Irish education was relatively informal until quite recently. Until the last quarter of the nineteenth century, a large majority of Irish teachers had received no training.
The Society for Promoting the Education of the Poor of Ireland, better known as the Kildare Place Society, was the first attempt to provide systematic non-denominational primary education. Founded in 1811, it trained several thousand teachers and supported schools throughout the country. Its personnel records from 1814 to 1854 are now held by the Church of Ireland College of Education, Upper Rathmines Road, Dublin 6, www.dcu.ie/church-of-ireland-centre/
Appendix 22 of the Irish Education Enquiry, 1826, 2nd Report. (4 Vols) lists all parochial schools in Ireland in 1824, including names of teachers and other details. It is published on CD-ROM by Eneclann (www.irishfamilyhistorycentre.com) and indexed in Schoolmasters and schoolmistresses in Ireland, 1826-1827, (1982), by Dorothy Rines Dingfelder, (NLI Ir 372 d 38).
The Enquiry itself was set up because of the objections of the Roman Catholic hierarchy to the non-denominational nature of the Kildare Place schools. Its outcome was the establishment of the Board of National Education in 1831, which ended state support for the Kildare Place schools and eventually placed control of elementary education in the hands of the local clergy in the form of National Schools, a system that is still in place.
The principal source for teachers in the National Schools is the series of Teachers Salary Books from 1834 to 1855 held by the National Archives. These are not particularly informative from a genealogical point of view, but sometimes include comments that can be of interest. They are organised by school, so it is necessary to know where your teacher was working.