Accounts of the history
of mathematics have generally paid little attention to the grass-roots
teaching, learning and applications of mathematics in the life of ordinary
people.** ^{3}** This paper is an a tempt to
give a brief overview of some of the significant social and political movements
in the eighteenth and nineteenth century which contributed to the development
of the mathematics taught outsi e the universities which led to the formation
of the ideological divisions within our contemporary school mathematics
curriculum.

The population of England
grew from six million in 1750 to nine million in 1800, and there was a
rapid change from urban to industrial life with all the problems that this
brings. With the belief hat industry could improve the conditions of life,
the philosophy of utilitarianism was developed intending that society should
provide the `greatest good for the greatest number'. By the mid eighteenth
century, new theories about the nature of man, of society, and of the acquisition
and purpose of knowledge, began to have far-reaching influences on political
and educational ideas. At the same time, "laissez-faire" economics was
promoted, allowing the free interplay of forces in economics and society
and monitoring their effects with minimal legislative interference. These
benefits and forces, it was assumed, would be controlled by educated people
making the `right' judgements about moral, political, and economic issues,
and the principle that education could influence human beliefs, attitudes,
morals and conduct was emphasised. In the later eighteenth century we see
a gradual development of the professionalisation and institutionalisation
of mathematics teaching. Set against the considerable changes in society
and economic development, the kind of mathematics, the people who taught
it, and the places where it was taught all underwent significant changes.
During a critical period from about 1750 to 1850 the gradual isolation
of the universities with their classical traditions from industrial growth
and technical improvement was evident, and the need for practical applications
of science and mathematics was answered by other sources and other institutions.
This division was closely linked to the class system and the established
English intellectual and social attitudes of the time.^{4}

In mathematics teaching two traditions can be identified from the sixteenth century. The "Liberal" tradition was based on translations of Billingsley's Euclid (1570) evolving into a formal style which continued until Playfair's Euclid of 1792 became the standard for the next hundred years.

The other "Vocational"
tradition is based on Robert Recorde's "Pathway to Knowledge" where the
principles of geometry were set out so they might "most aptly be applied
unto practise both for the use of instruments geometrical and astronomical
and also for projection of plats** ^{5}** in
every kind, and therefore much necessary for all sorts of men" Recorde
1551).

In this way geometry and arithmetic became popularised in many self-help books where detailed explanations and exhortations to the student accompanied the examples.(Fauvel,1989)

John Dee's "Preface"
to Billingsley's Euclid contains a comprehensive description of the "Mathematical
Arts" showing their universal usefulness and giving reasons for studying
mathematics at all levels. In Elizabethan times mathematics (which included
astronomy and astrology) was seen as the key to knowledge and the mysteries
of the universe.** ^{6}** This tradition continued
(Taylor, 1964, 1966) an in the 1780s at Woolwich Bonnycastle was writing
alternative treatments of geometry

"The Rights of Man" (Paine, 1798), was an attack on the established social order and its exploitation of the poor and working classes. The radicals saw the Church as the main obstacle to political r form in its reinforcement of the strong social stratification, and they replaced this indoctrination with rational education through their own schools, aiming to inform people of the reasons for the r condition and the state of society and industry, and placing instruction within the reach of everyone. Teaching methods encouraged self-confidence, and the capacity for clear self expression, and he organisers realised the importance of combining systematic education with mass political agitation. Books and newssheets were shared: an individual would take a book home, read a passage and prepare a talk for the next meeting; the book would then be passed on, and the process repeated. Many subjects, including some elementary mathematics were learnt in this way. As a result, men and women b came informed and critical leaders of the new working class movement, able to master and comprehend some of the most advanced political thinking. This was recognised as a threat by the establishment and in 1799 an Act of Parliament was passed ".for the more effectual suppression of societies for seditious purposes.." (Simon 1960,p.183)

By 1817 there were
popular demands for a rational secular education for all. Paine had demanded
the teaching of science which was directly applicable, to be regarded as
the cornerstone of a rationalist philosophy. These demands were of great
concern, and in 1817 a House of Lords Secret (*sic*) Committee reported
on the unprecedented circulation of "publications of the most seditious
and inflammatory nature, marked with a peculiar character of irreligion
and blasphemy, and tending not only to overturn the existing form of government
and order f society, but to root out those principles upon which alone
any government or any society can be supported." (Simon 1960 p. 131)

