Children have two basic goals in any day. The first goal is to save face and avoid humiliation. The second goal they would like to achieve before falling asleep at night is to feel successful about something that happened during their day. By the way, we as adults feel very much the same for ourselves: we would like to get through a day without embarrassment, and be able to feel that we were competent and successful about something we did in our day! For many children, these goals can feel elusive. It becomes incumbent upon us as parents and caring adults to support our children in this dual endeavor. One way to do this is to avoid correcting your children in public, or in a public manner. Children will remember the humiliation but not the lesson. Also, it is critical for us to remember that children will feel embarrassment in situations we may not feel they might. An example of this is a pre-teen going to school with a pimple on his or her nose. That child will "be sure" that everyone is looking.
To support the second goal, make sure to recount to your children something they did during the day that you noticed as having been done well. This could be a kindness you noticed, patience they demonstrated, politeness when they were not feeling particularly polite, a frustrating project that you saw them work on without quitting. Children are not great historians of their own successes. We become those historians for them.