Francisco DelgadoInstituto Tecnológico y de Estudios Superiores de Monterrey (ITESM) Campus Estado de México, A.P. 2, 52926, Atizapan, Edo. de Mexico, MEXICO.
In the past nine years, the teaching model of ITESM has rapidly evolved, taking into account the development of abilities, attitudes and values (AAV's) without forgetting the development of knowledge. The mathematics for architecture course was redesigned, using problem-based learning and an intensive application of computer technology to overcoming those difficulties. Now, the main purpose is to develop a mathematical, physical and technological culture in students of architecture to allow them to analyze and solve complex problems related to mathematics in architecture and design. The course was planned and implemented for the first semester of the architecture program and is actually related (through curriculum integration) to future courses which require specific mathematical applications and are now available online. The course is centered around five mathematical themes which give the students a panoramic of mathematics in architecture, from design to building engineering: - Functions and modeling;
- Derivatives and optimization;
- Applications of integrals in design, architecture and building engineering;
- Vectors and mathematics in physics;
- Three different types of geometry: Euclidean, spherical and fractal.
In this paper, we present a brief outline of the course, examples of student projects integrating architecture and mathematics, and experiences and statistical outcomes the course as regards the students.
- The program was very ambitious and seldom fulfilled the learning requirements of students;
- The course was not very attractive because mathematics teachers were not knowledgeable about applications of use in architecture and design; the ones taken into consideration artificial and unrealistic;
- The course did not have a purposeful continuity with the rest of the courses in the architecture program.
Still, even with the implementation of the new educative model (based on technology and techniques of meaningful learning) and the efforts to bring this philosophy into this course, advances were slight and did not help to solve the existing situation.
In this way, the following aims were established: - To guarantee that each concept introduced addressed a later curriculum requirement of the students, specifically those of applied physics, resistance of materials, structural systems and passive systems;
- To include applications derived from actual concepts but
with specific references to the areas of architecture and design;
To involve the Architecture faculty in the design of the course; - To include the use of computer technology for mathematics and architecture in order to arrive at the solutions of more complex problems related to both disciplines;
- To be consistent with the constructivist methodology followed in mathematics courses (normally using Problem Based Learning, or PBL) [De Graaff and Bouhuijs 1993; De Graaf and Cowdroy 1997] and that for architecture courses (using Project Oriented Learning, POL), this course is centered around complex scenarios tied to both disciplines, in which there is a mathematical.
- Vectors and their applications to physics and mechanics;
- Mathematical modeling (vector functions and multivariable functions);
- Optimization;
- Applications of integral (inertia momentum and mass centers);
- Introduction to differential equations;
- Emphasis on geometry.
On the other hand, some elements were identified in the original course which were excessively covered but lacked applications, such as: - Exhaustive study of functions;
- Limits and continuity;
- Treatment of derivatives and integrals centered in algebra;
- Integration methods;
- Plotting functions.
With this perspective, the new course replaced or simplified some subjects to include others that were more important. Five thematic units were constituted: - Functions and modeling;
- Derivatives and optimization;
- Applications of integrals in design, architecture and building engineering;
- Vectors and mathematics in physics;
- Three different types of geometry: Euclidean, spherical and fractal.
In addition, five complex (but generic) mathematical scenarios in architecture or design were determined for each unit. These scenarios provide a perspective for these disciplines.
- Parametric propagation of form in structural architecture;
- Optimal design of packages and containers;
- Ergonomics and design;
- Tensegrity;
- Fractal architecture.
All of them share the following characteristics: - They are centered on knowledge of the actual thematic unit;
- They use some of the concepts reviewed in previous units;
- They emphasize the necessity of new knowledge which will be learned in later units or later courses;
- They show the relationships between mathematics and architecture or design.
As an example, we will describe briefly the scenario corresponding to "parametric propagation of form" [Szalapaj 2001] in which, given a space to construct for a railway station, the student must design mathematically the form of the dome for it. Basically the activity is centered in designing a parametric function:
Fig. 1. Parametric propagation of form It is important to note that previous to this complex activity, the lectures as well as the exercises of this unit included analysis, modeling and construction of simple domes and cupolas, based on modeling with multivariable functions and vector functions. Another of the scenarios used in the course is integrated into the design of an electronic device, using geometry and calculus. In this activity, technical requirements such as capacity and size are combined with aesthetic requirements (fig. 2). Fig. 2. Ergonomics and design In order to arrive at these scenarios successfully, the course attempts to be a constructivist guide for developing skills. So, the student had previously been required to solve diverse problematic situations involved in the design of Kingdom Centre, the design of the Al Faisaliah tower, or with self-sustainable structures. These previous activities endow to the student with useful abilities which will be necessary in the more complex scenarios of PBL.
Fig. 3. Averages in department exams It is clear that dispersion has been reduced (in some sense this is to the general advantage of students in the whole architecture course), since the introduction of this course. Nevertheless, the finals grade for the course aren't significantly different. Another positive result is that students appreciate the professors who have shown a commitment to the course.
- the involvement of more professors who enrich the course with their own contributions of cases of analyses that can be used as scenarios;
- the deliberate use of architecture tools (ArchiCAD by example) must be considered, since its use predominates the other types of technologies not so able for a student of architecture as far as visualization 3D.
A pending aspect in the present version of the course is a revision of the contents. As he were mentioned at the beginning, does not exist a consensus on the mathematical contents that must learn an architecture student, neither in the national scope, nor in the international (still within the different profiles and directions that can be given to the formation of an architect). He is doubtless that in anyone of these contents that are defined, always will exist a lot of applications that can be useful like learning scenarios.
De Graaf, E. and R Cowdroy. 1997. Theory and practice of educational innovation. Introduction of Problem-Based Learning in architecture: two case studies. http://www.ijee.dit.ie/articles/999986/article.htm. Last revision 12 february, 1997. Delgado, F. 2003. Principia program; teaching mathematics to engineers with integrated curriculum, teamwork environment and use of technology, in the Proceedings of "Mathematics Education into the 21st century project", Brno, Czech Republic, 2003. Delgado, F., R. Santiago, and C. Prado. 2002. Principia program: experiences of a course with integrated curriculum, teamwork environment and technology used as tool for learning. In the Proceedings of 2nd International Congress of Teaching Mathematics; Crete, Greece, 2002. Polanco, R., P. Calderón, and F. Delgado. 2000. Effects of a Problem-based Learning program on engineering students' academic achievement, skills development and attitudes in a Mexican university. Paper presented at the 82nd. Annual Meeting of the American Educational Research Association. Seattle, April 10-14, 2000. Szalapaj, Peter. 2001. Parametric Propagation of Form. Architecture Week, 19 September 2001.
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