The NVORWO Commission on Standards for Mathematics Education
Fred Goffree, Chair
Frank van Merwijk
Jan Willem Oonk
Standards for Mathematics Education is a translation of a portion of 'Proeve van een nationaal programma rekenen-wiskunde & didactiek op de pabo', a Dutch publication containing standards for the subject area of mathematics education at institutes for primary teacher education in the Netherlands. This handbook was developed by a group of twelve Dutch mathematics educators, with the support of the National Institute for Curriculum Development (SLO) and the Freudenthal Institute. The core of this handbook consists of eighteen standards which describe the mathematics education program at institutes for primary teacher education. The remaining chapters may be regarded as a reservoir of ideas for educators. The complete table of contents of the Dutch-language publication is printed at the end of this booklet, so that readers may find the original Dutch version of the sections that have been translated here.
The Dutch Standards differ from their American counterparts, partly due to the difference in prior circumstances and partly to the differences between Dutch and American culture. The following information on the Dutch educational system is intended to orient the reader with the educational situation in the Netherlands in general and the Dutch standards in particular.
Approximately 15,000,000 people live in the Netherlands. The country contains 8,000 primary schools for children between the ages of 4 and 12. There are thirty-eight institutes for primary teacher education, which employ approximately 150 educators for the subject of mathematics education. Since 1968, all educators in mathematics education belong to a professional association. They meet one another at a number of annual conferences, and during workshops and courses. The continual development of instructional methods in mathematics education is the topic of discussion at these gatherings. Primary school teachers are educated at four-year institutes of 'higher vocational education' (as opposed to at universities). Every primary school teacher must be able to teach all subjects to students ranging in age from 4 to 12 years-old. At present, during the final year of the course, the student teachers choose a specialization, either in teaching the younger children (4 to 8 years-old) or the older children (8 to 12 years-old). The reform of primary school mathematics education in the Netherlands has a long history. Thanks to the influence of Hans Freudenthal, the (American) New Math movement did not take root in this country, and a different reform of mathematics education began to take place as far back as 1968. Mathematics curricula and sample lesson material were developed at the IOWO (now the Freudenthal Institute), which were then elaborated upon and incorporated into textbooks by teams of authors. A characteristic aspect of the implementation of realistic mathematics education in the Netherlands is the considerable influence exerted by textbooks on primary education. Six major realistic textbooks are published in The Netherlands: De wereld in getallen, Rekenen & Wiskunde, Pluspunt, Rekenwerk, Operatoir Rekenen' and Naar Zelfstandig Rekenen.
As is stated in the Standards for Mathematics Education, teacher education in the subject of mathematics is characterized by three pillars: reflection, construction and narration. These pillars were the deciding factor in the choice of sections to be included here. Sections 3 and 4 deal with reflection, and section 5 with narrative knowledge (narration) and section 9 with constructive analysis.
Why, in fact, did a national plan arise for teacher education? What inspired the NVORWO, in 1990 to approach the National Institute for Curriculum Development (SLO) regarding a project of this nature? The reasons were twofold. On the one hand, primary schools, in which textbooks based on realistic mathematics education were increasingly being introduced, were in need of well-educated teachers who could put the implementation of these textbooks into practice. On the other hand, the numerous institutes for primary teacher education developing at that time were organized in varying ways - some quite experimentally. The intrinsic change demanded by realistic mathematics education of teachers on a national level made it imperative that a nationwide approach to education be established - with quality at all locations.
A nationwide network of primary school educators for mathematics education has existed in the Netherlands since 1968. The majority of these educators has been involved in the development of realistic mathematics education at the primary school level. From the very start, the content and design of the courses to be offered in this subject area also formed a topic of discussion and development work. The amount of time in the entire course load allocated to the subject of mathematics education received attention as well. During the nineteen-seventies, in addition to Wiskobas blocks, the theoretical aspects of teacher education and a standpoint regarding this education were also established. Stated succinctly: "the creation of learning strands between children's subjective (informal) mathematics and objective (formal) mathematics is taught at the institutes for teacher education by using the students' own mathematical learning processes, their reflection on these processes, and an acquired basis of didactic orientation enables the students to examine the children's learning processes, to organize them, and thus to learn how to teach." (Goffree, 1979, p. 313) During the nineteen-eighties, this concept was elaborated upon further and developed into a complete curriculum (Goffree, 1985, 1992, 1993, 1994). The existence of this nationwide network of primary school educators (which can now be regarded as an integral part of the more comprehensive NVORWO infrastructure comprising school consultants, researchers, assessment developers and primary school teachers) also facilitated a national consensus on the quality of the courses offered.
The acknowledgment of the necessity for a national curriculum for mathematics education at institutes for primary teacher education and the prospect of creating such a curriculum in consultation with all the educators concerned inspired the NVORWO to submit the abovementioned proposal. A development project was initiated comprising a core group of prominent primary education educators and supported by as many of their colleagues as possible (Wijdeveld, January 14, 1991).
