How does American Headway deliver lessons that really do work in class?

Short answer: methodology.

Headway's balanced methodology combines the best of traditional approaches - solid grammar and practice, vocabulary development etc - and newer approaches, such as communicative role-play and personalization.

Authors Liz and John Soars have developed a set of principles which "... is a summary of all we believe when writing and teaching."

Here are three examples:
i) Grammar has a core place in language teaching and learning.
ii) A wide variety of practice tasks in all the four skills are essential to language learning.
iii) Everyday expressions, particularly of spoken English, also need a place in the syllabus. These can be functional, social, situational, or idiomatic.