The ten days of waiting. Jesus ascends into heaven (verse 9) after telling his followers to wait in Jerusalem for the power of the Holy Spirit. They are told when the Holy Spirit comes they will go on to be his witnesses in Jerusalem, Judea, Samaria and to the ends of the earth. After Jesus is taken from them they wait together for 10 days. I believe there was some time of prayer and I believe there was much conversation. I can only imagine, apart from selecting Judas' replacement, what they talked about. The disciples, the women, Jesus mother and brothers are all together in an Jerusalem upper room. It is speculated by many that it is the same room that was used for the last Supper.
Paintings often give us wrong pictures of what actually happened. Leonardo Da Vinci's famous painting of the last Supper, is an image of the last supper if Jesus lived in medieval Italy. The actual meal and the layout of the room would have been quite different (the table would have been coffee table height, there would have been no chairs but pillows to recline on, etc.). There seems to be a hang up in some denominations that only men are allowed to lead. They sometimes paint the picture of Jesus and the twelve apostles (all men) as the entire group of early Christ followers. Here in Acts 1 we get a much different picture. There are 120 people present. Chapter 1 of Acts specifically mentions Jesus mother and specifically includes "certain women" and Jesus' brothers (see Matthew 13:55). It is a pretty big and mixed group of men and women and, at this point in the journey, all Jews. Ten days later, in Chapter 2 on the day of Pentecost this same group of Christ followers has gathered. The Holy Spirit falls on all of them, everything that is said about this group is being said of men and women gathered under Jesus order to wait.
I once heard a sermon about how the only decision the Christ followers make before Pentecost is a total failure. The person preaching this message noted that after chapter 1, Matthias is never mentioned again. The speculation was that this decision, before the coming of the Holy Spirit, was not Spirit led and therefore did not survive or bear fruit. It is a lovely argument made on the dangerous platform of silence. For the record, Andrew, Thomas, Philip, Batholomew, James Son of Alpheaus and Simon the Zealot are never mentioned by name in the rest of the New Testament. Their lives and martyrdom are recorded in extra biblical sources of varying degrees of authenticity. One ought not to make speculative statements on what is not said. Jesus didn't say anything about a lot of stuff . . . do we argue that since he did not address it it must be OK? Or should we assume that since he did not address it it is not worth mentioning (rules in place are sufficient to cover). There is a St. Matthias church in my old neighborhood in Syracuse, so at least some people recognize his standing as the replacement Apostle.