Answer to Question #121002 in Geometry for Tina

Question

In measurement, what do you understand by the terms standard and none standard
units of measurements? Provide examples in each case.
1.3 De Villiers (2003:5 - 10) classifies five different functions of proof within mathematical
statements. With the aid of examples, discuss the implications of the five functions of
proof to teaching and learning geometry.

Answer 1

The standard is considered as a standard, template, sample, model which is taken as the basis for comparing other objects or properties with it.

For example, if you want to order a casket in an online store as a gift to a friend who collects things with elements of the so-called golden section, the store’s website will offer you publication sizes in inches or centimeters.

The idea that a deductive argument can provide additional insight and some form of new discovery. For example, an algebraic explanation of why the sum of a two-digit number and its inverse is always divisible by 11, can lead students to understand that the other factor is the sum of the digits of the original number, which they might not have noticed, considering only a few cases. Very rarely, any of them noticed this additional property in the empirical phase, and they would express a grateful surprise when they learned this later from the evidence when their attention was directed to it.

Instead of defining the evidence in terms of its verification function (or any other function in this regard), it is suggested that the evidence is rather defined simply as a deductive or logical argument that shows how a particular result can be obtained from another proved or alleged Results; no more, no less. It is not intended here that fidelity to the test function of evidence is generally sacrificed, but should not be increased to the defining characteristic of evidence.

Moreover, the verification function ought to be supplemented with other important functions of proof using genuine mathematical activities as described above. It is also not suggested that the preceding examples be directly implemented in a classroom as their success will depend largely on the past experience, expertise and ability of the audience, the classroom culture, as well as the skill of the teacher as a facilitator of learning. For example,  ‘for an argument to be considered a proof, the students need not only convince, but also to explain’. Then willproceeds to give an example of how this broader ‘didactical contract’ with respect to proof motivated students to actively engage in conjecturing, refuting and eventually developing a proof as a logical explanation through her continued insistence that they demonstrate why the pattern worked.