ZDM Mathematics Education (2007) 39:79–86 DOI 10.1007/s11858-006-0006-z
O R I G I N A L A RT I C L E
Proofs and hypotheses Hans Niels Jahnke
Accepted: 4 December 2006 / Published online: 2 February 2007 FIZ Karlsruhe 2007
Abstract On the basis of an analysis of common features and differences between general statements in every day situations, in physics and in mathematics the paper proposes a didactical approach to proof. It is centred around the idea that inventing hypotheses and testing their consequences is more productive for the understanding of the epistemological nature of proof than forming elaborate chains of deductions. Inventing hypotheses is important within and outside of mathematics. In this approach proving and forming models get in close contact. The idea is exemplified by a teaching unit on the angle sum theorem in Euclidean geometry.
1 Preliminary remark The following paper will point to one of Hans Georg Steiner’s scientific ideas. Other papers in this volume will illuminate other aspects of his work. Therefore a personal remark might be in order. I joined the Institut fu¨r Didaktik der Mathematik (IDM) of the University of Bielefeld as a young man. These were the early days of the Institute, and from the very beginning I experienced Steiner’s ability to bring together people from different parts of our discipline and different countries for an intense exchange of ideas. He felt a deep obligation to further young people and was tireless in H. N. Jahnke (&) Fachbereich Mathematik, Universita¨t Duisburg-Essen, 45117 Essen, Germany e-mail:
[email protected]
doing so. All over the years he constantly worked on developing and, if necessary, defending the way of academic communication he believed in. This shaped the minds and habits of many people, and I am grateful to him for the creation of this academic environment.
2 Proofs and empirical thinking Quite a few students of the secondary and even tertiary levels find measurements and examples more convincing than a mathematical proof. In an empirical study covering ca. 3000 British high achieving students of grades 8 and 9 more than 50% chose a ‘‘pragmatic argument’’ as warranty for the angle sum theorem for triangles. Following the terminology in (Balacheff 1988) such arguments comprise measurements and procedures similar to measurements as for example tearing out the vertices of a triangle and placing them side by side. Students attributed to these arguments a higher degree of persuasiveness. At the same time they were aware of the fact that their teachers favoured a deductive argument (Healy and Hoyles 2007, to appear). Coe and Ruthven summarise an earlier study of advanced students by saying: ‘‘The study reported here...finds that although...students are actively involved in developing mathematical ideas, this is within a restricted mode in which ideas of enquiry have become largely instrumentalised. In particular, validation is primarily and predominantly empirical, and there is little concern for illumination and systematisation.’’ (Coe and Ruthven 1994, p. 41). Similar results can be found in Bell (1976) and other studies. Particularly significant is the frequently quoted observation by Fischbein that some of his students had
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understood a proof and, nevertheless, asked for additional examples to get convinced of the truth of the theorem in question (Fishbein 1982, p. 16). To these findings the results of investigations fit that many school and university students and even teachers of mathematics have only superficial ideas on the nature of proof according to which the use of symbolic language is the decisive characterising feature of proof (Harel and Sowder 1998; Wittmann and Mu¨ller 1988). All these results need a careful analysis. In its entirety they show a basic preference for procedures of validation which, following the literature, can be designated as empirical. The fact that even high-performing students show this preference suggests the hypothesis that this phenomenon is not a mere consequence of a lack of mathematical competence. Rather, it seems that the usual teaching of mathematics is not successful in explaining the epistemological meaning of proof.
3 General statements The following two sections of this paper will develop a conceptual frame for the interpretation of the ways of thinking underlying the student behaviour described above. Since mathematical proofs aim at generally valid statements and since teachers of mathematics usually motivate proof by saying that it establishes the truth of a statement for all cases it is natural to investigate the status of generally valid statements and the ways how they are corroborated. We do this by comparing generally valid statements in different areas of knowledge, namely in everyday knowledge, in physics and in mathematics. What are general statements in the everyday world and how are they corroborated? Let us begin with fundamental and frequently offered examples. ‘‘Every morning sun rises.’’ and ‘‘Every human being is mortal.’’ Sensibly, these statements will not be doubted. The first statement is based on the continually repeated experience of the event of sunrise. Beyond this individual experience oral and written communications by other human beings in all times of history confirm that there has never been an exception to this statement. Thus, in establishing the statement no theoretical thinking, say of astronomy, is involved. Rather, astronomy is subject to the requirement that its results are only relevant if they do not contradict this statement. Similarly, things stand with the statement that all human beings are mortal. Again confirmation of its general validity consists of a combination of personal experience and evidence provided by other people.
