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Freudenthal and the Van Hieles’ Level Theory

A Learning Process

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Abstract

Freudenthal supervised dozens of PhD students. Most of them wrote their theses on subjects stemming from pure mathematics. On 4 July 1957 Pierre van Hiele (1909–2010) was the first student to obtain his doctorate on a subject from the didactics of mathematics. His wife Dina (Dieke) van Hiele-Geldof (1911–1958) got her doctorate on the same day, on a subject from the didactics of mathematics as well; her project was done under the supervision of Professor of Pedagogy Langeveld. This fact on its own makes it worthwhile devoting our attention to the work of the Van Hieles in this study. A further reason is that the results of both PhD projects not only had a strong influence on (research in the field of) didactics of mathematics in general, but certainly also on Freudenthal’s work in particular.

Ik heb geboft. Alwaar ik tegen een muur aankeek ging ten langen leste een deur open. Een van eigen inzicht, maar meestal andermans (zoals Van Hieles niveaus). (I have been lucky. Everywhere where I ended up looking at a wall a door opened at long last. A door of my own insight, but most of the time one of another person’s insight (such as Van Hiele’s levels).)

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

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References

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  7. In composing this and following paragraphs information has been taken from the following studies: P.M. van Hiele, De problematiek van het inzicht (Amsterdam 1957); D. van Hiele-Geldof, De didaktiek van de meetkunde in de eerste klas van het V.H.M.O. (Amsterdam 1957); De Moor, Van vormleer naar realistische meetkunde.

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  10. Freudenthal writes: “The procedures of this group (I may fairly speak of a school of teachers) depend heavily on the use of self-made material of a different kind, cards bearing instructions, models and construction pieces, like Meccano parts but of a still more flexible structure. This movement has originated from two different sources. One is the work of P.M. van Hiele and his wife, Mrs. D. van Hiele-Geldof.” in: Hans Freudenthal, ‘Initiation into Geometry’, The Mathematics Student 24 (1956), 83–97, q.v. 87.

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  22. Van Hiele to Freudenthal, 2 December 1948; RANH, Hans Freudenthal Papers, inv.nr. 37.

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  23. Interview with Van Hiele in: Goffree, Ik was wiskundeleraar 110.

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  24. Ibid., 113.

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  25. Ibid., 110.

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  27. Van Hiele to Freudenthal, 3 February 1957; RANH, Hans Freudenthal Papers, inv.nr. 37.

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  30. For an overview of all mathematical-didactical theses of the second half of the last century and the position the Van Hieles have in this overview, see also: F. Goffree, ‘Een halve eeuw onderzoek. Wiskundedidactiek in Nederland’, Nieuw Archief voor Wiskunde 5/3 (2002) 233–243 and F. Goffree, ‘De opbrengst. Wiskundedidactiek in Nederland’, Nieuw Archief voor Wiskunde 5/3 (2002) 333–345.

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  33. From Freudenthal's speech at the funeral of Mrs. van Hiele, July 1958; RANH, Hans Freudenthal Papers, inv.nr. 521.

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  34. He spoke for example about her “marvellous reports on classroom observations” and he wrote: “She took notes while teaching; and I was astonished how many more—and more important—things she observed than I was able to do.” In 1958 Mrs. van Hiele died. Freudenthal spoke in his speech at her funeral of “a teacher by the grace of God”, in: Hans Freudenthal, Revisiting mathematics education. China lectures (Dordrecht 1991) 93; Freudenthal speeched at the funeral of Mrs. van Hiele, July 1958; RANH, Hans Freudenthal Papers, inv.nr. 521.

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  35. Hans Freudenthal, ‘Traditie en Opvoeding’, Rekenschap 3 (1957) 95–103.

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  37. Freudenthal, ‘Traditie en Opvoeding’, Rekenschap 4 (1957) 95–103, q.v. 100.

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  38. Ibid., 100.

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  39. He made an exception for the (also used by among others the Van Hieles) two-dimensional approach of the project ‘paving stones’, in which the concept ‘fitting’ was central. See for example his descriptions of courses for initial geometry education in: Hans Freudenthal, ‘Het aanvankelijk meetkunde-onderwijs’, Faraday 26 (1956) 14–18, q.v. 17–18; Hans Freudenthal, ‘Initiation into Geometry’, The Mathematics Student 24 (1956), 83–97, q.v. 91–93.

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  40. Since the themes of both theses were closely connected, Freudenthal often spoke about ‘the work of the Van Hieles' in general, without distinguishing them specifically.

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  41. Van Hiele, De problematiek van het inzicht 8.

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  42. A more elaborate description of Piaget's work and ideas and (Freudenthal's) critique will be treated in paragraph 9.6.3.

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  43. Langeveld too, regularly published critiques on Piaget. His critiques were mainly based on—according to Langeveld—Piaget's poor insight in the social world of the child; B. Levering, ‘De betekenis of M.J. Langeveld voor de naoorlogse pedagogiek (met het accent op de periode 1945–1960)’, Pedagogisch Tijdschrift 16 (1991) 147–160.

