Appeared in PYTHAGORAS 38, pp. 3-4, Dec 1995.
(This journal is the official journal of the Association for Mathematics Education of South Africa (AMESA), PO Box 12833, Centrahil 6006, South Africa)
The handling of geometry definitions in school textbooks
Recently while out on teaching practice I was appalled by how poorly definitions of the quadrilaterals were handled in the high school textbooks our students were using. Although some textbooks occassionally provide different sketches (visual representations) of the same quadrilateral before exploring its properties by folding, rotation and measurement, they ultimately end up by just providing a single formal definition (eg. "A parallelogram is a quadrilateral with both pairs of opposite sides parallel"). The problem is that pupils are thereby given no opportunity to participate in the formulation and choice of these definitions. Such direct teaching I find very strange coming from authors and teachers who often profess to problem-centered teaching and constructivist learning theory.
Since earliest time progressive educators and philosophers have constantly argued against the direct imposition of ready-made, formal knowledge into the mind of the learner, insisting that the learner should take responsibility and participate in the (re) construction of their own knowledge and understanding.
An excellent example from classical time (400 BC) is Socrates' various discussions with the slave Meno and his students. By careful questioning, Socrates aids the learners in arriving at their own understanding. The humanist Jean-Jacques Rousseau writes in Emile (1762) about the necessity for participation by the learner and restraint from direct teaching by the educator:
"Teach your scholar to observe ... you will soon raise his curiosity. Put the problems before him and let him solve them himself. Let him know nothing because you have told him, but because he has learned it for himself.
Undoubeedly the notions of things thus acquired for oneself are cleare and much more convincing than those acquired from the teaching by others ... "
The Palistinean philosopher and poet Khalil Gibran eloquently writes in The Prophet (1923) about Teaching in a way which clearly seems to reflect some of the basic tenets of modern Constructivism, namely, that learning is ideosyncratic and that knowledge simply cannot be transferred directly from one person to another:
"The teacher who walks in the shadow of the temple, among his followers, gives not of his wisdom, but rather of his faith and his lovingness. If he is indeed wise he does not bid you enter the house of his wisdom, but rather leads you to the threshold of your own mind.
The astronomer may speak to you of his understanding of space, but he cannot give you his understanding.
The musician may sing to you of the rhythm which is in all space, but he cannot give you the ear which arrests the rhythm, nor the voice which echoes it... "
So the list goes. In 1908 Benchara Blandford writes (quoted in Griffiths & Howson, 1974: 216-217):
"To me it appears a radically vicious method, certainly in geometry, if not in other subjects, to supply a child with ready-made defintions, to be subsequently memorized after being more or less carefully explained. To do this is surely to throw away deliberately one of the most valuable agents of intellectual discipline. The evolving of a workable definition by the child's own activity stimulated by appropriate questions, is both interesting and highly educational."
Hans Freudenthal (1973: 417-418) also strongly criticizes the traditional practice of the direct provision of geometry definitions as follows:
"... most definitions are not preconceived but the finishing touch of the organizing activity. The child should not be deprived of this privilege ... Good geometry instruction can mean much - learning to organize a subject matter and learning what is organizing, learning to conceptualize and what is conceptualizing, learning to define and what is a definition. It means leading pupils to understand why some organization, some concept, some definition is better than another. Traditional instruction is different. Rather than giving the child the opportunity to organize spatial experiences, the subject matter is offered as a preorganized structure. All concepts, definitions, and deductions are preconceived by the teacher, who knows what is its use in every detail - or rather by the textbook author who has carefully built all his secrets into the structure."
Furthermore, according to the Van Hiele theory, understanding of formal definitions only develop at Level 3 (Ordering). From research experience it appears that most children in Std 6 (Grade 8) are at Van Hiele levels 1 and 2, so that the focus of activities should be on visualisation and analysis of the properties of the quadrilaterals. (In fact, research by De Villiers & Njisane (1987) has shown that only about 45% of Std 10 (Grade 12) pupils then taking mathematics in KwaZulu had mastered Level 2). Formal economic definitions like those normally given in Std 6 (Grade 8) textbooks are therefore completely out of place. Children at Van Hiele Level 1 would tend to give visual definitions (eg. a rectangle has all angles 90° and two long and two short sides) and those at Van Hiele 2 would tend to give correct, uneconomical definitions (eg. a rectangle is a quadrilateral with opposite sides parallel and equal, all angles 90°, diagonals equal, half-turn symmetry, two axes of symmetry through opposite sides, two long and two short sides, etc.)
