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New Math and the Modernization of Mathematics Education

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Abstract

After the realization of the 1958 curriculum in the mid 1950s a period of peace and quiet seemed to have begun in mathematics education. After all, this curriculum was unique since it was supported by both the establishment in the mathematics education and by the faction of the members of the Wiskunde Werkgroep who were known as persistent reformers. A curriculum so broadly based simply had to lead to deep satisfaction and solidarity. The opposite was true: years of great turmoil in the world of mathematics education had come, fed by the wish for a drastic change of course, based on various causes, both inside and outside of the Netherlands. It was as if the whole world of mathematics education had been struck by a fever blown over from the United States: the ‘New Math’ fever, most tersely characterized by the notion ‘modernization’. In those years Freudenthal plays the role that lingers in tradition as that of ‘the lonesome opponent of New Math’.

Ich bin kein ausgeklügelt Buch, ich bin ein Mensch mit seinem Widerspruch.

(I am not a sophisticated book, I am a man with my own inconsistencies.)

Hans Freudenthal in Schrijf dat op, Hans, 1987 [1]

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References

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  24. Jean Piaget (1896–1980) was a developmental psychologist from Switzerland, whose ideas—in particular in the didactical quarter these were not completely undisputed—about the development of the thinking power of children had a strong influence on many mathematical-didactical theories, especially in the New Math quarter. In the 1970s Freudenthal regularly criticized the interpretation of Piaget's studies. 9.5.3 See also paragraph 9.5.3 and 9.6.3.

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  27. On the effect of the New Math development in, among other countries, Belgium, see also paragraph 8.5.1.

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  28. In the Netherlands OEES: Organisatie voor Europese Economische Samenwerking.

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  29. The OEEC at the end of the 1950s consisted of: Belgium, Canada, Denmark, Germany, France, Greece, Great-Britain, Ireland, Italy, Iceland, Luxemburg, the Netherlands, Norway, Austria, Portugal, Turkey, the United States, Sweden and Switzerland.

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  30. In the Netherlands OESO: Organisatie voor Economische Samenwerking en Ontwikkeling.

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  47. Ibid., 3.

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  48. Ibid.

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  50. Ibid.

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  51. Ibid. Here the title and the tenor of his later articles can already be recognized, such as: Freudenthal, ‘Enseignement des mathématiques modernes ou enseignement moderne des mathématiques?’.

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  53. Gass to Freudenthal, 28 April 1959; ibid.

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  55. Ibid.

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  56. Ibid.

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  57. Freudenthal to Kwantes, 8 May 1959; ibid.

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  58. Ibid.

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  59. Freudenthal to A.J. Piekaar, 16 July 1959; ibid.

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  60. Ibid.

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  61. Ibid.

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  65. Hans Freudenthal, ‘Algèbre linéaire et géométrie élémentaire By Jean Dieudonné’, The American Monthly 74 (1967) 745–748, q.v. 745.

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  66. Leeman et.al., ‘Verslag van het seminarium “New thinking in school mathematics” van de OEES’, Euclides 35 (1960) 218–229, q.v. 228.

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  67. Interview with Vredenduin in: Goffree, Ik was wiskundeleraar 163.

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  69. Hans Freudenthal, ‘De begrippen axioma en axiomatiek in de Wis- en Natuurkunde’, Simon Stevin 39 (1955) 156–175; q.v. 164. See also: Hans Freudenthal, ‘Axiom und Axiomatik’, Mathematisch-physikalische Semesterberichte 5 (1956) 4–19.

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  71. Ibid., 166. Also in his lecture ‘ Axiomatiek in het wiskunde-onderwijs bij het V.H.M.O.’, delivered during the summer course of the MC in 1962 Freudenthal spoke about axiomatizng as the essence of modern mathematics; Hans Freudenthal, ‘Axiomatiek in het wiskunde-onderwijs bij het V.H.M.O.’ summer course MC 1962, 4 pages; RANH, Hans Freudenthal Papers, inv.nr. 537.

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  75. Hans Freudenthal, Wiskunde in wetenschap en dagelijks leven (Hilversum 1967). The international series ‘Wereldakademie’ was the result of a cooperation of seven European and one American publishers and an international Advice Council. The purpose of the Wereldakademie was publishing a series of informative books that would allow both ‘specialists' and ‘interested laymen’ to familiarize themselves with the most modern scientific insights and were written by authors with ‘a worldwide reputation’. Freudenthal's Wiskunde in wetenschap en dagelijks leven was translated and published in English, German, Swedish, Spanish, Italian, French, Portugese, Japanese and Russian; ibid., 1.

