# The teaching of mathematics in Britain in the Nineteenth Century.

This century saw the waning of the Church's power over schooling and education in Scotland. The Scottish Act of 1803 placed the examination of schools in the hands of the presbyteries and government inspection, which had been gaining support since the 1770's, was instituted in 1833. The final blow to the Church's power was the Education (Scotland) Act of 1872 which transferred the work of administering education from the Church to secular bodies. These were then encouraged to dedicate themselves to methods and activities which would eradicate illiteracy in the towns and cities of Scotland, much of which was due to the industrial revolution and the mass migration of workers from rural areas. Children were once again enjoined to go to school, but could leave the educational system once they could read, write and had an elementary knowledge of Arithmetic. It was now generally accepted that some level of understanding of Mathematics was absolutely necessary for modern life, and there were few schools who did not give Mathematics a place in a student's timetable of classes.

Mathematics was also becoming easier, thanks to efforts made abroad to formulate Arithmetic into a series of easily understandable rules and operations. There was also a lot of work done on when and how a pupil should be taught. Pestalozzi's work of 1803 influenced many who then incorporated his reforms into their own schools. The most fundamental of these were that children should start learning Mathematics and basic Arithmetic as soon as they entered school and that this work should be based on perception and the properties of physical objects. He also promoted a knowledge of the names of numbers and the simplest of Arithmetic operations before the introduction of figures and notation. What is perhaps most important is the position of prominence that Mathematics was placed in his school, no other subject was deemed more important at this stage of a pupil's education.

Such advances took time to filter into the more traditional education establishments. Much of Britain's institutions were still very conservative. Not the least of which were the two English Universities. The first half of the 19th century saw a revival of Mathematics education at Cambridge due in part to the efforts of Peacock, Babbage, and John Herschel who had formed the Analytical Society at Cambridge during their undergraduate years. The main aim of this club was to persuade the university to adopt the continental practise, and they translated Lacroix's

A short while later in 1826 a new institution advertising itself as being free from religious prejudice was founded. The University of London aimed to provide an education in Mathematics and Physical Science, Classics and Medicine. Despite his young age (less than 22) De Morgan got the chair of Mathematics at the newly opened university and it is through his efforts that methods for lecturing in Mathematics (and other subjects) improved in leaps and bounds. At the end of his lectures, De Morgan handed out sheets of problems based on that day's lecture. He then expected the students to attempt these problems and to hand in their solutions later so that he might mark them before handing them back with corrections. Hopefully they would learn from their mistakes, and he would find out where students were having difficulty understanding the ideas that he presented, and thereby improve his course for the following year.

De Morgan also commented on the quality of mathematics teaching in the schools. Despite the advances made in Europe many teachers had reverted to the old use of rote learning found in the Mediaeval ages, and little effort was made to ensure that pupils understood the underlying principles of what they were learning. Geometry especially was taught following the lines laid out in Euclid's Elements despite De Morgan's calls for different methods whereby the pupil learnt the vocabulary of the subject, with many illustrations of the diagrams and constructions that they would later work with when studying the axioms and theorems. Teacher training schools were proposed in 1839, and started up in an attempt to improve this state of affairs, but the reaction against this project was so great that it was rapidly dropped. Some schools were up and running already, but privately as for example David Stow's Glasgow Normal Seminary of 1836, but these were rare. Examination and certification of teachers, started by the SSPCK (Society in Scotland for Propagating Christian Knowledge) Schools before 1795, was not put into general practice.

In the 1820's a new type of school started appearing. These included Liverpool Institute (1825), the London University, later renamed the London College School, and Kings College School in 1829. These schools were less expensive due to being funded by a committee and, because they had been aimed at providing education for the middle classes, they sought more utilitarian ends. Because of its importance in the world of trade, commerce and industry, Mathematics (and other sciences) was given a pride of place in the syllabus.

In 1837 a select Committee of Parliament was appointed to consider how 'useful education' might best be provided for the children of the poorer classes in England. As a by-product of this, and in imitation of efforts in Scotland, the Committee of the Privy Council on Education proposed in 1839 that an Inspectorate of schools should be set up. Despite the work of the earlier Committee, there were no further movements towards compulsory education, although attempts were made to broaden state-assisted education and to improve its quality. In fact there started a counter movement to that which advocated education for all. Andrew Bell (who founded Madras College in St Andrews where EFR received his school education) wrote that the utopian scheme for universal diffusion of general knowledge would confound the distinctions between ranks and classes of society on which the general welfare hinges. This movement only had a limited effect and in 1859 a Public Commission was established to inquire into the state of popular education in England and report what measures, if any, are required for the extension of sound and cheap elementary instruction to all classes of people.

