I have argued, or tried to argue, for some time that modern mathematics education is not very much an ”enlightenment” phenomenon, but to the contrary much more the consequence of a reaction against the enlightenment, that took place around the middle of the 19th century in Germany. This not only means that mathematics education is not primarily concerned with emancipation and science, but also that it has a deeply ”religious” core. What I argue is that many of the central tenets of mathematics education can be traced back to certain forms of protestant Christianity, even though all religious language has now disappeared, since the movement of progressive education in the early 20th century.
My thesis is thus similar to what Max Weber and others have argued in relation to modernity at large: that its rationale can and should be understood as a secularized form of Christian theology. But I will be more specific, and talk only about mathematics education, as a combination of a set of practices, a way of talking about these practices, and an institutional form that gives it a role and function in society. One can say that I view mathematics as what Louis Althusser called an ”ideological state apparatus”.
What I want to do here is to give some empirical evidence for this thesis. I will do that by translating extracts from and commenting upon a book, published in 1979, by a German named Rolf Braun about the educator and philosopher August Wilhelm Grube. After Adolph Disterweg, Grube was one of the most influential writers in the emerging field of mathematics education in Germany. In particular, he was known for the ”monographic method”, where teaching is focused on one number at the time. Grube’s only book on mathematics education, Leitfaden Fur Das Rechnen in Der Elementarschule. Nach Den Grundsatzen Einer Heuristischen Methode, was originally published in 1842. It became immensely popular and came out in (at least) six further editions, the last one in 1881. While his method was widely used (and is to some extent still used today!), it was also the subject of severe criticism. Importantly, however, the impact of Grube’s Leitfaden can be seen in his firm position in the discourse throughout the latter half of the 19th century, and to some extent even in the first decades of the 20th. Eduard Jänicke, a chronicler of the field of mathematics education as it was consolidated in the 1880’s, wrote:
It has been some time since Grubes ideas took hold and set fresh new roots into the soil of school mathematics (Schulrechnen). Only with the second edition of the Leitfaden (1852) did one begin to think through his thoughts, talk about them, implement them. Diesterweg gratefully welcomed the reform proposals; aspiring teachers sought to try them out; others used the given impetus and hastily wrote a new book on arithmetic à la Grube. On the agenda of teacher conferences and on the front of the teachers magazines was only one issue that stirred the spirits, which became the shibboleth and needed to be settled: Grube pro et contra. (Rolf Braun, August Wilhelm Grube – Mathematikunterricht und Erziehung, Peter Lang, 1979, p. 66 – henceforth, if no other reference is given, page numbers refer to this book).
Maybe a good starting point is the state of the discussion of education in the 1850’s, as characterized by Adolf Diesterweg. There were two sides, one who
wants to educate (erziehen) the human so that she fits with the state, the church, the family, etc. as these now happens to be, and thus take the impermanent and changing social institutions as the point of departure for the education. (p. 46)
the other side however wanted to
use psychology to determine the nature and proper state of the human and then base the aim and means of education on the result of this research. They expect from a generation educated in the spirit of the eternal concept of humanity, a constant activity of reform of social institutions. (p. 46)
We have here basically an opposition between on the one hand a conservative view of education as a means of introducing and adapting young people to society as it is and on the other hand a ”progressive” view of education as a motor of social change. It should be noted that we are in the 1850’s, well before the ”movement” of progressive education. And an important difference is that the progressive at this point in time was not yet secularized, but more or less deeply christian. Grube positioned himself between these two camps, and Braun suggests that this is the reason why he got such a big following in the 1850’s and 1860’s. (p. 47)
Introduction to Grube’s Educational Theory
The basic point of departure for Grube was to find a middle path between what was then called ”formal” and ”material” education. He found the way to do this in a principle of ”aesthetic education”, inspired by indirectly by Friedrich Schiller. In the form Grube found it, the aesthetic education was focused on a ”feeling for” and ”theoretical insight in” language. Grube however applied these ideas on mathematics education.
Grube was dissatisfied by the tendencies of his day in education. He was critical of ”modern education” that created ”a rift, an inner gulf, a screaming disharmony”. He meant that modern education tears body and mind apart and ”makes one-sidedness a rule and necessity in life”. The consequence of industrialization and specialization, Grube wrote, ”was that people are intellectuals, artists, civil servants, craftsmen, farmers – all kinds of people, only not humans in the full sense of the word”. (p. 51, my emphasis)
And contributing to this was also the atheism and materialism of the times. Grube wanted to mitigate the effects of all these tendencies with his ”aesthetic” mathematics education.
