Three past mathematicians’ views on history in mathematics teaching and learning

In the HPM (History and Pedagogy of Mathematics) literature, the three mathematicians Henri Poincaré (1854–1912), Felix Klein (1849–1925), and Hans Freudenthal (1905–1990) are often mentioned in relation to having opinions about the role of the history of mathematics in the teaching and learning of mathematics. Such mentioning often appears without any further account of their viewpoints. In the paper, we seek to dig more deeply, asking the questions of what these three mathematicians’ views on history in mathematics education were, how they themselves expressed these, and what the similarities and differences might be. Our departure point was the known texts by the three mathematicians in which they touch upon aspects of the history of mathematics in relation to teaching and learning, or mathematics education in general. When possible, these written works were accompanied by secondary accounts. Finally, we draw perspectives concerning the issue of how the views of these three mathematicians may or may not align with some of the trends in modern day HPM research.


Introduction
In order to establish itself as an autonomous field of research and to maintain its conceptual coherence, any science in the humanities needs permanent investment in its own history.This is the most necessary as HPM, although being a comparably old field of activity, is only a small 'department' in the field of mathematics education research.This paper is a contribution to such work, and should be seen in the tradition of, for example, studies on Florian Cajory (1859-1930) (Fried, 2014, pp. 675-677) and Otto Toeplitz (1881-1940) (Schubring, 1978, ch. III.1;Fried & Jahnke, 2015).A similar example from the history of mathematics is the work of Lützen and Purkert (1994).It goes without saying that none of these studies-including the present one-can be considered as the final word.In the following, we take a closer look at three past mathematicians' writings on matters related to the role of history of mathematics in mathematics teaching and learning, namely, Henri Poincaré (1854Poincaré ( -1912) ) from France, Felix Klein (1849Klein ( -1925) ) from Germany, and Hans Freudenthal (1905Freudenthal ( -1990) from the Netherlands.
Poincaré and Klein were contemporaries, and for a time worked in the same mathematical field, whereas Freudenthal is of a generation 50 years later.We have chosen these three mathematicians as a starting point for our investigations into past prominent mathematicians' views on HPM related matters for two reasons.First, they are often mentioned in writings about the role of history of mathematics in mathematics education, and thus, they still seem to offer much inspiration to educators and researchers of today.Yet, they are often mentioned without explanation of their points of view and without going into further details.Second, despite the different periods in time, the pedagogical questions that these three mathematicians had to face were, in a certain sense, comparable, insofar as in both periods the deductive and formal structure of mathematics was a much-discussed educational issue.In the first period, this concerned the 'arithmetization of mathematics', and in the second period the influence of Bourbaki's structuralist view on the New Math movement basing the entire field of mathematics, especially the concept of number, upon set theory (e.g., Beckers, 2019;Phillips, 2015).

Research questions and methodology
In relation to our study of these three past mathematicians, a conjoint interpretative analysis of conceptions of different authors required the following: (1) the reconstruction of their individual conceptions; (2) identifying relations (similarities and differences) between the individual conceptions within the group under study; and (3) relating these views to present day research.Consequently, our study is guided by (a bundle of) the following three research questions: 1. What were these three mathematicians' views on the role of history of mathematics in mathematics education?2. What are the similarities and differences of the conceptions of these three mathematicians?3. How are the views of these three mathematicians aligned (or not) with some of the trends in modern day HPM research?
The first question is answered in the respective sections on the individual mathematicians, while the second and third questions are addressed in the final, discussion, Sect.6.
In relation to methodology, our departure point was the known texts by these three mathematicians in which they touched upon aspects of the history of mathematics in relation to mathematics education.For Poincaré, this entails especially his two publications in the Swiss journal L'Enseignement Mathématique from 1899 and 1904, but also parts of his philosophical essays The Value of Science and Science and Method (the English translations-in which also parts of the articles in L'Enseignement are translated into English).For Klein, our departure point was his published works.Yet, in his case, it has been possible to also consider the courses offered at the time at Göttingen Mathematical Institute for pre-service teachers, as well as in-service courses from 1892 to 1912.It was not possible to carry out such investigations in the cases of Poincaré and Freudenthal.For Freudenthal, we trawled his published books, in English, for any mention of 'history', thereby identifying several instances in which he addressed the role of the history of mathematics in mathematics education.
The reader should be aware of the fact that these three mathematicians only on a few occasions wrote about the relation between history of mathematics and its teaching and learning, and this was done in a rather unsystematic way.Hence, interpretations are necessarily delicate.
Furthermore, hermeneutics tells us that interpreting texts requires a careful consideration of language and of the way an author expresses ideas (Jahnke, 1994).This is even more necessary in the present study, since the general intellectual climate of the time around 1900 massively influenced the way that Poincaré and Klein expressed themselves and, to a certain extent, is still visible in Freudenthal's writings.This leads us to a last qualification: we would like to stress that we compare conceptions and points of views and do not make any statement about mutual influences.

Poincaré
Henri Poincaré (1854Poincaré ( -1912) ) is well known both as a scientist and a philosopher.1 His views on mathematics education and the role of history in the teaching of mathematics are published in his essays, in particular in "La logique et l'intuition dans la science mathématique et dans l'enseignement" from 1899.As the title indicates, his ideas of teaching and education were closely linked to the roles he attributed to intuition and logic in mathematics.
