The Legacy of Felix Klein pp 169180  Cite as
Klein’s Conception of ‘Elementary Mathematics from a Higher Standpoint’
Abstract
This chapter studies Klein’s conception of elementarisation; it is first put into the context of other approaches for mathematics teacher education in Germany. Then, approaches in mathematics education and in history of education to conceive of the relation between academic knowledge and school disciplines are discussed. The wrong translation of Klein’s German term “höher” in the long time prevailing American translation is commented on, in preparation for the analysis of the concept of “element” in the history of science. Klein’s practice and his introduction of the term “hysteresis” to emphasise the independence of school mathematics are discussed. The last section reflects the consequences of the hysteresis notion for integrating recent scientific advances into school curricula.
Keywords
Felix klein Elementarisation Elements Apollonius d’Alembert Hysteresis Set theory12.1 Introduction
My main issue here is the notion of elementarisation. There is a widespread misunderstanding to conceive of this term within connotations like “simple”, merely a didactical category as the exact opposite to scientific and academic knowledge. For Klein, however, these are completely misleading connotations; rather, deep philosophical and epistemological meanings are revealed to be implied.
12.2 A Differing View of Elementary Mathematics
Actually, the term “elementary mathematics” was not new in Klein’s times; there existed a wellknown publication for mathematics teachers and mathematics students—likewise in three volumes—that used this term and Klein expressed his profound disagreement with this work. It was the Encyclopaedia of Elementary Mathematics, published from 1903 with various reeditions, by Heinrich Weber and Josef Wellstein, both mathematics professors at the University of Straßburg.
Thus, we find in his book detailed and abstract discussions of the concept of number, limit concept, number theoretic issues, etc., while the elements of calculus remain disregarded, although the author supports their teaching in schools. (ibid., p. 300)
I shall indicate at once certain differences between this work and the plan of my lecture course. In WeberWellstein, the entire structure of elementary mathematics is built up systematically and logically in the mature language accessible to the advanced student. No account is taken of how these things actually may come up in school teaching. The presentation in the schools, however, should be psychological – to use a ‘catch word’  and not systematic. (ibid., p. 4)

Another difference between WeberWellstein and myself has to do with delimiting the content of school mathematics. Weber and Wellstein are disposed to be “conservative”, while I am “progressive”. […] We, who use to be called the “reformers”, would put the function concept at the very centre of teaching, because, of all the concepts of the mathematics of the past two centuries, this one plays the leading role wherever mathematical thought is used.

As opposed to these comparatively recent ideas, WeberWellstein adhere essentially to the traditional limitations of the subject matter. In this lecture course I shall of course be a protagonist of the new conception (ibid., pp. 4–5).
12.3 Differing Views of the Relation Between Academic Mathematics and School Mathematics
In fact, the basic epistemological issue implied by the concept of elementarisation is the relation between school mathematics and academic mathematics. This relation is far from being evident or easily resolvable. This is documented by two extreme recent positions in mathematics education and history of education about this relation. Both poles are represented by French researchers: Yves Chevallard and André Chervel.

“Objet de savoir”—subject of knowledge;

“Objet d’enseigner”—subject to be taught: the academic knowledge becomes teachable knowledge by the efforts of mathematics educators (their community being called “noosphère”);

“Objet d’ enseignement”—teaching subject (Chevallard 1985, 39).
As has been criticized by several researchers, the explanation offered by the transposition notion conceives of a unilateral process: it has as its starting point a pole designed as advanced, the academic or university knowledge and as its final point another pole inferior to it, made at school and involving the teacher in the classroom.
The other extreme is represented by the research area history of school disciplines. In fact, researchers of this area typically work on subjects such as literature, the humanities, the native language, history and geography, religion, and even philosophy. Thus, the focus of their approach is the socializing function of school and hence, in particular, of school disciplines. The school culture and school subjects are thus characterized by autonomies: it is believed that school disciplines enjoy autonomy with respect to the other disciplines (Chervel 1988, 73), while the example of mathematics shows that the set of all disciplines influence strongly the status, the level and the views of school mathematics (Schubring 2005). Moreover, Chervel emphasises the generative nature of the school, which results in creating, due to its character understood as relatively autonomous, school disciplines (see Vinao 2008).