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  76. Uitgeverij Meulenhoff to Freudenthal, 6 September 1963; RANH, Hans Freudenthal Papers, inv.nr. 42.

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  80. Freudenthal, ‘Was ist Axiomatik, und welchen Bildungswert kann sie haben?’, Der Mathematikunterricht (1963) 5–29, q.v. 13.

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  81. Freudenthal, ‘Enseignement des mathématiques modernes ou enseignement moderne des mathématiques?’ 29.

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  82. Freudenthal, ‘Logical analysis and critical survey’, in: Hans Freudenthal, ed., Report of the Relations between Arithmetic and Algebra 20–41, q.v. 23.

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  83. Freudenthal, ‘Was ist Axiomatik, und welchen Bildungswert kann sie haben?’, Der Mathematikunterricht (1963) 5–29, q.v. 16.

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  84. Freudenthal, ‘De algebraïsche en de analytische visie op het getalbegrip in de elementaire wiskunde’ 106–121.

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  85. Freudenthal, ‘Was ist Axiomatik, und welchen Bildungswert kann sie haben?’, Der Mathematikunterricht (1963) 5–29, q.v. 22. Freudenthal followed the same line of thought in his article ‘Logica als methode en als onderwerp’ from 1960, ‘Logical analysis and critical survey’ from 1962, and also in ‘Enseignement des mathématiques modernes ou enseigement modernes des mathématiques?’ from 1963; Freudenthal, ‘Logica als methode en als onderwerp’, Euclides 35 (1960) 241–255, q.v. 255; Freudenthal, ‘Logik als Gegenstand und als Methode’, Der Mathematikunterricht 13 (1967) 7–22; Freudenthal, ‘Logical analysis and critical survey’ 25–26; Freudenthal, ‘Enseignement des mathématiques modernes ou enseignement moderne des mathématiques?’ 30–31.

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  86. Freudenthal was familiar with the importance of the thought experiment in particular via the studies of G. Polya; Freudenthal, ‘Was ist Axiomatik, und welchen Bildungswert kann sie haben?’, Der Mathematikunterricht (1963) 5–29, q.v. 14–15; Freudenthal, ‘Enseignement des mathématiques modernes ou enseignement moderne des mathématiques?’ 32; Freudenthal, ‘Logica als methode en als onderwerp’ 255.

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  87. Freudenthal, ‘Enseignement des mathématiques modernes ou enseignement moderne des mathématiques?’ 34.

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  88. Freudenthal, ‘Was ist Axiomatik, und welchen Bildungswert kann sie haben?’, Der Mathematikunterricht (1963) 5–29, q.v. 13, and also 21.

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  89. Hans Freudenthal, ‘De betekenis van de wetenschappelijke basis voor de leraar’, De wetenschappelijke basis van de leraarsopleiding, mede in verband met de ontwikkeling van de exacte wetenschappen in de twintigste eeuw. Verslag van het zestiende congres van leraren in de wiskunde en de natuurwetenschappen, gehouden te Utrecht op 18 april 1966 (1966) 5–11, q.v. 9.

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  90. Ibid.

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  91. Hans Freudenthal, lecture at the international colloquium ‘Modern curricula in secondary mathematical education’, held in Utrecht, 19–23 December 1964 (unpublished) 7 pages; q.v. 1q; RANH, Hans Freudenthal Papers, inv.nr. 1831.

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  92. Freudenthal described this example in the following articles: ‘Integratie achteraf of vooraf’ (1961), ‘Enseignement des mathématiques modernes ou enseignement moderne des mathématiques?’ (1963) and ‘L’intégration après coup ou à la source’ (1969). Freudenthal used the example of the counter clerk some time before in his lecture ‘Opvoeding tot denken’ (unpublished 1945); Hans Freudenthal, ‘Integratie achteraf of vooraf’ (unpublished) 14 pages, q.v. 1, lecture at the WVO conference ‘Op zoek naar integratie’, 11 November 1961; RANH, Hans Freudenthal Papers, inv.nr. 536; also in: Freudenthal, ‘L’intégration après coup ou à la source’, Educational Studies in Mathematics 1 (1969), 327–337, q.v. 327; Freudenthal, ‘Opvoeding tot denken’ (unpublished 1945) 29 pages; RANH, Hans Freudenthal Papers, inv.nr. 466; Freudenthal, ‘Enseignement des mathématiques modernes ou enseignement moderne des mathématiques?’ 36.

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  94. Hans Freudenthal, lecture at the international colloquium ‘Modern curricula in secondary mathematical education’, held in Utrecht, 19–23 December 1964 (unpublished) 7 pages; q.v. 4; RANH, Hans Freudenthal Papers, inv.nr. 1831.

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  95. See also paragraph 9.5.3.

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  96. More about the use of set theory in education in the paragraphs 9.5.3 and 9.6.1.

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  98. Ibid.

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  99. Hans Freudenthal, ‘Het staartje van de mammoet’, De Groene Amsterdammer (28 juli 1962); RANH, Hans Freudenthal Papers, inv.nr. 1570. 1570.

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  100. J.C.H. Gerretsen, ‘Doelstelling van het wiskundeonderwijs’, Euclides 34 (1958) 90–94, q.v. 93–94.

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  102. ‘Uit de openingstoespraak van de voorzitter van WIMECOS tot de algemene vergadering van 28 december 1961’, Euclides 37 (1962) 264–267, q.v. 266.

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  104. There were, apart from the Royaumont Seminar in 1959, for example; another international colloquium in Aarhus in 1960 followed by a meeting of experts in Yugoslavia and annual Belgium congresses for teacher in Arlon and several courses for teachers in Brussels and Amsterdam; F. Loonstra en P.G.J. Vredenduin, ed., Modernization of mathematical teaching in the Netherlands (Groningen 1962) 5.

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  105. Wansink, Didactische oriëntatie voor wiskundeleraren. Deel II 405, Deel III 20.

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  106. Freudenthal, Schrijf dat op, Hans 137. In a letter to Howson Freudenthal wrote that he liked neither Gattegno nor Papy and that his active involvement in the CIEAEM only started when the Polish Anna Zofia Krygowska (1904–1988) became chairwoman; Freudenthal to Howson, 19 July 1983; RANH, Hans Freudenthal Papers, inv.nr. 38.

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  107. But Beth's cooperation with Piaget would be the result of the activities of this group.

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  108. R. Holvoet, ‘Over de opbouw van de methode Papy’, in: Wansink, Didactische oriëntatie voor wiskundeleraren. Deel II 405–449, q.v. 445.

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  109. G. Noël, ‘Pourquoi, pour qui enseigner les mathématiques? Une mise en perspective historique de l’évolution des programmes, au xxe sciècle, en Belgique’, Zentralblatt für Didaktik der Mathematik 34 (2002) 110–119.

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  110. G. Krooshof, ‘Moderniseren—nieuwbouw of verbouw?’, Euclides 42 (1966) 193–203, q.v. 194.

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  111. P.G.J. Vredenduin, ‘Het experiment Papy’, Euclides 42 (1966) 167–182, q.v. 171.

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  112. From 1968 onwards the Centre also had its own periodical: NICO: Revue Périodique du Centre Belge de Pédagogie de la Mathématique.

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  113. Interview with Vredenduin in: Goffree, Ik was wiskundeleraar 163.

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  114. P.G.J. Vredenduin, ‘Een opzienbarend boek’, Euclides 39 (1963/64) 237–247; Euclides 42 (1966/67) 90–94, 161–166. Papy's Mathématique moderne would eventually consist of six volumes.

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  115. F. Loonstra en P.G.J. Vredenduin, eds., Modernization of mathematical teaching in the Netherlands (Groningen 1962) 5.

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  116. Freudenthal was co-author of the volumes II ‘Geometrie’, III ‘Analysis' en IV ‘Praktische Methoden und Anwendungen der Mathematik’; H. Behnke, ed., Grundzüge der Mathematik für Lehrer an Gymnasien sowie für Mathematiker in Industrie und Wirtschaft I-V (Göttingen 1958–1966); RANH, Hans Freudenthal Papers, inv.nr. 1747.

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  117. Announcement in Euclides; Euclides 35 (1959) 112. At this conference Freudenthal lectured on ‘Logica als onderwerp en als methode’, Euclides 35 (1959) 241–255.

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  118. Morsch, Met de moed van de hoop 222–258.

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  119. S.J.C. Freudenthal-Lutter, De Jenaplanschool. Een leef- en werkgemeenschap (Utrecht 1970), 5. Langeveld wrote the preface.

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  120. M. van Essen en J.D. Imelman, Historische pedagogiek. Verlichting, romantiek en ontwikkelingen in Nederland na 1800 (Baarn 1999) 112.

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  121. In the Netherlands in 1975 there were about 120 schools organized according to Jenaplan Education. The most striking characteristic of these schools was the lack of the usual division in classes. Instead there were the ‘multi-age groups' (the so-called vertical division) which made ‘repeating a class' non-existent. Furthermore developing community spirit, ‘educating to critical thinking’ and paying attention to the child as an individual were considered of paramount importance.

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  122. In Assen at present is the national Jenaplan library, the ‘Suus Freudenthal-Lutterbibliotheek’, mainly a collection Mrs. Freudenthal built up via her work as secretary of Pedomorfose.

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  123. See also paragraph 9.5.3.

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  124. Oral tradition evidences that Freudenthal was initially asked as an advisor for this series, but he would have passed this on to De Iongh; De Moor, Van vormleer naar realistische meetkunde 503.

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  125. An invitation to Berkeley came to nothing, since there was no money available. But preceding his stay in New Haven he held a ‘summer course’ there; RANH, Hans Freudenthal Papers, inv.nr. 392.

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  126. Freudenthal, Lincos. Design of a language for cosmic intercourse. Part 1 (Amsterdam 1960).

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  127. ‘H. Freudenthal’, Vrij Nederland (1976) 232–247, 244. Freudenthal says about this: “We can explain nature, the universe with the help of mathematics. So when there is another nation which has also developed science, than they also have to know mathematics, it cannot be otherwise.”

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  128. Vredenduin reviewed Lincos in Euclides 36 (1961) 252–254. During his year in the United States Freudenthal lectured among other things about Lincos at Brown University (Rhode Island) and Brandeis University (Massachusetts).

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  129. Documents from the ‘Yale’- period; RANH, Hans Freudenthal Papers, inv.nr. 316, 392, 393, 1884.

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  130. Documents related to Freudenthal's rectorship; RANH, Hans Freudenthal Papers, inv.nr. 394.

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  131. ‘Commissie modernisering leerplan wiskunde’, Euclides 37 (1961) 144–151, q.v. 144.

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  132. Dodde, Het Nederlandse onderwijs verandert 139, 167 et seq.

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  133. Grotenhuis, Op zoek naar middelbaar onderwijs 157 et seq.

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  134. Mathematician A.F. Monna worked from 1961 at the Utrecht University. In 1965 he became professor of pure mathematics. From 1963 to 1974 he was also a member of the NOCW; Kleijne, ‘NOCW vijftig jaar’, Nieuw Archief voor Wiskunde 5 (2004) 308–313, q.v. 313; Adriaan Monna and Marius van der Put, ‘Ambtenaar en wiskundige’, Nieuw Archief voor Wiskunde 5 (2004) 136–146, q.v. 141, 144.

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  145. This application was eventually turned down in 1968 referring to the imminent plans for a nationwide organization for curriculum development, in which the CMLW would have to be absorbed. See also paragraph 9.3.1: E. Wijdeveld, H. Verhage, G. Schoemaker, ‘Van CMLW tot Freudenthal Instituut’, in: Goffree et al., eds., Honderd jaar wiskundeonderwijs 355–374, q.v. 359 et seq.

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  148. Minister Rutten had already proposed a reorganization of the secondary education in 1951, the so-called ‘Plan-Rutten’; Grotenhuis, Op zoek naar middelbaar onderwijs 155.

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  153. Commissie Modernisering Leerplan Wiskunde, Toelichting op het leerplan wiskunde (1968).

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  158. G. Krooshof, ‘Moderniseren—nieuwbouw of verbouw?’, Euclides 42 (1966) 193–203, q.v. 196.

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  159. Namely: the Katholiek Pedagogisch Bureau, the Christelijk Pedagogisch Studiecentrum and the Onderwijskundig Studiecentrum; Brandenburg, ‘De modernisering van het wiskundeonderwijs’, in: Wansink, Didactische oriëntatie voor wiskundeleraren. Deel III 15–49, q.v. 27.

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  161. Ibid.

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  164. Freudenthal, Schrijf dat op, Hans 351.

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  165. For the history of the initial geometry education, the controversy around it and movements that played a role see: De Moor, Van vormleer naar realistische meetkunde.

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  166. ‘Interimrapport van de commissie modernisering leerplan wiskunde’, Euclides 42 (1966) 129–134, q.v. 130.

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  168. M. Bijpost, J.K. Timmer, Met passer en liniaal 1. Eenvoudig leerboek der vlakke meetkunde (Zutphen 1957) 52, 76.

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  169. Other methods such as Inleiding in de meetkunde by Brandenburg and Schrier from 1961 also followed this approach; W.J. Brandenburg, L. Schrier, Inleiding in de meetkunde 1 (Groningen 1961) 100.

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  170. H.J. Jacobs e.a., Moderne wiskunde voor voortgezet onderwijs. Deel 1 voor de brugklas (Groningen 1972) 126–130.

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  171. For that matter, this approach to the theorem appeared in the more progressive text books even before 1968, see for example: D.N. van der Neut, A. Holwerda, Meetkunde. Met de beginselen der goniometrie. Eerste deel (Groningen 1959) 28; P. Wijdenes, Planimetrie. Eenvoudig schoolboek voor het eerste onderwijs in de vlakke meetkunde (Groningen 1931), second edition 1933, 22–23.

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  172. Commissie Modernisering Leerplan Wiskunde, Toelichting op het leerplan wiskunde (1968) 19.

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  173. This lecture has been published both in English and in Dutch (in a somewhat adapted version); Freudenthal, ‘Initiation into Geometry’, The Mathematics Student 24 (1956) 83–97; Freudenthal, ‘Het aanvankelijk meetkunde-onderwijs’, Faraday 26 (1956) 14–18, q.v.14–15.

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  174. Hans Freudenthal, ‘Het aanvankelijk meetkunde-onderwijs’, Faraday 26 (1956) 14–18, q.v. 14–15.

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  175. The most striking example is perhaps his contribution to the discussion with Ehrenfest; Ehrenfest-Afanassjewa and Freudenthal, Kan het wiskunde-onderwijs tot de opvoeding van het denkvermogen bijdragen?

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  176. A similar passage Freudenthal also wrote in ‘Traditie en opvoeding’ and in ‘Tradition and education’; Freudenthal, ‘Het aanvankelijk meetkunde-onderwijs’, Faraday 26 (1956) 14–18, q.v. 15; Freudenthal, ‘Traditie en opvoeding’, Rekenschap (3) (1957) 95–103; Freudenthal, ‘Tradition and education’, The New Era in Home and School 37 (1956) 127–132.

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  177. Hans Freudenthal, ‘Het aanvankelijk meetkunde-onderwijs’, Faraday 26 (1956) 14–18, q.v. 15.

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  178. E. Wijdeveld, ‘Omzien in verwondering. Ontstaan, werkwijze en effecten van de Commissie Modernisering Leerplan Wiskunde en het Instituut Ontwikkeling Wiskunde Onderwijs (1960–1970)’, Euclides 78 (2003) 218–225, q.v. 223. See also the printed version of the HKRWO lecture with the same title, 25 May 2002.

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  179. Freudenthal, ‘Preface of the editor,’ in: Hans Freudenthal, ed., Report on methods of initiation into geometry 5.

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  180. Freudenthal, ‘Het aanvankelijk meetkunde-onderwijs’, Faraday 26 (1956) 14–18, q.v. 15.

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  181. In ‘Initiation into Geometry’ he also mentioned the studies of Bos and Lepoeter, and the study of Van Albada; Freudenthal, ‘Initiation into Geometry’, The Mathematics Student 24 (1956) 83–97; E. Castelnuovo, Geometria intuitiva (Firenze 1948, second revised version 1952); Freudenthal, ‘Het aanvankelijk meetkunde-onderwijs’, Faraday 26 (1956) 14–18, q.v. 15–18.

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  182. Freudenthal, ‘Initiation into geometry’, The Mathematics Student 24 (1956) 83–97, q.v. 91–92.

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  183. Ibid., 87–93. A good example of working with the concept of ‘tiling’ in the initial geometry education could be found in the studies of the Van Hieles, who elaborated on the original concept of Van Albada.

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  184. Freudenthal, ‘Het aanvankelijk meetkunde-onderwijs’, Faraday 26 (1956) 14–18, q.v. 16.

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  185. Ibid., 16.

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  186. Hans Freudenthal, ‘Axiomatiek in het wiskunde-onderwijs bij het vhmo’, lecture during the summer course of the MC, 1962 (unpublished) 4 pages, q.v. 2; RANH, Hans Freudenthal Papers, inv.nr. 537.

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  187. Freudenthal, ‘Was ist Axiomatik, und welchen Bildungswert kann sie haben?’, Der Mathe-matikunterricht (1963) 5–29, q.v. 26.

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  188. Ibid.

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  189. Hans Freudenthal, ‘Axiomatiek in het wiskunde-onderwijs bij het VHMO’, lecture during the summer course of the MC, 1962 (unpublished) 4 pages, q.v. 3; RANH, Hans Freudenthal Papers, inv.nr. 537.

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  190. Ibid.

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  191. Ibid., 4.

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  192. Hans Freudenthal, ‘Bemerkungen zur axiomatischen Methoden im Unterricht’, Der Mathematikunterricht (Stuttgart 1966) 61–65. This is a summary of the discussion after Freudenthal's lecture ‘Was ist axiomatik…’ at the ICMI Seminar in Aarhus in 1960.

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  193. With this point of view he gained the support of Streefkerk who referred to Freudenthal's statements about axiomatics in education in 1964; H. Streefkerk, ‘Problemen rondom de vernieuwing van het wiskunde-onderwijs’, in: Stemmen uit de praktijk. Vernieuwing van het wiskundeonderwijs. Inleidingen en verslagen van de conferentie wiskunde, georganiseerd door de afdeling VHMO van het Christelijk Paedagogisch Studiecentrum (werkgroep wiskunde) op 6 en 7 maart 1964 (Den Haag 1964) 7–19, q.v. 18.

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  194. Freudenthal, ‘Logical analysis and critical survey’ 32. In somewhat other wording he also mentioned this subject in his lecture at the international colloquium ‘Modern curricula in secondary mathematical education’, held from 19 to 23 December 1964 in Utrecht.

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  195. Freudenthal, ‘Was ist Axiomatik, und welchen Bildungswert kann sie haben?’, Der Mathematikunterricht (1963) 5–29, q.v. 28.

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  196. Freudenthal, ‘Report on a comparative study of methods of initiation into geometry’, Euclides 34 (1957) 289–306, q.v. 299.

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  197. Ibid. Good examples how to realize this could be found in the studies of Van Albada and those of the Van Hieles according to Freudenthal.

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  198. Correspondence between Erven F. Bohn and Freudenthal, 1955; RANH, Hans Freudenthal Papers, inv.nr. 29; Hans Freudenthal, Exacte logica (Haarlem 1961).

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  199. Freudenthal, Exacte logica V.

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  200. Freudenthal, ‘Logica als methode en als onderwerp’, Euclides 35 (1960) 241–255, q.v. 255; Freudenthal, ‘Logik als Gegenstand und als Methode’, Der Mathematikunterricht 13 (1967) 7–22; Freudenthal, ‘Logical analysis and critical survey’ 20–41, q.v.25–26; Freudenthal, ‘Enseignement des mathématiques modernes ou enseignement moderne des mathématiques?’ 28–44, q.v. 30–31.

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  201. Freudenthal, Exacte logica V.

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  202. Ibid., VI.

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  203. Ibid., 56–57.

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  204. Lecture at the weekend conference of the Wiskunde Werkgroep of the WVO, November 1959, in 1961 published in Euclides; Freudenthal, ‘Logica als methode en als onderwerp’, Euclides 35 (1960) 241–255.

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  205. Ibid., 244.

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  206. Ibid., 249.

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  207. ∧ Ù and ∨ are today often written as ∀ (‘for all’) en ∃ (‘there exists’), respectively.

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  208. Freudenthal, ‘Logica als methode en als onderwerp’ 251.

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  209. Ibid., 252.

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  210. Ibid., 252–253. Freudenthal took his example from the collection ‘250 opgaven’ (‘250 exercises’), which was published in Euclides 32 (1956) as support of the 1958 curriculum.

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  211. Hans Freudenthal, ‘Functies en functie-notaties’, Euclides 41 (1966) 299–304, q.v. 304.

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  212. Hans Freudenthal, ‘Logica als methode en als onderwerp’ 255.