According to Poincaré, two different kinds of minds are at work when mathematicians do mathematics.One is occupied by logic and the other is guided by intuition: It is impossible to study the works of the great mathematicians, or even those of the lesser, without noticing and distinguishing two opposite tendencies, or rather two entirely different kinds of minds.The one sort are above all preoccupied with logic; to read their works, one is tempted to believe they have advanced only step by step, after the manner of a Vauban who pushes on his trenches against the place besieged, leaving nothing to chance.The other sort are guided by intuition and at the first stroke make quick but sometimes precarious conquests, like bold cavalrymen of the advance guard.(Poincaré, 1907, p. 15) These two ways of working are neither decided by the mathematical discipline in question, nor by the training of the mathematician, they are inherent in the individual mathematician, since "the mathematician is born, not made."(ibid., p. 15) However, the mode of intuition is associated with geometry, and logic with analysis; so, Poincaré observed, "it seems he [the mathematician] is born a geometer or an analyst."(p.15).
Poincaré pointed out that both sorts of minds are essential for scientific progress, as are both analysis and synthesis.He used Euclid's Elements as an example where one "without much effort, recognizes the work of a logician" though "each piece is due to intuition" (ibid., p. 17).However, contrary to what one could then think, he emphasised that minds do not change, but ideas about mathematics do, and it is the readers, he explained, who have pushed for compromises.The reason for this is the demand for rigour, which intuition cannot provide, and which became more and more in focus in Poincaré's time.He mentioned the well-known pathological example of a continuous function, which is nowhere differentiable: "Nothing is more shocking to intuition than this proposition which is imposed upon us by logic" (p.17)-a function which we cannot imagine in intuition.Poincaré (1907) was critical towards the consequences of rigour for the image of mathematics.He stressed: "[I]n becoming rigorous, mathematical science takes a character so artificial as to strike every one; it forgets its historical origins; we see how the questions can be answered, we no longer see how and why they are put.[…] intuition must retain its role as complement, I was about to say, as counterpoise or as antidote of logic" (p.21).
He was not against rigour as such, what he warned against was losing sight of intuition, which he found had a huge role to play both in research and in teaching.He emphasised that the intuitive notions of our ancestors, even if we no longer subscribe to them, have influenced the way notions are logically constructed in our time, so they still have an imprint.This is necessary knowledge for the "inventor", as it is for "whoever wishes to really comprehend the inventor" (ibid., p. 23)-that is, both for researchers and for students.In this sense, both logic and intuition are necessary for mathematics.Logic gives certainty and intuition paves the way for inventing new knowledge.In Poincaré's view, logic is the instrument of demonstration and intuition is the instrument of invention.
As also noted by Poincaré, there seems to be a contradiction here.This is because he on the one hand claimed that there are two kinds of mathematical minds (logicians and intuitionists), and on the other hand associated logic with analysis and intuition with geometry.Does this mean that analysts cannot make inventions and geometers cannot produce results free from uncertainty?"No."Poincaré mentioned two 'tools' used by logicians that guide them in making generalisations, one is mathematical induction and the other is analogy.That is, he did not exclude analysts as inventors, but he considered them a rare species: "We must admire them", he wrote, "but how rare they are!" (ibid., p. 25).
Poincaré's thoughts about the two different kinds of minds among working mathematicians had an impact on his thoughts on mathematics education as he discussed it in his 1899 essay in L'Enseignement Mathématique.Here he pointed towards the different roles of logic and intuition, with logic for the sake of rigour and intuition for the sake of inventing new mathematics, and he emphasised that rigour had been achieved in mathematics at the cost of mathematics moving away from reality.He illustrated this trend with the appearance of what he called "a whole host of bizarre functions … which seemed to strive to resemble as little as possible honest functions which are useful for something."(Poincaré, 1899, p. 158), e.g., the function mentioned above that is continuous and nowhere differentiable.Such functions, Poincaré claimed, are invented "to put at fault the reasoning of our fathers, and they serve no other purpose" (p.158).
