Last year I got to attend a workshop on the Toyota Kata by Hakan Forss and I could immediately see how it could be useful but I didn't have an obvious place to apply it until recently.
So what is the Toyota Kata? I see it as an alternative to retrospectives. It lets you define a collection of goals and run experiments to see if you can get progress towards that goal. Unlike retrospectives you can define goals that may seem distant (release to customers on every passing commit), may be simply something you want to maintain (have zero bugs in production) or be impossible (have zero wait time in moving a story from backlog to done). Once you've built this collection of measurable goals you decide what you're going to try and your expected outcome (we will to TDD and have 50% less bugs in production). You can define what the period is before you check in again and decide on the next step. But as with anything Agile; shorter is better. You can work it into a normal two week scrum cycle or if improvements are important enough then spend a small time on it every day. Next time you get together you can assess how the experiment(s) did and what you want to try and do next.
I really like how it's focused on doing small experiments rather than committing yourself to something to fix the problem immediately. Especially for those big issues it frees you up to try anything because you'll know soon if it helped and can try something else. With normal retrospectives it can be hard to get people to conduct experiments.
So my team recently went from doing 2 week sprints to a Kan-ban style flow of work. I had some concerns that it could lead to a range of problems and wanted it to be a positive change, So we figured out some measurable goals where we wanted improvement and things we to maintain. The improvement ended up being focused around cycle time as I felt sprints encouraged us to get things done because a sprint was ending even though it's an artificial deadline. If a task lasted more than a sprint we were able to quickly see it as an issue. Other measures were around velocity (measured in weeks instead of sprints) and team comfort (do they like the change or would they rather go back).
So far things have been great. Our cycle time is getting better and everyone is comfortable with the process. It's not a clear winner where people love the new process over the old. And our goals for cycle time seem lofty compared to what we ended up measuring. But it's given us a structure to keep experimenting and keep chipping away at these things. I absolutely would recommend it to anyone facing issues they feel cannot be resolved in a normal retrospective cycle. It can compliment or replace retrospectives as you see fit.