Like many things in education, it's going to come down to the location, race, and gender of a particular child and when in the 1800s you're thinking. That is, the closer to the 1700s you are, the more likely European and White American boys, enrolled in formal education, would have been studying the classical curriculum, which focused on Greek, Latin, rhetoric, logic, and some math and some science in service to that education. Education for girls looked different based on class, location, and timing. This close look at the history of "computers" goes in-depth on their education.

The other variable that influenced the nature of the math and science a young person might encounter was their future plans. That is, if a young person was interested in pursuing (or had been told by his parents he was going to pursue) a primarily religious education, they likely only encountered arithmetic as a minor area of focus or related to their religious studies. If a young person was attending a common school in the later half of the 1800s, their math education was primarily arithmetic. You can get a good sense of what that would look like here. There were likely pockets of more complicated and sophisticated mathematics, but odds are likely a young person who was interested in such studies was working with a tutor or enrolled in college.

Bookkeeping was a fairly common approach to mathematics for children of all genders during the later half of the 1800s. Girls who continued onto High School in America often came across the course in their studies and in many cases, it was framed as a useful skill for a wife to posses. Likewise, boys who attended rural high schools would have likely come taken some form of bookkeeping math in their studies as it was seen as a useful skill for farmers and merchants alike.

Regarding sciences, the demarcation of science branches is fairly modern. In the 1800s, especially the first half, a teacher would be most likely to focus on the natural sciences. That is, the things a young person could observe with their eyes and see in their books. These included things like zoology, geology, and biology. Writing about things a child had seen in nature was a fairly common writing activity. However, by the late 1800s, when the National Education Association was taking stock of which content was most common in American high schools, there was wide disagreement across the country about content, order, significance, and necessity. Some educators pushed hard for physics to be taught to young children, others wanted zoology to be the predominate science. There was a fairly vocal group of educators who felt earth science (though it wasn't called that yet) and the ability to understand the weather and seasons was really the only science a young person needed.

There was, though, a fair amount of consensus that science needed to be laboratory-based. So, in effect, the closer you get to 1900, the more likely a young person's study of science in school would be based in what we think of as "hands-on" science.

Which is to say, there was no standardization of the math and science curriculum in the 19th century.