- quality and structure of mathematical concepts and theories
- symbols and symbolization in mathematics
- role of diagrams in mathematics
- notational systems
- abstraction and generalization
- idealization and reification
- metaphors in mathematical language
- process-object duality
- epistemological obstacles (in the sense of Bachelard)
- methods of concepts formation
- mathematization and applications of mathematics
- specificities of the mathematical discourse, e.g. the metamathematical conventions like "Let A be...", "now we consider a..."
- hidden conventions and tacit assumptions
- proofs: types, their role and relevance
- historical development of mathematical concepts and theories
- social and societal importance and role of mathematics and its applications
- role of cultural tools, means and media
- maths and maths education are usually organized in separated institutions without any formal channels of communications
- the education of experts in the two communities is as well strongly separated in most places
- there are almost no common conferences or the like or shared journals
- one does not read texts from the respective other field
- there prevail negative opinions and prejudices about the respective other field and its representatives though not in an outspoken way
- characteristic features of the respective other discipline are viewed as deficits or weaknesses.

ICMI

Bulletin No. 49

December 2000

**I. Introduction**

Mathematics education as a scientific discipline is devoted to the study of how people learn and do mathematics of any kind and of how this learning and doing can be influenced and fostered by teaching. Thus the object of mathematics education clearly is a certain area of human activities whose content, object and goal is mathematics at different levels and in different forms. The latter should thereby not be taken to suggest an existence of mathematics outside and independent of the respective activities. Keeping this in mind, maths education can metaphorically be considered as studying the relationships between mathematics and human beings both taken in their whole variety. In this sense maths education has to be interested in the early counting activities of children and in the production of a proof in analytic number theory as well. Acceptance of this description of maths education clearly leads to attributing to mathematics a very special role for mathematics education as a scientific discipline. Very likely, the case is different for the role of (professional) mathematicians although many of them are involved in the teaching of mathematics at the university level. Yet, they themselves consider their main task to be to develop mathematical theories and/or to apply them in various contexts. This should/could give rise for maths education to investigate the mathematical activity of mathematicians, its conditions, forms, means, goals, intentions, etc. Yet, this is a passive role as the object of research studies. If there is more to be expected I will consider later.

**II. Mathematics as an object of research**

First, I will concentrate on the possible role of mathematics for maths education, its opportunities and obstacles. Thereby, I take as mathematics that which in the course of history has evolved as the product of the activity of mathematicians and has to a great extent been standardized, conventionalized and corroborated by extended experience and manifold practical usages. It is the notions, concepts, methods, notations, basic assumptions, whole theories, etc. which rather unanimously are considered to be mathematical. And, I completely refrain here from trying to describe what mathematics is. But, I only point to the fact, that there are many, even mutually excluding standpoints, regarding this question. For instance, the opinions are already very diverse on the nature of the objects studied in mathematics (e.g. platonism, realism, empirism, nominalism, mentalism, instrumentalism, fictionalism, etc).

Whatever the background philosophy or ontology is, in a first and uncritical approach, maths education has to take mathematics in the above sense as a kind of given and its central task then is to offer ways to the learner into the respective mathematical practices and discourses. I admit that possibly those pathways might differ depending on basic views about the nature of mathematics. Yet, the general tenet thereby is not weakened: in maths education one has to study and investigate mathematics as it is currently practiced, produced and used in all its forms. This, of course, is not a mathematical study but it is a meta-study of mathematics like musicology is the study of music. Also musicology has no choice than to analyze music as a given human phenomenon and activity and its products as well. Possibly, this aspect of maths education could be termed "mathematicology" and never should be permitted to lose sight of the other pole of the relationship which maths education is about: the human being. This kind of "mathematicology" thus has to investigate mathematics as one of the two sides of mathematical activity and not detached from the people who carry it out. Topics of this kind of research could be:

In short, mathematicology should try to develop theoretical descriptions and models of mathematical activities and processes. By this, what mathematicians and learners of mathematics do or not do is reflected upon, is made conscious and explicit and thereby amenable to being monitored, changed and influenced. Differing descriptions will arise which then have to compete for being considered viable and applicable. Of course, there is already available a comprehensive body of research of this kind which possibly is not taken account of to a sufficient extent and which should be extended in a systematic way.

