Mathematical Instruction Article

Didactics of Mathematics as a Mathematical Discipline:

A Portuguese reflection upon a Kleinean challenge

Elfrida Ralha, CMAT, Universidade do MinhoJaime Carvalho e Silva, CMUC, Universidade de CoimbraJose Manuel Castanheira, DME, Universidade da Madeira

Suzana Napoles, CMAF, Universidade de Lisboa

The famous German mathematician Felix Klein (Dus-seldorf, 25th April, 1849 - Gottingen, 22nd June, 1925)is well known both as a distinguished researcher (non-euclidean geometry, group theory and theory of func-tions) and as a remarkable teacher. According toHedrick and Noble5, he combined familiarity with allthe fields of mathematics and the ability to perceive themutual relations of these fields; and he made it his no-table function, as a teacher, to acquaint his studentswith mathematics, not as isolated disciplines, but asan integrated living organism. . . He endeavored to re-duce the gap between the school and the university, torouse the schools from the lethargy of tradition, to guidethe school teaching into directions that would stimulatehealthy growth; and also to influence university attitudeand teaching toward a recognition of the normal func-tion of the secondary school.

Klein was appointed professor at Erlangen6 in 1872and he joined the Technische Hochschule, at Munich,in 1875, and the University of Leipzig, in 1880. By1886 he accepted a chair at the University of Gottingenand there he established a research center which be-came a role model for the best mathematical researchcenters all over the world: he promoted weekly discus-sion meetings, he arranged for a mathematical readingroom with a mathematical library and he even man-aged to give fame to the journal Mathematische An-nalen and to convince David Hilbert to join in his re-search center. By then the distinction between pure andapplied mathematics began to fade away and the intel-lectual partnership between Klein and Hilbert provedto be, in Gottingen, a decisive orientation for othersin exploring the relationship among mathematics, sci-ence in general and technological disciplines. From thenon, Klein became also interested in mathematical in-

struction in schools and in 1905 he was to play a de-cisive role in formulating a plan recommending thatthe mathematical concept of function as well as thebasics of differential and integral calculus should betaught in secondary schools. Many countries aroundthe world gradually implemented this recommendationand, by 1908, Felix Klein became elected the first chair-man of the International Commission on MathematicalInstruction (ICMI), at the Rome International Mathe-matical Congress.

By 1908, Klein’s famous notes “Elementarmathematikvom hoheren Standpunkte aus” (Tome I) were pub-lished, in Leipzig, for the first time.

In this volume dedicated to “Arithmetik, Algebra andAnalysis” Klein would then write on its preface that:

The new volume which I herewith offer to the mathe-matical public, and specially to the teachers of mathe-matics in our secondary schools, is to be looked upon asa first continuation of the lectures “Uber den mathema-tischen Unterricht an den hoheren Schulen”. . . At thattime our concern was with the different ways in which

5E. R. Hedrich (Vice President and Provost) and C. A. Noble (Professor of Mathematics, Emeritus), from the University of California,were the translators (from German to English) and the authors of the preface of Klein’s books on “Elementary Mathematics from anadvanced standpoint”.

6Felix Klein is particularly linked to his “Erlangen Program” which he first presented, to a restricted audience, when he was appointedas professor for the Faculty of Philosophy and for the Senate of the University of Erlangen, in Germany.

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the problem of instruction can be presented to the math-ematician. At present my concern is with developmentswith subject matter of instruction. I shall endeavorto put before the teacher, as well as the ma-turing student, from the view-point of modernscience, but in a matter as simple, stimulatingand convincing as possible, both the content andthe foundations of the topics of instruction, withdue regard for the current methods of teaching.

Two other volumes on the same theme of “Elementar-mathematik vom hoheren Standpunkte aus” were tofollow: Part II, on “Geometry” (1909) and Part III,published posthumously by Springer, on “Precision andApproximation Mathematics” (1928).

