2   Standards for mathematics education at institutes for primary teacher education This chapter defines eighteen standards that are intended to be viewed as signposts. They describe where an educator should look to find quality at an institute for preservice primary teacher education. By formulating these criteria, a statement is thus made regarding the interpretation of the concept ‘quality of the primary teacher education’. This is a concept that has not been precisely determined up to now, nor always described with clarity. What actually determines quality? Is the quality of the institute determined by the difficulty or ease with which students receive a diploma? Or by the degree to which students collaborate with one another? Or by the amount of support students receive from their teachers in obtaining their diploma? By the number of students that complete the course within four years? By the ease with which a new primary school teacher adapts to an already existing team of teachers? Or by the degree to which the institute formulates new developments in the field? Discussions on quality have been based on criteria that are not necessarily explicit. We use these eighteen standards to elucidate our view of the quality of mathematics education at institutes for teacher education. The standards have been provided with subtle distinctions which offer the opportunity to focus attention on interesting details.

The Standards as Spotlight

The eighteen standards can also be viewed as:
 

  • 1. A spotlight to be focused on certain aspects of the educational environment. This spotlight function can enable the educator and student to view the quality of a given area at each point in the education. When all eighteen spotlights are ‘on’, the entire educational process becomes brightly illuminated.
  • 2. Reference framework for educators. The educators can use the standards as a frame of reference for examining the courses available.
  • 3. Gauge for educators. Educators can use the standards as a gauge for measuring the education of their own institute and as indicators for reflective observance of their own instruction. They can thereby examine whether their instruction sufficiently contributes to the development of the student’s professionalism. Is the education broad enough, for instance, and are all areas being dealt with sufficiently? The standards can inspire the educator to reconsider the content of the institute’s courses and the educational concept. Should the education prove at any point insufficient, the standards can then serve as indicators for designing new areas of the curriculum.
  • 4. Hallmark for external evaluators. Others, too, can use these standards to assess the education and the educators. For external evaluators, the standards can serve to illuminate vital areas of the curriculum.

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    Eighteen Standards for Mathematics & Didactics

    1 The mathematics education for Pabo3 students reveals characteristics of realistic mathematics education.

    1.1 Mathematics education is characterized by a positive atmosphere of collaboration and enjoyment.

    1.2 Interaction in the mathematics classes at the institute are characterized by the following: articulating one’s thought processes, listening to the solutions and explanations of others, negotiating, convincing and being convinced.

    1.3 Students are problem-oriented, involved, motivated, and take responsibility for their work.

    1.4 Mathematics is carried out in recognizable situations and familiar contexts. The students therefore have a natural approach that utilizes their common sense; they take time to construct a plan and pay attention to how it is organized.

    1.5 The educator is aware that the self-esteem of many female students is undeservedly low in the area of mathematics, whereby they have ceased to use their common sense.

    1.6 When doing mathematics, the teacher pays special attention to those students who have low self-esteem, an inappropriate faith in rules and who experience a sense of security in indiscriminately copying the educator or fellow students.

    1.7 When working on mathematics and studying instructional theory, the students devote attention to crystallizing their ideas, sketching situations, designing diagrams, making and using models. (In specific situations, students develop a model of that situation. This model can then be generalized to apply to many other situations. The character of the model of the first, specific, situation thereby changes, and becomes a general model for many situations, forming the foundation of the development of a more formal mathematical knowledge.)

    1.8 In mathematics education, the students are offered the opportunity to construct situations, reflect upon them, and elucidate them.

    1.9 Students acquire insight into their own repertoire of strategies and approaches to mathematics; they take into account their own strong and weak points and those of others, and are constantly involved in their own development.

    1. 10 Mathematics education at the institute is characterized by an accessibility to the educators’ and students’ available knowledge and expertise.

    1.11 Help is promptly given to slow students and attention is also paid to quick students by, among other things, involving them all in the educational process.

    1.12 Education in teaching the subject of arithmetic is characterized by clarity with respect to the expectations of the educator.

    1.13 In mathematics education, students work both independently and under the guidance of the educator.

    1.14 Mathematics education does not stand in isolation at the institute; other primary school subjects and areas of development are linked to the subject of mathematics education.

    2 The mathematical subject matter is studied on one’s own level and thereby placed in mathematical educational perspective.

    2.1 By working on one’s own basic numeracy during the first phase of the course, a natural approach and reflective capabilities are revealed and stimulated.