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Thus, in everyday thinking generally valid statements are established by regularly repeated personal experiences and oral or written communications from others. Statements of such generality are only of secondary importance in everyday thinking, perhaps since they are a matter of course. More important are statements of the following type: ‘‘Every evening Alfred returns back home at 6.00 p.m.’’ Ute, Alfred’s 12-year-old daughter, finds this statement confirmed by regularly repeated personal experience. In her mind the statement implicitly implies a proof. ‘‘Since the office closes at 5.00 p.m. and since tram and bus need one hour, Dad is back home at 6.00 p.m.’’ This is a germ of a proof. Ute explains to her friend Karin that Alfred’s office closes at 5.00 p.m. and shows her the timetables of trains and busses from which follow that Alfred is back home at 6.00 p.m. Alfred is a reliable person with fixed habits. Usually, he comes back home at 6.00 p.m. However, Ute does not suppose that this event will happen every evening without exceptions. Rather, it is an obvious implicit assumption associated with the general statement ‘‘Dad comes back home at 6.00 p.m.’’ that there might be circumstances which would cause Alfred not to be back home at 6.00 p.m. The train might be too late or Alfred might see a friend and go with him to a pub or might become tired of his fixed habits and go to Paris for higher amusement. In everyday situations the general statement ‘‘Alfred comes back home at 6.00 p.m.’’ is made under the not explicitly mentioned but always present assumption that there might occur circumstances which prevent the event from happening. Nevertheless, the statement is general. Also, the timetable argument connecting the closing of the office with Alfred’s homecoming has general validity. However, this argument proves a theorem which ‘‘admits exceptions’’ or, more precisely, a theorem whose limits of validity are not precisely specified. As a rule Alfred comes back home at 6.00 p.m., but not always. Note, that this type of statement is fundamentally different from a statistical law in physics. This is the typical case of a general statement in everyday thinking. They are true as a rule, but suffer exceptions if modifying conditions occur. It is characteristic of everyday thinking that these modifying conditions are usually not made explicit simply because this would be unpractical and uneconomical. In a fundamental sense a complete specification of these conditions is even impossible because they would form an infinite set which never could be completely given. For easier communication general statements of this type whose domain of validity is not specified will be called ‘‘open general statements’’ in this paper.
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In a second step we consider general statements in the empirical sciences, for the sake of clarity in physics. There are a number of common features and of differences with everyday thinking. The first common feature is that general statements occur in physics which are established by repeated experiences and whose validity is on this basis beyond doubt. Philosophers of science call these statements observational statements. Increasing the temperature of gas by 1F leads to a certain increase of the product of volume and pressure. This repeated experience is made by the researcher in his laboratory if he conducts the appropriate experiments, and the statement is confirmed by oral or written communications of his colleagues. As in everyday situations a general statement in physics is confirmed by a combination of personal experience and communications of other human beings. However, in contrast to everyday thinking physicists make a great effort to specify as precisely as possible the conditions which limit the domain of validity of a general statement. The law that the product of volume times pressure is proportional to temperature is valid only for a certain interval of temperature and only for so-called ‘ideal gases’, and the latter concept is only an abbreviation for a whole bundle of modifying conditions. The specification of these conditions is an indispensable requirement for the technical application of physics which rests on the reliable prediction of effects produced by a bundle of conditions. Whereas in everyday thinking practical reasons render the specification of the limits of validity of a general statement superfluous, practical reasons require such a specification in physics. Physics shares with everyday thinking the basic limitation that, for principle reasons, a complete specification of the modifying conditions for the validity of a general statement is impossible. Physics is about a reality independent of us which we are not able to completely represent in our theories. A physicist is always aware of the possibility that new phenomena might be discovered which falsify his theory in its present form and require a modification of some laws. In principle, scientific theories are open to revision. Although physicists intend to specify the domain of validity of natural laws as precisely as possible and although they are very successful with this, physics shares with everyday thinking the principle limitation that its statements are open general statements. Physics is different from everyday thinking in another important regard. Physicists build theories. According to Duhem a theory is ‘‘a system of mathematical theorems which can be derived from a small number of principles and which aim at representing as
