Support for Piagets theory is also expressed in the Montessori emphasis for exploration so children can learn at their pace.
At the core of Piagets theory is an assertion that "children construct, or create, logic and number concepts from within rather than learn them by internalization from the environment" (Kamii 2009 citing Piaget 1971, Piaget and Szeminska 1965, Inhelder and Piaget 1964, and Kamii 2000). Piaget distinguished three kinds of knowledge: physical knowledge, social knowledge, and logico-mathematical knowledge (Kamii 1996: 99). Piaget taught that the logico-mathematical knowledge is only partly acquired from objects because, for instance, the similarity between two blocks of different colours is not observable but is deduced by an individual through putting things in relationships with the relationships earlier discovered (Kamii 1996: 100). In other words, for Piaget, individuals or children use a logico-mathematical framework within their minds to acquire knowledge (Kamii 1996: 100). According to Kamii (1996: 100-101), through what Piaget described as logico-mathematical framework, a learner or student or child acquires knowledge through:
Applying Piagets teaching, Kamii concluded that children create their own arithmetic or mathematics in acquiring logico-mathematical knowledge using their ability to think and, thus, the goal of math education is to assist learners or children invent procedures for solving mathematical problems as well as in constructing "a network of numerical relationships" (Kamii 1996, 101). As pointed out by Piaget, mathematical knowledge is different from physical knowledge because the former is not observable while the latter is (Kamii 1996: 102). Following Piaget, Kamii said that "there is no such thing as addition fact" because sums are internalized or constructed from within (1996: 102). In illustrating Piaget thinking, Kamii said