Garrow, Alan J.P. "Chapter Five: The Peri Layer: 6.1-3; 7.1a, c, e, 4a; 9.1-5a; 11.3a, 4-6; 16.1-6, 8-9." The Gospel of Matthew's Dependence on the Didache. New York: T&T Clark International, 2004, 93-106.
Gorrow theorizes a redactional layer in the Didache which features frequent use of περὶ δὲ, hence called the "peri" layer. This layer, he thinks, sheds light on the eucharistic prayers of chapters 9-10 (Garrow 2004, 93).
There is a striking parallel between Didache 7.1a and 9.1. Both passages introduce a concept with περὶ δὲ followed by a genitive of a conceptual noun. Garrow thinks, however, that due to the phrasing of baptismal formulas (7.1, no article, 7.3, article, otherwise "the name of the Lord" rather than a trinitarian statement), he can discern different layers of redaction (Garrow 2004, 94). Scholars do disagree whetehr the various statements show differences in time of composition, as well as possibly indicating a move from allowing anyone to baptize to having specifically designated ministers (Garrow 2004, 94-95). Garrow takes the statements to be an insertion, made clumsily, as evidenced by the change from a plural to a singular address and the lack of instructions about the length of a fast (Garrow 2004, 95).
Garrow considers the peri layer not to have begun with 7.1, but that the statement in 7.1 indicates a change of topic. This suggests to Garrow that there is a link reaching back into the Two Ways materials. For this, he goes to 6.1-3 (Garrow 2004, 97). He then argues that it would be unusual to use two different types of teaching together (Jewish ethics and baptismal instructions) without inserting material that could suggest a relationship (Garrow 2004, 99). To him, this is the redactional layer which justifies the importance of baptism. Garrow's analysis gradually expands material he fits into the peri layer, as he includes anything which could be understood as coordinating two unconnected ideas (Garrow 2004, 100).
Based on this conclusion, Garrow moves to 11.3a, 4-6, dealing with apostles (Garrow 2004, 101). Again, he finds some statements which could seem disjointed, particularly in 11.3-6. Verses 7-12 are only loosely arranged, but do not seem as startling as the earlier passage.
Garrow effectively takes the instructions about apostles and prophets to have been awkward and require sommothing out and also to have been smooth, thus becoming more awkward when material is inserted. The community understanding was not rightly expressed in terms of apostles and prophets, so the text was changed to partially reflect a change of attitude (Garrow 2004, 102-103). In my opinion, this entire line of reasoning is needlessly complicated.
Garrow finally ties the peri layer to chapter 16, as he observes a similarity between 6.1-3 and 16.1-2, which urge diligence in pursuit of godliness (Garrow 2004, 105). He does foreshadow subsequent chapters in his book in which he will delve into the various ideas in more depth.