
06571523
j
2016c.00575
Lee, KoSze
Students' proof schemes for mathematical proving and disproving of propositions.
J. Math. Behav. 41, 2644 (2016).
2016
Elsevier, New York, NY
EN
E53
proof
proof schemes
mathematical reasoning
counterexamples
examples
mathematical proving
doi:10.1016/j.jmathb.2015.11.005
Summary: Classification of students' proof schemes offers insights into students' mathematical proving. However, previous studies revealed little about finer progressions in students' proving and barely examined their proof schemes involving counterexamples. This study examined students' proof schemes through their considerations of examples and counterexamples and reasoning in Proof Constructions. From 480 proofs constructed by 60 Singapore secondary three (1415 years old) students, classifications of schemes for Deductiveproof Construction and Proofbycounterexample Construction were established as a progression through four phases. The former consisted of seven levels, spanning from irrelevant inferences to deductive proofs using formal representations. The latter consisted of six levels, spanning from irrelevant inferences to construction of a general set of counterexamples. The refined classifications revealed students' progression through their sophisticated use of examples, counterexamples, and deductive inferences and that Proof Construction was knowledgedriven. Comparisons to previous classifications and the role of counterexamples in Proof Construction were discussed.