The p-norm of circulant matrices JMM 2022Ludovick Bouthat (Université Laval)The p-norm of circulant matrices JMM 2022 4 / 26Permutation-invariant matrix normfor every permutation matrix P and Q. ExampleEvery matrix norm induced by a permutation-invariant vector norm is a permutation-invariant matrix norm.Ludovick Bouthat (Université Laval)The p-norm of circulant matrices JMM 2022 6 / 26

The bedrock of mathematical activity is undoubtedly the concept of proof. Considering the abstract and conceptual nature of mathematical proofs, the act of proving is sometimes "thought to be the aspect of mathematical activity most resistant to the influence of technological change". After a careful reading of "Proof Technology in Mathematics Research and Teaching", however, one is convinced that the reality is quite different. In recent decades, development of tools such as automatic theorem provers has changed the work of mathematicians to the point of them questioning the very notion of mathematical demonstration. However, there is relatively little of this technological upheaval reflected in mathematics education.This book is intended for specialists in mathematics education with an interest in computer science advances as well as for researchers in the field of automated reasoning with an interest in the pedagogical and didactic implications of their work. It provides a valuable contribution to mathematics education by initiating a process of in-depth reflection on the educational value of new technological tools such as automatic theorem provers and dynamic geometric environments. Considering the wide range of subjects covered, few people will read this book from cover to cover with an unfaltering enthusiasm. There is no doubt, though, that there is something here to satisfy everyone's mathematical appetite.The first chapter of the section titled "Automatic Theorem Provers" sheds light upon the reasons why the great advances made in automated reasoning have been slow to be reflected in the teaching of mathematics. The proofs produced by the automatic theorem provers are often deemed terse to the point of being unreadable. This explains why there are still very few mathematicians making widespread use of proof assistants. In order to mitigate the impact of the significant gap between the way human express their mathematical thinking and the write-ups produced by the machine, the authors committed themselves to identify a possible source of this communication incompatibility. For humans, the raison d'être of a mathematical proof is not only to confirm the veracity of a statement. When a mathematician produces or consults a proof, he seeks above all an explanation; he seeks to base his understanding upon more elementary statements. Hence, clarity and readability are cardinal virtues. This is why the

In 2003, two school-based programs for teaching reading in English were introduced in Québec: the Accelerated Development of Reading (ADOR) program for first graders and, for ADOR pupils with weak reading skills, the Intensive Intervention in Reading (IIR) support program. Explicit and direct instruction provides the framework for ADOR and IIR. Two quasi-experimental studies were conducted to measure the effects of the ADOR and IIR programs compared to regular reading instruction and remediation. In both studies, the combined effects of ADOR and IIR were measured on two separate cohorts of grade 1 pupils over a full school year. This study shows that the ADOR and IIR programs result in superior reading performance and the largest effects were on subtests measuring reading comprehension.

In 2003, two school-based programs for teaching reading in English were introduced in Québec: the Accelerated Development of Reading (ADOR) program for first graders and, for ADOR pupils with weak reading skills, the Intensive Intervention in Reading (IIR) support program. Explicit and direct instruction provides the framework for ADOR and IIR. Two quasi-experimental studies were conducted to measure the effects of the ADOR and IIR programs compared to regular reading instruction and remediation. In both studies, the combined effects of ADOR and IIR were measured on two separate cohorts of grade 1 pupils over a full school year. This study shows that the ADOR and IIR programs result in superior reading performance and the largest effects were on subtests measuring reading comprehension.