Abstract
An unsuspecting reader might believe that it would not be asking too much to describe the goals of a certain piece of mathematical material like “Grains on the Chessboard” or even of the whole of mathematics education as it is pursued by Wiskobas. It can be presumed that those who develop any such kind of material will proceed in an orderly fashion and have definite aims in mind: i.e., first one would determine the objectives and then construct the means to achieve them.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
For the topic of goal-means see for example: MacDonald-Ross, M.: Behavioral objectives: A critical review’, in Instructional Science 2 (1973), 1–52.Wise, R. J.: The use of objectives in curriculum planning. A critique of planning by objectives’, Curriculum Theory Network 5 (1976), 280–290.
Lists of terms are found in, among others: McAshan, H. H.: The goals approach to performance objectives, Philadelphia 1974, p. 30.
For references concerning the overview in the first sub-section of this chapter we refer to the relevant sections in the following chapters.
Corte, E. de, Geerlings, C. T., Lagerwey, N. A. J., Peters, J. J., and Vandenberghe, R.: Beknopte didaxologie, Groningen 1972, p. 35.
Tyler, R. W.: Basic principles of curriculum and instruction, Chicago 1970.
A typical quote in this connection: The Mathematics, Architecture and Science Society at Leyden with the device “Mathematics is the mother of science” awarded gold to a paper in 1797 with the motto: Knowledge of geometry is the first step toward becoming a reasonable man. Dapperen, D. van: Vormleer,Amsterdam 1825, p. 34. An extensive profile of these, developments can be found in the classic: Klein, F. and Schimmack, R.: Der Mathematische Unterricht an den Höheren Schulen I, Leipzig 1907, pp. 71 ff.
E. W. Beth held an inquiry amongst members of the mathematics working group of the “Werkgemeenschap tot Vernieuwing van Opvoeding en Onderwijs” where, among others, the points quoted were enumerated See
Beth, E. W.: Doel en zin van het meetkundeonderwijs, Euclides 14 (1939), 236–244. Opponents of this type of reflection on objectives as held by Dijksterhuis, H. J. E. Beth and Verrijp in the 1920–1940 period, include Mannoury and Van Dantzig. Arithmetic education was ascribed the same formal values. For an overview see: Turkstra, H. and Timmer, J. K.: Rekendidactiek, Groningen-Djakarta 1953.
Turkstra, H.: Psychologisch-didaktische problemen bil het onderwijs in de wiskunde aan de middelbare school, Groningen 1934, p. 19.
Cuypers, K.: Het aankweeken van het wiskundig denken, Antwerpen 1940, p. 29, K.: “Het aankweeken van het wiskundig denken, Antwerpen 1940, p. 29. ” F. (initial only): ’De wiskunde op de MMS’, Euclides 14 (1938), p. 31.
Cuypers, K.: Her aankweeken van her wiskundig denken, Antwerpen 1940, p. 193.
Turkstra, H.: Psychologisch-didaktische problemen by het onderwijs in de wiskunde aan de middelbare school, Groningen 1934, p. 34.
Reindersma, W.: Over her inleidend onderwijs in de meetkunde,Groningen — The Hague 1926, pp. 16 ff.
See the views of Ehrenfest-Afanassjewa in: Kan het wiskunde-onderwijs tot de opvoeding van her denkvermogen bijdragen?,Purmerend 1951.
These are mainly the views of Van Hiele-Geldof and Van Hiele. See Hiele, P. M. van: De problematiek van her inzicht,Amsterdam, pp. 88–102.
For the views of Kohnstamm on the learning of methods of solution, and Langeveld’s theory on knowledge domains see
Kohnstamm, Ph.: Keur uit her didactisch werk, Groningen 1952. Langeveld, M. J.: Inleiding tot de studie der paedagogische psychologie van de middelbareschoolleeftijd, Groningen 1954. x Sluis, A. van der: ’Computerkunde bij het AVO, Euclides 46 (1970), 81–92.
For the meaning of number systems see the theme “The Land of Eight” in Chapter IV. In short, the notation for the binary system is: 10101’in the decimal system is 21 In the binary system one has two digits at one’s disposal (0 and 1). In base 3 system there are three (0, 1 and 2), and so on.
This view is found in: Beth, E. W.: Doel en zin van het meetkunde onderwijs, Euclides 14 (1939).
Goffree, F. and Wijdeveld, E. J.: ‘Een praktikum wiskunde’, Euclides 44 (1966), 193–219.
Much attention to the socialising aspect has been paid by Wiskivon (mathematics in secondary education). See: Sweers, W. (ed.): Leerplanontwikkeling onderweg I, IOWO publication, Utrecht 1977.
As indicated in Chapter I (see Note 12) this refers to the 6–12 age group. Kindergarten education has been paid more attention since 1975. See articles by Jeanne de Gooijer-Quint in the Wiskobas Bulletin.
See: Proeve van een leerplan for het basisonderwijs B: Rekenen, Kohnstamminstitute, Groningen 1968, p. 7.
Dam, P. R. L. van: Sommetjes in hokjes. Einddoelstellingen van het rekenonderwijs op de basisschool, CITO publication, Arnhem 1975.