Development work for vocational education
Curriculum development for institutes of primary teacher
education has its own, specific history. Although attention to the requirements
of the teaching profession has increased during the past hundred years,
the dominant role has traditionally been played by the subject matter.
This is not particularly surprising, considering the fact that primary
school teachers must deal with a great amount of subject matter. And, of
course, in order to be able to teach it, they must have mastered it themselves.
An extra problem arises, however, for the subject of mathematics.
For some students (and educators), even learning primary school subject matter in this subject area requires considerable effort. At times it appears that the entire course of study could be filled by acquisition of this basic knowledge.
During the nineteen-fifties and sixties in the Netherlands this was, in fact, not unusual. One may assume that the presence of so many subject areas at primary schools and the extensive amount of subject matter involved has made it impossible to raise the curriculum development for instructional theory at the institutes for teacher education above the subject matter itself. Little attention has ever been paid either to the profile of the ideal teacher or to insight into the development of his or her professional skill. In the area of mathematics education, however, a number of impulses in this direction did begin to crop up in the nineteen-seventies (see Goffree, 1992).
If one looks at how most curricula for vocational education are currently developed, one can see that, in most cases, the following steps are followed (Nijhof, Franssen, Hoeben and Wolbert, 1993): A professional profile is constructed, in which the necessary qualifications for practicing that profession are identified, and from which the attainment targets (i.e. knowledge, skills and attitudes) are derived for the course of study in question. The subject matter is then chosen, based partly on the students' own intellectual abilities at the start of the course. This subject matter is then organized alone, the lines of instructional theory and supplemented by potential educational activities. Finally, points of verification are introduced and tests, or guidelines for assessment, are designed.
This could be described as a rational approach. Regarded from a purely logical perspective, and viewed against the backdrop of traditional vocational education, this is indeed the way to design a vocational course of study. Upon closer examination, however, doubts may arise. A different development strategy can be seen in the manner in which problem-based learning was taken as the starting point for designing vocational education (for example, at the University of Limburg) (Schmidt, 1982). It is true that here, too, the point of departure was the study and analysis of the profession in question; the focus of investigation, however, was the core problems that would surface on the job, rather than the qualifications.
The curriculum was then designed around these core problems, and constant attention was devoted to the way the students, through working on the problems, could acquire the requisite professional skills. In the case of the course of study at the University of Limburg, the focus was not on the solutions to these problems but, rather, on the knowledge and skills that could be acquired through collaboration while searching for information with which to tackle the problems. And, of course, the goal was for students to develop a positive attitude with respect to problems and problem solving.
The rational model, described first, may indeed be suitable for fairly uncomplicated vocational profiles, that is, for professions in which instrumental knowledge is extensively used for conducting certain frequent (routine) activities. In such cases, the requisite knowledge is simple to map out and the necessary skills can easily be divided into sub-skills, each of which can fill a section of the curriculum.
Development work for institutes of teacher education
This rational model is not satisfactory, however, when it comes to educatig primary school teachers. For one thing, an approach based on profile and qualifications would merely lead to an instrumentally educated teacher, i.e., one who simply implements that which is available from, for instance, the educational publishers. But it is also unsatisfactory because of the fact that, in this approach, the student teacher remains entirely invisible. In contrast both to this top-down strategy according to the rational model and to the collection of core problems according to the second model, the project group (PUIK) posed an approach to development work inspired by realistic mathematics education.
In this approach, the students themselves and their professional development are the focus of attention. The concrete points of attention are subjective ideas, the students' own productions, reflection and interaction. The profile of the ideal teacher forms a desirable perspective for both student and educator. One advantage of this approach to development work is that both the institute as a research terrain and the educator/developer's practical knowledge in the area of mathematics education can be exploited. The project group thus chose an approach to curriculum development that lies close to the students, between the ideal profile of a beginning teacher and prevailing educational practice, and supported by an explicit viewpoint on how to educate primary school teachers in the subject area of mathematics.
An educational concept: three pillars
This viewpoint on educating teachers in the domain of mathematics, which arose in the development work of the nineteen-seventies and eighties, was further elaborated upon in the Puik-project (Puik, 1992, 1993, 1994). Currently, there are three pillars - reflection, construction and narration - upon which mathematics education can be built. The principles according to which realistic mathematics education is instructed at primary school form the foundation supporting these pillars (Treffers, De Moor & Feijs, 1989). This does not mean, of course, that instruction at institutes for teacher education is conducted in the same manner as at primary school; nevertheless, it is conducted according to the same principles and in the same spirit.
The similarity between primary school education and teacher education is the clearest in the area of reflection. Just as mathematics is learned by doing, so can the professional skills of the mathematics teacher be acquired through the performance of a great number of activities. These activities occur on three levels, namely, the level of the primary school subject matter, the level of teaching activities involving this subject matter, and the level of theoretical activity in the domain of the theory on mathematics education. At each level, activity will only lead to expertise if time is taken for reflection. Reflection on an activity, moreover, is an appropriate foothold for a course of study on a theoretical level. After all, a theory arises through the reflective ability of experts in the field and researchers. Throughout the course of study, a line of development is created that runs between practical activities, reflection on these activities, observation of the practical activities of others and examination of their reflections, examination of the theory as an extension of the reflections, and reflections on the theory behind one's own and others' activities (PUIK, 1993).
