Transition through mathematical tasks

The transition to university level mathematics is often problematic for students. Clark & Lovric (2008) have written about some of the differences between mathematics at school and at university, including the type of mathematics taught and the way mathematics is taught. Students at this stage also have to contend with social and cultural changes. As part of a project on task design, ten first year students at two different universities in Ireland were interviewed. In this paper, we will discuss their experiences of mathematics at school and university. In particular, we will consider the differences in the types of mathematical tasks encountered at both levels and the students' views of the influences of such tasks

Critical evaluation and design of mathematics tasks: pre-service teachers

This paper reports on a research project undertaken with a group (n=19) of Irish pre-service student teachers (PSTs) during the third year of a four-year undergraduate education course. A series of workshops were carried out on the critical evaluation and design of mathematics tasks. The research is presented as a case study using mixed methods to gather data. Through critically evaluating and designing mathematics tasks PSTs developed knowledge of cognitive demand, pedagogical design capacity and showed evidence of developing curriculum-making competences. The research highlights the need for PSTs to work together on evaluating and designing tasks

Better maths adds up to a stronger economy

Abstract included in tex

The use of mathematical tasks to develop mathematical thinking skills in undergraduate calculus courses – a pilot study

Mathematical thinking is difficult to define precisely but most authors agree that the following are important aspects of it: conjecturing, reasoning and proving, making connections, abstraction, generalization and specialization. In order to develop mathematically, it is necessary for learners of mathematics not only to master new mathematical content but also to develop these skills. However, undergraduate courses in Mathematics tend to be described in terms of the mathematical content and techniques students should master and theorems they should be able to prove. It would appear from such descriptions that students are expected to pick up the skills of (advanced) mathematical thinking as a by-product. Moreover, recent studies have shown that many sets of mathematical tasks produced for students at the secondary-tertiary transition emphasize lower level skills, such as memorization and the routine application of algorithms or procedures. In this paper we will consider some suggestions from the literature as to how mathematical thinking might be specifically fostered in students, through the use of different types of mathematical tasks. Efforts were made to interpret these recommendations in the context of a first undergraduate course in Calculus, on which large numbers of students may be enrolled. This itself constrains to some extent the activities in which the teachers and learners can engage. The tasks referred to here are set as homework problems on which students may work individually or collaboratively. We will report preliminary feedback from the students with whom such tasks were trialled, describing the students’ reactions to these types of tasks and their understanding of the purposes of the tasks

Mathematical Thinking and Task Design

Mathematical thinking is difficult to define precisely but most authors agree that the following are important aspects of it: conjecturing, reasoning and proving, abstraction, generalization and specialization. However, recent studies have shown that many sets of mathematical tasks produced emphasize lower level skills, such as memorization and the routine application of algorithms or procedures. In this paper we survey the literature on the design and use of tasks that aim to encourage higher level aspects of mathematical thinking in learners of mathematics at all levels. The frameworks presented here aim to guide task designers when writing a set of exercises

Better maths adds up to a stronger economy

Abstract included in tex

Weighted Trudinger-type inequalities

We establish sharp inequalities of Trudinger-type (with non-standard Young functions) on general domains Ω with respect to measures ν, μ. Our results are new even when is a Euclidean ball, and ν, μ are defined in terms of powers of distance to the boundary. In the Euclidean case, sharpness results are proved for the Young functions involved and the class of domains considered. New characterizations of QHBC domains are given

The use of unfamiliar tasks in first year calculus courses to aid the transition from school to university mathematics

Research has shown that mathematics courses at university often focus more on conceptual understanding than those at secondary school (Clark & Lovric, 2008). Moreover, the literature reports that the types of tasks assigned to students affect their learning. A project was undertaken by the authors in which tasks were designed and presented to first-year undergraduate Calculus students with the aim of promoting conceptual understanding and developing mathematical thinking skills. Here we present data from interviews with five students; they reported an increased emphasis on conceptual understanding at university, and found the tasks assigned beneficial in the development of conceptual understanding. We suggest that unfamiliar tasks are useful in the transition from school to university mathematics

The Development of Mathematics Resources

In this paper we give a brief overview of the ongoing development of Re-usable Learning Objects in mathematics and statistics at the National University of Ireland Maynooth. We focus on the materials that are being developed in collaboration with other third level institutions and supported by the NDLR. We briefly discuss how the topics were selected, the type of resources being developed, the technologies being used and how the resources can be integrated with the wide range of existing materials available. The resources are designed to complement, not replace traditional methods of teaching

"The backwards ones?" - Undergraduate students' reactions and approaches to example generation exercises

As part of a project exploring the design and use of mathematical tasks to promote conceptual understanding of Calculus concepts, first year undergraduate students were assigned homework problems which required them to use various processes including generalising, conjecturing, evaluating statements, analysing reasoning and generating examples. In subsequent interviews with five students, a number of them spontaneously referred to the example generation problems posed as being the "backwards ones" or requiring them to work backwards as well as forwards. In this paper, we will report on the students' reactions to a particular example generation exercise, the strategies they adopted in an effort to solve such problems, and what they feel they learnt in the process