    2.2 Various areas of instructional theory are often introduced through mathematical work at the students’ own level or by focusing on collaboration with fellow students.

    2.3 The characteristic construction of realistic learning strands (for, for instance, counting, adding and subtracting to 100, multiplication tables, long division and fractions) is also presented by using problems on the student’s own level.

    2.4 Devising an explanation to certain problems, whether for children or fellow students, generally begins by reflectively solving the problem on one’s own level.

    2.5 Designing education for specific subject matter is often initiated by one’s own mathematical activities based on this same material.

    3 Students acquire insight into children’s learning processes in the area of mathematics.

    3.1 Children’s mathematical activities (whether written, verbal or on videotape) are analyzed from various perspectives.

    3.2 Students develop activities themselves in order to acquire insight into children’s learning processes.

    3.3 The students regularly talk with individual children (in clinical interviews) about specific problems and their solutions.

    3.4 Students study a given method (such as the method of Kwantiwijzer) introducing diagnostic interviews with children and then hold interviews in accordance with it.

    3.5 Learning processes in the area of mathematics are a frequent topic of lectures, small group work and reading assignments.

    3.6 How to increase the level of understanding of both children and students is a topic of mathematical educational research.

    3.7 Children’s own mathematical productions provide study material for small group work on mathematics education and also serve as illustrations of knowledge transfer.

    4 Students acquire theoretical knowledge of mathematics education through actual practice.

    4.1 Theoretical opinions are always illustrated with examples from actual practice.

    4.2 Students’ narratives on education (from intemships) are placed in a theoretical framework. 4.3 Students’ experiences - both in primary school instruction and in their own education - are subjected to a theoretical examination.

    4.4 The educator chooses his narratives on education on the basis of personal experience, but also on the basis of the theoretical content.

    4.5 Students acquire a great deal of their theoretical knowledge through paradicmatic examples taken from educational practice (narratives).

    4.6 Students personalize the instructional theory through theoretical reflections on their own experiences and by applying the theory to actual practice.

  • 5 Students develop a rich repertoire of mathematics instructional theory

    • 5.1 Material available in this area from the educational publishers can be perused at the institute for preservice primary teacher education.

    5.2 Mathematics textbooks are viewed as a source of inspiration and are studied in order to develop strong ideas on mathematics education.

    5.3 Students are familiar with the major long-term learning strands for mathematics in terms of exploratory contexts, concept formation, introductory problems, core concepts, use of models, educational goals, staces, and potential final levels.

    5.4 Students become familiar with promising and respected inventions for mathematics education (such as, for instance, the empty number line and they gain some experience with these inventions in the primary school classroom and are able to critically evaluate them.

    5.5 Students have a collection of highlights from the primary school curriculum at their disposal.

    5.6 Students view videotapes and read or hear reports of prototypical interactive lessons. At both the primary school and the institute they practice their own variations of such prototypical interactive lessons.

    5.7 Certain tried and tested educational projects, form the canvas for the students’
    own designs.

    5.8 Different ways of explaining, posing questions and guided discovery are supplied with actual examples.

    6 Students develop a broadly applicable diagnostic repertoire

    6.1 Aid to individual children are recorded as case studies.

    6.2 Clinical interviews and diagnostic discussions are given the necessary attention with respect to design and depth.

    6.3 Special attention is paid to the educational perspective in diagnostic interviews.

    6.4 The special help available to children at the internship schools is a topic of study.

    6.5 Tests and test lessons that accompany a specific textbook are used by the students to develop their own testing material.

    6.6 Familiar sticking points in mathematics education are afforded both theoretical and practical attention.

    6.7 Remedial textbooks, are available for further analysis.

    6.8 When at all possible, students work with a remedial teacher at the primary school.

    6.9 The students are included in designing a plan of treatment for a child who is falling behind.

    6.10 The phenomenon of (extremely) gifted children does not escape attention.

    7 The students become familiar with the realistic mathematics textbooks now available.

    7.1 Evidence of realistic mathematics education found in the textbooks is discussed.

    7.2 The most recent textbooks are compared with older ones within the framework of the development of instructional theory.

    7.3 Textbooks are studied in the light of desired goals.

    7.4 Successful aspects of teacher’s guides to textbooks are examined in order to enrich the diagnostic repertoire and the repertoire of teaching strategies.

    7.5 Textbooks and their accompanying teacher’s guides are analyzed with respect to the perspective on realistic mathematics education propounded by the authors in the teacher’s guide.