If logic and rigour were our only guide to teaching, Poincaré reasoned, the teacher should begin with these weird functions, the weirdest ones in fact, since these are the most general.Poincaré tells us, this is what absolute logic would like to condemn us to; and he asked the question-rhetorical for him-"must we make this sacrifice?"to which he answered, "No."Teaching is for students, not for the teacher, so, in Poincaré's opinion, the teacher must endure teaching reasoning, which however does not satisfy him entirely.Instead, he must occupy himself with the minds of students.This has the consequence, Poincaré wrote, that instead of logic, when it comes to teaching, history of science must be our guide.The task of the teacher is to let the student's mind go through the same stages as its ancestors.He argued for this with reference to the genetic principle from zoology, that what counts for the development of the embryonic also seems to hold true for the development of the human mind (Poincaré, 1899, p. 159).Here Poincaré invoked what is called the 'biogenetic law' or the 'recapitulation principle' (Fried, 2014, p. 676).A principle that has been widely discussed in the mathematics education literature relating to HPM (e.g., see Fried & Jahnke, 2015;Furinghetti & Radford, 2002;Schubring, 1978).Poincaré (1908) was not against rigour in mathematics education, but his main concern was understanding.Or as phrased by Gray (2013), "The question of why something is true was much more important for Poincaré than the question of what is true" (p.12).In teaching, rigor should be introduced when it forces itself upon the student, so to speak.When the student is mathematically more advanced, mature enough, doubts will arise by themselves, Poincaré claimed, and then the teacher's proofs will be welcome, but not in the beginning.Because, he wrote,, "it's not enough to doubt everything, one also has to know why one doubts" (Poincaré, 1899, p. 160).Sticking to formal logic as our guide in teaching raises the question of why we chose these particular combinations rather than others: The logician cuts up, so to speak, each demonstration into a very great number of elementary operations; when we have examined these operations one after the other and ascertained that each is correct, are we to think we have grasped the real meaning of the demonstration?Shall we have understood it even when, by an effort of memory, we have become able to repeat this proof by reproducing all these elementary operations in just the order in which the inventor had arranged them?Evidently not; we shall not yet possess the entire reality; that I know not what which makes the unity of the demonstration will completely elude us.Pure analysis puts at our disposal a multitude of procedures whose infallibility it guarantees; it opens to us a thousand different ways on which we can embark in all confidence; we are assured of meeting there no obstacles; but of all these ways, which will lead us most promptly to our goal?Who shall tell us which to choose?We need a faculty which makes us see the end from afar, and intuition is this faculty: it is necessary to the explorer for choosing his route; it is not less so to the one following his trail who wants to know why he chose it.(Poincaré, 1907, pp. 21-22) The choice, he argued, can only be explained by remembering the intuitive notions that were replaced by these choices; if we do not have these memories, the choice will seem unjustified.Instead, we should use history as a guide, Poincaré argued, by following the path of our ancestors, being exposed to the trial and errors that led them to choose the path they took instead of another, will provide understanding.If we give the final form that logic imposes upon it right away, he claimed, we do not reach understanding; students will not really understand it and they will not be able to retain it unless they learn it by heart.
In 1904, he published yet another essay, "Les définitions générales en mathématiques" where he further discussed his views on teaching and understanding of mathematics.
Here he also gave concrete examples regarding the consequences of his ideas for teaching.The study of differential calculus, for example, can be approached in two ways, he said, that of Lagrange, which he found to be that of Newton, and that of Leibniz.Both approaches should be known; the question he addressed was with which one to begin.It had varied, he said, over time, whether one first introduced the derivative or differentials in the teaching, but he warned against introducing differential calculus in secondary school through differentials.Before getting exposed to differentials, students should first learn to think in derivatives, preparing the definition for the students by using concrete examples of tangents and velocity, and these are, he said, "not to be despised since the first was the starting point of Fermat and Roberval, the second that by Newton"2 (Poincaré, 1904, p. 277).Only when students are familiar with derivatives, and know how to calculate and handle derivatives, can they be exposed to differentials, he wrote.
The same goes for the definition of the integral, he said, following "our fathers [who] inscribed in a plane area a series of rectangles and obtained as a limit of these rectangles an integral that represented this plane area" (ibid., p. 278).The problem is, he said, that today this reasoning does not satisfy us, and so "to define an integral, we take all kinds of precautions; we distinguish between continuous functions and discontinuous ones, those which have derivatives and those that don't" (p.278).This is fine at university, but it would be "detestable" in high schools, he wrote, the students would never come to know what an integral is if we did not show it first.So, he said, […] what remains to be done is quite simple: define the integral like the area between the x-axis, two ordinates and the curve, show that when one of the ordinates moves, the derivative of this area is precisely the ordinate itself.This is Newton's reasoning, it's like that the integral calculus was born, and for better or for worse we must go through what our ancestors went through.(p.279) As we have seen, intuition plays a central role in Poincaré's philosophy of mathematics; it is required for mathematical knowledge.In his talk at the International Conference of Mathematicians in 1900, he spoke about intuition, introducing it as a faculty of the mind, which is necessary in order to create new results.Understanding in mathematics is not only about following logical deductions, he said, another kind of ability, mathematical intuition, is required.As we saw above, according to Poincaré, students need to be exposed to, follow, the historical origin of mathematical concepts for the learning of mathematics.Taking the above quote into account, an essential issue for Poincaré was the creation of mathematics, mathematical practice.Hence, in an educational setting, he was of the opinion that students should somehow follow the "trail of the explorer" to cultivate the faculty of intuition.
In addition, Klein, as elaborated below, expressed the opinion that "it is not possible to treat mathematics exhaustively by the method of logical deduction alone, but that, even at the present time intuition has its special province", referring to Poincaré as one of the "most active mathematical investigators of the present day … [who] originally made use of intuitive methods" (Klein, 1895, p. 246).
It seems to us that Poincaré's views on teaching are intimately connected to and founded on his philosophical thinking.Concerning mathematics proper, Poincaré was an intuitionist, in the sense that he was of the opinion that a formal proof in itself did not provide mathematical knowledge (McLarty, 1997, p. 97).Mathematics is more than a set of formal expressions, it deals with something, and in arithmetic he saw this something in the sequence of whole numbers.With this standpoint, teaching cannot be based on formal proofs alone, students (as researchers) need to think of something.On a more general level, according to Three past mathematicians' views on history in mathematics teaching and learning: Poincaré,… 1 3 Heinzmann and Stump (2021), Poincaré "always supports a single position aimed at a reconstruction of the process of understanding scientific theories".Understanding was a central issue in Poincaré's philosophy of mathematics, especially its historical elements (ibid.).