Interestingly, Grube saw all of these tendencies of his day as effects of one underlying cause, namely ”abstracting thinking” (abstrahierenden Denkens), in that it abstracts from the reality of things, and then only acknowledges the ”law of thought”. Grube argues that this emphasis on thinking results in a ”raw enjoyment in thinking” in that it makes ”the subject to world”: ”when the mind is intoxicated by its power, it replaces God himself”. He blames philosophy, and I suppose that it is philosophers such as Johann Gottlob Fichte that he is (implicitly) referring to. Grube sees the problems in politics, religion, in art and philosophy as caused by lack of depth of feeling and lack of unity between ”heart and reason”. With his aesthetic education Grube wanted to achieve harmony between the spiritual and corporeal existence of the human being. He meant that this unity was reached in an ”aesthetic state”. (p. 52)
Grube thought that this goal could only be reached through an object by which the pupil is moved, and it is in light of this idea that we can understand the rationale of the monographic method. I quote Rolf Braun:
The task of all teaching is thus for Grube to put the pupil in an aesthetic state in and through teaching. According to him, this is only possible when ”each stage of the teaching rises to a finished and accomplished whole, so that the idea that permeate and enliven the teaching object can be felt”. Only then will the disposition [Gemüt] of the pupil be moved and not only individual capacities and powers of the pupil be addressed. Then the pupil will be put in an aesthetic state by the teaching object, that is, the teaching will be educational [bildungswirksam].
With this concept of the aesthetic, that Grube puts in the center of his educational theory [Gemütspädagogik], he wants to acknowledge the necessity of letting the teaching object move the subject, the pupil, in her totality, that is, the pupil must be addressed ”holistically” by the teaching object, so that all of her capacities are activated. However, the teaching object must in itself be suitable for causing this aesthetic effect in the pupil.
What objects are suitable for having a ”holistic” effect on pupils? What objects can ”move the core” of a pupil? In fact, at the time, many different ”objects” were used in teaching for similar purposes as those that Grube describes here, for instance stories from the Bible, and historical biographical sketches, that conveyed messages building character, or, in Grube’s terminology, moving the disposition of the pupil towards a unified whole. It is in light of this purpose of teaching, and the idea that the pupil should be moved by the object, that we should understand the monographic method and its focus on individual numbers. The numbers are to be treated in a way the puts the pupil in an ”aesthetic state” and moves her by its inner properties.
But what is the teacher going to do if it is by the ”object” that the pupil is to be moved? The teacher is to ”lead” the pupil to let herself be moved. Some of the terms used at the time to talk about this was ”the heuristic method”, ”the socratic method” or the ”katechetic method”. Its characteristic feature is that the teacher ”guides” the pupil (as Socrates did with the slave Meno) by means of well chosen questions, rather than ”lecturing” about the subject. One of the educators that influenced Grube wrote:
The form of the educating (erziehenden) teaching is developing, and to this end the teacher stimulates (erregt) the pupils, so that they enhance their power by their own efforts of searching for the answer. […] he thus puts the pupils in self-activity, in which they learn out of themselves. […] The teacher supports the drive for learning (Bildungstrieb), that is, the desire and attention of the pupils, at the same time as he, perhaps unnoticed, always leads the way. (p. 58)
We can here see a specific instance of what is sometimes called the ”pedagogical paradox”, arising from the incompatibility of two doctrines: on the one hand that the pupil should ”develop”, freely, through ”self-activity”, and on the other hand that this development should lead to a pre-determined goal of ”Bildung”, character, moral and later also useful knowledge. The teacher has the complex task of ”leading the way”, while never actually ”showing” the pupils the way. The purpose of teaching is not the ”movement”, but the effort needed to move oneself – it is the power of self-movement that is the be developed in education, and the teacher has to ensure that what is developed is a power with direction.
For Grube, and in connection to elementary mathematics education, this general doctrine concerning education meant that:
not technique and virtuosity in calculation can be the main thing, but rather, through calculation, the achieving of clarity of perception, integrity of judgement (Selbsttärigkeit im Beobachten), freedom in construction (Kombinieren) – put shortly, a mathematical education (Bildung), that is more than just skill. (p. 59)
Grube wanted the pupils to become independent, to get a ”power” through the school and through the teaching, to manage the everyday life. But this could not be achieved through the already then common ”word problems” of calculation applied to practical life. To the contrary – and somewhat paradoxically – Grube meant that education can only be ”practical” insofar as it is based on the principle of the ”formation of disposition”, what in German was called Gemütsbildung, and on morality. Only when the education transforms the inner core of the pupil in the direction of morality would it be ”practical”, as Grube understood this word.
This, now, is what I am getting at – that the discourse on mathematics education, its doctrines and theories, did not emerge as means to promote the goals of today, of ”creativity”, ingenuity, effectiveness, entrepreneurship and the like – but to the contrary, as Grube here exemplifies, to make mathematics education into a rather conservative if not reactionary force, promoting morality and ”depth of feeling”. Grube confuses matters by calling the possession of such capabilities ”practical”; what he refers to is basically the capacity to stand back from practical life and perceive it in its ”truth”, to see the ”core” of things themselves, beyond what Heidegger would call their ”readiness-to-hand”, and this core was for Grube, as for many of his contemporaries, deeply Christian.
