Abstract
In this chapter the three-dimensional goal description that has been introduced previously is explained and discussed on the basis of the “Freckleham” theme.
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References
Referred to here are material aids including written assignments and texts such as the themes presented which are supplied to the pupils so that they may carry out learning activities. They can be accompanied by teachers’ manuals. In the text we often use the term “pieces of mathematics instruction”.
Baker, E. L.: The technology of instructional development’, in Travers, R. M. W. (ed.), Second handbook of research on teaching, Chicago 1973, p. 249.
The term curriculum development can refer to a wide range of planning, directed towards instruction as a whole, school programmes, and series of lessons. Here we areespecially interested in planning lessons by means of learning packages including projects, themes, partial curricula, such as exemplified in this study.
This terminology is found in: Krathwohl, D. R. and Payne, D. A.: `Defining and assessing educational objectives’, in Thorndike, R. L. (ed.), Educational measurement,Washington 1971, p. 21.
Teunissen, J. M. F.: Handelingsmodellen voor de constructie van onderwijsleersystemen’, in Creemers, B. (ed.), Bijdragen tot de onderwijskunde, Den Bosch 1973, pp. 65–111.
Walker, D. F.: `Curriculum development in an art project’, in Reid, W. A. and Walker, D. F. (eds.), Case Studies in Curriculum Change,London 1975, p. 99. Loc. cit. p. 95:
In their first few weeks of discussions the Project staff made several decisions that shaped their subsequent work. They decided to prepare some lessons immediately, rather than, say, prepare flow charts of the whole Project’s work, or write a rationale or other planning document. Also they decided to do lesson writing outside staff meetings, working individually or in teams of two, reserving for general discussion only questions that arose in the outside work. This decision meant that the work would be divided into two ongoing parts, production of lessons and support materials, and discussions,or as I later came to call it, deliberation
Loc. cit., p. 108.
Loc. cit., p. 133.
The working manner followed in the “Kettering Project” described by Walker, is closely similar to that of Wiskobas. A few people have certain basic ideas on education (the subject area) and the appointment of the other project members is made on the basis of the likelihood that they will be able to work as a team within that educational philosophy.
Loc. cit., p. 133.
For the terms integration phase, exploration phase, curriculum, see Sub-section 2 of Chapter I and Note 12.
The framework plans were never published. The six books of 40 pages each served as an overview for the designers of pieces of mathematics education for the various grades.
More about these points can be found in Notes 28, 29 and 30.
Bruggen, J. van: `Abacus en leerpsychologie’, in De Abacus, IOWO curriculum development publication 6, Utrecht 1977, p. 105.
The following can serve as information: The relation “is longer than” is indicated by D and is as long as by A statement about triples of objects (people in this case) such as ... D ... has the property: if a D b and b D c,then a D c This property is called transitivity. It also holds for the relation A statement about pairs of the kind.has the property: if ab,then b •- a. This property is called symmetry. It does not hold in general for file relation “ D ” between pairs.
About greetings see Ortega y Gasset, J.: De mens en de mensen,The Hague 1958, pp. 195ff.
An enquiry was held among some one hundred teachers taking an in-service course. A large majority reacted favourably to “Freckleham” and said the same for their pupils. In less than 10% the reaction was that the problem at issue was unrealistic. Experience has shown that in a few cases the context story had been somewhat modified. We had expected teachers to show more reserve towards this kind of theme.
Allerdings ist durchaus möglich, Guilford’s dreidimensionale Intelligenzfaktoren als Lehrziele zu betrachten, was beispielsweise auch in der pädagogischsychologiaschen Kreativitätsforschung geschieht (see also Mühle and Schell 1970 ). Dabei ist dann auf vorbildliche Weise das Problem gelöst, wie man vom Lehrziel zum Lehrstoff and zu lehrzielorientierten Testaufgaben gelangen kann.Ein diesem formal sehr ähnliches Konzept wird weiter unten vorgestellt, wobei die Lehrziele durch eine Klasse von Testaufgaben definiert and gleichzeitig nach dem Inhalts-and Verhaltenaspekt (evtl. auch nach dem Produkt-aspekt) bestimmt werden. (Klauer, K. J.: Methodik der Lehrzieldefinition and Lehrstoffanalyse, Düsseldorf 1972, p. 24.)
The four dimensions as suggested by De Corte are: subject matter dimension, content-information dimension, product-information dimension and operation dimension. Klauer takes De Corte’s dimensions one and two together, which in our opinion is meaningful. Corte, E. de: Onderwijsdoelstellingen. Bijdrage tot de didaxologische theorievorming enaanzetten voor het empirisch onderzoek over onderwijsdoelen,Leuven 1973, p. 146.
Wheeler, D. K.: Curriculum Process, London 1967, p. 113.
Block, A. de: Taxonomie van leerdoelen,Antwerp 1975, pp. 55ff.
Johnson, D. A. and Rising, E. R.: Guidelines for teaching mathematics, Belmont 1969. Dormolen, J. van: Didactiek van de wiskunde, Utrecht 1974.
Pikaart, L. and Travers, K. J.: `Teaching elementary school mathematics: a simplified model’, The Arithmetic Teacher 20 (1973), p. 334.
A lesson transcript elucidating the objectives pursued can be found in the following IOWO publication of theWiskivon team: Sweers, W. (ed.): Leerplanontwikkeling onderweg. IOWO publication Utrecht 1977, pp. 12–21.
