[RUME] Functions - recent research?

[Martti Pesonen]

Dear RUME experts,
I try to teach the concept of function (in Bourbaki sense, emphasizing the role of domain and co-domain) to our students,
who will become either secondary school subject teachers in mathematics or even mathematicians of some kind.
My institute is the University of Joensuu, Department of Physics and Mathematics, situated in the Eastern Finland.

The scope is from the calculus point of view to discrete structure functions
(naive set theory and logic, probability, graph theory)
and also the algebraic "internal binary operation" together with "external binary operation (scaling)" in linear algebra.

I know there is a lot of research done on the teaching and learning of these concepts.
However, I would like to know more obout two aspects:

1) the most recent knowledge on the function and binary operation educational research, experiences, experiments, publications

2) what the university freshmen are supposed to know and master about functions, and what is the reality

My own experiments are that the university maths freshmen here are ready to solve given equations,
can handle one real variable functions in calculus sense
(basic derivatives/primitives, finding minima/maxima and such standard procedures)
but very few are capable to analyse whether a given rule really means a function,
at least if it is explained with words or if the rule gives multiple results or if some restrictions have to be
set for the domain or the co-domain (surjectivity, bijectivity).
Also a lot of weakness is seen when dealing with restricting functions to subsets.

My own knowledge about the educational function concept research is limited to the nineties, when learning about modern pedagogies.
>From those times I remember works of Tall, Vinner and Dubinsky.

Martti E. Pesonen, Senior lecturer
Department of Physics and Mathematics
University of Joensuu, P.O.Box 111
SF-80101 Joensuu, FINLAND
Martti.Pesonen at Joensuu.Fi
http://www.joensuu.fi/mathematics/