In this chapter the content of mathematics to be taught in the early period of schooling and the points of evaluation of mathematical knowledge are dis-cussed from the perspective of mathematical science. It is rather difficult to answer the question what mathematics is. Mathematics occupies a special place in the family of sciences and school subjects. The complexity of the question is shown by the fact that this is a scientific problem still studied in a field of philosophy, called mathematical philosophy (see Ruzsa & Urbán, 1966, Rényi, 1973, Hersh, 1997, Gardner, 1998). Before we intend to an-swer the question in a way easy to understand we exclude several “misbe-lieves”, and views being misleading but widely accepted. First of all, mathe-matics is not “arithmetic”, what is more, is not “the science of quantities and space” as considered long time ago. For centuries, the subject of mathemat-ics has been much wider than that.
Mathematics is often considered as a field of the natural sciences which might have several reasons, but regarding its development, research meth-ods and internal structure, it differs, for example, from biology, physics and chemistry to a large degree (Bagni, 2010). Mathematics is not a natural
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ence since it does not examine the substances, the phenomena, etc. appear-ing in nature, and its methods are also completely different from those of physics, chemistry and biology, since in these latter ones the basic method of gaining knowledge and checking the knowledge is observation and ex-perimentation. As to its subject and methods, mathematics entirely differs from all other sciences. Mathematics is a science discovering the abstract nature and the interrelations of the structures studied by other sciences and those created by its own internal development, and it arrives at new knowl-edge by the axiomatic-deductive approach, that is, by using the strict rules of formal (mathematical) logic. Note that some dispute even this definition.
What is more, some people even doubt the reality of mathematical objects and formulas outside the human brain.
It is evident that mathematics is a part of human culture. Its results had been used in every historical age, while many important mathematical theo-ries came into being due to problems raised by other disciplines. Earlier it was mainly physics that had great impact on the development of mathemat-ics. Nowadays, it is the enormous progress of information technology that gives mathematical research a stimulus. We are also witnessing the growing demand in social science (economics, sociology, psychology, educational sciences) and also in biology for a strict mathematical foundation. The sys-tem of interrelations between mathematics and the other sciences is how-ever much richer than it might be evident from what has been mentioned so far. There are a number of examples that theories, results produced by the in-herent development of mathematics had seemed for a long time – perhaps for centuries – completely “useless” outside mathematics, and then they turned out to be exactly what were needed in physics or in information tech-nology. Many people, among them also mathematicians, find relationships between mathematics and arts. For the great majority of mathematicians the different proofs, results and theories have aesthetic value, too, and the smarter, the newer a proof, a result or a theory is and the deeper the argu-ments and ideas in them are, the higher their aesthetic value is. This aesthetic value is at least as important from the viewpoint of the development of mathematics as its (momentary or perceived) “utility”. According to the standpoint of UNESCO, mathematics – as a factor in addition to mother tongue literacy – is the basis of civilization.
Mathematics has a unique position as a school subject, as well, and works of many kinds from literature to the contemporary empirical pedagogical
in-vestigations tried to reveal its special status (Mérõ, 1992). Looking back to history, mathematical calculations and astronomical observations based on mathematics had tremendous practical importance in various societies (river civilisations; van der Waerden, 1977). This period is considered as the birth of mathematical science – and of other sciences – when mathematics (together with all other fields of sciences which had been differentiated since that time) was closely attached to philosophy.
Initially, teaching mathematics was intertwined with the study of mathe-matical science. Among the written finds, the Rhind papyrus of Egypt was with no doubt prepared for the clerical social stratum, while the text of the Anastasia I papyrus underlines the importance of proficiency in arithmetic.
Ancient mathematics education provided the highest possible scientific knowledge of that time – for the few who could have access to it at all.
In the European culture, after the prosperity of mathematics in ancient Greece, the development of mathematics was hallmarked by the results of Arab mathematicians, later by the scholar monks of the monasteries, and in the period of the Renaissance by mathematicians bringing the rebirth of sci-ences (Sain, 1986). In the schools of the monasteries mathematical educa-tion was built around two fields of the seven free arts, arithmetic and geome-try (belonging not to the “trivial” part but to the quadrivium). Both fields were needed for practical reasons, like e.g. the efficiency of work or astro-nomical problems. From the age of the Renaissance on, the books published met the practical needs of the strengthening bourgeoisie and presented the application of mathematical results by means of examples from trading and from real life, in general, see e.g. Treviso arithmetic (Verschaffel, Greer, &
de Corte, 2000).
From the 16th century on, mathematics had got an important role in all curricula which were introduced in the frame of unifying processes of the education (Szebenyi, 1997). According to Smolarski (2002), the Jesuit world curriculum, theRatio Studiorumborn in 1599 had not only devoted a significant role to mathematics education, but the didactic guidelines pre-pared for teachers contained proposals for the use of teaching methods which are still in use even today.
It has become widespread from the 16th–17th centuries that the outstand-ing mathematicians of the era had regular dialogues at personal meetoutstand-ings and via correspondence, and the quantity and depth of mathematical knowl-edge was growing so rapidly that the cutting knowl-edge research and the school
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