There are some notable persons who criticized formal logic in favor of informal logic for various reasons, like Schiller. So what is the school of thought or tradition that incorporates or adheres to the maxims asserting the much importance of informal logic or practical logic.
Brouwer was famous for thinking that the principles of reasoning should not be formalized because language is an inadequate tool for representing mathematical thought.
For this reason, he never formalized intuitionistic logic and it wasn't until his student Heyting formalized it that we got a formal system codifying the principles of intuitionistic logic.
You can find good discussion of this aspect of Brouwer's philosophy of mathematics, here. Brouwer's (somewhat obscure) presentation of these views can be read in his Cambridge Lectures on Intuitionism.