Wheat ripens in May, suggesting it was May when the disciples were plucking ears of grain soon after the beginning of Jesus' public mission (Mark 2:23). On this evidence, Jesus began his public mission around the time of the Passover, and of course his crucifixion took place at the time of the Passover. We can know the period of Jesus' public ministry if we know the number of annual Passover celebrations that occurred during this period.
There is general agreement that John's Gospel mentions the annual Passover celebration on at least three separate occasions. The first of these (John 2:13) is consistent with the event mentioned in Mark 2:23. Then, John 6:4 speaks of a second Passover on which Jesus did not travel to Jerusalem, as it occurred during the feeding of the five thousand. A third Passover (John 11:55) marks Jesus' final entry to Jerusalem. A possible fourth Passover is alluded to in John 4:54, where Jesus returns from Judea to Galilee. This would mean that his ministry took between two and three years.
Although John places the feeding of the five thousand at the time of the Passover, the synoptic gospels make no mention of this. Moreover, in Mark's Gospel, the story of feeding the five thousand, narrated in Mark 6:32-44, seems to occur only shortly before Jesus begins his final journey to Jerusalem, and there can not be two Passovers in the same year. The brevity of the synoptic accounts, especially in Mark's Gospel, and the absence of any journey to Jerusalem until the final, fateful trip suggests that Jesus' mission lasted about one year, but the parallel between John 6:4 and Mark 6:32-44 makes it at least possible that the public mission could have been for just two years.
The three Synoptic gospels refer to just one passover during his ministry, while the Gospel of John refers to three passovers, suggesting a period of about three years. However, the Synoptic gospels do not require a ministry that lasted only one year.
Paula Fredriksen, in 'The Historical Jesus, the Scene in the Temple, and the Gospel of John', published in John, Jesus, and History, Volume 2 page 250, says that in Matthew, Mark and Luke, the implied period is about one year and that in John the journeys to Jerusalem require a period of over two years.