
893:Brook Taylor’s Work on Linear Perspective
994:, Robert Addis translator, pages 133,4,
710:Scientific Papers of James Clerk Maxwell
386:, which was translated and published by
371:include exercises that are exemplars of
202:A distinction between an exercise and a
1029:Exercises for students from age 5 to 15
594:
692:R. Lidl & H. Niederreitter (1986)
434:In Europe before 1900, the science of
369:Nine Chapters on the Mathematical Art
98:gains many of its exercises from the
82:have based exercises on relations of
7:
1225:
885:New Principles of Linear Perspective
646:Fundamental Algebra and Trigonometry
444:New Principles of Linear Perspective
66:. Extensive courses of exercises in
794:The Chinese Roots of Linear Algebra
757:& I.S. Sominski, translated by
295:in universities, Schoenfeld noted:
1015:Exercises in Experimental Geometry
779:Problems in Differential Equations
737:D.O. Shklansky, N.N. Chetzov, and
341:Problems in Differential Equations
25:
1051:Jim Hefferon & others (2004)
923:, M. Ackerman translator (1979)
845:Compendio de Algebra de Abenbéder
546:Warwick reports that in Germany,
399:Compendio de Algebra de Abenbéder
304:Such exercise collections may be
219:He advocated setting challenges:
1224:
1215:
1214:
1052:
365:Book on Numbers and Computation
743:The USSR Olympiad Problem Book
471:described preparation for the
333:The USSR Olympiad Problem Book
259:chose their exercises freely:
1:
887:, Preface, p vi, as found in
670:Calculus and Its Applications
351:In China, from ancient times
318:Russian School of Mathematics
911:Taylor p vii, Andersen p 153
765:, W.H. Freeman & Company
238:zone of proximal development
114:. The standard exercises of
34:is a routine application of
970:University of Chicago Press
775:Aleksei Fedorovich Filippov
745:, W. H. Freeman and Company
516:examinations instituted at
181:. An artificially produced
110:variable or application of
1267:
763:Problems in Higher Algebra
698:Cambridge University Press
644:Marvin L Bittinger (1981)
337:Problems in Higher Algebra
1210:
866:University of Wales Press
329:W. H. Freeman and Company
126:of specified functions.
100:trigonometric identities
78:. Various approaches to
996:D. Van Nostrand Company
603:The Mathematical Magpie
562:. As he wrote in 1968,
280:language of mathematics
42:to a stated challenge.
964:Andrew Warwick (2003)
716:editor, page 216, via
432:
173:. Another exercise is
154:
1251:Mathematics education
1095:Mathematics education
708:J. C. Maxwell (1890)
583:Worked-example effect
436:graphical perspective
255:Authors of a book on
189:described this type:
175:completing the square
148:
32:mathematical exercise
18:Mathematical exercise
883:Brook Taylor (1719)
791:Hart, Roger (2010).
728:Schoenfeld 1988 p 82
611:Simon & Schuster
518:Cambridge University
473:entrance examination
204:mathematical problem
179:quadratic polynomial
44:Mathematics teachers
895:, p 152, Springer,
548:Franz Ernst Neumann
514:Mathematical tripos
477:École Polytechnique
1192:Cognitively guided
1038:James Alfred Ewing
1033:IMAGINARY platform
1011:Tatyana Afanasyeva
988:Stephen Timoshenko
622:Alan H. Schoenfeld
556:Stephen Timoshenko
414:The Ground of Arts
208:Alan H. Schoenfeld
163:elementary algebra
155:
1238:
1237:
1177:Modern elementary
1152:Three-part lesson
873:978-0-7083-2526-1
851:98:466,7 (#2465).
488:Sylvestre Lacroix
453:Taylor continued
424:Tutor's Assistant
327:and published by
135:Schaum's Outlines
16:(Redirected from
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118:involve finding
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648:, 2nd edition,
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607:Clifton Fadiman
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395:Arabic language
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293:problem solving
187:Stephen Leacock
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131:worked examples
94:. The topic of
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373:linear algebra
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378:In about 980
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149:According to
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712:, volume 2,
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552:problem sets
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511:
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467:
460:
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443:
440:Brook Taylor
433:
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383:
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357:rod calculus
350:
340:
339:(1965), and
336:
332:
322:
314:Kumon method
303:
290:
274:
254:
251:
242:
235:
232:challenging.
226:
218:
206:was made by
201:
183:word problem
156:
151:Lev Vygotsky
130:
128:
96:trigonometry
70:extend such
31:
29:
1142:Traditional
921:Felix Klein
864:, Cardiff:
714:W. D. Niven
520:, he notes
469:Felix Klein
461:The use of
306:proprietary
215:disservice.
171:polynomials
120:derivatives
52:subtraction
40:mathematics
946:, page 201
605:(1962) by
589:References
382:wrote his
310:Toru Kumon
165:calls for
72:arithmetic
1157:Singapore
1126:Australia
1102:Geography
799:JHU Press
609:(editor)
578:Algorithm
525:mediocre.
442:wrote in
428:textbooks
417:in 1543.
390:in 1996.
248:chapters.
195:fractions
124:integrals
104:functions
92:triangles
38:or other
1245:Category
1220:Category
1202:Critical
1187:Informal
1147:Exercise
1135:Approach
1114:New York
927:, p 59,
830:Al-Sijzi
572:See also
508:context.
380:Al-Sijzi
367:and the
343:(1963).
335:(1962),
312:and his
265:textbook
198:romance.
141:Overview
116:calculus
112:theorems
88:segments
80:geometry
64:integers
60:division
48:addition
1230:Commons
1040:(1911)
1027:(2004)
1013:(1931)
990:(1968)
942:(1816)
891:(1992)
843:(1917)
824:(1996)
761:(1965)
361:suanpan
347:History
325:Russian
36:algebra
1167:Reform
975:
899:
871:
849:Nature
805:
680:
656:
632:
403:Nature
363:. The
269:proofs
90:, and
84:angles
68:school
58:, and
1197:Ethno
1162:Saxon
1044:from
1017:from
847:from
177:in a
159:digit
106:of a
973:ISBN
897:ISBN
869:ISBN
803:ISBN
678:ISBN
654:ISBN
630:ISBN
499:way.
236:The
122:and
108:real
1182:New
1031:at
828:by
479:as
475:of
393:An
169:of
74:to
62:of
1247::
968:,
951:^
801:.
797:.
676:,
652:,
405:.
375:.
331::
282::
210::
86:,
54:,
50:,
30:A
1087:e
1080:t
1073:v
1048:.
1021:.
811:.
20:)
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