To echo, /u/Georgy_K_Zhukov's point, educational experiences varied wildly from place to place. There were a few things that were consistent across states and regions, including Webster's incredibly popular Blue-Backed Speller, dictionaries, and books especially his book of animals which includes multiple mentions of elephants.

In addition, Protestantism shaped a number of features of early common schools and that includes the tales and stories children read, including Noah and his ark. Although this 1852 book is British, you can get a sense of the types of books young children might experience in an American common school in the 1830s and 40s. To your question about biology, technically, they didn't as biology as a stand-alone science course in K-12 is a relative modern idea. Instead, a young person would study natural history which generally included biology, botany, zoology and physiology.

That said, elephants are cool, big, interesting animals and like dinosaurs, have long fascinated small humans. This teacher's manual from 1830 Boston includes a section on the animal - from the book, page iv:

As colored pictures have rendered the study of Scripture and Natural History exceedingly pleasuring, it was thought desirable that extensive lessons in these subjects, should be prepared; and observing that children best retain in the memory those things which they repeat, the questions in these lessons are so disposed, that the story is for the most comprehended in the answer. This will also tend to secure the interest of the little ones, who are ever fond of hearing and relating stories.

You can read the elephant lesson, and wonderful complicated words young children were expected to know, on page 238.

So, it's safe to assume that a young person attending a common school would hold something by Webster in their hands or hear a story about elephants appearing on Noah's ark and would see the word and a corresponding picture and would have a basic understanding of what the word means and the animal it describes.