In the mathematical activity the accent will often be put differently as to the language aspect, the use of models and concept formation. Thom rather emphasises the requirement of “meaningfulness” for the mathematical activity. Wiskobas’ efforts are marked, among other things, by a high degree of attention to the meaningful reality that the mathematical activities should have for the pupils. This appears, among other things, in the construction of “worlds”, but “meaningfulness” also always emerges in the theoretical reflections. In his reflections on didactical phenomenology, Freudenthal stresses the need to depart from the experiences lived through to arrive at the constitution of mental objects. Van den Brink, responsible for the Wiskobas programme in the lower grades, often refers in his papers in Wiskobas Bulletin to “realising” by which he means that the pupils should be able to imagine a world béhind the problems at issue: the problems must be meaningful to them. In fact what Goffree calls “semi-transparent mirror reflections” with a view to the teacher can also be applied to pupils: the pupil must be able to “see through” the problem, but also see his own mirror image in it. The themes included in this study can be taken as examples of “meaningful” mathematics instruction. Next to the work By Thom we also refer to Swenson’s book which considers initial instruction in arithmetic from the point of view of meaningful instruction. She says: Meaning is experience. Children learn to know and understand what they have experienced. ... Meaning is context. Children learn best within a rich context of meanings.... Meaning is intent. Children’s purposes and intentions are as important as adults’ purposes and intentions for them.... Meaning is organization. If the results of learning are to be useful, they must be organized.
From research by Inhelder, Piaget, Bruner, and others, the designer of “Freckleham” knew that third graders, generally speaking, are capable of reproducing, completing, and re-arranging a two-way order. It was not known, however, that children of this age are capable of constructing a two-way order themselves. See: Inhelder, B. and Piaget, J.: The early growth of logic in the child. Classification and seriation, London 1964.
The question of transitivity reasoning has been examined frequently in educational and developmental psychology. The results are often conflicting. The heart of the problem lies in the essence of reasoning by transitivity. How does it work? Suppose that for four individuals A, B, C and D it is known that A is heavier than B, B heavier than C and D is heavier than C; who is the heaviest; and who is the heaviest in the group A, B, C and D? And if the relation “is heavier than” is substituted by “is faster than” or “is older than” or “is further ahead of” or “is longer than”, will this make any difference for the reasoning? If in the last case the lengths are drawn or represented by straws, will this change matters? To make clear the problem we will give four illustrations, on the basis of which we will try to answer these kinds of questions.
This was established for some 50 teachers in in-service courses, who were presented with the problem of the bicycle tracks in the snow (see Note 28).
Frédérique Papy, especially, has stimulated the use of arrow-language for young children. The fact that children must learn to use this language is also noticed in a study by Wallrabenstein: older children are not capable of filling in arrow-diagrams on their own. See: Papy, F.: Graph Games, New York 1971. Wallrabenstein, H.: Development and signification of a geometry test’, in ducational Studies in Mathematics 5 (1974), 81–91.
See also Note 18 for this chapter.
This became evident at conferences and in-service courses. First of all a number of teachers and teacher trainers feared that the problem would not appeal to the children; and secondly they had difficulty in coming to grips with the objectives. At the first in-service course in Hilversum, it was seen that the instruction progressed poorly because of the lack of holistic goal description. The first teacher-participants, who had to work without these descriptions and explanations, were therefore less contented with the theme. Later, when the theme was disseminated with a three-dimensional goal description attached, this changed. We learned particularly from conferences that a clear goal description is considered to be of great importance.
See Chapter I, Sub-sections 2.2, 2.3, 4.1, 4.2 and 4.3.
See for example: Stake, R. E.: `A theoretical statement of responsive evaluation’, Studies in Educational Evaluation 2 (1976), 19–23.
Attempts toward “Three-dimensional” educational evaluation within mathematics education are not known to us, although the USMES-project (Unified Science and Mathematics for Elementary Schools) does point in this direction. See Shapiro, B. J.: The notebook problem. Report on observations of problem solving activity in USMES and control classrooms,USMES-publication 1972. Shapiro, B. J.: USMES evaluation report on classroom structure and interaction patterns,USMES publication 1974.
This idea was heard from school guidance counsellors in Arnhem. So-called “Indian Cards” are also known in mathematics education.
See: Bruggen, J. van: `Abacus en leerpsychologie’, in De abacus, IOWO curriculum development publication 6, Utrecht 1977, pp. 96–112.
From IOWO circles we refer to: Freudenthal, H.: Mathematics as an educational task, Dordrecht 1973, vi. Streefland, L.: Breuk in ontwikkeling. Een orientatie in psychologie, internal IOWO publication, Utrecht 1977.
Bruggen, J. C. van: Leerpsychologische vergelijkingen. Een literatuurstudie naar het nut van leerpsychologisch onderzoek voor de leerplanontwikkeling ten diente van wiskunde onderwijs, internal IOWO publication, Utrecht 1976.
The international group, presided over by Fishbein, was installed in 1977. This group is specially concerned with the psychology of mathematics education (PME). The first meeting was held in Utrecht, August 1977.
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Treffers, A. (1987). Three-Dimensional Goal Description. In: Three Dimensions. Mathematics Education Library, vol 3. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-3707-9_5
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