Text only works by having mobile data on

My wife sent me texts while I was out.  I have mobile data turned off.  I did not get the texts until I got home and she texted again.  I don't want text to eat up the mobile data. There was no graphics or data other than text.  It didn't show as being an error and she assumed it went through.  I shouldn't need to have data on to get simple texts.  I don't think I should need unlimited data plan just to get reliable texts.