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  44. In this thesis too Freudenthal played a (small) role: in his preface Koning thanks Freudenthal for his help with the statistical part of his thesis; J. Koning, Enige problemen uit de didactiek der natuurwetenschappen in het bijzonder van de scheikunde (Dordrecht 1948) 7, 218.

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  45. M.J. Langeveld, Inleiding tot de studie der paedagogische psychologie van de middelbare schoolleeftijd (Groningen, fifth revised edition 1954, first edition 1937) 459.

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  48. From: P.M. van Hiele, ‘De niveau's in het denken, welke van belang zijn bij het onderwijs in de meetkunde in de eerste klasse van het V.H.M.O.’, Paedagogische Studiën 32 (1955) 289–297, as quoted in: P.M. van Hiele, De problematiek van het inzicht 127.

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  49. Van Hiele, De problematiek van het inzicht 91–98.

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  50. Ibid., 100.

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  51. Freudenthal, Schrijf dat op, Hans 352, 359.

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  52. This is a wish Freudenthal expressed at several occasions in lectures and articles, for example in: ‘Rekendidaktiek’ (unpublished 1944); RANH, Hans Freudenthal Papers, inv.nr. 465; ‘En nu… de wiskunde’, De Groene Amsterdammer (2 December 1950); RANH, Hans Freudenthal Papers, inv.nr. 1439; also in ‘De leraarsopleiding’, De Groene Amsterdammer (25 October 1952); RANH, Hans Freudenthal Papers, inv.nr. 1469: about “raising education and teaching to the same level other techniques in our society have” in Hans Freudenthal, ‘Traditie en Opvoeding’, Rekenschap 4 (1957) 95–103, q.v. 98–99; finally about the urgency of a scientific-pedagogic study into mathematics education in the introduction to: ‘Het wiskunde-programma voor het V.H.M.O. Een ontwerp van de Wiskunde- Werkgroep van de W.V.O.’, Euclides 28 (1953) 206–226, q.v. 208.

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  53. Hans Freudenthal, ‘Rekendidactiek’ (unpublished 1944) 101–102; RANH, Hans Freudenthal Papers, inv. Nr. 465.

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  56. Freudenthal, Schrijf dat op, Hans 352.

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  57. Hans Freudenthal, ‘De algebraïsche en analytische visie op het getalbegrip in de elementaire wiskunde’, Euclides 24 (1948) 106–121, q.v. 110, 114.

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  58. Hans Freudenthal, ‘Initiation into Geometry’, The Mathematics Student 24 (1956), 83–97; and the Dutch abridged version: Hans Freudenthal, ‘Het aanvankelijk meetkunde-onderwijs’, Faraday 26 (1956) 14–18.

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  59. Hans Freudenthal, ‘Traditie en Opvoeding’, Rekenschap 4 (1957) 95–103.

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  60. ‘Het wiskunde-programma voor het V.H.M.O. Een ontwerp van de Wiskunde- Werkgroep van de W.V.O.’, Euclides 28 (1953) 206–226, q.v. 208.

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  61. Freudenthal, ‘Traditie en Opvoeding’, Rekenschap 4 (1957) 95–103, q.v. 97–98. This phrase is also found in ‘Het aanvankelijk meetkundeonderwijs' (1956) and in ‘Initiation into geometry’ (1956).

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  62. About ‘patience’ as the most important virtue of the educator he also wrote in: Freudenthal, ‘De leraarsopleiding’, Vernieuwing van Opvoeding en Onderwijs 109 (1956) 173–180, q.v. 176; Freudenthal, ‘Het aanvankelijk meetkunde-onderwijs’, Faraday 26 (1956) 14–18, 16.

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  63. Freudenthal, ‘Traditie en Opvoeding’, Rekenschap 4 (1957) 95–103, q.v. 100–101.

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

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

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  66. Hans Freudenthal, ‘Preface of the editor’, in: Hans Freudenthal, ed., Report on methods of initiation into geometry 5–7, q.v. 6–7. In 1987 he says about this report: “The report derives its great historical meaning from the contribution of the Van Hieles—in fact composed by me from their several publications. The contribution was the first internationally accessible explanation of their level theory …” In: Freudenthal, Schrijf dat op, Hans 347–348.

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  67. In subsequent years he mentioned the theory of the Van Hieles in, among other texts, the following ones: Hans Freudenthal, ‘Preface of the editor’, in: Freudenthal, ed., Report on methods of initiation into geometry 5–7; Hans Freudenthal, ‘Report on a comparative Study of Methods of Initiation into Geometry’, Euclides 34 (1959) 289–306, Hans Freudenthal, ‘A Comparative Study of Methods of Initiation into Geometry’, l’Enseignement Mathématique 5 (1959) 119–139; Hans Freudenthal, ‘Logica als Methode en als Onderwerp’, Euclides 35 (1960) 241–255; Hans Freudenthal, ‘Logical analysis and critical survey’, in: Freudenthal, ed., Report of the relations between arithmetic and algebra (Groningen 1962) 20–41; Hans Freudenthal, ‘Enseignement des mathématiques modernes ou enseignement moderne des mathématiques?’, L’Enseignement Mathématique 9 (1963) 28–44; Hans Freudenthal, ‘Was ist Axiomatik, und welchen Bildungswert kann sie haben?’, Der Mathematikunterricht (1963) 5–29; Hans Freudenthal, ‘Logik als Gegenstand und als Methode’, Der Mathematikunterricht 13 (1967) 7–22; see also: the lecture ‘Integratie achteraf of vooraf’ from 1961 and the lecture Freudenthal gave for the MC: ‘Axiomatiek in het wiskunde-onderwijs bij het VHMO’ in 1962; RANH, Hans Freudenthal Papers, inv.nrs. 536, 537.

