Department of Education in Technology & Science, Technion - Israel Institute of Technology
Haifa 32000, Israel
A number of universities and colleges have developed mathematics courses based on the relationship between subjects such as the course described by Jay Kappraff . However, only minimal information is available on the educational aspects of these courses [Banerjee and De Graaf 1996]. There is a need for a comprehensive survey and empirical studies of mathematics curricula for architecture education. This would help to reduce a gap between supporters and detractors of mathematics education in architecture.
This paper reports a study of learning mathematics in professional context in one of the architecture colleges in Israel. The effect of integrating architectural and structure design problems in the calculus curriculum on students' achievements and attitudes was examined. A previous paper of ours  presented results of the study with focus on assessment and educational research. Here we will consider in more detail applied contents and learning activities in the course, and our way forward in order to discuss them with the NEXUS community.
CASE STUDY FRAMEWORK
The shortcomings of the curriculum were found through the long-term experience of teaching it in one of the architecture colleges. First, the curriculum provided only partial knowledge of that required for other disciplines and design activities. In particular, the lack of mathematical analysis skills hampered the students' achievements in mechanics and structural design. Furthermore, it did not provide motivation for learning mathematics and recognizing its importance in architecture.
A revised mathematics curriculum was developed in the college
in order to eliminate these deficiencies and the study of calculus
was added. Initially, when the course was taught in a conventional
way, the students had difficulties in applying mathematical concepts
to other disciplines. Subsequently, the need for interdisciplinary
connections and applications was recognized.
The achievement test included five mathematical problems in architectural design:
In addition to the test we conducted a questionnaire. It examined the student's attitudes towards:
The results of the pilot study indicated that many college and university students could not cope with the test. On the other hand, the questionnaire results showed high student awareness of the importance of mathematics in design.
The interviews with architects reflected their view that mathematics
belongs to the core of architecture education. However, in their
own practice the architects tended to use graphical procedures
instead of analytical methods because of the lack of mathematical
APPLICATION ORIENTED CURRICULUM
In the first variant of the course, the topics presented in Table 1 were taught by a disciplinary approach, closely following standard college mathematics texts. The ability of college students to apply their knowledge of calculus in the architectural context was assumed. However, teaching practice indicated the uncertainty of this assumption. Therefore, we developed an alternative variant of the course which integrated calculus concepts and their applications. In this integrated curriculum the students were instructed to apply calculus to actual problems of architecture design.
In our study the two variants of the calculus course, one based on a disciplinary curriculum and another on an integrated curriculum, were taught in two parallel college groups by the same teacher. Learning achievements of the groups were compared.
Applied Contents. Here we considered the applied contents of the integrated curriculum and how it was delivered to students.
As the first step in designing the integrated curriculum, we formulated instructional objectives from the applications domain that should be added to objectives pertaining entirely to calculus. The following applied mathematical and attitudinal skills were included:
Next, applied contents corresponding to the instructional objectives were developed and fitted into the calculus topics of the curriculum. The applied problem solving activities were as follows.
The last step in designing the integrated curriculum was making decisions on how to combine applied topics with the calculus topics and deliver them to students. An approach to integration of mathematical and applied contents proposed by Niss  and Alsina  was accepted. Accordingly, in the integrated curriculum, calculus topics were considered as dominant and were studied in their conventional order, as presented in Table 1. The applications were studied through learning practice and concerned with constructive interpretation of the mathematical concepts.
Learning Sessions. The applied contents of the curriculum were delivered through a problem-oriented learning activities. Applied problems were solved through a common scheme: perception of a real object and defining the mathematical problem, solving the problem and applying the solution.
Each type of applied problems mentioned in the previous section was studied through a tutorial, a workshop and a homework assignment sessions. In the tutorial session the teacher presented a sample problem and discussed its solution. Then, in the workshop session the students solved applied problems in class (working in pairs). Finally, they solved and submitted problems included in the individual homework assignment.
For example, we considered a problem related to the analytical description of structure contours. One of architectural buildings shown to the students is given in Fig. 5a. The students were asked to verify that the roof edge contour has the form of a hyperbola (Fig. 5b), and to describe it analytically.
Fig. 5. Analytical description of a structure contour: a) the building example; b) the hyperbola which describes the roof edge contour
To solve the problem, the students were given the following:
The solution of the problem is presented in Fig. 5b. The students determined the coordinates of five points on the contour M1(6, 6), M2(0, 5), M3(-6, 6), M4(-12, 8.3), M5(-16,10.2). They described the contour by means of an equation
and calculated the values of parameters a = 9 , b = 5 using
points M1 and M2 . Then they checked that
the coordinates of points M3, M4 and M5
satisfied the equation.
EDUCATIONAL STUDY FRAMEWORK
The research sample included two groups of the architecture college first year students. The control group (N=30) studied calculus based on a disciplinary approach while the experimental group (N=30) studied an interdisciplinary course described in the previous section. Both groups studied the same mathematical concepts and methods, taught by the same teacher. However, practice in applying mathematical knowledge to design was offered only to the experimental group.
The instruments that were used for the purposes of this study were application skills tests and attitude questionnaires (pre-course and post-course), and student interviews at the end of the course, as detailed below.
The achievement pre-test examined the mathematical knowledge of both groups before the course. It was decided to use the same achievement test as that used in the pilot study in order to validate its relevancy to the research population.
