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de Villiers M.
(1997)
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The authors have attempted an ambitious project in designing three teaching experiments to investigate different contexts for geometry theory development; the role of the teacher in classroom interaction and the evolution of theorems in terms of their initial conjecture, formulation and eventual proof (explanation). Although they have clearly widely consulted the available literature in the area, the theoretical framework unfortunately appears a little unfocussed and haphazard at times. The authors correctly identify students' motivation to proof as a major problem and distinguish two important functions of proof, namely, verification (dealing with the truth of a statement) and explanation (dealing with understanding why it is true), but neglect to distinguish the discovery function (compare De Villiers, 1995). They appear to distinguish two hierarchical levels with verification at the first level and explanation at the second level; in other words, understanding the verification function appears to be a prerequisite for understanding the explanation function. For example, they claim that "the uncertainty status of the truth of a statement is crucial for the initial construction of the meaning of theorems". This is problematic as I believe that these functions may vary dramatically from context to context, and according to students' general cognitive development and experience. For example, letting students construct a kite using reflection on Sketchpad or Cabri in order to investigate its properties may lead more naturally to their explanation via symmetry, than to their actual verification, as students tend to have little doubt having dynamically investigated them. Of course, it is extremely important to develop this verification meaning of proof amongst students and that one identifies appropriate contexts to do so, but it is potentially restrictive to regard it as a necessary prerequisite for an initial introduction to proof. For example, consider the excerpts from the work of the two students in relation to the "table-and-ball" problem. In my opinion, these excerpts can just as easily be interpreted as their explanations for why their solutions are correct. It is certainly not conclusive that the students themselves interpreted their arguments purely as justifications, which is what the researchers seem to want to attribute to them. In order, to try and assess the meaning students give to their own arguments, the researchers should have attempted to formulate the question differently (eg. to ask students to "Prove the method ..." already assigns an a priori meaning of justification, since it means "the proposal of very safe arguments"). Alternatively, they could have probed deeper by asking further questions, for example: "Are you convinced that your solution is correct? Why?"and "Can you explain why your solution is correct?" It is possible to conjecture that students would interpret their own arguments as both justifying and explaining at the same time. It is also not clear whether students had any prior experience in solving similar real world problems; eg positioning objects on an actual table or a small model of it (not just a representation). Either way, it would be important to try and determine what the effect of the presence, or absence, of such prior experience might be on the outcome. For example, one may find that in the real world situation some students may instead use the lines of symmetry (through the midpoints of the opposite sides) of the rectangular shaped table to locate the centre. There is furthermore not enough detailed information to assess the transition from reality to the presentation of reality by perspective drawing, and the precise role of the teacher and the involvement of the students in this process. In the Sunshadows paragraph on the 9th page, the researchers refer to "proving that a condition is necessary". Logically, this is a little confusing, as proving strictly deals with establishing the sufficiency of certain conditions. For example, in proving a theorem true, we establish that the condition p is sufficient for the conclusion q . We do not really "prove" conditions necessary in mathematics (unless the conditions are necessary and sufficient, in which case we have an equivalence and have to prove the foward and backward implications, ie. and . Of course, one can explore whether certain sufficient conditions are really minimal (necessary) by leaving out some of them. For example, for a quad to be a rhombus it is clearly a sufficient condition to have diagonals perpendicularly bisecting each other. To explore whether this is also a necessary condition, one could try leaving out some of the conditions; eg check whether perpendicular diagonals or bisecting diagonals are sufficient on their own. It is this kind of exploration (rather than formal proving) that the researchers appear to be referring to on the 10th page under C). A significant finding in the Sunshadows paragragh was the close connection between the production of the conjecture and the eventual construction of a proof. However, this finding may be context specific and one must be careful not to overgeneralize. According to the researchers, the main purpose of investigating geometric constructions within a Cabri environment was to introduce students to an axiomatic-deductive approach to geometry. By selecting certain statements as axioms, the system is slowly built up by adding new constructions. This kind of axiomatization can be called constructive axiomatization where one starts out from a small number of axioms and logically deduce the one theorem after the other (compare Krykowska, 1971:129). However, from a historical perspective, Euclidean geometry did not initially develop in this way, but was only re-organized in this way by Euclid. The latter kind of axiomatization can be labelled as descriptive axiomatization (compare Krykowska, 1971:129-130) which means that after a certain set of statements have been already been discovered, known and used for a while, one starts analysing the logical relationships between them, first locally and then more generally. Finally, a subset of these statements are selected as axioms, and the remaining statements are re-structured into a deductive frame. Following a descriptive axiomatization approach, one could therefore easily let students use angle bisector, perpendicular bisector and other tools within Cabri or Sketchpad without necessarily first posing them with the problem of designing the tool themselves, explaining why they work or proving that they work. In this sense, constructions could simply be viewed as tools to explore and investigate interesting geometric relationships such as the concurrency of the angle bisectors of a triangle, etc. At a later stage, one could then come back to the logical systematization of the underlying theory on which the construction tools are based. From an epistemological view, it is therefore not essential ("correct" as the researchers seem to claim) that one has to develop and introduce constructions in a constructive axiomatic way. More generally, one could develop a substantial body of geometric knowledge before axiomatizing locally, and then more generally (compare Freudenthal, 1973:451). For example, one could first let students discover that the midpoints of the sides of a quad determine a parallelogram, and then use as explanation the (unproved) statement that the line segment connecting the midpoints of two sides of a triangle is parallel (and equal to half) the third side. Next one can again ask how we may prove this midpoint theorem in turn, and continuing to reason backwards in this way, one can arrive at appropriate axioms. Epistemologically, the researchers' analysis and implementation of the different ways in which mathematics develops and is systematized, therefore appears to be limited. ReferencesDe Villiers, M.D. (1995). An alternative approach to proof in dynamic geometry. MicroMath, Spring, 11(1), 14-19. Freudenthal, H. (1973). Mathematics as an Educational Task. Dordrect: D. Reidel. Krygowska, A.Z. (1971). Treatment of the Axiomatic Method in Class. In Servais, W. & Varga, T. Teaching School Mathematics. Penguin-Unesco, London, 124-150. |