    7.6 Small segments of certain textbooks are earmarked for constructive analysis, that is, they are used during the preparation of education for the internship school.

    7.7 When possible, review of a textbook by the staff at the internship school is attended by the students or simulated at the institute.

    7.8 Sticking points in the textbook are inventoried by the students in collaboration with their mentor at the internship school.

    7.9 Textbooks form a rich field of investigation for the study of learning strands, explanations, concept formation, differentiation, working self-reliantly, spotting problems, assessment, etc.

    7.1 0 Students articulate a personal evaluation of a textbook of their choice.

    8 Knowledge ofpedagogy, educational and developmental psychology and general instructional theory is applied to the field of mathematics education.

    8.1 Various approaches are explored and practiced during mathematics lessons.

    8.2 Use of manipulatives plays a role in realistic mathematics education against the backdrop of activity psychology.

    8.3 A foundation for the five fundamental educational tenets of realistic mathematics, such as the relation between construction and accommodation, is found in general educational psychology.

    8.4 Considerable attention is paid to the pedago-ic relationship with the children and to the pedagogic climate in the mathematics lesson.

    8.5 Realistic mathematics education for young children (K-2) is linked to other theoretical orientations, such as fundamental development and experiential learning.

    8.6 In addition to the considerable attention devoted to cognitive aspects of the learning process, affective aspects also receive attention.

    8.7 Familiar topics from the theoretical side of developmental psychology (such as Piaget’s phases of development and criticism of phenomena such as seriation and conservation), are discussed in relation to research in the area of mathematics.

    8.8 In certain circumstances, material and methods such as those of Maria Montessori, Peter Petersen, Celestine Freinet and Rudolf Steiner may provide the impulse for a comparative study.

    9 Students acquire skill and take pleasure in designing education and educational materialsfor mathematics.
     

    9.1 Students design their own education based on their own work on a rich variety of problems; core concepts here are inspiration and reflection.

    9.2 The designing of education offers the opportunity to contemplate local theories in the domain of application.

    9.3 When designing education, students begin to view existing textbooks, theories and general educational insights and skills from a new perspective.

    9.4 Designing educational material is seen as one’s own educational production, necessitating reflection upon what one has learned.

    9.5 Teaching something one has designed oneself has the nature of an educational experiment; the focus is therefore not only on the instruction.

    9.6 The student’s own design process, too, deserves further consideration; a logbook or design book can play a crucial role here.

    9.7 Design products of professional developers in this field are presented as examples and provide motivation.

    10 Links are created with otherprimary school subjects and their corresponding instructional theories.

    10.1 Small, clear projects are undertaken, in which the students deal with a number of subjects in relation to one another.

    10.2 Links with other educational areas are sought in existing mathematics textbooks.

    10.3 Mathematical activities in other subjects can be inventoried, such as, for instance, in geography, crafts and physical education.

    10.4 Integrated activities are emphasized, particularly in mathematics for kindergartners.

    10.5 At the end of the preservice education course, time is reserved for comparin. the instructional theories of various subjects.

    11 Collaboration between students is stimulated and rewarded.

    11.1 Students work together to solve mathematical problems and discuss approaches and solutions.

    11.2 The preparation of intemships, as well as the actual instruction and reflection, often takes place in small groups.

    11.3 In the small group sessions on mathematics education, students are encouraged to work on the problems together and to come to an agreement.

    1 1.4 The students themselves are partly responsible for the progress of the group.

    11.5 Preparation of final projects may include workshops and presentations, so that the students can take advantage of the expertise available in the group.

    11.6 Advanced students help beginning students acquire numeracy.

    12 The institute ensures that a great variety of situations and instances are available in which students can optimally develop as individuals in the professional sphere.

    12.1 Students experiment and practice at primary schools.

    12.2 Students attend staff meetings at their internship school.

    12.3 Students engage in discussions with future colleagues and school consultants.

    12.4 Students attend a parents’ evening and, if possible, provide a contribution.

    12.5 Students aid, stimulate and assess the children.

    12.6 If requested by school staff, students conduct a special study assignment; for instance, on comments made about a given mathematics textbook or on extra material for certain students.

    12.7 Students are challenged to contribute their own interests, knowledge and skills and to expand these by, for example: collaborating on a mathematics project for a certain group or for a number of groups simultaneously; designing a mathematical treasure hunt for the upcoming school field trip; writing an article on mathematics for the school newspaper; setting up a work corner in the workshop center for beginning students, who will soon be starting intemships in the lower grades of primary school.

