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Notes and references
The use of an inductive approach, by which the student may be motivated to experiment with mathematical situations on his own, has been recommended in later years at several meetings or conferences, e.g.: The international working session on New Teaching Methods for School Mathematics, Athens 1963. Mathematics to-day, OECD, Paris, 1964, pp. 290–308 (summary and resolutions), esp. pp. 294–96, 298, 300, 306–7. The Stanford Conference (December 1964) on The Learning of Mathematics by Young Children. Mathematics in primary education (Unesco Institute for Education, Hamburg, 1966, Report compiled by Z. P. Dienes), pp. 71–72. The Cambridge Conference on School Mathematics (1963). Goals for school mathematics, Houghton Mifflin Co., New York, 1964, pp. 13–23 (‘Pedagogical principles and techniques’), esp. pp. 13–16, 17–18. The Nordic Committee for the Modernization of School Mathematics (1960–67). New school mathematics in the Nordic countries (Translation in parts of the report and the recommendations of the committee, Nordisk udredningsserie 1967: 11, Stockholm 1967), pp. 34–37, 48–51.
The importance of giving better conditions for developing mathematical learning ability is recognized in nearly all experimental projects and by most international conferences. For the concretization of the vague ideas within this field the contributions in the Bulletin of the International Study Group for Mathematics Learning has been most valuable. From January 1968 the title of this periodical was changed to Journal of Structural Learning, Gordon and Breach, New York and London. In the issue 1 (1969) 2/3 the editorial indicates some of the activities with regard to the further study and research in the field of the learning of mathematics. In the new periodical Educational Studies in Mathematics the problems of the learning of mathematics has been an underlying theme in many articles. Attention is especially drawn to the following contribution in 1, No. 3, Jan. 1969: Burt A. Kaufman and Hans-G. Steiner, ‘The CSMP Approach to a Content-Oriented, Highly Individualized Mathematics Education’.
Below are mentioned contributions concerning the inductive approach in general. In many cases the titles, however, refer to “discovery”. Hence, I should like to emphasize that I have in my address advocated the use of the inductive approach to be of high importance in connection with mathematical instruction, while I have given no recommendation with regard to the use of the “discovery method” as a means in its own right. Van Caille, D. J., ‘Learning mathematics through discovery in the primary school’, Mathematics Teaching, 1969, no. 46. Courant, R., ‘Mathematics in the modern world’, Scientific American, Sept. 1964. Dewey, John, How we think, Heath and Co., Boston, 1919. Dienes, Z. P., ‘A new approach to the teaching of mathematical structures’, Bulletin of the International Study Group for Mathematics Learning, Summer-Fall, 1966. Dienes, Z. P., Building up mathematics (pp. 31–48). Hutchinson, London, 3rd impression, 2 ed., 1963. Dienes, Z. P. and Jeeves, M. A., Thinking in structures, Hutchinson, London, 1965. Hendrix, Gertrude, ‘Learning by discovery’, The Mathematics Teacher 54 (1961), 290–299. Hull, William P., ‘Learning strategy and the skills of thought’, Mathematics Teaching, 1967, no. 39. Nuffield Mathematics Project: I do, and I understand, Chambers and Murray, 1967. Polya, G., How to solve it. A new aspect of mathematical methods, Doubleday, New York, 2nd ed., 1957. Polya, G., Mathematical discovery. On understanding, learning and teaching problem solving, I–II, Wiley, New York, 1962–65. Polya, G., Mathematics and plausible reasoning, I–II, Princeton University Press, Princeton, N. J., 1954. Scandura, Joseph M., ‘Research in psychomathematics’, The Mathematics Teacher 61 (1968), 581–591. Steiner, Hans-G., ‘Einfache Verknüpfungsgebilde als Vorfeld der Gruppentheorie’, Der Mathematikunterricht 12 (1966), Heft 2. Steiner, Hans-G., ‘Examples of exercises in mathematization on the secondary school level’, Educational Studies in Mathematics 1 (1968), 181–201. Walter, Marion, ‘Mirror cards. An example of informal geometry’, The Arithmetic Teacher 13 (1966), 448–452. Zassenhaus, Hans, ‘Experimentelle Mathematik in Forschung und Unterricht’. Mathematisch-Physikalische Semesterberichte 13, Heft 2.
