History in Mathematics Teaching: Modern Times

History in Mathematics Teaching: Modern TimesAuthor(s): D. R. GreenSource: Mathematics in School, Vol. 5, No. 4 (Sep., 1976), pp. 5-9Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30211590 .

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HISTORi IN MATHEMATICS TEACHING:

M

TIMES by D. R. Green, CAMET, Loughborough University of Technology

In a previous article it was shown that despite the recommendations of many mathematics educators down the years, little attention was paid to the history of Mathematics, particularly in schools, prior to the advent of "modemn mathematics". In the 1960s school mathematics in Britain (and indeed elsewhere) underwent radical reappraisal and with it came the opportunity for history to be more seriously considered. A look at the texts and syllabuses which have been produced will show to what extent this happened.

SCHOOL TEXTS FOR 11-16 YEARS School Mathematics Project The now obsolete T and T4 books set the pattern which SMP were to follow. The only pieces of history in them worth recording are a brief discussion of Cantor (T, p.19) and Klein's Erlangen programme (T, p.46), a page or so relating the story of the Chevalier de Mere prompting Pascal and Fermat to study probability (T4, pp.144-5) and a brief mention of Monge's work in projective geometry (T4, pp.250-1). In all, the historical element is perhaps 1% of the texts and moreover is incidental rather than integral in most instances.

Books 1 to 5, and 3T are much the same. A typical incidental note is, from book 1, chapter 3: "Many people like lying in bed in the morning. Rene, Descartes, one of the greatest French mathematicians of the seventeenth century, was no exception: but he was not just being lazy, he found he could think better there! One day, when resting in bed, he solved the problem of describing the position of a point in a plane; his method, as you will see, was a development of the older idea of latitude and longitude." (p.33). The only follow-up the writer can find to this comes in Chapter 7: "(x,y) co-ordinates are called Cartesian co-ordinates after Rene Descartes whom we read about on p.33.'" (p.102) Although it would be wrong to claim much for such incidental historical allusions - and no mathematics is explained by the example cited - at least the term Cartesian has been presented in a meaningful way, which is to be applauded. But what about all the other terms - sine, root, zero, algebra .. .?

A somewhat more instructive historical interlude is provided in Book 2, Chapter 1. Setting aside consideration of whether

"Topology" is really "worth the candle", a question on the

Koenigsberg Bridge Problem really does link mathematics with real life, through its history. "Question 16... In 1737 the famous Swiss mathematician Leonard Euler, then working at the Court of Catherine the Great of Russia had his attention drawn to the problem of the bridges of Koenigsberg ... In the river Pregel, which runs through the town, are two islands; they are joined to the banks of the river and to each other as shown in Fig. 20. The problem was this: is it possible to take a walk so as to cross each of the bridges once and once only? Euler dealt with this problem by first reducing it to a problem about nodes (like Question 15). See if you can solve the problem that puzzled the best mathematicians in Europe two centuries ago, until Euler noticed the underlying principles. " (p.12).

Figure 20

5



The weaknesses in this particular presentation of a delightful and worthwhile problem are (i) its relegation to "question 16" and (ii) lack of follow-up. If, as was perhaps thought, it is rather difficult for Book 2, then what a pity the majority are to be denied its mastery altogether - why not postpone its presentation? Again there is more emphasis given to the (trivial?) finding of a solution than to discovering the "underlying principles" which should be the heart of the matter.

Books A-H are similar but have perhaps rather less history. Incidentally the current SMP 7-13 Project has as one aim the "Integration with other Subjects via projects" and the director has expressed his hope (1974) that the history of mathematics will receive more attention than it has in the numbered and lettered series, and more recently (1975) he wrote that "we intend to put in as much as we can". It will be interesting to see what can be achieved at this level.

Midlands Mathematics Experiment This project "took little note of the History of Mathematics" and this is evident in the texts. The course contains a full page on navigation (IA p.11) but only brief references elsewhere, generally along the lines of those in SMP but less frequently.

