Michael de Villiers 2000

de Villiers M. (2000)
developing understanding of proof within the context of defining quadrilaterals.

Contribution to: Paolo Boero, G. Harel, C. Maher, M. Miyazaki (organisers) Proof and Proving in Mathematics Education. ICME9 TSG 12. Tokyo/Makuhari, Japan.

© Michael de Villiers

Traditionally, teachers have presented proof in the geometry classroom only as a means of obtaining certainty; i.e. to try and create doubts in the minds of their students about the validity of their empirical observations, and thereby attempting to motivate a need for deductive proof. This approach stems possibly from a narrow formalist view that the main function of proof is the verification of the correctness of mathematical statements. It has also dominated most mathematics teacher education courses. In fact, De Villiers (1987) found that about 60% of prospective mathematics teachers at South African universities saw the function of proof only in terms of verification/justification/conviction, and were not able to distinguish any other functions of proof.
  However, proof has many other important functions within mathematics which in some situations are of far greater importance than that of mere verification. Some of these are (compare De Villiers, 1990; 1997):

o explanation (providing insight into why it is true)
o discovery (the discovery or invention of new results)

o intellectual challenge (the self-realization/fulfilment derived from constructing a proof)

o systematisation (the organisation of various results into a deductive system of axioms, concepts and theorems)

Although many recent publications and research relating to proof within dynamic geometry have focussed on both the conviction and explanation functions of proof (compare Movshovitz-Hadar (1988), Hanna (1989), Chazan (1990), Hadas & Hershkowitz, 1998), it seems that careful consideration of the other functions have unfortunately been neglected.
  It seems important from an epistemological perspective that mathematics teachers should also attempt to develop an understanding and appreciation of these other functions to make proof a more meaningful activity for their students. However, if teachers are expected to induct their own students early into the art of problem solving and posing, allowing sufficient opportunity for exploration, conjecturing, refuting, reformulating, discovering, explaining, systematizing, etc., teachers themselves should have been exposed to such approaches in their own learning of mathematics. It is therefore of fundamental importance in mathematics teacher education to devise innovative, new ways of expanding prospective teachers' views of proof to incorporate these aspects.
  Traditionally most teachers and textbook authors have simply provided students with ready-made content (eg. definitions, theorems, proofs, classifications, etc.) that they merely have to assimilate and regurgitate in tests and exams. Traditional geometry education of this kind can be compared to a cooking and bakery class where the teacher only shows students cakes (or even worse, only pictures of cakes) without showing them what goes into the cake and how it is made. In addition, they're not even allowed to try their own hand at baking!
  The direct teaching of geometry definitions with no engagement of students in the underlying process of defining has often been criticised by mathematicians and mathematics educators alike. For example, already in 1908 Benchara Blandford wrote (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 definitions, 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. Let us try to discover the kind of conception already existing in the child's mind - vague and crude it generally is, of course, otherwise what need for education? - let us note carefully its defects, and then help the child himself to refashion the conception ..."

The purpose of this paper will be to present and discuss some examples of activities which have been developed in a mathematics teacher education course to specifically focus on developing understanding of the systematization function of proof. In addition, it is based on a reconstructive teaching approach which means that the content is not directly presented to students in a finished form, but is re-constructed collaboratively by the students and the teacher in a typical mathematical manner (compare De Villiers, 1998). A theoretical distinction will also be made between two different kinds of defining, namely, descriptive and constructive defining of concepts.
  Linchevsky, Vinner & Karsenty (1992) have reported that many student teachers do not even understand that definitions in geometry have to be economical (contain no superfluous information) and that they are arbitrary (in the sense, that several alternative definitions may exist). It is plausible to conjecture that this is due to their past school experiences where definitions were supplied directly to them. It would appear that in order to increase students' understanding of geometric definitions, and of the concepts to which they relate, it is essential to engage them at some stage in the process of defining of geometric concepts. Due to the inherent complexity of the process of defining, it would also appear to be unreasonable to expect students to immediately come up with formal definitions on their own, unless they have been guided in a didactic fashion through some examples of the process of defining which they can later use as models for their own attempts.

Apart from developing some understanding of the systematization function of proof, actively engaging prospective mathematics teachers in the process of defining quadrilaterals, can assist them in realizing:

1) that different, alternative definitions for the same concept are possible;
2) that definitions may be uneconomical or economical;

3) that some economical definitions lead to shorter, easier proofs of properties

Note that for obvious reasons the activities given below have been greatly abbreviated and that the actual worksheets include a lot more technical detail and structure (eg. see De Villiers, 1999). Although Sketchpad is used, the activities can obviously be done with any other dynamic geometry software, as well as in paper and pencil contexts (although not as effectively).

Activity 1: Exploration of properties of isosceles trapezium

In this activity students use Sketchpad to first construct a dynamic isosceles trapezium by using reflection, and then explore its properties (eg. angles, sides, diagonals, circum circle). By dragging students also explore special cases (eg. rectangle, square).

o involves Van Hiele 1 (visualization) & Van Hiele 2 (analysis & formulation of properties)
o properties of isosceles trapezium are explained (proved) in terms of reflective symmetry

Activity 2: Describing (defining) an isosceles trapezium

Students select different subsets of the properties of an isosceles trapezium as possible descriptions (definitions), and first check whether they are necessary and sufficient by construction on Sketchpad, and then by logical reasoning (proof).

o involves Van Hiele 3 (local ordering)
o the explicit function of proof here is that of systematization (ie the deductive organization of the properties of an isosceles trapezium).

o involves the mathematical process of descriptive defining

References

Chazan D. (1990). Quasi-empirical views of mathematics and mathematics teaching. In: Hanna G., Winchester I. (Eds) Creativity, thought and mathematical proof. Toronto: OISE.
De Villiers M. (1987) Algemene beheersingsvlakke van sekere wiskundige begrippe en werkwyses deur voornemende wiskunde-onderwysers. SA Journal of Education 7(1), 34-41.
De Villiers M. (1990) The role and function of proof in mathematics. Pythagoras 24, 17-24.
De Villiers M. (1997) The Role of Proof in Investigative, Computer-based Geometry: Some personal reflections. In: Schattschneider D., King J. (1997). Geometry Turned On! Washington: MAA.
De Villiers M. (1998) To teach definitions or teach to define? In: Olivier, A.I & Newstead, K. (Eds). PME 22 Proceedings, Stellenbosch: South Africa.
De Villiers M. (1999) Rethinking Proof with Sketchpad. Key Curriculum Press Emeryville, CA: USA.
Hanna G. (1989). Proofs that prove and proofs that explain. Proceedings of PME13. Paris, 45-51.
Hadas N., Hershkowitz R. (1998) Proof in geometry as an explanatory and convincing tool. In Olivier, A & Newstead, K. (Eds.) (1998) Proceedings of 22nd PME-conference, Stellenbosch, South Africa, July, Vol 3, 25-32.
Linchevsky L., Vinner S., Karsenty R. (1992) To be or not to be minimal? Student teachers' views about definitions in geometry. Proceedings of PME 16 (New Hampshire, USA), Vol 2, pp. 48-55.
Movshovitz-Hadar N. (1988) Stimulating presentations of theorems followed by responsive proofs. For the Learning of Mathematics 8(2), 12-19;30.
Van Hiele P.M. (1973) Begrip en Inzicht. Purmerend: Muuses.