Historical sources in the classroom and their educational effects

International audienceLet History into the Mathematics Classroom is an upcoming book by the French Commission inter-IREM Histoire et Épistémologie. Starting from an example from this book, we will first endeavour to identify specific features of the output of this community of practice. On this basis, we will formulate a list of research questions that are not specific to the HPM community, so as to foster collaboration with mathematics education researchers working in other fields. In the second part of the paper, we will report on a recent experiment in primary school which sheds light on the extent to which meta-tasks – a notion introduced in the first part – can be successfully entrusted to young students.</p

ER membrane phospholipids and surface tension control cellular lipid droplet formation

Cells convert excess energy into neutral lipids that are made in the endoplasmic reticulum (ER) bilayer. The lipids are then packaged into spherical or budded lipid droplets (LDs) covered by a phospholipid monolayer containing proteins. LDs play a key role in cellular energy metabolism and homeostasis. A key unanswered question in the life of LDs is how they bud off from the ER. Here, we tackle this question by studying the budding of artificial LDs from model membranes. We find that the bilayer phospholipid composition and surface tension are key parameters of LD budding. Phospholipids have differential LD budding aptitudes, and those inducing budding decrease the bilayer tension. We observe that decreasing tension favors the egress of neutral lipids from the bilayer and LD budding. In cells, budding conditions favor the formation of small LDs. Our discovery reveals the importance of altering ER physical chemistry for controlled cellular LD formation

Membrane Curvature Catalyzes Lipid Droplet Assembly

Lipid droplet (LD) biogenesis begins in the endoplasmic reticulum (ER) bilayer, but how the ER topology impacts this process is unclear. An early step in LD formation is nucleation, wherein free neutral lipids, mainly triacylglycerols (TGs) and sterol esters (SEs), condense into a nascent LD. How this transition occurs is poorly known. Here, we found that LDs preferably assemble at ER tubules, with higher curvature than ER sheets, independently of the LD assembly protein seipin. Indeed, the critical TG concentration required for initiating LD assembly is lower at curved versus flat membrane regions. In agreement with this finding, flat ER regions bear higher amounts of free TGs than tubular ones and present less LDs. By using an in vitro approach, we discovered that the presence of free TGs in tubules is energetically unfavorable, leading to outflow of TGs to flat membrane regions or condensation into LDs. Accordingly, in vitro LD nucleation can be achieved by the sole increase of membrane curvature. In contrast to TGs, the presence of free SEs is favored at tubules and increasing SE levels is inhibitory to the curvature-induced nucleation of TG LDs. Finally, we found that seipin is enriched at ER tubules and controls the condensation process, preventing excessive tubule-induced nucleation. The absence of seipin provokes erratic LD nucleation events determined by the abundance of ER tubules. In summary, our data indicate that membrane curvature catalyzes LD assembly.Peer reviewe

Justifying a calculation technique in years 3 and 6

International audienceIn this progress report, we present the outline and the first results of a research project which aims to study the argumentative capacity of year 3 (age 8) to year 6 (age 11) students, in a numerical context involving natural and decimal numbers. This project centers on a technique for dividing natural numbers by two, a technique which appears in Al-Khwarizmi’s Kitāb al-ḥisāb al-Hind

From the historical text to the classroom session: analysing the work of teachers-as-designers

While classical studies have highlighted the many potential benefits of using original historical sources in the classroom, few studies have documented actual classroom practices outside of research contexts. In this case study, I aim to describe and explain how five French high school teachers autonomously designed and implemented classroom sessions starting from the same document, namely an excerpt from Euler's Elements of Algebra presenting an algorithm for square root approximation. From a methodological viewpoint, it enables me show how two general frameworks for the study of teachers' professional practices-the Documentational Approach to Didactics and the Didactic and Ergonomic Double Approach-can be tailored to fit the specific challenges of using historical sources. The empirical results provide fresh insights into the conditions for a mathematically rich use of historical sources in the classroom, and on the connections between this use and the integration of a historical perspective in the teaching of mathematics

Une architecture d'environnement de resolution de problemes en modelisation. Apports de l'Intelligence Artificielle

INIST T 76422 / INIST-CNRS - Institut de l'Information Scientifique et TechniqueSIGLEFRFranc

Mathematical analysis of informal arguments: A case-study in teacher-training context

International audienceWe will endeavour to show that some historical documents which are probably too difficult to be used in the classroom can nevertheless be fruitfully used in teacher-training, in order to provide teachers with tools for the analysis of informal or semi-formal justifications. We will analyze an extract from the Nine Chapters, so as to spell out tool for the analysis of three school documents bearing on the multiplication of decimal or general fractions