Stages of learning
It helps to have a rough framework of how learning proceeds, in order to know how to focus your efforts and what to watch out for. Below is such a framework that grew out of my experience. It is important to get a sense of this framework but not be rigid in considering it.
Stage I: Introducing the first two songs
In the first stage, you are focusing on introducing this new type of music to your child. If you haven’t already, complete the exercise in How to introduce the idea of what a rāgam is. Once your child is able to “pass the test” at the bottom of that article, you are ready for the lesson in this article. Be patient until your child is able to pass the basic test at the end of that article.
Stage II: Building up an abstract idea of what a rāgam is
However, your child might not yet have abstracted the idea of a rāgam yet, even if they can identify the two songs clearly. It might take 2-8 rāgams for them to be able to grasp this abstraction. And that is okay and expected. Between the second and say, eight rāgam or so, your focus is best spent on teaching them one rāgam at a time, covered in How to introduce each rāgam.
Stage III: Scaling, recognition, retention, and enjoyment
After the point where the concept of what a rāgam is is firmly grasped, the child will know exactly how the game is played, and will start developing their own mental tools for adding a new rāgam to their music vocabulary. Things will get much easier from there. The learning approach changes at that point, more focused on growing your child’s database of rāgams, on improving recognition of known rāgams, on retaining what has been learned, and—this is important—on learning to enjoy various aspects of Carnātic music.