86 6 Mathematics Education as a ‘Design Science’
As a systemic-evolutionary design science mathematics education can follow different paths. It is certainly not reasonable to develop it into a “monoparadigmatic”
form as postulated, for example, for the natural sciences. In a design science the simultaneous appearance of different approaches is a sign of progress and not of retardation as stated by Thommen (1983, p. 227) for management theory:
Because of a continuously changing economic world it is possible to (re-)construct an eco-nomic context within different formal frameworks, or models. These need not be mutually exclusive, on the contrary, they can even be complementary, for no model can take all prob-lems and aspects into account as well as consider and weigh them equally. The more models exist, the more problems and aspects are studied, the greater is the chance for mutual cor-rection. Therefore we consider the variety of models in management theory as an indicator for an advanced development of this field moving on in an evolutionary, not a revolutionary process in which new models emerge and old ones disappear.
As a design science mathematics education can be no exception from this rule.
That teachers take part in design can be no excuse for mathematics educators to refrain from this task. On the contrary: The design of substantial teaching units, and particularly of substantial curricula, is a most difficult task that must be carried out by the experts in the field. By no means can it be left to teachers, although teachers can certainly make important contributions within the framework of design provided by experts, particularly when they are members of or in close connection with a research team. Also, the adaptation of teaching units to the conditions of a special classroom requires design on a minor scale. Nevertheless, a teacher can be compared more to a conductor than to a composer or perhaps better to a director (“metteur en scène”) than to a writer of a play. For this reason there should exist strong reservations about
“teachers’ centers” wherein teachers meet to make their own curriculum.
We should be anxious to delineate teaching units of the highest quality from the mass of units developed at various levels for various purposes. These “substantial”
teaching units can be characterized by the following properties:
1. They represent central objectives, contents and principles of mathematics teach-ing.
2. They provide rich sources for mathematical activities.
3. They are flexible and can easily be adapted to the conditions of a special classroom.
4. They involve mathematical, psychological and pedagogical aspects of teaching and learning in a holistic way, and therefore they offer a wide potential for empir-ical research.
Typically, a substantial teaching unit always carries a name. As examples I men-tion “Arithmogons” by Alistair McIntosh and Douglas Quadling, “Mirror cards” by Marion Walter, “Giant Egbert” and other units developed in the Dutch Wiskobas project, and Gerd Walther’s unit “Number of hours in a year” (Walther1984, 72–
78). Other examples and a systematic discussion of the role of substantial teaching units in mathematics education are given by Wittmann (1984).
For the sake of clarity, one example of a substantial teaching unit is sketched below. In our primary project Mathe 2000 the following setting of arithmogons is used in grade 1:
A triangle is divided in three fields by connecting its midpoint to the midpoints of its sides. We put counters or write numbers in the fields. The simple rule is as follows: Add the numbers in two adjacent fields and write the sum in the box of the corresponding side (cf. Fig.2).
88 6 Mathematics Education as a ‘Design Science’
Fig. 2 Modified representation of arithmogons
Various problems arise: When starting from the numbers inside, the numbers outside can be obtained by addition. When one or two numbers inside and respectively two or one number outside are given, the missing numbers can be calculated by addition and subtraction. When the three numbers outside are given, we have a problem that does not allow for direct calculation but requires some thinking. It turns out that there is always exactly one solution. However, it may be necessary to use fractions or negative numbers.
The mathematics behind arithmogons is quite advanced: The three numbers inside form a vector as well as the three numbers outside. The rule of adding numbers in adjacent fields defines a linear mapping from the three-dimensional vectorspace over the reals into itself. The corresponding matrix is non-singular. One can generalize the structure to n-gons as shown in McIntosh and Quadling (1975).
The teaching unit based upon arithmogons consists of a sequence of tasks and problems that arise naturally from the mathematical context. The script for the teacher may be structured as follows:
1. Introduce the rule by means of examples and make sure that the rule is clearly understood.
2. Present some examples in which the numbers inside are given.
3. Present some examples in which some numbers inside and some numbers outside are given.
4. Present a problem in which the numbers outside are given.
5. Present other problems of this kind.
As can be seen, a substantial teaching unit is essentially open. Only the key problems are fixed. During each episode the teacher has to follow the students’ ideas in trying to solve the problems. This role of the teacher is completely different from traditional views of teaching. Teaching a substantial unit is basically analogous to conducting a clinical interview during which only the key questions are defined and the interviewer’s task is to follow the child’s thinking.
