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Project Details


112318-CP-1-2003-1 -IT-COMENIUS-C21


Project span

Project Coordinator

CAFRE Centro di Ateneo di Formazione e Ricerca Educativa
Università di Pisa

Contact person
Prof. Franco FAVILLI


Project Partners
(CZ) Univerzita Karlova v Praze

(DK) Skårup Seminarium

(FR) Institut Universitaire de Formation des Maîtres de Créteil

(IT) Università degli Studi di Firenze

(IT) Università degli Studi di Siena

(SK) Univerzita Mateja Bela

Teacher Training Systems
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Mathematics in the educational system

The role of mathematical education is very important. Mathematics should be considered as one part of human culture. Mathematics develops independent thinking of pupils by problem solving and logical thinking through its deductive build-up. With introducing concepts like variable, expression, equation, inequality, function etc., mathematics develops pupils' abstract thinking and contributes to their general intellectual development. Geometry encourages pupils to be precise through geometric constructions and helps them to develop logical thinking by proving correctness of constructions and by verifying correct number of solutions to a given problem. Its role is irreplaceable. It also helps pupils to develop spatial imagination by solving problems from solid geometry and functional thinking by solving construction problems based on congruent mappings and homothety. Mathematics also brings pupils to understanding of the concept of infinity both in the context of a sequence and its limit and in the context of geometry. In topics dedicated to combinatorial analysis, probability and statistics, mathematics contributes to the development of combinatorial and probabilistic thinking of pupils and encourages them to solve problems taken from life (transport, management, theory of prognoses). An important role is played by such problems that are used by teachers for explaining strategy of solution and process of converting a real problem into a mathematical one and vice versa and for leading pupils to a correct evaluation of results and their number and possibilities of using computer technology. Besides, mathematics is an important tool for other subjects.

Teacher training in mathematics

The future primary school teachers (from the first till the fifth grade) take a four-year study. They get acquainted with all fields of study taught at this stage of schools. Their knowledge of mathematics involves elementary arithmetic, set theory, relations, mappings, algebraic structures, mathematical logic, essentials of geometry etc. Besides mastering these mathematical theories they are instructed about methods and efficient procedures of teaching.

The future lower and upper secondary school teachers (from the sixth till the twelfth grade) take a five-year study. Mathematics is combined with other subject (e.g. physics, descriptive geometry, computer science, chemistry, biology, geography, arts, foreign languages). Faculties gained a certain autonomy after 1989, which influenced the non-unified system of teacher training. Most faculties have separated the lower and upper secondary level, some of them offer either lower secondary level or lower and upper secondary level.

The traditional way of training prospective teachers of mathematics emphasized scientific knowledge. Nowadays, facing the necessity of preparing teachers for a new flexible school system, we concentrate much more on the didactical aspects of teacher training.  The starting point is to state which parts of mathematics are necessary for future mathematics teachers.

To maintain the balance between humanization of education and the growth of demands for mathematical literacy in both children and adults, the graduate secondary school teacher should

  • have good and systematic knowledge of mathematical disciplines taught at basic or secondary school, esp. of algebra, geometry and calculus; basic knowledge of psychology, mathematical education and pedagogy; the extent, depth and structure of knowledge sufficient to encourage and fix self-confidence as an expert and teacher; understanding of mathematics as the integrator suitable and necessary for modeling and solving problems from life;
  • know how to continue and apply knowledge to solve standard and more difficult mathematical problems, be able to find the possibility of modeling mathematically a real problem, to choose the appropriate mathematical method for solving it and to interpret correctly the solution;
  • have basic knowledge of function and use of computers, of algorithmisation of problems and programming and know how to use them in teaching;
  • have basic knowledge of logical structure of mathematics and be able to use it;
  • be acquainted with the structure of school mathematics, together with connections to previous and following school levels;
  • have necessary knowledge of numbers, have functional thinking, developed geometrical plane and space imagination and have a good command over probabilistic and statistical methods;
  • be able to communicate in the oral and written form with experts as well as with pupils and to express mathematical ideas and procedures clearly and without mistakes in language;
  • realize the liaisons between the structure of mathematics as science and as teaching subject; be able to transform his scientific knowledge into his teaching in the way corresponding with his pupils' level;
  • have a basic knowledge of organization and management of the teaching procedure, diagnostics and ways of evaluation;
  • be ready to use alternative ways of teaching and differentiate between different groups of pupils;
  • have the basic survey of philosophy and history of mathematics and mathematical education;
  • understand the necessity to continue his studies in mathematics and to improve his pedagogical competence and be able to do it; follow the appropriate literature and foreign sources.
Not all faculties in the Czech Republic have moved towards this new system of teacher training, some of them stress only the pure mathematical content and do not care for the didactical aspects. Anyway, we can see the massive tendency towards creating didactical schools.

Module of the subject base - mathematics, 1st cycle (Year 1 - 3):

Elementary mathematics (52 hours)

Introduction to calculus (39 hours),
Calculus of functions of 1 variable (104 hours),
Sequences and series (39 hours),
Functions of several variables (39 hours)

Introduction to algebra (39 hours),
Linear algebra (52 hours),
Polynomial algebra (39 hours),
Algebraic structures (39 hours)

Elementary geometry (78 hours),
Conics (39 hours),
Analytic geometry (65 hours)

Computers and informatics (52 hours)

Problem solving (26 hours) 

Compulsory modules of the 2nd cycle (Year 4 - 5):

Module of Didactics of mathematics
Didactics of mathematics (104 hours),
Problem solving (26 hours),
History of mathematics (26 hours),
School practice (4 weeks)

Module of mathematics
Set theory (39 hours),
Probability and Statistics (52 hours),
Finite mathematics or Numerical methods (39 hours) 

Optional modules of the 2nd cycle (Year 4 - 5):
(8 courses to be chosen from the offer according to the student's preferences)

Module of mathematics
Differential equations, Functional analysis, Application of calculus, Number theory, Symmetries in algebra, Equations and their systems, Curves and surfaces, Geometry of Gauss plane, non-Euclidean geometries, programming, Mathematical software, Modern mathematics.

Module of Didactics of mathematics
Research in Didactics of mathematics, Diploma thesis seminar, Problem solving

Special offer in co-operation with the (on the Department of English language and literatureoffer:  CLIL - teaching mathematics in English)


Franco Favilli and Giuseppe Fiorentino, eds.

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