Normalising geometrical figures: dynamic manipulation and construction of meanings for ratio and proportion

Enlarging-shrinking geometrical figures by 13 year-olds is studied during the implementation of proportional geometric tasks in the classroom. Students worked in groups of two using ‘Turtleworlds’, a piece of geometrical construction software which combines symbolic notation, through a programming language, with dynamic manipulation of geometrical objects by dragging on sliders representing variable values. In this paper we study the students’ normalising activity, as they use this kind of dynamic manipulation to modify ‘buggy’ geometrical figures while developing meanings for ratio and proportion. We describe students’ normative actions in terms of four distinct Dynamic Manipulation Schemes (Reconnaissance, Correlation, Testing, Verification). We discuss the potential of dragging for mathematical insight in this particular computational environment, as well as the purposeful nature of the task which sets up possibilities for students to appreciate the utility of proportional relationships.     

An Experiment in Teaching Elementary Ideas of Vectors

For a long time schools have been finding it difficult to appoint sufficient numbers of teachers of mathematics of the quality they would like. Qualifications of the teachers of arts subjects in a school are usually a good deal higher than those teaching sciences or mathematics. At the same time a revolution in the syllabus is taking place, and although the ‘Modern’ Syllabus is being encouraged there are very few competent teachers of it available.

In many schools the choice of a new textbook is handicapped by a lack of knowledge. This project, under the auspices of CAMET, is an attempt to provide a remedy.     

Understanding the hierarchical classification of quadrilaterals through the ordered relation according to diagonal properties

This article's aim is to suggest a supplementary learning environment to understand the hierarchical classification of quadrilaterals for high school or higher degree learners. Three diagonal properties, ‘being congruent’, ‘being perpendicular’ and ‘dividing each other in particular ratio,’ and all possible combinations of these properties, were used to construct the quadrilaterals in a dynamic geometry environment. According to the diagonal properties, 15 quadrilaterals could be constructed and an order relation was constituted on 16 quadrilaterals including the quadrilateral that did not have any diagonal property. The definition of order relation is ‘any quadrilateral Q_i is included by another quadrilateral Q_j, if and only if Q_i has all diagonal properties of Q_j.’ According to this relation, an ordered relation diagram was created, and it was found that this relation was not well ordered. After the dynamic geometry construction of each quadrilateral, observations about the diagonal properties of special quadrilaterals were noted. Furthermore, the conditions under which a quadrilateral can be concave are examined. This alternative approach to the construction of quadrilaterals provided an opportunity to define quadrilaterals with more economical and less confusing way than using angle and side properties. For example, ‘a Kite is a quadrilateral whose diagonals are perpendicular and at least one of the diagonals bisects the other’ and ‘a Trapezoid is a quadrilateral whose diagonals divide each other in same ratio.’     

On the expectations of the mathematical knowledge of first-year undergraduates

In a recent survey Sutherland and Dewhurst itemized the extent to which various types of departments expected selected mathematical topics in the background of incoming undergraduates. Using the results of diagnostic tests it is possible to estimate the actual capability of students with given prior qualifications to provide a ‘probable preparedness’ in specific topics. This paper compares expectations with probable preparedness in a range of topics. It is found that while some departments have realistic expectations about the background of their incoming students, others seem over-optimistic about the mathematical knowledge of today's students. Sometimes, topics that the department actually requires must in fact be covered ab initio with the students that they recruit.     

Convergence for Fourier series solutions of the forced harmonic oscillator

Harmonic oscillator equations of the form ÿ + ?²y = h(t) where ? is a real constant and h(t) is a continuous, piecewise smooth, periodic ‘forcing’ function are considered. The exact solution, obtained through the Laplace transform is cumbersome to handle over long t intervals, and thus solving ‘term-by-term’ by replacing h(t) by its Fourier series is an attractive and accurate alternative. But this solution is an infinite series involving sums of sine and cosine terms, and thus one should worry about convergence of a solution in this form. In the article, it is shown that such a series solution indeed converges uniformly over the entire real line and is twice continuously differentiable, the derivatives being calculated ‘term-by-term’. Only results commonly available in the undergraduate literature are used to verify this and in so doing, a non-trivial application of these results is given. Also included are some interesting problems suitable for undergraduate research.     

A tale of mathematical myth-making: E T Bell and the ‘arithmetization of algebra’

Throughout E T Bell’s writings on mathematics, both those aimed at other mathematicians and those for a popular audience, we find him endeavouring to promote abstract algebra generally, and the postulational method in particular. Bell evidently felt that the adoption of the latter approach to algebra (a process that he termed the ‘arithmetization of algebra’) would lend the subject something akin to the level of rigour that analysis had achieved in the nineteenth century. However, despite promoting this point of view, it is not so much in evidence in Bell’s own mathematical work. I offer an explanation for this apparent contradiction in terms of Bell’s infamous penchant for mathematical ‘myth-making’.     

