Clearing search will not let the user enter a new search directly
When the user clicks the x that clears the search-field in order to make a new search, the cursor is not placed in the search field. The user has to click in the search field in order to write something. The way this works it is quicker to just select the text in the search field and erase it by hitting the Back space.
Sofia Birden commented
While I do agree that it would be more convenient if the curser was already in the box, after hitting the x, I have to disagree that it is quicker to select the text and delete. It is much faster to hit the x then put the cursor in the box. Having the cursor auto show in the box is better since a lot of our students start typing their search as soon as they hit the x only to find out that nothing typed because of the cursor.