Contemporary School Mathematics Of the five main texts for this 'O' level course, two show some

sympathy for the history of mathematics. "Mathematics 1", for

example, has a quasi-historical account of how counting may have originated (p.21) and then (p.22), under "Projects", is found: "1. Find out all you can about the history of numbers and in

particular find out why "Zero" was introduced into number

systems. 2. Make a small abacus and learn how to use it.

3. Make a list of the symbols of the numbers one to ten which were used in the following civilisations:

(i) Egyptian; (ii) Babylonian; (iii) Greek; (iv) Roman; (v) Chinese; (vi) Hindu; (vii) Any ancient European civilisation, and (viii) Make up a completely new set of symbols for life on Mars."

Book 2 has some rather dubious "history" but in an exercise

is presented some of the interesting background of -r (pp.29- 30), a topic often favoured for an historical approach: "EXERCISE 8

1. Look up in your Bible, 1 Kings V/ll 23, 11 Chronicles IV 2. What values of r are suggested?

256 2. The Egyptians used the approximation8 . Work this out

to three decimal places. 3. In A.D. 510 an Indian said

If a circle has a diameter of 20,000 paces its circumference can

be found by adding 4 to 100 multiplying this by 8 and adding 62,000.

Circumference Find the circumference and the ratio diamneter

4. In 1656 Wallis made the approximation r=2(2

22 44 66 8..." 133557 7....

(How do you think the pattern continues?) Find a to 3 decimal places. 5. Many years before this Archimedes made a 96 sided polygon to circumscribe a given circle and a similar polygon was inscribed in the circle. The circumference of the circle

clearly must be between the perimeters of these two polygons.

From this he calculated that rr lies between

1 10 3- and 3

Work out each of these to 4 places of decimals. 6. Calculate

1 1 1 1 1 1 1 1 1

and find the square root of your answer.

7. A suggested value for -r was -/10. Find it to 4 decimal places.

4 8. 12 12

11+ 2+32 2+52

Work upwards from the bottom to find its value.

9. 7r=4 1 1 1 1 1 1 1

3 5 7 9 11 13 15

Calculate this to 3 places of decimals. This series was attributed to Gregory and also to Leibniz. .. ."

Perhaps rather more of the history would bring better

perspective, but half a loaf is better than no bread. There are

objections which may be raised about the misuse of the =

sign (somewhat surprising in view of the series editor's recently expressed views in "Uncle Geoffrey's aptitude test' (Ref. 5), and elsewhere (Ref. 6)!) and the word'ing of question 4 reinforces the impression that the authors were rather too casual about equality and approximation. No wonder rr is so

frequently 2! This exercise on rr, completed with Buffon's needle problem, does indicate what might be done but it falls short of being adequate, in the writer's view.

Taken as a whole, the published books and booklets do not match even this modest standard and history is rarely evident.

Scottish Mathematics Group The first seven books, equivalent to an 'O' Level course, have

very few historical references, many less than SMP. The only non-trivial historical material comes in Book 6 where counting systems are discussed.

Shropshire Mathematics Experiment This is perhaps one project in which the lack of history is not attributable to lack of awareness by the text writer. Mr. Herit-

age, himself knowledgeable of the subject, is of the opinion that the history of mathematics is generally irrelevant to the aims of the SME project, namely the development of certain

conceptual structures and abilities in problem solving. The lit-

tle history which is found in the texts is included for "secondary" motivational reasons.

Nevertheless one striking passage is found in Book 3M, being an extract from Robert Recorde's famous early Arith- metic book "The Ground of Artes": "Addition ... The easiest way in this arte, is to adde but two summes at ones togyther: how be it, you maye adde more, as I wi/ tel you anone. Therefore whenne you wy//e adde two summes, you shall fyrste set downe one of them, it forceth not whiche, and then by it draw a lyne crosse the other lynes. And afterwarde sette downe the other summe, so that that lyne maye be betwene them: as if you woulde adde 2659 to 8342, you must set your sumes as you see here.

(8342) (2659)

And then if you list, you maye adde the one to the other or

els you may adde them bothe together in a new place..." (pp.32-33)

6



Deciphering the "English" and also the diagram are interesting exercises in which the mathematics itself can play an important part, although admittedly careful handling of this "research" would be necessary.