Unique factorization and the fundamental theorem of arithmetic

The fundamental theorem of arithmetic is one of those topics in mathematics that somehow ‘falls through the cracks’ in a student's education. When asked to state this theorem, those few students who are willing to give it a try (most have no idea of its content) will say something like ‘every natural number can be broken down into a product of primes’. The fact that this breakdown always results in the same primes is viewed as ‘obvious’. The purpose of this paper is to illustrate with a number of examples that the ‘Unique Factorization Property’ is a rare property and the fact that the natural numbers possess this property is ‘fundamental’ to our understanding of this number system.     

New opportunities for encouraging higher level mathematical learning by creative use of emerging computer aided assessment

This article defines ‘higher level mathematical skills’ and details an important class: that of constructing instances of mathematical objects satisfying certain properties. Comment is made on the frequency of higher level tasks in undergraduate work. We explain how such questions may be assessed in practice without the imposition on staff of an onerous marking load. Included are examples which have been implemented on a free computer aided assessment system. Lastly we report an investigation of students’ reactions to these questions and discuss their design and impact.     

A brief historical introduction to solitons and the inverse scattering transform–a vision of Scott Russell

The objective of this study was to evaluate biomathtutor by (i) investigating the impact of biomathtutor on the mathematics skills competencies of bioscience undergraduates, and (ii) assessing students’ and tutors’ reactions to biomathtutor, identifying whether and how tutors might integrate it into their curricula and blend it with more traditional teaching practices to enhance their students’ learning experiences. A multi-method approach was adopted in which a quasi-experiment and non-experimental evaluation of biomathtutor were used to collect both quantitative and qualitative data, using mathematics tests, questionnaires, tutor interviews and student focus groups. Eighty-nine bioscience undergraduates and eight tutors participated in the study. A comparison of student performance in the quasi-experiment, which adopted a pre-test-intervention-post-test methodology, revealed no significant difference between pre-test and post-test scores for either the ‘control’ group (no intervention) or for any of the mathematics learning support interventions used, including biomathtutor. Despite the limitations of the quasi-experiment which are discussed, tutors’ and their students’ reactions towards biomathtutor were very positive, with both groups agreeing that biomathtutor represents a very well designed and useful learning resource that has a valuable role to play in supporting mathematics learning within bioscience curricula. Students felt that using biomathtutor had helped them acquire new biological and mathematical knowledge and had increased their competence and confidence in mathematics, with many students confirming that they would use biomathtutor again. Tutors felt it would be useful to embed biomathtutor, where possible, into their curricula, perhaps linking it to assessment strategies or integrating it with their current more traditional teaching practices. Students indicated that they too would like to see biomathtutor embedded within their curricula, primarily because it would motivate them to use the resource. Modifications to biomathtutor, which may need to be considered in light of any potential further development of this resource, are discussed.     

Number concepts with Number Worlds : thickening understandings

This study focuses on the nature of preservice elementary school teachers’ understandings of several concepts in elementary number theory that are evoked by a computer-based microworld called Number Worlds. In particular, the focus is on the concepts of factor, multiple and prime number. The notion of ‘thickness’ is examined with respect to understanding mathematical concepts and alludes to various attributes of Number Worlds that may be responsible for provoking different layers and depths of understanding, termed ‘thick understanding’.     

Axioms for a fuzzy measure of inequality

It has been argued that the concept of inequality is inherently imprecise. A difficulty with standard inequality measures is that they generally make no allowances for this, and when they do, it is by dropping the ‘completeness’ axiom in ranking social states (e.g. the Lorenz criterion). It is argued here that the erring axiom is not ‘completeness’ but ‘exactness’ which, being implicit, tends to escape notice. A fuzzy measure of inequality, along with a set of necessary and sufficient axioms, is established. The new measure has several attractive properttes: It allows for tentative judgements and doubts. It is easy to interpret and compute, and the Gini ranking turns out to be a nearest exact approximation of it.     

Mathematics,Science of Possibility ‐‐ A Critical Review

This paper supports the view that search for the applicative standpoint in mathematical education has yet to be sufficiently exploited. Mathematical propositions are considered to have both an ‘internal’ and ‘external’ role and thus a system of ‘potential models’ is evolved. Despite the stress on the applicative nature of the subject it is argued in conclusion that the position in the body of the paper is compatible with the synthetic apriority of mathematics.

     

Stability and Shapley value for an n-persons fuzzy game

The aim of the paper is to explain new concepts of solutions for n-persons fuzzy games. Precisely, it contains new definitions for ‘core’ and ‘Shapley value’ in the case of the n-persons fuzzy games. The basic mathematical results contained in the paper are these which assert the consistency of the ‘core’ and of the ‘Shapley value’. It is proved that the core (defined in the paper) is consistent for any n-persons fuzzy game and that the Shapley values exists and it is unique for any fuzzy game with proportional values.     

Enactment of school mathematics curriculum in Singapore: whither research!

