Changing E and B

In this conversation we will attempt to go into the depths of students' understanding of electromagnetism and try to unlock, or at least loosen, a common mindset of Science and Engineering students. We have already had a conversation on electric, E, and magnetic, B, fields. Taking this to the next stage, a common sentence found in many Physics and Mathematics textbooks is  "a changing electric field creates a changing magnetic field and vice versa". Is this correct? Does the changing electric field itself actually create a changing magnetic field (and vice versa) ?