We have to take it into account that the topics of the curriculum formulate not simply teaching objectives but at the same time involve the personal de-velopment enhancing the learners’ mathematical thinking. Some aspects justify the discussion of the mathematical topics parallel to each other (but not in a mosaic like way). For example the development of the contents
“number of pieces” and “number of measurement” of a natural number goes together with the development of the concept of an operation and they strengthen each other. Both can be confirmed by the appearance of the topic of series first as a new problem situation. Parallel to this a number series as an object can provide an efficient starting internal image for the learners to the conceptualization of the set of natural numbers. Covering the topics to-gether can also offer several didactical implications.
Naturally, the inherent logic of the didactic construction of the topics makes it necessary for the learners to devote a longer period of time to each topic. In this way they can perceive the special internal logic, “play field” of a given topic.
The didactic method of merging the topics, which was one of the basic el-ements of Tamás Varga’s didactic concept seems to be implemented in the education practice (Halmos & Varga, 1978). This method makes it possible to present the unity of mathematics, the interrelation of topics according to the development of learners’ thinking already from early school age. The small steps of the development of thinking do not make it possible to make a big progress in a topic. The simultaneous development, however, ensures that learners progress in the building of all mathematical topics until the
point allowed to them by the development level of thinking. On the other hand it guarantees that leaps in thinking and forced progress at the level of concepts should not be imposed on the learners. The progress in the abstrac-tion level of thinking makes it possible to use the symbols of higher and higher level and to develop reasoning about mathematical objects and to im-prove the verbal formulations.
The progress learners are making in various topics is related to each other.
For example before the development of the appropriate level of thinking in terms of proportions it is totally incidental if students can perform a conver-sion of units. This can also hinder students in the function type interpretation of relations between geometrical quantities.
The methods of mathematical thinking gained in small steps can provide an impetus to the elements of knowledge. This provides the fabric which in-tegrates the disintegrated cognitions into competencies and makes the grow-ing child able to get to higher and higher level of abstraction, to focus on the essence, and to recognize structural frameworks. Thus after completing grade 6 young learners will be more and more able to use mathematical rea-soning.
References
AMS (2010). Letters to the Editor,Notices of the American Mathematical Society, 57(7), 822–823.
Bagni, G. T. (2010). Mathematics and positive sciences: A reflection following Heidegger, Educational Studies in Mathematics, 73(1), 75–85.
Bán, S. (1998). Gondolkodás a bizonytalanról: valószínûségi és korrelatív gondolkodás [Thinking about the uncertain: probabilistic and correlational reasoning]. In B. Csapó (Ed.).Az iskolai tudás(pp. 221-250). Budapest: Osiris Kiadó.
Beke, M. (1900).Számtan és mértan. A gazdasági ismétlõ iskoláknak. Budapest: Wodiander és Fiai könyvkiadó.
Beke, M. (1911). Vezérkönyv a népiskolai számtani oktatáshoz. Budapest: Magyar Tudomány-Egyetem nyomda.
Beke, M., & Mikola, S.(1909).A középiskolai matematikai tanítás reformja, [The reform of secondary mathematics education], (p.10.) Budapest: Franklin Társulat.
Bildungstandards in Fach Mathematik für den Primarbereich. Beschlüsse der Kultusministerkonferenz. Luchterhand- Wolter Kluwe: München, Neuwied, 2005 Borel, A. (1998). Twenty-Five Years with Nicolas Bourbaki, (1949–1973).
http://www.ega-math.narod.ru/Bbaki/Bourb3.htm.
Burkhardt, H. et al. (1984).Problem Solving – A World View. Proceedings of problem solving theme group ICME 5.Adelaide
Byers, W. (2007).How mathematicians Think?. Princeton and Oxford: Princeton University Press.
C. Neményi, E., Radnainé Szendrei, J., & Varga, T. (1977). Matematika 1–8. osztály [Mathematics for grades 1-8]. In: Az általános iskolai nevelés és oktatás terve.
114/1977. (M.K.11.) OM számú utasítás. Országos Pedagógiai Intézet.
Csapó, B. (2000). A tantárgyakkal kapcsolatos attitûdök összefüggései [Students’ attitudes towards school subjects].Magyar Pedagógia, 100(3). 343–366.
Csapó, B. (1994). Az induktív gondolkodás fejlõdése [Development of inductive reasoning].
Magyar Pedagógia, 94(1-2). 53–80.
Csíkos, C. (1999). Iskolai matematikai bizonyítások és a bizonyítási képesség [Proofs in school mathematics and the ability to construct proofs]. Magyar Pedagógia,99(1).
3–21.
Csíkos, C. (2003). Matematikai szöveges feladatok megoldásának problémái 10–11 éves ta-nulók körében. [The difficulties of comprehending mathematical word problems in 10-11-year-old]Magyar Pedagógia, 103(1). 35–55.
