Carson, D.A., and Douglas Moo An Introduction to the New Testament - Second Edition. Grand Rapids: Zondervan, 2005. "Matthew" Carson & Moo pp. 134-168
The discussion of the outline of a book of the Bible has always seemed curious to me. Carson & Moo are no exception in this chapter. They review various means of outlining the book, seeing that some authors have identified geographic frameworks, some have looked at christological development, and some have looked at a series of multiple discourses. It strikes me that we do best to observe important features of a text but not worry too much about what framework the author had in mind if it is not patently obvious.
Authorship of books of the Bible is a difficult issue. After all, many of the texts in Scripture have no internal ascription. Scholars of a more liberal bent will frequently suggest that false claims to authorship were rampant in antiquity so where we have a claim to authorship in the Bible we should assume it is false. It does appear that the Gospels had the names of the evangelists attached to them prior to A.D. 140 and that the identification of authors may well have been present from very early on. After all, we don't seem to have any question about which Gospel is attributed to which author, as might have happened if there were a lengthy period of doubt. There are suggestions that Matthew may have originally been written in Hebrew or Aramaic. However, this claim may have been misplaced, as it may also be a claim made about another writing from the apostolic age. The place of authorship is not known. It would seem from the text that it was written in a place where there was both a significant Jewish cultural foundation and a substantial number of Gentiles who would not be completely familiar with Jewish customs.
Many scholars consider that Matthew was influenced by Mark, dating Matthew at A.D. 80 or later. Yet a number of scholars are now dating all the synoptics prior to 70 and suggesting that Mark may not have had primacy. The early church fathers viewed Matthew as the first of the evangelists to write.
We should hesitate to assume that the gospels were written to incite conversion to Christ. They seem aimed at those who believe on Jesus and need ongoing encouragement in their faith. Matthew seems to be aimed at a predominantly Jewish audience. Yet he explains many of the Jewish customs, which may indicate that he intended the gospel to be useful to Gentiles as well.
Matthew does not state why he wrote the Gospel. There are many different themes in the text. So it's difficult to say what his purpose was. We do see among the primary themes a description of events in the life of Jesus, as well as a good deal of Old Testament justification for the actions of Jesus. Jesus is clearly presented as the Messiah promised by God, the one who is rejected by Jewish leaders, and the one who is bringing in his eschatalogical kingdom, reigning over and protecting all who trust him.
The text of Matthew is well attested. There are places where attempts to harmonize Matthew with other Gospels will be frustrating. Yet the text itself is stable and understandable.
ADOPTION INTO THE CANON
Matthew was universally accepted in antiquity except by Marcion, who rejected all things which seemed Jewish. The book's canonical status has never been seriously questioned.
MATTHEW IN RECENT STUDIES
Matthew's Gospel tended to be ingnored by English-language scholars until the latter part of the 20th Century. In recent years there have been more major commentaries. Text-critical studies have been fairly common until recently, as scholars wish to see how the text may have been pasted together from various other sources. But in recent years more commentators have been looking at the text as a cohesive whole.
THE CONTRIBUTION OF MATTHEW
Matthew contributes to the canon especially in his preservation of "large blocks of Jesus' teaching" (p. 163), by giving us the story of the incarnation from Joseph's point of view, by emphasizing Jesus' life and work as the fulfillment of Old Testament prophecies, by viewing Jesus as the one who fulfills the law of God, by looking forward to the relationship between Israel and the Church, and by emphasizing roles of Jesus such as "Christ, the Son of David, the Son of God, the Son of Man, the Servant of the Lord" (p. 164) etc.