ABSTRACT
Due to the national and international dominance of the figure of Hans Freudenthal, other Dutch researchers in the field of mathematics education became somewhat overshadowed. In this contribution we describe the life and work of Pierre van Hiele. His major role, both nationally and internationally, remained underexposed to date in the literature on the history of mathematics education.
Pierre van Hiele (1909–2010) was not only a secondary school mathematics teacher and textbook author, but first and foremost an outstanding mathematics didactician. Based on his PhD investigations – and these of his wife, Dieke van Hiele-Geldof (1911–1958) – he developed the level theory of mathematical thinking. We pay attention to these research activities, with an emphasis on the level theory. This theory consists of five levels:
Level 0 At this ‘ground’ level, the student gets an idea, as concrete as possible, of the field to be learned.
Level 1 At this level, often called the ‘visual’ level, student’s thinking focusses on the properties of an object, and if relevant, of comparable objects.
Level 2 At this level, often called the ‘descriptive’ level, the structure developed at Level 1 is the object of student’s thinking. The student organizes this structure in a logical sense that he uses to make chains of arguments.
Level 3 At this level, often called the ‘theoretical’ level, the student abstracts from Level 2 reasonings and focusses on the logical structure of these reasonings.
Level 4 is often called the ‘theoretical-deductive’ level. Thinking at this level focusses on special aspects of reasoning, for example, necessary and sufficient conditions, the conversion of a proposition, prove by contradiction.
In this presentation, we focus on the genesis of Van Hiele’s theory in the Dutch “milieu” for mathematics education before and just after WWII with Tatjana Afanassjewa (1876–1964) as its central figure, who inspired both Freudenthal’s and Van Hiele’s thinking about geometry education.
SELECTED BIBLIOGRAPHY
Ehrenfest-Afanassjewa, T. (1931). Uebungensammlung zu einer geometrischen Propaedeuse. Den Haag: Martinus Nijhoff.
La Bastide-van Gemert, S. (2015). All positive action starts with criticism. Hans Freudenthal and the didactics of mathematics. New York, NY: Springer.
Smid, H. J. (2016). Formative years: Hans Freudenthal in prewar Amsterdam. History and Pedagogy of Mathematics. Montpellier, France.
Van Hiele, P. M. (1957). De problematiek van het inzicht, gedemonstreerd aan het inzicht van schoolkinderen in de meetkunde-leerstof. Amsterdam: J.M. Meulenhoff; Purmerend: J. Muusses; Groningen: N.V. Erven P. Noordhoff; Den Haag: N.V. Uitgeverij Nijgh & Van Ditmar; Leiden: Spruyt, Van Mantgem, & De Does N.V. (Dissertation)
Van Hiele-Geldof, D. (1957). De didaktiek van de meetkunde in de eerste klas van het V.H.M.O. Amsterdam: J.M. Meulenhoff; Purmerend: J. Muusses; Groningen: N.V. Erven P. Noordhoff; Den Haag: N.V. Uitgeverij Nijgh & Van Ditmar; Leiden: Spruyt, Van Mantgem, & De Does N.V. (Dissertation)