# The teaching of mathematics in Ancient Greece.

Education varied greatly from state to state in Ancient Greece. Spartan youths were sent to militaristic institutions and were trained and taught to become highly moral soldiers. The Athenians had a much more private education at home. They were taught music and gymnastics from an early age so that they might attain the goal of graceful perfection both physically and mentally. Within the various forms of education endemic to Greece the study of Mathematics also differed, but what was taught had a very different structure to the present. Possibly the main difference is that Arithmetics and Geometry were considered to be separate subjects. Even within Arithmetic itself there were two forms, the first of which was taught to the middle and artisan classes and was very much a calculation based subject. This learning was specific to their occupation and was to be mirrored in the Middle Ages by the Trade Guilds. The second form, the science of numbers, was the preserve of a few of the upper classes who had the time and money for a more lengthy education.

Instruction for the upper class individuals started at home under the guidance of their parents or an educated slave. Most, if not all, of the upper classes learned the minimum which seems to have included Letters, Music, Gymnastics and only a modicum of Arithmetics or Geometry. At the age of 12 the boys were moved to a school where they then learnt Grammar and the basics of Logic and Rhetoric. By the end of this stage many did not go further, but those that chose to continue breached the preserve of the science of numbers. There were two routes to further learning for these initiates. One route was to employ a sophist, as was done in Rome at this time, but the second route was to attend one of the colleges or academies set up by people like Plato, Aristotle or Pythagoras.

Pythagoras's school was set up in 518 BC in Croton and it was here that much of the science of numbers and many advances in geometry were made and discussed. The science of numbers, which was essentially the consideration of such things as perfect, abundant and square numbers and their properties, became the belief that everything in the world and universe can in some manner be mathematically expressed. It is partly through this and Pythagoras's observations of a vibrating string that Music came to be considered as one of the Mathematical Sciences. Pythagorean's also believed that the human soul could rise towards the divine through philosophical thought as a manner of purification and so they practised a strict code of living. It is possibly due to this extreme view of the role of Mathematics in life, and the violent end to the society, that so many Educationalists have advised against prolonged consideration of mathematical ideas since they believed that it led to too great a level of abstraction and drew the mind away from the realities of the world they lived in.

Plato's Academy (an institution which lasted over 900 years until it was closed down by Emperor Justinian in 529AD as a 'pagan' establishment) was set up to educate the future politicians and statesmen of Athens. Plato's ideas of Mathematics in life and in education seem far less extreme than those touted by Pythagoras, as can be seen in Plato's Laws. Mathematics was then considered the basis from which to move into philosophical thought and as such Plato proposed that studying mathematics should occupy the student for the first ten years of his education. This he believed provided the finest training for the mind since they were then able to understand relations that cannot be demonstrated physically. Since clear logical thinking was prized not only in philosophical discussions but also in the political arena, Plato encouraged his students to train in mathematics because he thought that it encouraged the most precise and definite kind of thinking of which humans are capable. Plato's

Aristotle's Lyceum had a much broader curriculum to the Academy and dealt more with the natural sciences. It is worth pointing out that this was much more extensive, although not as advanced, as what was taught centuries later in the British Universities. The manner of instruction in the Lyceum was the same as that in the Academy and also the Pythagorean School years before. Groups of students would gather around and ask questions of a more learned master who would, in turn, attempt to answer them and then a discussion would commence on the subject. This casual conversational style of instruction has not really been developed and the contrast with teaching methods of Europe in later centuries is marked.

Instruction for the upper class individuals started at home under the guidance of their parents or an educated slave. Most, if not all, of the upper classes learned the minimum which seems to have included Letters, Music, Gymnastics and only a modicum of Arithmetics or Geometry. At the age of 12 the boys were moved to a school where they then learnt Grammar and the basics of Logic and Rhetoric. By the end of this stage many did not go further, but those that chose to continue breached the preserve of the science of numbers. There were two routes to further learning for these initiates. One route was to employ a sophist, as was done in Rome at this time, but the second route was to attend one of the colleges or academies set up by people like Plato, Aristotle or Pythagoras.

Pythagoras's school was set up in 518 BC in Croton and it was here that much of the science of numbers and many advances in geometry were made and discussed. The science of numbers, which was essentially the consideration of such things as perfect, abundant and square numbers and their properties, became the belief that everything in the world and universe can in some manner be mathematically expressed. It is partly through this and Pythagoras's observations of a vibrating string that Music came to be considered as one of the Mathematical Sciences. Pythagorean's also believed that the human soul could rise towards the divine through philosophical thought as a manner of purification and so they practised a strict code of living. It is possibly due to this extreme view of the role of Mathematics in life, and the violent end to the society, that so many Educationalists have advised against prolonged consideration of mathematical ideas since they believed that it led to too great a level of abstraction and drew the mind away from the realities of the world they lived in.

Plato's Academy (an institution which lasted over 900 years until it was closed down by Emperor Justinian in 529AD as a 'pagan' establishment) was set up to educate the future politicians and statesmen of Athens. Plato's ideas of Mathematics in life and in education seem far less extreme than those touted by Pythagoras, as can be seen in Plato's Laws. Mathematics was then considered the basis from which to move into philosophical thought and as such Plato proposed that studying mathematics should occupy the student for the first ten years of his education. This he believed provided the finest training for the mind since they were then able to understand relations that cannot be demonstrated physically. Since clear logical thinking was prized not only in philosophical discussions but also in the political arena, Plato encouraged his students to train in mathematics because he thought that it encouraged the most precise and definite kind of thinking of which humans are capable. Plato's

*Republic*gives a different level of mathematical learning to the one just described. A learning reduced to the elementary which was possibly inspired by the public pressure from the Romans who had a very different opinion of the worth of Mathematics in Education.Aristotle's Lyceum had a much broader curriculum to the Academy and dealt more with the natural sciences. It is worth pointing out that this was much more extensive, although not as advanced, as what was taught centuries later in the British Universities. The manner of instruction in the Lyceum was the same as that in the Academy and also the Pythagorean School years before. Groups of students would gather around and ask questions of a more learned master who would, in turn, attempt to answer them and then a discussion would commence on the subject. This casual conversational style of instruction has not really been developed and the contrast with teaching methods of Europe in later centuries is marked.

**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.