Då detta var en liten, kvalitativ, undersökning så öppnar detta upp för många uppföljningsstudier. Exempelvis så kan det utföras en likadan eller snarlik undersökning fast i större omfattning och med ett bredare urval för att få ett mer signifikant underlag. En sådan studie skulle då även kunna involvera icke svensktalande finska elever, om undersökarna talar finska eller undersökningen utförs på engelska. Undersökningen kan givetvis även genomföras mellan andra länder än Sverige och Finland, dessa var de länder som intresserade författarna.
En kvantitativ uppföljande studie vore givande för att få ett pålitligt och statistiskt signifikant resultat. Detta skulle exempelvis kunna ske med hjälp av enkäter som skickas ut istället för intervjuer som sker i person. Att använda en annan metod för datainsamling påverkar givetvis undersökningens natur och bör has i åtanke vid eventuell utformad samt eventuellt utförande.
43
7 Referenser
Andrews, P. (2014). The emperor’s new clothes: PISA, TIMSS and Finnish mathematics. I A.-S. Röj-Lindberg, L. Burman, B. Kurtén-Finnäs & K. Linnanmäki (Red.), Spaces for
learning: past, present and future (s. 43-65). Vasa: Åbo Akademi University.
Artigue M. (2001) What Can We Learn from Educational Research at the University Level?. I D. Holton, M. Artigue, U. Kirchgräber, J. Hillel, M. Niss, A. Schoenfeld (Red.), The
Teaching and Learning of Mathematics at University Level (s. 207-220). New ICMI Study
Series, 7. Dordrecht: Springer. doi: 10.1007/0-306-47231-7_21
Astala, K., Kivelä, S. K., Koskela, P., Martio, O., Näätänen, M., & Tarvainen, K. (2006). The PISA survey tells only a partial truth of Finnish children’s mathematical skills. Matilde, 29, 9.
Boyce, W., & DiPrima, R. (2013). Elementary Differential Equations and Boundary Value
Problems. Hoboken, New Jersey: John Wiley & Sons.
Boyd, R. (2002). Scientific Realism. I Stanford Encyclopedia of Philosophy. Tillgänglig:
https://plato.stanford.edu/archives/sum2010/entries/scientific-realism/
Byrd, K. L., & MacDonald, G. (2005). Defining College Readiness from the Inside Out: First-Generation College Student Perspectives. Community College Review, 33(1), 22-37.
doi:10.1177/009155210503300102
Conley, D. (2008). Rethinking College Readiness. New directions for higher education,
2008(144), 3-13. doi:10.1002/he.321
Cornu, B. (2001). Limits. I D. Tall (Red.), Advanced mathematical thinking (s. 153-166). Dordrecht: Springer.
Dahlgren, L. O., & Johansson, K. (2011). Fenomenografi. I B. Fejes, A., & Thornberg, R. (Red.), Handbok i kvalitativ analys (s. 122-135). Stockholm: Liber AB.
Denscombe, M. (2016). Forskningshandboken för småskaliga forskningsprojekt inom
samhällsvetenskaperna. Lund: Studentlitteratur AB.
Fernández-Plaza, J. A., & Simpson, A. (2016). Three concepts or one?: Students’
understanding of basic limit concepts. Educational Studies in Mathematics, 93(3), 315-332. doi:10.1007/s10649-016-9707-6
Gillham, B. (2008). Forskningsintervjun: Tekniker och genomförande. Lund: Studentlitteratur AB.
Güçler, B. (2013). Examining the discourse on the limit concept in a beginning-level calculus classroom. Educational Studies in Mathematics, 82(3), 439-453. doi:10.1007/s10649-012-9438-2
Hanna, G. (2000). A critical examination of three factors in the decline of proof.
