There are three worlds of mathematics according to David Tall: the world of conceptual embodiment, the world of symbolic calculation and manipulation, and the world of axiomatic formalism. This classification is based on how mathematical concepts/objects developed. It is important for us teachers to be at least aware of these three worlds. It tells us that math ideas are not formed in the same way therefore we can’t teach all math topics in the same way. The use of real-life contexts, the use of concrete materials, may afford learning of some concepts but may hinder the learning of others.
World of Conceptual Embodiment
According to Tall, the world of conceptual embodiment grows out of our perceptions of the world and consists of our thinking about things that we perceive and sense, not only in the physical world, but in our own mental world of meaning. The world includes the conceptual development of Euclidean geometry and other geometries that can be conceptually embodied such as non-Euclidean geometries and any other mathematical concept that is conceived in visuo- spatial and other sensory ways. A large part of arithmetical concepts also developed via conceptual embodiment (see Figure below).
World of Symbolic Calculation
The second world is the world of symbols that is used for calculation and manipulation in arithmetic, algebra, calculus and so on. The ‘development’ of the objects of this world begin with actions (such as pointing and counting) that are encapsulated as concepts by using symbol that allow us to switch effortlessly from processes to do mathematics to concepts to think about. But the focus on the properties of the symbols and the relationship between them moves away from the physical meaning to a symbolic activity in arithmetic. My post Levels of understanding of function in equation form describes the development of the idea of equation from action to object conception.
World of axiomatic formalism
The third world is based on properties, expressed in terms of formal definitions that are used as axioms to specify mathematical structures (such as ‘group’, ‘field’, ‘vector space’, ‘topological space’ and so on). It turns previous experiences on their heads, working not with familiar objects of experience, but with axioms that are carefully formulated to define mathematical structures in terms of specified properties. Other properties are then deduced by formal proof to build a sequence of theorems. The formal world arises from a combination of embodied conceptions and symbolic manipulation, but the reverse can, and does, happen.
Read the full paper Introducing the Three Worlds of Mathematics by David Tall.