Several German editions of parts I and II were publishedand for the 3rd edition, Felix Klein, himself, would in-corporate more material to the original text, adding thefollowing justification to its preface:

Two volumes on “Elementary Mathematics from an Ad-vanced Standpoint”. . . were to bring the attention ofsecondary school teachers of mathematics and sciencethe significance for their professional work of their aca-demic studies. . . But during the sixteen years whichhave elapsed since the first publication, science hasadvanced, and great changes have taken placein our school system, changes which are still inprogress. This fact is provided for in the appendiceswhich have been prepared.

Meanwhile several translations of this work were tak-ing place, from which we enhance the Spanish version(Madrid, 1927 and 1931) which was coordinated underthe direction of Rey Pastor:

And we also enhance the English version, translatedfrom the 3rd German Edition by E. R. Hedrick andC. A. Noble (Tome I, in 1932, and Tome II, in 1939)and published by The MacMillan Company, New York.By 2004 Dover Publications offered us an unabridgedrepublication of this translation.

On the one hand, we keep in mind that, after onehundred years, this unique work is still a remarkabletext for both mathematicians and mathematics edu-cators and we are pleased to acknowledge that theCentro de Algebra and the Centro de Matematica eAplicacoes Fundamentais, from the University of Lis-bon, are preparing the first Portuguese translationwhich is to be presented as the next three issues (thefirst of them, on Aritmetica, is now completed) of thecollection “Leituras em Matematica”, from SociedadePortuguesa de Matematica.

On the other hand if, in the very words of Klein him-self, many changes, in our school systems, had occurredin sixteen years, what can we say about such a realityin this new century? A century that has, for example,enthusiastically embraced the so-called Modern Mathe-matics and, not long after that, has completely bannedsuch teaching principles and mathematical content fromour schools.

Wishing to reflect upon some ideas for the teachingand learning of mathematics in the XXI century weshare - through the organization of a Workshopon Didactics of Mathematics as a Mathemat-ics Discipline, next October, at the Universityof Madeira - our worries with a specialized panel ofmathematicians, historians of mathematics and mathe-matics educators who celebrating the 100th anniversaryof Klein’s publications, in a partnership between IMU

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(International Mathematical Union) and ICMI, havedecided to re-visit Klein’s ideas for, in the words ofMichele Artigue:

(It is intended as) a stimulus for mathematics teachers,so to help them make connections between the mathe-matics they teach, or can be asked to teach, and the fieldof mathematics, while taking into account the evolutionof this field over the last century.

And, according to Bill Barton:

The objectives are (· · · ) to produce a readable butprofessional book that conveys the connected-ness, growth, relevance, and beauty of the dis-cipline of mathematics from its big ideas to thecutting edge of research and applications.

As referred in the ICMI web sitehttp://www.mathunion.org/index.php?id=805, the KleinProject will have three outputs: a book simultaneouslypublished in several languages, a resource DVD to assistteachers wishing to bring some of the ideas to realisa-tion in their classes, and a Wikipedia-based web-siteseen as a vehicle for the many people who will wish tocontribute to the project in an on-going way.

Within the prospective languages of the Klein project, aPortuguese version of those outputs from its beginningwill serve directly the community of mathematics teach-ers in Portugal, in Brazil, in Africa or in the rest of theworld that use Portuguese as their primary language,

making it the 5th language in the world by number ofnative speakers.

The Mathematics Education of the present times is,undoubtedly, affected by many factors, some of whichKlein himself might had never dreamed of, but the part-nership among those who deal with mathematics andwith education is, 100 years after Felix Klein defendedit, still in need of some debate. For example it mightbe useful to reflect upon:

- Which “advanced” mathematics is more prone tobe introduced in our actual school curricula?

- Which mathematics is, nowadays, more excitingand more alive?

- Which influences, both within mathematics andexternally, are we getting in our mathematicsclasses?

- Which ways do contemporary mathematics edu-cators prefer?

- Within a universal discipline such is mathematics,which educational factors are national and whichare international?

- What kind of knowledge can we withdraw frompast experiences?

In summary: what are the new mathematical challengesto be faced by our school systems in a global societywhere, according to MSC (Mathematical Subject Clas-sification), ninety seven categories/mathematical fields(with several thousands of sub-fields) are nowadays tobe taken into consideration?

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