Interchange, 31(1), 21–33. doi:10.1023/A:1007630729662
Hautamäki, J., Laaksonen, S., & Scheinin, P. (2008). Level and balance of achievement. I J. Hautamäki, E. Harjunen, H. Airi, T. Karjalainen, S. Kupiainen, S. Laaksonen, J. Lavonen, E. Pehkonen, P. Rantanen, & P. Scheinin (Red.), PISA 2006 Finland — analyses, reflections
44 Hemmi, K., Lepik, M., & Viholainen, A. (2013). Analysing proof-related competences in Estonian, Finnish and Swedish mathematics curricula—towards a framework of developmental proof. Journal of Curriculum Studies, 45(3), 354–378.
doi:10.1080/00220272.2012.754055
Holmlund, H., Häggblom, J., Lindahl, E., Martinson, S., Sjögren, A., Vikman, U., & Öckert, B. (2014). Decentralisering, skolval och fristående skolor: resultat och likvärdighet i
svensk skola (IFAU-rapport, 2014:25). Uppsala: Institutet för arbetsmarknads- och
utbildningspolitisk utvärdering.
Jerkert, J. (2017). Scientific theory and practise: Six introductory texts. Stockholm: Division of philosophy. Department of philosophy and history. KTH Royal Institute of Technology.
Johansson, B. (1998). Förkunskapsproblem i matematik? Nämnaren, (4), 20-22. Tillgänglig: http://ncm.gu.se/pdf/namnaren/2022_98_4.pdf
Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn
mathematics. Washington, DC: The National Academies Press.
Kroksmark, T. (2007). Fenomenografisk didaktik: En möjlighet. Didaktisk Tidskrift, 17(2-3), 1-50.
Kungliga Tekniska Högskolan. (2015). Kursplan. SF1625 Envariabelanalys. Stockholm: Kungliga Tekniska Högskolan, SCI/Matematik.
Kvale, S. (2013). Doing Interviews. London: SAGE Publications.
Kvale, S., & Brinkmann, S. (2014). Den kvalitativa forskningsintervjun. Lund: Studentlitteratur AB.
Larsson, S. (2011). Kvalitativ analys - exemplet fenomenografi. Lund: Studentlitteratur. Martio, O. (2009). Long Term Effects in Learning Mathematics in Finland - Curriculum Changes and Calculators. The Teaching of Mathematics, 12(2), 51-56.
Marton, F. (1981). Phenomenography - Describing Conceptions of the World Around Us.
Instructional Science, 10(2), 177-200. doi:10.1007/BF00132516
Marton, F. (2005). Phenomenography: A Research Approach to Investigating Different Understandings of Reality. I R. Sherman & B. Webb (Red.), Qualitative Research In
Education: Focus And Methods (s. 140-160). London: RoutLedgeFalmer.
Nationalencyklopedin [NE]. (2017a). beroende variabel. Tillgänglig:
https://www.ne.se/uppslagsverk/encyklopedi/lång/beroende-variabel
Nationalencyklopedin [NE]. (2017b). definitionsmängd. Tillgänglig:
http://www.ne.se/uppslagsverk/encyklopedi/lång/definitionsmängd
Nationalencyklopedin [NE]. (2017c). funktion. Tillgänglig:
http://www.ne.se/uppslagsverk/encyklopedi/lång/funktion
Nationalencyklopedin [NE]. (2017d). gränsvärde. Tillgänglig:
http://www.ne.se/uppslagsverk/encyklopedi/lång/gränsvärde-(övreplusundrematematik)
Nationalencyklopedin [NE]. (2017e). oberoende variabel. Tillgänglig:
45 Nationalencyklopedin [NE]. (2017f). värdemängd. Tillgänglig:
https://www.ne.se/uppslagsverk/encyklopedi/lång/värdemängd
OECD (2016), PISA 2015 Results (Volume I): Excellence and Equity in Education, OECD Publishing, Paris. doi:10.1787/9789264266490-en
Parameswaran, R. (2007). On understanding the notion of limits and infinitesimal quantities. International Journal of Science and Mathematics Education, 5(2), 193-216. doi:10.1007/s10763-006-9050-y
Persson, A., & Böiers L-. C. (2005). Analys i flera variabler. Lund: Studentlitteratur AB. Persson, A., & Böiers L-. C. (2010). Analys i en variabel. Lund: Studentlitteratur AB. Pinter, C. (2010). A Book of Abstract Algebra. (2. uppl.). Mineola, New York: Dover Publications.
