See the notes at Matthew 18:1-5. Compare Mark 9:33-38.

There arose a reasoning - Εισηλθε δε διαλογισμος, A dialogue took place - one inquired, and another answered, and so on. See this subject explained on Matthew 18:1 (note), etc.

(9) Then there arose a reasoning among them, which of them should be greatest.
(9) Ambition results in dishonour, but the result of modest obedience is glory.

Then there arose a reasoning among them,.... The Vulgate Latin, Syriac, and Arabic versions read, "a thought entered into them"; suggested very likely by Satan, which broke out into words, and issued in a warm dispute among them; and this was in the way, as they were travelling from Caesarea Philippi, to Capernaum; see Mark 9:33.
Which of them should be greatest; that is, "in the kingdom of heaven", as in Matthew 18:1 in the kingdom of the Messiah, which they expected would be a temporal one: wherefore the dispute was not about degrees in glory, nor in grace; nor who should be the greatest apostle and preacher of the Gospel; but who should be prime minister to the king Messiah, when he should set up his monarchy in all its grandeur and glory.

STRIFE AMONG THE TWELVE WHO SHOULD BE GREATEST--JOHN REBUKED FOR EXCLUSIVENESS. (Luke 9:46-48)
(See on Matthew 18:1-5).

Which of them should be greatest. See notes on Matthew 18:1-6 and Mark 9:33-50. For notes on the question of John and the Lord's answer, see Mark as above.

And there arose a reasoning among them - This kind of reasoning always arose at the most improper times that could be imagined.