A teaching method comprises the principles and methods used by teachers to enable student learning. These strategies are determined partly on subject matter to be taught and partly by the nature of the learner. For a particular teaching method to be appropriate and efficient it has to be in relation with the characteristic of the learner and the type of learning it is supposed to bring about.
Teachers are highly respected in the Islamic tradition.
The Qur'an does not talk about teachers too much, perhaps because part of the reform that Islam was bringing was an educational reform, and so, during the time of the Prophet (S), the society was transitioning to one with more focus on learning. However, one can see all the prophets discussed in the Qur'an as teachers.
Furthermore, the Prophet (S) emphasised teaching and learning, for instance, when he freed some literate prisoners of war, rather than requesting money in exchange for returning them, he requested that they teach others to read first. Islam has been referred to as 'the world's largest literacy program' and I think that is an apt description.
Additionally, there are many mentions in hadith of the importance of teaching, necessity and value of learning, and honour of the teacher, such as:
* It is narrated that the Prophet (S) said, ‘The best form of charity is for a man to gain knowledge and then teach it to his fellow brethren.'
* It is narrated that Imam Ali (A) said, ‘Everything decreases with giving away except knowledge.’
* It is narrated that Imam al-Baqir (A) said, ‘For the teacher of good, all the animals on the land and the fish in the sea seek forgiveness on his behalf, as do all creatures great and small in Allah’s earth and sky.’
* It is narrated that Imam al-Sadiq (A) said, 'Everything has a zakat, and the zakat of knowledge is to teach it to those who are worthy of it.’
If you have an interest in this subject, I would recommend the following book:
Desire of the Aspirant: On the Etiquette of the Teacher and the Student
by al-Shahid al-Thani [a classical Muslim scholar of the Shi'i persuasion]
translated by Alexander Khaleeli
Yes there are many teaching methodologies is Quran. Just example : the conversation between Al-Khidr and Moses which shows the methods of teaching.
Also the method of teaching in the life of Luqman and his son.