The official curriculum for mathematics in Singapore schools is based on a framework that has mathematical problem solving as its primary goal. It is detailed and one may say that the gap between the designated curriculum and teacher intended curriculum is often very narrow. This is so as the main source of instructional materials is textbooks which are very closely aligned with the official national curriculum. There is a dearth of research on the enactment of the curriculum in Singapore schools, with the few research studies done so far appearing to cover only a narrow focus. The author’s view is that, even though only a few such studies have been published, schools have always been engaged in small-scale investigations, the findings of which are necessary to guide decisions on matters related to choice of textbooks and pedagogies for improved student learning. Considering all the published research and the investigative work undertaken by educators in Singapore, it may be said that the conceptual model proposed by Remillard and Heck is rigorous. In addition, the issues in this particular issue of ZDM offer educators, both classroom teachers and others, very good perspectives for research on the enactment of the school mathematics curriculum.     

Mathematical under-preparedness: the influence of the pre-tertiary mathematics experience on students’ ability to make a successful transition to tertiary level mathematics courses in Ireland

Internationally, the consequences of the ‘Mathematics problem’ are a source of concern for the education sector and governments alike. Growing consensus exists that the inability of students to successfully make the transition to tertiary level mathematics education lies in the substantial mismatch between the nature of entrants’ pre-tertiary mathematical experiences and subsequent tertiary level mathematics-intensive courses. This paper reports on an Irish study that focuses on the pre-tertiary mathematics experience of entering students and examined its influence on students’ ability to make a successful transition to tertiary level mathematics. Brousseau's ‘didactical contract’ is used as an intellectual tool to uncover and describe the contract that exists in two case mathematics classrooms in Irish upper secondary schools (Senior Cycle). Although the authors are professional mathematics educators and well informed about classroom practice in Ireland, they were genuinely surprised by the very restrictive nature of this contract and the damaging consequences for students’ future mathematical education.     

Choosing more mathematics: happiness through work?

This paper examines how A-level students construct relationships between work and happiness in their accounts of choosing mathematics and further mathematics A-level. I develop a theoretical framework that positions work and happiness as opposed, managed and working on the self and use this to examine students' dual engagement with individual practices of the self and institutional practices of school mathematics. Interviews with students acknowledge four imperatives that they use as discursive resources to position themselves as successful/unsuccessful students: you have to work, you have to not work, you have to be happy, you have to work at being happy. Tensions in these positions lead students to rework their identities or drop further mathematics. I then identify the practices of mathematics teaching that students use to explain un/happiness in work, and show how dependable mathematics and working together are constructed as ‘happy objects’ for students, who develop strategies for claiming control over these shapers of happiness.     

A new theoretical approach to sample problems and deductive reasoning in Sanskrit mathematical texts

Geometry is the earliest recorded branch of Indian mathematics. Mathematics of the Vedic period consists of those geometric techniques needed for the construction of the altars and fire-places described by the priestly hereditary class for the performance of their rites. The link between geometry and ritual suggests that mathematical accuracy was considered of the utmost importance in this context. The study of rational figures in the Sanskrit work Ga?ita Sāra Sa?graha, to which Mahāvīrācārya, a ninth-century ce Jaina mathematician, dedicates a special treatment, reveals striking parallelism with the earlier geometry developed in connection with the Vedic sacrifice. Mahāvīra makes extensive use of the udde?aka or ‘sample problem’, and I suggest a new way of interpreting the udde?aka as a significant device for constructing an ‘actual proof’ which validates and links a mathematical rule to its unmentioned premises and provides a system of knowledge based on deductive syllogism.     

Relative arithmetic

In nonstandard mathematics, the predicate ‘x is standard’ is fundamental. Recently, ‘relative’ or ‘stratified’ nonstandard theories have been developed in which this predicate is replaced with ‘x is y ‐standard’. Thus, objects are not (non)standard in an absolute sense, but (non)standard relative to other objects and there is a whole stratified universe of ‘levels’ or ‘degrees’ of standardness. Here, we study stratified nonstandard arithmetic and the related transfer principle. Using the latter, we obtain the ‘reduction theorem’ which states that arithmetical formulas can be reduced to equivalent bounded formulas. Surprisingly, the reduction theorem is also equivalent to the transfer principle. As applications, we obtain a truth definition for arithmetical sentences and we formalize Nelson's notion of impredicativity (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

It's not the done thing: social norms governing students’ passive behaviour in undergraduate mathematics lectures

Students often play a passive role in large-scale lectures in undergraduate mathematics courses: they observe the lecturer demonstrate mathematical procedures, but they rarely engage in authentic mathematical activity themselves. This study uses semi-structured interviews of undergraduate students to investigate the implicit and explicit social norms and expectations that influence students to maintain their passive roles during lectures. Students were aware that their passivity was influenced by social norms, but perceived these norms as necessary for allowing the lecturer to get through the content in the allotted lecture time, while enabling students to avoid being publicly embarrassed in the lecture. However, the students appreciated opportunities to work on examples in small groups during lectures. We argue that the success of small group interactions during large-scale lectures depends on students and lecturers establishing supportive social norms, and adjusting their lecture goals from ‘covering the content’ to ‘developing mathematical understanding’.