The achievement post-test was also given to the control and experimental groups and consisted of five open mathematical problems:
The pre-course questionnaire consisted of four closed items identical to those used in the pilot study. The post-course consisted of two sections. Section A included eleven closed items. The first two items were selected from the pre-questionnaire. Items 3-7 related to the impact of learning mathematics on understanding certain design activities: optimization, contour calculations, numerical consideration and Computer Aided Design (CAD). Items 8-11 dealt with evaluation of applications for understanding mathematical concepts such as the derivative, the integral, the graph of function and the extremum. Section B was given only to the experimental group and examined the impact of the course on motivation, understanding, creativity and interest in learning mathematics.
The results of the pre- and post-questionnaires (section A) were analysed using the Wilcoxon rank-sum test (Siegel, 1988) to examine differences of attitudes in the control and experimental groups.
A number of students from the experimental group (high and low achievers) were interviewed at the end of the course in order to get more detailed feedback on the aspects mentioned in the post-questionnaire and support its validity.
The validity of the achievement tests and the questionnaires was examined by consulting experts (architects). The relevance of the content items was testified by the Pearson correlation analysis. The Cronbach was applied to examine the reliability of the tests and questionnaires.
SUMMARY OF RESULTS
Initially, backgrounds of the experimental and control groups were compared by five independent variables: student gender, age, level of mathematics studied in high school, mathematics grade in the matriculation certificate, and Psychometric test grade. A salient conclusion was that the differences between the groups before the course were non-significant. The psychometric grades of both groups were higher than average for architecture students in the college. Mathematical pre-test achievements were low. In contrast to the pre-test, the groups' post-test results were significantly different. The post-test mean grade of the experimental group 73.2% was significantly higher than that of the control group 61.0%. Further more, the percentage of post-course failures in the experimental group (10%) was much lower than in the control group (43%).
The dependence of the post-test grade on the factors represented
by the five variables and the teaching method factor was analyzed
by a stepwise regression. Only two of these factors, namely the
level of mathematics studied at school and the teaching method
were found to be significant predictors. Through analysis of
variances of the two factors we found that the proposed teaching
method contributed to students with heterogeneous mathematical
Attitudes. The answers to the pre-course questionnaire given by the experimental and control groups reinforced the pilot study results. In particular, 43% of the students believed that the architect should be involved in the mathematical aspects of design. After the course, the number of students who held this opinion rose to 80% in both groups, due to the study of mathematics. Before the course, 37% of students considered mathematics a necessary tool for architecture studies. The post-course questionnaire indicated that this opinion was shared by 47% of the control group students vs. 80% in the experimental group. The Wilcoxon test comparison of pre- and post-course results showed that the attitudes changed significantly only in the experimental group.
The difference between the groups in the post-course questionnaire was also significant for the effect of mathematics studies on abilities to calculate dimensions of structural elements, and to understand conputer-aded design operations. Practice in contour design contributed to better understanding of the mathematical function concept.
Section B of the post-course questionnaire examined attitudes of the experimental group towards the proposed integrated curriculum. It was found that 67% of the students recognized the importance and relevance of studying applied problems for their architecture studies. The students also mentioned that this approach stimulated their motivation for learning mathematics (60%), interest in this subject (70%) and creativity (30%). For 40% of the students, the integrated approach reduced learning difficulties. In addition, 87% of the respondents became interested in continuing mathematics studies by this approach.
Student interviews. The students interviewed from the experimental group pointed out that they were quite impressed by the new approach of the course. They especially mentioned the value of practice in mathematical analysis of real architectural artifacts presented visually in the course. The students independently defined and solved applied problems in their structural design project, which was parallel to the course. They stated that, thanks to the course, they developed a better understanding of the statics concepts and recognized the role of mathematics as a design tool.
THE WAY FORWARD
With regard to an architecture education program in general, it is required to continue studying mathematics after the course towards developing mathematical thinking in the context of creative design and professional communication in architecture. To answer the need, we started a follow-up research considering mathematical aspects of the architectural design education. The goal of our new research is to develop a studio environment for design projects, which inspires students to think mathematically, and examine its value for learning mathematics and architecture.
This learning environment will provide experience with a hierarchy of architectural objects from drawing basic geometrical figures to analytic design and building physical models of composed 3D structures. Project assignments will require students to rely on the following design factors: dimensions, symbolism and expression, efficiency and functionality, steadiness, aesthetics, optimal planning, and diversity of shapes.
The learning population includes second-year students in the college, who studied mathematics in the first year of their studies, following the integrated curriculum. Each of the students will perform three different project assignments. The first project will deal with designing a combination of basic geometrical figures of harmonic proportions, to face a given surface. In the second project the student will design structure contours using mathematical functions, and build their physical models. The third project will focus on spatial morphology, dimensions and phisical modelling of complex structures.
Our study will apply ethnographic research methods, including interviews with designers and observations of their work in the architectural studios, as well as interviews with students and observations of their design experiences. The focus will be on employing mathematical concepts and skills.
This approach can also increase the motivation and confidence of the students in studying mathematics and encourage them to use mathematical methods in their personal design activities.
The proposed calculus curriculum can serve as a module for a new mathematics programme for architecture colleges.
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ABOUT THE AUTHORS
Sarah Maor is
a doctoral student in the Department of Education in Technology
and Science, Technion and a
Copyright ©2003 Kim Williams Books