I would define a “formal manner” as it pertains to teaching as a more rigid strict approach that relies heavily on a set of conventional rules. The rules clearly define what is right and wrong and may not give much flexibility. A formal manner may also use a more traditional style and in terms of language I would associate a formal manner more with structural methods of teaching language.
I think there is certainly a place for teaching grammar in a formal style. Correct grammar is based on a fairly strict set of rules to define what is the proper way to construct sentences. Since the rules can be defined clearly (although perhaps not simply) then it makes sense to teach them in a structured formal manner. There isn’t much room to change or bend rules if a person is to learn to speak proper English. For this, a formal structured approach will be beneficial to students seeking to know the meaning behind why we speak the way we do. It will also help them to make a decision on speaking correctly when they aren’t certain about a particular phrase or sentence.
But I think teaching grammar solely in a formal manner may not adequately prepare a student to function in an English speaking environment. There are many more aspects of the English language that are possibly learned better in a less formal manner. The communicative approach can be a better way to practice using language correctly. Understanding the sound and flow of correct grammar is a crucial part of learning the language. Some scientific studies have shown the human brain has a natural inclination to learning languages without the need for grammatical explanation.
This natural inclination to absorbing grammar rules is not a result of formal teaching but a result of hearing, listening, reading, and speaking the language. Over time the brain/person learns the rules informally of the language. They may not understand what present perfect progressive means but they know how and when a sentence should use present perfect progressive verbs.
This is not to say that there is anything wrong with formal teaching of grammar in any way. But a more communicative approach will certainly be beneficial to the students absorption of the rules of grammar. Over time as the student can be surrounded by the language through a variety of forms. The student can listen to audio CD’s of people speaking English. They can engage in conversations with the teacher and other students. They can read and write papers. This constant absorption of materials in English will start to train the student to recognize proper grammar. Over time the student will get a sense for if a sentence or spoken language is grammatically correct because it won’t “sound right”.
In conclusion, I think the best approach to teaching grammar is a combined approach that focuses on formal training as well as a more casual communicative approach so as to surround the student with the language so they can absorb it and also learn the formal rules behind what they are hearing, speaking, reading and writing.