Mathematics for the Majority The Schools Council Working Paper 14, which was the forerunner to this project, does refer to history once or twice and the project's teacher's book "From Counting to Calculat- ing" commends a study of early counting systems and calculating devices as helpful in the teaching of arithmetic. How- ever, the Continuation Project team find that "history is not obviously connected with any part of our material and has not been included in the objectives."

Kent Mathematics Scheme This material bank has "some material in the lower levels deal- ing with Early Number from Primitive Times to Hindu-Arabic numerials but "History" does not appear as a specific topic." However, future units may be added which will be more ex- plicitly historical.

Other Texts The general picture here is much the same. However, one series with substantial historical introductions to chapters is "New Mathematics" by Snell & Morgan (C.U.P., 1962). The interested reader is referred to these books as indicating what might be considered the minimum coverage of history likely to be of benefit in mathematics texts. It is difficult to do justice to Snell & Morgan's books, but the following, taken from the introduction to book IV (which goes beyond 'O' level in completing the Jeffery syllabus), indicates the flavour of the approach:

"One of the more fascinating periods in the history of mathematics is roughly A.D. 500-1300, out of which has come such scanty information that writers tend to dismiss it as a period of stagnation. And yet, at the beginning, Greek mathematics had largely stifled itself by its neglect of algebra, and, at the end of the period we find the ground prepared for the lightning advances of the sixteenth and seventeenth centuries.

Undoubtedly the Hindus and the Arabs did much useful work, some of which may well have been inspired by Greek sources. Foremost amongst the Arabs in his work on algebra was al-Khwarizmi, author of Hisab-al-Jabr w-al-Muqabalah. AI-Khwarizmi, Muhammad ibn-Musa (A.D. 780-c.850) was a

principal figure in the history of Arabic mathematics. He was one of the great scientific minds of early Islam, and his text-book on algebra was used until the sixteenth century as one of the standard mathematical works in many European universities. He gave a method for solving quadratic equations which runs as follows: to solve x2+5x =25, let AB represent the value of x and construct the square ABCD. (See Fig. a.) Produce AD to H and AB to K so that DH=BK=2.5 cm. (One half of the coefficient of x.)

A

B

K

D H

P

R 0

C

Complete the square AKRH. Its area is x2+ 2.5x+2.5x +(2.5)2=(x + 2.5)2 But the given equation is x2+5x=25.

. (x+2.5)2 = 25+(2.5)2 = 31.25, .. x+2.5 = /31.25

= 5.59

. x = 3.09

The method which will be used in chapter 1 of this book for the solution of quadratic equations is based on al-Khwarizmi's work, and is known as the method of completing the square." (p.xiii)

Incidentally our words algorithm and algorism come from the name of Al-Khwarizmi and the word algebra comes from this word Hisab-al-Jabr...

Discussion Clearly there has been almost no British project or series of texts which has taken seriously the incorporation of historical material into its mathematics and it must be concluded that virtually no pupils at school have met the history of mathematics through their school texts. The level of history in mathematics teacher training (discussed later) suggests that pupils are unlikely to meet it in the classroom.

EXAMINATION SYLLABUSES FOR 11-16 YEARS

CSE Syllabuses The only two CSE boards in 1970 to offer a Mode 1 syllabus involving the History of Mathematics (North West and South- ern) both dropped the option soon afterwards. It is virtually impossible to check on all the Mode 3 work but the Boards think the incidence of history is very slight. Just four of the 14 Boards mentioned Mode 3 work when enquiries were made by the writer in 1974. ASLEB said "perhaps a very little"; East

Midlands guessed that "about 300 pupils" did some history as part of coursework; South Western had "177 candidates" whose assessment included a small amount of history; W. York & Lindsey estimated that "150 pupils" attempted history questions in Mode 3 paper. The results of a similar survey in 1976 were much the same, just 3 Boards together mentioning about 500 candidates out of a total CSE mathematics candida- ture of about 330,000 for all 14 Boards, with "10%" the estimated percentage of history on the particular papers mentioned. The general reaction from the Boards was that the small amount done, usually in projects, was of little value.

