The decline in education in Europe and Britain especially continued through into the Mediaeval years, especially during the 11th century and the Norman conquest of England. King Harold's concerns about the English nobles' views of education and even the clergy's diminishing ability in Latin prose made little impact on the situation. Ability in warfare and defending ones land was more important at that time and the noblemen spent their time learning the skills of the fighter, rather than the scholar. Since there were few others with the time and the money to study, the level of education and mathematical knowledge of the populace dropped still further. By this time schools were reduced to little or no arithmetic, it is doubtful whether few knew more than basic counting and finger reckoning.
During the 12th century the education in Scotland was beginning to improve thanks to efforts made by the Church, although the sciences were still neglected for the most part. The country was divided into 11 dioceses by the end of this century (although this number increased later) and schools attached to the new churches and cathedrals were set up much in the same manner as those south of the border. These schools aimed to teach people to read and write in order to aid their ability to understand the Scriptures. Many of these schools also started taking in the poor, rather than just the rich and those wanting to take up Orders.
In Europe the situation was little better than England, although major centres of learning still existed in places like Paris where the University was founded during the 12th century, and this is where British students of ability went to study. This practise continued for many years in Scotland especially after the War of Independence with England, but English students were restricted by Henry II in 1167 when he banned them from attending the University of Paris. Cambridge and Oxford were then the only possible alternatives and these two rapidly grew into universities worthy of admiration. Despite this John of Holywood, known better as Johannes de Sacrobosco, not only studied at the University in Paris but also lectured there helping to promote Fibonacci's work and in 1220 wrote on Astronomy and Geometry, indicating that some of the Quadrivium still survived in Europe.
Fibonacci's texts, Liber abbaci, Practica geometria and others, were written between 1202 and 1228. These helped the spread of the Hindu-Arabic number system, which Fibonacci had learnt while travelling in his youth, and prompted the spread of Algebra and the knowledge of Greek texts saved in Arabia but lost to Europe for centuries. One of the first of these to make a great impact was the translation of Euclid's Elements into Latin from Arabic, and many other texts were soon being studied.
By the 14th century the level of Mathematics required to graduate from such universities as Paris and Prague was considerable. For the Bachelor's Degree they were advised to read Sacrobosco's Tractatus de Sphaera. For the Masters Degree they were to be 'acquainted' with the first six books of Euclid, Optics, Hydrostatics, the Theory of the Lever and Astronomy (although exactly what level of knowledge 'acquainted' infers is unknown). A similar system was supposed to be in place in Oxford and Cambridge in the 13th century, although Roger Bacon's account of his time there questions this. Despite the fact that the three years of further study required for the Masters degree were solely dedicated to the Quadrivium, Bacon doubted that few if any had read more Geometry than Euclid's definitions and the first five propositions of Book 1. The list of courses in the 14th century seems more extensive and includes Algorism, Ptolemaic Astronomy, Perspective, Proportion, Measurement of Surfaces, as well as the ubiquitous finger reckoning that was still required by many when they first entered University. All of these advances were in spite of the fact that the English nobles were now feeling the pressures of the Hundred Years War against France, much in the same way as the Scottish had before them, and the antipathy towards book learning had set in.
While there were still no Universities in Scotland, the levels of education also rose during the 13th and 14th centuries. The Dominicans had more of a presence, whereas before this the majority of the monasteries had been Benedictine or Cistercian run. These latter two Orders had little interest in educating the population beyond obtaining further members of the clergy, while the Dominicans started actively opening schools and taking in the poor children of the area. With the increased amount of Mathematics being taught in universities, more and more of the clergy were becoming mathematically literate and were then passing this knowledge on to their students. However, their teaching style left much to be desired. A form of dialogue between the pupil and the master had developed. The pupil was required to learn the questions to ask of the master, who then answered them also by memory. This method led to rote learning rather than true understanding, so further study by an individual would have been hampered.
Any further knowledge that was required by an individual could be obtained from a scrivener, a private tutor of mathematics (mostly concerned to give tuition on keeping accounts and book-keeping), who would also do the accounts of others to supplement their own wages. This form of private tuition grew during the Renaissance until it reached its peak in the 17th century before waning in the 18th century with the increase in the number of schools and night classes.
By far the most advanced mathematics teaching during the Middle Ages was that done by the trade guilds. This was an apprenticeship of seven years to a master of a trade who then taught you all that he thought you should know. The lifestyle that these artisan and merchant classes led was not conducive to further study had they wanted to. The trades which specifically dealt with parts of Geometry and Arithmetic included builders and architects, merchants and traders, and the early forms of money lenders in the larger cities and towns.
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.