Many students encounter difficulty in their transition to advanced mathematical thinking. Such difficulty may be explained by a lack of understanding of many concepts taught in early school years, especially multiplicative reasoning. Advanced mathematical thinking depends on the development of multiplicative reasoning. The purpose of this study…

This study addressed one aspect of scientific reasoning, the control of variables reasoning scheme. We explored whether a short intervention aimed at accelerating this reasoning scheme by CASE lessons would improve students' ability to apply this scheme in problems related to the biology curriculum. About 120 students from grade nine were assessed…

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…

The author reflects on Piaget's 1972 article, "Intellectual evolution from adolescence to adulthood," addressed to questions regarding what he alleged to be the final, most advanced level of cognition in his developmental stage theory--formal operations, as described in his 1958 volume coauthored with Inhelder, "The growth of logical thinking from…

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,…

This article explores relationships between designing and making in the work of children within the age range 5-11 when engaged in practical modelling tasks. The notion of the model is explored from the perspective of concrete representations. It is suggested that concrete models may be used as hypotheses from which to test ideas about the nature…

Executive functions have been defined in the neuropsychological literature as those cognitive processes that underlie future-oriented, goal-directed behavior such as planning, working memory, inhibition, and self-monitoring. Piaget's final stage of cognitive development, formal operations, involves a systematic approach to problem solving that…

This paper has two purposes: (1) to explain briefly in terms of Piaget's theory why relationships are fundamental for constructivist teachers; and (2) to show how constructivist teachers can think about relationships in classroom activities. In a nutshell, the message is that the process by which children are constructing their intelligence,…

The undergraduate liberal arts program is assumed to promote sophistication in thinking. At the same time, certain levels of cognitive ability are assumed to be present in students accepted into selective undergraduate colleges. Beginning in the fall of 1991, a long-term research study was undertaken at a small residential liberal arts college…

Jean Piaget's classic theory of cognitive development would imply that the higher-order items on the Kentucky state assessment would only be possible for students well into concrete operations or beginning formal operations. The implication would be that Kentucky fourth graders who are not fully concrete yet may be hitting a developmental ceiling…

This paper discusses ways to use worked examples in teaching mathematics. It is argued that neither investigative teaching such as discovery learning nor lecturing and starting from the abstract are helpful as they are based on emotive associations with general labels rather than precise details of pedagogic strategies. (KHR)

This paper describes part of a study undertaken to examine the relationship between dispositions to critical thinking, procedural knowledge in science, and formal reasoning. Three tests were administered to 346 grade 7 students at the beginning and at the end of the school year: California Critical Thinking Dispositions Inventory, Group Assessment…

Reports on the results of using, in the context of secondary education in Pakistan, an intervention program which earlier was shown in British schools to have a substantial effect over a two-year period on the cognitive development of pupils. (Author/CCM)

Extends prior theory and research by postulating the existence of an intermediate class of concepts called 'hypothetical'. Investigates the hypothesis that three kinds of scientific concepts exist by constructing and administering a test on concepts introduced in a college biology course. Supports the hypothesis that intellectual development…