Kloppenborg, John S. "Chapter Six: The Use of the Synoptics or Q in Did. 1:3b-2:1." in Van de Sandt, Huub (editor). Matthew and the Didache: Two Documents from the Same Jewish-Christian Milieu? Minneapolis: Fortress Press, 2005, 105-129.
Kloppenborg observes that recent scholarship has moved from considering the Didache dependent on canonical texts to its independence (Kloppenborg 2005, 106). Upon comparison with various other early Christian church orders, Kloppenborg considers Didache 1:3b-2:1 to be of great importance, as numerous texts present the greater context of the Two Ways material, but only some include 1:3b-2:1 (Kloppenborg 2005, 107). The discussion of dependence and its direction is complicated, and has led various scholars to even contradictory conclusions (Kloppenborg 2005, 108-109).
At issue in interpretation is not the fact that there are differences between the Didache and the Synoptic texts, but evaluation of the significance of those differences (Kloppenborg 2005, 110). In Kloppenborg's view, this discussion is made more challenging if the Synoptic texts were fluid in nature for some time before arriving at a final form. Not only the fact that a text could change, but also the fact that the Didache tends to make allusions rather than quotations causes difficulties (Kloppenborg 2005, 111). It is very difficult to identify the definitive source of an allusion. Kloppenborg suggests the most fruitful way of evaluating the dependence would be to identify areas where the Didache specifically uses material which must by its nature have been revised by a redactor (Kloppenborg 2005, 112).
Didache 1:3b-2:1 may be of importance as we consider the relationship between the Didache and Matthew's Gospel (Kloppenborg 2005, 113). In this passage, Kloppenborg considers the material related to Matthew to be heavily reworked. The material, compared with the statement of Jesus included in Matthew, has been expanded. This has suggested to some, including Koester, to assign it to the work of a different redactor than other parts of the Didache (Kloppenborg 2005, 114). While Kloppenborg recognizes that the critical edition of a reconstruction of Q is a hypothetical document, he also observes that it was prepared without reference to the Didache (Kloppenborg 2005, 115). The passage, however, does line up with material in the Q edition. The admonition to pray for enemies occurs both in Q and in Didache 1:3b. Kloppenborg finds developments in the Didache when comparing it to the material he sees as a common source for Luke and Q (Kloppenborg 2005, 120). However, in his subsequent discussion he doesn't find entirely conclusive evidence for the way dependence or wording may be confirmed. The purpose of prayer for enemies in Didache 1:3b is to win them over. This differs from many of the New Testament passages, where there may be a motive of gaining God's favor (Kloppenborg 2005, 123). The clearest evidence Kloppenborg finds of a reference specifically to Matthew rather than Q or Luke is the statement of the "right" cheek in Didache 1:6. Here, though, many commentators take Matthew to be dependent on the Didache, which may further complicate the situation (Kloppenborg 2005, 124). The remaining possible piece of evidence is a command to give freely, even in the case of a request for a theft of garments (Kloppenborg 2005, 126). Here the Didache and Luke are more similar, while Matthew deviates.
Kloppenborg concludes that there is no concrete evidence for the Didache's dependence on a specific source, whether Matthew, Luke, or Q. It has significant differences from other works (Kloppenborg 2005, 129). The compiler may well have known Luke, and possibly knew Matthew and Q, but we cannot be entirely certain.