G.C.E. 'O' Level Syllabuses No 'O' level syllabus since 1956 has involved the history of mathematics. However, the feasibility study on a Common Examination at 16+ mounted by SMP in conjunction with the Oxford & Cambridge Board and the Southern Board has included optional topic work on the History of Mathematics, and this is a most interesting development for the future.

Discussion It is doubtful whether more than a very few schools offer unexamined history of mathematics courses below the Sixth Form except, interestingly enough, in the history of calculating devices as part of computer appreciation courses. The project textbooks and examination syllabuses lead one inevitably to conclude that the history of mathematics is really non-existent in the vast majority of schools for pupils aged 11-16, and visits to schools tend to confirm this.

THE SIXTH FORM Currently no GCE 'A' Level board features the history of mathematics on any syllabus. The only board in recent years to do so was Cambridge who set one question per 'A' Level Mathematics or Pure Mathematics paper, in the years 1961-72. Why was this discontinued? "The reason... was because there was such a poor response from the candidates. Indeed, the examiners reported that with very few exceptions the attempts at the questions... were very poor indeed." The reasons for this are perhaps not hard to discover: the low priority given to the history of mathematics with its inevitable rejection by the more able; inadequate teacher preparation; the examination system; the lack of integration with the rest of the 'A' Level course; lack of readily available suitable material.



A Mathematical Association pamphlet (1967) indicated that some non-examined history was to be found in the sixth form, and Holt and Mclntosh (1965) reported such a course in "Mathematics Teaching", but these are very much the exceptions. A telling statistic comes from a survey report (1969) on college of education entrants for 1965 in the Manchester ATO. Of the 1663 students in the sample only eight (i.e. 1/2%!) had studied the history of mathematics while at school.

Perusal of Sixth Form mathematics texts shows that historical notes are even scarcer for 'A' Level than they are for pre- Sixth Form courses. However, the Leverhulme Research Pro- ject, designed to modify a new Open University course on the History of Mathematics for use in Sixth Forms (and elsewhere), may be a significant new development and interested teachers and lecturers should write to the Open University (see Ref. 8).

PRIMARY SCHOOLS As a gauge of this age range the Nuffield Guides and Fletcher's "Mathematics for Schools" Series will be referred to. Most of the Nuffield guides do not have historical references but an exception is "Computation and Structure 2" which devotes some space to the history of counting (pp.2-5). Reference is made to Chinese, Babylonian, and Roman systems as well as to our own. In the summary is found

"We have been concerned with ... the historical development of natural numbers for (a) interest and (b) a visual perception of the abstract notion of a number. This work can help to bring out thedifference between numerals and numbers. "

Further on in "Computation and Structure 2" is the historical development of length, weight, volume and the concluding booklist refers to at least eight history books.

Fletcher's books for pupils do not have historical references, although "tallying" (Level 2, book 1) comes close to doing so. The teachers' guides also have no emphasis on history, although a very brief mention of the history of Pythagoras' theorem is to be found (Guide to Level 2, book 9).

A survey of other popular Primary school texts shows that some elementary history may well be included for work on number and measure.

COLLEGES OF EDUCATION AND POSTGRADUATE CERTIFICATE COURSES It is particularly difficult at this time of college restructuring to estimate the extent and worth of the history of mathematics courses currently found in the colleges of education. At a guess perhaps 25% of specialist mathematics students and most of those destined for Primary schools will meet the history of mathematics to some extent. Colleges vary so much that generalisations may mean little. Some, with knowledgeable and enthusiastic staff, do much while others may completely ignore this area. (One interesting course, at Stockwell College, combines History and Mathematics as the two main subjects of study with the history of mathematics providing the bridge between the two disciplines.)

