You and your kids are learning a much larger history than I did as a child. "History" was once considered to consist exclusively of written records. The idea was that we could only know what happened in the past from the observations of someone who saw the past happen, and the only way to know that was if the observer wrote down what he knew. Thus, history was considered to start with Egypt, which until a few decades ago was thought to have the first writing. It's now known that Mesopotamia and Elam had writing even before Egypt, but either way, writing was only a little over 5,000 years old, and thus history started a little over 5,000 years ago. Our species, Homo sapiens sapiens, was known to be somewhere between 50 and 300 thousand years old, and genus Homo millions of years old, but the vast majority of that time was considered to be outside of history, consigned to the realm of archaeology and paleontology.
That sharp line between history and archaeology, or prehistory, has dissolved today. Archaeology made such huge strides in the 20th century, with more accurate dating and exponentially increasing collections of systematically catalogued artifacts, that it became silly to regard its conclusions about the past as less reliable than written records. Indeed, laboratory archaeology began to conclusively refute some written records; for example, carbon dating of the Shroud of Turin proved that it was not old enough to have been worn by Jesus Christ. As archaeology began to influence every part of our knowledge of history after writing was developed, it became hard to argue that it couldn't tell us about history before the development of writing.
Ignoring what archaeology, biology, and other disciplines tell us about life before writing also endangers our ability to understand history. For example, past historians explaining the origin of civilization have often taken as their starting point Thomas Hobbes and Jean-Jacques Rousseau, philosophers who explained civilization as replacing a "state of nature," where humans lived in a solitary manner with no society, no rules and no laws. Biology, archaeology, and paleontology now make it obvious that no human ever lived in a "state of nature"; we did not develop society and social rules and laws ourselves, but inherited them from our primate ancestors. Modern human societies are incredibly diverse and complex, but all are built on the same foundation laid by Ramapithecus millions of years ago.
So today, most teachers no longer pretend that history started with cuneiform writing. The fundamentals that shape the human species' story - fire, agriculture, the institution of marriage, the anti-incest taboo, social hierarchy, and many others - had all been laid long before 5,000 years ago. You have to understand them before you can understand what came afterward.