    12.8 Students draft an educational ‘contract-with-oneself’, in order to anticipate the potential of the institute’s fertile environment, and in order to take optimal advantage of this environment in a personal manner and according to individual interest.

    13 Students feel included at the institute and take personal responsibility for their own development in the area of mathematics education.

    13.1 Students are regarded from the very start as future colleagues and are treated as such.

    13.2 Students are stimulated to draft a ‘contract-with-oneself’ for their particular specialization at the end of the course.

    13.3 Students’ projects and own creations are regularly exhibited at the institute.

    13.4 The logbook plays an important role in assimilating the education.

    13.5 In their graduation projects the students reveal their ability to apply their knowledge and, on the basis of these projects, can expand their theoretical knowledge.

    14 Students develop as reflective practitioners.

    14.1 Time is regularly allotted for making reflective solutions, taking reflective notes and reflecting on the theories.

    14.2 Reflection increases as the level rises. It develops from the simple notation of events to analytical commentary grounded in a theoretical background and a developing perspective.

    14.3 During interactive lessons at the institute, students regularly articulate both their solution procedures and their thoughts on the activities.

    14.4 Reflection on actual practice increasingly becomes a point of assessment.

    14.5 Logbook notes are regarded as confidential communication between student and educator.

    14.6 Reflection forms an important aspect of the educational ‘contract-with-oneself’.

    14.7 Students also reflect upon their own individual learning style and transport this style to a higher level.


     

    15 The image of the primary school is constantly present in a variety of ways.

    15.1 Both during lectures and in small group work, primary education is regularly portrayed through narratives, student projects, textbooks and videotapes.

    15.2 Time is reserved for presenting the students’ essential experiences in actual teaching practice.

    15.3 Experiences gained during the internship are set against the backdrop of realistic mathematics education.

    15.4 Changing exhibitions in which students portray primary education are presented at the institute, for instance: a poster of children’s work, a photographic essay of a lesson, a slide show with sound on diagnostic work.

    15.5 Interesting events in the area of mathematics (ranging from a successful explanation to an arithmetic project involving the entire school) stimulate students to engage in their own creative productions.

    15.6 Students make so-called situation analyses within the framework of educational projects, in which the image of the practice school is evoked on a number of essential points.

    16 The profile of the primary school mathematics teacher functions at crucial moments of the course as a beacon and point of standardization.

    16.1 One can see in the mathematics education program how work on the professionalization of the future teacher is successively undertaken throughout the various segments of the course.

    16.2 At a certain point in the course, a checklist of elements for creating an ideal mathematics teacher is presented so that the list can fulfill a functional role in the course from that point on.

    16.3 When drafting an educational contract-with-oneself, the profile of both the ideal mathematics teacher and the ideal teacher of other subjects is taken into account.

    16.4 Certain aspects of the profile require personal interpretation by individual students; time is allotted for the reflection necessary here.

    16.5 The educator’s own choice of content and design for the profile reveals the rationale of the educational program as well.

    16.6 The profile of the ideal mathematics teacher provides a good foothold for collaboration with colleagues from education courses in other subjects so that, when useful, integration of the courses offered is possible.


     

    17 Working on the students’ own numeracy is the focus of attention during the entire course.

    17.1 Conclusion of the first phase of the course with a test on mathematical skills is re-arded as an important milestone on the way to professionalization.

    17.2 Mathematical skills and numeracy lie on one line; mathematical skills end with the 6th grade worksheets, while numeracy is always regarded from the perspective of theory on mathematics education.

    17.3 Beyond the lectures and group work, various signs presuming numeracy are also found in society at large.

    17.4 Concrete problems are used when presenting the problem of innumeracy.

    17.5 Students are called upon to interpret signals from society at large in a numerate manner.

    18 Students are given the opportunity to become familiar with software in the field of mathematics.

    18.1 Students evaluate educational software and also construct evaluation criteria for these programs.

    18.2 When possible, the leamin. processes intended by certain software programs are studied by the students at the internship school.

    18.3 Students investigate and develop the use of software in certain courses, thereby subjecting both the course and the software to a fundamental analysis.

    18.4 Interesting software acquires a special place in the course.

    18.5 Suitable mathematics software is always available for perusal at the workshop center.

    18.6 The use of educational software in primary school is subject to discussion both within a general framework and within the framework of instructional theory.

    18.7 Concise and meaningful software segments are used at the institute for introducing an educational topic (for instance, mental arithmetic and estimation).