Purpose: Appreciating the relationship between Sylvio Ceccato and Ernst von Glasersfeld, both as people and in their work. Approach: historical and personal accounts, archeological approach to written evidence. Findings: Ceccato’s work is introduced to an English speaking audience, and the roots of Glasersfeld’s work in Ceccato’s is explored. Flaws in Ceccato’s approach are indicated, together with how Glasersfeld’s work overcomes these, specially in language and automatic translation, and what became Radical Constructivism. Conclusion: Glasersfeld willingly acknowledges Ceccato, who he still refers to as the Master. But Ceccato’s work is little known, specially in the English speaking world. The introduction, critique and delineation of extension and resolution of Ceccato’s ideas in Glasersfeld’s work is the intended value of the paper.
Piagetian theory describes mathematical development as the construction and organization of mental operation within psychological structures. Research on student learning has identified the vital roles two particular operations – splitting and units coordination – play in students’ development of advanced fractions knowledge. Whereas Steffe and colleagues describe these knowledge structures in terms of fractions schemes, Piaget introduced the possibility of modeling students’ psychological structures with formal mathematical structures, such as algebraic groups. This paper demonstrates the utility of modeling students’ development with a structure that is isomorphic to the positive rational numbers under multiplication – “the splitting group.” We use a quantitative analysis of written assessments from 59 eighth grade students in order to test hypotheses related to this development. Results affirm and refine an existing hypothetical learning trajectory for students’ constructions of advanced fractions schemes by demonstrating that splitting is a necessary precursor to students’ constructions of three levels of units coordination. Because three levels of units coordination also plays a vital role in other mathematical domains, such as algebraic reasoning, implications from the study extend beyond fractions teaching and research. Relevance: The paper uses constructivist theories of learning, including scheme theory and Piaget’s structuralism, to study how students construct mature conceptions of fractions.