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completely and precisely as possible a connected group of experimental laws’’ (Duhem 1978, p. 20/21). In short, theories of physics are networks of natural laws connected by mathematical proofs. This fact has important consequences for the corroboration of general statements (see Jahnke 2005). Until now we have only considered the type of corroboration based on personal experience shared with others of a regularly repeated phenomenon. By building theories a completely new point of view enters our discussion. An isolated natural law which is not linked to other natural laws can only be corroborated by experiments which are directly related to it. In contrast to this, a natural law within a theoretical network is corroborated not only by direct measurements but also by all the measurements of all the natural laws belonging to the network. Thus, a mathematical proof deriving a statement (natural law) of the theory does not provide absolute certainty to this statement, but it will considerably enhance its certainty since the corroboration does not only come from an isolated set of direct measurements but also from all measurements related to the theory. Such a theoretical network of statements and measurements connected by mathematical proofs is the safest form of knowledge at our disposal, although this does not change its, in principle, preliminary character. The fact that theories in their entirety are corroborated or refuted by the whole set of phenomena to which it refers is called ‘holism’. The French physicist Pierre Duhem was the first who pointed to this fact (Duhem 1978). Further the work of Thomas Kuhn and Imre Lakatos might be seen in this context as well as the structuralist reconstructions of empirical theories (Sneed 1971, for the application to mathematics see Jahnke 1978). In a last step we consider general statements in mathematics and refer to the two theorems prominent in school mathematics that the sum of the first n odd numbers is equal to n2 and that the angle sum in triangles amounts to 180. Both theorems can be considered as equations which are true for infinitely many objects. The nature of these objects is determined by the system of axioms of the theory: in the first case by the axioms of natural numbers and in the second by the axioms of Euclidean geometry. The systems of axioms are themselves general statements which are valid for objects which do not have other properties than the only one that the axioms are true for them. Thus, the objects for which the two theorems are true without any exception are constituted by the systems of axioms. Thus, mathematics differs fundamentally from physics and everyday thinking in that a complete specification of conditions is not only striven for but
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in fact possible. This happens at the cost of the fact that mathematics cuts its relations to empirical reality. Statements valid without exception are possible in mathematics only because and insofar as mathematics keeps the set of conditions constant. As a consequence the objects of mathematics are generated by the theory. To say it with A. Einstein ‘‘Insofar as the theorems of mathematics refer to reality they are not certain, and insofar they are certain they do not refer to reality’’ (Einstein 1921; p. 123, my translation) Therefore, in mathematics the concept of a generally valid statement is connected to a fixed corpus of conditions; it is a theoretical concept and meaningful only in regard to a given theory. A statement valid without exceptions is only possible if it refers to hypothetical objects which exist in a universe constituted by a closed and unchangeable set of conditions. In a sense physics can be seen as a mediating stage between everyday thinking and mathematics. Physics shares with everyday thinking the empirical relationship and the fundamental openness and changeability of its statements. It shares with mathematics the intention of specifying the conditions of statements as much as possible and of building networks of statements. Spoken pointedly, physics would become mathematics if it would decide to stop experimentation. The delimitations between these three ways of thinking are ideal–typical. Of course, there are initial stages of (local) theory nets in everyday thinking, and, conversely, everyday thinking is present in mathematics, especially in school mathematics. In everyday thinking a person is going to build a theory net if she asks for non-obvious reasons of an event in her surroundings. If Alfred does not come back home at 6.00 p.m. Ute might ask for the reason and go through a series of hypotheses and compare them with some additional evidence she might have. In regard to the presence of everyday thinking in mathematics consider the example of the geometrical theorem that a triangle is uniquely determined by the lengths of its three sides. It is true in general and can be deduced from the axioms of Euclidean geometry. Obviously the theorem gets wrong if modifying conditions occur, for example, if the sum of the lengths of two sides is smaller than the third side. If necessary, the Greeks added such modifying conditions as so-called ‘determinations’ but in general they omitted them. In the present case it is easy to get a generally valid theorem by adding the condition as the so-called triangle inequality, and this is what modern mathematics tries to achieve as much as possible. That this is not done by Euclid and for good reasons not in school mathematics
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shows how much Euclid and school geometry are rooted in everyday thinking.