Przenioslo, M. (2004). Images of the limit of function formed in the course of mathematical studies at the university. Educational Studies in Mathematics, 55(1-3), 103-132.
doi:10.1023/B:EDUC.0000017667.70982.05
Saff, E., & Snider, A. D. (2014). Fundamentals of Complex Analysis Engineering, Science,
and Mathematics. London, Essex: Pearson Education.
Sfard, A., & Linchevski, L. (1994). The Gains and the Pitfalls of Reification: The Case of Algebra. Educational Studies in Mathematics. 26(2/3), 191-228. doi:10.1007/BF01273663 Sfard, A. (1991). On the Dual Nature of Mathematical Conceptions: Reflections on Processes and Objects as Different Sides of the Same Coin. Educational Studies in Mathematics. 22(1), 1-36. doi:10.1007/BF00302715
Sierpínska, A. (1987). Humanities students and epistemological obstacles related to limits.
Educational studies in mathematics, 18(4), 371-397. doi:10.1007/BF00240986
Sierpinska, A. (1992). On Understanding the Notion of Function. I E. Dubinsky & G. Harel (Red.), The concept of function: Aspects of epistemology and pedagogy (s. 25-58). Washington, DC: Mathematical Association of America.
Skolverket. (2011a). Matematik. Hämtad 2017-11-25 från
https://www.skolverket.se/laroplaner-amnen-och-kurser/gymnasieutbildning/gymnasieskola/mat
Skolverket. (2011b). Naturvetenskapsprogrammet. Hämtad 2017-11-25 från
https://www.skolverket.se/laroplaner-amnen-och- kurser/gymnasieutbildning/gymnasieskola/programstruktur-och-examensmal/naturvetenskapsprogrammet#anchor_2
Sparr, G., & Sparr, A. (2000). Kontinuerliga system: Övningsbok. Lund: Studentlitteratur AB.
Stadler, E. (2009). Stadieövergången mellan gymnasiet och universitetet: Matematik och
lärande ur ett studerandeperspektiv (Doktorsavhandling, Acta Wexionensia, 195). Göteborg: Intellecta Infolog. Tillgänglig:
https://www.diva-portal.org/smash/get/diva2:286375/FULLTEXT01.pdf.
Tall, D. (1980, Augusti). Mathematical intuition, with special reference to limiting
processes. Paper presenterat vid PME4, Proceedings of the fourth international conference for the psychology of mathematics education, Berkley, California, USA.
46 Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics,12(2), 151-169. doi:0013-1954/81/0122-0151
Thomas, M. O. J., de Freitas Druck, I., Huillet, D., Ju, M.-K., Nardi, E., Rasmussen, C., & Xie, J. (2012, Juli). Survey team 4: Key mathematical concepts in the transition from
secondary to university. Paper presenterat vid ICME12, the 12th International Congress of
Mathematical Education, Seoul, Korea.
Thunberg, H., & Filipsson, L. (2005). Gymnasieskolans mål och Högskolans
förväntningar: En jämförande studie om matematikundervisningen. (GMHF). Stockholm:
KTH.
Utbildningsstyrelsen. (2016). Grunderna för gymnasiets läroplan 2015. Helsingfors: Utbildningsstyrelsen.
Vetenskapsrådet. (2002). Forskningsetiska principer: inom
humanistisk-samhällsvetenskaplig forskning. Stockholm: Vetenskapsrådet.
Vretblad, A. (2005). Fourier Analysis and Its Applications. New York: Springer-Verlag New York Inc.
Zill, D., & Cullen, M. (2009). Differential Equations: With Boundary-Value Problems. Pacific Grove, Kalifornien: Brooks/Cole.
Åsberg, R. (2001). Ontologi, epistemologi och metodologi: en kritisk genomgång av vissa
grundläggande vetenskapsteoretiska begrepp och ansatser (Rev. uppl.) (IPD-rapport,
47
Bilagor
Bilaga 1 - Sierpinska
”U(f)-1: Identification of changes observed in the surrounding world as a practical problem to solve.”
”U(f)-2: Identification of regularities in relationships between changes as a way to deal with the changes.”
”EO(f)-1: (A philosophy of mathematics): Mathematics is not concerned with practical problems.”
”EO(f)-2: (A philosophy of mathematics) Computational techniques used in producing tables of numerical relationships are not worthy of being an object of study in mathematics” ”U(f)-3: Identification of the subjects of change in studying changes.”
”EO(f)-3: (Unconscious scheme of thought) Regarding changes as phenomena; focussing on how things change, ignoring what changes.”
”U(f)-4: Discrimination between two modes of mathematical thought: one in terms of known and unknown quantities, the other - in terms of variable and constant quantities.”
”EO(f)-4: (Unconscious scheme of thought) Thinking in terms of equations and unknowns to be [...]”
”U(f)-5: Discrimination between the dependent and the independent variables”
”EO(f)-5: (Unconscious scheme of thought) Regarding the order of variables as irrelevant.” ”EO(f)-6: (An attitude towards the concept of number) A heterogeneous conception of number.”
”U(f)-6: Generalization and synthesis of the notion of number.”
”EO(f)-7: (An attitude towards the notion of number) A Pythagorean philosophy of number: everything is number.”
”U(f)-7: Discrimination between number and quantity.”
”EO(f)-8: (An unconscious scheme of thought) Laws in physics and functions in
mathematics have nothing in common; they belong to different domains (compartments) of thought.”
”U(f)-8: Synthesis of the concepts of law and the concept of function; in particular,
awareness of the possible use of functions in modelling relationships between physical or other magnitudes.”
”EO(f)-9: (An unconscious scheme of thought) Proportion is a privileged kind of relationship.”
”EO(f)-10: (A belief concerning mathematical methods) Strong belief in the power of formal operations on algebraic expressions”
”EO(f)-11: (A conception of function) Only relationships describable by analytic formulae are worthy of being given the name of functions.”
”U(f)-9: Discrimination between a function and the analytic tools sometimes used to describe its law.”
”EO(f)-12: (A conception of definition) Definition is a description of an object otherwise known by senses or insight. The definition does not determine the object; rather the object determines the definition. A definition is not binding logically.”
”U(f)-10: Discrimination between mathematical definitions and descriptions of objects.” ”U(f)-11: Synthesis of the general conception of function as an object.”
”EO(f)-13: (Conception of functions) Functions are sequences.”
”U(f)-13: Discrimination between the notions of function and sequence.”
”EO(f)-14: (Conception of coordinates) Coordinates of a point are line segments (not numbers).”
”U(f)-14: Discrimination between coordinates of a point of a curve and the line segments fulfilling some function for the curve.”
”EO(f)-15: (Conception of graph of function) The graph of a function is a geometrical model of the functional relationship. It need not be faithful, it may contain points (x,y) such that the function is not defined in x.”
48
”U(f)-15: Discrimination between different means of representing functions and the functions themselves.”
”U(f)-16: Synthesis of the different ways of giving functions, representing functions and speaking about functions.”
”EO(f)-16: (A conception of variable) The changes of a variable are changes in time.” ”U(f)-17: Generalization of the notion of variable.”
”U(f)-18 Synthesis of the roles on notions of function and cause in the history of science: awareness of the fact that searches for functional and causal relationships are both expressions of the human endeavour to understand and explain changes in the world.” ”U(f)-19: Discrimination between the notions of functional and causal relationships.”
49