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Risba kot orodje za vpogled v matematično razumevanje
Alenka Lipovec in Manja Podgoršek
Polno besedilo (pdf) | Ogledi: 135 | Napisan v slovenščini. | Objavljeno: 9. oktober 2016
https://doi.org/10.20419/2016.25.452 | Citati: CrossRef (1)
Povzetek: Vizualne reprezentacije omogočajo osmišljanje pomena matematičnih pojmov, odnosov in procesov, zato imajo pomembno vlogo pri pouku matematike. V predstavljeni raziskavi smo pri udeležencih preučevali razumevanje osnovnih matematičnih pojmov s pomočjo risb. Pojem je bil podan v simbolni obliki (npr. 17 – 9), udeleženci pa so ga morali prikazati z risbo. Zanimalo nas je, ali dijaki oz. študenti in bodoči učitelji (N = 345) ustrezno (v skladu z matematično definicijo) narišejo podani matematični pojem. Podatke smo obdelali s kombinacijo kvantitativne in kvalitativne metodologije. Rezultati so pokazali, da udeleženci zahtevane pojme z risbo ustrezno prikazujejo, pri čemer je delež ustreznosti risb v povezavi z abstraktnostjo prikazanega pojma. Ugotovili smo tudi, da študenti 4. letnikov, ki se izobražujejo za poučevanje na razredni stopnji, pozitivno izstopajo. Po pregledu vzorca risb smo na osnovi vsebinske analize oblikovali dve temi, ki prikazujeta dva načina matematičnega razumevanja (instrumentalno in relacijsko) oz. dva tipa matematičnega znanja (proceduralni in pojmovni). Izsledki raziskave lahko služijo raziskovalcem pri oblikovanju novih raziskovalnih instrumentov za merjenje matematičnega razumevanja in učiteljem pri izbirah načina vpogleda v učenčevo razumevanje.
Ključne besede: matematika, razumevanje, vizualizacija, risba, poučevanje
Citiraj:
Lipovec, A. in Podgoršek, M. (2016). Risba kot orodje za vpogled v matematično razumevanje [Drawing as a tool for an insight into mathematical understanding]. Psihološka obzorja, 25, 156–166. https://doi.org/10.20419/2016.25.452
Seznam literature v članku
Arcavi, A. (2003). The role of visual representations in the learning of mathematics. Educational Studies in Mathematics, 52(3), 215–241. CrossRef
Archer, K., Savage, R., Sangera-Sidhu, S., Wood, E., Gottardo, A. in Chen, V. (2014). Examining the effectiveness of technology use in classrooms: A tertiary meta-analysis. Computers & Education, 78, 140–149. CrossRef
Antolin Drešar, D. in Lipovec, A. (2015). Slovenian pupils' conceptions of the mathematician profession. V O. Fleischmann (ur.), The teaching profession: New challenges – new identities (str. 21–29)? Zürich, Švica: LIT.
Badillo, E., Font, V. in Edo, M. (2015). Analysing the responses of 7-8 year olds when solving partitioning problems. International Journal of Science and Mathematics Education, 13, 811–836. CrossRef
Bakker, A. in Gravemeijer, K. (2004). Learning to reason about distribution. V D. Ben-Zvi in J. Garfield (ur.), The challenge of developing statistical literacy, reasoning, and thinking (str. 147–168). Dordrecht, Nizozemska: Kluwer.
Ball, D. L., Thames, M. H. in Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59, 389–407. CrossRef
Beitzel, B. D., Staley, R. K. in Dubois, N. F. (2011). The (in)effectiveness of visual representations as an aid to solving probability word problems. Effective Education, 3(1), 11–22. CrossRef
Binterová, B., Petrášková, V. in Komínková, O. (2014). The CLIL method versus pupils` results in solving mathematical word problems. The New Educational Review, 38, 238–249. Bruner, J. (1964). The course of cognitive growth. American Psychologist, 1(19), 1–15. CrossRef
Confrey, J. in Smith, E. (1995). Splitting, covariation, and their role in the development of exponential functions. Journal for Research in Mathematics Education, 26(1), 66–86. CrossRef
Crooks, N. M. in Alibali, M. W. (2014). Defining and measuring conceptual knowledge in mathematics. Developmental Review, 34, 344–377. CrossRef
Černela, R. (2014). Predstave učencev o matematikih (neobjavljeno diplomsko delo) [Childrens' conceptions of the mathematicians (unpublished bacelors thesis)] . Pedagoška fakulteta Univerze v Mariboru, Slovenija.
de Bock, D., van Dooren, W., Janssens, D. in Verschaffel, L. (2007). The illusion of linearity: From analysis to improvement. New York, NY, ZDA: Springer.
