Objectives and tasks The number of compulsory classes in mathematics is decreasing in the first four grades over the years, therefore the organisation of knowledge based on review becomes more important in Grade 5 than before. Special attention should be given to the establishment and profound understanding of concepts. This cannot be achieved without colourful activities and diverse exercises. Experiments and games does not disappear in the upper grades. The foregoing together with the age specific features of learning provide enough ground for giving a higher priority to arithmetic and algebra in the first two grades of the second stage of basic education, both in the timetable and the syllabus, than in the subsequent grades. A properly developed concept of what numbers are, understanding and practising operations with an expanding circle of numbers is a prerequisite of continued success.
The fundamental objectives are the following: improving thinking based on understanding, learning the two-way path between real situations and mathematical models, and gradually making pupils use modelling as a tool.
In connection with this, the task of teachers is to show pupils the quantitative an spatial relations found in the immediate environment, lay the foundations of am up-to-date and usable mathematical background, develop thinking and provide them with a mathematical knowledge and tools which is necessary for learning other subjects and coping with practical issues.
Studying mathematics should develop individual learning starting with empirical experience. It should develop a demand for independence in thinking and serve the formation of personal traits.
Teachers should take every effort to ensure positive motivation among pupils, increase their independence, make them work with precision and endurance, develop reasonable self-confidence, willpower, quality communication and the practice of using arguments to support ideas.
In the upper grades of primary school, there is growing focus on analytic thinking, justification in addition to problem solving, the ability to understand, recognise and individually formulate simple conclusions.
Some of the simplest forms of deduction must be acquired and applied through the individual interpretation of different types of information derived from different fields and by learning these information. At the same time the importance of induction does not decrease in the upper grades of primary school.
Whereas a certain part of mathematical knowledge becomes more abstract in this stage, a significant part of this knowledge remains related to concrete experiences. Therefor emphasis should be placed on diverse activities, awareness and various ways of recording, interpreting and organising experience, establishing relations. Learning mathematics in the upper grades are characterised by exploration, making pupils follow the path from raising the problem to solving the problem with growing independence, knowing that the path is not always free of mistakes.
Problem solving based on drawing conclusions and simple algorithms which are created and used by pupils are also priorities. At first these are done in concrete situations, then generalisations can be made building upon the concrete situations and considering the age specific features of the learning process.
In addition to gradual abstraction, the learning process should include frequent concretising. Generalisation must be supplemented by specialisation.
In line with the available facilities, teachers should promote the use of electronic devices and sources of information (pocket calculator, graphic calculator, computer, internet, etc.) for gathering information and making problem solving easier.
Teaching mathematics at primary school should provide a firm foundation for continuing studies in the field of science at the secondary level.
Developmental requirements A significant proportion of pupils progress from concrete to abstract thinking in these four years. This process determines how the requirements related to development are defined.
The application of acquired mathematical concepts
The development of a mathematical approach In the first part of this period, in the field of arithmetic and algebra, the concept of number is formed through practical activities. The range of numbers pupils work with is growing.
With respect to the four rules of arithmetic, the goal is to develop an established concept of operation and to further improve calculation skills. A calculator may be used to help understanding, working with and practising newly introduced operations.
The elementary concepts and relations of mathematics should be used in other classes and in daily life.
The study of relations between variables improves the skills related to working with functions. Simple functions and diagrams occurring in practice are taught.
In geometry, two and three dimensional orientation and skills are developed by using various tools.
Pupils arrive at the use of simple geometrical transformations through various activities. This will help to establish a dynamic view of geometry later.
Certain elements of mathematical logic (i.e. ‘and’, ‘or’, ‘not’, ‘every’ ‘there is’ must be used consciously when teaching mathematics. ‘If, then’ type conclusions are introduced in connection with the justification or refutation of simple guesses towards the end of this period.
Problem solving skills and logical thinking A great emphasis is placed on the development of understanding, writing open sentences on the basis of a text and solving them first by (systematic) trial and with algebraic methods later. At a later stage, the development of discussion skills, the ability to find alternative solutions, are facilitated by interpreting and analysing mathematical discourse.
Modelling is an important tool in mathematics, as it provides help to solve problems.
Appropriate level of attention must be given to measurements and construction with practical significance. This way pupils are made to apply in practice the concepts established visually, such as circumference, area, volume and the related methods of calculation.
Pupils are made to understand through various exercises that there are certain and impossible events, and there are events which may happen. A concept of probability derived from demonstrations is gradually built up.
The application of acquired learning methods and thinking Induction has a crucial role in learning mathematics for a considerable time. At this stage, proofs with a few steps and deduction can also be introduced.
It is important to use statements borrowed from a mathematical as well as an ordinary context when pupils analyse whether a statement is true or false. This way they can learn how to formulate guesses and regularities.
In the various exercises, pupils group, classify, sort and select elements according to certain criteria to develop a set theory based approach, which is important in the various fields of mathematics as well as in other topics.
With the help of the figures and simple diagrams which accompany the various exercises, pupils are made to understand the role of modelling.
The fundamental elements of statistics are present from the very outset in activities, such as collecting and recording data, making diagrams and the use of algorithms with a few steps improves the ability to analyse, characterise and represent data. This way pupils will become able to make algorithms with a few steps themselves.
