Даниел А. Романo
Универзитет у Бањој Луци
Природно-математички факултет

УДК 371.214.1:51
Оригинални научни рад
дои: 10.7251/НСК1311 046Р



У овој презентацији ријеч је о општем теоријском погледу на математички доказ. Понуђена је једна анализа директног и индиректног доказа која нам омогућава да сагледамо студентско поимање ове врсте доказа. Анализа се ослања на теоријскe конструктe ‘когнитивно јединство’ и ‘мета-когнитицно јединство’ организацијских, когнитивних и метакогнитивних фактора у процесу конструисања прихватљивог доказа. Сем тога, у циљу дубљег разумијевања успостављена је корелација са тeлеолошким, епистемиолошким и комуникативним аспектима. У овом моделу, процес доказовања се може описати посредством ‘ограничења епистемиолошке важности’, ‘ефикасности у вези са постизањем циља’ и ‘комуникацијом у складу са заједнички прихваћеним правилима’.

Kључне ријечи и изрази: доказ, аргументација, когнитивно јединство, метакогнитивно јединство


Antonini, S. and Mariotti, M.A. (2008). Indirect proof: What is specific to this way of proving? ZDM Mathematics Education, 40, 401-412. Arzarello, F., and Sabena, C. (in print). Semiotic and theoretic control in argumentation and proof activities. ESM Special Issue on Semiotics. Arzarello,F. and Sabena,C., (2011). Meta-cognitive unity in indirect proofs, CERME 7, WG1 Arzarello, F., Micheletti, C., Olivero, F., Paola, D., and Robutti, O. (1998). A model for analysing the transition to formal proofs in geometry. PME 22, Stellenbosh, South Africa, vol. 2, pp. 24-31. Bass,H., (2011). A Vignette of Doing Mathematics:A Meta-cognitive Tour of the Production of Some Elementary Mathematics, The Montana Mathematics Enthusiast, Vol. 8, nos.1&2, pp.3- 34 Blakey, E., and Spence, S. (1990). Developing metacognition. Eric Reproduction Services No. ED327218. Retrieved January 7, 2008 from EDRS Online:http://www.thememoryhole.org/edu/eric/ed327218.html. Balacheff, N. (1987). Processus de preuves et situations de validation. Educational Studies in Mathematics, 18(2), 147-176. Boero P., Garuti R., Lemut E., and Mariotti M. A. (1996). Challenging the traditional school approach to theorems: a hypothesis about the cognitive unity of theorems. PME XX.Valencia, Spain. Boero, P., Douek, N., Morselli, F., and Pedemonte, B. (2010). Argumentation and proof: a contribution to theoretical perspectives and their classroom implementation. In M.M.F. Pinto & T.F. Kawasaki (Eds.), Proceedings of the 34th Conference of the International Group for the Psychology of Mathematics Education (vol. 1, pp. 179204), Belo Horizonte, Brazil: PME. Boero, P., (2011). Argumentaion and proof: Discussing a “successful” classriim discussion, CERME 7, WG1 Chalmers, C., (2009).Group Metacognition During Mathematical Problem Solving, In R. Hunter, B. Bicknell, & T. Burgess (Eds.), Crossing divides: Proceedings of the 32nd annual conference of the Mathematics Education Research Group of Australasia, MERGA. Vol. 1, 105-112

Dominowski, R. (1998). Verbalization and problem solving. In D. Hacker, J. Dunlosky & A. Graesser (Eds.), Metacognition in educational theory and practice. (pp. 25-45). Mahwah, NJ: Lawrence Erlbaum.11 1 Duval, R. (1991). Structure du raisonnement déductif et apprentissage de la demonstration. Educational Studies in Mathematics, 22(3), 233-261. Flavell, J. (1976). Metacognitive aspects of problem solving. In L. B. Resnick (Ed.), The nature of intelligence (pp. 231-235). Hillsdale, NJ: Lawrence Erlbaum. Fortunato, I., Hecht, D., Tittle, C. K., and Alvarez, L. (1991). Metacognition and problem solving.TheArithmetic Teacher, 39(4), 38-40. Fujita, T., Jones, J. and Kunimune, S., (2010). Students’ geometrical constructions and proving activities: A case of cognitive unity; Procedeenings of 34th Congerence of the International Group for Psychology of Mathematics Education, Vol.3, 3-16. Gama, C. (2000). The role of metacognition in problem solving: Promoting reflection in interactive learning systems. Sussex, England: University of Sussex. Garofalo, J., and Lester, F. (1985). Metacognition, cognitive monitoring, and mathematical performance. Journal for Research in Mathematics Education, 16(3), 163-76. Garuti, R., Boero, P., Lemut, E., & Mariotti, M. A. (1996). Challenging the traditional school approach to theorems: a hypothesis about the cognitive unity of theorems, Proceedings of PME 20, vol. 2, pp. 113-120. Valencia, Spain: PME. Garuti R., Boero P., and Lemut E. (1998).Cognitive unity of theorems and difficulty of proof. http://www.mat.ufrgs.br/~portosil/garuti.html Gillies, R. (2000). The maintenance of cooperative and helping behaviours in cooperative groups. The British Journal of Educational Psychology, 70(15), 97–111. Goos, M., Galbraith, P., and Renshaw, P. (2002). Socially mediated metacognition: Creating collaborative zones of proximal development in small group problem solving. Educational Studies in Mathematics,49(2), 193-223. Hinsz, V. B. (2004). Metacognition and mental models in groups: An illustration with metamemory of group recognition memory. In E. Salas & S. Fiore (Eds.), Team cognition: Understanding the factors that driveprocess and performance (pp. 3358). Washington, DC: American Psychological Association. Hanna, G. (1989). More than formal proof. For the Learning of Mathematics, 9(1), 20-23. Harel, G. (2007). Students’ proof schemes revisited. In P. Boero (Ed.), Theorems in school: from history, epistemology and cognition to classroom practice (pp. 65–78). Rotterdam: Sense Publishers. Knipping,C.,(2012). The social dimension of argumentation and proof in mathematics classrooms, http://www.icme12.org/upload/submission/1935_F.pdf Mariotti, M.A. (2006).Proof and proving in mathematics education, In: A.Gutiérrez and P.Boero (eds): Handbook of Research on the Psychology of Mathematics Education: Past, Present and Future; Sense Publishers, The Netherlands Meijer,J., Veenman, M.V.J. and van Hout-Wolters, B.H.A.M., (2006), Metacognitive Activities in Text-Studying and Problem-Solving: Development of a taxonomy, Educational Research and Evaluation, Vol. 12, No. 3, 209 – 237

Moga (Maier), A., (2012). Metacognitive Training Effects on Students Mathematical Performance from Inclusive Classrooms, Phd Thesis, Babeș-Bolyai University, Cluj-Napoca, Faculty of Psychology and Educational Science, Cluj-Napoca Nikiforuk, L.A., (2009). What Are the Metacognitive Strategies That I Can Incorporate Into My Daily Teaching Practice and How Can I Have My Grade 4 Students Use These Strategies to Become More Aware of Themselves as Learners?, Master of Education, Faculty of Education, Brock University St. Catharines, Ontario Panaoura, A. and Philippou, G., (2005).The measurement of young pupls’ metacognitive ability in mathematics: The case of self/representation and self/evaluation, Paper presented at the Conference of European Society for Research in Mathematics Education. Sant Feliu de Guíxols. http://cerme4.crm.es/Papers%20definitius/2/panaoura.philippou.pdf. Panaoura, A. and Panaoura, G., (2004). Young Pupils´ Metacognitive Abilities in Mathematics in Relation to Working Memory and Processing Efficiency, In Proceedings of the International Biennial SELF Research Conference. Berlin. http://self.uws.edu.au/Conferences/2004_Panaoura_Philippou.pdf. Panaoura, A. and Panaoura, G. (2006). Cognitive and metacognitive performance on mathematics, In Novotná, J., Moraová, H., Krátká, M. & Stehlíková, N. (Eds.). Proceedings 30th Conference of the International Group for the Psychology of Mathematics Education, Vol. 4, pp. 313-320. Prague. Panaoura, A. (2007). The impact of recent metacognitive experiences on preservice teachers’ self-representation in mathematics and its teaching, CERME 5, 329-338 Pedemonte, B. (2007). How can the relationship between argumentation and proof be analysed? Educational Studies in Mathematics, 66(1), 23-41. Pugalee, D. (2001). Writing, mathematics, and metacognition: Looking for connections through students’ work in mathematical problem solving. School Science and Mathematics, 101(5), 236-246. Rodríguez, O.H. and Cepeda, W.V (2008).Cognitive and metacognitive processes of preservice mathematics teachers while solving mathematical problems, ICME11, Романо, Д.А. (2005). Основе математике, Дио Први – Увод у математичку логику; Mат-Кол (Бања Лука), Посебна издања, Број 3. Романо, Д.А. (2008). Mатематичка логика, Књига 1; Mат- Кол (Бања Лука), Посебна издања, Број 7. Романо, Д.А. (2009). Истраживање математичког образовања; ИМО, I, Број 1, 1-10 Романо, Д.А. (2009а). Шта је алгебарско мишљење? Mат-Кол (Бања Лука), XV(2), 19-29 Schoenfeld, A.H. (1994). Whatdoweknowabout mathematics curricula? Journal ofMathematical Behavior, 13(1), 55-80. Schoenfeld, А.H. (2008). Research methods in (mathematics) education. In: L.D. English (ed) Handbook of internationalresearch in mathematics education. 2nd edition, Taylor and Francis, NY, 467-519.

Schoenfeld, А.H. (2010). Namjere i metode u istraživanju matematičkog obrazovanja, IMO, III, Бroj 4, 23-34 Schraw, G. (2001). Promoting general metacognitive awareness. In H. Hartman (Ed.), Metacognition in learning and instruction: Theory, research and practice (pp. 3368). -Dordrecht, The Netherlands: Kluwer Academic Publishers.

Daniel A. Romano



Starting from a general discussion on mathematical proof, a structural analysis wascarried out in this presentation, leading to the construction of a model within whichdirect and indirect proofs can bedescribed and how our students consider these proofs. The model shows itself a good interpreting tool to identify and explaincognitive and didactic issues, as well to precisely formulate research hypothesesconcerning students’ difficulties withdirect and indirect proofs. We discuss the theoretical construction of ‘cognitive unity’ and ‘meta-cognitive unity’, which may give reason of success and difficulties in indirect proofs. Key words and phrases: proof, argumentation,cognitive unity and meta-cognitive unity Math. Subject Classification (2010): 97B50, 97C70 ZAM Subject Classification (2010): B50, C30, C80, D20, D70

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