id: 06010932
dt: j
an: 2012f.00537
au: Arzarello, Ferdinando; Dreyfus, Tommy; Gueudet, Ghislaine; Hoyles, Celia;
Krainer, Konrad; Niss, Mogens; Novotná, Jarmila; Oikonnen, Juha;
Planas, Núria; Potari, Despina; Sossinsky, Alexei; Sullivan, Peter;
Törner, Günter; Verschaffel, Lieven
ti: Do theorems admit exceptions? Solid findings in mathematics education on
empirical proof schemes.
so: Eur. Math. Soc. Newsl. 82, 50-53 (2011).
py: 2011
pu: European Mathematical Society (EMS) Publishing House, Zurich
la: EN
cc: E50
ut: mathematical education; mathematical thinking; theorems; proofs; empirical
proof schemes; concept of formal proof
ci: ME 2009f.00383
li:
ab: From the text: One of the goals of teaching mathematics is to communicate
the purpose and nature of mathematical proof. {\it H. N. Jahnke} [ZDM
40, No. 3, 363-371 (2008; ME 2009f.00383)] pointed out that, in
everyday thinking, the domain of objects to which a general statement
refers is not completely and definitely determined. Thus the very
notion of a “universally valid statement" is not as obvious as it
might seem. The phenomenon of a statement with an indefinite domain of
reference can also be found in the history of mathematics when authors
speak of “theorems that admit exceptions". This discrepancy between
everyday thinking and mathematical thinking lies at the origin of
problems that many mathematics teachers encounter in their classrooms
when dealing with a universal claim and its proof. The solid finding
(the term “solid finding" was explained in the previous issue of this
newsletter) to be discussed in this article emerged from results of
many empirical studies on students’ conceptions of proof. In a
simplified formulation, the finding is that many students provide
examples when asked to prove a universal statement. Here we elaborate
on this phenomenon.
rv: