Abstract
In our proposal (§ 2.4) we stated our expectations of what this research would produce, both practically and theoretically.
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Notes Chapter 3
See bibliography.
The progress of such a process cannot be entirely reconstructed rationally; firstly, because the process itself is far from rational (see Kuhn, 1972); secondly, because of the number of people involved and, thirdly, because of the dilemma of distance-participation. See also Treffers (1987).
The levels distinguished here are related to Van Hiele’s levels for learning mathematics; See, in connection with this, Treffers (1987). One could state that the point, in this case, is one of relevant levels for producing (learning to produce) a course for the area of fractions. The characteristics distinguished by Goffree (1986) of how young and mature students leam namely: anticipate and reflect, respectively correct and restructure are obvious here.
Later in this study the reflective function of the free individual productions will be concretized further (from chapter 4). Here we suffice by once again referring to Treffers (1987).
See, in connection with this, for example, Bidwell (1982), Kieren et al (1985), Streefland (1987b).
See, for example, Streefland (1986b). In response to Streefland (1984c), Payne (1986) deals with the matter of children’s own constructions. While appreciating the advocated standpoint, he — justifiably —has his doubts with regard to the manageability of the teaching process and the necessary abilities of the teacher. ‘Empirische ergebnisse zu diesen Fragen enthält das erwähnte Heft jedoch nicht’ Translation: The booklet as referred to does not contain empirical data with respect to these questions. He concludes (loc. cit. p. 56). ‘Empirical data’ should be understood, moreover, as coming from the tradition of research in North America. Seen from our standpoint, the publication in question is bursting with raw empirical material.
See, in connection with this, for example, Treffers and Goffree (1985) and Streefland (1986b).
See, in addition to the publications mentioned in note 7, Freudenthal (1984) and Treffers (1987).
In the words of Steffe and Cobb (1983), this means a constant’ shifting’ from the micro to the macro-scheme, as indicated in, respectively, the long-term group process and the individual learning process.
‘Part of education is bringing your knowledge up for discussion, both among colleagues and in the outside world.’ (V. Aalst, 1986, p.21) On the student’s level this signifies doing justice to the educational principles of interaction and individual contribution through constructions and productions. On the researcher’s level the point is that his discoveries are brought up for discussion in a variety of ways in the spirit of the above quote (see also §3.6).
Only then, when specific learning theories for other subjects are available as well, can there be any possibility of a specific learning theory for the beta sciences, let alone a general learning theory having the necessary degree of veracity.
See Streefland (1986b) (1987b).
Indications were indeed present to warrant the assumption that the assignment of creating individual free productions could turn out to be a fruitful educational principle holding a central function in the course to be produced. The author learned from his experiences with construction assignments that students often went beyond the bounds of the assignment and produced a great deal more than had in fact been requested. Van den Brink’s experiences with very young children in a productive role also played a part in the considerations, as did De Lange’s experiences with two-stage tasks in the HEWET project. Van den Brink (1985) (1986), De Lange (1987).
The experiences gained from a course intended for this purpose were satisfactory but not convincing. The researcher sensed intuitively that there was a gap in the outline between the concrete activities and the algorithmizing process. Something was needed to fill it! But what? Acceptance of the provisional correctness of such a discovery would therefore hold direct consequences for the outline (and concrete elaboration) of the course in question. See Streefland (1982c) (1983a).
To avoid misunderstandings, we reiterate here that this had to do with the realization of course according to realistic criteria. Although a wide selection of data and didactic tips can be found, nationally and internationally, in educational textbooks, practical articles, educational reports and broader research publications, these are almost exclusively based on divergent views or theories. They are certainly not useless for our research project as potential sources of inspiration but not, however, suitable as direct suppliers of ideas or elements for the course to be produced.
This sort of glance back is inherent to realistic mathematics education, in which one also attempts to supply the vertical mathematization process at set intervals with a (temporary) finality.
The longitudinal character of the enterprise excludes in advance the alternative of the laboratory experiment. In the first place, the test subjects would at a certain point be disturbed by the influence of the simultaneously continuing normal fraction instruction. Moreover, experiences with individual students cannot simply be transferred as a generalization to the classroom situation.
Development research is interpreted in a variety of ways. Post-development research, whether or not in the service of new developments, and research tied to developments are also called development research. The model of development research presented by Terwel (1984, p.177), for instance, is fundamentally different than the framework as outlined in this study. In Terwel’s definition, the researcher remains out of bounds during the creation of a first version of educational course materials. In the first phase of his model, the problem orientation at school, the first version of the material is already present. His researcher, as an outsider, forms hypotheses during the second phase while, in our opinion, the development of ideas, the formation (and rejection) of hypotheses and the entire didactic deliberation in which the origin of the first version is embedded in fact form the heart of the scientific activities. Moreover, in our ‘model’ the theory plays a constructive role. In Terwel’s model the hypotheses are tested in a second round of evaluation while, in our view, the formation of hypotheses and theories should be at the service of the course to be developed right from the beginning in order to be able to push back the boundaries of the available knowledge; these boundaries are in fact the cause of having to begin with an unfinished product. In other words: in Terwel’s model theory is constructed, that is, the researchers’ implicit theory. Our approach produces constructive theory. The research performed by us concurs as a matter of fact with Lakatos’ heuristic model for mathematical discoveries (Lakatos 1977); see also Davis and Hersh (1985). Lijnse’s standpoint is comparable to Terwel’s model, although compelled by circumstances (Lijnse 1987). In his view, the primary function of didactic research, following the conclusion of the PLON-project, is reflection on preceding developments. (PLON: Project Leerpakket Ontwikkeling Natuurkunde; Project for the development of learning materials for physics). With regard to research into concept and attitude formation by the PLON curriculum, Lijnse states (loc. cit. p.317): ‘It therefore fails to astonish me that, with respect to comparative cognitive learning results/yield in the area of physically correct concept formation, no noticeable difference occurs between PLON-education and traditional education. It is in fact the research into ‘alternative frameworks’ which clearly shows that, although the PLON-curriculum does indeed appear to be more closely linked to the students’ environment, a great number of problems with respect to the physical-conceptual link remain unsolved for obvious reasons’, (italics L.S.)
The above quote contains an important argument for carrying out development research as defined in this study. In our case, we are dealing with the mathematical-conceptual link between teaching and learning which, in all research projects familiar to us, left a great deal to be desired. See further Streefland (1988) (1989).
Some care should be taken here. After all, the students remain a closed book due to everything they do not write down, construct or produce.
Because the group involved in the research project was of a manageable size, it was possible for each individual student to have a turn within a fairly short time span. This was necessary in order to be able to continue certain activities together quite soon after each round of interviews. Particularly towards the end of the research project did interviews take place with this goal in mind, because only then could one begin to clearly discern the contours of the course with regard to its various aspects.
See De Jong (1986).
Here experiences come to mind which took place during exploration of distribution situations and during the algorilhmizing process for addition and subtraction of dissimilar fractions while applying the ratio table as a means of progressive schematization and abbreviation. Streefland (1982c).
The ‘weak’ arithmetic students are in fact the ones who, through their difficulty with fractions, can point the way to didactic solutions of the problem. Moreover, one may expect that the reaction of exactly these weak students to free production assignments will be of decisive significance for the feasibility of the methodical application of said assignments. See Streefland (1987b) (1988).
With regard to the condition of realistic mathematics education, the Fatima Jozef school played a pioneering role in the renovation of arithmetic and mathematics education within the Wiskobas project of the IOWO. This school took part in developing various continuing education courses for teachers, introduced an entirely new arithmetic/mathematics program as participatory school in the Wiskobas project under guidance of members of the Wiskobas team, and acted as a testing station for previously executed research into fractions. See, in connection with this, Wiskobas team (1977) (1978) (1980b) and Streefland (1983a).
See also chapter seven, in which the comparative research is described.
The research project began in September 1983. During the research period the legal amendment regarding primary education was brought into force (on August 1, 1985). For the Fatima Jozef school this involved joining with the Fatima kindergarten under one roof. In the new terminology class (or grade) four becomes group 6.
This has to do with the ‘weights’ 1; 1.25 and 1.9 (1.40 and 1.70). for the significance of these figures see chapter 7, §7.2, where the research group is compared with the control group that participated in the comparative research.
See article 6 of the formation amendment W&O, Staatsblad, 1985, p.337.
During the school-year 1984-85 the research group was tested by the School Pedagogical and Psychological Service in Hilversum. The instruments used were parts of the Differential Ability Test (Number Series series A and Spatial Insight series A), supplemented by the Service’s ‘own’ material for number series, word significance, reasoning with words and sense of sentence construction. The PMT-K of Hermans was also used. The results were (according to stanine standards): Classification of the research group in percentages on the PMT-K of Hermans during the school-year 1984-85. These figures, regarded globally, agree with the expectations held by the group’s teachers.
This group consisted of thirteen students, six girls and seven boys. In long-term research of this type, some turnover can hardly be avoided. If we include the ‘part-timers’, a total of eighteen students participated in the research project. The ‘core’ of the group therefore consisted of thirteen students. Where relevant, mention is made in the description of the study of contributions from students who departed early or arrived late.
During both school-years, the research group was part of a combination class so that, during the experimentation hours, the other side of the combination had to be taught by other teachers than their own. These were W. Haverkort and J. Baay, the latter also director of the school.
Terminology of the British psychologist-mathematician R.R. Skemp (1976) (1979). One can speak here of continual interim reflection.
See Ten Have (1977), Terwel (1978), Goffree (1979, §4.1 pp.187-203) and Terwel (1984). An alert but passive observer may not, however, be able to avoid involuntarily influencing the research situation. See, in connection with this, Adda (1982).
See, for example, Boenders (in conversation with Thomas Kuhn) (1978, p.53).
See Boenders (in conversation with Karl Popper) (1978, p.15).
In the ‘mutual observation’ method, one presents the interviewee with (partial) protocols in order to verify what has registered as well as the possible presence of biased observations on the part of the interviewer. See Van den Brink (1981).
See, in connection with this, Streefland (1984d) (1985c).
See chapter 2, §2.3.
In this way was development research into a course for subtraction and addition of dissimilar fractions according to progressive schematization using the ratio table presented at the fifth international congress for the psychology of mathematics education (PME) in Grenoble and publicized in the Proceedings of the congress. See Streefland (1981c). A number of publications of varying size and distance from educational practice appeared concerning the course. Streefland (1982c) (1983d) (1983c) (1987b).
A number of teacher trainers and educational advisors were consulted with regard to this at the yearly Wiskobas fall conference.
Bidwell (1982), Kieren et al (1985).
Advancement of the advocated scientific discussion based on the demands on development research as stated by Treffers is imperative due to the need for theoretical clarity. ‘And clarity signifies that which can be shown not to be able to fulfil the claim of generality should be referred back to specificity.’ Streefland (1986b, loc.cit. p.19). The above quote refers to Gagné’s three-phase model for solving mathematical problems in application, and which demonstrably belongs to the basic structuralistic concept and not to the realistic one. (Gagné 1983).
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Streefland, L. (1991). The Context of the Research. In: Fractions in Realistic Mathematics Education. Mathematics Education Library, vol 8. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-3168-1_4
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