A criticism of some courses has been that the students are

essentially lectured at or, if required to submit work, need do little more than copy it from the nearest book. The ATCDE report on mathematics Main Courses in Colleges of Education (1967) referred several times to the history of mathematics and emphasised that:

"The proper study of the history of mathematics must always involve constructive and creative endeavour on the part of the individual students and student groups. "

This undoubtedly valid assertion is, however, not so easy to implement. One recent (as yet uncompleted) attempt to improve the availability and awareness of source material, and so encourage a more meaningful study of the history of

mathematics, is Leo Roger's "Resourcefile' entitled "Finding Out in the History of Mathematics" which lists commonly available books, articles, films, etc.

It is doubtful whether the already hopelessly overcrowded P.G.C.E. courses find time to introduce the history of mathematics in any real sense. However, the Mathematics Teacher Education Project, based at Leeds University, is aim- ing to produce some historical material for use in the P.G.C.E. cou rses.

UNIVERSITIES A substantial number of our mathematics teachers are edu- cated in our Universities and a survey of history in Under- graduate mathematics courses can indicate the kind of experience which secondary school mathematics teachers may get. All English and Welsh universities and University Colleges were surveyed in 1974 and some again in 1975 and all again in 1976. Of 48 institutions circulated, information was received from 43.

Of these 43, eighteen are offering history of mathematics in some form to some undergraduate students. Details are as follows (best treated as a guide only):

Birmingham: Bradford: Cambridge: Keele:

Lancaster:

Leeds:

London: Bedford:

Imperial: Kings:

Royal Holloway: Westfield:

Nottingham: Sheffield:

Southampton:

Wales:

Warwick:

3rd year option, 30 hours, commencing 76-77. Included in a "complementary studies" course. A short optional unexamined course. For Arts Students - Subsidiary Course, 1 year, 72 hours. Included in a History of Science course open to all, also touched on in a 3rd year mathematics option (Mathematical Education). Part of History of Science course, also briefly in a Mathematical Education option for mathematics students.

3rd year hons. option, 24 hours (Geometrical ideas). 3rd year option, 26 lectures. A course entitled "The Development of Mathe- matical Ideas" includes history. One lecture per week for six terms spread over all three years. 3rd year option, 40 lectures. For 1st year mainly, 35 lectures. First run in 1974-75, being repeated 75-76. Year 1, term 1, six lectures (Applied Mathematics). For combined honours students as part of a history and philosophy of science option, 2nd year. A 3rd year honours option, also as part of a 2nd year History of Science option. A voluntary 3-day residential course for 2nd year honours students, for all Colleges combined. In addition: Aberyswyth: 3rd year option, 25 lectures. Bangor: 3rd year option, 20 lectures Swansea: A short unexamined course. Seminars in conjunction with Open University broadcasts open to all students (about 20 hours). Also essay options in all three years.

Others planning such courses include Loughborough (for Edu- catipn & Mathematics students), Salford and Surrey.

What emerged from this survey was that many University

MVathematics Departments would like to offer the History of Mathematics but feel prevented from so doing by lack of expertise or of time. No clear pattern emerges, except a pref- erence for its inclusion "late in the course". Although some future teachers obviously follow these courses, clearly most do not. (A rough estimate based on the available evidence would be 10% of those produced.)

THE OPEN UNIVERSITY The Open University has introduced a 1/2 credit second level course (AM289) which began in February 1976. In view of the large numbers who may follow this course (or eavesdrop!) and the high percentage of teachers among them, this could prove a very effective development. The Leverhulme Research Pro- ject (mentioned in Section 4) is preparing materials relating to the OU history of mathematics course designed specifically for school use, and the recently published "History of Mathema- tics. Topics for Schools" by W. Popp is an integral part of this.



JOURNALS OF MATHEMATICAL EDUCATION The 'Mathematical Gazette' As a previous article (Vol. 5 No. 3) has indicated the incidence of historical articles in the Gazette declined in the period 1900-1959, perhaps reflecting a similar decline in interest by mathematical educators over that period. Analysis of the Gazette of 1960-1975 completes the picture (Table 1).

Number of Pages devoted Approximate Gazette Years Historical to Historical % of total

Numbers Covered Articles Material pages

347-363 1960-64 13 124 6% 367-386 1965-69 3 24 1% 387-406 1970-74 12 59 4% 407-411 1975-Mar 76 3 17 41/2%

TABLE 1 It can be seen that there has been some pick-up, perhaps with "modern" mathematics in all its many aspects demanding slightly less of the Gazette's pages.

