Can teachers and mathematicians communicate productively? The case of division with remainder

Is it possible that a meeting of mathematicians and primary school teachers will be productive? This question became intriguing when one professor of mathematics initiated a professional development course for practicing primary school teachers, which he taught alongside a group of mathematics Ph.D. students. This report scrutinizes the uncommon meeting of these two communities, who have very different perspectives on mathematics and its teaching. The instructors had no experience in primary school teaching, and their professed goal was to deepen the teachers’ understanding of the mathematics they teach, while teachers were expecting the course to be pedagogically relevant for their teaching. Surprisingly, despite this mismatch in expectations, the course was considered a success by teachers and instructors alike. In our study, we analyzed a lesson on division with remainder for teachers of grades 3–6, taught by the professor. The framework used for the data analysis was mathematical discourse for teaching, a discursive adaptation of the well-known mathematical knowledge for teaching framework. Our analysis focuses on the nature of the interactions between the parties and the learning opportunities they afforded. We show how different concerns, which might have hindered communication, in fact fueled discussions, leading to understandings of the topic and its teaching that were new to all the parties involved. The findings point to a feasible model for professional development where mathematicians may contribute to the education of practicing teachers, while they are gaining new insights themselves.     

Gödel’s proof

In 1931, the great Austrian mathematician Kurt Gödel proved “all consistent axiomatic formulations of number theory include undecidable propositions”. This discovery of Gödel and its proof had enormous repercussions in mathematics and computer science. The proof hinged upon the writing of a self-referential mathematical statement, in the same way as the liar’s paradox — I am lying — is a self-referential statement. In this three-part article, we describe Gödel’s discovery and his famous proof.     

Stichwort: Unsicherheit — Ungewissheit — Entscheidungen

Ohne Zusammenfassung      

Accounting for young children’s competence in educational research: New perspectives on research ethics

Educational researchers working with young children face ethical issues when researching the talk and interactions of young children. Issues around the competence of children to participate in research pose challenges to educational researchers and to the young participants and their families, within what are seen as increasingly risky and regulated research environments. This paper examines some of these issues in light of recent sociological perspectives that account for children as competent practitioners of their social worlds. Drawing on research investigating the governance of the lives of young children in Australia, we examine the rights of children to be both seen and heard as competent research participants. These sociological directions afford opportunities to reconsider the ethical issues around research with young children. Such an approach breaks new ground in early childhood education research.