After this stage the changes and improvements to education, and the place that Mathematics held in the common curriculum, became much more rapid with several Government backed experiments, such as the Fife Mathematics Project of the 1960's and others, leading the way to better methods and syllabuses.

Mathematics was also becoming easier, thanks to efforts made abroad to formulate Arithmetic into a series of easily understandable rules and operations. There was also a lot of work done on when and how a pupil should be taught. Pestalozzi's work of 1803 influenced many who then incorporated his reforms into their own schools. The most fundamental of these were that children should start learning Mathematics and basic Arithmetic as soon as they entered school and that this work should be based on perception and the properties of physical objects. He also promoted a knowledge of the names of numbers and the simplest of Arithmetic operations before the introduction of figures and notation. What is perhaps most important is the position of prominence that Mathematics was placed in his school, no other subject was deemed more important at this stage of a pupil's education.

Such advances took time to filter into the more traditional education establishments. Much of Britain's institutions were still very conservative. Not the least of which were the two English Universities. The first half of the 19th century saw a revival of Mathematics education at Cambridge due in part to the efforts of Peacock, Babbage, and John Herschel who had formed the Analytical Society at Cambridge during their undergraduate years. The main aim of this club was to persuade the university to adopt the continental practise, and they translated Lacroix's

*Differential and Integral Calculus*book into English to aid this. By 1817 they had partial success as Peacock was appointed moderator in the university examinations and introduced the new notation this way. By 1823, while Augustus De Morgan was at Cambridge, the analytical methods and notation of differential calculus made their way into the course. However, it is obvious that even during De Morgan's time the examinations at Cambridge were still very narrow, and students who, like De Morgan, read more widely than the strict syllabus did not get recognised for their ability.A short while later in 1826 a new institution advertising itself as being free from religious prejudice was founded. The University of London aimed to provide an education in Mathematics and Physical Science, Classics and Medicine. Despite his young age (less than 22) De Morgan got the chair of Mathematics at the newly opened university and it is through his efforts that methods for lecturing in Mathematics (and other subjects) improved in leaps and bounds. At the end of his lectures, De Morgan handed out sheets of problems based on that day's lecture. He then expected the students to attempt these problems and to hand in their solutions later so that he might mark them before handing them back with corrections. Hopefully they would learn from their mistakes, and he would find out where students were having difficulty understanding the ideas that he presented, and thereby improve his course for the following year.

De Morgan also commented on the quality of mathematics teaching in the schools. Despite the advances made in Europe many teachers had reverted to the old use of rote learning found in the Mediaeval ages, and little effort was made to ensure that pupils understood the underlying principles of what they were learning. Geometry especially was taught following the lines laid out in Euclid's Elements despite De Morgan's calls for different methods whereby the pupil learnt the vocabulary of the subject, with many illustrations of the diagrams and constructions that they would later work with when studying the axioms and theorems. Teacher training schools were proposed in 1839, and started up in an attempt to improve this state of affairs, but the reaction against this project was so great that it was rapidly dropped. Some schools were up and running already, but privately as for example David Stow's Glasgow Normal Seminary of 1836, but these were rare. Examination and certification of teachers, started by the SSPCK (Society in Scotland for Propagating Christian Knowledge) Schools before 1795, was not put into general practice.

In the 1820's a new type of school started appearing. These included Liverpool Institute (1825), the London University, later renamed the London College School, and Kings College School in 1829. These schools were less expensive due to being funded by a committee and, because they had been aimed at providing education for the middle classes, they sought more utilitarian ends. Because of its importance in the world of trade, commerce and industry, Mathematics (and other sciences) was given a pride of place in the syllabus.

In 1837 a select Committee of Parliament was appointed to consider how 'useful education' might best be provided for the children of the poorer classes in England. As a by-product of this, and in imitation of efforts in Scotland, the Committee of the Privy Council on Education proposed in 1839 that an Inspectorate of schools should be set up. Despite the work of the earlier Committee, there were no further movements towards compulsory education, although attempts were made to broaden state-assisted education and to improve its quality. In fact there started a counter movement to that which advocated education for all. Andrew Bell (who founded Madras College in St Andrews where EFR received his school education) wrote that the utopian scheme for universal diffusion of general knowledge would confound the distinctions between ranks and classes of society on which the general welfare hinges. This movement only had a limited effect and in 1859 a Public Commission was established to inquire into the state of popular education in England and report what measures, if any, are required for the extension of sound and cheap elementary instruction to all classes of people.

After this stage the changes and improvements to education, and the place that Mathematics held in the common curriculum, became much more rapid with several Government backed experiments, such as the Fife Mathematics Project of the 1960's and others, leading the way to better methods and syllabuses.

**Article by:***J J O'Connor*and*E F Robertson*based on a University of St Andrews honours project by Elizabeth Watson submitted May 2000.