Regarding geometry, Poincaré is known for his geometrical conventionalism.Heinzmann and Stump (2021) interpreted this, based on Gray (2008), "as a kind of hybrid analytical-synthetic expressions 'guided' by experience." The main point here is that geometry lies on the border with physics.Theories of physics, however, are not uniquely determined by experience.To a given set of measurements might fit several different theories.Thus, concerning scientific theories it does not make sense to speak of 'truth' in the traditional sense.Rather, scientists agree in their discourse on the theory they want to follow.Writing about language, Gray (2013, p. 7) explained that in Poincaré's view, "we share the usage of key terms and we strive for objectivity through discourse, not for truth."This then has consequences also for geometry.Regarding the continuum, "this, Poincaré said, was chosen by us as a matter of convenience without which science as he knew it could not proceed, but it was not forced upon us" (ibid.).Therefore, in constructing a theory, a scientist is dependent on the decisions on definitions and theories his ancestors have made.Hence, the history of a theory largely determines how it evolves, and understanding a theory implies necessarily to know a certain amount of its history; a point of view in line with having students learn from past mathematicians' creations of how and why they developed the ideas that led to the concepts students are supposed to learn.
Poincaré did not discuss his concept of 'history' in his essays on teaching, which is of course an important issue for the HPM community to reflect upon-the underlying concept of history, when it is integrated in the teaching and learning of mathematics.
In summary, for Poincaré we have seen that his ideas about teaching reflect his philosophical ideas of mathematics and science.The central notions are intuition, understanding, and geometrical conventionalism, as unfolded above.Intuition, as a special faculty of the mind necessary for creating mathematics, causes earlier intuitive notions of our ancestors to leave an imprint on the further logical construction of the notions in questions.Hence, Poincaré's philosophy of mathematics makes it a coherent standpoint to require that teaching and education take its guidance from history and strive to imitate the working mathematicians of the past.

Klein
For twenty years, Felix Klein (1849Klein ( -1925) ) was deeply involved in the reform of mathematics teaching at German secondary schools, and author of an excellent book on the history of mathematics in the nineteenth century (Klein, 1926).History of mathematics also plays an important part in volume 1 of his Elementary Mathematics from an Advanced Standpoint (Klein, 1908).In the following, we give some information on the development of Klein's involvement in teacher training and the courses for teachers of mathematics offered at the Göttingen Mathematical Institute and discuss his published programmatic statements on history and education.
Klein took up his professorship at the University of Göttingen in 1886.Around 1890 he became increasingly interested in problems of teacher training.When continuing education courses for teachers of science and mathematics were established in 1892 at the University of Göttingen, Klein took an active part in forming their mathematical component.One could expect that history of mathematics should have played a part in these courses if Klein had attributed any value to it.Nevertheless, this was not the case.Table 1 shows the mathematical themes that were treated between 1892 and 1912 (the year of Klein's retirement). 3n Table 1, one can observe trademarks of Klein's educational thinking, for example, an emphasis on applied mathematics and on models and instruments.At the time, topical developments such as Klein's reform ideas, condensed in the so-called 'syllabus of Meran 1905', played a role.Yet, there is no history of mathematics.
The picture changes slightly when we look at the programs of lectures of the Göttingen Mathematical Institute during the same period.Confining ourselves to mathematics in a narrow sense (leaving aside astronomy and theoretical physics), there was no course in the history of mathematics before 1908.In this and some subsequent years, however, there were the announcements displayed in Table 2.
Thus, only for some years after 1908, courses in history of mathematics were offered at the Göttingen Mathematical Institute.Klein had taught the course on Elementary Mathematics from a Higher Standpoint, which gave rise to the book (Klein, 1908) in the winter semester of 1907.It was this book in which he most intensely intertwined educational and historical considerations, and it seems very plausible that this work led him to his efforts of introducing historical courses into the mathematical syllabus.
We turn now to some of Klein's programmatic statements.On many occasions and in many papers Klein discussed his ideas related to training of teachers of mathematics.We confine ourselves to three texts.The first one is chapter 12 of the Evanston Lectures (Klein, 1894) which he delivered in 1893, that is, at the beginning of the period we are considering.The other two texts refer to the so-called 'biogenetic law' and are from 1895 and 1908.
In the Evanston Lectures, Klein reported that Göttingen mathematicians distinguish between 'general' and 'higher' courses.A general course was intended for the large majority of students who were planning to become teachers of mathematics and physics, whereas a higher course would address students whose "final aim is original investigation" (Klein, 1894, p. 94).Future teachers of mathematics had to pursue, in addition to lectures on pure mathematics, a "thorough course in physics.Astronomy is also recommended" (p.95).Klein himself "believed that the technical branches, such as applied mechanics, resistance of materials etc. would form a valuable aid in showing the practical bearing of mathematical science" (ibid.).Geometrical drawing and descriptive geometry were also part of the course for future teachers, and, as expected, there was no history of mathematics.
