In his book, The New Testament, Donald Hagner offers a nice summary of the various proposals for the genre of the Gospel of Matthew.
“A number of suggestions, not necessarily mutually exclusive, have been put forward. Matthew may be considered as a kind of midrash—theological interpretation—on the Gospel of Mark. Some have seen Matthew as a kind of lectionary, its composition determined by the lectionary year, so as to provide consecutive liturgical readings in accord with the Jewish festal year. Others have regarded Matthew as a kind of catechetical handbook because of its lengthy teaching discourses. In this view, one of Matthew’s main purposes is the building up of Christian discipleship through the instruction of its readers. Krister Stendahl went so far as to propose the hypothesis of a Matthean school, modeled on the rabbinic schools, that produced a teacher’s manual, a Christian equivalent to the Qumran community’s Manual of Discipline. Another suggestion is that Matthew is a book of church correctives, addressing various issues such as division in the community, spiritual malaise, disregard for the law, hypocrisy, and false prophets. Given the burden of this Gospel to demonstrate that Jesus is the Messiah, and especially Matthew’s fulfillment quotations, some have described it as basically missionary propaganda. Finally, it has been suggested that Matthew basically presents polemic against the rabbis concerning the true interpretation of Torah. In this view, Matthew’s Gospel draws on the traditions about Jesus and the Pharisees and then uses this material in the struggle with the Pharisaic Judaism of a later time. Matthew is thus a Jewish Christian counterpart of, and response to, the rabbinic at Yavneh.” (211-12)