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Looking at all aspects of Maths teaching begs the question: how was maths taught in the time of Plato and beyond?

## Maths Background in Ancient Greece

Education in general varied widely from state to state. Spartans placed great emphasis on morality while Athenians focused on graceful perfection of the physical and mental forms.

The study of Mathematics was taught very differently than today, mainly because **Geometry** and **Arithmetic** were separate subjects. The concept of Arithmetic was further broken down, into **calculations**, which was taught to **artisans** and others of middle class status.

Those who had time, money and inclination could pursue further studies into the **science of numbers**.

Sons started their education at home, under the guidance of a parent or a maths tutor. Early education included Letters, Music and Gymnastics, but little to no Arithmetic or Geometry.

Around age twelve, students enrolled in maths lessons, where they furthered their learning in Grammar, **Logic** and Rhetoric.

Those from working-class families generally ended their formal schooling at that point, but those students from high society would matriculate at colleges or academies set up by Plato, Pythagoras or Aristotle.

**Pythagoras** founded a school in 514 BCE. We owe much of our knowledge of geometry to studies made at that institution.

From the consideration of perfect, abundant and square numbers and their properties came the idea that everything in the universe can be expressed mathematically.

Thanks in part to those studies, music came to be considered a **mathematical science**.

**Plato**'s Academy lasted over nine hundred years. Meant to develop future statesmen, Plato's views on Mathematics were far more limited than Pythagoras'. He acknowledged that **training in Maths** led to clear, logical thinking, an attribute prized by politicians of the day.

He proposed that a student should study maths only during the first ten years of his education in order to best transition into the type of rational, philosophical thinking needed to succeed in politics.

His Academy was deemed a 'pagan' establishment and was shut down by Emperor Justinian in 529 CE.

Further evidence of Roman control over Plato's teachings is evident in **The Republic**, wherein he avers that only **elementary learning** of Mathematics is necessary.

## Maths Teaching in Ancient Rome

As did the Greeks, so did the Romans: early education at home, only for boys. Students engaged in formal maths learning around age twelve and studied Letters, Music and elementary Arithmetic, using their fingers and **abaci**.

Unless called for by social status or profession, scholars in Rome did not engage in further maths studies outside of what they **learned at home**.

Those who could apply themselves further attended lessons given by a **Mathematics Master**. Instances of such were rare, the exception rather than the rule of Roman scholarship.

Surviving documents from the period seem to indicate that Maths teaching was not wholly welcome or accepted, but seen as necessary in some instances.

- Vetruvius suggested students of a
**rchitecture**study Optics, Astronomy, Law, Geometry and Arithmetic - Galen recommended future
**physicians**study Medicine, Rhetoric, Music, Dialectics, Geometry and Arithmetic - Varro and Seneca also recommended the study of Geometry and Arithmetic, if only to improve or deepen
**logical thinking**abilities.

In Ancient Rome, professions requiring Logic, Rhetoric or Oratio were far more esteemed than those dealing with numbers and sciences. In fact, high society looked down on 'illiberals', those who had careers requiring **extensive maths** (or science) knowledge.

### Because of this prevailing attitude, the teaching of maths stalled for centuries.

Cassiodorus, a 6^{th} century Roman statesman passionate about education, promoted the need for teaching to the Roman Catholic church. He categorized learning into two broad categories: **Trivium,** which dealt in speaking and languages, and **Quadrivium**: the study of Arithmetic, Geometry, Music and Astronomy.

That same division exists in today's educational parameters, only with different names: **Humanities** and **Sciences**, with the exception that modern day educational philosophy treats music as an aspect of human culture rather than a science.

By the 9^{th} century, most** monasteries** included schools, where new members of the priesthood received education. While the Trivium was wholly embraced, to learn Maths and sciences were only taught at the highest **centres of teaching. **

The school attached to the great **Cathedral of York** was one of the few institutions in England which taught Mathematics.

The Church became the **driving force** of education in Europe. Maths instruction was unfortunately minimal.