(1) In his 1895 paper on arithmetization, Klein presented a view on mathematics as an interplay of logical rigor and arithmetization on the one hand, and geometrical intuition on the other.Only in a short paragraph at the end, he made some remarks on the pedagogical side of mathematics and argued that at universities the introductory mathematical lectures should start from intuition "because the learner naturally always goes through the same course of development on a small scale as science has gone through on a large scale" (Klein, 1895, p. 240).This, undoubtedly, was an enunciation of the 'biogenetic law', but Klein did not use this very term, nor did he mention the 'inventor' of the biogenetic law, Ernst Haeckel (1834Haeckel ( -1919)).Two years later, in a talk at the 1897 meeting of the German Mathematical Society (DMV) (Pringsheim, 1898), Alfred Pringsheim (1850-1941), polemicizing against Klein, defended the courses which based analysis on a rigorous theory of real numbers.He cited Klein's argument explicitly as Haeckels' principle of the agreement between phylogeny and ontogeny (Pringsheim, 1898, p. 74).After some critical discussion, Pringsheim concluded that "the principle cited by Mr. Klein does in no way appear valid and at the very least needs to be examined closely case by case" (p.75).Therefore, he stated a modified principle in the following way: "Every individual goes through essentially the same course of development as science itself, as long as he is not shown a better way."(ibid.)To be clear, although Pringsheim's whole discussion shows that he considered the principle as more or less irrelevant, he nevertheless did not refute it in toto, but restated it with a limiting side condition.
Klein extensively replied to Pringsheim at the DMV meeting of the following year (published 1899), and again one of his arguments was that history proves the indispensable role of geometrical intuition.Concerning the biogenetic law, Klein cited word-for-word the respective sentences in his arithmetization paper (Klein, 1899, p. 127), but used the term 'phylogenetic law' only as an indirect quotation from Pringsheim (1898, p. 128).Klein in no way, however, entered into a discussion or a defence of the biogenetic law, but simply ignored Pringsheim's remarks in this regard.Haeckel was a proponent of a strict and unconditioned recapitulation of phylogeny by ontogeny (e.g., Haeckel, 1906), and Klein might have felt very uneasy to have been put in this corner by Pringsheim, and would have preferred to express himself in the vague and general way in which he had done so before.Therefore, it seems plausible that Klein, like Pringsheim, did not think that recapitulation should be applied unconditionally in every case, but rather, as Pringsheim had said, believed that it had to be examined case by case.Of course, this tacit agreement did not change their sharp opposition about pedagogical matters.The whole controversy, documented in the papers by Klein (1895Klein ( , 1899) ) and Pringsheim (1898Pringsheim ( , 1899)), with many sarcastic and ironic remarks on both sides, would deserve a separate study.
(2) The second case where Klein referred to the biogenetic law was 13 years later in his Elementary Mathematics from an Advanced Standpoint, vol.1 (Klein, 1908, we quote Klein, 1932).This time, he explicitly used the term "biogenetic fundamental law" on the very last page of the book in a short pedagogical comment on set theory.He used it as a backing for his protest against treating such "abstract and difficult things" too early, and he added the remark that "Such thoughts have become today part and parcel of the general culture of everybody" (ibid., p. 268), not forgetting the limiting qualification that one should follow this law "at least in general" (ibid.).According to Klein, the law says that instruction should guide youth slowly to higher things and finally to abstract formulations.He then entered into a polemic against people "who, after the fashion of the medieval scholastics, begin their instruction with the most general ideas, defending this method as the 'only scientific one'".Klein opened this polemic with the remark that it be necessary "to formulate this principle frequently" (ibid.).
But how does this fit in with the fact that he himself did not cite it in the 267 pages preceding this last page?It therefore seems that the core of Klein's remarks lies in the following two sentences: An essential obstacle to the spreading of such a natural and truly scientific method of instruction is the lack of historical knowledge which so often makes itself felt.In order to combat this, I have made a point of introducing historical remarks into my presentation.By doing this I trust I have made it clear to you how slowly all mathematical ideas have come into being; how they have nearly always appeared first in rather prophetic form, and only after long development have crystallized into the rigid form so familiar in systematic presentation.It is my earnest hope that this knowledge may exert a lasting influence upon the character of your own teaching.(ibid.) The message here is quite different from 'applying' the biogenetic law in order to derive sequences of topics in teaching from sequences of events in history.Instead, Klein hoped that enhancing historical knowledge among teachers at schools and professors at university would exert a lasting influence on their teaching.This is a very indirect influence in the process of becoming an educated mathematician, and beyond very general lessons no concrete conclusions could be drawn from history in regard to teaching.
