"independent-thinking" children - an excellent topic!
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Having raised such a child successfully, & worked with other parents, in an UNschooling-program I founded, a progressive learning-adventure we all shared in, for our homeschooling community's enrichment, here are some ideas:
*
Model yourself an Independent-lifestyle 
, incldg. noticing details, & wondering, & questioning, be that in religion, medicine, history & politics, natural sciences, etc.
(a common mistake is to want "closure" to a question, hence too many interpretations & conclusions are reached pre-maturely, resulting in unfortunate mistakes; especially in medicine, using traditionally accepted science models) - Avoid that, by
continually Noticing Details, & asking uncommon
and open-ended questions, & doing your own research.
* Once you have taught your child
necessary Life-safety precautions, such as: what is food, & what is not, Electrical-outlets & fingers do not go together, what else can kill you, don't run into the street, yadda yadda, you get the drift - then, they're ready to Explore, & Discover the wonders of our world, more safely

excited & in fun... preserve the fun.
* Live cognizant & admit to yourself, and child/ren (when they are of pre-school age), that
you do not know it all. (iow, just because you are physically-taller & stronger, and have lived longer, does not equate with 'wisdom'. - We teach children a very few basics, before we then get out of their way, and allow them to spread their wings... and Invite their considerate input as their opinions are valid.
* When you give children plus their phenomenal Learning-skills
due respect, Children respect you far more, and you can live in a manner consentually. Does this make good sense?
* We each got our beginning-learnings, prenatally in the womb.
Beyond development & growth, our prenatal
environment also "imprinted/raised" each of us in more ways than most people considered. - As we, who work with people's challenges, afterwards, have time and again, helped them transform...
What positive parental experiences

have you had along these thoughts?