12.4 Implications of the Term “Advanced”
Kilpatrick had emphasised in his lecture at ICME11, 2008, in Mexico that the term “advanced” used by the American translators in the title is profoundly misleading and does not correspond to Klein’s conceptions of elementarisation (Kilpatrick 2008). In fact, the term “advanced” corresponds best to Chevallard’s conception of transposition.
The term “advanced” implies a fundamental misunderstanding of Klein’s notion of elementary and of Elementarmathematik. The term “advanced” means that elementary mathematics is somewhat delayed, being of another nature. It means exactly the contrary of what Klein was intending. By contrasting two poles, ‘elementary’ versus ‘advanced’, one would admit just that discontinuity between school mathematics and academic mathematics that Klein wanted to eliminate.
For Klein, there was no separation between elementary mathematics and academic mathematics. His conception for training teachers in higher education departed from a holistic vision of mathematics: mathematics, steadily developing and reforming itself within this process, leading to ever new restructured elements, and provides therefore new accesses to the elements. There is a widespread understanding of the term “elementary” as meaning it something ‘simple’ and not loaded with a conceptual dimension—perhaps even approaching ‘trivial’. Connected, in contrast, with the notion of element, ‘elementary’ means for Klein to unravel the fundamental conception. What is at stake, hence, is the concept of elements.
12.5 The Concept of Elements
Beyond mere factual information, with his lecture notes Klein led the students to gain a more comprehensive and methodological point of view on school mathematics. The three volumes thus enable us to understand Klein’s farreaching conception of elementarisation, of the “elementary from a higher standpoint”, in its implementation for school mathematics: the elements are understood as the fundamental concepts of mathematics, as related to the whole of mathematics and according to its restructured architecture.
According to Heath, the elementary nature of the first four books distinguishes them from the rest by the “fact that the former contain a connected and scientific exposition of the general theory of conic sections as the indispensable basis for further extensions of the subject in certain special directions, while the fifth book is an instance of such specialization…”. Heath also calls the first four books a “textbook or compendium of conic sections,” and the last four books “a series of monographs on special portions of the subject.” (ibid., pp. 58–59)^{1}
Toomer adopts the same kind of image when he writes, “[Apollonius’] aim was not to compile an encyclopedia of all possible theorems on conic sections, but to write a systematic textbook on the ‘elements’ and to add some more advanced theory which he happened to have elaborated.” (ibid., p. 59)
Fried and Unguru agree in particular with the distinction of “elements” from an encyclopedia but disagree with Heath’s assessment as a compendium. They insist on the systematic character of exposition and on the connected and scientific exposition as indispensable basis for refinements and extensions.
“On appelle en général élémens d’un tout, les parties primitives & originaires dont on peut supposer que ce tout est formé”. (d’Alembert 1755, 491 e)^{3}
“Ces propositions réunies en un corps, formeront, à proprement parler, les élémens de la science, puisque ces élémens seront comme un germe qu’il suffiroit de développer pour connoitre les objets de la science fort en détail”. (d’Alembert 1755, 491 d)^{4}
An extensive part of the entry is dedicated to the reflection on elementary books—livres élémentaires, such as schoolbooks, which are essential, on the one hand, to disseminate the sciences and, on the other, to make progress in the sciences, that is, to obtain new truths. In his reflection on elementary books, d’Alembert emphasised another aspect of great importance regarding the relationship between the elementary and the higher: he underlined that the key issue for the composition of good elementary books consists in investigating the “metaphysics” of propositions—or in terms of today, the epistemology of science.