Reflection is a prerequisite for learning, particularly for learning from one's own activities. The skill of reflection, therefore, should be one of the attributes of a beginning mathematics teacher.
By the time they enter higher education, students have already gained a variety of learning and teaching experiences, including in the area of mathematics. Most students, therefore, as a result of their experiences involving subject matter, teachers and fellow students, already subscribe to a certain viewpoint, both with respect to mathematics itself and to learning and teaching math. This viewpoint is usually unintentional and almost always delivered implicitly with the subject matter. Neither the educators nor the students, themselves, are particularly conscious of the presence of this viewpoint. Educators can learn a great deal from reading students' essays on their own mathematical history. On the whole, students who were weak at school in mathematics usually have more to recount than the others, although one must read between the lines to find the viewpoint.
The students' personal assimilation and coloration of what is taught at the institute for teacher education is based on these experiences and viewpoints. Partly through new experiences gained both at their internship school and in their own course work, they will construct their own expertise and corresponding viewpoint. Should one, during the training course, wish to adjust and alter the existing preconceptions of beginning students in favor of a view on mathematics that fits the realistic approach, one should not commit the error of ignoring the students' initial circumstances. This is even important during the first phase of the course, in the section on basic numeracy. In this section teacher educators aim to improve student teachers own individual mathematical strategies and approaches. If one presents this course in a purely product-oriented manner and believes that new educational approaches for primary school children will be sufficient in themselves for remedying students' deficiencies, then one is simply reinforcing preconceptions that run counter to the principles of the available practice material. The necessary adjustments of the student teachers beliefs on mathematics education will then become even more difficult.
Not only does this constructivistic approach require special attention at the start of the course of study, but the final phase, too, must be regarded with more than the usual attention. One must at least ascertain whether the education has been successful in approximating the ideal profile of the beginning teacher and, if so, that this has not been exclusively in terms of knowledge and skills. The aspects of attitude and viewpoint are, after all, at least as important. The graduating students, too, must know what kind of mathematics teacher they have become. In order to guide such development along the right paths during their four years at the institute, students must frequently be offered the opportunity to express and discuss their individual viewpoints. This will provide them with important insights, as discussions bring up values and norms that affect both the subject and the teachers.
During the introduction of kindergarten math, a third pillar, that of narration, is added to those of reflection and construction. A great deal of knowledge of education of student teachers and teachers gathered from among the children in the classroom, has been notated in the form of anecdotes. The anecdotes of the educational pioneers Thijssen and Ligthart, for instance, in which children and teachers are described in the smallest detail, are wellknown in the Netherlands. Researchers such as John Holt and Hans Freudenthal added reflections to their classroom observations, at times thereby creating a link with the underlying theory. And then there are also other Dutch development researchers, such as Van den Brink (1989), Streefland (1988) and Gravemeijer (1994). Their theoretical reflections are notated not infrequently in the form of anecdotes from the classroom. In other words, some primary school anecdotes have a theoretical charge and some do not. For teacher education, such narratives can play an important role in the area located between theory and practice, an area often mistakenly described in mathematics education as a gulf. Educators collect certain anecdotes as part of their repertoire and cherish the paradigms; these are the narratives that contain considerable exemplary properties. They represent numerous observations of a given phenomenon and are of a high theoretical caliber. Moreover, these narratives reveal essential aspects of (realistic) mathematics education (such as phases, intuitive ideas, levels and level advancement in learning processes) and are easy to remember. This is known as narrative knowledge (Puik, 1992).
Standards for quality
An investigation into the quality of the courses offered for mathematics education could commence with these three pillars. The focus of such an investigation is then on what the students construct for their education and what they do with it, their ability to reflect on their own activities in actual practice, and what they, as experts in this practice, have to report.
A researcher examining the quality actually looks at what the student is learning or has learned from the courses offered; more aptly stated, the researcher observes what the student is making or has made of the education. This is an excellent method of evaluation, albeit difficult methodologically and extremely labor intensive. Those who prefer more direct and cheaper research will lower their sights and simply observe what it is the students are being offered. The researcher will in the latter case focus attention on essential aspects of the learning environment, findthe salient features of the course, and observe in a fair amount of detail what is taking place. In order to attach significance to these observations, they are placed in the encompassing framework of the philosophy of mathematics education and the concept of teacher education. This, then, initiates the moment of reflection (Inspectie van het onderwijs, 1989). These standards for a national plan for mathematics education at institutes for primary teacher education provide criteria for quality and a reference framework for examining the course of study. The assumption of the project group was that this investigation would be primarily conducted by the educators themselves, for the benefit of quality control and in order to improve their own courses in mathematics education.