If text is editable, how important is to show it (appearance wise) inside a text box?

A new approach we adopted for one of our applications is that all our editable text such as name, a description that was once filled out to be border-less. To edit the text the users simply click on top of the text again and it becomes editable.

The border shows in these instances.

• Border shows on roll over
• Border shows while editing the text

My question is, will this make sense to users? Should I keep this method? Is there anything else I can do to improve this?