curriculum became separated from each other. Until today, however, it has remained an important aspect in determining the school curriculum that it should be scientifically correct and it should prepare for future (eventual) higher level mathematical studies. In the relationship of mathematics as a science and mathematics as a school subject, the thoughts of Dewey (1933), where he weighs the growing number of the fields of sciences and the in-creasing knowledge of these fields against the – comparatively less chang-ing – learnchang-ing capacities of children, are especially valid nowadays.
Thus in the framework of the subjects of modern public education, it has become a basic issue for mathematics education to outline a body of knowl-edge which is correct and coherent from the point of view of mathematical science, and, simultaneously, is in line with the age-group characteristics of learners. These efforts are made difficult by the fact that the basic notions of
maintained in other countries: for example, the name of one of the faculties of the University of Vienna (Universität Wien) was also Faculty of Mathematics and Science several years ago. (Today there are separate faculties: Faculty of Mathematics, Faculty of Physics, etc.).
Mathematics is not a natural science studying phenomena occurring in re-ality, since it works with abstract concepts, and it reveals the relations be-tween them allowing drawing only very strict conclusions. Mathematical theories and concepts certainly come from reality, and this is why mathe-matics can be successfully used in various areas, and frequently, the same mathematical results are applied in very diverse fields.
It is a natural consequence of the development of natural sciences that the outdated, refuted theories together with all the “scientific results” achieved until that time are deleted, declared unscientific, etc. – just remember how the geo-centric world view was replaced by the heliogeo-centric one. The theories serve for the explanation of the repeatable experiments and observable phenomena, and a theory, in general, is considered valid if its “explanation is the best”.
In contrast to this, it comes from the special theme and methods of mathe-matics that the mathematical knowledge consists of “ideas” created by hu-man beings, which ideas do not loose their validity through the centuries, therefore they should not be and need not be “thrown away”. Although, it is true that in the course of the development of mathematics, some of the ear-lier concepts and theories need to be improved in preciseness (see e.g. natu-ral numbers, real numbers, and the Euclidean geometry and the theory of sets, respectively), and some topics become more fashionable so to speak in certain periods of time than others. However, every age builds on the mathe-matical knowledge of the previous ages, and develops it further. This ex-plains why the mathematical knowledge taught in the majority of primary and secondary schools of the world – at least in the core curriculum – was al-ready known in the ancient times. The most important thing is that they still create the basis of the mathematical science. One of the big challenges of teaching mathematics is to find new topics for young children, as well as to teach this fundamental mathematical knowledge by methods which make it possible to bridge the huge gap – at least in some important areas – between the ancient level and the contemporary mathematical knowledge in higher education curricula containing serious mathematics.
It is very important for mathematical science that this cognitive, theoreti-cal perception have an unbroken arc being as high as possible, since an ever
growing number of employees have to apply consciously the most up-to-date mathematical achievements. The educational and workplace context of the PISA surveys reflects these requirements (OECD, 2009).
What we want to emphasize is that the development of abstract thinking and the use of abstract thinking are simultaneous processes in teaching mathematics. The everyday work of mathematics teachers is pervaded by this paradox, and they have to give authentic and comprehensible new infor-mation to students by dancing between these ropes.
Similarly to the natural sciences, a strong specialization has taken place in the discipline of mathematics since the ancient times until today: new fields were born partly as a result of the internal development of mathematics, partly on “external” influence that is based on the demands of end-users.
The biggest reviewing monthly in mathematics, theMathematical Reviews lists annually more than 75 thousand scientific articles in mathematics clas-sified according to their subject. The latest classification of the mathemati-cal topics consists of 47 pages, where the number of the main areas is more than 60 and they are divided into sub-topics in two further steps (MSC 2010).
Table 3.1 shows that the contextual areas indicated in the mathematical evaluation frames integrate naturally into the main areas of mathematical science.
Table 3.1 Main areas of mathematics
Primary content areas Main areas according toMathematics Subject Classification
Numbers, operations, algebra 11: Number theory
12: Field theory and polynomials
(further topics of abstract algebra: 06, 08, 13-22) Relations, functions 26: Real functions
(further topics of analysis and differential equations:
28-49)
Geometry 51: Geometry
(further topics of geometry and topology: 52-58) Combinatorics, probability
calculation, statistics
05: Combinatorics
60: Probability theory and stochastic processes 62: Statistics
Methods of mathematical logic 03: Mathematical logic and foundations
The areas with higher serial numbers not mentioned on the right hand side of the table (e.g. 65: Numerical analysis, 68: Computer science) integrate into the topics mentioned on the left hand side through the areas mentioned here. Similarly, the topics of the primary content areas fit well into the clas-sification by the Mathematical Reviews of the main disciplinary areas of teaching mathematics. These are the following: 97 Mathematics education;
97E Foundations of mathematics; 97F Arithmetic, number theory; 97G Ge-ometry; 97H Algebra; 97I Analysis; 97K Combinatorics, graph theory, probability theory, statistics.
Thus we can conclude that on the whole the evaluation frames of mathe-matics correspond to the present research fields of mathematical science.
This is one of the reasons for their selection. The other reason is that the de-velopment of mathematical thinking can be implemented through these top-ics as basic material with the help of modern educational methods. We will see later that the counterparts of these contextual elements appear in the Na-tional Core Curriculum, as well as in the mathematics curriculum of grades 1–6. The system outlined here is in harmony with the historical-cultural tra-ditions, as well as with the frames of the mathematical assessment prepared in Germany – a country being in a similar position by the PISA1surveys – as a result of the educational reforms induced by the poor results. The Bildungstandard(2005) accepted by the Conference of the German Minis-ters of Provinces specifies the following content areas in the systemization of requirements demanded by the end of grade 4: numbers and operations;
space and form; pattern and structure, quantities and measures; data, fre-quency and probability. The handling of classical geometry and measure-ments as separate areas is widely spread in the educational systems all over the world, and this differentiation has been found in the surveys of IEA2 or-ganization, too, since the beginning.
1PISA:Programme for International Student Assessment, is an international programme under the guidance of OECD in order to assess the students’ understanding of texts, knowledge in mathematics and natural sciences.
2IEA: International Association for the Evaluation of Educational Achievement, organization managing the international assessment of students since the 1960s. A TIMSS (Trends in International Mathematics and Science Studies) survey has been evaluating the mathematical knowledge with four year regularity since 1995.