Pierre van Hiele, David Tall and Hans Freudenthal: Getting the facts right

TL;DR

This paper clarifies the historical and conceptual misunderstandings surrounding Van Hiele's theory, emphasizing its broader application beyond geometry and correcting misattributions by Freudenthal and Tall.

Contribution

It provides a detailed correction of the misinterpretations of Van Hiele's work by Tall and Freudenthal, highlighting the wider scope of Van Hiele's theory in mathematics education.

Findings

Van Hiele's theory has broader applications beyond geometry.

Tall misread Van Hiele's original work.

Freudenthal misrepresented Van Hiele's ideas and adopted them for his own.

Abstract

Pierre van Hiele (1909-2010) suggested, both in 1957 and later repeatedly, wide application for the Van Hiele levels in insight, both for more disciplines and for different subjects in mathematics. David Tall (2013) suggests that Van Hiele only saw application to geometry. Tall claims that only he himself now extends to wider application. Getting the facts right, it can be observed that Tall misread Van Hiele (2002). It remains important that Tall supports the wide application of Van Hiele's theory. Tall apparently didn't know that Freudenthal claimed it too. There appears to exist a general lack of understanding of the Van Hiele - Freudenthal combination since 1957. Hans Freudenthal (1905-1990) also suggested that Van Hiele only saw application to geometry, and that only he, Freudenthal, saw the general application. Freudenthal adopted various notions from Van Hiele, misrepresented those, gave those new names of himself, and started referring to this instead of to Van Hiele. The misrepresentation may clarify why Tall didn't recognise Van Hiele's theory. Freudenthal mistook Van Hiele's distinction of concrete versus abstract for the distinction of reality versus model (applied mathematics). Freudenthal's misconception of "realistic mathematics education" (RME) partly doesn't work and the part that works was mostly taken from Van Hiele. This common lack of understanding of Van Hiele partly explains the situation in the education in mathematics and the research on this. Another factor is that mathematicians like Freudenthal and Tall are trained for abstraction and have less understanding of the empirics of mathematics education.