In the first phase of the French Revolution, the composition and publication of livres élémentaires constituted a key issue of the concerns for building a new society. The elaboration of livres élémentaires was conceived of as essential for instituting the new system of public education; practically the first measure for this task was to organise a concours for composing these textbooks (Schubring 1984, pp. 363 f.; Schubring 1988).^{5} It is highly characteristic that in the later Napoleonic period the emphasis on livres élémentaires was replaced by a policy of creating livres classiques, focussing on the humanities (Schubring 1984, p. 371).
12.6 Klein’s Practice
I shall by no means address myself to beginners, but I shall take for granted that you are all acquainted with the main features of the most important disciplines of mathematics. (Klein 2016a, b, p. 1 ff.)
And it is precisely in such summarising lecture courses as I am about to deliver to you that I see one of the most important tools. (ibid., p. 1)
My task will always be to show you the mutual connection between problems in the various disciplines, these connections use not to be sufficiently considered in the specialised lecture courses, and I want more especially to emphasize the relation of these problems to those of school mathematics. In this way, I hope to make it easier for you to acquire that ability which I look upon as the real goal of your academic study: the ability to draw (in ample measure) from the great body of knowledge taught to you here vivid stimuli for your teaching. (ibid., p. 2)
I should remark here that, given the methodological task of these lecture courses, Klein evidently did not aspire to elaborate any teaching unit or to propose a didactical sequence—as he always emphasised, this should be the exclusive task of the teacher, given his autonomy with regard to teaching methods. Klein was therefore always distant from what became later the dominant practice of socalled Subject didactics (StoffDidaktik) in Western Germany (see Schubring 2016).
The normal process of development […] of a science is the following: higher and more complicated parts become gradually more elementary, due to the increase in the capacity to understand the concepts and to the simplification of their exposition (“law of historical shifting”). It constitutes the task of the school to verify, in view of the requirements of general education, whether the introduction of elementarised concepts into the syllabus is necessary or not. (Klein and Schimmack 1907, p. 90)
The historical evolution of mathematics entails therefore a process of restructuration of mathematics where new theories, which at first might have been somewhat isolated and poorly integrated, become well connected to other branches of mathematics and effect a new architecture of mathematics, based on reconceived elements, and thus on a new set of elementarised concepts.
A retardation of the effect when the forces acting upon a body are changed (as if from viscosity or internal friction); esp.: a lagging in the values of resulting magnetization in a magnetic material (such as iron) due to a changing magnetizing force.
In this connection I should like to say that it is not only excusable but even desirable that the schools should always lag behind the most recent advances of our science by a considerable space of time, certainly several decades; that, so to speak, a certain hysteresis should take place. But the hysteresis, which actually exists at the present time is in some respects unfortunately much greater. It embraces more than a century, in so far as the schools, for the most part, ignore the entire development since the time of Euler. (Klein 2016a, b, pp. 220–221; my emphasis, G.S.) (Fig. 12.1).
12.7 Modernism and the Challenge by Set Theory
Set theory was a case for Klein where this theoretical development was too fresh, and not yet accomplished and even further from having matured to the point of having induced an intradisciplinary process of integration and restructuration. The concepts of set theory did not (yet) provide new elements for mathematics—hence Klein’s polemic against Friedrich Meyer’s schoolbook of 1885 who’s intention had been, in fact, to use set theory as new elements for teaching arithmetic and algebra (see Klein 2016a, b, p. 289, note 181). Meyer, mathematics teacher at a Gymnasium in Halle and friend of Cantor, introduced there the notions of set theory—not yet fully developed then by Cantor—as foundations for the number concept. Klein had sharply criticised this schoolbook in his first edition, but softened his critique in subsequent editions. In Klein’s times, mathematics had not achieved the level of architecture established by Bourbaki—and hence not of “modern math”.
Given that set theory has been almost identified with modernism in mathematics, I need to comment somewhat on the book by Herbert Mehrtens: Modeme—Sprache—Mathematik (1990), where he models Göttingen mathematics as bipolar: Hilbert representing “modernism” and Klein representing “countermodernism”. I was always critical of this book and Mehrtens’ assessments, since he misrepresents both mathematicians: Hilbert was not that theoretician and formalist who freely created abstract theories whom Mehrtens compared with the artists of that time as no longer bound by any claim to represent reality. And depicting Hilbert as “antiintuitive” (1996, p. 521) deeply misunderstands Hilbert’s vision and practice of mathematics. Klein is, on the other hand, denounced by Mehrtens to be tied to reality and to intuition largely because German Nazi mathematicians later abused these notions.
