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Final Report of the Tempus Phare Grant No IMG-97-CZ-2025
Josef Dalik
Department of Mathematics,
Faculty of Civil Engineering, Technical University of Brno, Czech Republic
CONTENTS
Introduction
School Education in Ireland
1. Primary and Secondary Education in Ireland
2. Third Level Education in Ireland
The St. Patricks College, Maynooth
1. History
2. Mathematical Education Programs at NUI Maynooth
Education at Irish Universities
1. Teaching
2. Examinations
Mathematics in the Civil Engineering Programs
1. School of Engineering of NUI Dublin
2. University of Dublin
3. Faculty of Civil Engineering of TU Brno
Comparisons and Conclusions
1. Department of Mathematics of NUI Maynooth
2. Transition to Third Level
3. Methodology of Teaching at the Irish and Czech Universities
4. Mathematics and Theoretical Basis in the Civil Engineering Programs
Literature.
I. Introduction
The following main reasons 1, 2 lead myself to the application for this project.
To learn completely the methodology and content of teaching introductory mathe-matical courses within the civil engineering programs at a stable western university:
On the road from the real socialism to democracy, the prestige of engineering education programs at Czech universities has been decreasing and the secondary schools have been reducing the teaching of natural-scientifically and technically oriented courses with a varying intensity. That is why the average students entering the Technical University of Brno (briefly TU Brno) are not prepared for the study of the mathematical, physical, chemical and geological courses satisfactorily. We will call these courses the theoretical basis (of engineering programs). The result is that the contents of courses of the theoretical basis is very hard to understand for an essential part of students. Some of them give up their studies for this reason as a whole. We devote much effort to the modification of these courses in order to facilitate their studies. We have assumed that there is a big diversity in the knowledge of students entering the engineering programs at the western universities too and that the teaching methods correspond to this situation.
To get a complete and reliable information concerning programs for the degrees in civil engineering at some universities abroad:
There are intensive discussions at TU Brno devoted to the decrease of amount of time which the students spend in the classroom, to the definition of new proportions between basic components of the engineering programs and to other important problems in the last time. In these discussions, information concerning engineering programs at universities from abroad is often used as an argument as well. This information is very often misunderstood because of its incompleteness. It is mostly taken from the presentations on internet or from occasional personal contacts with colleagues from abroad.
The aim of this report is to compare:
The transition from secondary schools to the study of civil engineering at universities in Ireland and in the Czech Republic.
The content and methodology of teaching the theoretical basis, with emphasis on mathematics, within the civil engineering programs at some Irish and Czech universities.
In order to get the necessary knowledge, I have spent four weeks at the Department of Mathematics of the National University of Ireland (abbreviated by NUI) Maynooth. As this university has no engineering program, I paid my attention to the methodology of teaching mathematics on the level corresponding to the engineering mathematical courses at Maynooth. All the knowledge obtained later at the universities in Dublin indicated that the system of teaching mathematics is essentially the same at all Irish universities. In the second half of my stay in Ireland, a series of consultations with educators (mathematicians and civil engineers) engaged in the civil engineering programs of NUI Dublin and University of Dublin took place. Because it was impossible to get any systematic information concerning other disciplines than mathematics at these consultations, I have restricted my interest to the following subjects:
An overview of the mathematical courses from civil engineering programs.
The form and concrete content of their examinations.
The extent of other theoretically oriented courses in the civil engineering programs.
Relative part of the mathematical and theoretical courses in the whole civil engineering programs.
Chapter II contains basic data concerning primary and secondary schools as well as a brief overview of Irish universities. After a notice about the history of the St. Patricks College, Maynooth, three basic mathematical programs taught at NUI Maynooth are described in Chapter III. In this description, information concerning methodology is prefer-red to the exact content of the courses. Chapter IV is devoted to the methodology of teaching mathematics at Irish universities. It is based on the data from Chapter III and on consultat-ions with the staff of Department of Mathematics of NUI Maynooth, but the basic facts are valid in Ireland generally. Both the extent and the contents of mathematical courses within the civil engineering programs at NUI Dublin, University of Dublin and TU Brno are des-cribed in Chapter V. The data from the Chapters II Vare discussed in the last Chapter VI.
II. School Education in Ireland
1. Primary and Secondary Education in Ireland
Education at the Irish schools is compulsory for children since four years. They attend the eight-year primary school and then, usually since twelve, they attend the secondary school. The compulsory school education finishes at the age of sixteen. The program of secondary school is divided into the following three parts:
1. Lower Level. It corresponds to the school-years 1, 2, 3 and it is closed by examinations from all the courses studied and by awarding the so-called Junior Certificate for each of them.
2. Transition. It takes the school-year 4 and it is practically oriented. Instead of attending the school, pupils are employed in Ireland or abroad, they study foreign languages etc.
3. Higher Level. It consists of the school-years 5 and 6 and it is closed by examinations from all courses which a given pupil attends. For every course, the so-called Leaving Certificate (abbreviated by L.C.) is awarded. The final grades depend on a number of points computed from the results of several partial examinations. It is essential that the examinations as well as the computation of points are the same at all secondary schools, so that the L.C. is an objective appreciation of the knowledge. For example, there are two examination papers on mathematics identical for all secondary schools in Ireland and the examinations take part at the same time. The results are examined centrally and anonymously; some examination papers for the Higher Level L.C. from mathematics are enclosed in [8]. Maximal attainable number of points is 100 and the individual grades are related to the points in the following way:
Number of points for the
Grade Higher Level Lower Level
A1 100 60
A2 90 50
B1 85 45
B2 80 40
B3 75 35
C1 70 30
C2 65 25
C3 60 20
D1 55 15
D2 50 10
D3 45 5
Since the sixties, secondary education in Ireland is generally free by law. This change is one of the reasons, why the amount of young people with complete secondary education has been increasing. This concerns practically all teenagers in the last years. Afterwards, ap-proximately one half of them start their studies at a university or an institute of technology.
2. Third Level Education in Ireland
There are four universities in Ireland. The oldest one was founded in 1592. It is the Dublin University, well known under the name Trinity College. Degrees from the Dublin Institute of Technology are also recognized by this school. The NUI has been existing since 1908 and still younger are the University of Limerick, oriented technically, and the Dublin City University with the St. Patricks Teacher Training College of Drumcondra.
NUI is the largest university in Ireland. Its only central institution is the Senate, whose sole authority is to elect an External Examiner for the examinations of programs in the responsibility of each department of the following four parts of NUI:
NUI Cork, former University College Cork and Queens College; founded in the year 1850.
NUI Dublin, former names have been the Catholic University, Royal University and University College of Dublin consecutively; it exists since 1850.
NUI Galway, former University College Galway and Queens College; founded in 1850.
NUI Maynooth, former St. Patricks College, Maynooth appeared in 1795.
All these schools received the official statute of a university in the year 1997. Before, the first three have been the so-called Constituent Colleges of NUI and the name of the predecessor of NUI Maynooth was a Recognized College of the St. Patricks College, Maynooth. The meaning of this name was that examinations at this school have been recognized as the examinations at NUI.
Programs for the Bachelor degrees at Irish universities take three or four years. They can be prolonged to programs for the Master degree by one or two years. The graduates from secondary schools can also study at one of twelve Irish institutes of technology (former regional technical colleges), where the programs are from two to four years long.
III. The St. Patricks College, Maynooth
1. History
The St. Patricks College, Maynooth was founded in the year 1795 on the basis of a decision of the National Parliament of Ireland in Dublin. From beginning, Arts, Science, Philosophy and Theology have been taught at this school. In 1896, the Pontifical University was created by the theological and philosophical programs. Examinations of several programs of the St. Patricks College, Maynooth have been recognized by NUI since 1910. In this way, a new school called the Recognized College of the St. Patricks College, Maynooth has been established. For almost two centuries of its existence, the St. Patricks College, Maynooth was exclusively a seminary for future priests of the Roman Catholic religion. In this way, the school contributed to an increase of prestige of the Irish nationality in a hard period of its existence. Only since 1966 it admitted lay students.
2. Mathematical Education Programs at NUI Maynooth
NUI Maynooth consists of 22 departments which operate in the following faculties:
Faculty of Arts, Faculty of Science, Faculty of Philosophy, Faculty of Celtic Studies.
Up to some exceptions, teaching of the Department of Mathematics concerns the Faculties of Arts and Science. Mathematical courses particulate in sixteen Bachelor programs among which there are eleven three-year programs and five four-year programs. The collection of mathematical courses as taught in each of these programs is a suitable extension of one of the basic mathematical programs I, II, III, which we describe in what follows.
I. 1st Double Maths is an intensive one-year program which requires a very good knowledge of mathematical foundations - students with L.C. on a Higher Level C or better are permitted only. This program represents approximately one half of the students duties in the first school-year. It consists of the courses
122 Calculus in One Variable 3 units Semester I
141 Finite Mathematics (Number theory, Vectors and Geometry in 3D, Matrices, Combinatorics and Probability) 3 units Semester I
123 Calculus in Several Variables 3 units Semester II
131 Linear Algebra 3 units Semester II
151 Mathematical Computing 1 hour laboratory Semester I, II
tutorials 1 hour Semester I, II
and is closed by the examinations
E1 8 questions 1-4 (122), 5-8 (123) 100% = 6 questions
E2 8 questions 1-6 (141), 7,8 (151) 100% = 6 questions
E3 8 questions 1-6 (131), 7,8 (151) 100% = 6 questions.
In each of these exams, each student can receive at most 10% for elaboration of homeworks and 10% for successful solution of the Christmas test. He can gain or loose 10% for his work in the course 151.
II. 1st Maths Studies (school-year 1) and 2nd Maths Studies (school-year 2). Basic part of this program consists of three 3-unit one-semester lectures on Calculus, with one-hour tutorials. This three-semester program, which is standard in the USA, is extended by lectures on Linear Algebra, Statistics, Number Theory and Mathematical Computing in the second school-year. The content of the courses is broader than that in the programs I, III. It is designed in a way being suitable for those who wish to teach mathematics or to apply it. This program is taught at the Faculty of Arts exclusively as a part of combinations of mathematics with other subjects. The program 1st Maths Studies consists of the courses
103 Calculus I (Differential Calculus and Introduction to the Integral Calculus)
3 units Semester I
104 Calculus II (Integral Calculus in One Variable, Infinite Series) 3 units Semester II
tutorials 1 hour Semester I, II
and is closed by the examinations
E1 8 questions 1-8 (103) 100% = 6 questions
E2 8 questions 1-8 (104) 100% = 6 questions.
In each of the exams, every student can receive at most 12,5% for the homeworks and 12,5% for the Christmas test. The program 2nd Maths Studies (school-year 2) consists of the courses
205 Calculus III (Vectors, Analytic Geometry in 3D, Calculus in Several Variables) 3 units Semester I
222 Introduction to Statistics 3 units Semester I
213 Linear Algebra with Applications 3 units Semester II
224 Number Theory 3 units Semester II
207 Mathematical Computing 1 hour laboratory for the time of one Semester
tutorials 2 hours Semester I, II
and is closed by the examinations
E1 8 questions 1-4 (205), 5-8 (222) 100% = 3 questions from 1-4 and 3 questions from 5-8
E2 8 questions 1-4 (224), 5-8 (213) 100% = 3 questions from 1-4 and 3 questions from 5-8.
In each of the exams, every student can receive at most 7,5% for the homeworks, 7,5% for the Christmas test and 15% for the work in the course 207.
III. 1st Science General (school-year 1) and 2nd Science General (school-year 2) is the basic mathematical program of the Faculty of Science. 1st Science General is compulsory for all Bachelor programs at this faculty and 2nd Science General is compulsory for the one-subject programs in Mathematics and Computer Science as well as for all combinations of Mathematics with another subjects. This program represents at least one quarter of the students duties in school-year 1 and one third of their duties in school-year 2. The program 1st Science General (school-year 1) consists of the courses
101 Differential Calculus in One Real Variable
2 units Semester I
111 Vectors and Matrices 2 units Semester I
102 Integral Calculus 2 units Semester II
112 Introduction to Probability 1 unit Semester II
105 Number Theory (optional) 1 unit Semester II
tutorials 1 hour Semesters I, II
and is closed by the examinations
E1 8 questions 1-4 (101), 5-8 (102) 100% = 6 questions
E2 9 questions 1(101-compulsory), 2-5(111), 6,7(112), 8,9(105)
100% = 6 questions.
In each of the exams, every student can receive at most 12,5% for the homeworks and 12,5% for the Christmas test. The program 2nd Science General (school-year 2) consists of the courses
203 Multivariate Calculus 2 units Semester I 1 unit Semester II
211 Linear Algebra 2 units Semester I
220 Introduction to Statistics 2 units Semester II
207 Mathematical Computing 1 hour laboratory for the time of one Semester
204 Infinite Series (optional) 1 unit Semester II
tutorials 1 hour Semesters I, II
and is closed by the examinations
E1 8 questions 1-6 (203), 7,8 (204) 100% = 6 questions
E2 8 questions 1-4 (211), 5-8 (220) 100% = 3 questions from 1-4 and 3 questions from 5-8.
In each of the exams, every student can receive at most 6,25% for the homeworks, 6,25% for the Christmas test and 12,5% for his work in the course 207.
Exact contents of all the courses as mentioned above, mathematical Bachelor programs which are extensions of the programs I, II, III as well as an overview of the prescribed and recommended reading texts can be found in [8].
The Master programs of the Department of Mathematics are extensions of the Bachelor programs. They are the same for the Faculty of Arts and Faculty of Science and require at least one year of intensive full-time study. Students who prepare their own theses need two years for this degree as a rule. The postgraduate doctoral mathematical programs require at least three years of full-time study and research.
IV. Education at Irish Universities
1. Teaching
Each department at an Irish university publishes a yearly information for students - see [8] as an example - containing content of programs, names of their co-ordinators, prescribed and recommended textbooks and a list of teachers with their pedagogical orientation and a brief scientific characteristics.
As it is obvious from Chapter III, Sect.2, lectures are the dominating form of teaching. Because in the described programs there is no sufficient amount of time for a systematic practice of the theory presented at lectures, the lectures have to be very easy to understand and the role of an independent study of lectures and textbooks prescribed or recommended by the department is very important. As far as I could see, the prescribed textbooks are well suited for the study of undergraduates. Every week, the students are required to give over one homework. The questions in the homework are the same for all students. For the first-year programs, the questions are prepared by co-ordinators of the programs. Already at the beginning of the academic year, students can buy the list of homework papers for a symbolic fee. This Problem Book - see [10,12,13] - includes also the deadlines for the transmission of homeworks and requirements concerning their form. In the higher school-years, there is usually one lecturer for the whole program and he announces the homework questions during the semesters. Beside lectures, students take part in small working groups (at most 15 persons) called tutorials. In 2nd Maths Studies, tutorials take two teaching hours a week. In all other programs, there are one-hour tutorials. Their attendance is compulsory and the content depends on the tutor (the conductor of tutorials) to a large extent. Most often, the tutor solves problems of the same types as in the homeworks or examination papers in collaboration with students and answers questions concerning the content of lectures. The tutor corrects the homeworks and returns them to the students of his group. The tutor sends those students who demonstrate serious drawbacks in the secondary-school mathematics to the so-called foundations. This is a one-hour course, in which students study the basic mathematical topics from the secondary school by means of the book [14]. Before Christmas, students pass the so-called Christmas test. At this time, students who have not been attending tutorials regularly receive a cautionary letter from the department to their home addresses. This develops a certain pressure in favour of improving the systematic work of students during the school-year. At the end of Semester II, the tutor writes a final report containing an overview of the presence of students in tutorials, results of the solutions of homeworks and the result of the Christmas test as well as his opinion whether each of the students will be successful at the examinations or not. Besides tutorials, students have the opportunity to consult the content of lectures
in the consultation hours of their lecturer or tutor,
in the consultation hours of a consultant, who does not take part in the teaching.
The last possibility has been given to the students because they had made use of the consultation hours of their teachers in a minimal extent. The teaching of the course Mathematical Computing takes place in a computing laboratory. Also the work of students in this course is under control and evidence.
Examinations
The department prepares a first version of the examination paper and sends it to the External Examiner who either approves its content or suggests certain changes of it. The final version of the examination paper hands the department over to the Supervisor of Examinations. Several examples of used examination papers are included in [9,11]. The Supervisor of Examinations determines the time and place of the exam, admissible maximal amount of students and publishes deadlines as well as other rules for applications. He makes an evidence of the applications and assures the correct course of the exam in collaboration with the Examination Office. The examination is a written one only and usually takes three hours. Students have to obey exact rules and are under strict control. One supervisor is responsible for 8-10 students, all papers are stamped with a date of the exam etc. Breaking of these rules may have very unpleasant consequences for the student. The students products go back to the department, whose teachers correct them and give these corrected products to the External Examiner. After his positive approval, the department publishes the results. The Supervisor of Examinations prepares a complete overview of the results for the Examination Board, among whose members representatives of all departments appear. This Examination Board decides which students finish their studies in the given academic year successfully. The External Examiner prepares a report concerning all examinations in the academic year for the Senate of NUI. If any problems appear, the Senate asks the President of the university for an improvement and the President informs Senate about the result of this improvement. In this way, the institution of an External Examiner controls the level of degrees offered by the Irish universities. Every examination has one date between 15 May and 6 June and another date in September for repeating. Each admission to an examination is charged.
In each examination paper, the number of questions is determined which students can choose and whose correct answers are evaluated by 100(. A student is successful if he receives 40% for his answers at least. About one week after the date of every examination, the department gives the students an opportunity to look at their corrected products and to consult the resulting number of points. If a student is not satisfied with the correction of his product, he can write a requirement for an explanation and discussion concerning the result of the exam with the department staff. This service is charged.
V. Mathematics in the Civil Engineering Programs
In this Chapter, we describe the extent, content and forms of examinations of the four-years Bachelor civil engineering programs at NUI Dublin, University of Dublin and of the first four years of the Master civil engineering program at TU Brno.
1. School of Engineering of NUI Dublin
The School of Engineering is a part of the Faculty of Engineering and Architecture. Students with L.C. on the Higher Level B or better are admitted to the study of engineering at NUI Dublin only. There is one program in the first school-year common for all students of engineering. After this first year, students choose one of the following disciplines with the following numbers of places:
Agricultural and Food Engineering 20 places,
Chemical Engineering 36 places,
Civil Engineering 50 places,
Electronic and Electrical Engineering 77 places,
Mechanical Engineering 66 places.
In what follows, we describe mathematical courses from the common program in the year 1 and from the civil engineering program in the years 2, 3, 4.
1st year
EM.1.01 Mathematics 4 units, each unit 25 lectures (12 credits)
Unit 1: Sets, functions, continuity, differentiation, curve sketching, optimization.
Unit 2: Definite and indefinite integration, techniques of integration, applications of integration, first and second order ordinary differential equations.
Unit 3: Infinite series, power series, Taylors theorem and series, partial differentiation of functions of two or more variables, Lagrange multiplier method.
Unit 4: Euclidean 3-space, inner products, cross products, complex numbers, determinants, matrix algebra, simultaneous linear equations, eigenvalues.
tutorials 2 hours whole year (optional)
This course is closed by two written 3 hour examinations at the end of Semester II. Both examination papers consist of 10 questions. Students can gain 90( for 7 complete answers and 10( for a successful Christmas test.
2nd year
EM.2.01 Mathematics 4 units (8 credits)
Unit 1: Vector spaces, basis and dimension. Linear transformations and matrices. Change of basic matrices. Reflection and rotation matrices. Orthogonal and perspective projections. Further theory of determinants. Row, column and determinantal rank of matrices and their equivalence. Nullity. Systems of equations. Eliminants. Resultants. Sylvesters determinantal criterion for two polynomials to have a common root. Discriminants. Further properties of eigenvalues and eigenvectors.
Unit 2: Symmetric bilinear forms. Quadratic forms. Positive definitness. Inner products. Orthonormal bases. Gram-Schmidt process. Diagonalizability of matrices. Preservation of the characteristic polynomial. Eigenvalues determinant and trace under similarity. Principal axis theorem. Classification of quadratic forms. Simultaneous reduction of a pair of forms, one being positive definite.
Unit 3: Functions of two or more variables. Graphs, contours, continuity. Partial derivatives. Linear approximations. Tangent planes and normals to surfaces. Differentiability. Gradient, directional derivative, grad, div and curl. Laplacian. Taylors theorem for functions of two or more variables. Critical points and their classification. Hessian matrix. Lagrange multipliers. Line integrals, introduction to double integrals and Greens theorem.
Unit 4: Counting procedures. Probability spaces. Events, conditional events, independent events. Bayes theorem. Random variables. Density and distribution functions. Mean and variance. Basic discrete and continuous distributions: Uniform, binomial, geometric, Poisson, exponential and normal. Random samples. Confidence intervals. Chi-square distribution. Students t-distribution. Use of tables. Hypothesis testing. Engineering examples.
tutorials 1 hour whole year (optional)
This course is closed by two written 3 hour examinations at the end of Semester II. Each exam consists of 10 questions and students can gain 100( for 6 correct answers.
3rd year
MP.3.01 Engineering Computations 1 unit (2 credits)
Error analysis, numerical solution of algebraic and transcendental equations, matrix inversion, determination of eigenvalues and eigenvectors, numerical differentiation and integration, application of finite difference methods to ordinary and partial differential equations, interpolation and sampled data and finite element techniques. Non-linear optimization.
EM.3.01 Mathematics [LT-FS-CV or CofV] 1 unit (2 credits)
Laplace transform (LT). Heavyside-step and Dirac-Delta functions, convolution, solutions of differential equations, engineering examples.
Fourier series (FS). Hilbert spaces, partial differential equations (wave and heat), engineering applications. Introduction to calculus of variations (CofV). OR
Introduction to theory of functions of a complex variable (CV). Cauchy-Riemann equations. Harmonic functions. Conformal mappings. Cauchy integral formulae.
EM.3.02 Mathematics (Integral Calculus) 1 unit (2 credits)
Further advanced calculus. Scalar and vector fields over curves and surfaces. Grad, div and curl. Change of variable. Curvilinear coordinates. Integration over domains, curves and surfaces. Divergence theorem. Stokes theorem. Applications.
MP.3.02 Mathematical Physics (Differential Equations) 1 unit (2 credits)
Ordinary differential equations. Isoclines. Linear systems of first order. Phase plane. Singular points. First order p.d.e. How they arise. Sketching characteristics. Solutions. Second order p.d.e. How they arise. The Cauchy problem. Characteristics. Difference schemes for Laplace. Wave and diffusion equations. Motivation for different initial or boundary value problems. Properties of solutions.
There is one written 3 hours examination of the course MP.3.01 and one written 3 1/2 hour examination of the courses EM.3.01, EM.3.02, MP.3.02.
4th year
Mathematical courses EM.4.01 and EM.4.02 belong to twelve one-year, one-hour courses, two of which has each student to choose. The content of these mathematical courses is declared by the Department of Mathematics at the beginning of every academic year. Students choose the mathematical courses exceptionally.
Homeworks are ordered at the lecturers in form of numbers of problems from the Problem book - see [12,13]. The purpose of the homeworks is to help students in their preparation for examinations. They are neither controlled nor corrected systematically.
The dates of examinations, whose number is 9,11,9,7 in the year 1, 2, 3, 4 respectively, are concentrated to the time from 15 May to 6 July. There is one date for each exam. Generally, a student is successful at an exam if he obtains 40(. If he is not successful, he has the opportunity to repeat in one date during September. The number of credits available for the mathematical courses, theoretical courses (they include mathematics, physics, chemistry and geology) and all courses can be found in the following table:
yearmathematical coursestheoretical coursesall courses11234602886038126040060
Table 1: Number of credits at NUI Dublin.
University of Dublin
Engineering programs belong to the Faculty of Engineering and Systems Sciences. Students with L.C. on the Higher Level C or better are admitted to the study of engineering at the University of Dublin only. During the first two years, the program consists of basic courses in engineering and is common for all students of engineering. After the completion of these two years, students choose one of the following special programs:
civil, structural and environmental engineering
mechanical and manufacturing engineering
electronic engineering
computer engineering
electronic/computer engineering
electronics/optoelectronics
There are about 240 engineering students in all these special programs. We describe the basic mathematical courses from the years 1,2 and the mathematical part of the program of civil, structural and environmental engineering in the year 3.
1st year
1E1 Pure mathematics 2 units (9 credits)
Linear algebra, discrete mathematics, series and limits, differentiation and applications, integration and applications.
tutorial 1 hour whole year
1E2 Applied mathematics 2 units (9 credits)
Vectors, vector analysis, concepts, laws, ordinary differential equations, kinematics, dynamics, oscillatory motion.
tutorial 1 hour whole year
2nd year
2E1 Pure mathematics 2 units (8 credits)
Calculus of functions of several variables, matrix analysis, eigenvalues and eigenvectors, further ordinary differential equations, Laplace transforms, vector analysis.
tutorial 1 hour whole year
2E2 Applied mechanics 2 units (8 credits)
Dimensional analysis, damped harmonic oscillators, particle motion in two and three dimensions, Lagranges equations, applications to scientific and engineering problems.
tutorial 1 hour whole year
3rd year
3E1 Mathematical methods 2 units (10 credits)
Linear Programming (6 lectures), Non-linear Programming (6 lectures), Dynamic Analysis (6 lectures), Variational Calculus (4 lectures), Fourier Series (10 lectures), Partial Differential Equations (12 lectures).
tutorial 1 hour whole year
3E2 Numerical Methods 1 unit (5 credits)
Errors, roots of equations, differential equations, numerical integration, interpolation and curve fitting, numerical linear algebra.
3E3 Statistics 1 unit (5 credits)
Elementary probability. Binomial distribution. Normal distribution. Quality assurance. Simple reliability models. Statistical estimation of means, proportions. Interval estimates. Simple linear regression.
Examinations in the first and fourth years are held within a three-week period beginning in the middle of May. At this time, one date for each course is available. During September, each exam is repeated for students not having been successful in spring. In the second and third years, examinations are held partly at the end of Semester I and partly at the end of Semester II. Again, each exam is repeated in September for students not having been successful during the school-year. At these examinations, 40( are required for success and no points can be obtained for the work during the year. The number of credits available for mathematical courses and theoretical courses in comparison with the complete amount of credits can be found in the following table:
Yearmathematical coursestheoretical coursesall courses11836632161664320206040060
Table 2: Number of credits at the University of Dublin.
3. The Faculty of Civil Engineering of TU Brno
At this faculty, there are about 1,000 students in the first year. Mathematical prerequisites of the students of civil engineering programs of TU Brno are on a relatively low level. Students can choose one of seven different civil engineering programs, one of which is taught in English only. The courses in the first two years, where the basic mathematics is taught completely, are common for five of these programs. We describe the common mathematical courses from the years 1,2.
1st year
Mathematics (1) 3 units Semester I (42 lectures up to 45 min)
Matrix operations, Gauss elimination, differential and integral calculus of functions in one variable, applications.
practice 3 hours Semester I
Mathematics (2) 3 units Semester II
Inverse matrices, determinants and their applications, eigenvalues and eigenvectors of matrices, vector operations and analytic geometry. Differential calculus of functions in several variables. Differential equations of order 1 and linear differential equations of higher orders.
practice 3 hours Semester II
2nd year
Mathematics (3) 2 units Semester I
Fourier series and the Fourier method for partial differential problems. Integration of functions in two and three variables. Integration over curves and surfaces. Applications.
practice 2 hours Semester I
Mathematics (4) 3 units Semester II
Random variables and probability, independent events, basic discrete and continuous distributions. Hypothesis testing and parametric tests. Numerical methods for the solution of one equation in one real unknown, eigenproblems of matrices (the power method), solution of systems of linear and nonlinear equations by iteration. Interpolation and approximation of functions. Numerical integration and differentiation. Numerical solution of boundary value problems for ordinary differential equations of second order.
practice 1 hour Semester II
Because of a large number of students, there are several parallel lectures read by different teachers. Each semester consists of a fourteen-week period of teaching followed by a five-week period reserved for examinations. During the second period, students try to pass through examinations from the courses taught in the first period; the number of exams is five at most. If a student fails in an exam, he has two additional attempts which he can choose from numerous dates as offered by the Department of Mathematics during the whole acad-emic year. The examinations are written only and students can work for two hours. There is no choice and for a success, students have to answer at least 50( of questions correctly. 70( can be gained for the solution of computational problems and 30( for the theoretically oriented questions. The numbers of credits available for mathematical and theoretical cours-es are compared with the complete amount of credits from the first four years in Table 3.
yearmathematical coursestheoretical coursesall courses11326602812603006040060Table 3: Number of credits at the Faculty of Civil Engineering of TU Brno.
VI. Comparisons and Conclusions
1. Department of Mathematics of NUI Maynooth
In order to get information necessary for this project, I visited the Department of Mathematics of NUI Maynooth. My only preliminary knowledge of this academic working place originated from its presentation on internet. Although there are no engineering programs at NUI Maynooth, the Department of Mathematics of NUI Maynooth appeared to be a very good choice for the following reasons:
The staff of this department dedicates much effort to the improvement of quality of teaching. This can be observed during extensive pedagogical discussions inside of the department and also under the following circumstances which are not usual at all university departments:
A large number of own textbooks offered to the students - see [19-22].
Teaching of the course Mathematical Computing.
The course of Foundations, organized for first-year students with weak knowledge of the basic secondary-school mathematics.
Appointment of an independent consultant for the students of mathematical courses.
Each of the teachers has been willing to answer my questions anytime.
Discussions with the Head of Department, Professor Anthony G. OFarrell, M.Sc., Ph.D., M.R.I.A., devoted to the pedagogical and scientific work of teachers as well as to the conduction of academic mathematical working places, have been very inspiring. Prof. OFarrell offered myself very good working conditions, strong support and extensive information. He also initiated my consultations at the universities of Dublin.
I hope that connections of the Department of Mathematics and Descriptive Geometry of TU Brno with this working place, which is of a high pedagogical and also scientific quality, will continue in a suitable form. For example a participation of young teachers from TU Brno in the teaching of mathematical courses at NUI Maynooth for a suitable period of time would bring them valuable experience in the area of both pedagogical and scientific work.
2. Transition to Third Level
Thanks to the L.C., Irish universities have a guaranteed objective information about the applicants knowledge of all the courses which he or she attended in the higher level of secondary school. Up to some exceptions including applicants from abroad and adults, Irish universities accept new students on the basis of L.C. and do not organize any own entrance examinations.
In the Czech Republic, secondary schools are finished by leaving examinations from the Czech language, a foreign language and from two another subjects at least. There are no central instructions concerning the level of knowledge related to the marks. For the sake of objectivity, heads of the leaving examination commissions are teachers from other secondary schools. Under the communist government, topics taught at the secondary school have been prescribed completely. Hence the knowledge of graduates from all secondary schools has been on the same level and the marks obtained for the leaving examinations have been comparable. But after 1989, secondary schools have changed essentially. In agreement with their pupils, they have concentrated more on humanitary than on natural-scientific or technical education. For this reason, the number of teaching hours devoted to foundations of the theoretical basis has decreased in general, but with strongly varying intensity. Hence the level of knowledge of the theoretical basis has decreased and there exist big differences between knowledge evaluated by the same marks at different secondary schools. Therefore, Czech universities have to organize their own entrance examinations.
In the last year of secondary school, pupils pass through a so-called state comparative examinations from ten subjects. The examination papers are identical for all pupils and their results are corrected by a computer. The purpose of state comparative examinations is to compare the quality of Czech secondary schools.
Hence in the Czech Republic, knowledge of graduates from secondary schools applying for a university is examined in the following three ways:
by the leaving examinations,
by the state comparative examinations,
by entrance examinations at universities.
Description of systems of entry to third level in eight other countries is available in [5].
Conclusions. In Ireland, transition to the third level is optimal from the economic point of view, because one exam from one subject is sufficient, and also from the point of view of students, because their knowledge is examined objectively in a long period of time under conditions which minimise the danger of stress. On the contrary, graduates from Czech secondary schools have to pass through three different procedures for the examination of their knowledge if they apply for a university. This is not only expensive for the state but also unfair to the students because they are accepted to the universities on the basis of entrance exams basically. The entrance examinations give an incomplete information depending on the frame of students mind in one moment only.
I propose to announce the results of state comparative examinations to the Czech universities in order to accept students on the basis of this information. Publication of the comparison from [5] extended by the situation in the Czech Republic could contribute to a reasonable solution of the problem concerning the transition to third level in the Czech Republic.
3. Methodology of Teaching at the Irish and Czech Universities
There is a big difference between the Czech and British systems of education. In this comparison, we concentrate to the level of basic engineering mathematics. We first point out concrete differences from the previous chapters and then make conclusions.
a) Each university department in Ireland provides students with all the relevant information. There is no information on this level available in the Czech universities.
b) The extent of time devoted to tutorials (practice) is much smaller in Ireland than in the Czech Republic. Comparison of the quotient
time for tutorials (practice)
((((((((((((((((((
time for lectures
valid for engineering mathematical courses of school-years 1,2 at NUI Dublin, University of Dublin and TU Brno can be seen in Table 4. The Irish tutor helps students to overcome the
yearNUI DublinUniversity of DublinTU Brno10.50.5120.250.50.6
Table 4: The time for tutorials (practice) versus the time for lectures.
most serious problems in their study. The teacher in the Czech practice goes through all the typical problems and explains their solution by means of the theory presented in lectures. Both in the Irish tutorials and in the Czech practice, teachers struggle for an active role of students.
c) The homeworks are identical for all students of the same program in Ireland. They are prepared by the co-ordinator and published in advance in a form of a Problem Book. The homeworks given to the Czech students are in responsibility of the conductor of practice usually.
d) The matter studied by the Irish students in a school-year is examined by ten examinations on average. One fixed date during the school-year is determined for each exam. Usually, all the dates are concentrated in a three-week period beginning in the middle of May. If a student fails in a moderate number of exams, he obtains an opportunity to attend repeating in September. Questions in the Irish examination papers require neither complicated computations nor exact formulations of definitions or theorems. Instead, students have to demonstrate the ability to formulate typical questions in a correct mathematical form, to understand the basic ideas from lectures and to apply them to simple concrete situations. Typically, students have a choice; it means that each student can select a prescribed number of questions such that 100( can be obtained for their correct answers. Generally, 40( are sufficient for a success.
On the contrary, Czech students can arrange the dates of examinations in a way enabling them a complete preparation immediately before each exam. If they are not successful, they can choose two additional attempts from a large number of dates during the school-year. The number of examinations at the Czech universities corresponds to the number of examinations at the Irish universities. At the Czech mathematical examinations, there is no choice, the lower bound for a success is higher than in Ireland and the examinations are either written only or both written and oral. Usually, theoretical questions play a role which is more important than in Ireland.
In Ireland, a big attention is devoted to the preparation of examination papers, to the correct pass of the exams and to the correction of the students products. Teachers are engaged in the professional parts of examinations only. Moreover, through the institution of an External Examiner, Irish universities guarantee a systematic control of the quality of examination papers as well as of the products of students. Czech teachers are responsible for the complete pass of examinations. In the Czech universities, there is no control of the adequacy or quality of examinations.
Conclusions. All the information from above illustrates that the Czech and British systems of education are essentially different. Their basic components result from a long tradition on both sides and are connected with a specific system of secondary education. We point out some of the basic differences.
In Ireland, students are better informed about the programs they study and about their teachers. The fact that the extent of Czech practice is much greater than the extent of Irish tutorials is essential. In the Irish lectures, more attention has to be devoted to computations because there is not enough time for systematic explanations in the tutorials. According to my experience, presentation of computations in a lecture cannot substitute an active work of students under the teachers direction. Especially in the first year, the Czech practice is of an essential help to the students. The quality of homeworks given to the Irish students is guaranteed by the co-ordinator of each course. Unlike the Czech students, the Irish students cannot leave the preparation for an examination to the time immediately before the exam because they pass through three or four exams within a week. This fact implies that
the Irish students have to study systematically during the year and, therefore, the knowledge remains longer in their minds and
criteria for a success at the Irish exams are weaker than those at the Czech exams.
I highly appreciate the British institution of External Examiner. It is a simple and therefore cheep system which offers the universities a satisfactory outline of the quality of examinations of different academic departments active in the same area and enables them to influence the examinations.
I can see in general, that the Irish universities have a high respect to the activities of teachers - they do not let them make jobs which do not exploit their qualification - and that the teachers have a high respect to the students in the sense that they devote much more attention to the precise preparation of lectures, homeworks and examinations than the Czech teachers do. Instead of non-effective contact with students, they devote more care to an increase of the efficiency of their pedagogical activity.
Mathematics and Theoretical Basis in the Civil Engineering Programs
In this section, we compare
mathematical prerequisites of incoming students,
relative extent of mathematical courses in civil engineering programs and
content of mathematical courses in civil engineering programs
between NUI Dublin and University of Dublin on one hand and TU Brno on the other hand. In some cases, we make conclusions from these comparisons.
a) Engineering programs belong to the most prestigious in Ireland. In correspondence to this fact, very strong requirements on the mathematical prerequisites from the secondary school, namely L.C. higher level B or better and L.C. higher level C or better have to be fulfilled by the students accepted to the study of civil engineering at NUI Dublin and at the University of Dublin respectively. On the contrary, engineering programs at the Czech universities have a low prestige. Approximately one half of applicants for the civil engineering programs come from secondary schools oriented to the preparation of technicians for the civil engineering industry. At these schools, the extent of teaching foundations of theoretical basis is very low. The remaining applicants are graduates from grammar schools which provide their pupils with a better prerequisites for the theoretical basis. But the ambitious graduates from grammar schools prefer the study of a lot of other subjects to the study of engineering.
b) In Table 5, we compare relations of the extent of mathematical courses to the extent of the whole civil engineering programs (in () at NUI Dublin, University of Dublin and Faculty of Civil Engineering of TU Brno. In this table, the same comparison of the extent of theoretical courses including mathematics, physics, chemistry and geology, appears. These comparisons have been computed from the tables 1, 2, 3 in a straightforward way. According to these tables, intensity of teaching mathematics in the first two years of the compared universities is on the same level. This intensity is preserved in the third year at the Irish universities, but no mathematics is taught in the third year at TU Brno.
Mathematicstheoretical coursesNUI Dublin11,67(22,5 (University of Dublin21,86( 29,15(TU Brno8,75( 15,83(
Table 5: The relative part of credits for mathematics and for theoretical basis.
c) As we can see from Table 5, the relative part of teaching mathematics at TU Brno is much lower than that at Irish universities under comparison. Similar relation concerns the content of the mathematical courses. All the mathematical topics taught at TU Brno are included in the mathematical courses at the Irish universities under comparison. The following topics are taught both at NUI Dublin and University of Dublin, but they are not taught at TU Brno.
Linear algebra, especially transformation of coordinates in vector spaces, diagonalization of matrices and applications.
Infinite series.
Laplace transform and its applications.
Introduction to the variational calculus.
Extensive area of the derivation, physical meanings, basic properties, classification and traditional mathematical methods for the solution of partial differential equations.
Differences in the content of basic mathematics between NUI Dublin and University of Dublin can be found in Chapter V. One can see there, that at NUI Dublin, much attention is paid to linear algebra and at the University of Dublin, linear and non-linear optimisation are taught.
Literature
NUI Maynooth Calendar 1997-1998. National University of Ireland Maynooth 1997.
The University of Dublin Calendar 1997-8 - Part 1. Trinity College, Dublin 1997.
3. Engineering (Session 1998/99). NUI Dublin 1998.
Information for Applicants to Undergraduate Degree Courses (98/99). NUI Dublin 1998.
Commission on the Point System, Consultative Process - Background Document. The Stationery Office, Government Publications Office, Dublin 1997, pp 67-77, 171-174.
Memorandum of Faculty of Engineering and Architecture. NUI Dublin 1998.
Lecture Time Schedule - School of Engineering 1997/98. NUI Dublin 1997.
Student Handbook (Academic Year 1997/98). Department of Mathematics, NUI Maynooth 1997.
Examination Papers 1997. Department of Mathematics, NUI Maynooth 1997.
1st Science General Problem Book 1997/98. Department of Mathematics, NUI Maynooth 1997.
First Year Summer Examinations (5 years). Department of Mathematics NUI Dublin 1998.
First Year Problem Book. Department of Mathematics NUI Dublin 1997.
Second and Third Year Problem Book for Engineering Students. Department of Mathematics NUI Dublin 1998.
A. Croft, R. Davison: Foundation Maths, Second Edition. Addison-Wesley 1997.
H. Anton: Calculus with Analytic Geometry, 6th Edition. John Wiley and Sons, New York 1997.
H. Anton, Ch. Rorres: Elementary Linear Algebra, Applications Version, 7th Edition. John Wiley and Sons, New York 1994.
D. W. Lewis: Matrix Theory. World Scientific, Singapore 1991.
G. Polya: How to Solve It, 2nd Edition. Penguin Books 1990.
J. Love: Introduction to Mathematical Computing. Lecture Notes of the Department of Mathematics, NUI Maynooth 1998.
F. OCairbre: Differential Calculus and Integral Calculus. Lecture Notes of the Department of Mathematics, NUI Maynooth 1998.
A. G. OFarrell: Infinite Series. Lecture Notes of the Department of Mathematics, NUI Maynooth 1998.
D. B. Redmont: Linear Algebra Notes. Lecture Notes of the Department of Mathematics, NUI Maynooth 1998.
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