Developing the right attitude towards learning Pupils must get used to estimating the result prior to calculation and measurement and checking result after solving a problem. Pupils must learn to accept only realistic results. This can be achieved through the above listed activities and by practising rounding up and down, which is essential for practical calculations.
Pupils must get used to making a solution plan or, in certain cases, a sketch before solving a problem. They must learn how to write down the solution. Pupils must learn to formulae sentences with precision and focus on the main points in the final years of primary school education.
Pupil must use their mother tongue and the terminology of mathematics in the classroom, with the level of precision that can be reasonably expected from them at this age. The system of symbols is gradually expanded.
Pupils must first understand the content of a concept and comprehend the concept itself. Definition must follow this stage. In the upper grades of primary school, requirements include the application of the definitions learnt in class. The use of different procedures and certain axioms as tools in problem solving is an important area of development.
The continuous development of reasoning, refutation and discussion skills, correct communication is a constant task.
Pupils need to learn how to use textbooks, exercise books, statistics and encyclopaedia - at a later stage. They need to learn how to use multimedia devices, if possible. The interactive use of these devices activates pupils, facilitates learning, makes their mathematical approaches more efficient.. Positive motivation may arise their interest in the curiosities and history of mathematics. The life and achievements of a few Hungarian and foreign mathematicians may be demonstrated as a supplement to the material being taught.
Number of teaching hours per year: 148 Methods of thinking
PUPILS SHOULD KNOW THE FOLLOWING
Improving acquisition skills.
Learning to learn (methodology of learning mathematics).
The requirements of improving methods of thinking are made concrete in other topics.
Improving communication skills.
Curiosities from the history of mathematics. Library work, using IT devices.
Using the logical elements of language correctly. Not only in statements with mathematical content.
Interpreting and using expressions needed for comparison (e.g. equal, smaller than, larger than, more than, less than, at least, at most, not, and, or, every there is).
Improving probability and statistics based attitude.
The concept of certain, possible and impossible.
Improving interpretative and analytic reading, recognising relations, recording relations with simple symbols.
multiplication, division in the case of positive fractions and decimals fractions (role of 0 in multiplication and division);
multiplication and division by 10, 100, 1000.
Addition, subtraction, multiplication, division by two digits within the circle of natural numbers. Addition and subtraction of positive fractions with a single-digit denominator (in case of decimal fractions up to thousandths) in case of two terms, and checking if result is correct..
Improving discipline and consistence.
Order of operations.
Knowing the correct order of operations in the case of the first four rules of arithmetic.
Improving estimation skills.
Rounding up and down, checking.
Improving deduction skills. Interpretative and analytical reading, improving, problem solving skills.
Solving simple equations of the first degree and inequalities by deduction, breaking down, checking by substitution. Proportionate conclusions (standard units of measurement and exercises in connection with their conversion), simple problems expressed verbally.
Solving simple equations and problems by deduction.
Mathematical relations, functions and sequences
PUPILS SHOULD KNOW THE FOLLOWING
Determining position, finding points with given properties.
Continuum of numbers, graphic image of number intervals, reading figures.
The orthogonal co-ordinate system.
Determining location in practical situation and concrete cases. The Cartesian orthogonal co-ordinate system.
Graphic image of concrete points, the co-ordinates of a given point.
Improving ability to establish correlation. Interpreting tables and graphs; finding correlation between quantities on the basis of graphic representation. Recording correlation. Making a graph for a table / table for a graph. Preparation of a function based approach. Determining elements on the basis of a known rule; formulating rule(s) in case of known elements. Looking for alternative solutions.
Correlation between variable quantities. Table / graph of simple linear correlation - adding missing elements to table according to known or recognised rule. Changes of sum, difference, product, quotient.
PUPILS SHOULD KNOW THE FOLLOWING
Improving the way of looking at space; making bodies.
Constructing bodies, their properties.
Improving the set theory based approach.
Observing properties (e.g. symmetry).
Grouping bodies according to properties. The properties / lattice of cube, cuboid. Parallelism, perpendicularity, convexity. Reciprocal position of spatial configurations. Visual concept of plane figures and polygons, study of properties.
Proper use of compasses and rulers. Drawing parallel and perpendicular lines with a pair of rulers.
Improving problem solving skills by geometric construction.
Rectangles, triangles and their properties. Visual concept of distance, locating points with given properties. Visual concept of circle, sphere, their occurrence in practice.
Copying segment, measuring given distance.
Points at equal distance from two points. Perpendicular bisector of a segment Perpendicular line crossing a line at a given point. Constructing a triangle out of three sides.
Visual concept of perpendicular bisector.
Proper use of protractor.
Concept of angle, measuring angle, types of angle.
Practising how to calculate circumference, area, surface and volume.
Improving calculation skills.
Measurements in practice.
Using measuring devices.
Improving estimation skills.
Circumference, are of rectangle (square), surface and volume of cuboid (cube) expressed in selected and standard units of measurement.
Exercises with calculation.
Standard units of measurement and conversion (length, area, volume, cubic capacity, time, weight).
Calculating circumference and area of rectangle (square), surface and volume of cuboid (cube) in concrete cases.
Standard units of length and area, simple cases of conversion in concrete, practical exercises. Unit of volume, cubic capacity, time, weight.