Proof-related reasoning in upper secondary school: characteristics of Swedish and Finnish textbooks

References

• Alfredsson, L., Bråting, K., Erixon, P., & Heikne, H. (2011). Matematik 5000: Kurs 2c, blå Lärobok. Stockholm: Natur & Kultur. Swedish.
   Google Scholar
• Alfredsson, L., Bråting, K., Erixon, P., & Heikne, H. (2013). Matematik 5000: Kurs 5, blå Lärobok. Stockholm: Natur & Kultur. Swedish.
   Google Scholar
• Almeida, D. (2001). Pupils’ proof potential. International Journal of Mathematical Education in Science and Technology, 32(1), 53–60.
   Google Scholar
• Balacheff, N. (1988). Aspects of proof in pupils’ practice of school mathematics. In D. Pimm (Ed.), Mathematics, teachers and children (pp. 216–235). London: Holdder& Stoughton.
   Google Scholar
• Baylis, J. (1983). Proof — the essence of mathematics. International Journal of Mathematical Education in Science and Technology, 14(4), 409–414.
   Google Scholar
• Bell, A. W. (1976). A study of pupils’ proof-explanations in mathematical situations. Educational Studies in Mathematics, 7(1–2), 23–40.
   Google Scholar
• Bergwall, A. (2015). On a generality framework for proving tasks. In K. Krainer & N. Vondrová (Eds.), Proceedings of the ninth conference of the European society for research in mathematics education ( CERME9, 4–8 February 2015) (pp. 86–92). Prague, Czech Republic: Charles University in Prague, Faculty of Education and ERME.
   Google Scholar
• Bergwall, A. (2017). Conceptualizing reasoning-and-proving opportunities in textbook expositions: cases from secondary calculus. In T. Dooley & G. Gueudet (Eds.), Proceedings of the tenth congress of the European society for research in mathematics education ( CERME10, February 1–5, 2017) (pp. 91–98). Dublin, Ireland: DCU Institute of Education and ERME.
   Google Scholar
• Bergwall, A., & Hemmi, K. (2017). The state of proof in Finnish and Swedish mathematics textbooks–capturing differences in approaches to upper secondary integral calculus. Mathematical Thinking and Learning, 19(1), 1–18.
   Google Scholar
• Boesen, J., Helenius, O., Bergqvist, E., Bergqvist, T., Lithner, J., Palm, T., & Palmberg, B. (2014). Developing mathematical competence: From the intended to the enacted curriculum. The Journal of Mathematical Behavior, 33, 72–87.
   Google Scholar
• Davis, J. D., Smith, D. O., Roy, A. R., & Bilgic, Y. K. (2014). Reasoning-and-proving in algebra: The case of two reform-oriented US textbooks. International Journal of Educational Research, 64, 92–106.
   Web of Science ®Google Scholar
• De Villiers, M. D. (1990). The role and function of proof in mathematics. Pythagoras, 24, 17–24.
   Google Scholar
• Finnish National Board of Education. (2004). National core curriculum for upper secondary schools 2003. Vammala: Vammalankirjapaino Oy.
   Google Scholar
• Glasnovic Gracin, D. (2018). Requirements in mathematics textbooks: A five-dimensional analysis of textbook exercises and examples. International Journal of Mathematical Education in Science and Technology, 49(7), 1003–1024.
   Web of Science ®Google Scholar
• Hanna, G. (2018). Reflections on proof as explanation. In A. Stylianides, & G. Harel (Eds.), Advances in mathematics education research on proof and proving (pp. 3–18). Cham: Springer.
   Google Scholar
• Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. In A. Schoenfeldt, J. Kaput, & E. Dubinsky (Eds.), Research in collegiate mathematics education III (pp. 234–283). Providence, RI: American Mathematical Society.
   Google Scholar
• Harel, G., & Sowder, L. (2007). Toward comprehensive perspectives on the learning and teaching of proof. In F. K. Lester Jr (Ed.), Second Handbook of research on mathematics teaching and learning (pp. 805–842). Charlotte, NC: Information Age Publishing.
   Google Scholar
• Hemmi, K. (2008). Students’ encounter with proof: The condition of transparency. ZDM, 40(3), 413–426.
   Google Scholar
• 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.
   Web of Science ®Google Scholar
• Hemmi, K., & Ryve, A. (2015). Effective mathematics teaching in Finnish and Swedish teacher education discourses. Journal of Mathematics Teacher Education, 18(6), 501–521.
   Web of Science ®Google Scholar
• Hiebert, J., & Grouws, D. A. (2007). The effects of classroom mathematics teaching on students’ learning. In F. K. Lester Jr (Ed.), Second Handbook of research on mathematics teaching and learning (pp. 371–404). Charlotte, NC: Information Age Publishing.
