Since the sixties, differentiation has been an important issue in the discussion of and decision-making about education. However, for a long time the empirical research on differentiation has been characterized by a strong focus on the influence of grouping on the results of learning. The conclusions of this research were not univocal and often incomparable. In empirical research little or no attention has been paid to other aspects of differentiation, for example those dealing with the instructional process, and only from the eighties on have these aspects been regular subjects of investigation. So at the beginning of our study little was known about the didactical action of teachers of mathematics with respect to classroom-differentiation (i.e. their classroom-differentiation-behaviour).

For those training prospective teachers of mathematics this meant that the question what opportunities we should offer them to prepare themselves on classroom-differentiation, could not be answered on the basis of much empirical evidence. One of the aims of the present study is to fill up that lack of knowledge by extending our information about the actual classroom-differentiation-behaviour of mathematics teachers, the types in which that behaviour can be classified, and the factors which have an influence on it. From the viewpoint of relevance of education we have mainly dealt with the factors `differentiation methods of schoolbooks on mathematics' and `grouping'. Other factors have only been incorporated in our investigation as covariates of these two. For reasons of feasibility and check we have limited the main research to the mathematics education in schools for general secondary and pre-university education with a one-year transition course. We have formulated the following research questions:

(a) To what extent and in which types do mathematics teachers show classroom-differentiation-behaviour in the one-year transition course of schools for general secondary and pre-university education?
(b) What influence do the factors `differentiation methods of mathematics schoolbooks' and `grouping' have on the classroom-differentiation-behaviour of mathematics teachers?
(c) Which other factors like teacher, group, school and situation characteristics do also have an influence on the classroom-differentiation-behaviour of mathematics teachers?

To investigate these research questions we have developed a theoretical framework on the basis of educational and subject didactical assumptions, in which we have differentiated between two methods of classroom-differentiation, a process-oriented one and a product-oriented one. In the first method the learning process is the central element, and time- and task-bound differences between pupils function as criteria for the differential course of the teaching-learning process. In the second method the products of learning are the central elements and these are used as explicit and implicit differentiation criteria, together with differences in personal characteristics between pupils that are related to them. Connected with these methods of classroom-differentiation we have discriminated between two types of classroom-differentiation-behaviour of mathematics teachers, knowing a) process-oriented and b) product-oriented. Furthermore we have presupposed the possibility that teachers can avoid classroom-differentiation actively (non-differentiation-behaviour). Because these three types of classroom-differentiation-behaviour can operate on different aspects of the teaching-learning process, it was not possible to postulate correlations between them in advance.

We have made the present study from an action-theoretical point of view. This means that the teacher is not seen as a black box who shows overt behaviour that is only influenced by circumstances, but as a professional who acts on purpose on the basis of his subjective theory on education. Part of this subjective theory is also the knowledge the teacher has of his own didactical actions. Therefore in our investigation on the classroom-differentiation-behaviour of mathematics teachers we have made use of the reports of teachers about their own behaviour, because that knowledge is an integrated part of their didactical actions. We also collected complementary information about the overt classroom-differentiation-behaviour by means of the perceptions of the pupils who experienced that behaviour. For both kinds of information we developed almost identical questionnaires. From the comparison of those two kinds of information we have been able to draw conclusions with regard to the didactical actions of teachers.

With respect to the factor `differentiation methods of mathematics schoolbooks' we could also use the distinction between the process-oriented and the product-oriented differentiation method. A strategy of analysis was designed to investigate the differentiation method of mathematics schoolbooks, which made use of accompanying publications written by the authors, as well as learning material from the textbooks. For the analysis of the learning material we assumed a number of critical features which could be connected to the differentiation methods for theoretical reasons.

For reasons of relevancy for educational practice the factor grouping has been limited to the variations `ability grouping' (with two ability levels) and `mixed-ability grouping' practised in the transition year of many schools for general secondary and pre-university education. As such the first treatment of the factor grouping was one of educational practice on school-level. A stringent definition of heterogeneity versus homogeneity in terms of a greater and smaller amount of variance of objectively measurable characteristics of pupils in groups has been left behind, not only for practical reasons but also on action theoretical grounds. On the contrary, the concepts of heterogeneity and homogeneity were treated with the teacher's subjective perception as point of departure. This subjective perception was supposed to play a mediative role in relation to the influence of the factor grouping on classroom-differentiation-behaviour. The subjective perception of the heterogenity has been described in terms of the amount of variance of the teacher's own scores of the learning results of his pupils on his selfmade tests and in the form of grades. Comparable assumptions have been made for the factor `general group level' which can be seen as a derivative of the grouping method. Mathematics teachers evaluate that level in the first place as classroom subject level using their own tests and grades. This means that the factor grouping and its derivative `general group level' are mainly interpreted in terms of educational practice on school-level, but that checks have been made with respect to the subjective perceptions and assessments of the teachers to declare its influence on classroom-differentiation-behaviour.