The Stamp Act (1817)
required the registration and taxing of all newspapers and journals, and
as a result, the radical newspapers were forced underground.^{8}

Richard Carlile's "Address
to Men of Science" (1821) also demanded a curriculum which contained reading,
writing, the use of figures, elements of astronomy, geography, natural
history and chemistry so that children may "at an early period of life
form correct notions of organised and inert matter, instead of torturing
their minds with metaphysical and incomprehensible dogmas about religion"
(Carlile 1821 p.22 ) He believed that science, best studied by observation
and experiment, was the key to knowledge and freedom, and promoted a materialist
psychology, and demanding social and moral education by example.

By the late 1780s, to counter the radical political literature that was freely circulating, Sunday Schools were established for the poor, their major purpose being to indoctrinate pupils in the principles of religion and the duties of their state in life. Here, if you were lucky, it was possible to learn reading, writing, elementary arithmetic, and the catechism. However, due to the teachers' concern for the health and welfare of their pupils they unwittingly `created thought in the unthinking masses'.(Simon, 1960 p.183)

In the 1830s we begin
to see the establishment of the English Public School system. The amount
of mathematics and science taught in these schools was very variable and
schools like Eton, Harrow and Rugby** ^{9}**
did not appoint mathematics masters until challenged by some of the newly
founded institutions. Substantial reforms were made to preserve the establishment,

The Private or `Dissenting'
Academies were the places where Non-conformists could be educated** ^{12}**
The earliest of these was Warrington Academy, founded in 1757, and the
subjects taught had obvious practical applications. Manchester College
of Arts and Sciences, founded in 1783, taught sciences and practical arts
on four evenings a week. Its syllabus contained classical languages, grammar
and rhetoric, mathematics (including trigonometry), mechanics, natural
philosophy, (including astronomy and chemistry) English composition, French,
commercial and economic geography, history, politics, writing, drawing,
book-keeping and shorthand. Subjects like these became the standard curriculum,
and most of the important cities of this time developed similar educational
institutions. There was a great demand for applied science, and "mixed
mathematics".

The Literary and Philosophical
Societies also supported the development of Mechanics Institutes, which
became another focus for working class self-education. They introduced
science, literature and the arts; deliberately excluded politics and religion,
and provided lectures, evening and day classes, and libraries. There was
a substantial demand for reading scientific (and clandestinely also political)
texts and reading rooms and loan systems were established. The curriculum
was based on what was "useful" to workers, and lectures were related to
practical applications and local engineering and manufacturing problems.** ^{14}**
Advanced classes were given in a selection of subjects like Grammar, French,
Latin; Science, Chemistry, Electricity; Mixed Mathematics, Algebra and
Mensuration. Provision for science also meant that collections of apparatus
began to be built up, and lecturers established courses, developed curricula,
and wrote texts. (Inkester 1975, Royle, 1971)

The Royal Naval Academy,
founded at Portsmouth in 1722, (renamed the Royal Naval College in 1806)
transferred to Greenwich in 1873. After undergoing similar problems and
reorganisations to its military counterpart, from 1885 the Academy taught
ballistics for gunnery and torpedo officers, mechanics and heat for engineers,
and dynamics for ship construction. Thus it was that by the end of the
century clear, practically focused and vocationally relevant courses had
evolved for the training of military and naval personnel.

No women were admitted
to Oxford or Cambridge before the beginning of this century; the Victorian
attitude to the mental capabilities of women, and their low social status,
together ensured that a y opportunities for further education were severely
limited. However, this was to change slowly with the publication of the
"Educational Times" in 1847, where subjects like the importance of women
in society, and the qualities of women's minds were intelligently discussed.
The College of Preceptors, founded in 1846, played a major role in supporting
women, and from the 1860s we find a growing movement for the elimination
of sex differences in education, particularly in mathematics and science.
From the mid nineteenth century, higher education for women began to develop.
Queens College** ^{16}** was founded in London
in 1848, the Ladies College Bedford Square in 1849, and by 1878 University
College became the first co-educational institution where women and men
were examined together.