However, with the resurgence of war, focus shifted once again away from teaching altogether, until Gerbert D'Aurillac discovered mathematical texts written by Boethius.

By the time Gerbert became Pope in 999 CE, he had written his own copy of Boethius' documents and proclaimed it **essential teaching** within the Church.

Maths teaching was in vogue, and remained a **core requirement **in schools for five centuries.

During this time, across the world, great advances were made in the study of numbers and relations: interest computation, algebra, differential calculus and factorization, among others. New advances were added to maths curriculum almost as soon as they were proven.

In Britain, a church movement separate from Roman Catholicism was gaining power. That body wished to rid itself of any association with 'pagan' ways and ideals. That meant that any Mathematics studies fell out of favor in this country.

## The History of Maths in the Renaissance Period

The Renaissance brought forth new ways of thinking about education. More than simply learning what one must to do their job, emphasis was placed on students learning **life skills**.

During this period, Euclid's *Elements* was translated into many languages and widely disseminated, thanks to the newly-invented** printing press**. That meant that maths scholars in Germany, France and Italy were using the same materials to study, with or without teachers.

### The works of more modern mathematicians were being printed too.

In England, Robert Recorde wrote one of the first series of **maths textbooks**, designed to give ordinary citizens a better understanding of addition, subtraction and equations.

It was Recorde who gave us the '=' symbol. In formulating his texts, he reasoned there was nothing more equal than two parallel lines of the same length.

Mid-sixteeth century, Ramus proposed a return to the original seven liberal arts studies, but organized them by three 'laws':

- only true and necessary facts may be included
- only facts belonging to the subject in question may be included
- general things must be dealt with generally; particular things particularly

He then proceeded to write entire **textbooks** on each subject.

Remus and Recorde's textbooks awakened an interest in Mathematics, not just in scholars and well-to-dos but among the middle class. As society's hunger for maths knowledge grew, private Maths Tutors were soon earning a comfortable living.

### It took a bit longer to coordinate advanced maths teaching in schools.

Upon graduation from Cambridge, John Dee proclaimed the need for improving maths in Education. Not that there was much practical or commercial use for maths at that point, but because of its ability to 'lift the heart to the heavens!' - reflecting Pythagoras' declaration of centuries before.

In spite of assertions by Dee and others, universities maintained the **Medieval Curriculum** – teaching math, geometry and astronomy only minimally; focusing on the Trivium.

## The History of Mathematics in the Nineteenth Century

Much focus was given to when and how a pupil should be taught maths:

Children should start learning basic Arithmetic as soon as they enter school, and their learning should be based on perception of physical objects.

In spite of progressive thinking, many of Britain's schools were still very conservative. The foundation of the Analytical Society of Cambridge served to persuade that institution to embrace broader Maths studies. By 1823, **differential calculus** had made its way into the curriculum.

Soon, a new type of school opened, one where maths and other sciences were given pride of place.

The **University of London** would provide an education in Mathematics and Physical Sciences. The first chair of Mathematics was given to __Augustus De Morgan__, who revolutionized the way maths was taught.

After lecture, he would hand out **test papers** based on that day's lesson. The students were to return the completed exams, from which he would discern areas of understanding and where more explanation was needed. The next day's lecture incorporated what had not been grasped the previous day.

## Learning Maths Today

Although great advances have been made in the field of Mathematics, the **art of teaching** maths at primary school or GCSE maths revision, remains essentially unchanged since De Morgan's time.

Even private tutoring of maths harkens back to ancient times, except that you can teach maths online.

Government and organizations have invested tremendous resources in developing teachers and teaching methods. Expanded common core maths curriculum includes the study of:

- Mathematical concepts
- Algebra
- Geometry
- Calculus
- Trigonometry
- The study of equations of all types: quadratic, linear, differential, parametric – among others.

The fact remains that it has always been **teachers** who, through their very love of the subject material, ignite a desire to understand and explore the exciting arena of maths.

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