The idea is not a new one, and was popular in geometry instruction, searching for a “definitive” pseudo-deductive treatment. Names include Reindersma, Wolda, EhrenfestAfanassjewa, Van Hiele-Geldof and Van Hiele. The work of the last two especially was theoretically based and resulted in a distinction of learning levels. See Hiele, P. M. van: Begrip en inzicht, Werkboek van de wiskundedidaktiek,Purmerend 1973.
A very strict vertical planning according to the spiral idea is used by Dienes. See Dienes, Z. P.: Les six étapes du processus d’apprentissage en mathématique,Paris 1970.
The problems will arise without any doubt, since modern arithmetic and mathematical methods like ‘Elementair Wiskundig Rekenen’, Hoi, Rekenen, Getal in beeld’, and Taltaal’ Operatoir Rekenen’, `De Wereld in Getallen, Rekenwerk and Rekenen en Wiskunde’, which differ from existing methods in several ways, will probably do well in the coming years.
Kirsch, A.: `Vorschläge zur Behandlung von Wachstumsprozessen und Exponentialfunktionen im Mittelstufunterricht’, Didaktik der Mathematik 4 (1976), 257–285.
A treatment of similar ideas is found in a paper by the Schools Council Sixth Form Mathematics Project in Mathematics Applicable,London 1975.
For an extensive profile see Moor, E. de and Treffers, A.: `Het aanvankelijke meetkunde-onderwijs I, II, and III’, Euclides 50 (1974), 41–61, and 81–99.
An extensive analysis of text books of the “New Math” trend is found in Wiskobas Bulletin 7 (1978), Nr. 4.
For an overview see Pollak, H. O.: `The interaction between mathematics and other school subjects’, paper for the Third Congress on Mathematics Education, Karlsruhe 1976.
An extensive reflection on these matters is found in: Measurement in School Mathematics’, Yearbook of the National Council of Teachers of Mathematics, Reston 1976.
For the language aspect of mathematics see Brinke, J. S. ten: Moedertaalonderwijs en toch geen “Nederlands”’, Euclides 45 (1970), 327–336.
A clipping from Het Parool May 15th 1973. Articles on population increase are quite frequent.
For an interesting reflection on the grains problem and the use of calculators see: Papy, G.: Schaakbord en zakrekenmachine’, Nico, Nr. 20 (1976), 67–80. For other possibilities in working with a calculator see Blij, F. van der: `Voor minder dan twee tientjes rekenpret’, Wiskrant, Nr. 9 (1977), 10–11.
Similar historically coloured reflections on general objectives for mathematics education in West Germany and Great Britain, respectively, are found in:
Lenné, H. Analyse der Mathematikdidaktik in Deutschland,Stuttgart 1969. McNelis, S. and Dunn, J. A.: Why teach mathematics?’ International Journal of Mathematical Education in Science and Technology 8 (1977), 175–184. n For other reflections on general objectives see
Bigalke, H.: Zur “gesellschaftlichen Relevanz” der Mathematik im Schulunterricht. Aufgabe und Ziele des Mathematikunterrichts’, Zentralblatt für Didaktik der Mathematik 8 (1976), 25–34.
Braunfeld, P. and Kaufman, B.: `Mathematical education: A viewpoint’, International Journal of Mathematical Education in Science and Technology 3 (1972), 287–291.
Christiansen, B.: `Induction and deduction in the learning of mathematics and in mathematical instruction’, Educational Studies in Mathematics 2 (1969), 139–160.
Johnson, D. A. and Rising, G. R.: Guidelines for Teaching Mathematics, Belmont 1969. Kratz, J.: `Aufgaben und Möglichkeiten des heutigen Mathematikunterrichts an den allgemeinbildenden Schulen’, Zentralblatt für Didaktik der Mathematik 6 (1974), 116–120.
Lenné, H. Analyse der Mathematikdidaktik in Deutschland,Stuttgart 1969.
Servais, W.- Objectives de l’enseignement de la mathématique (mimeo), 1975.
Watson, F. R.: `Aims in mathematical education and their implications for the training of mathematics teachers’, International Journal of Mathematics Education in Science and Technology 2 (1971), 105–119.
Winter, H.: Vorstellungen zur Entwicklung von Curricula für den Mathematikunterricht in der Gesamtschule’, in Beiträge zum Lernzielproblem, Ratingen 1972.
Winter, H.: Allgemeine Lernziele für den Mathematikunterricht’, Zentralblatt für Didaktik der Mathematik 6 (1974), 106–116. Wittenberg, A. I.: Bildung und Mathematik, Stuttgart 1963.
A more extensive treatment, directed especially to the problems in developing countries, is found in: d’Ambrosio, U.: `Overall goals and objectives for mathematical education’ (mimeo), 1976; published in the Proceedings of the Third International Congress on Mathematics Education,1977.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1987 D. Reidel Publishing Company
About this chapter
Cite this chapter
Treffers, A. (1987). One-Dimensional Goal Description. In: Three Dimensions. Mathematics Education Library, vol 3. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-3707-9_3
Download citation
DOI: https://doi.org/10.1007/978-94-009-3707-9_3
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-8160-3
Online ISBN: 978-94-009-3707-9
eBook Packages: Springer Book Archive