Probing interactions in exploratory teaching: a case study

This study investigates an exploratory teaching style used in an undergraduate geometry course to help students identify an ellipse. We attempt to probe beneath the surface of exploration to understand how the actions of teachers can contribute to developing students’ competence in justifying an ellipse. We analyse the complex interactions between student, content, and teacher, and discuss explicit pedagogical strategies that help students develop a higher level of geometric reasoning. The findings indicate that students engaged in guided explorations by the teacher and in group discussions with peers were able to identify an ellipse and justify their reasoning.     

The activity structure of technology-based mathematics lessons: a case study of three teachers in English secondary schools

ABSTRACT

This article examines patterns of classroom organisation and interaction associated with the use of a particular type of digital technology – the dynamic software GeoGebra – in the lessons of an opportunity sample of three English secondary-school mathematics teachers. The concept of activity structure is used to organise this study, further informed by the concept of instrumental orchestration. While the case study analysis identifies structures already reported in those earlier papers, it also draws attention to the prevalence of a Predict-and-test format in tasks carried out by students at the computer. This study also shows how synthesising the activity structure and instrumental orchestration frameworks may be productive.     

Group algebras with units satisfying a group identity II

We classify group algebras of torsion groups over a field of characteristic with units satisfying a group identity.

     

Exceptional units in a family of quartic number fields

We determine all exceptional units among the elements of certain groups of units in quartic number fields. These groups arise from a one-parameter family of polynomials with two real roots.

     

A case study of pedagogy of mathematics support tutors without a background in mathematics education

This study investigates the pedagogical skills and knowledge of three tertiary-level mathematics support tutors in a large group classroom setting. This is achieved through the use of video analysis and a theoretical framework comprising Rowland's Knowledge Quartet and general pedagogical knowledge. The study reports on the findings in relation to these tutors’ provision of mathematics support to first and second year undergraduate engineering students and second year undergraduate science students. It was found that tutors are lacking in various pedagogical skills which are needed for high-quality learning amongst service mathematics students (e.g. engineering/science/technology students), a demographic which have low levels of mathematics upon entering university. Tutors teach their support classes in a very fast didactic way with minimal opportunities for students to ask questions or to attempt problems. It was also found that this teaching method is even more so exaggerated in mandatory departmental mathematics tutorials that students take as part of their mathematics studies at tertiary level. The implications of the findings on mathematics tutor training at tertiary level are also discussed.     

Retention of differential and integral calculus: a case study of a university student in physical chemistry

This paper reports a study on retention of differential and integral calculus concepts of a second-year student of physical chemistry at a Danish university. The focus was on what knowledge the student retained 14 months after the course and on what effect beliefs about mathematics had on the retention. We argue that if a student can quickly reconstruct the knowledge, given a few hints, this is just as good as retention. The study was conducted using a mixed method approach investigating students’ knowledge in three worlds of mathematics. The results showed that the student had a very low retention of concepts, even after hints. However, after completing the calculus course, the student had successfully used calculus in a physical chemistry study programme. Hence, using calculus in new contexts does not in itself strengthen the original calculus learnt; they appeared as disjoint bodies of knowledge.     

The MACSI summer school: a case study in outreach in mathematics

To encourage the study of mathematics in Ireland, the Mathematics Applications Consortium for Science and Industry (MACSI) organizes a summer school once a year. The different aspects of this summer school are presented. Students are selected depending on their motivation, academic abilities, gender and geographical origins. Instruction and supervision is provided by academics, post-doctoral fellows and post-graduate students. The teaching programme evolves every year and reflects the interests of the people involved. Feedback from participants has been almost uniformly positive. Students favour interactive sessions and enjoy the residential aspect of the summer school. Food and accommodation are however the most costly aspects of this summer school. In this respect the support of Science Foundation Ireland has been invaluable.     