The results of the present study are threefold. In the first place a questionnaire about classroom-differentiation-behaviour for teachers and pupils has been developed with which the three types of classroom-differentiation-behaviour can be measured in a relatively simple way. The scores for process-oriented classroom-differentiation-behaviour turned out to be independent of the scores for product-oriented classroom-differentiation-behaviour and non-differentiation-behaviour. These last two scores showed a negative correlation. Occurring discrepancies between the high alphas of the subquestionnaires in the construction inquiry and the lower alphas in the main investigation can be explained in terms of the way in which both research groups have been formed.

Secondly the present study has generated a strategy for the analysis of the differentiation methods in mathematics schoolbooks. With the aid of this strategy it will be possible to decide which mathematics schoolbooks use a more or less process-oriented or product-oriented differentiation method. With regard to the transition class, the analysis of two much-used mathematics schoolbooks named Getal-en-Ruimte (Number-and-Space) and Moderne-Wiskunde (Modern-Mathematics) have led to a clear comparison of their differentiation methods.

In the third place the present study has resulted in more insight into the influence of certain factors on the classroom-differentiation-behaviour of mathematics teachers in transition classes of schools for general secondary and pre-university education. It has been shown that the differentiation methods of the two used schoolbooks on mathematics influence the classroom-differentiation-behaviour of mathematics teachers in the expected direction. We also have found an independent positive influence of the process-oriented method in mathematics schoolbooks on process-oriented classroom-differentiation-behaviour of mathematics teachers. With respect to product-oriented classroom-differentiation-behaviour a comparable independent influence of the product-oriented differentiation method of mathematics schoolbooks has not been detected. These results are based on the scores of both teachers and pupils.

With respect to the influence of the factor grouping on classroom-differentiation-behaviour the results of the investigation are less clear. Although the heterogeneity of the groups has not been detected in a objective way, teachers in mixed-ability groups percieve subjectively a greater heterogeneity than teachers in ability groups, however, only with respect to long-term learning processes expressed in the form of grades. Generally speaking we have not found enough support for the assumption that a greater heterogeneity in classes leads to more classroom-differentiation-behaviour and to less non-differentiation-behaviour. The results in fact point for non-differentiation-behaviour into the reverse direction: in mixed-ability groups teachers show more non-differentiation-behaviour than in ability groups. Also in mixed-ability groups more process-minus-product-oriented classroom-differentiation-behaviour has been found. The perception of teachers and groups however differentiated strongly with regard to the influence of the factor grouping. In the perception of the teachers process-oriented classroom-differentiation-behaviour has positively been influenced by a greater heterogeneity, in the perception of pupils this relation was altogether negative. In the perception of teachers non-differentiation-behaviour and process-minus-product-oriented classroom-differentiation-behaviour have positively been influenced by greater heterogeneity, while in the perception of classes no differences have been detected.

With respect to the factor `general group level' we have found that teachers in low ability groups show less process-oriented and less process-minus-product-oriented classroom-differentiation-behaviour than teachers in high ability groups. The first of these two effects occurs only in the scores of the teachers, the second one in the scores of both teachers and pupils. With respect to product-oriented classroom-differentiation-behaviour and non-differentiation-behaviour no differences occurred between low and high ability groups.

The research on covariates has not brought forward a variable which could declare the foregoing results. Rather, an independent effect of the factor 'percentages of girls in classes' has been found. This factor has a positive influence on the process-oriented classroom-differentiation-behaviour of mathematics teachers both in the perception of teachers as well as pupils.

In the discussion chapter we suppose that the lack of more classroom-differentiation-behaviour in mixed-ability groups in comparison with ability groups is perhaps a consequence of the choice of the school not to stress differences between pupils in an early stage of their school career. This school vision may and can have an influence on the attitude of teachers not to stress differences between pupils in the classroom and so to avoid certain aspects of differentiation-behaviour. Therefore the role which the school vision on grouping plays in the behaviour of teachers should be examined in further research.

The present study shows that classroom-differentiation-behaviour of mathematics teachers could be seen as an important mediating variable between the input-variables of the teaching-learning process and the output-variables in the form of the learning results of the pupils. Classroom-differentiation-behaviour as such will be examined as an important issue in further research on the effects of classroom-differentiation. In connection with our investigation we have discussed the possibility of developing a teacher training program in which all professional aims and skills of classroom-differentiation-behaviour with respect to the practice of mathematics education will be treated either in preservice or in in-service training on the basis of the different phases in the learning needs of prospective-teachers and teachers. Finally the present study offers some possibilities for mathematics teachers, sections, and authors and publishers of mathematics schoolbooks to improve the quality of classroom-differentiation in mathematics education.