Background: Shayer, Ginsburg, and Coe (2007) showed that children leaving primary school in Y6 entered secondary school with much lower levels of understanding of the physical conservations than in 1976. It seemed desirable to investigate cognitive development in the first three years of secondary education. Aims: By using two Piagetian tests of…

This paper focuses on the changes in thinking involved in the transition from school mathematics to formal proof in pure mathematics at university. School mathematics is seen as a combination of visual representations, including geometry and graphs, together with symbolic calculations and manipulations. Pure mathematics in university shifts…

One of the most challenging daily experiences of teaching young adolescents is helping them transition from Piaget's concrete to the formal operational stage of cognitive development during the middle school years. Students who have reached formal operations can design and test hypotheses, engage in deductive reasoning, use flexible thinking,…