Although Plato did deliver his lectures there, "the metaphysical theories of the director [Plato] were not in any way 'official' and the formal instruction in the Academy was restricted to mathematics" (Chermiss 1980, p.18). In other words, the Academy was an association of highly educated people engaged in independent research and studies.
As the founding father of the Academy, Plato became its first Head (scholarch) and remained at this position for forty years. Plato's immediate successor as the Head of the Academy was his 40-year old nephew Speusippus (347-339 BC) who, after his uncle's death in 347BC, remained scholarch for the next eight years. According to his contemporaries, Speusippus was a proliferate writer who produced many works written in the form of treatise and dialogues. Unfortunately, we have only few of the texts attributed to him: the information on Speusippus ideas and doctrines is primarily retrieved from third party sources (Dilon 2003).
The scarce information available these days makes it clear that despite his familial connection with Plato, Speusippus could barely be named as the continuator of his uncle's major ideas. In particular, Speusippus rejected the famous Theory of Forms developed by Plato during his years at the Academy. Besides, Speusippus believed that the Good was secondary while Plato maintained the Good was ultimately primary. Also, Speusippus claimed it was not possible to have comprehensive knowledge of any thing without understanding of the properties that distinguish this thing from other things. These views differed substantially from the principles argued by Plato in his broad philosophic doctrine. And only in his works dedicated to ethics, Speusippus abided by the views similar to those of Plato: he further elaborated Plato's ideas of citizenship, justice and legislation as well as opposed the hedonistic theory of the value of life proposed by Aristippus of Cyrene, founder of the Cyrenaic school. Speusippus also studied the science of numbers and, contrary to Plato, rejected the theory of ideal numbers (the Platonic Forms of numbers) which further led him to rejection of the Platonic concept of ideas in general (Smith 2007, III, pp. 893-894).
In 339BC, Xenocrates of Chalcedon, a renowned philosopher and mathematician, became the next scholarch of the Academy. Xenocrates did abide by the views of Plato and further elaborated the ideas and theories of his famous predecessor often using mathematical instruments and elements in doing so. In particular, Xenocrates contributed significantly to the studies of demonology (Platonic theology). The scholar believed that duality and unity were gods which ruled the world and pervaded all things while the human soul was a self-moving number and that the demonical powers that also existed in the universe acted as an intermediate link or channel between the divine and the soul: "Xenocrates distinguishes three cosmological causes: the Forms as original patterns, the demiurge, and matter. Via allegories he combines the philosophical worldview with mythical religion" (Ricken 1991 pp. 119-120). Xenocrates' interpretations of Plato's principles (particularly his attempt to order philosophic principles into logic and mathematics) seriously supplemented the foundations of Platonist philosophy.
Polemon of Athens became the Head of the Academy in 314