Depaepe, F., Torbeyns, J., Vermeersch, N., Janssens, D., Janssens, R., Kelchtermans, G. in Van Dooren, W. (2015). Teachers' content and pedagogical content knowledge on rational numbers: A comparison of prospective elementary and lower secondary school teachers. Teaching and Teacher Education, 47, 82–92. CrossRef
diSessa, A. A. (2004). Metarepresentation: Native competence and targets for instructions. Cognition and Instruction, 22(3), 293–331. CrossRef
Dixon, J. K., Andreasen, J. B., Avila, C. L., Bawatneh, Z., Deichert, D. L., Howse, T. D. in Turner, M. S. (2014). Redefining the whole: Common errors in elementary preservice teachers' self-authored word problems for fraction subtraction. Investigations in Mathematics Learning, 7(1), 1–22.
Doerr, H. M. (2006). Teachers' ways of listening and responding to students' emerging mathematical models. ZDM Mathematics Education, 38(2), 255–268. CrossRef
Doorman, L. M. (2002). How to guide students? A reinvention course on modelling motion. V L. Fou-Lai (ur.), Common sense in Mathematics education (str. 97–114). Taipei, Taiwan: National Taiwan Normal University.
Dreyfus, T. (1995). Imagery for diagrams. V R. Sutherland in J. Mason (ur.), Exploiting mental imagery with computers in mathematics education (str. 3–19). Berlin, Nemčija: Springer.
Eldén, S. (2012). Inviting the messy: Drawing methods and children's voices. Childhood, 20(1), 66–81. CrossRef
English, L. D. (1993). Children's strategies for solving two- and three-dimensional combinatorial problems. Journal for Research in Mathematics Education, 24(3), 255–273. CrossRef
Freudenthal, H. (1991). Revisiting mathematics education: China lectures. Dordrecht, Nizozemska: Kluwer Academic.
Guillemin, M. (2004). Understanding illness: Using drawings as a research method. Health Policy & Services, 14(2), 272–289. CrossRef
Güler, G. in Çiltaş, A. (2011). The visual representation usage levels of mathematics teachers and students in solving verbal problems. International Journal of Humanities and Social Science, 1(11), 145–154.
Hesse-Biber, S. N. in Leavy, P. (2004). Distinguishing qualitative research. V S. N. Hesse-Biber in P. Leavy (ur.), Approaches to qualitative research: A reader on theory and practice (str. 1–15). New York, NY, ZDA: Oxford University Press.
Hiebert, J. in LeFevre, P. (1986). Conceptual and procedural knowledge in mathematics: an introductory analysis. V J. Hiebert (ur.), Conceptual and procedural knowledge: The case of mathematics (str. 1–27). Hillsdale, MI, ZDA: Lawrence Erlbaum Associates.
Hill, H. C., Rowan, B. in Ball, D. L. (2005). Effects of teachers' mathematical knowledge for teaching on student achivement. American Educational Research Journal, 42, 371–406. CrossRef
Hsieh, H.-F. in Shannon, S.E. (2005). Three approaches to qualitative content analysis. Qualitative Health Research, 15(9), 1277–1288. CrossRef
Isik, C. in Kar, T. (2012). An error analysis in division problems in fractions posed by pre-service elementary mathematics teachers. Educational Sciences: Theory and Practice, 12(3), 2303–2309.
Kearney, K. S. in Hyle, A. E. (2004). Drawing out emotions: The use of participant-produced drawings in qualitative inquiry. Qualitative Research, 4(3), 361–382. CrossRef
Kokol Voljč, V. (1995). Razvoj matematičnih pojmov - empirično ali teoretično? [Development of the mathematical concepts – empirical or theoretical?]. Educa, 5(1–2), 21–28.
Leikin, M., Waisman, I., Shaul, S. in Leikin, R. (2014). Brain activity associated with translation from a visual to a symbolic representation in algebra and geometry. Journal of Integrative Neuroscience, 13(1), 35–59. CrossRef
Levstek, T., Bregant, T. in Podlesek, A. (2013). Razvoj aritmetičnih sposobnosti [Development of arithmetical abilities]. Psihološka obzorja, 22, 115–121. CrossRef
Lipovec, A. in Antolin Drešar, D. (2015). Shematic and pictorial representations of exponentiation. V O. Fleischmann (ur.). The teaching profession: New challenges - new identities? (str. 137–144). Zürich, Švica: LIT.
