Abstract
The previous chapter described the permanent goals of mathematics education that are pursued by Wiskobas.
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References
Measuring
Geometry
Probability and Statistics
Relations and Functions Groot, A. D. de: Over fundamentele ervaringen: prolegomena tot een analyse van gesprekken met schakers’, Pedagogische Studien 51 (1974), p. 332. 8 Popham, W. J., Eisner, E. W., Sullivan, H. J., and Tyler, Z. L.: Instructional objectives, Chicago 1969, p. 35.
Loc. cit., p. 37.
Loc. cit., p. 33.
Loc. cit., p. 45.
Loc. cit., p. 43.
Lic. cit., p. 134.
Loc. cit., p. 8.
For the use of the terms introducet, equip with, confront with and unfold see Oudkerk Pool, T.: Van zaakvak tot wereldverkenning’, Resonans 7 (1975), 183–188.
Popham, W. J., Eisner, E. W. et al.: Instructional objectives, Chicago 1969, p. 28.
Loc. cit., p. 16.
Loc. cit., p. 17.
Loc. cit., p. 26.
Loc. cit., p. 61.
Loc. cit., p. 62.
However, the terms are often not interpreted in the same way. This can be demonstrated by the term “specific”. A large number of Dutch speaking authors (De Corte, Stroomberg, Huber, Pilot and others) use the term “specific” as a synonym for “concrete”. But sometimes the term is reserved for less concrete objectives (Westrhenen), or in reference to a different dimension and taken as the opposite of general objectives, similarly to abstractconcrete: Corte, E. de: Onderwijsdoelstellingen. Bijdrage tot de didaxologische theorievorming en aanzetten voor het empirisch onderzoek over onderwijsdoelen, Leuven 1973, p. 17. Stroomberg, H. P.: `Onderwijsdoelstellingen en doelstellingenonderzoek’, Pedagogische Studien 50(1973), p. 512.
Gronlund, N. E.: Stating behavioral objectives for classroom instruction, London 1970, p. 53. Gerlach, V. S. and Sullivan, H. J.: Constructing statements of outcomes, E.glewood 1967, p. 5. Walbesser, H. H.: An evaluation model and its application: Second Report, Washington 1968, p. 7.
Sullivan, H. J.: Objectives, evaluation and improved learner achievement’, in Popham, W. J. (ed.), Instructional Objectives, Washington 1969, p. 83.
Eisenberg, T. A.: Behaviourism: The bane of school mathematics’, International Journal of Mathematics Education in Science and Technology 6(1975), p. 164.
Klauer includes the concept of probability in the description of an objective: “Lehrziel ist die Erreichung einer bestimmten Lsungswahrscheinlichkeit bei Aufgabenklassen.” See Klauer, K. J.: Methodik der Lehrzieldefinition and Lehrstoffanalyse, Dsseldorf 1974, p.42.
It is quite a problem to discover what is the relationship between an objective and the set of tests by which it can be represented. Roughly speaking, if a formulated objective is taken as the starting point, the problem arises as to how this objective can be unfolded in sub-objectives that together cover the total objective, and how to refine these sub-objectives so that they become susceptible to measuring such that the envisaged thing is covered by what is actually measured. De Groot refers to this as the coverage problem, which in his opinion has not been solved by the common methods of measurement. He draws attention to the importance of students’ reporting and to formulations of objectives like “I have learned that”. This learning can apply to oneself as well as to the world around one. De Groot gives an important addition to common methods of measurement, which can also be applied to mathematics. Groot, A. D. de: `Over fundamentele ervaringen: Prolegomena tot een analyse van gesprekken met schakers’, Pedagogische Studien 51(1974), 329–349.
National Advisory Committee on Mathematical Education: Overview and analysis of School Mathematics Grades K-12, Washington 1975, p. 51.
McAshan, H. H.: The goals approach to performance objectives, Philadelphia 1974, p. 2.
Gronlund also makes a two-way division. He speaks of instructional objectives and behavioural objectives. Terms like knowing, understanding and applying can be used for Gronlund, N. E.: Stating behavioural objectives for classroom instruction, London 1970.
Published earlier on this subject: Treffers, A. and Wijdeveld, E. J.: Over operationele doelstellingen, Wiskobas Bulletin 2(1973), 627–636.
Statewide Mathematics Advisory Committee: Mathematics framework for California public schools. Kindergarten through grade eight, Sacramento 1972, p. 109.
Eisner, E. W.: Emerging models for educational evaluation’, School Review 80(1972), 573–590.
Oudkerk Pool, T.: `Leerdoelen, wat doe ik ermee?’, Onderwijs en Opvoeding 26(1975), 223–229. Eisner, W. W.: Epilogue, in Popham, W. J. (ed.), Instructional Objectives, Washington 1969, p. 131. Kieviet, F. K.: Open and gesloten curricula, Groningen 1976, pp. 10–11.
Loc. cit., p. 28.
Ibid.
Ibid.
Bruner, J. S Toward a theory of instruction, New York 1966, 72. Shulman, L. S.: Psychological controversies in the teaching of science and mathematics, in Crosswhite, F. J. (ed.), Teaching mathematics: Psychological foundations, Worthington 1973, p. 19.
Examples of the mathematical thinking process can be found in: Krutetskii, V. A.: The psychology of mathematical abilities in schoolchildren, Chicago 1976.