As can be seen above, their own definitions would also tend to be partitional at Std 6; in other words, they would not allow the inclusion of squares among the rectangles (by stating 2 long and 2 short sides). Although Std 6 pupils may memorize given formal definitions and regurgitate them in the examination, they are unlikely to accept, understand and appreciate them. (I'm not sure why there is this obsession with textbook authors to provide formal definitions for the quadrilaterals in the primary and junior secondary school. We do not necessarily need formal definitions for every concept that we work with - is it really necessary to provide a formal definition to a child of the concept "table" for the child to know what a table is?). In fact, a teaching experiment in 1977/78 (reported in Human & Nel et al, 1989) seems to indicate that the question of formal definitions should not be approached before about Std 8, and then only if pupils are actively involved in the process of constructing definitions for the various quadrilaterals.
Another problem with just providing pupils with ready-made definitions is that the misconception is invariably created that there is only one correct definition for each concept. The possibility of correct alternative definitions simply does not arise in such a situation. However, if pupils are themselves posed with the problem, as was done in the 1977/78 teaching experiment, of providing short, accurate descriptions (definitions) over the phone or by telegram to someone else, different descriptions (definitions) naturally arise. Furthermore, since many of their own definitions are incomplete (containing too little information) or uneconomical (containing too much information), a discussion of necessary and sufficient conditions naturally arises out of this. Therefore, children learn much more about the nature of definitions and the process of defining concepts, and they begin to realize where definitions come from and why certain ones may be preferred to others.
Understanding the need for definitions is also not addressed by just providing pupils with ready-made definitions. One of the activities used in the above teaching experiment was to challenge pupils for example to try and prove all the properties of a rhombus. The problem is that when they try to do this they invariably land up with varying forms of circular arguments. In order to avoid such a circular argument, one therefore has no choice but to accept one of the properties without proof, ie. as a definition and use it to prove the other properties. This then brings home powerfully the message that it is impossible to prove all statements in mathematics and that we have to accept certain statements without proof as definitions (or axioms) to avoid circularity.
Often the presentation of formal definitions is preceded by an activity whereby pupils have to compare in tabular form various properties of the quadrilaterals, eg. to see that a square, rectangle and rhombus have all the properties of a parallelogram. The purpose clearly is to prepare them for the formal definitions later on which are hierarchical. (In other words, the given definitions provide for the inclusion of special cases, eg. a parallelogram is defined so as to include squares, rhombi and rectangles). However, research reported in De Villiers (1994) show that many pupils, even after doing tabular comparisons and other activities, if given the opportunity, still prefer to define quadrilaterals in partitions. (In other words, they would for example prefer to define a parallelogram as a quadrilateral with both pairs of opposite sides parallel, but not all angles or sides equal).
A constructivist approach in contrast, would allow pupils to formulate their own definitions irrespective of whether they are partitional or hierarchical. By then discussing and comparing in class the relative advantages and disadvantages of these two different ways of classifying and defining quadrilaterals (both of which are mathematically correct), pupils may be led to realize that there are certain advantages in accepting a hierarchical classification (for more details, see De Villiers, 1994). Simply ignoring pupil views and perceptions regarding definitions as is done in most South African high school textbooks at present is to pay mere lip service to constructivism and sound educational practice.
Perhaps even more disconcerting than the direct presentation of the definitions of the quadrilaterals is their poor mathematical treatment in some textbooks. For example, in one textbook the following glaring inconsistency was noticed. For example, both a kite and trapezium are partitionally defined, respectively as quadrilaterals with "two pairs of adjacent sides equal, but opposite sides not equal" and "exactly one pair of opposite sides parallel". However, on the very next page a diagram is given where a rhombus is shown to be a special case of a (convex) kite and a parallelogram as a special case of the trapezia.
Defining concepts accurately in mathematics is certainly not an easy task, and is only developed after lots of experience and practice. However, the educational experience is worth the trouble, and I would like to encourage our authors and teachers out there to seriously rethink their treatment of geometry definitions.
Michael de Villiers, Editor
De Villiers, M.D. & Njisane, R.M. (1987). The development of geometric thinking among black high school pupils in KwaZulu. Proceedings of the 11th PME conference, Montreal, Vol 3, 117-123.
De Villiers, M.D. (1994). The role and function of a hierarchical classification of the quadrilaterals. For the Learning of Mathematics, 14(1), 11-18.
Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht: Reidel.
Griffiths, H.B. & Howson, A.G. (1974). Mathematics: Society and Curricula. Cambridge University Press.
Human, P.G. & Nel, J.H. In medewerking met MD de Villiers, TP Dreyer & SFG Wessels. (1989). Alternatiewe aanbiedingstrategiee vir meetkunde-onderwys: n Teoretiese en empiriese studie. ENWOUS verslag no. 2, Universiteit Stellenbosch.
Human, P.G. & Nel, J.H. In co-operation with MD de Villiers, TP Dreyer & SFG Wessels. (1989). Appendix A: Curriculum Material. RUMEUS Curriculum Material Series No. 11, University of Stellenbosch.