id: 06460746
dt: j
an: 2015d.00031
au: Kjeldsen, Tinne Hoff; Lützen, Jesper
ti: Interactions between mathematics and physics: the history of the concept of
function ‒ teaching with and about nature of mathematics.
so: Sci. Educ. (Dordrecht) 24, No. 5-6, 543-559 (2015).
py: 2015
pu: Springer Netherlands, Dordrecht
la: EN
cc: A30 I20 M50
ut: function concept; history of mathematics; mathematics and physics; nature
of mathematics
ci:
li: doi:10.1007/s11191-015-9746-x
ab: Summary: In this paper, we discuss the history of the concept of function
and emphasize in particular how problems in physics have led to
essential changes in its definition and application in mathematical
practices. Euler defined a function as an analytic expression, whereas
Dirichlet defined it as a variable that depends in an arbitrary manner
on another variable. The change was required when mathematicians
discovered that analytic expressions were not sufficient to represent
physical phenomena such as the vibration of a string (Euler) and heat
conduction (Fourier and Dirichlet). The introduction of generalized
functions or distributions is shown to stem partly from the development
of new theories of physics such as electrical engineering and quantum
mechanics that led to the use of improper functions such as the delta
function that demanded a proper foundation. We argue that the
development of student understanding of mathematics and its nature is
enhanced by embedding mathematical concepts and theories, within an
explicit-reflective framework, into a rich historical context
emphasizing its interaction with other disciplines such as physics.
Students recognize and become engaged with meta-discursive rules
governing mathematics. Mathematics teachers can thereby teach inquiry
in mathematics as it occurs in the sciences, as mathematical practice
aimed at obtaining new mathematical knowledge. We illustrate such a
historical teaching and learning of mathematics within an explicit and
reflective framework by two examples of student-directed,
problem-oriented project work following the Roskilde Model, in which
the connection to physics is explicit and provides a learning space
where the nature of mathematics and mathematical practices are linked
to natural science.
rv: