28,914 research outputs found
Students' difficulties with vector calculus in electrodynamics
Understanding Maxwell's equations in differential form is of great importance
when studying the electrodynamic phenomena discussed in advanced
electromagnetism courses. It is therefore necessary that students master the
use of vector calculus in physical situations. In this light we investigated
the difficulties second year students at KU Leuven encounter with the
divergence and curl of a vector field in mathematical and physical contexts. We
have found that they are quite skilled at doing calculations, but struggle with
interpreting graphical representations of vector fields and applying vector
calculus to physical situations. We have found strong indications that
traditional instruction is not sufficient for our students to fully understand
the meaning and power of Maxwell's equations in electrodynamics.Comment: 14 pages, 11 figure
The history of the concept of function and some educational implications
Several fields of mathematics deal directly or indirectly with functions: mathematical
analysis considers functions of one, two, or n variables, studying their properties as well as
those of their derivatives; the theories of differential and integral equations aim at solving
equations in which the unknowns are functions; functional analysis works with spaces made
up of functions; and numerical analysis studies the processes of controlling the errors in the
evaluation of all different kinds of functions. Other fields of mathematics deal with concepts
that constitute generalizations or outgrowths of the notion of function; for example, algebra
considers operations and relations, and mathematical logic studies recursive functions.
It has long been argued that functions should constitute a fundamental concept in secondary
school mathematics (Klein, 1908/1945) and the most recent curriculum orientations clearly
emphasize the importance of functions (National Council of Teachers of Mathematics, 1989).
Depending on the dominant mathematical viewpoint, the notion of function can be regarded
in a number of different ways, each with different educational implications.
This paper reviews some of the more salient aspects of the history of the concept of
function,1 looks at its relationship with other sciences, and discusses its use in the study of
real world situations. Finally, the problem of a didactical approach is considered, giving
special attention to the nature of the working concept underlying the activities of students and
the role of different forms of representation
A versatile approach to calculus and numerical methods
Traditionally the calculus is the study of the symbolic algorithms for differentiation and
integration, the relationship between them, and their use in solving problems. Only at
the end of the course, when all else fails, are numerical methods introduced, such as the
Newton-Raphson method of solving equations, or Simpson’s rule for calculating areas.
The problem with such an approach is that it often produces students who are very well
versed in the algorithms and can solve the most fiendish of symbolic problems, yet
have no understanding of the meaning of what they are doing. Given the arrival of
computer software which can carry out these algorithms mechanically, the question
arises as to what parts of calculus need to be studied in the curriculum of the future. It
is my contention that such a study can use the computer technology to produce a far
more versatile approach to the subject, in which the numerical and graphical
representations may be used from the outset to produce insights into the fundamental
meanings, in which a wider understanding of the processes of change and growth will
be possible than the narrow band of problems that can be solved by traditional symbolic
methods of the calculus
Canonical endomorphism field on a Lie algebra
We show that every Lie algebra is equipped with a natural -variant
tensor field, the "canonical endomorphism field", naturally determined by the
Lie structure, and satisfying a certain Nijenhuis bracket condition. This
observation may be considered as complementary to the Kirillov-Kostant-Souriau
theorem on symplectic geometry of coadjoint orbits. We show its relevance for
classical mechanics, in particular for Lax equations. We show that the space of
Lax vector fields is closed under Lie bracket and we introduce a new bracket
for vector fields on a Lie algebra. This bracket defines a new Lie structure on
the space of vector fields.Comment: 18 page
Students’ Evolving Meaning About Tangent Line with the Mediation of a Dynamic Geometry Environment and an Instructional Example Space
In this paper I report a lengthy episode from a teaching experiment in which fifteen Year 12 Greek students negotiated their
definitions of tangent line to a function graph. The experiment was designed for the purpose of introducing students to the
notion of derivative and to the general case of tangent to a function graph. Its design was based on previous research results on
students’ perspectives on tangency, especially in their transition from Geometry to Analysis. In this experiment an instructional
example space of functions was used in an electronic environment utilising Dynamic Geometry software with Function
Grapher tools. Following the Vygotskian approach according to which students’ knowledge develops in specific social and
cultural contexts, students’ construction of the meaning of tangent line was observed in the classroom throughout the
experiment. The analysis of the classroom data collected during the experiment focused on the evolution of students’ personal
meanings about tangent line of function graph in relation to: the electronic environment; the pre-prepared as well as
spontaneous examples; students’ engagement in classroom discussion; and, the role of researcher as a teacher. The analysis
indicated that the evolution of students’ meanings towards a more sophisticated understanding of tangency was not linear. Also
it was interrelated with the evolution of the meaning they had about the inscriptions in the electronic environment; the
instructional example space; the classroom discussion; and, the role of the teacher
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