The dominance of a classical curriculum as the norm for a Western-style education was coming to end at the turn of the 20th century. In effect, the answer to your question would vary wildly based on a particular child's gender, race, social class, or disability status. For example, the son of an titled or wealthy British man living in a colony would likely still be working on a classical curriculum - Latin, possibly Greek, logic, rhetoric, maths, and some sciences - with a tutor or a British run boy's school set up for the sons of British men in country. The boys may have studied history but it would be more a hobby or an incidental occurrence while studying logic and rhetoric. Odds are good their fathers intended for them to return to England for secondary or tertiary school. If a father was more interested in a "modern" education, a boy might be studying literature, sciences, maths, possibly Latin, and history. Their sisters, though, would likely have a very specific education depending on her social class and her parents' future plans for her. There are exceptions, of course, but generally speaking the daughter of a white British man would be receiving an education intended to prepare her for society, and being a mother or wife. In both cases, a modern education or a wealthy girl's education - the study of history would be dependant on class and could range from learning a personal family history related to England and the colony, to a general history of both countries. That education would likely cast England as civilized and "normal", and the colony as in need of civilization and exotic.

Let's say, though, you're the son of a man native to the colony who has access to power - e.g., a job in government, a military leader, an office or clerical job for a large shipping company - your education would likely be similar to that given to the sons of British men in country. Your sister's education would follow a similar path as British daughters and the nature of studying history you'd experience would be tied up in class. There are, again, exceptions.

If we look at the education of children native to the colony outside the umbrella of power, a great deal would be tied up in who is providing that education. If the British presence in your community or village was religious in nature, your experiences of history might be related to the evolution of the Church of England. If you attended a secular, British-run school you would like be expected to memorize the names of all of the Kings and Queens of England (a very common recitation task in English-speaking schools around the world.) and would likely learn English using primers or textbooks that referenced British historical figures. If you were indigenous or aboriginal, you may find yourself attending a school intent on educating you out of your family history, culture, language, and norms - in that case, you'd like be taught a very British-centric history, in which England was cast as a helpful, nobel nation bringing you "civilization."

All of that said, so much about a child's life was determined by perceptions of their race, gender, class, their colony's history, and the nature of the British presence, it's difficult to make generalizations as a group. If you have a question about an individual country, I'm happy to dive back in!