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Book Reviewscation” (by Jean-Luc Dorier and Katia Mass), “Problemsolving in mathematics education” (by Manuel Santos-Trigo) and “Instrumentation in mathematics education”(by Luc Trouche). They can also present theories or conceptsused in mathematics education like “Activity theoryin mathematics education” (by Wolff-Michael Roth)and “Didactic contract in mathematics education” (byGuy Brousseau). The content of the Encyclopedia actuallyrepresents the major results obtained in mathematicseducation over more than 40 years, with a variety ofperspectives (epistemological, cognitive, socio-cultural,etc.) developed by its international group of authors.Visiting the letter “M”An Encyclopedia can be a useful tool for answering aprecise question. It can also be viewed as a place to wander…Let’s try a brief random walk within the letter “M”of the Encyclopedia – one of the richest letters, with 25entries! (This is perhaps not surprising, since Mathematicsis the central focus here.) The first entry is entitled“Manipulatives in Mathematics Education” (by Maria G.Bartoloni Bussi and Francesca Martignone). This articlementions various kinds of manipulatives, ranging fromhistorical ones like Napier bones to very recent digitaltools, like the Bee-bot floor robot; it discusses the differencesbetween concrete and virtual manipulatives, froman educational perspective; it identifies critical issues,linked with the use of manipulatives, like students’ autonomyor their age (why are high school teachers oftenreluctant to use manipulatives in class?); it also presentsa theoretical approach, the semiotic mediation, whichis especially relevant to studying the learning-teachingprocesses when manipulatives are involved.After the “Manipulatives” entry, there starts a longlist of “Mathematical…” entries: “Mathematical ability”,“Mathematical approaches”, “Mathematical functionsteaching and learning”… Let’s read this one (by MogensNiss), which is the first article we meet in the list with afocus on a specific mathematical theme. Within the hugebody of research on this topic, the author retains a focuson students’ difficulties. Functions can have diverse representations:algebraic, graphical, tabular etc.; this causesseveral specific difficulties that have been clearly identifiedand has led to the design of teaching interventionsusing special software supporting the articulation ofseveral representations. Another dimension of complexityis that functions have different aspects: a simple correspondencelinking “every element in a given domainto one and only one element in another domain” is forthe learner very different from a tool intervening in themodelling of extra-mathematical situations, for example.“Mathematical Modeling and Applications in Education”and “Mathematical representations” are other entriesunder the letter M that can usefully complement thearticle about functions. “Mathematical Proof, Argumentationand Reasoning” also faces the challenge of synthesisingmultiple research works on the subject. In this article,Gila Hanna recalls that “a proof is much more thana sequence of logical steps that justifies an assertion” andthat it can play various roles in mathematics practice, likeestablishing connections and suggesting new hypotheses.It can also take different forms, remaining informal butproviding a high level of reliability. Teachers have to introducestudents to these different kinds of proofs and, atthe same time, teach them the rules of reasoning as wellas presenting patterns of argument. This is a delicate task,indicating the need for adequate teacher education (preserviceand in-service) – this connects us directly with the“Mathematics Teacher Education Organization” entry, afew steps further in the Encyclopedia… This article (byJarmila Novotná, Hana Moraová and Maria Tatto) offersan international view of the multiple existing organisationsfor teacher education but also discusses the skills,abilities, knowledge and attitudes that students graduatingfrom teacher preparation programmes should master.The reader interested in teacher education can go onand read the “Models of In-service Mathematics TeacherEducation” and “Models of Preservice MathematicsTeacher Education” entries and can naturally switch tothe letter T, where they will find, for example, “TeacherEducation Development Study – Mathematics (TEDS-M)”. The challenge here might be to stop reading theEncyclopedia!Final word (or not)Let us go back to the foreword of the Encyclopedia, writtenby Jeremy Kilpatrick:“This encyclopedia represents a major step forwardin the field of mathematics education, bringing to everyonewith a professional interest in mathematics educationaccess to the latest and best thinking in the field.”(Kilpatrick, Foreword, vi).Naturally, I fully support this enthusiastic statement.Moreover, this major step is not a final step, since theonline version should permit regular updates and discussionbetween authors and readers. For all your questionsabout research in mathematics education, you will findelements of answers in the Encyclopedia of MathematicsEducation and you can contribute with your commentsto a continuous improvement of its content!Ghislaine Gueudet is a professor of mathematicseducation at the ESPE Bretagne(School for Teacher Education). Shehas represented the French Associationfor Research in Mathematics Education(ARDM) on the French ICMI sub-commission(CFEM) since 2008 and also representsthe CFEM on the EMS Educational Committee.Her research concerns university mathematics educationand the design and use of educational resources (digitalresources in particular).EMS Newsletter December 2014 59