4 Proof-novices as ‘theoretical physicists’ Let us come back to the question how the thinking of the students can be understood who find empirical evidence more convincing than a mathematical proof. Students are moulded by everyday thinking and they try to make accessible the meaning of mathematical proof out of this perspective. Arguments like the one which deduces Alfred’s homecoming at 6.00 p.m. from the closing of his office at 5.00 p.m. serve as paradigms for their search of the meaning of proof and will guide their behaviour in proof situations. Our previous analysis shows that they will run at least into two difficulties of a general nature. 1.
The mathematical notion of a generally valid statement contrasts with the everyday thinking of the students. They understand a general statement in the sense of an ‘open general statement’. In this frame it is not obvious why definitions of concepts are needed and why conditions should be made explicit as far as possible. To the contrary, such demands seem to be unpractical and uneconomical. Therefore, students should be led by substantial examples to the insight how the use of definitions and the specification of conditions limits the openness of a general statement and increases the quality of a prediction. This is exactly the situation in physics.
The demand of a complete specification of conditions cannot be imposed on the students from one day to the next. With necessity this would lead to the result that they consider proof as a mere ritual. The claim that mathematical statements are safer than those of ordinary thinking would remain mysterious to them. As mentioned there is quite a number of mediating forms of thinking between everyday thinking and working in mathematical theories, and it is an important educational task to show a feasible way to the students from everyday thinking into the theoretical world of mathematics. Our analysis suggests the idea that inventing conditions and experimenting with conditions provides a more adequate approach to proof than the elaboration of long chains of deductions. The experimenting should happen in an environment (like physics) in which students and their teachers can seriously work on and discuss the question why and on which grounds a statement is generally valid.
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Obviously, students follow everyday routines when they attribute to measurements and to the words of their teachers the decisive importance for the validation of a statement. They suppose rightly that a general statement is corroborated by personal and intersubjective evidence and not by theoretical deductions. The everyday routines are fundamental and do not get obsolete by transition to scientific thinking. This explains their persistence. Physics transcends everyday thinking not by negating these fundamental forms of validation but by building theories. As described earlier, theories establish links between statements and links between measurements, and this lends a new quality to empirical corroboration. The epistemological motivation of proof is not to be founded on the idea that proofs in contrast to measurement provide absolute certainty, but on the idea that proofs open new and more complex possibilities of empirical corroboration. In short, in an empirical environment proofs do not replace measurements but make them more intelligent.
All in all, it is obvious that physics taken as a paradigm for the empirical sciences and the empirical dimension of thinking in general may serve as an important mediating stage between everyday thinking and mathematics. This is true in an analytical as well as a constructive perspective. The analytical perspective would imply to model and understand the thinking of proof novices by comparing it with scientific procedures in physics. Metaphorically spoken, one considers a novice in proof as behaving like a theoretical physicist in the hope to understand better the inner logic of his thinking. On the other hand, the perspective of the physicist opens a genetic approach from everyday thinking to mathematical proof. This will be elaborated in the last part of the paper.
5 Elements of a genetic approach The metaphor of the theoretical physicist suggests to look for a genetic approach to proof which does not put aside the empirical thinking, but takes it serious and, precisely this way, shows the power of theoretical thinking (Jahnke 1978). The approach would consist of three phases, a first phase of informal thought experiments (grade 1+), a second phase of hypotheticodeductive thinking (grade 7+) and a third phase of autonomous mathematical theories. Students of the third phase would work in closed theories and only
