Introducing the structure of proof in lower secondary school geometry: a learning progression based on flow-chart proving

ArticleProceedings of the 12th International Congress on Mathematical Education (ICME-12)．COEX, Seoul, Korea, 2012-7-8/15, 2012, 2858-2867．conference pape

Supporting students to overcome circular arguments in secondary school mathematics : the use of the flowchart proof learning platform

ArticleProceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education．Ankara, Turkey, 2011-7-10/15, 2011, 2:353-360．conference pape

Functions of Open Flow-chart Proving in introductory lessons of formal proving

ArticleLiljedahl, P., Oesterle, S., Nicol, C., & Allan, D. (Eds.). (2014). Proceedings of the Joint Meeting of PME 38 and PME-NA 36 (Vol. 4):225-232. Vancouver, Canada: PME.conference pape

Torus fibrations and localization of index II

We give a framework of localization for the index of a Dirac-type operator on an open manifold. Suppose the open manifold has a compact subset whose complement is covered by a family of finitely many open subsets, each of which has a structure of the total space of a torus bundle. Under an acyclic condition we define the index of the Dirac-type operator by using the Witten-type deformation, and show that the index has several properties, such as excision property and a product formula. In particular, we show that the index is localized on the compact set.Comment: 47 pages, 2 figures. To appear in Communications in Mathematical Physic

"Waiting" in the World of Becoming : Based on Antoine de Saint-Exupery\u27s insights into "replete life"

The present article attempts to explicate the nature of "waiting", especially in the world of becoming. In Chapt. I, the manifold structure of "waiting" is illustrated through two articulations; 1) into "hope", "patience", etc., on the subjective aspect of "waiting" (IA1), and, 2) into the worlds of "nature", "instrument-machinery", and "becoming", to which the objective aspect of "waiting" belongs (IA2). Nextly, "waiting" in the world of becoming is characterized by 1) the indefiniteness of the objective aspect, 2) the larger context including manipulation, and, 3) the mutually dependent relationship between the subjective and the objective aspects. (IB). In Chapt. I, "demarche (walking)" is outlined which plays a central role to the description of "replete life" in Citadelle, written by Antoine de Saint-Exupery. This "demarche" is found to satisfy the three characteristics of "waiting" in the world of becoming (IIA). Then "demarche" is further featured with respects to its temporal structure (IIB) and to its other moments: the relationships between others and self, and, between objects and self (IIC). Lastly, the implications of the above explication are drawn on to the problem of "errors" in the world of becoming. (IID)

Study of Genetic Effects of Sulphur Mustard Gas on Former Workers of Ohkunojima Poison Gas Factory and Their Offspring

General health examination and one-dimensional electrophoretic examination to detect mutations at the protein level were conducted in order to elucidate potential genetic effects of sulphur mustard gas on children of the former workers of Ohkunojima Poison Gas Factory. In the general examination, no disease which was definitely considered to be caused by genetic effects was observed, and no examination values obtained for the children proved to be significantly abnormal compared with those for their parents. Blood samples from 456 children were electrophoretically examined for 30 protein systems. A total of 36 protein variants were detected in 10 protein systems, and the frequency of variants was 2.63 per 1,000 tests. Family study was completed for 32 of these variants, all of which were confirmed to be genetic variants. In 29,868 locus tests, mutation occurred in germ cells of parents could not be detected. Among 36 variants, two PGM2 variants and one GPI variant were detected for the first time in this study

Developing a Curriculum for Explorative Proving in Lower Secondary School Geometry

ArticleProceedings of the 13th International Congress on Mathematical Education (ICME-13)．Hamburg, Germany, 2016-7-24/31．conference pape