One of the mainstays of modern evolutionary biology is that evolution is highly sensitive to initial conditions and that, according to the Butterfly Effect from Chaos Theory, it is impossible to predict evolution in any way. But Chaos Theory is more than the Butterfly Effect, and there are aspects of chaos that - in complete contradiction to the Butterfly Effect - are insensitive to initial conditions. Also known as Universality within Chaos Theory, there is a phenomenon known as a Period-Doubling Cascade, or Feigenbaum Cascade, which is ubiquitous in iterative non-linear dynamic processes with limited resources. And iterative non-linear dynamic processes with limited resources are not exactly uncommon in the universe. Evolution is just such a process.This study examines the spontaneous creation of innovations in Intentional Teaching among animals and early humans. Such innovations occurred during Cultural Evolution, which followed Physical and Biological Evolution. Each of the teaching innovations represents nothing less than a new form of evolution. If one accepts that transmission of cultural information is as legitimate a form of transmission of acquired information as transmission of DNA during biological reproduction, then the innovations in teaching methods are potentially highly significant. There are seven new teaching methods identified by cognitive archaeologists. Each of them transmits a new kind of information that is on a higher cognitive level than previous methods. The co-evolution of tool technology, teaching, and biology, led eventually to spoken language.Examination of the time intervals between these seven evolutionary stages shows that they are compatible with a Feigenbaum Cascade from Chaos Theory. Physical and Biological Evolution also converge to the time pattern of the Cascade, as do the development of information technology in the form of written language, the printing machine, and the computer. This forms the basis of a theory of Big Evolution, which - like Big History - covers the whole of time.