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Science and Mathematics Education

Some Reflections on the Van Hiele theory

Profile image of Michael de VilliersMichael de Villiers

Abstract

This paper gives a review of research on the Van Hiele Theory over the past 30 years, and highlights some important issues regarding theoretical implications for designing learning activities in dynamic geometry contexts, as well as issues of further research such as hierarchical class inclusion. Links to a recorded video of this talk is also given.

FAQs

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What are the implications of the Van Hiele theory for geometry education?add

The Van Hiele theory suggests that students must progress through distinct thought levels sequentially, which implies a need for specific instructional strategies tailored to each level. For instance, only 10-15% of pupils were at Level 2 by Grade 5 in Russian studies, indicating lower geometry proficiency due to earlier curriculum gaps.

How does the Russian geometry curriculum relate to the Van Hiele theory?add

The traditional Russian geometry curriculum's two-phase approach aligns with the Van Hiele levels, beginning with intuitive learning and progressing to formal deductive reasoning. Research showed that only 10-15% of students reached Level 2 before entering complex Level 3 tasks.

What does the evidence suggest about students' transitions between Van Hiele levels?add

Studies indicate that progression from Level 1 to Level 2 involves significant restructuring of relationships between geometric concepts, not merely verbalization of knowledge. For example, only 45% of South African pupils in Grade 12 achieved Level 3 proficiency despite the curriculum's expectations.

What are the characteristics of thought levels in the Van Hiele theory?add

Each of the first four levels in the Van Hiele theory exhibits distinct characteristics; for example, Level 2 students analyze properties yet fail to interrelate them, leading to misconceptions about class inclusions. In contrast, students at Level 3 demonstrate hierarchical thinking, understanding broader concepts through logical deduction.

How does terminology impact students' understanding in the Van Hiele framework?add

The Van Hiele theory emphasizes that acquisition of technical terminology is critical for progressing from Level 1 to Level 2 in geometry education; insufficient understanding can hinder concept transitions. This was evident in findings where transition challenges were magnified for second-language learners due to technical language demands.

Figures (9)
halfturn of the grey triangle around the midpoint of the side AB maps angle A onto angle
halfturn of the grey triangle around the midpoint of the side AB maps angle A onto angle
Invited plenary presented at the 4 Congress of teachers of mathematics of the Croatian Mathematical Society, Zagreb, 30 June —2 July 2010, as well as at the National Mathematics Congress in Namibia, Swakopmund, 21-23 May 2012. Also presented as public lecture in the Program Post-Graduate Studies in Mathematics Education, Pontifical Catholic TInivercity af San Panlo Rrazil 19 Anril 9010 In the first case, pupils can again see that angle A = angle D due to a saw being formed
Invited plenary presented at the 4 Congress of teachers of mathematics of the Croatian Mathematical Society, Zagreb, 30 June —2 July 2010, as well as at the National Mathematics Congress in Namibia, Swakopmund, 21-23 May 2012. Also presented as public lecture in the Program Post-Graduate Studies in Mathematics Education, Pontifical Catholic TInivercity af San Panlo Rrazil 19 Anril 9010 In the first case, pupils can again see that angle A = angle D due to a saw being formed
error) to ‘discover’ that they always add up to 180°.
error) to ‘discover’ that they always add up to 180°.
Invited plenary presented at the 4" Congress of teachers of mathematics of the Croatian Mathematical Society, Zagreb, 30 June —2 July 2010, as well as at the National Mathematics Congress in Namibia, Swakopmund, 21-23 May 2012. Also presented as public lecture in the Program Post-Graduate Studies in Mathematics Education, Pontifical Catholic University of Sao Paulo, Brazil. 19 April 2010.
Invited plenary presented at the 4" Congress of teachers of mathematics of the Croatian Mathematical Society, Zagreb, 30 June —2 July 2010, as well as at the National Mathematics Congress in Namibia, Swakopmund, 21-23 May 2012. Also presented as public lecture in the Program Post-Graduate Studies in Mathematics Education, Pontifical Catholic University of Sao Paulo, Brazil. 19 April 2010.
Formal Van Hiele Level 3 definitions in textbooks are often preceded by an activity
Formal Van Hiele Level 3 definitions in textbooks are often preceded by an activity
was an important factor (e.g. calling a square a special rectangle).
was an important factor (e.g. calling a square a special rectangle).
dragging, students also explore special cases (rhombus, square). Investigate
dragging, students also explore special cases (rhombus, square). Investigate
Construct and connect the midpoints of the sides of a kite. (The result generalizes to any quadrilateral with perpendicular diagonals).
Construct and connect the midpoints of the sides of a kite. (The result generalizes to any quadrilateral with perpendicular diagonals).

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