I hasten to say that this mathematicology is not to be identified with maths education research. But I consider it as a fundamental branch of the whole complex and interdisciplinary endeavor of the latter. For facilitating the learning of mathematics a deepened understanding of mathematics as a human activity is an indisputable and indispensable precondition. This deepened understanding is in my view not just knowledge of mathematics per se but the kind of knowledge about mathematics which I want to indicate by the above features of mathematicology. Appropriate parts of mathematicology have to form an integrated part of the expertise of maths education researchers, of teachers and also of students. This corresponds to the idea that knowledge has to be organized and regulated by meta-knowledge. I emphasize that this kind of analyses termed by me mathematicology is not what runs as research on the foundations of mathematics like in logic, proof theory, set theory, model theory. These are equally mathematical activities in need of being reflected upon along the lines sketched above. In my sort of mathematicology the focus is on the human subjects as the agents of the mathematical activities. Foundational research to the contrary excludes the human actor completely. The methods employed in those studies can vary from theoretical analyses of mathematical texts to empirical studies of the mathematical activities of individuals and groups. An overarching goal has thereby to be to gain a deepened understanding of mathematics as a human activity.

A possibly bad analogy is with sports and the current scientific studies of physical activities: anatomic conditions, nutrition, training methods. Taking only the positive aspects one could say that also in sports we have a human activity which is investigated with the aim of finding better ways for becoming an expert in this activity. For that, the complex processes regulating physical behavior have to be understood in a viable way. Another analogy which might better clarify my intentions regarding the role of mathematicology for mathematics education is the relationship of linguistics to spoken languages and to language acquisition. Linguistics is the scientific study of language as a human, a psychological, a societal, a neurophysiological, etc. phenomenon and as such correlates with mathematicology whereas mathematics corresponds to the actually spoken language. As linguistics endeavors to investigate and understand the human activity of language use, production and acquisition, mathematicology should serve the same purposes for the human activity of mathematics use, production and acquisition. In both cases research results can shed new light on related educational processes and problems.

Well, suffice these remarks to convey the central intention of a "mathematicology".

Which use can be made of the outcomes of this kind of research about mathematics? First, they can inform and motivate further research in maths education like empirical studies about concrete learning and teaching processes. Those investigations need regulation and structuring from a theoretical basis. Foregoing analyses of the above kind permit to ask relevant research questions appropriately. Too many empirical studies, in my view, have an ad hoc character leaving open why just that is investigated. Thereby no strict and univalent derivation of research problems can be expected but the epistemological insights from mathematicology will have to be interpreted in the respective contexts and transformed into answerable questions. In a similar way findings of mathematicology can be used to interpret and analyze phenomena found in teaching and learning processes. A well known example is the relation of learning obstacles to epistemological obstacles.

Second, curricular decisions can be informed and legitimated by a deeper understanding of mathematics and mathematical activities. At least, here one can find arguments and reasons for discussions and deliberations about content and organization of curricula. Again, nothing is deductively determined but rational discourse is enhanced and supported to transcend ideological positions and subjective taste.

Third, mathematical activities including research can capitalize on "mathematicology" and its outcomes. Reflected understanding of tools and means used hitherto in a more unconscious and routinized way might change and broaden their usage. For instance, models of processes of mathematization can be used to organize, plan, design and monitor applications of mathematics. Knowledge about the semiotics of symbol use can guide deliberate symbolization.

Fourth, mathematicology permits to talk about mathematics as a human activity by supplying notions, concepts, theories and empirical results. This not only is very important for mathematics education as a research discipline but should also form part of classroom discourse. Offering views on one's own activities in mathematics will help to demystify mathematics in schools and thereby possibly lower some psychological barriers. For instance, I think here of a discussion of the epistemological and psychological status of so-called mathematical objects. Students thereby can be offered a chance to form a rational understanding of their own mathematical activity by developing a consistent view on what they are talking about in mathematics.

Fifth, epistemological analyses of mathematics of the sort mentioned above finally might lead to a substantiated critique of certain practices in mathematics. I will not go into any details here and only point to the fact that in the course of history didactical needs and stipulations have had ever again a strong influence on developments within mathematics. This can be seen for instance from Babylonian and Egyptian texts and the exactification of calculus in the early 19th century. Here the need to make the pertinent maths accessible to a broader community lead to a clarification of fundamental concepts like continuity.

**III. Mathematics and mathematics education - two cultures**

It is notorious that between the communities of mathematicians and maths educators there are very few functioning and mutually enriching relationships, notwithstanding some notable exceptions. Indicators for this lack and deficit are in my view especially the following points:

I am aware of these statements being of a very generalizing quality which is only intended to mark basic trends in the relationships between the two fields. Of course the concrete situation varies from country to country and there again from institution to institution. Thus, to each point raised above exist counterexamples. Yet, contrary to mathematics, in the social sciences counterexamples do not invalidate assertions. This is a feature which again contributes to the mutual misunderstandings between the two considered scientific fields. I further believe that the distinction and separation indicated here for maths and maths education are not equally sharp and abrupt in other fields of academic inquiry like, say, biology, sociology, humanities, language studies. This can be expected from a comparison of the respective scientific cultures along the lines as carried out below for maths and maths education.