We do not know whether Klein (or Pringsheim for that matter) really had read any of Haeckel's writings.We only know that in 1877, when he was a young man of 28 years, Klein had heard a talk by Haeckel (Tobies,202,p. 181).But by the end of the nineteenth century, Haeckel and his biogenetic law were surely known to nearly any educated person in Germany (Gould, 1977, ch. 4), and might have been spread more by newspapers and party talk than through actual readings of Haeckel's writings.Thus, when Klein used the term 'biogenetic law', as he also stressed himself, he could count on a certain basic understanding by the public.It is plausible that the 'law' functioned more as a model of what genesis or development of mathematics could mean than as a theory that was actually applied.This is confirmed by looking at how Klein (1908) intertwined historical and educational arguments (Jahnke, 2018(Jahnke, , 2020)).In the third part on analysis, Klein (1932, pp. 144-161) gave a detailed discussion of the elementary transcendental logarithmic/exponential and trigonometric functions (Jahnke, 2018, pp. 228-235;2020, pp. 8-11).Klein criticised the usual practice in German secondary schools of gradually extending the meaning of powers and logarithms from natural to negative, fractional and irrational numbers, thus finally arriving at a definition of the logarithmic and exponential functions.This was a standard procedure of early analysts, but Klein objected that this procedure required a number of definitions and restrictions that cannot be explained to the learner and, therefore, must appear as arbitrary "authoritative conventions" (Klein, 1932, p. 145).
To motivate his own proposal, Klein entered into a long historical digression (ibid., pp. 146-154) from which he concluded that the "simple and natural way" for introducing logarithms at school is its definition as the integral of the hyperbola between the ordinates 1 and x.In fact, however, this was not at all a consequence of history, but the application of a principle taken from the newest mathematics of his time, namely elliptic functions.
The first principle is that the proper source from which to bring in new functions is the quadrature of known curves.This corresponds, as I have shown, not only to the historical situation but also to the procedure in the higher fields of mathematics, e.g., in elliptic functions.(ibid., p. 156) Hence, it was the "higher standpoint" of complex function theory from which Klein considered school mathematics.He even went so far as to introduce trigonometric functions by the integral (Klein, 1932, p. 78), its inversion leading to the sine function (see also Jahnke, 2020, p. 10).
Overall, for Klein (as 13 years earlier for Pringsheim), it was the mathematics of their time that determined their views on educational problems.Nevertheless, it is impressive to see how much Klein got involved in the history of mathematics, when he delivered his lecture on "Elementary Mathematics from a Higher Standpoint" in 1907-1908, and the subsequent edition of the lecture notes (Klein, 1908).This might explain why in the following years some lectures on the history of mathematics appeared in the syllabus of the Göttingen Mathematical Institute.Hence, one cannot deny that the history of mathematics at that time of his life played an important role in his thinking about mathematics and educational matters.But how can we describe this role?There is no conclusive answer to this question.In the preface to the first edition of his Elementary Mathematics…, Klein (1908) wrote that "it has given me especial pleasure to follow the historical development of the various theories in order to understand the striking differences in methods of presentation which parallel each other in the instruction of today" (Klein, 1932, p. iv).And in the introduction, he wrote on the general intention of his lectures: My task will always be to show you the mutual connection between problems in the various fields […] 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 there put before you a living stimulus for your teaching.(Klein, 1932, pp. 1-2) Hence, a historical background might help in understanding differences and connections in the various fields and thus to transform "the great body of knowledge" to a "living stimulus" for teaching.
In summary, concerning the teaching of mathematics, Klein made only casual and rather unspecific use of historical arguments, such as, intuition "proceeds" or "is more important than" logical rigor.Furthermore, only in 1907/8 when he intensely worked on the historical roots of school mathematics did he provide the personal experience of how inspiring history could be and how it might help to transform mathematical knowledge into a living organism.This changed his original opinion on the matter and from then on, he considered history of mathematics to be a worthwhile component of teacher training.

Freudenthal
From a mathematics education point of view, the Dutch mathematician Hans Freudenthal (1905Freudenthal ( -1990) ) is known as the father of the Realistic Mathematics Education (RME) approach to mathematics teaching and learning (e.g., la Bastide- van Gemert, 2006;van den Heuvel-Panhuizen, 2021).A central element here is Freudenthal's idea of 'guided reinvention', of which we already find the early seeds in his Mathematics as an Educational Task: To acquire knowledge is re-discovering not what others knew before me but rather what I myself knew when my soul stayed in the realm of the ideas.We need not devour Socrates to the last morsel and we need not share his belief in pre-existence.What then remains is learning by re-discovery, where now the "re" does not mean the learner's pre-history but the history of mankind.It may seem as though the learner is repeating the development of his ancestors in rediscovering what they knew.Therefore I would prefer to call it re-invention, but this is an unimportant point of terminology.(Freudenthal, 1973, p. 102) Surely, the "history of mankind" is a subtle reference to the biogenetic law.Yet, Freudenthal did not believe that the learner should follow the exact trail of our ancestors.He stated, "It is not the historical footprints of the inventor we should follow but an improved and better guided course of history" (ibid., p. 103).Freudenthal expressed the idea more clearly in Revisiting Mathematics Education-China Lectures: History teaches us how mathematics was invented.I asked the question of whether the learner should repeat the learning process of mankind.Of course not.Throughout the ages history has, as it were, corrected itself, by avoiding blind alleys, by cutting short numerous circuitous paths, by rearranging the road-system itself.We know nearly nothing about how thinking develops in individuals, but we can learn a great deal from the development of mankind.Children should repeat the learning process of mankind, not as it factually took place but rather as it would have done if people in the past had known a bit more of what we know now.