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  68. Freudenthal, ‘Report on a comparative Study of Methods of Initiation into Geometry’, Euclides 34 (1957) 289–306, q.v. 306.

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  69. He gave this lecture at the weekend conference of the WW: Freudenthal, ‘Logica als Methode en als Onderwerp’, Euclides 35 (1960), 241–255. This is identical to the lecture, ‘Logik als Gegenstand und als Methode’, he gave at the ICMI seminar in Aarhus in 1960, also published as article: Freudenthal, ‘Logik als Gegenstand und als Methode’, Der Mathematikunterricht 13 (1967) 7–22.

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  70. Freudenthal, ‘Logica als Methode en als Onderwerp’, Euclides 35 (1960), 241–255, q.v. 255.

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  71. Hans Freudenthal, ‘Trends in Modern Mathematics’, ICSU Review 4 (1962) 54–61. This article was also published in German: ‘Tendenzen in der modernen Mathematik’, Der Mathe­matische und Naturwissenschaftliche Unterricht 16 (1963) 301–306.

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  72. Freudenthal, ‘Trends in Modern Mathematics’, ICSU Review 4 (1962) 54–61, q.v. 55.

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  73. Hans Freudenthal, ‘Logical analysis and critical survey’, in: Freudenthal, ed., Report of the relations between arithmetic and algebra; Freudenthal, ‘Enseignement des mathématiques modernes ou enseignement moderne des mathématiques?’ 28–44.

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  74. Freudenthal, ‘Logical analysis and critical survey’ 23–24; italics mine.

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  75. Freudenthal, ‘Enseignement des mathématiques modernes ou enseignement moderne des mathématiques?’ 32.

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  76. Ibid., 39.

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  77. Ibid., 34.

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  78. Ibid., 41; italics mine.

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  79. This same passage can also be found in: Freudenthal, ‘Enseignement des mathématiques modernes ou enseignement moderne des mathématiques?’ 34; Freudenthal, ‘Logical analysis and critical survey’ 27; see also: ibid., 23, 26.

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  80. Freudenthal, ‘Logical analysis and critical survey’ 23.

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  81. Ibid., 27.

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  82. Dr. P.M. van Hiele and Dr. D. van Hiele-Geldof, ‘Een fenomenologische inleiding tot de meetkunde’, Euclides 33 (1957) 33–47, q.v. 45; NB.: the Van Hieles consistently used “matematiseren” instead of “mathematiseren”.

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  83. Freudenthal, ‘Logical analysis and critical survey’ 32.

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  84. Freudenthal, ‘Enseignement des mathématiques modernes ou enseignement moderne des mathématiques?’ 43.

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  85. She wrote: “If the mathematical language is used too early and the teacher has not started from everyday speech, the mathematical language will be learned without the corresponding mathematical understanding”, in: D. van Hiele-Geldof, De didaktiek van de meetkunde in de eerste klas van het V.H.M.O. 44.

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  86. Freudenthal, ‘Enseignement des mathématiques modernes ou enseignement moderne des mathématiques?’ 41.

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

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  88. More about this article in paragraph 8.6.3.

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  89. The background to this tendency and Freudenthal's role in these developments will be treated extensively in chapter 8.

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  90. Freudenthal, Schrijf dat op, Hans 348.

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  91. Freudenthal, Revisiting mathematics education. China lectures 96–102.

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  92. By his own account he had written ‘Logical analysis' at the end of the 1950s, ‘Enseignement' dated according to him from 1961.

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  93. Freudenthal, ‘Enseignement des mathétiques modernes ou enseignement moderne des mathématiques?’ 33.

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  94. Freudenthal, ‘Logical analysis and critical survey’ 27.

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  95. Freudenthal, Revisiting mathematics education. China lectures 98.

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  96. Ibid., 99.

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  97. Freudenthal, ‘Het aanvankelijk meetkunde-onderwijs’ 16; Freudenthal, ‘Initiation into Geometry’, 92.

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  98. Goffree, Ik was wiskundeleraar 101; John C. Moyer to Freudenthal, 16 March 1977, and answer; RANH, Hans Freudenthal Papers, inv.nr. 54.

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    Sacha la Bastide-van Gemert

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la Bastide-van Gemert, S. (2015). Freudenthal and the Van Hieles’ Level Theory. In: All Positive Action Starts with Criticism. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9334-6_7

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