I'm not sure I understand your question. I'll answer anyway. I am a native of the U.S.
In most places in the U.S. children are first taught to write "manuscript." Manuscript consists of separate letters. It looks like the sans serif type used here on italki. We are taught to write both upper-case and lower-case letters.
At about age eight or nine, the schools then teach "cursive." Cursive letters are connected. Cursive is often called simply "handwriting." Learning to write well is called "penmanship."
Nowadays, most people do most of their writing on a keyboard. As a result, many of us (including me) have lost our skill at penmanship. My cursive writing is almost illegible. I only use it for my signature. When I am forced to write with a pencil or pen, if it is for my own use I write separate letters, "manuscript," upper- and lower-case.
Even this is hard for other people to read. though. When it is important for other people to read my writing, for example when I am addressing a letter, I use "block letters" (separated letters, uppercase only).
There is starting to be controversy about the teaching of cursive. Some school systems in the United States think it is a waste of time. They are no longer teaching it. Others continue to think it is important and continue to teach it.