Csíkos C. (2007).Metakogníció – A tudásra vonatkozó tudás pedagógiája[Metacognition – The pedagogy of knowing about knowledge]. Budapest: Mûszaki Kiadó.
Dávid, L. (1979).A két Bolyai élete és munkássága[The life and work of the two Bolyais], 2nd edition. Budapest: Gondolat Könyvkiadó.
Dehaene, S. (2002). Verbal and Nonverbal representations of numbers in the human brain. In A. M. Galaburda, S. M. Kosslyn, & Y. Christen (Eds.).The languages of the brain, (pp.
179–190). Cambridge: Harvard University Press.
Dewey, J. (1933).How we think. Boston: D. C. Heath and Co.
Dossey, J., Csapó, B., de Jong, T., Klieme, E., & Vosniadou, S. (2000). Cross-curricular competencies in PISA. Towards a framework for assessing problem-solving skills. In The INES Compendium. Contributions from the INES Networks and Working Groups.
(pp. 19–41). Tokyo: Fourth General Assembly of the OECD Education Indicator Programme. Paris: OECD.
Dudley, U. (2010). What is mathematics for?Notices of the American Mathematical Society, 57(5). 608–613.
Even, R., & Tirosh, D. (2008). Teacher knowledge and understanding of students’
mathematical learning and thinking. In L. D. English (Ed.):Handbook of international research in mathematics education. (pp. 219–240) London: Routledge.
Freudenthal, H. (1980a).Weeding and sowing.Dordrecht: D. Reidel Publishing Company.
Freudenthal, H. (1980b). Mathematics as an educational task. Dordrecht: D. Reidel Publishing Company.
Freudenthal, H. (1991). Revisiting mathematics education. China Lectures. Dordrecht:
Kluwer Academic Publishers.
Gardner, M. (1997, October 12). Book review on What is mathematics, really? by Reuben Hersh.Los Angeles Times, Oct 12. p. 8.
Ginsburg, H. P. (1998). Toby matekja [Toby’s math]. In R. J. Sternberg & T. Ben-Zeev (Eds.).A matematikai gondolkodás természete,(pp. 175-199), Budapest: Vince Kiadó.
Gray, E. M., & Tall, D. O. (1994). Duality, ambiguity and flexibility: a proceptual view of simple arithmetic.Journal for Research in Mathematics Education, 26(2). 115–141.
Halmos, M., & Varga. T. (1978). Change in mathematics education since the late 1950’s – ideas and realisation Hungary.Educational Studies in Mathematics9. 225–244.
Hersh, R. (1997).What is Mathematics, Really?Oxford: Oxford University Press.
ICMI, 1908. http://www.icmihistory.unito.it/portrait/klein.php
Klein, S. (1987).The effects of modern mathematics. Budapest: Akadémiai Kiadó.
Kosztolányi, J. (2006). On teaching problem-solving strategies. PhD thesis, Debrecen:
University of Debrecen.
Lemaire, P., & Lecacheur, M (2004). Five-rule effects in young and older adults’ arithmetic:
Further evidence for age-related differences in strategy selection.Current Psychology Letters[Online], 12(1), http://cpl.revues.org/index412.html.
Mérõ, L. (1992). Matek, torna, memoriter.Café Babel,5–6. 69–76.
Mullis, I. V. S., Martin, M. O., Olson, J. F., Berger, D. R., Milne, D., & Stanco, D. M. (2008).
TIMSS 2007 Encyclopedia. A guide to mathematics and science education around the world.Volume I, A. Boston: Boston College.
NCTM (National Council of Teachers of Mathematics) (2000). Principles and standards for school mathematics. Reston, VA: NCTM.
Nemzeti alaptanterv (2007). [National Core Curriculum]
http://www.nefmi.gov.hu/ letolt/kozokt/nat_implement_090702.pdf
A Nemzeti alaptanterv kiadásáról, bevezetésérõl és alkalmazásáról szóló 243/2003 (XII. 17.) Korm. rendelet (a 202/2007. (VII. 31.) Korm. rendelettel módosított, egységes szerke-zetbe foglalt szöveg)
http://www.okm.gov.hu/kozoktatas/ tantervek/nemzeti-alaptanterv-nat (2010. 06. 13.) OECD (2009).Learning mathematics for life: A perspective from PISA. Paris: OECD.
Pegg, J., & Tall, D. (2010). The fundamental cycle of concept construction underlying various theoretical frameworks. In B. Sriraman & L. English (Ed.). Theories of mathematics learning – Seeking new frontiers.(pp. 173–192.) New York: Springer.
Pethes, J. (1901).Vezérkönyv a számtantanításhoz. Tanítók és tanítónövendékek számára.
[Guidebook on teaching arithmetic. For teachers and student teachers] Nagykanizsa:
Fischel Fülöp könyvkiadása.
Péter, R., & Gallai, T. (1949).Matematika a gimnázium I. osztálya számára.[Mathematics for grammar schools. Grade I.] Budapest: Tankönyvkiadó.
