233 research outputs found
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
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
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 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
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
A Phenomenological Exploration of the Childfree Choice in a Sample of Australian Women
Choosing not to have children is considered a deviation from cultural norms, particularly the dominant pronatalist discourse; this is especially so for women. However, little research has documented the experience of Australian women who have consciously chosen to remain childless. Ten voluntarily childfree women participated in unstructured interviews about their choice and its ramifications. The data analysis revealed three broad themes – the experiences and processes of making the choice; the ongoing effects of their choice, ranging from support and acceptance to pressure and discrimination; and no regret as the women described engaging in meaningful, generative activities that contributed to society
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