Raths, J.: Onderwijzen zonder specifieke leerdoelen als uitgangspunt’, Onderwijs en Opvoeding26 (1975), 211–218.
Stenhouse, L.: An introduction to curriculum research and development, London 1975. Arnold, W. R.: `Management by learning activities: An alternative to objectives’, The Arithmetic Teacher25 (1977), 52–56.
This interpretation of process goals is in accordance with what is often encountered in literature on the subject. However, a few other meanings are attributed to the term process goals.
Becker, H., Haller, H. D., Stubenrauch, H., and Wilkending, G.: Das Curriculum. Praxis, Wissenschaft and Politik, München 1974, p. 27.
Goffree, F.: Doorkijkspiegelingen’ (Semi-transparent mirror reflections) (Fifteen reflections on teaching learning material), Wiskobas Bulletin3 (1974), 474–495.
McAshan, H. H.: The goals approach to performance objectives, Philadelphia 1974, p. 2. K.auer, K. J.: Methodik der Lehrzieldefinition and Lehrstoffanalyse, Düsseldorf 1974, p. 42.
Many of the authors quoted agree on this point: Klauer, McAshan, De Corte and De Groot. See also Note 49.
Wood, R.: `Objectives in the teaching of mathematics’, in Ashlock, R. B. and Herman, W. L. (eds.), Current research in elementary school mathematics, New York 1970, pp. 22–45.
Taxonomies that show agreement with Wood’s are: “Educational Testing Service” (USA), “The Indian National Council of Educational Research Classification” (India), the classification of “The Schools Mathematics Study Group” and “The International Study of Achievements in Mathematics” (Husn, Sweden). Here is a short explanation of Wood’s classification:
Finch, C.: Walt Disney. From Mickey Mouse to Disneyland, Amsterdam 1975.
For this terminology see Rasche, H.: De functie van doelstellingen in een leerplan’, Pedagogische Studin 50 (1973), p. 530.
See Notes 39 and 43 and also: Philp, H.: `Mathematical education in developing countries; some problems of teaching and learning’, in Howson, A. G. (ed.), Developments in mathematical education, Cambridge 1973, pp. 154–181.
For such a strict approach to product goals see: Corte, E. de, and Janssens, A.: Praktische leidraad voor het formuleren van leerdoelen, Leuven 1974.
The inadequacy of this strict approach is most clearly evident in Klauer. See the remarks in Note 27.
See for example: Greenberg, H. J.: `The objectives in mathematics education’, The Mathematics Teacher 67 (1974), 639–644. Steiner, H. G.: `Mathematics curriculum development in the USA. A look at the past twenty years’, Zentralblatt fiir Didaktik der Mathematik 8 (1976), 136–141.
Block, A. de: Taxonomie van leerdoelen, Antwerp 1975, p. 163.
Wilson, J. W.: `Evaluation of learning in secondary school mathematics’, in Bloom, B. S., Hastings, J. T., and Madaus, G. F. (eds.), Handbook on formative and summative evaluation of student learning, New York 1971, pp. 643–697. Freudenthal, H.: `Lernzielfindung im Mathematikunterricht’, Zentralblatt für Pädagogik 20(1974), 719–739.
This source of misunderstanding is also found in the above mentioned work by Wilson: placing certain test items under certain categories is, disregarding the actual instruction given, very arbitrary. Sullivan has also referred to this. See Sullivan, H. J.: `Objectives, evaluation, and improved learner achievement’, in Popham, W. J. (ed.), Instructional objectives, Chicago 1969, p. 94.
Brink, J. van den: Autobusproblemen, internal IOWO publication. Brink, J. van den and Wijdeveld, E.: De kamping, IOWO curriculum development publication 8, Utrecht 1978.
Wijdeveld, E.: Vierkubers, IOWO curriculum development publication 5, Utrecht 1977. Goffree, F.: `Kijk op kans. Proefwerk nieuwe stijl’, Wiskobas Bulletin2 (1973), 907–919. b0 Popham, W. J., E.sner, E. W. et al.: Instructional Objectives, Chicago 1969, p. 35.
De Block on “expressive objectives”: It is clear that we are concerned here with a description of the learning process, not the learning objectives. Of course this does not exclude that sometimes we do not know what the actual result of certain intended objectives will be. Nor does this mean that the learning process (subject matter, methods and media) is of no importance. Eisner does not make sufficient distinction between the learning objectives and the learning process and thus comes to his highly disputable theses. (Block, A. de: Taxonomie van leerdoelen, Antwerp 1975, p. 157.)
De Groot does not use the term process goals, but what he says about the subjective aspect of objectives, the pursuit of general objectives and the fact that the affective domain is no separate domain, but is of a cognitive nature, fits within the terms of process and product goals. His ideas on evaluation can therefore well be applied to the evaluation of process and product goals. We have in mind especially his “student reporting” in the form of “I have learned that See: Groot, A. D”. de: Hoe stelt men eindtermen op? Universiteit en Hogeschool 20 (1974), 213–233.
Groot, A. D. de: `Over fundamentele ervaringen: prolegomena tot een analyse van gesprekken met schakers’, Pedagogische Studien 51 (1974), 329–349.
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Treffers, A. (1987). Two-Dimensional Goal Description. In: Three Dimensions. Mathematics Education Library, vol 3. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-3707-9_4
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