Theories of mathematics education: European perspectives, commentaries and viable research directions

We briefly comment on different perspectives on (1) the role of mathematics education theories, (2) the issue of plurality and healthy heterogeneity versus consolidation; (3) underlying inquiry systems or the implicit role of philosophy in theories of mathematics education. This paper also outlines developments within the European research scene on theory usage in mathematics education research, which complement the discussion at the 29th PME research forum in Melbourne.     

Theory usage and theoretical trends in Europe: a survey and preliminary analysis of CERME4 research reports

The 29th PME research forum on theories included only one European perspective on mathematics education. In order to convey trends in theory usage in Europe we compile, survey and analyze a large subset of the research papers from the 4^th European Congress on Mathematics Education (CERME4). That is, this paper includes a discussion of trends seen within CERME4 reports¹ on theory usage by European researchers inseven of the fourteen working groups and (a) Outlines similarities and differences in theory usage and (b) takes a futuristic stance on ways in which researchers from different traditions could understand each other. Such an enterprise would further Hans-Georg Steiner's vision of bridging theoretical traditions which are independently formulated in different regions of the world.     

A mild deprotection strategy for allyl-protecting groups and its implications in sequence specific dendrimer synthesis

A mild deprotection strategy for allyl ethers under basic conditions in the presence of a palladium catalyst is described. Under these conditions, aryl allyl ethers can be cleaved selectively in the presence of alkyl allyl ethers. These conditions are also effective in the deprotection of allyloxycarbonyl groups. The utility of the current methodology in sequence specific dendrimer synthesis is demonstrated.     

Analysis of the nonisothermal crystallization kinetics in three linear aromatic polyester systems

Melt or cold crystallization kinetics has a strong bearing on morphology and the extent of crystallization, which significantly affects the physical properties of polymeric materials. Nonisothermal crystallization kinetics are often analyzed by the classical Johnson–Mehl–Avrami–Kolmogorov (JMAK) model or one of its variants, even though they are based on an isothermal assumption. As a result, during the nonisothermal (e.g. constant heating or cooling rate) crystallization of polymeric material, different sets of model parameters are required to describe crystallization at different rates, thereby increasing the total number of model parameters. In addition, due to the uncorrelated nature of these model parameters with the cooling or heating rate, accurate modeling at any intermediate condition is not possible. In the present work, these two limitations of the conventional approach have been eliminated by exhibiting the existence of a functional relationship between cooling or heating rate and effective activation energy during nonisothermal melt or cold crystallization in three linear aromatic polyesters. Furthermore, it has been shown that when the JMAK model is used in conjunction with this functional relationship, it is possible to precisely predict the experimental nonisothermal melt or cold crystallization kinetics at any linear cooling or heating rate with a single set of model parameters.     

Kinetics of thermal degradation of poly(-caprolactone)

The thermal degradation/modification dynamics of poly(-caprolactone) (PCL) was investigated in a thermogravimetric analyzer under non-isothermal and isothermal conditions. The time evolution of the molecular weight distribution during degradation was studied using gel permeation chromatography. Experimental molecular weight evolution and weight loss profile were modeled using continuous distribution kinetics. The degradation exhibited distinctly different behavior under non-isothermal and isothermal heating. Under non-isothermal heating, the mass of the polymer remained constant at initial stages with rapid degradation at longer times. The Friedman and Chang methods of analysis showed a 3-fold change (from 18 to 55–62 kcal mol⁻¹) in the activation energy from low temperatures to high temperatures during degradation. This suggested the governing mechanism changes during degradation and was explained using two parallel mechanisms (random chain scission and specific chain end scission) without invoking the sequential reaction mechanisms. Under isothermal heating, the polymer degraded by pure unzipping of specific products from the chain end.     

Mediated action in teachers’ discussions about mathematics tasks

This paper presents analyses of teachers?? discussions within mathematics teaching developmental research projects, taking mediation as the central construct. The relations in the so-called ??didactic triangle?? form the basic framework for the analysis of two episodes in which upper secondary school teachers discuss and prepare tasks for classroom use. The analysis leads to the suggestion that the focus on tasks places an emphasis on the task as object and its resolution as goal; mathematics has the role of a mediating artefact. Subject content in the didactic triangle is thus displaced by the task and learning mathematics may be relegated to a subordinate position.