ERIC Number: EJ1051699
Record Type: Journal
Publication Date: 2015-Feb
Engaging with Constructive and Nonconstructive Proof
Garofalo, Joe; Trinter, Christine P.; Swartz, Barbara A.
Mathematics Teacher, v108 n6 p422-428 Feb 2015
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.
Descriptors: Mathematics Instruction, Mathematical Logic, Validity, Secondary School Mathematics, Preservice Teachers, Secondary School Students, Problem Solving, Mathematics Skills, Word Problems (Mathematics)
National Council of Teachers of Mathematics. 1906 Association Drive, Reston, VA 20191-1502. Tel: 800-235-7566; Tel: 703-620-3702; Fax: 703-476-2970; e-mail: email@example.com; Web site: http://www.nctm.org/publications/
Publication Type: Journal Articles; Reports - Descriptive
Education Level: Secondary Education; Higher Education; Postsecondary Education
Authoring Institution: N/A
Grant or Contract Numbers: N/A