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  213. Ibid., 255 Freudenthal followed the same line of thought in his article ‘Logical analysis and critical survey’ from 1962 and also in ‘Enseignement des mathématiques modernes ou enseigement modernes des mathématiques?’ from 1963; Freudenthal, ‘Logical analysis and critical survey’ 25–26; Freudenthal, ‘Enseignement des mathématiques modernes ou enseignement moderne des mathématiques?’ 30–31.

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  214. Freudenthal in the interview: Bert Vuysje, ‘De wiskundeles verandert: 3 + 2 = 1’, HP Magazine (24 December 1969–6 January 1970); RANH, Hans Freudenthal Papers, inv.nr. 1847.

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  215. Freudenthal, Schrijf dat op, Hans 351.

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  216. Freudenthal's publications of the period 1956–1966 in Euclides: Freudenthal, ‘Report on a comparative study of methods of initiation into geometry’, Euclides 34 (1957) 289–306; Freudenthal, ‘Logica als methode en als onderwerp’, Euclides 35 (1960) 241–255; Freudenthal, ‘Functies en functie-notaties’, Euclides 41 (1966) 299–304; Freudenthal, ‘Modernisering Leerplan Wiskunde. Antwoord aan Prof. De Bruyn’, Euclides 43 (1968) 321–322.

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  217. Freudenthal to R.C. Kwantes, 25 March 1959; RANH, Hans Freudenthal Papers, inv.nr. 1829.

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  218. Freudenthal to Albert Legrand, Institut de l’Unesco pour l’Education, 9 februari 1968; RANH, Hans Freudenthal Papers, inv.nr. 365.

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  219. Ibid.

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  220. See for example—besides similar thoughts in his unpublished article ‘Oude en nieuwe universiteiten’—the article: Freudenthal, ‘De algebraïsche en de analytische visie op het getalbegrip in de elementaire wiskunde’ 106–121.

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  221. Willem Jan Brandenburg (1921) studied mathematics and physics in Amsterdam and after that became a mathematics teacher. After studying pedagogy in Groningen (which he finalized in 1958) he subsequently defended his mathematical-didactical thesis, in which he described the problems arising after the modernization of the school mathematics of 1967, with Prof. J.C.H. Gerretsen and Prof. L. van Gelder in 1968. From 1963 onwards he worked as a senior lecturer at the department Lerarenopleiding (Teacher education) of the Pedagogisch Instituut in Groningen; Brandenburg, Modernisering van het wiskunde-onderwijs.

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  222. Adriaan D. de Groot (1914) studied mathematics and psychology. In 1946 he defended his thesis entitled Het denken van de schaker. Een experimenteel-psychologische studie. (The thinking of the chess player, and experimental-psychological study) He had been a mathematics teacher for some years, when he from 1950 onwards was employed as professor at the University of Amsterdam, where he specialized in psychological-educational research. One of his most cited publications was Vijven en zessen. Cijfers en beslissingen: het selectieproces in ons onderwijs. At the end of the 1950s a comparative study into methods for initial geometry education was performed under his supervision (Euclidean geometry versus transformation geometry); A.D. de Groot, Vijven en zessen. Cijfers en beslissingen: het selectieproces in ons onderwijs (Groningen 1966); F. Goffree, ‘Een halve eeuw onderzoek. Wiskundedidactiek in Nederland’, Nieuw Archief voor Wiskunde 5/3 (2002) 233–243.

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  223. See also: J. Vos en K. van der Linden, Waarvan akte. Geschiedenis van de M.O.-opleidingen 1912–1987 (Assen 2004); Klaske Blom, ‘Van de acten van bekwaamheid. De middelbaar-onderwijs akten KI en Kv nader bekeken’ (unpublished 1998).

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  224. Wansink, Didactische oriëntatie voor wiskundeleraren. Deel I 26–28. In 1952 for the academically educated teachers, 1958 for the certificate teachers and in 1961 for the engineer-teacher. See also paragraph 6.5.

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  225. Vos and Van der Linden, Waarvan akte. Geschiedenis van de M.O.-opleidingen 1912–1987 116.

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  226. Freudenthal also lectured on ‘Geschiedenis van de wiskunde’; ‘Berichten Mathematisch Instituut der Rijksuniversiteit Boothstraat 1C, 6 en 17, Utrecht’, Euclides 41 (1965) 52.

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  227. See also Freudenthal on the education of teachers in paragraph 6.5.

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  228. Freudenthal, Exacte logica v.

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la Bastide-van Gemert, S. (2015). Method Versus Content. In: All Positive Action Starts with Criticism. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9334-6_8

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