Education will be essential to the continued viability of any permanent settlement. Even if there is a constant influx of educated colonists from Earth, the long term health of the colony will depend on a well educated population. Skilled workers are needed for equipment maintenance and engineering. Continuous education will be an important part of settler's life. Example of this is the continuous education of military personnel on ships during long postings, or the modern emphasis on continuous education in professions.
Earth based lecture may be possible, but face an important limitation from communication latency. The round trip delay of 8 to 30+ minutes will make interactions practically impossible. Pre recorded lectures will be available as with modern university services.
Children's education will require personnel, facilities such as kindergartens and schools and eventually universities.
Although the immigrant population is likely to be highly motivated, educated and intelligent, their children will tend to average out. The settlement will need to provide high quality education to maintain its 'edge' and give the young members of the settlement good opportunities on Mars.
If transportation is sufficiently inexpensive, Earth universities may develop subsidiaries on Mars. A martian doctorate might become a significant academic advantage, and the nature of the martian environment will create academic opportunities for research and discovery.