The investigations of George Cantor, the founder of this theory, had their beginning precisely in considerations concerning the existence of transcendental numbers. They permit one to view this matter in an entirely new light.
In connection with this, there has arisen, finally, a still more farreaching entirely modern generalisation of the function concept. Up to this time, a function was thought of as always defined at every point in the continuum made up of all the real or complex values of x, or at least at every point in an entire interval or region. But since recently the concept of sets, created by Georg Cantor, has made its way more and more to the foreground, in which the continuum of all x is only an obvious example of a “set” of points. From this new standpoint functions are being considered, which are defined only for the points x of some arbitrary set, so that in general y is called a function of x when to every element of a set x of things (numbers or points) there corresponds an element of a set y. (Klein 2016a, b, p. 220)
Let me point out at once a difference between this newest development and the older one. The concepts considered under headings 1. to 5. have arisen and have been developed with reference primarily to applications in nature. We need only think of the title of Fourier’s work! But the newer investigations mentioned in 6. and 7. are the result purely of the drive for mathematical research, which does not care for the needs of exploring the laws of nature, and the results have indeed found as yet no direct application. The optimist will think, of course, that the time for such application is bound to come. (ibid.)
Provided that a deep epistemological need exists, which will be satisfied by the study of a new problem, then it is justified to study it; but if one does it only to do something new, then the extension is not desirable. (Klein 2016a, p. 157)
commenting:Obviously one can then operate with a, b, c,…, precisely as one ordinarily does with actual numbers. (Klein 2016a, b, p. 14)
The tendency to crowd intuition completely off the field and to attain to really pure logical investigations seems to me not completely realisable. It seems to me that one must retain a remainder, albeit a minimum, of intuition. (ibid., p. 15)
I have felt obliged to go into detail here very carefully, in as much as misunderstandings occur so often at this point, because people simply overlook the existence of the second problem. This is by no means the case with Hilbert himself, and neither disagreements nor agreements based on such an assumption can hold. (ibid., p. 16)
The second problem, which Klein is emphasising here, was put by him as the epistemological aspect of the task of justification of arithmetic—and his intention was to say that it should not be overlooked when researching upon the logical aspect of justification of arithmetic.
12.8 Concluding Remarks
In fact, school mathematics will always be confronted with the tension between logical and epistemological aspects; there can be no definite solution. But Klein’s concept of hysteresis offers a viable approach to realising an elementarisation that puts school mathematics into a productive relation with the progress of mathematics.
The attractiveness of Klein’s lecture notes is due to his epistemological understanding of the elements, and to not falling into the trap of practicing elementarisation as a simplification but as the challenge to understanding the connectivity and coherence of the branches and specialities of mathematics.
Footnotes
 1.
Quotes from ibid, p. lxxvi and lxxvi–lxxvii.
 2.
Alain Trouvé has studied contributions to the notion of element by philosophers, scientists and pedagogues, since Antiquity until the early 19th century (Trouvé 2008). Essentially, it is a documentation of positions taken, without a deeper analysis. Trouvé understood “élémenter” in the traditional sense, as “simplifying the contents of teaching”, a first form of what would later be called “transposition didactique” (ibid., p. 93).
 3.
In general, one calls elements of a whole the primitive and original parts, of which one might suppose that this whole is formed.
 4.
These propositions, united in one body, will properly constitute the elements of science, since these elements will be like a germ, which it would be sufficient to develop in order to know the objects of science in great detail.
 5.
Recently, Barbin has proposed a seemingly related notion: élémentation. It means, according to her, “the process by which a science is organized in view of its presentation or its teaching, and especially in the case oft he writing of a textbook” (Barbin 2015, p. 41). This notion is rather near to transposition.
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