For many years, learning the competences to teach mathematics in early education at university has been associated with the ability to reproductively apply methodological guidelines. Currently, however, the need to not only understand the mathematical meanings given by teachers, but also students of the specialty, are seen to be important. This article attempts to engage in an interpretive line of thinking with regard to mathematics education, coming from the perspective of students learning to be early education teachers. Their understanding of the contexts for learning mathematical concepts, as well as their sensitivity to the processes of constructing mathematical knowledge by very young pupils, being a way of predicting what educational activities will be undertaken in the classroom in the future. This text is the result of qualitative analyses of written essays of early education students, where respondents had to make conceptualizations of their beliefs by justifying the selection of particular declarative statements. Students’ mathematical meanings were also uncovered in their strategies for solving mathematical problems for very young pupils. Moreover, the results of this analyses provides a context for reading the students’ understanding of mathematics learning processes.
The debate between Ludwik Fleck (microbiologist and philosopher of science) and Tadeusz Bilikiewicz (historian and philosopher of medicine) took place shortly before the outbreak of World War II and remained virtually unnoticed until 1978. A wider recognition of their exchange was possible only after the English and German translations appeared. Basically, the polemics concerned understanding of the concept of style and influence that the environment exerted on scientific activity and its products. The polemic started with the review of Bilikiewicz’s book Die Embryologie im Zeitalter des Barock und des Rokoko (1932) where the historical account of the development of embryology in the early and late Baroque period was interwoven with bold sociological remarks. The commentators of the debate were quick to notice that the claims made by Fleck at that time were crucial for understanding of his position, especially because they let to interpret his views in a non-relativist way. While the importance of the controversy was univocally acknowledged, its assessment so far has been defective for two reasons. First, for decades the views of Bilikiewicz were known only from the short and rather critical presentation given by Fleck and this put their discussion into an inadequate perspective. Second, for over 40 years it remained a complete puzzle what prompted their exchange of views. This paper closes these gaps. Thus, on the one hand, I reconstruct the central issue of the disputation between Fleck and Bilikiewicz and situate it within the context of Bilikiewicz’s views. On the other hand – and this is more important – I try to explain the origin of their debate by quoting some recently discovered and unpublished archival materials. A review of their correspondence gives me an opportunity to advance some hypotheses about the aims and hopes connected with their project but also possible reasons for its failure.
The distinction between primary and secondary qualities, most famously outlined by Galileo, and subsequently supported, inter alia, by Descartes and by Locke, has widely been considered one of the crucial factors in the development of modern idealism. In its contemporary form, the distinction identifies some of the perceived properties as mental phenomena due to their content and structural dependence on the mind. However, this account of the primary/secondary distinction is largely different from its original version developed by the above-mentioned philosophers, within whose work the mental being of the perceived qualities was demonstrated objectively, from the conceptually-derived nature of matter, and not subjectively, by referring to the mind’s participation in the cognitive process. It was only at the next stage of the early modern subjectivisation of sense perception, best exemplified by such philosophers as Arnold Geulincx and Richard Burthogge, that the creative role played by the mind in sensation and, consequently, the mind-dependency of the sensible qualities was recognised – a turn influenced by the reinterpretation of Aristotelian philosophy offered by Jacopo Zabarella and the Paduan school, as well as by anti-Aristotelianism of the kind developed in Netherlands. Furthermore, the two different approaches to the primary/ secondary distinction can be linked with two main types of post-Cartesian idealism, i.e. Berkeleian and Kantian – a claim for which illustrative evidence from British philosophy, namely from Berkeley’s and Burthogge’s respective theories, can be drawn.