JOURNAL OF MATHEMATICAL BEHAVIOR 13, 129-132 (1994)

Nicolas HerscovicsJanuary 12, 1935-January 3, 1994

It is with regret that we announce the premature death of Nicolas Herscovics,professor in the Department of Mathematics and Statistics at Concordia Univer-sity, Montreal . He left us on January 3, 1994, on the eve of his 59th birthday,after a long battle with cancer.

Professor Herscovics was an active member of several groups whose aimswere the study and improvement of mathematics teaching and learning . In Cana-da, he was involved in the CMESG (Canadian Mathematics Education StudyGroup), AMQ (Association mathematique du Quebec), GDM (Groupe des didac-ticiens des mathematiques), and APAME (Association pour l'avancement desrnathematiques a l'elementaire); in the United States we can mention NCTM(National Council of Teachers of Mathematics), and PME-NA (the North Ameri-can Chapter of PME), of which he was one of the founders . On the internationalscene he was active in CIEAEM (la commission internationale pour 1'ctude etl'avancement de 1'enseignement des math€matiques), also in PME (the Interna-tional Group for the Psychology of Mathematics Education)-another group ofwhich he was a founding member .

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In the too short 15-year period of his researcher's life, he presented over 70papers at the different annual meetings of these associations where lie liked todiscuss results from his research and reflections . He never missed an occasion topush his colleagues and his students into joining the different associations andattending their meetings. His main research interest centered on the elaborationof models useful in describing the understanding of conceptual schemes, believ-ing that such models could constitute a more scientific basis for evaluation .remediation . and pedagogical intervention . He did not have much credence in theexistence of a general model applicable to every aspect of mathematical activity .such as concept formation, problem solving, proofs, and so on- which is whyhe thought that we should start by describing what understanding means forparticular notions . This was the essence of many of his publications, whichreport studies of the notion of point, straight line, natural numbers, numeration .fractions, the four arithmetic operations, variable, function, and equations .

Although at first glance we might be inclined to think that his work was purelytheoretical, it is important to mention that fundamentally he aspired to be emi-nently practical . From the very beginning his ambition was to change what wasgoing on in the classroom . In fact, our first grant was on the integration ofresearch in the preserviee and inservice training of teachers, a project that wasbased on his idea that all innovation must pass through the teacher . He foundmuch pleasure in working with these teachers, helping them to acquire a moreglobal vision of the concepts they were teaching-for hint, it was a way to fighttheir tendency to rely on "recipes ." The essence of his ideas can he found in theproceedings of the different meetings of PME . PME-NA, and CMFSG- It isworth mentioning, as well, two articles, which were published in French (Ber-geron, J,C ., lerscovics, N ., & Sinclair, IF . 1992, Contribution a la genese dunombre, Archives de Psychologie, 60, 147-170 ; Bergeron . .CC ., & Herscovics,N., 1992 . Une approche epistdmologique pour l'etude de la construction desdebuts de l'arithmetique chez les enfants de 5 a 6 ans, The Alberta Journal ofEducational Research) . They present the description and use of the model ofunderstanding that we developed together over the years . The model, called theTwo-Tiered Model, is used to describe the understanding of the concept ofnatural number with 5-6 year olds .

I have described other aspects of the personal and professional life of NicolasHerscovics in a recent memorial text appearing in the Spring 1994 Newsletter ofthe International Group for the Psychology of Mathematics Education (PME) .Permit me to reproduce it here .

On January 3, 1994 . PME lost one of its founding members, ProfessorNicolas Herscovics. Nicolas was a faculty member of the Department of Mathe-matics and Statistics at Concordia University in Montreal . Canada . When heattended his first ICME meeting in Karlsruhe, Germany, in 1976, he deplored thefact that researchers interested in the teaching and learning of mathematics, and

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who had so much to discuss, were able to gather only once every four years .Therefore, he asked the participants who were interested in forming a subgroupof ICME, one that would meet every year in order to promote research in thisdomain, to endorse his proposal . The signatures he gathered, such as those ofFreudenthal, Fischbein, Karplus, Bauersfeld, and Skemp, were sufficiently nu-merous and illustrious to lead to the immediate creation of IGPME (InternationalGroup for the Psychology of Mathematics Education)-the former name ofPME .

Thereafter, Nicolas attended almost all of PME's annual meetings where heaccomplished much underground work in order to convince colleagues of thenecessity of not staying in isolation-one way being to recruit people in theircountry or even organize a PME meeting . He was also instrumental in foundingPME-NA-the North American chapter of PME . With Carolyn Kieran andmyself, he co-organized two meetings in Montreal : PME-NA V in 1983 andPME-XI in 1987 .

His career as a researcher was relatively short, only 15 years-the last two ata reduced pace because of his illness . In spite of such a short period, he neverthe-less published over 70 papers (most of then paired with an oral communication),directed many masters degree students, co-directed three doctoral students withme, and co-directed (with me) a research group funded by Quebec's Ministry ofEducation during the last 13 of these 15 years . Concurrently, he almost alwayshad a full teaching load in mathematics and mathematics education!

For 12 years we worked on the problem of describing the understanding ofearly arithmetic concepts by 5 to 9-year-olds . We proposed a descriptive modelof understanding that was constantly improved in the light of experimental evi-dence. With minor adaptations it was possible to apply it to more advancedconcepts . Nicolas was also interested in the learning and teaching of algebra,which he investigated with other colleagues . His last interest centered on pre-algebra, which he researched with Liora Linchevski from Israel, and which untilvery recently, he was discussing with Gerald Goldin .

A brief glance at his life path reveals why he was a latecomer to research . Hisparents, who were Jewish-Rumanian, migrated to Belgium . During World War IIthey were arrested and deported to German labor camps, while Nicolas and histwin sister, aged 5, were kept by a Catholic couple and also by a nun . In his ownwords, he was one of the rare Jews who served mass . His mother did not survivethe labor camps . After the war, his father returned to Belgium where he married awidow with five sons and returned to his former trade, tailoring . Nicolas, nowaged 10. had to quit school and work in his father's store and learn the trade .When he was 16, the entire family moved to Montreal where he worked in theclothing industry and then in the plumbing industry, all the while attendingevening school . After graduating from university with a first degree in physicsand a second in mathematics, he left the business world to become a university

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professor-at a substantial cut in salary!When he was asked to take charge of the in-service program for secondary

level mathematics teachers, he felt the need to acquire some knowledge ofpedagogy and psychology. So he registered as a PhD student at the University ofMontreal, from which he graduated in 1979 at the age of 44 . 1 had the pleasureof being not so much his thesis "director" as his "accompanist ." Far from endingwith his graduation, our collaboration lasted until 1991 when he learned he hadcancer .

A dozen years of almost daily work with him entitle me to attest to his keenintelligence, creativity, organizing ability, and inexhaustible energy . He was verydemanding of both his students and his collaborators, but it was never more thanhe required of himself . When debating ideas he could be harsh and without muchconcern for his opponents' feelings, an attitude that sometimes brought himfierce opposition . Nevertheless, after any such episodes, he always ended upbeing very positive and most generous with his time and with his ideas . Finally,he was also extremely honest with regard to giving credit for any ideas he used-even when these originated with his students .

In conclusion, 1 think that the world of mathematical didactics has lost a star,a star that alas crossed our sky for only a short instant . He often told me howmuch he would have liked to accomplish if only he had another 10 or 15 yearsleft . 1 can only wish that other researchers or students (it cannot be 1 because I amretiring in a few months) feel the need and the taste to keep on exploring the newavenues that he traced .

Jacques C . BergeronDcpartemcni de didactique

Universite de Montreal