Pólya, G. (1945).How to solve it.Princeton: Princeton University Press.
Pólya, G. (1954).Mathematics and plausible reasoning(Volume 1, Induction and analogy in mathematics; Volume 2, Patterns of plausible inference). Princeton: Princeton University Press.
Pólya, G. (1981).Mathematical Discovery(Volumes 1 and 2). New York: Wiley.
Rapolyi, L. (2005, Ed.). Wigner Jenõ válogatott írásai [Selected writings of Eugene Wigner]. Budapest: Typotex Kiadó.
Radnainé Szendrei, J. (1983). A matematika-vizsgálat [The mathematics study].Pedagógiai Szemle. 32(2). 151–157.
Rátz, L. (1905).Mathematikai gyakorlókönyv 1–2. kötet[Mathematics exercise book]. Bu-dapest: Franklin.
Rényi, A. (1973).Ars mathematica.Budapest: Magvetõ Kiadó.
Rickart, C. (1998). Strukturalizmus és matematikai gondolkodás [Structuralism and mathematical reasoning]. In R. Sternberg & T. Ben-Zeev (Eds.).A matematikai gondol-kodás természete. Vince Kiadó, Budapest, 279–292.
Robitaille, D. F., & Garden, R. A. (1989).The IEA Study of Mathemtics II: Contexts and outcomes of school mathematics.Pergamon Press, Oxford.
Ruzsa, I., & Urbán, J. (1966).A matematika néhány filozófiai problémájáról[On some philosophical problems of mathematics]. Budapest: Tankönyvkiadó.
Sain, M. (1986).Nincs királyi út! Matematikatörténet[History of mathematics]. Budapest:
Gondolat Kiadó.
Schoenfeld, A. H. (Ed.). (1994).Mathematical thinking and problem solving.New Jersey:
Lawrence Erlbaum.
Senk, S. L. (1989). Van Hiele levels and achievement in writing geometry proofs.Journal for Research in Mathematics Education, 20. 309–321.
Smolarski, D. C. (2002). Teaching mathematics in the seventeenth and twenty-first centuries.
Mathematics Magazine, 75. 256–262.
Szebenyi, P. (1997). Tagoltság és egységesítés – tananyagszabályozás és iskolaszerkezet [Division and unifacation – Content control and school structure].Magyar Pedagógia, 97(3-4). 271–302.
Szendrei, J. (2002). Matematika. Az Eötvös József Szabadelvû Pedagógiai Társaság NAT 2002 tervezete [Mathematics. The Core Curriculum proposal of the József Eötvös Liberal Pedagogical Society].Új Pedagógiai Szemle.52(12). Supplement. 33–46.
Szendrei, J. (2005). Gondolod, hogy egyre megy? Dialógusok a matematikatanításról.
[Dialogs on mathematics teaching]. Budapest: Typotex Kiadó.
Szendrei, J. (2007). When the going gets tough, the tough gets going problem solving in Hun-gary, 1970–2007: research and theory, practice and politics. ZDM, Mathematics Education, 39. 443–458.
Tversky, A., & Kahneman, D. (1983). Extensional versus intuitive reasoning: The conjunction fallacy in probability judgment.Psychological Review, 90. 293–315.
Vancsó, Ö. (2010). Mathematical logic and statistical or stochastical ways of thinking – an educational point of view. Session 3F ICOTS-8 Ljubljana 2010. www.icots8.org.
van der Waerden, B. L. (1977).Egy tudomány ébredése. Egyiptomi, babiloni és görög mate-matika[Science Awakening I: Egyptian, Babylonian and Greek Mathematics]. Gondo-lat Kiadó, Budapest.
van Hiele, P. M. (1986).Structure and insight: A theory of mathematics education. Orlando:
Academic Press.
Varga, T. (1972). A matematika tanításának várható fejlõdése [The expectable development of mathematics]. In A. Cser (Ed.).A matematikatanítás módszertanának néhány kérdé-se.Budapest: Tankönyvkiadó. 303–349.
Varga, T. (1988). Mathematics education in Hungary today. Educational Studies in Mathematics. 19. 291–298.
Verschaffel, L., Greer, B., & De Corte, E. (2000).Making sense of word problems.Lisse:
Swets & Zeitlinger.
[NKR] Nemzeti Kutatás-nyilvántartási Rendszer https://nkr.info.omikk.bme.hu/cerif/
tudomany.htm 2010.10.15.
[NEFMI] Nemzeti Erõforrás Minisztérium, Felsõoktatási szakok tudományági besorolása, http://www.nefmi.gov.hu/letolt/felsoo/szakok_tudagi_ besorolasa_20040226.pdf 2010.10.15.
[MSC 2010] Final Public Version [Oct. 2009], American Mathematical Society; electronic form to be downloaded: http://www.ams.org/mathscinet/msc/ msc2010.html
[MR] Mathematical Reviews, American Mathematical Society; electronic form:
http:// www.ams.org/mathscinet/