The article presents a theoretical analysis of the concept of reflection. The author argues in favour of the necessity of conceiving the concept of reflection in a broad sense, and not using the concept in the meaning of introspection. To grasp reflection in its complexity and as a core process in organisational learning it is necessary to…

A one-credit seminar course for junior-senior mathematics majors is described. The topic for the seminar is continued fractions, taught by a modified Moore Method, where the focus is on students creating their own mathematics. The students make conjectures after completing computational problems, then work on refining the needed hypotheses,…

In this paper, two classes of graphs of arbitrary order are described which contain unique Hamiltonian cycles. All the graphs have mean vertex degree greater than one quarter the order of the graph. The Hamiltonian cycles are detailed, their uniqueness proved and simple rules for the construction of the adjacency matrix of the graphs are given.…

This article describes the experiences of a group of primary-grade students and their teacher as they explore negative numbers over the course of two years. It is just one example of how a problem-solving environment contributes to discussion and understanding in mathematics. (Contains 3 figures.)

This article is about the contributions children make in clinical interviews. This issue is studied by re-analysing a selection of the empirical excerpts used by Piaget in his seminal book The Child's Conception of the World. The focus is on how children use language non-literally, and especially on how they use meta-communicative markers ("as…

With the introduction of the latest national curriculum, using and applying mathematics retained its attainment target within the level descriptions, but the programme of study was integrated into the programmes of study for curriculum content (number, shape, space, and measures). This brought it in line with the national numeracy strategy, which…

The normal equations discussed in this paper for a least-squares parabolic fit have a unique solution if and only if there are at least three different x-values in the observations. This requirement is satisfied by most real sets of quantitative observations. For particular data sets, the appropriateness of parabolic fits should be assessed with…

In the seventh century, around 650 A.D., the Indian mathematician Brahmagupta came up with a remarkable formula expressing the area E of a cyclic quadrilateral in terms of the lengths a, b, c, d of its sides. In his formula E = [square root](s-a)(s-b)(s-c)(s-d), s stands for the semiperimeter 1/2(a+b+c+d). The fact that Brahmagupta's formula is…

Adults and children often encounter situations where they have to make judgments about "how much" or "how long" or "how many." The significance of estimation as an ordinary, everyday, and natural aspect of measurement needs to be conveyed to students through their mathematical experiences. Many students, however, tend…

Older adults' preferred partners for collaborative everyday problem solving and the types of everyday problems solved alone and with others were examined in a sample of community dwelling older adults (N = 112, M age = 71.86 yrs., SD = 5.92 yrs.). Family members (i.e., spouses, adult children) were the most frequently nominated partners for…

Performing inquiry should enhance the intellectual development of students on numerous levels. We evaluate whether inquiry-based activities elicit inquiry behaviors by comparing these activities with traditional review sessions. We find that both inquiry-learning and teaching behaviors increase during inquiry-based activities, but that teachers…

The article describes how one Educational Psychology Service in the UK developed a service delivery based on self-organised learning (SOL). This model is linked to the paradigms and discourses within which educational psychology and special educational needs work. The work described here is dedicated to the memory of Brian Roberts, academic, close…

The main aim of this article is to discuss the attitudes to mathematics of students taking a basic mathematics course at a Swedish university, and to explore possible links between how well such students manage to solve tasks about limits of functions and their attitudes. Two groups, each of about a hundred students, were investigated using…

At the start of a freshman calculus course, many students conceive the classical definition of limit as the most problematic part of calculus. They not only find it difficult to understand, but also consider it of no use while solving most of the limit problems and therefore, skip it. This paper reformulates the rigorous definition of limit, which…

This article presents the results of research about students' and teachers' use of an interaction framework (THINK) to guide group communication about problem solving. Students who used the THINK framework demonstrated greater gains in problem-solving achievement than students who did not use the framework.