1,852 research outputs found
Proof in dynamic geometry environments
This article suggests that there is a range of evidence that working with dynamic geometry software affords students possibilities of access to theoretical mathematics, something that can be particularly elusive with other pedagogical tools. Yet the paper concludes that further research into the use dynamic geometry software to support the development of students’ mathematical thinking could usefully focus on the nature of the tasks students tackle, the form of teacher input, and the role of the classroom environment and culture
Decisions and disease: a mechanism for the evolution of cooperation
In numerous contexts, individuals may decide whether they take actions to
mitigate the spread of disease, or not. Mitigating the spread of disease
requires an individual to change their routine behaviours to benefit others,
resulting in a 'disease dilemma' similar to the seminal prisoner's dilemma. In
the classical prisoner's dilemma, evolutionary game dynamics predict that all
individuals evolve to 'defect.' We have discovered that when the rate of
cooperation within a population is directly linked to the rate of spread of the
disease, cooperation evolves under certain conditions. For diseases which do
not confer immunity to recovered individuals, if the time scale at which
individuals receive information is sufficiently rapid compared to the time
scale at which the disease spreads, then cooperation emerges. Moreover, in the
limit as mitigation measures become increasingly effective, the disease can be
controlled, and the rate of infections tends to zero. Our model is based on
theoretical mathematics and therefore unconstrained to any single context. For
example, the disease spreading model considered here could also be used to
describe social and group dynamics. In this sense, we may have discovered a
fundamental and novel mechanism for the evolution of cooperation in a broad
sense
Connecting Mathematics and the Applied Science of Energy Conservation
To effectively teach science in the elementary classroom, pre-service K-8 teachers need a basic understanding of the underlying concepts of physics, which demand a strong foundation in mathematics. Unfortunately, the depth of mathematics understanding of prospective elementary teachers has been a growing and serious concern for several decades. To overcome this challenge, a two-pronged attack was used in this study. First. students in mathematics courses were coupled with physical science courses by linking registration to ensure co-requisites were taken. This alone improved passing rates. Secondly, an energy conservation project was introduced in both classes that intimately tied the theoretical mathematics base knowledge to problems in physical science, energy efficiency, and household economics. These connections made the mathematics highly relevant to the students and improved both their theoretical understanding and their grades. Together, the two approaches of tying mathematics to physical science and applying mathematical skills to solving energy efficiency problems have shown to be extremely effective at improving student performance. This five-year study not only exhibited record improvements in student performance, but also can be easily replicated at other institutions experiencing similar challenges in training pre-service elementary school teachers
Applications of Flow Network Models in Finance
In this thesis we explore the applications of flow networks in practical problems in finance. After introducing basic definitions and background information, we first survey some known applications of flow networks in theoretical mathematics. We also briefly comment on their potential applications in the setting of financial flow networks. We then construct networks from practical financial flows and present the construction, reasoning, and known applications. Lastly, we show a design of financial flow networks that takes time into consideration and discuss its applications
The Informal Logic of Mathematical Proof
Informal logic is a method of argument analysis which is complementary to
that of formal logic, providing for the pragmatic treatment of features of
argumentation which cannot be reduced to logical form. The central claim of
this paper is that a more nuanced understanding of mathematical proof and
discovery may be achieved by paying attention to the aspects of mathematical
argumentation which can be captured by informal, rather than formal, logic. Two
accounts of argumentation are considered: the pioneering work of Stephen
Toulmin [The uses of argument, Cambridge University Press, 1958] and the more
recent studies of Douglas Walton, [e.g. The new dialectic: Conversational
contexts of argument, University of Toronto Press, 1998]. The focus of both of
these approaches has largely been restricted to natural language argumentation.
However, Walton's method in particular provides a fruitful analysis of
mathematical proof. He offers a contextual account of argumentational
strategies, distinguishing a variety of different types of dialogue in which
arguments may occur. This analysis represents many different fallacious or
otherwise illicit arguments as the deployment of strategies which are sometimes
admissible in contexts in which they are inadmissible. I argue that
mathematical proofs are deployed in a greater variety of types of dialogue than
has commonly been assumed. I proceed to show that many of the important
philosophical and pedagogical problems of mathematical proof arise from a
failure to make explicit the type of dialogue in which the proof is introduced.Comment: 14 pages, 1 figure, 3 tables. Forthcoming in Perspectives on
Mathematical Practices: Proceedings of the Brussels PMP2002 Conference
(Logic, Epistemology and the Unity of the Sciences Series), J. P. Van
Bendegem & B. Van Kerkhove, edd. (Dordrecht: Kluwer, 2004
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