Grube searched for a middle path between the ”camps” in education, described above referring to Diesterweg. One way to describe these camps is in terms of subjectivism and materialism. There were then the two evils of present day mathematics education for Grube. The subjectivism derived its tenets from philosophy, from what in philosophical terms is called subjective idealism, that when applied to education wanted the subject to ”make its own world”, through a kind of expansion of the mind propelled by ”self-activity”. The materialism, on the other hand, could rather be seen as a force originating in the practical life and in what today would be called ”economy”. It took the existing world, rather than the subject, as its point of departure, and had as its goal the adaption of the subject to the presently existing.
It is interesting to note how well this dichotomy fits with our present. The ”subjectivism” of today is ”progressive” education, in particular in the form it sometimes took in the 1970’s, influenced by the student revolts of 1968. In Swedish it is called ”flumskola”, with its focus on the ”inner” development of the children, unconcerned with the harsh realities of work and economy. The ”materialism” is of course also massively present in the pressure from the economic sphere, wanting to shape education in its image.
Most important, however, is to note that it is actually the ”middle path”, of Grube and others, that dominates education today. It is a paradoxical and complex combination of the respective ”cores” of on the one hand a subjective idealism that wants to expand the power of the subject indefinitely, and on the other hand, a materialist determinism, that takes as its sole point of departure the ”necessities” of the already existing outside world of the market. This, I think, is why mathematics education – and the education system at large – stands to firmly in modern society. It is a ”compromise formation”, making it (seemingly) possible to have two opposite things at the same time: on the one hand emancipation of the human subject – as expressed in educational theories deriving from German idealism and its romantic successors – on the other hand a practical reinforcement of social structures and hierarchies.
What we can see in Grube is the formation of an educational doctrine fitting this double and somewhat impossible goal. Grube’s innovation was to use numbers as the ”objects” that were to move the core of the pupils. It is not difficult to see how they can fit very well with the intention of finding a middle path between the subjective and the material. On the one hand, mathematics was already associated with the inner core of humanity. A great philosophical predecessor is Baruch Spinoza, who in his Ethics considered ”mathematical thinking” to be the only truly free thinking that humans were capable of. Before him, Descartes had said that when we have mathematical ideas in our mind, we have them, in some way, ”in God”, that is, when we think mathematically, we think ”like Gods”. Kant made geometry and arithmetic into ”bridges” between the subject and the phenomenal world, and thus positioned mathematics equally in the core of humanity and as the essence of the (human) world. He did this in an attempt to make sense of the physics of Newton, and more generally, the idea of mathematics somehow ”connecting” something deeply human with something essential of reality, is a founding idea of modernity. What is often forgotten is that this idea was, by way of ”common sense”, fitted into a christian cosmology up to the very end of the 19th century. Thus, the inner core of humanity, the essence of the world, as well as the ”bridge” between them, until then, were usually interpreted in relation to the presence and power of some kind of Christian divinity. From this perspective, mathematics could enter the education system as part of a ”humanistic” curriculum, siding with the classical languages. On the other hand, mathematics has been conceived of as a tool for instrumental action. Thus its place in education could be argued for in two complementary ways, partly as a means of formation of the inner core of the pupil, to an ideal of ”harmony” with something truly human, fitting the essence of what is truly human with the world, and partly as a ”tool”, that can be used to ”solve problems”, to manage the everyday and professional life. In this sense mathematics entered the education system in connection with industrialization and economy, as part of a package of new ”useful” school subjects also containing the sciences and modern languages.
Grube’s educational theory intended to make sense of this ”double nature” of mathematics. But he was firmly focused on the first aspect, which he saw as primary. If only the ”inner core” of the mathematical objects were made to move the core of the pupils, he thought, they would surely also be able to perform the petty task of solving problems.
Grube conceived of the numbers as learning objects, in a way not too dissimilar to how this concept is used today, and these were to be approach aesthetically. He wanted to replace the ”extensity of the many” with the ”intensity of the example”, and called this the ”principle of the examplary”. The simplicity and unity of the thing itself should form the point of departure for the teaching method. Then, Grube thought ”the pupil will be capable of subsuming the many instances of the everyday life under the in this way appropriated concept”. (p. 63)
But while the method would surely, Grube thought, put the pupils in position to solve their everyday life problems, it was morality that was its main end. Grube wanted to pupils to ”empathize” (einzuleben) with the objects around which the teaching circle, feeling and enjoying, in relation to these objects, their own knowledge and power grow, thus influencing their desire. ”According to Grube, the teaching is morally effective when the learning leads to a will to learn”. Interestingly, this means that the moral effect of mathematics education does not lie in the transmitting of ”moral or religious” doctrines, but in that ”the performance of calculations leads to a will to perform calculations”. The attentive and emphatic activity of working with numbers is thus a moral end in itself, in its contrast with the superficial and fragmented life of industrialized modernity.
Rolf Braun, August Wilhelm Grube – Mathematikunterricht und Erziehung, Peter Lang, 1979