Izvleček
V članku obravnavamo pomen dokaza v matematiki. Po kratkem zgodovinsko-teoretičnem uvodu se dotaknemo pomena dokaza v srednji šoli. Omenimo tudi Pitagorov izrek, ki velja za enega glavnih rezultatov v matematiki, in navedemo dva dokaza, ki temeljita na geometrijski podlagi. Obravnavamo tudi glavne tehnike dokazovanja (ki so primerne za srednješolski nivo): dedukcija, dokaz s protislovjem, kontrapozicija, protiprimer in indukcija. Pri vsaki tehniki podamo tudi (srednješolskemu nivoju) ustrezne primere. Omenimo tudi eno najslavnejših domnev – Collatzovo domnevo, katere tezo lahko brez težav razumejo celo osnovnošolci, dokaz (ali protiprimer) pa še vedno čakamo.
Abstract
Mathematical Proof in Upper Secondary School
This article covers the significance of proof in mathematics. It starts with a brief historical-theoretical background before highlighting the role of proof in the context of upper secondary school mathematics. We also refer to the Pythagorean theorem, one of the most important mathematical concepts, and provide two proofs based on geometric foundations. Further, we examine the main proof techniques at the secondary-school level: deduction, proof by contradiction, counterexample, contrapositive and induction, along with suitable examples. We also look at one of the most famous conjectures, i.e., the Collatz conjecture, whose thesis is simple enough for primary school students to understand but whose proof (or counterexample) has not yet been provided.