Most methods seem to assume grammatical knowledge you cannot expect at that age. Which methods are recommended for this age and knowledge?
There is one thing that clearly works: immersion. However, children tend not to tolerate it when a family member speaks in a different language to them which that person doesn't speak on a near native level. And moving to an Esperanto-speaking country or sending the child to an Esperanto-speaking nursery, is probably out of the question.
I know one little child who was growing up in a television-free household in Germany and learned English simply by being allowed to watch one or two Peppa Pig episodes in English on a computer every night instead of a bedtime story - and never any other films. When this child moved to the UK at the age of 5, she could communicate instantly in her new nursery, even though at home she never used English other than the episode titles (which she always pronounced flawlessly). Presumably, Esperanto songs and nursery rhymes, Esperanto picture books and Esperanto bedtime stories could be combined for a similar effect.
What is crucial for effective language learning at a young age, and in fact for efficient language learning at any age, is to have situations that satisfy the following conditions:
- the learner wants to communicate
- there is a natural reason why the communication has to be in Esperanto (something that doesn't work: when a parent suddenly speaks in a foreign language)
- the learner can understand enough to make the effort worthwhile for intrinsic reasons (i.e. for reasons other than wanting to learn a language).
For a child who doesn't know Muzzy in Gondoland in another language yet, you should be able to achieve this by letting them watch Mazi en Gondolando. And in fact, Pasporto al la tuta mondo should work as well. Watch these together with the child, only one episode or less at a time, depending on the child's age. Briefly discuss each episode with the child in the family language. Don't forget that young children enjoy repetition, which is quite fortunate given the scarce supply of Esperanto material.
My answer so far was influenced by the wording of the question, which has 'method' in it. However, without having tried it, I suspect that an even better approach could be joining Pasporta Servo to ensure a steady stream of interesting foreigners with whom communication naturally happens in Esperanto. Or even taking the child to (preferably international) Esperantist meetings. This approach is probably less effective in English-speaking countries than in countries with a greater percentage of Esperantist visitors who don't speak the local language well.
For children in the 5-10 age range, the Esperanto Association of Britain offers a course called 'Urso-Kurso'. It is best if children can already read, but sympathetic adults can help a great deal. The course includes detailed explanations for the helping adults, who don't have to know Esperanto themselves, on how to present the material, and how to exploit it further for the child's general education. The materials provide basic vocabulary and structures (without getting bogged down in 'grammar'!), through the use of lots of varied activities - drawing, colouring, puzzles, games, songs...
Perhaps a fun way to encourage immersion could be to sing with the children in Esperanto. There are some quite high-quality songs for kids on YouTube by Babelo Filmoj. The kids might be able to passively learn the lyrics to these songs which is a way to learn some words and phrases without having to consciously study.
Mi trovis tiun videojn.
Mi ne scias se gxis estos.