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This article presents several mathematics problems on the same theme that can be used in class or for independent study. It is very difficult to show students how mathematics research is done by mathematicians, so the author first introduces a problem that can be tackled solely by arithmetic and then gives an idea of how a mathematician might tackle such a problem. Additionally, attention is drawn to how students can engage in mathematical work, including experimenting, collecting data, simplifying problems, checking work, proving, extending, and generalising.

This is the third article in the "On the Shoulders of Giants" series that takes an outstanding mathematician, tells something of their life and work, and provides an item or two based on their achievements that can be used in a primary class. This article focuses on the work of Cheryl Praeger, with an activity based upon her work with mathematics and weaving.

In this paper, we list some of the areas of the Australian curriculum that have links with the concept of infinity. We do this in order to promote a discussion about what aspects of infinity should become familiar to primary teachers. From our viewpoint, infinity has connections with Algebra, Art, Geometry and Measurement, Probability, Science and Technology and is an essential ingredient in the teaching of mathematics in primary school.This work was first motivated by the concern that for many young children, infinity appears to be mysterious and mythical (see Pehkonen and Hannula 2006 based on a survey of 300 students aged from 11 to 14 years old), but there is no reason why this should be the case. Then, as we looked further into the curriculum, we saw that infinity was closely linked to many areas of both the primary and secondary curriculum and that an understanding of the concept of infinity could improve students' learning in a number of topics.We organise the paper by first considering the nature of infinity. In the four sections following infinity, we note that there are important parts of curriculum mathematics that are fundamentally affected by infinity in some way. In these sections, we discuss places where infinity and the collateral concept of convergence are critical for certain aspects of Geometry, Number (decimals and measurement), Algebra and Probability. These are followed by a discussion of the relevance of these to teaching and what understanding of infinity in these concepts might be worthwhile for teachers to know in order to help their students' search for knowledge and to gain a deeper understanding of specific topics in the curriculum.

With the impact of the COVID pandemic, many students have been learning mathematics online. This article shows how emojis can be used as a virtual manipulative in online and digital learning platforms.

Duncan Symons and Derek Holton discuss the different types of mathematical reasoning and what each of these might look like in the classroom. By suggesting language that can be used to describe the different methods of reasoning, they hope to provide teachers with the tools they need to better recognise and assess student reasoning.

The authors illustrate three basic types of reasoning used in mathematics by showing how they operate in practical and mathematical situations. The importance and function of the different types of reasoning in each situation is outlined. As a consequence the authors note that while introducing new techniques by example is good from a pedagogical viewpoint, several examples by themselves do not provide a proof that the techniques are correct. (Contains 3 figures and 1 footnote.)

Before we consider the German tertiary system, we review the education system and consider other relevant background details. We then concentrate on the tertiary system and observe that the mathematical enrolments are keeping up with the overall student enrolments. At the same time, the first year mathematics enrolments for women are greater than that for men, although more men are still studying mathematics at university. Finally, we note that the German economy seems to play a role in mathematics enrolments though not necessarily to its comparative detriment. (Contains 7 notes and 8 figures.)

In this article, we start by considering the relevant parts of the French educational system and the data relating to science at the end of secondary school and the early years of university. We show an increase, till 1995, and then a decline in the study of mathematics at baccalaureate and university level. However, our main conclusions relate to a decline in the study of all sciences, rather than just mathematics. The reason that we have to consider all sciences is because of the difficulty in getting accurate figures for mathematical enrolments at the university level. We do, however, have accurate figures for high school level mathematics enrolments. We consider the reasons for this downturn in enrolments, which appear to be largely political and social. It is also noted that there have been declines in the numbers of students in areas other than science. Finally, we show that for French tertiary students, well-paid secure jobs are at least as important as subject choice. This is dramatically illustrated by the numbers of applicants applying for competitive positions as secondary teachers. (Contains 8 figures and 9 notes.)

Here, we look at the enrolments of students in undergraduate mathematics courses in a number of countries. The data show various increases and decreases and we suggest some common reasons for the fluctuations. These include students' goals of a secure and well-paid job, government actions and the state of the economy in the country concerned. We consider several ways in which departments have successfully approached downturns in numbers by their interactions with students by introducing new teaching approaches, using technology and establishing mathematics centres. (Contains 5 figures.)

Here, we investigate what loci are produced when a square of side-length one is allowed to rotate around a square of side-length n, where n is a whole number. We find that if i = 1, 2, 3 or 4 (mod 4), the loci obtained for n [congruent to] i (mod 4) all have the same symmetry and we show how the perimeter of each class can be determined. We also analyse the case of rotation of a square of side-length one about a rectangle of dimensions m x n when m and n are whole numbers. A similar result is obtained for classes of rectangles modulo 4 to that obtained in the square case. Finally, we suggest that this is a useful investigation to undertake with secondary school pupils. We discuss how we have used this investigation with such a group.

This paper proposes an expanded conception of scaffolding with four key elements: (1) scaffolding agency--expert, reciprocal, and self-scaffolding; (2) scaffolding domain--conceptual and heuristic scaffolding; (3) the identification of self-scaffolding with metacognition; and (4) the identification of six zones of scaffolding activity; each zone distinguished by the matter under construction and the relative positioning of the participant(s) in the act of scaffolding. These key elements are illustrated with empirical examples drawn from a variety of research studies. Scaffolding, thus conceived, brings together several theoretical domains, and by situating metacognition within a framework derived from the social activity of scaffolding, a bridge is formed between the instructional support a teacher might provide and the learner's self-control of the learning process. With regard to instruction and the role of the teacher, it is the authors contention that a major object of instruction is the progressive relocation of scaffolding agency in the direction of the learner with the long-term goal of equipping the learner to take control of their own learning.

In a future that is likely to be increasingly dominated by new technology, it is important to consider what mathematics is and how it might be taught. This article discusses these and related issues in the context of both universities and schools.

Here we give an example of a problem that could be beneficially investigated by AS/A level students. It is a geometry problem that they can profitably tackle by geometric (especially geometry software) and algebraic means. Such problems naturally lead students to the need for proof--an essential part of mathematics that is often lacking in current courses. We encourage students and staff to send this journal the results of similar examples that they themselves have investigated.

Describes the Six Circle problem which consists of the numbers 1-6, six circles, and asks whether it is possible to put the numbers in the circles--which are configured in a triangle--so that the sums of the three numbers on either side are the same. (NB)