Lipovec, A. in Ferk, E. (2012). Matematično znanje za poučevanje [Mathematical knowledge for teaching]. Pedagoška obzorja, 27(1/2), 53–70.
Lipovec A., Podgoršek, M. in Antolin Drešar, D. (2015). Grafične predstavitve nekaterih elementarnih matematičnih pojmov [Graphic representations of some mathematical concepts]. Pedagoška obzorja, 30(3/4), 19–35.
Lowrie, T. in Diezman, C. M. (2011). Solving graphics tasks: Gender differences in middle-school students. Learning and Instruction, 21(1), 109–125. CrossRef
MacDonald, A. (2013). Using children's representations to investigate meaning-making in mathematics. Australasian Journal of Early Childhood, 38(2), 65–73.
Marjanovič Umek, L. (2011). Otroška risba [Children's drawing]. V L. Marjanovič Umek in M. Zupančič, M. (ur.) Razvojna psihologija: Izbrane teme (str. 127–155). Ljubljana: Znanstvena založba Filozofske fakultete.
Maxwell, E. A. (1995). Fallacies in mathematics. Cambridge, Združeno kraljestvo: Cambridge University Press.
Mitchelmore, M. in White, P. (2004). Abstraction in mathematics and mathematics learning. V M. J. Hoines in A. B. Fuglestad (ur.), Proceedings of the 28th conference of the International Group for the Psychology of Mathematics Education: Vol. 3 (str. 329–336). Bergen, Norway: Bergen University College.
Mitchelmore, M. C. in White, P. (1995). Abstraction in mathematics: Conflict, resolution and application. Mathematics Education Research Journal, 7(1), 50–68. CrossRef
Nelsen, R. B. (2000). Proofs without words: Exercises in visual thinking. Washington, DC, ZDA: Mathematical Association of America.
Picker, S. H. in Berry, J. S. (2000). Investigating pupils image of mathematicians. Educational Studies in Mathematics, 43(1), 65–94. CrossRef
Pirie, C. in Kieren, T. (1994). Growth in mathematical understanding: How can we characterise it and how can we represent it? Educational Studies in Mathematics, 26(2/3), 165–190. CrossRef
Presmeg, N. C. (1992). Prototypes, metaphors, metonymies and imaginative rationality in high school mathematics. Educational Studies in Mathematics, 23(6), 595–610. CrossRef
Rittle-Johnson, B., Schneider, M. in Star, J. R. (2015). Not a one-way street: Bidirectional relations between procedural and conceptual knowledge of mathematics. Education Psychological Review, 27(4), 587–597. CrossRef
Rittle-Johnson, B. in Siegler, R. S. (1998). The relation between conceptual and procedural knowledge in learning mathematics: A review. V C. Donlan (ur.), The development of mathematical skills (str. 75–110). London, Združeno kraljestvo: Psychology Press.
Rivera, F. D. (2014). From math drawings to algorithms: Emergence of whole number operations in children. ZDM Mathematics Education, 46(1), 59–77. CrossRef
Rose, G. (2014). On the relation between 'visual research methods' and contemporary visual culture. Sociological Review, 62(1), 24–46. CrossRef
Saban, A. in Akbulut, M. G. (2012). An investigation of primary school student's perceptions of violence revealed through their drawings. Turkish Journal of Education, 1(1), 21–37.
Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20–26.
Sowell, E. J. (1989). Effects of manipulative materials in mathematics instruction. Journal for Research in Mathematics Education, 20(5), 498–505. CrossRef
Star, J. R. (2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36(5), 404–411.
Šali, B. (1961). Rorscachova psihodiagnostika [Rorschach psychodiagnostic]. Kranj, Slovenija: Zavod za izobraževanje kadrov in proučevanje organizacije dela.
Thom, J. S. in McGarvey, L. M. (2015). The act and artifact of drawing(s): Observing geometric thinking with, in, and through children's drawings. ZDM Mathematics Education, 47(3), 465–481. CrossRef
Van den Heuvel-Panhuizen, M. (2003). The didactical use of models in Realistic Mathematics Education: An example from a longitudinal trajectory on percentage. Educational Studies in Mathematics, 54(1), 9–35. CrossRef
Van Steenbrugge, H., Lesage, E., Valcke, M. in Desoete, A. (2014). Preservice elementary school teachers' knowledge of fractions: A mirror of students' knowledge? Journal of Curriculum Studies, 46(1), 138–161. CrossRef
Zhang, Y. in Wildemuth, B. M. (2009). Qualitative analysis of content. V Wildemuth, B. (ur.), Applications of social research methods to questions in information and library science (str. 308–319). Westport, CT, ZDA: Libraries Unlimited.
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