lasts 5 years. It is a first stage of compulsory nine-year education (named "Basic school"). Mathematics represents about 20 % of the content of education. Other subject: Czech (reading, writing, literary education), Mathematics (arithmetic, geometry), Science, Foreign language (English, German, Russian, Spanish from the 3rd Grade), Technology, Art, P. E., Music. One teacher in 1st and 2nd Grade usually teaches all subjects. Later are some subjects exempt Czech and Mathematics can be taught by other teacher (who is more experienced) according the conditions in concrete school.
Reading and writing of numbers up to 1 million. Comparison of numbers.
Decimal notation.
Mathematical notations and expressions. Mathematical symbols.
Fractions (halve, quarter, third).
Basic arithmetical operations. Mutual relationships between operations.
Properties of arithmetical operations. Real meaning of addition, subtraction,
multiplication and division. Addition, subtraction, multiplication and division of natural numbers by heart and written with paper and pencil.
Round off. Estimations of results and check of correctness of calculations.
Simple and complex word problems.
Orientation in plane and space.
Basic plane's and shape's geometrical units, their elementary properties.
Relations between geometrical units (position, size, order). Perimeter and area of plane's geometrical units.
Measurement and estimation of the length. Units of length and area. Relation between the units ( transfer of units).
Drawing and picturing of geometrical shapes.
Numbers' and geometrical patterns.
Sorting a set of objects.
Collecting, recording, representing and evaluating of data. Record a set of data in a table and extract specific pieces of information from tables, lists, diagrams and graphs.
Orientation in the plan, calendar and time table (schedule).
Investigate number of possibilities.
Drawing according a model.
Solving of practical problems from real situations.
Evaluation of situations, selection of solving method.
| Grade | Number | Arithmetical operations | Geometry |
|---|---|---|---|
| 1. | to at least up to 20. Write the digits. Use numbers for expression of quantities and order. Compare numbers on the base of quantity or order. Solve the word problems leading to addition or subtraction and relationship for n-more/n-less in the field up to 20. | Know and use addition and subtraction facts up to 10 (including 0). Add or subtract where the numbers involved are not greater then 20 (type: 12 + 7, 19 - 7). Comutativity of addition. Signs +, - , =. | Recognise circle, sphere, square, cube and triangle. Count geometrical shapes ( ….). Orientations in space (right, in front of, ..) Smaller, bigger, equal. Abacus - illustration of the problems. |
| 2. | Count, read, write and order numbers to at least up to 100. Illustrate the concrete sets with given number of elements up to 100. Units and tens in two digits' number. Grouping units into tens. Comparison of numbers and note of the result of comparison. Marks <, >. Round off numbers up to 100. | Addition and subtraction in the field of 20. Addition and subtraction up to 100, examples: 20 + 30, 50 - 20; 20 + 5, 25 - 5; 23 + 5, 28 - 5; 28 + 5, 33 - 5. Understanding of multiplication and division. Know and use multiplication number facts in 2, 3, 4, and 5 multiplication tables. Multiplication by 10. Comutativity of multiplication. Solve word problems involving multiplication and division where one or two calculations are needed. | Abscissa, pointed line, curved line. Triangle, quadrangle, pentagon, oblong. Cylinder, ashlar. Construct buildings from cubes according a schema. Number of squares or cubes needed for building of the shape. Symmetrical shape. |
| 3. | Count, read, write and order numbers to at least up to 1 000. Deepen the idea of decimal notation and use it in comparison of numbers. | Add or subtract mentally two 2-digit numbers. Know multiplication facts up to 10 x 10. Written algorithm of addition and subtraction. Estimation and check of the results. | Buildings from cubes according the ground plan (with given number of cubes in each column). Jigsaws from squares and triangles. Drawing of abscissas with given lengths. Pyramid, cone, cube, ashlar. Vertex, side, edge, .. number at a solid. Radius and diameter of the circle. Division of the shape into parts. |
| 4. | Count, read, write and order numbers to at least up to one million. Comparison of numbers according their notation. Round of to thousands, hundreds, and tens. Whole, part, fraction, numerator, and denominator. Mark graphically the fraction of the whole. Add fractions with the same denominator. | Multiplication and division up to 100. The division with rest. Written multiplication by 2-digit numbers. Written division by 1-digit number. | A circular line - circle - drawing. Equal circles, axis of the abscissa. Line, half-line, perpendiculars and parallels. Division of the plane and circle. Perimeter and area, cm , m, dm, mm, km, l, kg, g. Views on solids. |
| 5. | Count, read, write and order numbers greater than million. Milliard. | Multiplication/division by suitable 2-digit numbers. Written division by 2-digit number. Write, model, add, subtract and round off given decimal numbers (tents, hundredths). Multiply and divide decimal number by ten. | Equilateral and isosceles triangle. Drawing with the help of circular - line. Drawing of oblong and square. |
| Using mathematics 1st - 3rd grade. | Numerical and geometrical patterns. Relationships and algorithms. Handling data. Orientation in space and time. Problems from economy (discounts, debts, savings, fraction, …) Measurement and the idea about size. Reading of the temperature. Evaluation of the situations. Investigate the number of possibilities. |
|---|---|
| Using mathematics 4th - 5th grade. | Numerical and geometrical patterns. Relationships and algorithms. Purchase (discount, without reduction). Handling data. Extract specific pieces of information from tables, yearbooks. Orientation in the space and time. Problems from economy (discounts, debts, savings, …) Measurement and the idea about size. Reading of the temperature. Evaluation of the situations. Investigate the number of possibilities. |
Primary school: In Germany, primary education comprises the grades 1 to 4 (with the exception of some experimental schools comprising grades 5 and 6 additionally). The number of lessons per week varies normally between 19 and 25 ("school hours" of 45 minutes each); sometimes supporting lessons are included in this total number of lessons per week. During third grade (the grade of the planned teaching experiment in Germany) 23 - 24 lessons per week are provided (1. grade: 19 - 20 lessons; 2. grade: 21 - 22 lessons; 4. grade: 24 - 25 lessons), and out of them normally 5 lessons mathematics teaching. The number of students per class varies between 16 and 32.
