Problem set 3

The article presents several mathematical problems concerning Euclidean geometry

Higher order reasoning produced in proof construction: how well do secondary school students explain and write mathematical proofs?

The culture of mathematical explanations and writings based on conceptual understanding in proof construction is on the focus of the paper. We explore students’ attempts to explain construction of mathematical proofs after reading them and write mathematical proofs after working out their own constructions. Two examples of proofs, by induction and by contradiction, are discussed in detail to highlight students’ difficulties in proving and possible ways for their resolving

Using materials from the history of mathematics in discovery-based learning

This paper reports on attempt to integrate history of mathematics in discovery-based learning using technology. Theoretical grounding of the idea is discussed. An exploratory environment on triangle geometry is described. It is designed to support and motivate students' activities in learning through inquiry. Conjectures about properties of Lemoine point and Simson line are produced and proved by students using e-learning textbook

Inquiry activities in a classroom: extra-logical processes of illumination vs logical process of deductive and inductive reasoning. A case study

The paper presents results of the research, which was focused on studying students’ inquiry work from a psychological point of view. Inquiry activities of students in a classroom were analysed through the evaluation of the character of these activities within learning process with respect to mathematician’s research practice. A process of learning mathematical discovery was considered in detail as a part of inquiry activities of students in a classroom

Problem set 4

This problem set includes a selection of probability problems. Probability theory started essentially as an empirical science and developed on the mathematical side later. Pascal and Fermat were among those who suggested the basics of probability theory as we now know it. The problems featured in this issue demonstrate diversity of ideas and different concepts of probability, in particular, they refer to Laplace and Bernoulli models as well as to geometric probability

On development of students' abilities in problem posing: a case of plan geometry

The paper reports on results of the training, which was aimed at the formation of skills and habits of posing problems of different complexity levels in the course of plane geometry using the drawing as the primary source for students’ activities in problem posing process. The paper describes and analyses some tasks, which were developed to enable the researchers to look into the thinking processes used by students when they are involved in problem posing activities. The author stresses role of students’ skills to inquiry work and important features of the use of technology in the different stages of the training

Using the history of mathematics for mentoring gifted students: Notes for teachers

The paper presents a theoretical framework, methodology and practical implications for the work with gifted students using history of mathematics. A teaching-learning model, where history of mathematics is integrated in problem-solving activities, is described. Didactical material based on the concepts of triangle geometry is given in the scope of this model. A beautiful and intriguing piece of geometry – the Lemoine point is the focus of consideration. Its properties are investigated through appropriately designed activities for students. Different examples show the importance of history of mathematics for the development of students’ mathematical thinking

About a constructivist approach for stimulating students' thinking to produce conjectures and their proving in active learning of geometry

The paper describes processes that might lead secondary school students to produce conjectures in a plane geometry. It highlights relationship between conjecturing and proving. The author attempts to construct a teaching-learning environment proposing activities of observation and exploration of key concepts in geometry favouring the production of conjectures and providing motivation for the successive phase of validation, through refutations and proofs. Supporting didactic materials are built up in a way to introduce production of conjectures as a meaningful activity to students

Problem set 10

This article introduces a regular problem section in the Australian Senior Mathematics Journal. It notes that the section aims to give readers an opportunity to exchange interesting mathematical problems and solutions. It adds that the set in each issue will consist of up to five problems

Network of Hydrogen Bonds as a Medium for DNA Interaction in Solvents

We suggest that the DNA molecules could form the cholesteric phase owing to an interaction mediated by the network of the hydrogen bonds (H-network) in the solvent. The model admits of the dependence of the optical activity of the solution on the concentration of the PEG, and the change in the sense of the cholesteric twist due to the intercalation by the daunomicyn. Using the experimental data for the cholesteric phase of the DNA dispersion, we obtain a rough estimate for the energy given by our model, and show that it should be taken into account as well as the energy due to the steric repulsion, van der Waals, and electrostatic forces, generally used for studying the DNA molecules. The elastic constant of the H-network generating the interaction between the DNA molecules is determined by the energy due to the proton's vibration in the hydrogen bonds.Comment: 12 pages, Latex, 2 figure