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

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

We Never Did This: A Framework for Measuring Novelty of Tasks in Mathematics Textbooks

Textbooks are an important resource in Irish mathematics classrooms, which can have both a positive and negative impact on teaching and learning. The Project Maths initiative is prompting teachers and students to cross boundaries and interact with mathematics in ways that had not been considered previously. Publishers have produced new texts in response to the expectations of the revised curriculum and the changed needs of the classroom. This paper presents a framework to consider the degree of novelty presented in tasks found in mathematics textbooks. Novelty is something that has been referred to, yet not addressed directly, in existing frameworks for the analysis of mathematical tasks. A particular strength of our framework is that it takes into account the experience of the solver, as opposed to just focusing on how a task has been structured. Sections of textbooks currently being used in Irish classrooms at second level have been analysed using this framework and the results indicate that while all textbooks incorporate a significant level of novelty, there is still room for more novel tasks to be included

A consideration of familiarity in Irish mathematics examinations

In this paper, we focus on the idea of familiarity and the differing levels of it that are apparent in Irish mathematics end of school state examination questions. We provide the results of an analysis of recent Higher Level and Ordinary Level Leaving Certificate mathematics examinations in terms of familiarity. Our findings do not indicate any particular recurring pattern evident in the levels of familiarity measured but generally not more than 20% of marks are allocated to unfamiliar questions

Measuring Students’ Persistence on Unfamiliar Mathematical Tasks

182 students responded to a number of Likert-scale items regarding their persistence on mathematical tasks. Rasch analysis was then used to construct a measure of persistence from their responses and to assign persistence scores to each student. The same students, all of whom were enrolled in the first year of a third-level programme, also completed a 30-minute test involving mathematics items from PISA. The latter, although commensurate with the students' level of mathematical education, represented largely unfamiliar tasks to the students and required the transfer of previously learned mathematical knowledge and skills to a new context. The students' performance on these items was used to construct a second measure of persistence. Initial findings indicate that although the correlation between the self-reporting measure and the evidence provided by the PISA-type test is statistically significant, there are some inconsistencies between the self-reported data and observed behaviour

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 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 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

Dilemmas experienced in lecturing undergraduate calculus

We consider a set of accounts written by two university lecturers describing incidents that took place during their first-year Calculus modules. Analysis of these accounts revealed that the lecturers had to make some difficult decisions while teaching. These situations sometimes involved choices between two or more alternatives each of which had disadvantages. We labelled these choices ‘dilemmas’. Here we present and discuss the three most common types of dilemma evident from our data: namely, balancing good practice in teaching with students’ feeling of discomfort; balancing the needs of students with different backgrounds; balancing time constraints and active participation by students