By Michael de Villiers, 3 August 2008 at profmd@mweb.co.za
"... 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.
Philosophically, and from personal experience doing some original mathematics, I don't at all believe that mathematics always starts with definitions. Even more importantly, I don't believe that it is good practice educationally to introduce mathematical concepts by means of formal definitions. (Though some terminological ones are obviously unavoidable like what is meant by a 'saw' more commonly known as 'alternate angles').
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 association, children gradually learn that a table for example can have different numbers of legs, different shaped tops, and be made from different materials.
Similarly, I strongly believe young children can best learn to understand the concept of 'rectangle' informally when given say a 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. Mastery of the one level is a prerequisite for the next. So they first have to master the Visual level (Level 1 - shape, size & orientation), then comes the Analysis level (Level 2 - exploring of special properties, e.g. sides, angles, etc.), and last comes the Formal level (Level 3 - definitions & proof), when some of these properties are selected to act as definitions.
So when one should start with activities like the Introductory one in Investigation 1 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. 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 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 letting them observe 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' later on when encountering formal definitions which allow class inclusion.
The interested reader is encouraged to further consult my paper TO TEACH DEFINITIONS IN GEOMETRY OR TEACH TO DEFINE?
Of interest too might be my joint paper with Rajen Govender, which evaluated some of the learning activities from my Rethinking Proof book regarding defining a rhombus (which lead to some improvement in the next edition in 2003) CONSTRUCTIVE EVALUATION OF DEFINITIONS IN SKETCHPAD
Several other papers of mine on geometry education, definitions, proof and the Van Hiele theory are available at Mathematics Education Articles or Archive of earlier Mathematics Education Articles