My position now is to acknowledge the deeply rooted cultural differences between the two communities of maths and maths education and to turn them first into a topic of reflection and investigation and possibly then into a source of mutual understanding and common development. Without trying to be exhaustive and in a more naive way let me state some of the dividing traits which lurk beneath the apparent separation of mathematics and mathematics education as scientific communities (the order is by chance).

1. Maths education is primarily concerned with actual and living human beings. Thus it has to pay attention to the needs, wishes, anxieties, expectations, feelings, etc. of concrete people, be that students, teachers or researchers. Even more, all that becomes a research topic. Further, the social context, the individual and social histories interfere with mathematical learning processes. None of that occurs in mathematics: quite to the contrary, there the human being and its subjectivity have to be eliminated, or at least one strives for that. Even in physics now the role of the observer is recognized. But mathematics is presented as if it were not devised by people. And in both sciences this respective trait is considered as the ultimate criterion of high quality. A slight change appears to develop in applied mathematics where increasingly the role of the human agents is recognized in determining the success or failure of a specific case of application. Here the situation is very similar to that in software engineering. But still it is not the mathematics itself which incorporates the human individuals and social groups but in the process of application the latter get more actively involved and respected. As it turns out, a mathematical optimum not necessarily will always present a socially acceptable course of action.

2. Maths education is by its very nature a highly multidisciplinary field again contrary to the monolithic edifices of mathematics. This is also reflected in the broad variety of research methods employed in maths education contrasting with the very restricted methods of mathematical research. On the other hand applied mathematics necessarily has an interdisciplinary character since one of its main functions is to provide operational models for processes and situations either from other scientific fields or from concrete practice. In any case it never will be the mathematics by itself which determines a solution and its quality.

3. Mathematical concepts are well defined within the respective theories. Many notions central to maths education (like meaning, understanding, thinking, generalization) are inherently vague and open. In a way, the respective research can be understood as the endeavor to explicate and develop a sensible meaning of those terms. Again, similar features can be found in areas of applied maths where concepts are rather determined by the reference to the context of application than by formal-logical definitions. An example might be furnished by the finite-element-method for which only after a long time of successful use a formal justification by an exact theory has been developed.

4. Maths education has a reflective and cybernetic relationship to its objects of research. Those are not viewed as immutable but the wider goal of research is even to enable their change and development. Further, human beings react autonomously and in not foreseeable ways to outside influences. This has to be respected in maths education research and its practical implementations as well. Needless to say that nothing of this sort can be found in pure mathematics, but to some extent can be observed in applications of mathematics to economy, politics, and society.

5. Mathematics is (viewed as) universal and context-independent. By its very nature maths education invariably is embedded in the wider culture and its history. Even its success and viability to a large extent depend on conditions and restrictions beyond its influence. There is the eminent problem of the dissemination of research outcomes which is hampered by ideological barriers, economic interests and organizational structures. Maths education is part of politics in this sense and has to reflect this role consciously. Mathematicians on the other hand due to the detached quality of their field do not yet feel the need for reflecting on the social status and role of mathematics. Up to now this attitude even proved advantageous for the scientific community of pure mathematicians. That any application of mathematics is highly context-dependent and even so on different layers of contexts needs not to be stressed: solutions to "real" problems never can have a universal validity.

This list of traits describing a structural gulf between pure mathematics and mathematics education as scientific disciplines could be extended but I will close it now. The aim of this is not critique but to exhibit the differences which possibly threaten and impede communication and exchange between the communities. Recognizing this is in my view the first step to understanding each other without abandoning one's identity. It would be important to realize the strengths and weaknesses of the respective other discipline to open up the route to more cooperation. Given the sketched traits, mathematics educators should be more inclined and motivated to make the initial moves. For that, I propose to look for partners in the fields of applied mathematics which shows some features analogous to aspects of mathematics education. One of those I have already mentioned, namely the recognition of the role of human agents. Another is that successful applications of maths as a rule demand team work by experts from different fields. And, like maths education, applied maths has to strive for social acceptance, for an understanding of its proposed models and problem solutions and for a broader usefulness and viability of its methods and theories. In other words, applied maths has no more the universality, homogeneity and detachedness as it is pretended by traditional pure mathematics. What here and now might be a viable solution to a problem will possibly be inadmissible there and then. Thus, the notion of (viable) mathematization could be a linking theme between maths education and socially aware applied mathematics.

Willi Dörfler

Universität Klagenfurt

Klagenfurt, Austria

Willi.Doerfler@uni-klu.ac.at