How can I help my child succeed?
I think the heart of this question can be answered by one of Dr. Suzuki’s founding principles. Learning is not about talent, destiny, or even intelligence. It is about environment. There are many things you can do that fall within this realm, but the bottom line is, if you want them do their best, surround them with the right influences. Understand that your child may opt for music to be a hobby in their life, that is OK. What is important is that they have the skills to make it a career, if that is the path they choose.
I am not a parent myself, so the suggestions listed below are from my perspective as a student and teacher.
Practice with them. For the beginner, you are essential. I realize this is a tall order for a family with multiple children, but where there is a will, there is a way.
Insist they practice everyday! Make it part of the daily routine. This may not make you popular at times, but every musician I know states they had a parent that forced the issue for at least some period while they were growing up. Designate a specific time each day for practice, and stick to it.
Eliminate distractions during practice time. Practice time is valuable, and should be treated like a treasure. I am a fan of modern technology. It provides such easy access to resources like audio/video recordings, practice helps from worthy players, etc., that were not available even for me not that long ago. But, if not used with proper discipline, phones especially create more problems than they solve. Studies show that when someone is focused on a task and get interrupted by a phone call or text, it takes on average 20 minutes to get back the same level of focus following the distraction. In our fast-paced era, that is a lot of time. Put the cell phone on silent, or better yet, leave it in another room during practice time. If you object to me taking calls or texting during your child’s lesson, you should feel the same about it happening during their practice.
Take them to concerts! They may not always like it, but they need to be exposed to it. Let them witness in person what others are doing on their instrument, both professionals and amateurs. They will hear more music that inspires them than not.
Build your classical audio library. Listening to their specific study material is vital, but the pillars of music in the repertoire need to be heard as well. For me, the Mendelssohn violin concerto gave me something to aspire to, and that was good motivation to practice.
Set the example. If it is important to you, they will respect it, and it is likely to rub off, at least to a degree.