Hanna Gila, Reid David A., de Villiers Michael. Proof Technology in Mathematics Research and Teaching

Springer International Publishing, 2019. — 379 p. — (Mathematics Education in the Digital Era 14). — ISBN 978-3-030-28482-4, 978-3-030-28483-1This bookpresents chapters exploring the most recent developments in the role of technology in proving. The full range of topics related to this theme are explored, including computer proving, digital collaboration among mathematicians, mathematics teaching in schools and universities, and the use of the internet as a site of proof learning.
Proving is sometimes thought to be the aspect of mathematical activity most resistant to the influence of technological change. While computational methods are well known to have a huge importance in applied mathematics, there is a perception that mathematicians seeking to derive new mathematical results are unaffected by the digital era. The reality is quite different. Digital technologies have transformed how mathematicians work together, how proof is taught in schools and universities, and even the nature of proof itself. Checking billions of cases in extremely large but finite sets, impossible a few decades ago, has now become a standard method of proof. Distributed proving, by teams of mathematicians working independently on sections of a problem, has become very much easier as digital communication facilitates the sharing and comparison of results. Proof assistants and dynamic proof environments have influenced the verification or refutation of conjectures, and ultimately how and why proof is taught in schools. And techniques from computer science for checking the validity of programs are being used to verify mathematical proofs.
Chapters in this book include not only research reports and case studies, but also theoretical essays, reviews of the state of the art in selected areas, and historical studies. The authors are experts in the field.Proof Technology: Implications for Teaching (Gila Hanna, David A. Reid, Michael de Villiers
A Fully Automatic Theorem Prover with Human-Style Output (M. Ganesalingam, W. T. Gowers)
A Common Type of Rigorous Proof that Resists Hilbert’s Programme (Alan Bundy, Mateja Jamnik)
SMTCoq: Mixing Automatic and Interactive Proof Technologies (Chantal Keller)
Studying Algebraic Structures Using Prover9 and Mace4 (Rob Arthan, Paulo Oliva)
Didactical Issues at the Interface of Mathematics and Computer Science (Viviane Durand-Guerrier, Antoine Meyer, Simon Modeste)
Issues and Challenges in Instrumental Proof (Philippe R. Richard, Fabienne Venant, Michel Gagnon)
The Contribution of Information and Communication Technology to the Teaching of Proof (Maria Alessandra Mariotti)
Journeys in Mathematical Landscapes: Genius or Craft? (Lorenzo Lane, Ursula Martin, Dave Murray-Rust, Alison Pease, Fenner Tanswell)
Using Automated Reasoning Tools to Explore Geometric Statements and Conjectures (Markus Hohenwarter, Zoltán Kovács, Tomás Recio)
Computer-Generated Geometry Proofs in a Learning Context (Pedro Quaresma, Vanda Santos)
Using 3D Geometry Systems to Find Theorems of Billiard Trajectories in Polyhedra (Heinz Schumann)
Learning Logic and Proof with an Interactive Theorem Prover (Jeremy Avigad)
Web-Based Task Design Supporting Students’ Construction of Alternative Proofs (Mikio Miyazaki, Taro Fujita, Keith Jones)
Reasoning by Equivalence: The Potential Contribution of an Automatic Proof Checker (Christopher Sangwin)
Virtual Manipulatives and Students’ Counterexamples During Proving (Kotaro Komatsu, Keith Jones)
Proof Technology and Learning in Mathematics: Common Issues and Perspectives (Nicolas Balacheff, Thierry Boy de la Tour)