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Assimilation and Accommodation are the two complementary processes of Adaptation described by Piaget, through which we internalize the outside world information. They are inseparable and exist in a dialectical relationship.
In Assimilation, we perceive and incorporate an outside information without changing the structure of that internal world. When new information that we come across fits out beliefs or, more abstract, fits those fields in our internal world, then it can be assimilated without any trouble(however, might cause Confirmation bias).
In Accommodation, the internal world has to accommodate itself to the evidence with which it is confronted and thus adapt to it, which can be a more difficult and painful process. It often follows with Cognitive dissonance because of conflict with pre-existent beliefs that don’t fit in our world-view.
Any example of betrayal, of discovering a new fact that shatters previous beliefs - are examples of such accommodation.
The terms are also used to describe forms of knowledge in Kolb’s Experiential Learning Cycle.