'Mathematics in School' Certainly this journal has given space to a number of interesting and one hopes helpful articles - from the pen of Michael Holt, John Dubbey and others. There is, in the writer's opinion, a need for further articles relating more closely to the individual lessons in the classroom so that the teacher is encour- aged to incorporate the material without feeling himself inadequate or the material too peripheral or indeed irrelevant. A new series of topic-based articles is being planned to meet this need.

Other Journals "Mathematics Teaching" has carried an occasional historical article but somewhat fewer than those of the Mathematical Association journals. Even fewer historical articles are to be found in other journals such as "Educational Studies in Mathematics" and "International Journal of Mathematical Education in Science and Technology".

Comment It is almost certainly true that the rather limited coverage of history of mathematics is not due to editorial decisions but to lack of copy - itself a reflection of the status of the subject in the mathematical education world. It is also significant that very few indeed of the articles discuss the pedagogical aspects, nearly all being straightforward historical accounts.

HAS THE HISTORY OF MATHEMATICS ANY FUTURE? "How can we expect teachers to plan a curriculum, choose texts, explain mathematics to their colleagues and to parents, or to give a sound picture of it to their students, unless they have some knowledge and experience of the history and philosophy of mathematics and its role in contemporary culture? Yet we rarely offer our students courses in the foundations, history or philosophy of mathematics." K O May, 1972.

It must be recorded that the role of the history of mathematics in the teaching of mathematics is a largely unexplored field ... the harvest is plentiful but the labourers are few. The regular inclusion of historical topics in "Mathematics in School" is one encouraging sign. Also there seems to be a general upsurge of interest in the history of mathematics at the moment and with the reorganisation of colleges of education, the new proposals for school examinations (Common Exami- nation at 16+; CEE: N & F .. .), the Dip HE, and the enthusiasm of the polytechnics and universities for the new degree structures, we can expect some improvement in the position of the history of mathematics. Let us hope that this is to the benefit of mathematics teaching!

In a subsequent article some ideas on using the history of mathematics in the classroom will be discussed.

REFERENCES 1. Association of Teachers in Colleges and Departments of Education Teaching Mathematics. Main Courses in Colleges of Education. 1967. 2. Holt, M. J. and Mclntosh, A. J. A Mathematics Course forArts Specialists. Mathematics Teaching No. 32. Autumn 1965. 3. Manchester University The Mathematical Ability and Background of Entrants to Colleges of Education. School of Education, Manchester. 1969. 4. Mathematical Association Experiments in the Teaching of Sixth Form Mathematics to Non- specialists. Pamphlet No. 2. Bell. 1967. 5. Matthews, G. "Uncle Geoffrey's aptitude test'. Mathematics in School. Vol. 4 No. 6. November 1975. 6. Matthews, G. and Bausor, J. "Models and Measurement". int. J. Math. Educ. Sci. Techno/. Vol. 3 No. 3. July 1972. 7. May, K. O. "Teachers should know about Mathematics". Int. J. Math. Educ. Sci. Techno/. Vol. 3 No. 2. April 1972. 8. Nicholson, J. S. A. (Mrs) Research Officer, History of Mathematics, Faculty of Arts, The Open University, Milton Keynes MK7 6AA. 9. Popp, W. History of Mathematics. Topics for Schools. Transworld. 1975. 10. Rogers, L. F. Finding Out in the History of Mathematics. Preprint, January 1975. Acknowledgements The author wishes to thank the following for permission to reproduce material from books published by them: Cambridge University Press: (a) SMP texts. (b) New Mathematics: a unified course Book IV K. S. Snell and J. B. Morgan, 1962. Penguin Books Limited Learning Mathematics Book 3M. pp.32-33. R. S. Heritage and W. I. Lewis, 1968. Edward Arnold Ltd. Contemporary School Mathematics Books 1 and 2. H. D. Ellerby and A. J. Sherlock, 1971.

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