In 1826 University
College was founded with the support of those who were excluded from Oxford
and Cambridge, liberal politicians, and Jeremy Bentham, the humanist philosopher.
In 1828, after demands to provide a religious foundation in London, King's
College was founded. In 1828 De Morgan was appointed the first professor
of mathematics at University College. He was a thoughtful, idealistic a
d energetic educator whose text books and pedagogical writings show a deep
concern for the problems of learning and teaching. His motives for writing
On the Study and Difficulties of Mathematics (1 31) are to help `tutorless'
students, with the areas of elementary mathematics which give most difficulty,
describing their nature without emphasising routine operations. De Morgan
takes the view t at mathematics is a necessary part of a liberal education,
and that it is useful, being the key to other sciences. Much of his work
was serialised through the "Society for the Diffusion of Useful Knowledge"
(SDUK).^{17}

Meanwhile Whewell at Cambridge, aimed to place mathematics in the curriculum of every student of the university, reinforcing the "Liberal" view: "I believe that the mathematical study to which men are led by our present requisitions has an effect, and a very beneficial effect on their minds: but I conceive that the benefit of this effect would be greatly increased, if the mathematics thus communicated were such as to dissipate the impression, that academical reasoning is applicable only to such abstractions as space and number." (1836, p.44)

As the century progressed,
university mathematicians seemed less inclined to spend their time educating
the masses; growing professionalism motivated more `pure' mathematical
interests and since, from 1850, Cambridge required a knowledge of Euclid
for its entrance exams, other universities followed suit.^{18}

The products of industry shown in the Exhibition of 1851 were based more on the freelance initiatives of innovators than any government sponsored organisation, and it later became clear to government that economic advantage rested not only on technical education but also a good primary education. The 1870 act ensured that education up to age 13 was available to all, and while the attempts to devise differentiated schooling on a class based system had failed, these attitudes prevailed in the secondary, technical and grammar schools that evolved. It is here that we see the ideological divide; the establishment provided for its own in continuing the liberal tradition in the Public Schools and using mathematics to control the `gateway' to Universities where `pure' mathematics flourished at the same time invoking the utility of `vocational' mathematics to train the industrial workforce in technical schools and colleges.

Looking at the more recent past, this conflict has been compounded by issues involving pedagogy as well as style and content, but the expectations of the two ideologies can be detected in the nature and mode of presentation of the curriculum materials of today. Dowling (1998) locates these ideologies in a detailed analysis of contemporary school texts; in this brief presentation I am attemptin to show their social and historical roots.

2. Here I give the general background to the argument. Earlier discussion of this subject can be found in Rogers (1979 and 1981). A longer version of this paper will be available on request.

3. There are exceptions; see Taylor's work on the Mathematical Practitioners (1964 and 1966) and Yeldham (1926 and 1936) but there is little attention to social issues.

4. For some general background to this period and its social, economic and political detail see Simon (1960) and Hobsbawn (1968).

5. A "plat" was a plan used for building or surveying; a map or chart for finding direction or navigation; or an explanatory table or diagram.

6. It is interesting to note that the editions of Euclid up to Leek and Serle (1661) all contained Dee's "Preface" but after this it disappears from further editions. Thus Euclid clearly becomes part of the "Liberal" tradition.

7. Bonnycastle (1810 p. ii) carries a list of practical texts of arithmetic, geometry, mensuration, astronomy, plane and spherical trigonometry, some having gone through numerous editions.

8. A significant figure in all this was James Mill (not to be confused with J.S.Mill the philosopher) who was educated at Edinburgh university and came to London in 1802 as a journalist. His politic l and educational theory can be found in the Westminster Review particularly during 1824 - 1826.

9. Leading the reform was Dr. Thomas Arnold, appointed as Headmaster of Rugby in 1828, who reformed the school and created the ideal school for the Victorian upper middle class.

10. The schools concerned at this time are the so-called `nine greats': Charterhouse, Eton, Harrow, Merchant Taylors, Rugby, Shrewsbury, St Pauls, Westminster, and Winchester.

11. Among its early members were Boulton, Watt, Priestly, Galton and Erasmus Darwin (the grandfather of Charles Darwin).

12. Oxford and Cambridge would only admit those who were prepared to acknowledge the King as head of the Church of England.

