What do Hindu scriptures or Gurus say on How to Introduce Hinduism to Children?
Let me start from the last question, YES, there is prescribed ways in our texts. Shodasa Sanskara is the answer for your question.
The parenting actually starts before the birth of your child. Because it is scientifically proven that the child learn when it is in the womb. The thoughts of mother at the time of pregnancy will deeply influence the child. Take the case of Prahlada, Rishi Narada told stories of Vishnu while Prahlada was in womb of his mother, which helped him to become a great bhaktha of Lord Vishnu.
So according to Hindu culture there are some practices to be followed until the birth of the child when one women is pregnant. Actually the time of conceiving a child and mindset of husband and wife will influence the child.Most Brahmins communities used to follow these rituals in connection with major events in their lives, such as pregnancy, childbirth, education, marriage, and death. These are known as Shodasa Sanskara or 16 Samskaras.
The very first of these Sanskara have an important role in deciding your child's behavior,
Literally it means, gifting the womb, is the act of conception. This is the first sacrament which followed immediately on every matrimonial union. There are a number of rites performed before conception. Brahmin community usually follow Shanthi Muhurtha, which is decided based on the astrology.When one child is conceived at a Uttama Shanti Muhurtha, he will have it's benefits and qualities.
This is the second of the 16 Sanskaras and starts after the 2nd or 3rd month after the Garbhadhana. As the food of mother will effect and influence child, this sanskara instruct what food items and medicines to have during pregnancy. The pregnant woman should have a calm and quite mind. It instruct not to be angry, not to be listen or involve in any bad karmas. (Now in the modern era, they must avoid watching violence in films, serials, and any other activities including angry, sorrow, etc.).
This is performed for the right growth of the child and to have the pureness of the mind of pregnant woman. It starts at the 4th month of pregnancy. At this stage, child will form his/her body parts. During this period hearing good songs or keerthana will help the child to be an expert in art. This is like growing a plant, if you give enough water and fertilizers, you get beautiful flowers:).
These are just a small summary of these practices, very short in fact. In Gharbadana Sanskara itself there are numerous practices such as eating sattwik food items, need of Brahmacharya before the marriage, etc. The above practices are related to parents rather than the children, so first of all parents must be aware of such practices and follow them. Fortunately, our ancestors was very keen in developing a good society by instructing to follow such practices.
Now I can ask you why was Vivekanada born in India? Why was so many great saints and intellectuals born in India? You have your answer, don't you?
Another question, Why was Brahmin people not involved in any violence compared to other communities? Because now also they follow Shodasa Sanskara practices as such, so the children are born with intelligence and good qualities. Yes, India have the power to change the world in it's culture.
There was a one day class about these 16 Sanskaras in our area and it was taken by acharya Dr.Sreenath Karayatt(clinical hypno terapist(RMP) at Psychology), Kozhikode, Kerala. He works with Sreshtacharasabha in Kozhikode. It was very useful for all people, especially for young people who are yet to marry. He has represented India in international conferences regarding this subject. Here is his facebook profile, you may contact him for the details. I will request him to provide a detailed answer in the stackexchange if possible.
And for the young children around you,
Let them hear stories from their grand parents, let them hear stories from our puranas like Ramayana, Srimad Bhagavata, Mahabharatha,etc. Panchathanthra will be also helpful for small ones. It will make them capable of identifying what is good and bad in their life. It is not just for introducing them to Hinduism. Our stories introduce them to the Life.
Let them develop the habit of reading books, introduce great saints like Swami Vivekanda through small books will be great. Then eventually give them the books like Bhagavat Geeta.(Keep them out of mobile and internet until they are capable of using them wisely. If they fall in love with the books, you don't need worry about that much).
- Let them know the science in our culture. Make them aware of the benefits of Yoga. Now most of the western countries insist to follow yoga for physical and mental health in schools. We don't need to look to west for that, but we need to realize ourselves and should follow such practices in our daily life.
Teach them to practice vrathas and fast from the childhood. Explain them the benefits of those practices, you may not need to explain them first as children will not ask for them. When they ask explain them.
As @sai has mentioned, teach them to chant Mantras like Gayathri Manthra. Introduce great words from Vedas and upanishads like 'Ahimsa Paramo Dharma' , 'Sathymeva Jayathe',etc. when possible.
Most importantly SANSKRIT, help them to learn SANSKRIT
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