It is hard to improve something you cannot see.
Lean visual management is a principle used to make knowledge work visible. We use Kanban boards to make the work in process visible to the whole team, as well as anyone that happens to glance at it.
The following figure shows an example process using Trello.com as the tool. It could just as easily be a physical whiteboard with columns and sticky notes.
In this short fictional process the steps are:
Add a Backlog or To Do column with everything, here fictional task A, task B, and so on. Then use a column for Design and a column for Development. Note that the example also uses ready queues between process steps to help everyone see bottlenecks more easily.
Kaizen is about continual improvement. When the team sees the board every day for the 15-min daily synchronization meeting, everyone can see the current state of the project and spot bottlenecks as easily as you can tell the highway has a traffic jam while on the way to work.
The visual nature of the Kanban board is its strength. Everyone can see the current status, so everyone has the information needed to recommend improvements. More heads are better than one when it comes to improvements and getting well from project problems.
If you notice no one is making improvement suggestions, coach the team about the expectation.
If the leadership also has access to the board, it may reduce Lean transaction waste for reporting progress. They can see the board any time they want and see the current state without interrupting anyone on the team for an update for report X or Y or Z.
Ideally, you convince them to look at the Kanban board at whatever frequency they want. However if your management has many projects and your set of projects is just a smaller subset of them, they may not change their methods of tracking progress just for you. You may have to frame it as an experiment for them to observe to what degree Kanban provides the status information they need.
This is probably why many in software now talk about their goal to have the entire enterprise be Lean-Agile.