13. That is, practical geometry, measurement, arithmetic and sometimes fluxions. Nicholson (1823) is a typical text in this genre. See also Cook (1981)

14. The Prospectus of the Sheffield Mechanics Institute (1832) states; "The object of this Institute is to supply, at a cheap rate, to the classes of the community, those advantages of instruction, in the various branches of Science and Art which are of practical appli ation to their diversified avocations and pursuits." (Inkester 1975)

15. Even then, mathematics was not regarded as a subject really suitable for girls neither in the `liberal' sense nor in the `vocational' sense. (see Harris 1997 particularly chapters 3 and 4)

16. De Morgan taught at Queens college, but only for a year, apparently feeling that the ladies were not of a sufficiently high standard; and as a member of the London Mathematical Society, showed an interest in the attempts to reform the teaching of school geometry. (Rice 1996)

17. The SDUK was founded in 1826 by Henry Brougham and other liberal politicians as an alternative to the radical press, and through its publications intended to give a `suitable direction' to working class thinking. The Differential and Integral Calculus (1842) was originally published in the Penny Cyclopaedia in forty two weekly parts.

18. Further discussion of the development of the mathematics curriculum and its pedagogy in the latter part of the century can be found in Price (1983).

19. Perry was an engineer and his syllabus provided a new paradigm that came from outside the school tradition. (DSA 1899)

20. Cayley as chief examiner for Cambridge entrance insisted on keeping Euclid

Bonnycastle, J. (1780) The Scholar's Guide to Arithmetic, or a Complete Exercise Book for the use of Schools London

Cook, I. (1981) `Geometry
for a Carpenter in 1800, *Mathematical Gazette*. **65** (433) October
1981 (193 - 195)

De Morgan, A. (1831)
*On the Study and Difficulties of Mathematics* London

Dowling, P. (1998)
*The Sociology of Mathematics Education: Mathematical Myths / Pedagogic
texts*, London, Falmer

DSA (Department of Science and Arts) Practical Mathematics: Summary of Six Lectures delivered to Working Men by Professor John Perry London HMSO

Fauvel, J. (1989) "Platonic
Rhetoric in Distance Learning: How Robert Recorde Taught the Home Learner."
*For the Learning of Mathematics* **9** (1) February 1989 (2 -
6)

Guicciardini, N. (*1989)The
Development of the Newtonian Calculus in Britain 1700 - 1800* Cambridge,
Cambridge University Press

Hobsbawn, E.J. (1968) Industry and Empire, London

Inkester, I. (1975)
`Science and the Mechanics Institutes 1820 - 1850; the case of *Sheffield'
Annals of Science*** 32** (5) 1975 (451 - 474)

Mehrtens, H., Bos, H., Schneider, I. (1981) Social History of Nineteenth Century Mathematics Boston, Birkhauser

Nicholson (1823) A Popular Course of Pure and Mixed Mathematics for the use of Scholars and Students London

Paine, T. (1798) The Rights of Man, London

Price, M.H. (1983)
`Mathematics in English Education 1860 - 1914: some questions and explanations
in curriculum history' *History of Education* 12 (4) 1983 (271 - 284)

Price, M. H. (1994) Mathematics for the Multitude? Leicester, The Mathematical Association

Recorde, R. (1591)*A
Pathway to Knowledge*, London

Rice, A. (1996) `Mathematics
in the Metropolis: A Survey of Victorian London' *Historia Mathematica*
**23** (4) 1996 (376 - 417)

Rogers, L. F. (1979) `Pure Mathematics and School Mathematics in Nineteenth Century England:

Comparison and Contrast' in Bielefeld IDM (1979) (231 - 242)

Rogers, L.F. (1981) `A Survey of Factors Affecting the Teaching of Mathematics Outside the Universities in Britain in the Nineteenth Century' in Mehrtens et. al. (1981) (149 - 164)

Simon, B. (1974) *The
Two Nations and the Educational Structure 1780 - 1870*, London, Lawrence
& Wishart

Taylor, E.G.R. (1964)
*The Mathematical Practitioners of Tudor and Stuart England*, Cambridge,
Cambridge University Press

Taylor, E.G.R. (1966)
*The Mathematical Practitioners of Hanoverian England*, Cambridge,
Cambridge University Press

Whewell, W. (1836)
*Thoughts on the Study of Mathematics as Part of a Liberal Education*,
Cambridge

Yeldham, F. A. (1926)
*The Story of Reckoning in the Middle Ages*, London, Harrap &
Co.

Yeldham, F. A. (1936)
*The Teaching of Arithmetic Through Four Hundred Years*, London, Harrap
& Co.