id: 06514361
dt: j
an: 2016a.00448
au: Garofalo, Joe; Trinter, Christine P.; Swartz, Barbara A.
ti: Engaging with constructive and nonconstructive proof.
so: Math. Teach. (Reston) 108, No. 6, 422-428 (2015).
py: 2015
pu: National Council of Teachers of Mathematics (NCTM), Reston, VA
la: EN
cc: E53
ut: proving; constructive proofs; nonconstructive proofs; counterexamples
ci:
li: http://www.nctm.org/Publications/mathematics-teacher/2015/Vol108/Issue6/Engaging-with-Constructive-and-Nonconstructive-Proof/
ab: Summary: One method of proof is to provide a logical argument that
demonstrates the existence of a mathematical object (e.g., a number)
that can be used to prove or disprove a conjecture or statement. Some
such proofs result in the actual identification of such an object,
whereas others just demonstrate that such an object exists. These types
of proofs are often referred to as constructive and nonconstructive,
respectively. In this article, the authors share four tasks that they
use to encourage secondary school students and preservice mathematics
teachers to consider the conditions under which an example or
counterexample, or even the logical demonstration that an example
exists, can serve as a proof. The authors have regularly observed that
students and others working through these tasks expand their approaches
to proving statements and solving nonroutine mathematical problems.
Thoughtful use of the tasks presented in this article can help students
develop mathematical power and proficiency. (ERIC)
rv: