History in Mathematics Teaching: Before the Advent of Modern MathematicsAuthor(s): D. R. GreenSource: Mathematics in School, Vol. 5, No. 3 (May, 1976), pp. 15-17Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30211568 .

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History in mathematics

teaching: before the advent of

modem mathematics by D. R. Green, CAMET, Loughborough University of Technology

Since the study of history only found its way substantially into schools at the end of the last century, it may seem surprising to find emphasis given to the history of mathematics in the popular texts on the teaching of mathematics (and in some on mathematics itself) of that period.

Several nineteenth century books written by R. Potts of Trinity College, Cambridge, set the scene. His Elemen- tary Arithmetic (Potts, 1876) began with 26 pages of his- tory, which he considered "most important and most neglected" and throughout the book were historical sec- tions - notably fourteen pages on the history of logarithms. Similar was his Elements of Geometry (Potts, 1845), and his Elementary Algebra (Potts, 1879) which began with 112 pages of historical introduction! These books were originally sold by instalments (by mail order usually) in a dozen or so separate sections. Potts' contents list to his Elementary Arithmetic and the introduction to his Elementary Algebra [below] indicate the subject matter covered and also show that "very unsatisfactory" examination results were a problem in 1880 and not of more recent invention!

ELEMENTARY ARITHMETIC, WITH BRIEF NOTICES OF ITS HISTORY. BY R. POTTS, M.A., TRINITY COLLEGE, CAMBRIDGE.

In twelve Sections, demy 8vo. CONTENTS AND PRICES

PRICE SECTION I. Of Numbers, pp. 28 3d. SECTION II. Of Money, pp. 52 6d. SECTION III. Of Weights and Measures, pp. 28 3d. SECTION IV. Of Time, pp. 24 3d. SECTION V. Of Logarithms, pp. 16 2d. SECTION VI. Integers, Abstract, pp. 40 5d. SECTION VII. Integers, Concrete, pp. 36 5d. SECTION VIII. Measures and Multiples, pp. 16 2d. SECTION IX. Fractions, pp. 44 5d. SECTION X. Decimals, pp. 32 4d. SECTION XI. Proportion, pp. 32 4d. SECTION XII. Logarithms, pp. 32 6d.

An American series of four booklets by W. W. Rupert (1900) was somewhat similar to those produced by Potts, being also available in a single book. The first of the four booklets in particular attempted to present geometrical theorems in an historical manner. Rupert's prefatory remarks included: "by giving the history of a few of the most celebrated geometrical theorems and problems (the author) might place a'light in the win- dow' which may throw a cheerful ray down the long and sometimes dusty pathway that leads to geometrical truth."

Part 1 of Rupert's first booklet began: "THEOREM 1. The sum of the three angles of every plane triangle is equal to two right angles.

The mathematical truth enunciated in the above theorem is not new. It has been known for more than two thousand years. Thales, one of the seven sages of Greece, who was born about 640 B.C., must have been aware that the sum of the angles of a triangle is equal to two right angles.

Our reason for believing that Thales was not ignorant of the theorem under consideration is found in the beautiful demonstration by which he proved that every angle in a semi-cricle is a right angle. This appears to have been regarded as the most remarkable of the geometrical achievements of Thales, and it is stated that on inscribing a right-angled triangle in a semi-circle he sacrificed an ox to the immortal gods.

Before giving the proof by which Thales probably established the truth that every angle inscribed in a semi-circle is a right angle, it will be well to consider the geometrical capital which this Grecian mathematician had at his command..."

And so he continued - integrating mathematics with its history as best he could.

Articles referring to the historical approach to mathematics were also in evidence in those days. G. Heppel, in Nature in 1893, argued for the history of mathematics to improve the teaching of the subject. He did not favour any deep commitment, but rather the anecdotal, with definitely no examination. For Heppel the mathematics itself had to come first and its history a very poor second. Again, writing in Science a year later, J. V. Collins (1894) advocated the study of the history of mathematics by the teacher to broaden the knowledge of the teacher, and also to enliven the subject matter. Collins considered this approach to be a means to improving the teaching of mathematics itself. A few years later, at the historic 1901 British Association meeting, W. N. Shaw, a mathematician employed at the Meteorological office, went so far as to remark:

"I would venture to say that in ... the teaching of (applied) mathema- tics encouragement should be given to the study in the original text of the masterpieces of applied mathematical literature." (Perry, 1901.)

Although many writers down the years have called for an historical approach to mathematics teaching, there is little evidence that schools have exposed their pupils to the history of mathematics except in isolated cases. Calls for reform have generally come from Universities, often with a view to the improvement of teacher train- ing. If we look at some very popular textbooks of the last hundred years, we can perhaps see one factor behind the lack of history in mathematics teaching. The algebra books of Todhunter, Chrystal, Hall & Knight, Pendlebury's Shilling Arithmetic, for example, or the many books by Durell, show a complete absence of historical references or allusions. These authors led the field and as well as directly influencing teaching with the wide adoption of their books, they were emulated by

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less well-known contemporary writers. It is interesting to speculate what might have happened to school mathematics had Durell incorporated historical material into his textbooks!

It is true to say that the history of mathematics has in the main been advocated for teachers rather than for their pupils, and many references to it can be found in books on the teaching of mathematics. The books by Smith (1900) and Young (1907), for example, are well- known for their advocacy of the study of history, and Benchara Branford's remarkable book (1908) postulated that the development of mathematical knowledge in the individual parallels the historical development of mathematics itself. Branford included, as an appendix, an essay on the history of arithmetic written by Conant which makes interesting reading. The "genetic theory" which Branford espoused was also alluded to by the famous mathematical educator T. P. Nunn, who, as author of a Board of Education Report (1912) while Vice-Principal of the London Day Training College, wrote: "The student in training must think his way through the mathemati- cal curriculum from the genetic standpoint; logic, psychology, and the history of the science being his guides."

Despite this promising assertion, Nunn's proposed two-year syllabus for teachers made no mention of the history of mathematics at all! However, in his well- known book on the teaching of algebra, Nunn (1919) did refer to the history of mathematics on at least thirty occasions, albeit briefly each time.

Another famous mathematical educator of the past, F. W. Westaway (1931), wrote: "Few boys know anything about the great mathematicians except by name. 'No time' says the mathematics master, a statement that cannot be denied. Yet, in order to include at least a little of the history of mathematics, I would sacrifice something else. Let a few of the really great mathematicians live once more ..." a theme which today's teachers can usefully consider.

The famous names of Godfrey and Siddons appeared together as authors of a book for mathematics teachers. Interestingly, it was published in 1931, some seven years after Godfrey's death, and his contribution was actually written in 1911. In the book Siddons wrote: "I have placed the idea of solving problems first because it comes first historically" when discussing the introduc- tion of algebra, but the only further historical allusion is in the introduction to indices where one finds: "It is instructive to show a boy how, by finding 10= V'10=3-162 and taking the square root again 10"=1.779 he can construct a table of the indices of 10 corresponding to a lot of numbers ... But we need not waste time over this. Napier and Briggs did this for us 300 years ago."

These remarks are indicative of the attitude adopted in school textbooks and by teachers themselves. Although history received mention here and there in manuals of teaching mathematics, little attempt was made to incorporate it into the textbooks.

A favourable attitude to historical material is to be found in a publication of the Incorporated Association of Assistant Masters (1957), where a quite uncompromis- ing statement is made: "The incorporation of history in the teaching of mathematics is essen- tial if, to quote from the introduction to this report, the ideas of 'its purpose, its structure, its wonder, its creativeness' are to be aroused in the child."

It seems unlikely that many children receive the mathematical education referred to here if history is indeed so important!

There have, of course, been many books on the history of mathematics itself. Some have been scholarly works but many which have tried to cover too much ground have fallen victim to inadequate research with inevita- ble inaccuracy. Popularizers have found the myths more tempting than the facts - Men of Mathematics by E. T.

Bell (1953) often comes in for criticism from historians of mathematics for the many dubious statements which it contains. How important this is considered to be depends on one's attitude to mathematics, to history, and to education. Some authors of histories of mathema- tics have seen their works as performing specific func- tions. F. Cajori (1896), for example, introducing his his- tory said that ".. certainly the experience of many instructors establishes the importance of mathematical history in teaching", and he wrote his book "with the hope of being of some assistance to my fellow teachers". D. E. Smith, in the preface to the history by Vera San- ford (1930), said that the aim of her book was to show mathematics "as a moving stream instead of a stagnant pool" and he went on to suggest that by studying such histories the teacher would see what was of value and what no longer fulfilled any need in the school syllabus. Smith thus saw histories of mathematics as a means of promoting curriculum development.

Some of the influential reports published this century by the Mathematical Association make reference to the history of the subject. (For example: Report on Primary and Secondary Schools ..., 1919; Report on the Teach- ing of Trigonometry in Schools, 1950; Mathematics in Secondary Modern Schools, 1959.) Generally, the his- tory references have been as footnotes or a last chapter or an appendix and little real emphasis has been placed on the subject. This is reflected by the fact that of all the reports published by the Mathematical Association, none has been devoted to the history of mathematics.

