Under Sensorimotor:
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Object permanence is the awareness that an object continues to exist even when it is not in view.

This was our first clue that DD was advanced within Piaget's stages. She had severe separation anxiety when she was 3-4 months old. Her Dr. freaked out and said that she was very very early for this phase. Most babies don't exhibit this until at least 8 months.

Under concrete:
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Categorical labels such as "number" or "animal" are now available to the child.

DD understood this way before age 2. She understood animals were different and could categorize them from an early age. And numbers beyond rote counting was right before age 2. I don't exactly remember when but when it came to classification of breeds within the dog category such as the example in the article ... she just seemed to understand. Dogs were not just dogs but clearly different. She was under age 3 when she figured that out.

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Piaget determined that children in the concrete operational stage were fairly good at the use of inductive logic. Inductive logic involves going from a specific experience to a general principle. On the other hand, children at this age have difficulty using deductive logic, which involves using a general principle to determine the outcome of a specific event.

I joke that logic is DD's middle name. She has no problem deducing either.

As for Formal Operational Stage, I think she is showing signs of this but I wouldn't say she is fully into this stage. Back when I didn't really see her advancing with noticeable work, such as jumping into multiplication or really reading beyond her starting phase ... I did see the conceptional side advancing quickly. She is very much about imagination and could always play for long periods entertaining herself with whatever scene she had created in the moment, but during this period of no real academic growth her thought process advanced like crazy. She has always had advanced speech but during this period her ability to really communicate and asked relevant questions to the topic of the moment showed her abstract thinking.