What is the meaning of “meaning”? A case study from graphing

This article raises questions about the meaning of “meaning,” which often is understood in terms of the referent or interpretant (sense) of mathematical signs. In this study, which uses data from an interview study with scientists who were asked to read graphs from their own work, a phenomenologically grounded approach is proposed with the intent to contribute toward a more appropriate theory of meaning. I argue that graphs accrue to meaning — which always arises from already existing, existential understanding of the world more generally and the workplace in particular — rather than having or receiving meaning from some place or person. We experience graphs as meaningful exactly at the moment when they are integral to a world that we already understand in an existential but never completely determinable way.     

Reinventing solutions to systems of linear differential equations: A case of emergent models involving analytic expressions

An enduring challenge in mathematics education is to create learning environments in which students generate, refine, and extend their intuitive and informal ways of reasoning to more sophisticated and formal ways of reasoning. Pressing concerns for research, therefore, are to detail students’ progressively sophisticated ways of reasoning and instructional design heuristics that can facilitate this process. In this article we analyze the case of student reasoning with analytic expressions as they reinvent solutions to systems of two differential equations. The significance of this work is twofold: it includes an elaboration of the Realistic Mathematics Education instructional design heuristic of emergent models to the undergraduate setting in which symbolic expressions play a prominent role, and it offers teachers insight into student thinking by highlighting qualitatively different ways that students reason proportionally in relation to this instructional design heuristic.     

Development of a mathematical ability test: a validity and reliability study

The identification of talented students accurately at an early age and the adaptation of the education provided to the students depending on their abilities are of great importance for the future of the countries. In this regard, this study aims to develop a mathematical ability test for the identification of the mathematical abilities of students and the determination of the relationships between the structure of abilities and these structures. Furthermore, this study adopts test development processes. A structure consisting of the factors of quantitative ability, causal ability, inductive/deductive reasoning ability, qualitative ability and spatial ability has been obtained following this study. The fit indices of the finalized version of the mathematical ability test of 24 items indicate the suitability of the test.     

Incentivizing advanced mathematics study at upper secondary level: the case of bonus points in Ireland

Secondary level mathematics education in Ireland has recently experienced a period of significant change with the introduction of new curricula and the addition of an incentive to study upper secondary mathematics at the most advanced level (Higher Level). This incentive, typically referred to as ‘bonus points’, appears to have aided a significant increase in the number of students studying upper secondary mathematics at Higher Level. However, thematic analysis of interviews with experienced upper secondary mathematics examiners and exploration of mathematics diagnostic test data outlined in this paper suggest that the difficulty of the Higher Level upper secondary mathematics final examination in Ireland has reduced since the introduction of the bonus points initiative. The sharp increase in students attempting this examination coupled with a policy of maintaining a consistent proportion of students achieving passing grades was identified as a key reason for this possible reduction in standards.     

Match twice and stitch: a new TSP tour construction heuristic

We present a new symmetric traveling salesman problem tour construction heuristic. Two sequential matchings yield a set of cycles over the given point set; these are then stitched to form a tour. Our method outperforms all previous tour construction methods, but is dominated by several tour improvement heuristics.     

The many facets of a definition: The case of periodicity

This paper was triggered by an authentic conversation between two mathematics teacher educators who debated whether a constant function is a periodic function, within the framework of a professional development program for secondary mathematics teachers. Their initial conversation led to deep mathematical and pedagogical musing surrounding mathematical definitions. In this paper, we present various aspects of a mathematical definition, including the role and nature of definitions in school mathematics, critical versus preferable features of a definition, and the arbitrariness underlying the choice of definition. We discuss the interplay between logical and pedagogical considerations with respect to definitions, drawing on the definition of a periodic function as an example.     

Portfolio construction on the Athens Stock Exchange: a multiobjective optimization approach

In this research article, our purpose is to propose a single-period multiobjective mixed-integer programming model for equity portfolio construction, in order to generate the Pareto optimal portfolios, using a variant of the well-known ε-constraint method. The decision maker's investment policy, i.e. constraints regarding the portfolio structure, is strongly taken into account. An illustrative application in the Athens Stock Exchange market is also presented.     