It is interesting to look at the incidence of historical articles in the early years of the Mathematical Gazette compared with the 1950s. The table below covers roughly five-year periods (six issues per year, latterly four issues per year) but the somewhat exploratory first ten issues are treated separately:

Gazette Approximate Number of Pagesdevoted Approximate Numbers YearsCovered Historical to Historical % of total

Articles Material pages 1-10 1894-1897 9 38 30%

11-40 1897-1903 - - 0% 41-70 1903-1908 7 45 7% 71-100 1908-1912 7 63 8%

101-130 1912-1917 17 102 10% 131-160 1917-1924 10 53 6%

307-326 1950-1954 7 35 2% 327-346 1954-1959 3 24 11/2%

It can be seen that after a spate of early articles, and then a dearth, the Gazette settled down early this cen- tury to about 8% of pages referring to historical matters, but by the 1950s there had been a falling away in per- centage terms although it must be realised that the Gazette had got steadily larger (8 pages in 1894, 12pp. in 1895, 24pp. in 1896, then 32pp.... and 80 pages in the 1950s). Few of these articles discussed pedagogy, being in the main straightforward historical accounts, and the writer is led to conclude that the history of mathematics as a teaching aid did not really find much support within the Mathematical Association before 1960. There is a possibly significant achievement in promoting interest in the history of mathematics credited to the Mathematical Association, namely the establishment of the Mathematical Association Diploma with an annual examination (first held in November 1961 and alas to be terminated after November 1977); the Diploma was designed for (non-graduate) teachers of mathematics and in the past has qualified the successful candidates for a slightly enhanced salary. The examination consists of four 3-hour papers in (1) and (2) Pure Mathematics, (3) Mechanics and Statistics, (4) the History and Ideas of Mathematics. The syllabus for this last paper covers the

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following: "Symbols and technical terms. Great mathematicians. Development of number, geometry, analytical geometry and calculus. Study of motion and of the heavenly bodies. Abstraction, generalisation. Non- Euclidean geometries."

Some evidence which indicates the difficulty of estab- lishing the history of mathematics in school teaching relates to the inclusion of it as part of the London Uni- versity "O" level (Alternative "A") Mathematics Syl- labus, in 1952. The "O" level examination consisted of three papers chosen from seven, one of which was on the history of mathematics. However, lack of candidates caused the withdrawal of the paper after 1956, and Freebury's book (1958) written with the London syl- labus in mind somewhat missed the boat! The NUJMB also had some history included in their mathematics at about this period.

Having noted the very limited nature of the history of mathematics at the school level of the period to 1960, it is all the more surprising that the important Ministry of Education Pamphlet 36 (1958) entitled Teaching Mathematics in Secondary Schools should have a whole twenty-one page chapter on "The History of Mathema- tics and its Bearing on Teaching". This outlines the didactic and cultural arguments for including history and refers to many topics which can be profitably taught historically and deserves close attention by mathema- tics teachers. Its obvious enthusiasm does not seem to have rubbed off on many!

The advent of "modern mathematics" which has brought such innovations during the last fifteen years, and the parallel reappraisal of the mathematics cur- riculum, have taken much of the mathematical educator's attention in recent years. In a subsequent article the review of the attitude towards history of mathematics will be continued and its place today will

be considered.

The reproduction from the book by Potts appears by courtesy of the Cambridge University Press.

References (1) Bell, E. T. Men of Mathematics, 2 vols., Penguin, 1953. (2) Branford, B. A Study of Mathematical Education. O.U.P.,

1908. (3) Cajori, F. History of Elementary Mathematics.

Macmillan (N.Y.), 1896. (4) Collins, J. V. Plea for Teaching the History of Mathematics.

Science, Vol. 23, No. 573, Jan. 1894. (5) Freebury, H. A. A History of Mathematics for Secondary

Schools. Cassell, 1958. (6) Godfrey, C. and The Teaching of Elementary Mathematics,

Siddons, A. W. C.U.P., 1931. (7) Heppel, G. Nature, Vol. 48, 1893, pp. 16-18. (8) I.A.A.M. The Teaching of Mathematics. C.U.P., 1957. (9) Ministry of Teaching Mathematics in Secondary Schools.

Education Pamphlet 36, H.M.S.O., 1958. (10) Nunn, T. P. The Training of Teachers of Mathematics

Special Reports on Educational Subjects, Vol. 27: The Teaching of Mathematics in the United Kingdom. Part 2: Board of Education, 1912.

(11) Nunn, T. P. The Teaching of Algebra, Longmans Green, 1919.

(12) Perry, J. The Teaching of Mathematics (British Association Report) Macmillan, 1901.

(13) Potts, R. Elements of Geometry, C.U.P., 1845. (14) Potts, R. Elementary Arithmetic, C.U.P., 1876. (15) Potts, R. Elementary Algebra, Longmans, 1879. (16) Rupert, W. W. Famous Geometrical Theorems and Problems

with their History (issued in four parts) D. C. Heath, 1900-01.

(17) Sanford, V. A Short History ofMathematics, Harrap, 1930.

(18) Smith, D. E. The Teaching of Elementary Mathematics, Macmillan, 1900.

(19) Westaway, F. W. Craftsmanship in the Teaching ofElementary Mathematics, Blackie, 1931.

(20) Young, J. W. A. The Teaching of Mathematics, Longmans Green (N.Y.), 1907.

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