(Freudenthal, 1991, p. 48) Freudenthal defined the term 'invention' to be understood as "steps in learning processes, which [are] accounted for by the 're' in reinvention, while the instructional environment of the learning process is pointed to by the adjective 'guided'" (ibid., p. 46).Therefore, the student is the reinventor and the teacher the guide.To the question of how the teacher is to guide the reinvention of the student, Freudenthal replied as follows: [...] guiding reinvention means striking a subtle balance between the freedom of inventing and the force of guiding, between allowing the learner to please himself and asking him to please the teacher.Moreover, the learner's free choice is already restricted by the 're' of 'reinvention'.The learner shall invent something that is new to him but well-known to the guide.(p.48) Striking this delicate balance, however, is not an easy task, since it "depends on such a perplexing manifold of hardly retrievable and only vaguely discernible variables that it seems inaccessible to any general approach" (ibid., p. 55).Freudenthal went on to argue that the reinventor should be guided towards an 'activity' and that this activity should consist in reinventing 'mathematising' rather than mathematics.Without going deeply into Freudenthal's view on mathematising (and mathematical modelling), some further explanation of this relation to history may be found in his Didactical Phenomenology of Mathematical Structures: Our mathematical concepts, structures, ideas have been invented as tools to organise the phenomena of the physical, social and mental world.Phenomenology of a mathematical concept, structure, or idea means describing it in its relation to the phenomena for which it was created, and to which it has been extended in the learning process of mankind, and, as far as this description is concerned with the learning process of the young generation, it is didactical phenomenology, a way to show the teacher the places where the learner might step into the learning process of mankind.Not in its history but in its learning process that still continues, which means dead ends must be cut and living roots spared and reinforced.(Freudenthal, 1986, p. ix) The topic of the role of history in relation to the teacher was explicitly addressed by Freudenthal in a paper-"Should a mathematics teacher know something about the history of mathematics?"-inFor the Learning of Mathematics (an edited translation of a presentation at IREM dated June 17, 1977). 4Here he argued that future teachers should indeed learn more than they are required to teach, and that this 'more' can refer to quantity, but that it can also refer to 'profundity': "Can history contribute to profundity?Yes, provided that it means profundity to the trainer."Interestingly, from an HPM perspective, Freudenthal noted, History is worth being studied at the source rather than by reading and copying what others have read and copied before.Sources are nowadays easily accessible, though astonishingly few know this fact.Whoever is interested in the history of mathematics should study the processes rather than the products of mathematical creativity.(Freudenthal, 1981, p. 34) Yet, for Freudenthal the potential role of history of mathematics in mathematics education is rather different from the point of the trainer, i.e., teacher, as compared to that of the pupils (students).Addressing its potential role in school mathematics, Freudenthal stated, To the majority the past is an amorphous pulp where school instruction has scattered a few glass marbles.I believe it becomes man to understand the past of his race, of the Earth, of the Universe in a structured way, and I will try to contribute to this goal.This to my view is the use of the history of mathematics and adjacent areas: serving history rather than mathematics; rather than the comprehension of mathematics promoting that of history.As a bonus it can aid mathematics too.(ibid., p. 33) But how may this be accomplished?Unlike the case of the teachers, Freudenthal's suggestion is not to have pupils study the works of past mathematicians, but rather to take as the point of departure cultural and societal conventions, traditions, etc., for example, by asking why the day has 24 h, the hour sixty minutes, the minute sixty seconds.Or where the numbers come from, what they point to, what they mean, why we have ten numerals, etc.This is to say, inviting pupils to investigate what is behind these conventions, he suggested "… what is behind it can be worth being consciously experienced.Much of it can mean a start on long excursions, some of it as early as primary school, and most of it would be within a teacher's reach who has been taught how and where to look up data" (ibid., p. 34).Hence, history is not to serve in the role of a supplement, but as something integrated in the back catalogue of the teacher: This is how I understand history of mathematics and sciences in the classroom, and the teacher's intellectual baggage required: integrated knowledge.Integrated because familiar to the teacher and a cornucopia available for instruction, not hidden in drawers that are opened at pre-established moments.(ibid., p. 34) In fact, these viewpoints on the role of history as a means for developing profundity of the teachers, appears to be aligned with Freudenthal's personal experiences, as he expresses these in Didactical Phenomenology…: "I have profited from my knowledge of mathematics, its applications, and its history.I know how mathematical ideas have come or could have come into being" (Freudenthal, 1986, p. 29).
On several occasions, Freudenthal implicitly addressed the topic of the biogenetic law, and argued that such a use of history is not to be striven for-and that a knowledge of the history of mathematics, in fact, enables the educators to avoid this scenario: [...] history can be a guide-line that is not to be despised, albeit one with twists that need not be followed though [...] in the past some people believed that the individual repeats the history of mankind-in fact education and instruction take care that this neither does nor need happen.(Freudenthal, 1986, p. 516) In summary, for Freudenthal we have identified four main issues related to our first research question.Firstly, he integrated his allusion to the 'biogenetic law' into his concept of 'guided reinvention'.Secondly, Freudenthal (1986) pointed to the importance of the role of history in his phenomenological analyses of mathematical concepts.Thirdly, he justified the potential role of history in teacher training by the idea that history contributes to the profundity of our mathematical knowledge.Finally, he took the view that historical interventions in mathematics teaching at schools should contribute more to comprehending the general history (of culture) than to comprehending mathematics.