Using algorithmic thinking to design algorithms: The case of critical path analysis

Algorithmic thinking is emerging as an important competence in mathematics education, yet research appears to be lagging this shift in curricular focus. The aim of this generative study is to examine how students use the cognitive skills of algorithmic thinking to design algorithms. Task-based interviews were conducted with four pairs of Year 12 students (n = 8) to analyze how they used decomposition and abstraction to specify the projects, designed algorithms to solve scheduling problems by first devising fundamental operations and then using algorithmic concepts to account for complex and special cases of the problems, and tested and debugged their algorithms. A deductive-inductive analytical process was used to classify students’ responses according to the four cognitive skills to develop sets of subskills to describe how the students engaged these cognitive skills.     

A case for combinatorics: A research commentary

In this commentary, we make a case for the explicit inclusion of combinatorial topics in mathematics curricula, where it is currently essentially absent. We suggest ways in which researchers might inform the field’s understanding of combinatorics and its potential role in curricula. We reflect on five decades of research that has been conducted since a call by Kapur (1970) for a greater focus on combinatorics in mathematics education. Specifically, we discuss the following five assertions: 1) Combinatorics is accessible, 2) Combinatorics problems provide opportunities for rich mathematical thinking, 3) Combinatorics fosters desirable mathematical practices, 4) Combinatorics can contribute positively to issues of equity in mathematics education, and 5) Combinatorics is a natural domain in which to examine and develop computational thinking and activity. Ultimately, we make a case for the valuable and unique ways in which combinatorics might effectively be leveraged within K-16 curricula.     

Mathematical tasks,study approaches,and course grades in undergraduate mathematics: a year-by-year analysis

Students approach learning in different ways, depending on the experienced learning situation. A deep approach is geared toward long-term retention and conceptual change while a surface approach focuses on quickly acquiring knowledge for immediate use. These approaches ultimately affect the students’ academic outcomes. This study takes a cross-sectional look at the approaches to learning used by students from courses across all four years of undergraduate mathematics and analyses how these relate to the students’ grades. We find that deep learning correlates with grade in the first year and not in the upper years. Surficial learning has no correlation with grades in the first year and a strong negative correlation with grades in the upper years. Using Bloom's taxonomy, we argue that the nature of the tasks given to students is fundamentally different in lower and upper year courses. We find that first-year courses emphasize tasks that require only low-level cognitive processes. Upper year courses require higher level processes but, surprisingly, have a simultaneous greater emphasis on recall and understanding. These observations explain the differences in correlations between approaches to learning and course grades. We conclude with some concerns about the disconnect between first year and upper year mathematics courses and the effect this may have on students.     

Defining as a mathematical activity: A framework for characterizing progress from informal to more formal ways of reasoning

The purpose of this paper is to further the notion of defining as a mathematical activity by elaborating a framework that structures the role of defining in student progress from informal to more formal ways of reasoning. The framework is the result of a retrospective account of a significant learning experience that occurred in an undergraduate geometry course. The framework integrates the instructional design theory of Realistic Mathematics Education (RME) and distinctions between concept image and concept definition and offers other researchers and instructional designers a structured way to analyze or plan for the role of defining in students’ mathematical progress.     

Systemic risk assessment: a case study

Project Risk Registers have been used extensively for many years. However, they do not account for the interaction between risks, for example, the occurrence of one risk exacerbating other risks or portfolios of risks being more significant than the sum of the individual risks. This leads to the need to consider ‘risk systemicity’ as a part of risk analysis. This paper reports on a specific case for a large multinational project based organization, one that the authors had been involved with in the analysis of a number of projects that had massive cost overruns. Following these analyses the organization was persuaded of the importance of risk systemicity. The organization therefore engaged the authors to develop a ‘Risk Filter’. This filter is a tool for identifying areas of risk exposure on future projects and creating a framework for their investigation. The ‘Risk Filter’ is now used on all projects ever since its introduction; by the end of May 2003 it had been used by nine divisions, on over 60 major projects, and completed by 450 respondents. It is also used at several stages during the life of a project to aid in the risk assessment and management of each project, and contributes to a project database.