Hernán NottoliFaculty of Architecture, Urbanism and Design Centre of Mathematics & Design University of Buenos Aires 4433 Rivadavia Av. - Buenos Aires (1205) - Argentina
In this paper we will analyse some aspects that we consider relevant with respect to how knowledge is verified and ranked in evaluation tests. Although the mechanisms are analogous in other disciplines, this paper fundamentally analyses the case of mathematics in architecture and design schools. One particular characteristic of this area is that students in the design disciplines have a scarce affinity for the knowledge of the field of mathematics. Finally we will suggest some proposals (see Example 1), based on the steps of methodology set forth that could optimise the objectivity of the information being evaluated, as well as the results obtained, to impart knowledge to the students and to make possible the subsequent transfer of that knowledge to other disciplines.
The logical reasoning of mathematics is fundamentally based
on the process called
If we frame the theme that gives cause to this work with an
example in particular, it is feasible to reason how an inference
arises by means of deduction. In effect, if we consider as valid
the However, mathematical science has not considered abduction as valid for its reasoning, because it might lead to false conclusions arising from true premises or, inversely, true conclusions arising from false premises. Let us consider a non-scientific example so that we can validate this claim. The following Now let us see what happens when abduction is made with the
same example. The But it is possible to have the inverse situation, that is,
to reach a true conclusion starting from a false premise. The
Now, the methods used to determine if the knowledge of students of mathematics is satisfactory are based -- almost without exception -- on abduction, in spite of the fact, as we have just seen, that there are risks that such reasoning does not lead to absolutely valid conclusions, and in spite of the practically nonexistent use of such a way of reasoning in the process of mathematical logic. The epistemologist Charles Pierce has studied the abductive
process of inference, based on working with the following Therefore, the conclusion obtained through abduction is not highly reliable. However, this is the method used and recognized by every educational institution at present. And why is it reliable as an evaluation methodology? We shall now analyze the process known as induction, in order to arrive at a satisfactory explanation. Let us analyse another approach to the same example. Let us
modify the question to: The reiteration of these cases by induction constitutes the statistical justification that really supports the method of abduction as the way of evaluating, and makes predictable its reasonable degree of reliability. Two particularities are stressed in these previous considerations:
Let us now analyse the mechanism to determine the effectiveness of the method for evaluating the knowledge of the students who are taking examinations. In the course of mathematics of the School of Architecture of the University of Buenos Aires, a series of statistics has been worked out regarding the themes developed in the current syllabus of the subject, which also indicates the results that should obtained in the final examinations that the students have to take in order to pass the course. It is important to point out that the mode of evaluation was first written and then oral, so that it could be possible to evaluate the knowledge of the students in depth; that the examination included open, closed and mixed type questions (an example of which will be developed in the Appendix to this paper); and that electronic support, such as calculators and graphic computers, was allowed but written texts on the discipline were not. The data which is relative to the themes developed in the course, subdivided according to each heading and percentage of time allotted, is shown in graph 1. Graph 1
The data concerning the processed information [1] are shown in graph 2. Graph 2 The evaluated figures in percentiles in the graphs show that the students who study Architecture and Design have a higher affinity for themes that contain images (geometry, topography), or topological developments (graphs, symmetries), but not for calculus, which is a branch of mathematics that is always rejected by these students. In order to optimize the procedure for determining if the knowledge of the students is adequate to pass mathematics, it must be emphasized that it is necessary to include diverse modes of questioning in the examinations. Our experience with having included problems containing open, closed and mixed-type questions has been highly positive. Also, let us take into consideration that if the errors the
students may make in an examination are not extremely serious,
they should not be considered as such, if the students are warned
of the errors committed and are able to solve them by themselves.
This type of path is of great usefulness in museums, exhibitions, etc., where the ideal paths are those in which it is possible to go through all the rooms or stands, without having to travel by the same place more than once. The example of the Barcelona Pavilion designed by Mies van der Rohe in 1929 is the case of a building conceived to exhibit paintings, sculptures, drawings, etc.; its spaces are continuous and "multiple use" and the exercise proposed in this article consists in finding in that building an Eulerian path (there could be more than one solution). In general it is possible to determine a Eulerian path in diagrams of different configurations: [3]:
This type of topological and visual reasoning is, as we stated, very suitable for students of design schools, for it is easily applicable to this type of problem in design but not to others in which the fundamental component is calculation. These types of exercises are useful from another point of view, for those teachers who usually use algebraic logical reasoning or deductions in which pure mathematical nomenclatures and codes are only inferred.
[1] The statistics have been worked out over a total of 2,892 exercises, solved by 536 students. The author sincerely thanks the collaboration given by the Faculty of his Chair of Mathematics II of the School of Architecture, Design and Urban Planning of the University of Buenos Aires, for the compilation and processing of these data. return to text[2]
Gardner, H., Multiple intelligences: The theory in Practice,
Perseus Book Group. 1993. [3]
The name eulerian arises from the mathematician Leonard Euler
(1707-1783). See V. Spinadel and H. Nottoli, Mathematical tools
for Architecture and Design, Ed. FADU, 2005.
He has published several books and articles in his country and abroad, and he has presented many papers in a great quantity of congresses and scientific meetings around the world. His curriculum appeared in the Dictionary Of International Biography of Cambridge, England. Currently, Architect Nottoli is Tenure Professor of Mathematics in the Faculty of Architecture of the University of Buenos Aires, and Main Researcher in the Secretariat of Science and Technology in the same School of Design.
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