One of the most common questions which as an Agile Coach or a Scrum Master, I had to answer while working with teams was “What should be done to the unfinished User Stories at the end of the iteration?” Here arose the dilemma of Splitting a user story or not. It is understandable to want to allocate credit for the effort put into the work completed.
Usually, a team wants partial credit for the stories which are not finished in the sprint by splitting it into finished work vs unfinished work. Scrum takes the approach of no-partial-credit because working software is the primary measure of progress.
The most common recommendation is that an unfinished user story gets returned to the product backlog, keeping the original estimate same. During the next sprint planning event, the team has two options, either keep the estimate same or re-estimate considering the effort which is left to finish which was remained unfinished in the prior sprint. I have let the team take the approach which they are comfortable with and I have constantly coached the team to weigh-in the approach of re-estimating the user story rather than keeping the estimate same.
When a Scrum Master or Coach takes a not-partial-credit stance, the team, most likely, do not like it. They feel that they are losing the credit for the finished work. Hence there are two options for them to consider. Either keep the estimate same as original when pushing back to product backlog or re-estimate. They will definitely not going to choose the option # 2. Personally, I would let the team be in their comfort zone and constantly by picking any option out of the two and coaching them towards the mature practice. However, I will definitely take the stance of no-partial-credit to enforce the habit of committing the user stories which they can finish within a sprint.