Tall称具体化、符号化、形式化3种认知发展形式为数学三个世界。他在继承前人成果的基础上提出该理论，重新解释了人的认知发展过程。他认为抽象知识间是可以相互转化的，结晶体概念是转化的根源，并验证了其教学有效性。目前该理论处于结构稳定，内容不断完善阶段，已得到国际认可。%Tall called three cognitive development modes three worlds of mathematics. They were conceptual-embodied, proceptual-symbolicand axiomatic-formal.He showed the development process of three worlds of mathematics, and explained the process of cognitive development again by studying. He thought abstract knowledge could convert to each other; crystalline concepts were the origin of converting, and showed the validity of this theory. At present this theory is on the stage of matural structure and content perfected constantly and approved in the world.
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