The following overview shows the distribution of content topics for the four years of primary mathematics teaching (see KM, 1985, 30f.)
| Arithmetic | geometry | Measures |
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| Arithmetic | geometry | Measures |
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| Arithmetic | geometry | Measures |
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| Arithmetic | geometry | Measures |
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Teacher training: In Germany, pre-service teachers are trained in two phases; a first phase takes place at the university and a second phase at school related seminars (the so called Studienseminare). At the beginning of the university studies the teacher students choose the school levels (primary, lower or higher secondary level) as well as their future teaching subjects. Because of big differences in different federal states of Germany we will only focus on teacher training in North-Rhine-Westphalia. In this state teacher students who want to become primary teachers have to study three subjects. Obligatory subjects are »Mathematics« and »German language«, one of which is the student's chosen main subject with a total of 42 hours (so called »Semesterwochenstunden«, that means in one of six terms an average of 7 hours per week). The second further obligatory subject has to be studied on the basis of a total of 22 hours (»Semesterwochenstunden«). Additionally a third subject (of equally 22 hours extent) can be chosen according to the offer of the university (for example arts, music, sports, …). The studies in mathematics comprise disciplinary areas of mathematics (as for instance arithmetic, elementary geometry, elements of number theory) and diddactics of mathematics (as for instance didactics of arithmetic, or geometry, mathematics related to real world experiences). When choosing mathematics as the main subject (or as the further subject) 22 - 24 (12) hours are devoted to disciplinary studies and 20 - 22 (10) hours to didactical studies.
The duration of the university studies is normally of 6 semester length with one additional semester for examinations.
Understanding of learning and teaching: Learning in general and the learning of mathematics in particular is understood as the active acquisition of knowledge and accordingly teaching as the organization of learning processes: »When mathematics learning is seen as a constructive discovery process, then in an extraordinary way justice is done to the requirements and objectives of mathematics teaching. As a consequence teaching has to be organized in a way that children get as many as possible opportunities for self acting learning in all phases of the learning process« (KM 1985, 26). Although this basic principle of organizing teaching is established in the syllabus certainly one cannot speak of its overall realization. An exemplary way of possibilities and problems of realizing this principle of understanding learning as a self acting process of childrens' activities that are guided and organized by teaching will be illustrated by the planned teaching experiment (»From informal to formal strategies for addition and subtraction (3. grade)«).
KM - Der Kultusminister des Landes Nordrhein-Westfalen (Hg.; 1985). Richtlinien und Lehrpläne für die Grundschule in Nordrhein-Westfalen: Mathematik. [Syllabus and guidelines for primary education in North-Rhine-Westphalia: Mathematics] Köln: Greven Verlag
In Italy compulsory education is from age 6 to 15. Primary education is from 6 to 11 (i.e. from grade 1 to grade 5) and is subdivided in
two cycles (grades 1-2 and grades 3-4-5).
Different time slots in different schools (the range is from full time (Mon-Fri 8,30-16,30) to partial time (Mon-Sat 8,30-12,30).
Subjects:
Language and Mathematics are explicitly acknowledged as basic instruments to read and understand reality.
Mathematics (from "Programmi Didattici per la Scuola Primaria, D.P.R. 12 Febbraio 1985)
GENERAL LINES
Mathematics is mainly considered in its cultural aspect. The introduction is very significant.
It reads: "Mathematical education helps in forming a child's thought in its various aspects, as concerns intuition, imagination, the act of planning, of making a conjecture, of reasoning and of verifying an hypothesis. It specifically develops concepts, as well as methods and attitudes which enable the children to produce skills of ordering, quantifying and measuring real events and making a critical examination of reality."
"Experience has shown that it is not possible to arrive at abstract reasoning without first passing through the observation of reality, mathematical activity, and problem solving."
The main aim of mathematical education is to train pupils in approaching and solving problems, in making suitable representations, and in interpreting and verifying results. Briefly, mathematics teaching is, as far as problem solving is concerned, a systematic and progressive development. Furthermore, the Programs recognise explicitly that language and mathematics are basic instruments of knowledge. They are to be acquired by every learner, at every level of schooling.
OBJECTIVES AND CONTENTS
A firm connection between goals and contents is evident. Some objectives are fundamental, i.e. those concerning natural and decimal numbers, calculating ability and geometric knowledge; some others are to be completely attained in later education, such as those concerning logic, probability, statistics, and computer science. Compulsory instruction is seen in a global sense.
Specific subjects are grouped according to major themes: Problems, Arithmetic, Geometry and Measure, Logic, Probability, Statistic and Computer Science.
Subdivision per year is not provided, according to a trend which is also present in the programs for secondary education.
The emphasis on problem solving is very significant as far as its application in classroom practice is concerned. Problem-solving oriented activities must not proceed episodically, but rather, in the framework of a progressive organisation of mathematical knowledge.
Traditional subjects, such as Arithmetic and Geometry, are given new emphases.
Arithmetic teaching has a fundamental role. The Programs suggest that the approach to basic arithmetical notions has to be pluralistic. For example, since many studies confirm that the concept of number is quite complicated, it seems useful to learn it by means of several procedures, i.e. counting objects, repeating number sequences, comparing numbers, measuring sets.
The Programs don't provide a hierarchy of different approaches, even if the ordinal approach seems to be pre-eminent.
Mental computation and its strategies are important in mathematics teaching, mainly in order to train pupils to use computers skilfully and not mechanically. The concept of algorithm is of great educational value; analysing situations from an algorithmic point of view provides a good training in preparation for computer procedures.
The computer is seen as a powerful tool to explore the world of numbers: however pupils must not conceive it as an omnipotent instrument to solve problems.
Attention is also focused on the following concepts: prime numbers, ordering structures, arithmetic laws, and later the concepts of ratio, fraction, directed numbers and decimal numbers.
The importance of the number line as a didactic aid is pointed out.
An important innovation of the Programs is the fundamental role of geometry in mathematics teaching. Initially, the learning of geometry must be "a progressive acquisition of skills, from achieving a sense of direction, to recognising shapes and locating objects, in general leading to an acquisition of a progressive knowledge of space which can also be reached by means of using suitable points of reference".
Learning geometry means drawing a scheme of reality, even when it changes, consequently focusing on symmetries, translations, rotations, and similitudes. Measurement is another fundamental point. It realises the connection between reality, sometimes described in geometric terms, and the world of numbers. The act of measuring contributes to the knowledge of facts and phenomena; in turn one can consider other fields in which measure plays an important role: statistics, probability, arithmetics, natural and social sciences.
The introduction of international units of measure is suggested.
Logic, Probability and Statistics are new important topics. "The teaching of logic…has to be a topic for a great deal of thought and attention, in order to stimulate a child's cognitive development."
The use of sets, and consequent graphical representations, are very different from the way they were conceived in the sixties. Sets are introduced informally, through common language; the graphical representation of sets is used only as support in various mathematical activities.
As far as probability and Statistics are concerned, the Programs suggest the development of skills dealing with the statistical representation of facts and prediction.
"Great educational value must be recognised in concepts, principles, and ability which are linked to the statistical representation of facts, phenomena, and procedures and to the elaboration of judgements and predictions in unsolved situations."
The introduction of first elements of Probability, perhaps at the end of the primary course, is supposed to give pupils an intuitive base, on which analysis of unsolved situations by means of reasoning can be founded.
DIDACTIC SUGGESTIONS
The Programs contain didactic suggestions which are strictly linked to content.
These guidelines are enough to help teachers in organising classroom activities, without limiting their autonomy. The fundamental steps of mathematical teaching are pointed out:
The underlying philosophy can be summarised by the following conclusions:
"After all, the development of mathematical thinking and activity is mainly intended to give pupils a wide base of facts and procedures, upon which they can build intuitive knowledge, processes, computation, algorithms and elementary mathematical patterns."
"A better attitude towards mathematics is necessary, so that it is seen both as a strong instrument to know and to interpret reality. It is also seen as an exciting activity of human thinking."