Discussion
In this last section, we reap the fruits of our analyses above.We first ask for similarities and differences of the three mathematicians' individual conceptions (research question 2).In the texts we have considered, Poincaré and Klein both made a more or less vague use of the biogenetic law.Both argued that history teaches us a general priority or, even better, displays a dominance of intuition over logic.Both were aware that in the sense of temporal order (intuition before logic) there exist exceptions.Both also advocated the idea that there are two different types of mathematicians, namely, one preferring intuition and one preferring logic.In their view, this is a matter of one's genetic disposition or, as Poincaré put it, how mathematicians are born.(For similar ideas with Klein, see Mehrtens, 1990, p. 215ff.).This shows that Klein and Poincaré were living in similar intellectual cultures, where ideas of developmental biology and the rise of genetics played a growing role.
A difference is that Poincaré's reference to the 'biogenetic law' was rooted in his general conception of scientific epistemology, which is his focus on understanding and 'conventionalism' guided by experience.He did not explicitly say so, but we have good reasons for assuming this (see Sect. 3).This was not the case with Klein, who in this regard was a less systematic thinker.As shown, in working on the first volume of his Elementary Math-ematics…, during the winter term 1907/8, Klein took a step beyond the biogenetic law in realising that knowledge in history of mathematics in itself will change the image of mathematics, and help teachers at universities as well as in schools to transform mathematics into a 'living body'.Hence, from 1908 he promoted the offering of historical courses for teachers at the Göttingen Mathematical Institute.
Freudenthal too alluded to the biogenetic law and explicitly quoted Poincaré on this point, however with a subtle difference.Whereas Poincaré had argued that researchers, teachers and students should share the same imprecise, preliminary intuitions, Freudenthal spoke of 'guided reinvention'.On the one hand, this is due to his 'activist' conception of learning ('mathematics as activity').On the other hand, it takes into account the teacher, as well as the completely different situation of a student of today working on a problem compared to that of a past mathematician working within the same topic.According to our judgement, Freudenthal's (1981) paper contains new elements, which we did not find in Poincaré's papers, but which are similar to those of Klein in his later years.Freudenthal now argued that knowledge in the history of mathematics in and by itself could be a valuable component of the professional knowledge of teachers of mathematics.For explaining what history of mathematics could afford, he used the two terms of 'profundity' and 'integrated knowledge', which explain better than Klein's wording what the history of mathematics can contribute.In contrast to Klein, Freudenthal also supported the idea of including cultural-historical discussions, e.g., why a circle has 360 degrees, etc., into mathematics teaching at schools.Neither Freudenthal nor Klein, however, envisioned an actual integration of the history of mathematics per se into the mathematics classrooms at school.
How then is all this aligned (or not) to some of the basic trends in modern day HPM research (research question 3)?In a first approximation, the basic philosophy underlying modern day HPM research is characterised by the idea that it is more fruitful to stress the sometimes-considerable tension between the historical appearance of a mathematical topic and its modern counterpart, than trying to smooth this difference and adjust the historical concepts in question to the ideas a student already has.This is a strategy of 'cognitive dissonance' (Pengelley, 2011).A survey on the use of historical sources in the classroom used the concept of reorientation to express this new approach (Jankvist, 2014).History challenges the perception of the learner by making the familiar unfamiliar-the idea of dépaysement (Barbin, 1997)-and working on this difference causes a reorientation of the student's views and a deepening of the mathematical understanding (Jahnke et al., 2000).
HPM related research today thus offers a myriad of examples of including elements of the history of mathematics into teaching, not as integrated knowledge as Freudenthal suggested, but through the use of examples, problems, etc. from the history of mathematics (e.g., Fauvel & van Maanen, 2000) in K-12, etc.In particular, a new movement on the use of primary historical source material, i.e., works of past mathematicians, has gained ground within HPM during the past decades.Even though Freudenthal thought that history should be studied at the source, he only intended such endeavours for mathematics teachers.Current HPM research provides several examples of promising empirical results of having students read and work with primary historical material (e.g., Barnett et al., 2021;Glaubitz, 2011;Jankvist & Kjeldsen, 2011;Kjeldsen & Blomhøj, 2012).As to the use of the history of mathematics in teacher training, HPM is also capable of displaying positive results (e.g., Arcavi et al., 1982;Clark, 2012;Jankvist et al., 2020).Still, to the best of our knowledge, the history of mathematics is not a topic that permeates mathematics teacher education worldwide.
So, while some of the trends of modern HPM research can be considered to be in line with our three past mathematicians' views, there are others that are in contrast to them.As seen above, Poincaré, Klein and Freudenthal all referred to the biogenetic law.Yet, in practice, all three of them made only a very casual use of it.Our analyses reported in this paper to some extent dispel 'the myth' that the historical roots of the HPM perspective lay in 'recapitulation theories'.Furthermore, research in the modern field of HPM has surely gone beyond these traditional issues and their roles in teacher education and classroom teaching.A multitude of new issues and conceptions, both theoretically and empirically supported, have emerged within HPM over the last 40 years.
Funding Open Access funding enabled and organized by Projekt DEAL.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material.If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.To view a copy of this licence, visit http:// creat iveco mmons.org/ licen ses/ by/4.0/.