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then proof would mean what an educated mathematician would understand by proof. The first phase would be characterised by informal argumentations and would comprise what has been called ‘‘preformal proofs’’ (Kirsch 1979), ‘‘inhaltlichanschauliche Beweise’’ (Wittmann and Mu¨ller 1988) and ‘‘proofs that explain’’ in contrast to proofs that only proof. (Hanna 1989). These ideas are well implemented in primary and lower secondary teaching, in English speaking countries as well as in Germany. In the present paper the interest is on the second phase. In this phase the concept of proof should be explicitly taken as a theme. But this is a major difficulty and the main reason why teachers and textbook authors mostly prefer to remain implicit about the notion of proof. There is no easy definition of the very term ‘‘proof’’ since this concept is dependent of the concept of a theory. If one speaks about proof one has to speak about theories, and most teachers are reluctant to speak with seventh graders about what a theory is. On the other hand, this is unavoidable (compare for similar ideas, Bartolini Bussi et al. 1997). Thus, when introducing proof the seventh graders have to be provided with examples of theories, theories they are able to work with and to survey. The idea is to build local theories, that is small networks of theorems. This corresponds to Freudenthal’s notion of ‘‘local organization’’ (Freudenthal 1973, p. 458) but with a decisive modification. The idea of measuring should not be dispersed into a general talk about intuition, rather we should build small networks of theorems based on empirical evidence. The networks should be manageable for the pupils, deductions and measurements should be organically integrated. Frequently, educators propose to relate mathematical proof with procedures of validation in other areas and disciplines, as for example law and physics. In general, however, this is only done to get a background from which to differentiate mathematical proof proper. In a genetic perspective, however, it should be the other way round. Genetic means to start from the most natural and the most universal ways of cognition and to arrive at specific ways of knowing by specialisation. Therefore, we look for connections and embedding. An appropriate frame could be the notion of the hypothetico-deductive method which is fundamental for scientific thinking. By way of a deduction consequences are derived from a theory and these are checked against the facts which are to be explained. This basic approach is common to all sciences be they physics, sociology, linguistics or mathematics. Such an approach would agree with our analysis of the relation between everyday thinking and mathe-
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matical thinking. The path from the former to the latter requires a growing insight that statements are dependent of other statements and that we cannot speak about truth without specifying the conditions from which we start. Therefore, in school proving should be initiated by inventing conditions and experimenting with conditions. Instead of the term ‘condition’ we propose the term ‘hypothesis’ because it is more general, less technical and provides more connections to other fields of knowledge. It is important that the notions of ‘‘proof’’ and ‘‘hypothesis’’ are not only components of the background philosophy of the teacher, but are also used and practised explicitly in the classroom starting with grade 7. A reflective understanding of proof can only be developed if this concept is actually and repeatedly made a theme. It is a great failure of school mathematics if this is not done. The hope that an adequate understanding of proof would emerge solely out of examples is deceptive as experience shows. As a rule neither students nor their teachers can explain what a proof is and what its role in mathematics is. It should be noted that H. G. Steiner has frequently stressed the necessity to explicitly introduce the notions of definition, axiom and proof in the classroom (Steiner 1968).
6 Example: angle sum in triangles In this section some dimensions of our approach to proof will be clarified by discussing the example of the angle sum theorem. Frequently, the importance of this theorem is underestimated. Actually, it belongs to the deep theorems of school mathematics. It is not obvious, fundamental for geometry and paradigmatic for the greatest epistemological revolution in mathematics, the discovery of non-Euclidean geometries. In teaching it is worth making a real story out of it. Students are frequently asked to measure the angles of a triangle. After they find that their sum is always nearly equal to 180, they are told that measurements can establish this fact only for individual cases and that they will have to prove it if they really want to be sure that it is true for all triangles. As explained earlier this is wrong and will not lead to an adequate understanding of proof. For the students (and their teachers) the theorem is a statement about real (physical) space and, therefore, the theorem is true because it is corroborated by measurement. This is the intellectually honest answer. On this basis a teacher should explain why a mathematical proof is urgently desirable (see Hanna and Jahnke 1996, Sect. 3).
