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Cognitive learning and two-way learning are effective knowledge representations, which can simulate the human brain to learn concepts. The combination of both topics has achieved some results and improved conceptual evolution ability in fuzzy formal concept analysis (FFCA), but there are still some shortcomings: 1) FFCA does not consider the flexibility of concepts, which makes it difficult to select suitable concepts; 2) the existing necessary and sufficient concepts fail to learn directly from any information granules; 3) the cognitive learning mechanism ignores to integrate previously acquired knowledge into the present state in the process of dynamic concept learning. To tackle these issues, in this paper, we put forward a novel two-way concept-cognitive learning (TCCL) model based on three-way decision in fuzzy formal contexts. Firstly, we introduce the object and attribute operators to learn variable precision object induced three-way concept, where such concept has flexibility by adjusting thresholds. Then, to learn directly necessary and sufficient three-way concept from the given clues, we investigate a new TCCL model, which has low computation cost for concept learning. Furthermore, updating mechanism of three-way concept is discussed in dynamic learning environment. Finally, the conducted experiments explicate the effectiveness and feasibility of our proposed approach in the large-scale datasets. [ABSTRACT FROM AUTHOR]