By Michael de Villiers, created 3 August 2008; slightly updated 13 March 2025.
"... the Socratic didactician would refuse to introduce the geometrical objects by definitions, but wherever the didactic inversion prevails, deductivity starts with definitions. (In traditional geometry they even define what is a definition - a still higher level in the learning process.) The Socratic didactician rejects such a procedure. How can you define a thing before you know what you have to define?" - Hans Freudenthal (1973:416) in Mathematics as an Educational Task.
Likewise, from a philosophical points of view, as well as from some personal experience of doing a bit of original mathematics myself, I don't at all believe that research mathematics always starts with definitions. Even more importantly, I don't believe that it is good practice educationally to always introduce mathematical concepts by means of formal definitions. (Though some 'terminological' definitions are obviously unavoidable like what is meant by a 'saw' more commonly known as 'alternate angles').
In fact, it is a very unnatural way generally to learn 'via definition'. Young children do not, by means of a formal definition, learn what a table, chair or fork is, but by seeing many different examples of the same concept called a table, chair or fork. By means of visual association, children gradually learn that a table for example can have different numbers of legs, different shaped tops, and be made from different materials. So gradually, and through varied visual experience and association, they over time begin to develop a 'concept image' of what a table, car or house is, and not by us giving them any formal 'concept definition' of these objects.
Similarly, I strongly believe young children can best learn to understand the concept of 'rectangle', visually and informally, when given say a ready-made dynamic sketch of a rectangle as shown in the introduction in Investigation 1, and simply told that this figure is called a 'rectangle', nothing more. The formal definition is of course implicitly implied by the shape, but it is essentially meaningless to young children at Van Hiele level 1 as at this stage children are predominantly visually oriented. If children are now encouraged to drag this figure called a 'rectangle' into different orientations and shapes (including that of a 'square'), or at least shown how, this has better potential for developing a sound, robust 'concept image' of what a 'rectangle' really is than just memorizing a meaningless definition together with a static stereotype picture of a rectangle.
According to the Van Hiele theory, children's conceptual development is NOT age dependent at all, but depends entirely on their experience, and develops in levels or stages. Mastery of the one level is a prerequisite for the next. So they have to master the following levels in order:
a) the Visual level (Level 1 - shape, size & orientation)
b) the Analysis level (Level 2 - exploring of special properties, e.g. sides, angles, etc.)
c) the Formal level (Level 3 - definitions & proof), when some of these properties are selected to act as definitions.
"Geometry is the gate to Science. This gate is so small that one can only enter it as a child." - William Clifford (1845-1879), an English mathematician and philosopher
So when should one start with activities like the Introductory one in Investigation 1?
This depends entirely on the experience of the children, but should preferably be as young as possible. Unfortunately geometry is often neglected in the primary school, and then suddenly children are confronted with geometry materials often presented at a Formal level in the high school without them having had sufficient development in Van Hiele Levels 1 and 2.
Another problem with the teaching of geometry in the elementary phase is that learners are usually mostly presented only with manipulatives or pictures of the various quadrilaterals that are 'static' and 'rigid' instead of 'dynamic' - i.e. seeing a rectangle be transformed into a square plants the seed for an inclusive classification and formal definition later.
Initially, a teacher could perhaps first focus on the most well-known ones like squares, rectangles, rhombuses, and parallelograms in Investigation 1, then proceeding to the Analysis level in regard to these quadrilaterals in Investigation 2 before returning to Investigation 1 to also visually explore the other quadrilaterals like isosceles trapezoids, kites, etc.
In one important aspect though my suggested approach here is very different from the original Van Hiele theory, and that is in relation to strongly recommending visually exploring class inclusion early on as shown in the activities in Investigation 1. In the original theory, class inclusion is placed only at Level 3 (Formal definition & proof), but my experience has been that young children can easily be led to see that a square is a special rectangle by using dynamic geometry to let them observe for themselves that a rectangle can be dragged into the shape of a square.
In fact, I believe it is important to do this precisely when these concepts are informally introduced for the first time to young children. Otherwise, children are likely to develop static views of quadrilaterals as concepts which do not allow them to view some as special cases of others. For example, they may develop the view of a rectangle as a quadrilateral with 'two long and two short sides', which then does not allow a square to be seen as a special case. Research has indicated that once these 'partitional' views have formed they tend to fossilize and become very resistant to change, causing 'cognitive conflict' (or simply rejection) later on when encountering formal definitions which allow class inclusion.
Unfortunately there are some well-intentioned, but very misguided materials out there in relation to the educational use of dynamic geometry. In some materials and some research studies I have seen, learners are very early on tasked with constructing their own 'dynamic square', 'dynamic rectangle', etc. using dynamic geometry. In other words, learners are asked very early on to construct these 'dynamic quadrilaterals' with the software, and then to test their constructions by dragging to see whether it always remains a 'square' or 'rectangle'.
While probably inspired by a constructivist view of learning, namely that learners should construct their own knowledge rather being given it ready-made, asking learners to do such constructions are only possible once learners are on Van Hiele Level 3. In other words, they should already be conceptually at the level where they at the very least understand 'if-then' relationships. For example, they should understand that if we can construct a quadrilateral in a certain way (i.e. having say 3 right angles or two axes of symmetry through each pair of opposite sides), then it will remain a 'rectangle'. But unless Van Hiele Level 1 & 2 are already fully established, such an endeavour is conceptually doomed to fail.
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Interested readers are further encouraged to consult my 1998 PME paper TO TEACH DEFINITIONS IN GEOMETRY OR TEACH TO DEFINE? or the joint 2009 chapter in the NCTM Yearbook on Geometry DEFINING IN GEOMETRY.
Of interest too might be my joint papers with Rajen Govender, which evaluated some of the learning activities from my Rethinking Proof book regarding defining a rhombus (which lead to some improvement of the rhombus definition activities in the next edition in 2003) CONSTRUCTIVE EVALUATION OF DEFINITIONS IN SKETCHPAD - 2002 or CONSTRUCTIVE EVALUATION IN A DYNAMIC GEOMETRY CONTEXT - 2003
My book RETHINKING PROOF WITH SKETCHPAD - 1999-2012 (click link to download for free) also contain suggested activities for an isosceles trapezium ranging from Van Hiele Level 1 & 2 (Visualization & Properties) through to Van Hiele Level 3 (Constructing & Defining). Defining activities for a rhombus at Van Hiele Level 3 are also given in the book. Sketchpad 5 sketches associated with the activities from the book can be downloaded as a zipped file from at Rethinking Proof GSP 5 Sketches.
Also watch online a video, or download the video or a PDF of my plenary paper given in Brazil and Croatia (2010) SOME REFLECTIONS ON THE VAN HIELE THEORY
Several other papers of mine on geometry education, definitions, proof and the Van Hiele theory are available at Mathematics Education Articles.
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Free Download of Geometer's Sketchpad
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