In the discussions of the rôle of the first elements of logic in the teaching of elementary mathematics Patrick Suppes has made contributions of special interest. While the dealing with logic as such in the elementary curriculum in most cases is limited to work with truth-tables (and maybe a few rules of deduction) together with a rather careful treatment of solution of open statements, Patrick Suppes advocates that the able elementary school child should receive deep experiences with regard to elementary mathematical logic, in particular that portion of it which is concerned with the theory of logical inference. His views are expressed in the following two publications: Suppes, Patrick and Binford, Frederick, Experimental Teaching of Mathematical Logic in the Elementary School. (Cooperative Research Project No. D-005, Stanford University, Calif., 1964). Suppes, Patrick, Introduction to logic. Van Nostrand, Princeton, N. J., 1960.
By subsequent education under item (b) is meant education in mathematics as well as in other subjects, during schooldays as well as later in life. I have elaborated my views on the objectives of mathematics teaching in an article included in the report from the Nordic Committee (see [1]).
The epistemological aspects in the teaching of mathematics are considered explicitly only in few cases in the standing discussion of the didactics of mathematics. Implicitly these aspects are of significance in many contributions concerning on the one hand the general education through mathematics, and on the other hand the rôle of the axiomatic method in secondary school curriculum. The following items should be noted in this setting. Freudenthal, Hans, ‘Was ist Axiomatik, und welchen Bildungswert kann sie haben?’, Der Mathematikunterricht 9 (1963), Heft 4. Hasse, Helmut, ‘Mathematik als Geistenwissenschaft und als Denkmittel der exakten Naturwissenschaften’, Der Mathematikunterricht 9 (1963), Heft 2. Krygowska, Anna Zofie, ‘Teaching geometry in to-day's “unified mathematics”’. Mathematics Teaching, 1965, no. 30, 53–57; no. 31, 15–22. Steiner, Hans-G., ‘Wie steht es mit der Modernisierung unseres Mathematikunterrichts?’, Mathematisch-Physikalische Semesterberichte 11, Heft 2. Wittenberg, Alexander Israel, Bildung und Mathematik, Klett, Stutgart 1963. (See also the extensive review of this book by H. Coers in Mathematisch-Physikalische Semesterberichte 11, Heft 2.)
Mathematics in primary education (Unesco Institute for Education, Hamburg, 1966, Report compiled by Z. P. Dienes): In chapter 2 of this report a review is given of important theoretical considerations by a number of psychologists with regard to concept formation and learning of mathematics. Dienes, Z. P., The power of mathematics, Hutchinson, London, 1964, pp. 17–42 and pp. 104–132. Skemp, Richard R., ‘A three-part theory for learning mathematics’, in New approaches to mathematics teaching, Macmillan, London, 1966, pp. 40–47.
The results from the experimental teaching are now used as background for a series of textbooks starting from the middle of the 6th grade. These books are only available in the Danish language: Christiansen, Bent, Christiansen, Allan, and Lichtenberg, Jonas, Matematik 7, 1–2, Munksgaard, Copenhagen, 1967–68 (pp. 288–231; Teachers guide, p. 196). Christiansen, Bent and Pedersen, Johs., Matematik 8 G1, Munksgaard, Copenhagen, 1969 (p. 123; Teachers guide, p. 78).
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Christiansen, B. Induction and deduction in the learning of mathematics and in mathematical instruction. Educ Stud Math 2, 139–159 (1969). https://doi.org/10.1007/BF00303454
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DOI: https://doi.org/10.1007/BF00303454