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Abstractly spoken, an argument in favour of proof might run as follows. A proof of the angle sum theorem will not give absolute certainty, but it will enhance our belief in it since it connects the theorem with other geometrical theorems which can also be corroborated by measurements. Thus, the theorem is not only tested by measuring the sum of the angles of a triangle, but also by measuring corresponding angles, alternate angles or sums of angles in 4-, 5-, 6-gons. Thus, in a long school career the students come to know so many statements which are empirically testable and connected with the angle sum theorem that they understand why Euclidean geometry is the oldest and empirically best corroborated theory we have and why, for a long time, mathematicians and philosophers attributed absolute certainty to it. Legitimately, in nearly every textbook the Euclidean proof of the angle sum theorem (Fig. 1) is represented which reduces the theorem to the equality of alternate angles at parallels intersected by a third line. In many textbooks the latter theorem is ‘proved’ by means of transformations (rotations) in a way that ‘‘opposition is impossible’’. Obviously, the authors of textbooks intend to provide to the students the impression that by way of a mathematical proof absolute certainty can be attained. According to our explanations earlier this is not adequate. Mathematically, too, this procedure is highly questionable. The implication (equality of alternate angles at intersected straight lines parallelism) is a theorem of absolute geometry and does not need the axiom of parallelism. But the inverse implication (parallelism equality of alternate angles at intersected straight lines) which is needed in the previous proof presupposes the axiom of parallelism. This expresses the uniqueness of the parallel. The core of the following proposal consists in the idea to present the theorem about alternate angles, and this means nothing else than the axiom of parallelism, not as a proven statement, but as an empirically suggested hypothesis which the students find by themselves. From this hypothesis a small theory is developed whose
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Fig. 1 Euclidean proof of the angle sum theorem
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various consequences can be examined by measurements. Since the latter agree very well with the theoretical deductions the hypothesis and the theory are confirmed in the sense of physics. For the sake of brevity we present the teaching unit only in a schematised way. Teaching experiences will be described in a separate publication. 6.1 Step 1 After exploring a configuration with two intersecting straight lines the following worksheet with three intersecting lines is given to the students. 6.2 Worksheet Straight line h rotates counter clockwise around P (Fig. 2). Please, measure angles b and e for some positions of h and tabulate your results. Note your observations down. Students are supposed to observe: point Q will travel more and more outwardly when h rotates. e and b will become smaller and smaller. The differences between two consecutive values of e and b are the same. The difference between e and b is always 77, or b is the difference between e and 77. There is at least one position of h (when Q is far outward ‘‘in the infinite’’) where e = 77. h and g will not intersect in this position. It is plausible that the property of b to measure the difference between e and 77 will be true in the infinite, thus b = 0. This is plausible but we cannot check it directly. The observations suggest the alternate angle theorem as an empirical hypothesis. The very term ‘‘Hypothesis’’ is written down on the blackboard, students are asked whether they know this concept and what it might mean. Experiences show that students recognise the numerical regularities in the table very well although the measured values represent them only approximately. It is more difficult to connect the values with a geometrical idea and to imagine the travelling of point
Q to the infinite. In this age students are simply not familiar with this type of activity. 6.3 Step 2 Figure 1 is given to the students. They find a = a¢ and b = b¢ because of the hypothesis and thus a + b + c = 180 for a triangle. This result is checked by measurements. The measurements are taken serious that means students work with big and carefully drawn triangles and the results are written down in tables. Then they calculate the mean value of their measurements. Now they have two different values for the angle sum of triangles, the one derived from the hypothesis (180) and the mean value of their measurements, say 179.5. 6.4 Step 3 Students investigate by measurements and deductions the field of all (simple) polygons (Fig. 3). This is an exciting activity, and one can be sure that they get reasonable results. At the end they arrive at a proof of how the angle sums of polygons depend on the angle sum of triangles. If possible one should visualise the dependencies of the theorems in a tree to show the influence of the hypothesis of the alternate angles. 6.5 Step 4 Students predict the angle sum of a 10-gon. The ‘theoretical’ value gives 1,440, the measured mean value leads to 1,436. Then they measure the angle sum of 10-gons and calculate a mean value. In general the result will be that the measured mean value is nearer to the ‘theoretical’ predicted value than to the prediction based on the measurements of triangles. Therefore, the latter is not confirmed, the former seems to be better, but remains a hypothesis. 6.6 Step 5 Students get a sheet with information about the history of the axiom of parallelism and the attempts to check it by measurements. They read and discuss it.
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7 Experimenting with hypotheses 77°
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Fig. 2 Straight line h rotates counter clockwise around P
The dependence of statements from hypotheses should be made a theme in mathematics teaching again and again from grade 7 to grade 12. With teachers of other
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Fig. 3 Explore the angle sums of polygons!
disciplines this could be expanded to an interdisciplinary module on ‘‘scientific argumentation’’. In such a module mathematical proof would be sensibly integrated. Other opportunities for experiments with hypotheses are the so-called ‘‘Fermi-questions’’ (see Herget and Torres-Skoumal 2007, to appear) and, more generally, any sensible modelling activity (see e.g. Hanna and Jahnke 2002). Modelling is nothing else than inventing a hypothesis fitting to a concrete case. What might distinguish the present approach from others is the accent on explicitly introducing some terms of the meta-language of mathematics into teaching and making them germs of reflection. This is what H. G. Steiner among others had proposed many years ago.
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