The structural similarity between substantial teaching units on the one hand, and clinical interviews on the other, suggests an adaptation of Piaget’s method for study-ing children’s cognitive development to empirical research on teachstudy-ing units (cf., Fig.3). As a result we arrive at “clinical teaching experiments” in which teaching units can be used not only as research tools, but also as objects of study.
Fig. 3 Comparison of clinical interviews with teaching experiments
The data collected in these experiments have multiple uses: They tell us something about the teaching/learning processes, individual and social outcomes of learning, children’s productive thinking, and children’s difficulties. They also help us to eval-uate the unit and to revise it in order to make teaching and learning more efficient.
The Piagetian experiments were repeated many times by other researchers. Many became a focus of extended psychological research. Some even established special lines of study; for example, the “conservation” experiments. It is no exaggeration to say that Piaget’s experiments and the patterns he observed in children’s thinking survived much longer than his theories, in many cases until the present. In the same way, clinical teaching experiments can be repeated and thereby varied. By comparing the data we can identify basic patterns of teaching and learning and derive well-founded specific knowledge on teaching certain units. Much can be learned here from Japanese research in mathematics education (cf., Becker and Miwa1989).
In conducting such studies, existing methods of qualitative research can effectively be used, particularly those developed by French mathematics educators in connection with “didactical situations” and with “didactic engineering” (cf. Brousseau 1986;
Artigue and Perrin-Glorian1991; Arsac et al.1992). Concerning the reproducibility of results it is very instructive to look at the social sciences. Friedrich von Hayek, another Nobel prize winner in economics, has convincingly pointed out that empirical research on highly complex social phenomena yields reproducible results if directed towards revealing general patterns beyond special data (von Hayek1956). To admit that the results of teaching and learning depend on the students and on the teacher does not preclude the existence of patterns related to the mathematical content of a specific teaching unit (cf., also, Kilpatrick1993, pp. 27–29 and Sierpinska1993, pp.
69–71). Of course, we must not expect all these patterns to arise on any occasion nor under all circumstances. It is quite natural that patterns will occur, varying with the educational ecologies. One should be reminded here of the well-known fact that
90 6 Mathematics Education as a ‘Design Science’
Piagetian interviews also reveal recurring content-specific patterns which, however, do not occur with every individual child.
Research centered around teaching units is useful for several reasons. First, it is related to the subject matter of teaching (cf., the postulate of “relatedness” in Kil-patrick1993, p. 30). Second, knowledge obtained from clinical teaching experiments is “local”. Here we need to be more careful in generalizing over contents than we have been in the past. In the future we can certainly expect to derive theories covering a wide range of teaching and learning. But these theories cannot emerge before a variety of individual teaching units has been investigated in detail. For studying the mathematical theory of groups the English mathematician Graham Higman stated in the fifties “that progress in group theory depends primarily on an intimate knowledge of a large number of special groups”. The striking results achieved in the eighties in the classification of finite simple groups showed that he was right. In a similar way, the detailed empirical study of a large number of substantial teaching units could prove equally helpful for mathematics education.
Third, theory related to teaching experiments is meaningful and applicable. We should, however, be aware that, due to the inherent complexity of teaching and learn-ing, the data and theories that research might provide may never provide complete information for teaching a certain unit. Only the teacher is in a position to determine the special conditions in his or her classroom. Therefore there should be no sharp separation between the researcher and the teacher as stated earlier. As a consequence, teachers have to be equipped with some basic competence in doing research on a small scale. The writer’s experience in teacher training indicates that introducing student teachers into the method of clinical interviews is an excellent way towards that end (Wittmann1985).
In the writer’s opinion, the most important results of research in mathematics education are sets of carefully designed and empirically studied teaching units that are based on fundamental theoretical principles. It follows that these units should form a major part of the professional training of teachers. Teachers who leave the university should have in their baggage a set of substantial teaching units that represent the standards of teaching. From the experiences with our primary project Mathe 2000 it is clear that such units are the most efficient carriers of innovation and are well-suited to bridge the gap between theory and practice.