   Google Scholar
• Joutsenlahti, J., & Vainionpää, J. (2010). Oppimateriaali matematiikan opetuksessa ja osaamisessa[learning materials in the teaching and learning of mathematics]. In E. K. Niemi, & J. Metsämuuronen (Eds.), Miten matematiikan taidot kehittyvät? Matematiikan oppimistulokset peruskoulun viidennen vuosiluokan jälkeen vuonna 2008 [How do pupils’ mathematical skills develop? The learning outcomes in the end of the fifth grade in compulsory school]. (pp. 137–146). Helsinki: Opetushallitus [The Finnish National Board of Education]. Finnish.
   Google Scholar
• Knutsson, M., Hemmi, K., Bergwall, A., & Ryve, A. (2013). School-based mathematics teacher education in Sweden and Finland: characterizing mentor-prospective teacher discourse. In B. Ubuz, C. Haser, & M. A. Mariotti (Eds.), Proceedings of the eighth congress of the European society for research in mathematics education ( CERME8, February 6–10, 2013) (pp. 1905–1914). Ankara, Turkey: Middle East Technical University and ERME.
   Google Scholar
• Kontkanen, P., Lehtonen, J., Luosto, K., & Westermark, H. (2007). Ellips 6: Sannolikhet och statistik. Helsingfors: Schildts Förlags Ab. Swedish.
   Google Scholar
• Kontkanen, P., Lehtonen, J., Luosto, K., Savolainen, S., & Westermark, H. (2007). Ellips 8: Rot- och logaritmfunktioner. Helsingfors: Schildts Förlags Ab. Swedish.
   Google Scholar
• Lakatos, I. (2015). Proofs and refutations: The logic of mathematical discovery. Cambridge, UK: Cambridge University Press.
   Google Scholar
• Movshovitz-Hadar, N. (1988). Stimulating presentation of theorems followed by responsive proofs. For the Learning of Mathematics, 8(2), 12–30.
   Google Scholar
• National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
   Google Scholar
• National Council of Teachers of Mathematics. (2009). Focus in high school mathematics: Reasoning and sense making. Reston, VA: National Council of Teachers of Mathematics.
   Google Scholar
• Otten, S., Gilbertson, N. J., Males, L. M., & Clark, D. L. (2014). The mathematical nature of reasoning-and-proving opportunities in geometry textbooks. Mathematical Thinking and Learning, 16(1), 51–79.
   Web of Science ®Google Scholar
• Sevimli, E. (2018). Undergraduates’ propositional knowledge and proof schemes regarding differentiability and integrability concepts. International Journal of Mathematical Education in Science and Technology, 49(7), 1052–1068.
   Web of Science ®Google Scholar
• Stacey, K., & Vincent, J. (2009). Modes of reasoning in explanations in Australian eighth-grade mathematics textbooks. Educational Studies in Mathematics, 72(3), 271–288.
   Web of Science ®Google Scholar
• Stein, M. K., Remillard, J., & Smith, M. S. (2007). How curriculum influences student learning. In F. K. Lester Jr (Ed.), Second Handbook of research on mathematics teaching and learning (pp. 319–369). Charlotte, NC: Information Age Publishing.
   Google Scholar
• Stylianides, A. J. (2007). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38(3), 289–321.
   Web of Science ®Google Scholar
• Stylianides, G. J. (2009). Reasoning-and-Proving in school mathematics textbooks. Mathematical Thinking and Learning, 11(4), 258–288.
   Web of Science ®Google Scholar
• Stylianides, G. J. (2014). Textbook analyses on reasoning-and-proving: Significance and methodological challenges. International Journal of Educational Research, 64, 63–70.
   Web of Science ®Google Scholar
• Stylianides, G., Stylianides, A., & Weber, K. (2017). Research on the teaching and learning of proof: Taking stock and moving forward. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 237–236). Reston, VA: The National Council of Teachers in Mathematics.
   Google Scholar
• Swedish National Agency for Education. (2011). Läroplan, examensmål och gymnasiegemensamma ämnen för gymnasieskola 2011 [Curriculum, exam objectives and upper secondary school subjects 2011]. Stockholm: Skolverket. Swedish.
   Google Scholar
• Szabo, A., Larson, N., Viklund, G., Dufåker, D., & Marklund, M. (2012). Matematik Origo 2c. Stockholm: Sanoma Utbildning AB. Swedish.
   Google Scholar
• Szabo, A., Larson, N., Viklund, G., Dufåker, D., & Marklund, M. (2013). Matematik Origo 5. Stockholm: Sanoma Utbildning AB. Swedish.
   Google Scholar
• Thompson, D. R., Senk, S. L., & Johnson, G. J. (2012). Opportunities to learn reasoning and proof in high school mathematics textbooks. Journal for Research in Mathematics Education, 43(3), 253–295.
   Web of Science ®Google Scholar
• Weber, K. (2001). Student difficulty in constructing proofs: The need for strategic knowledge. Educational Studies in Mathematics, 48(1), 101–119.
   Google Scholar
• Weinberg, A., & Wiesner, E. (2011). Understanding mathematics textbooks through reader-oriented theory. Educational Studies in Mathematics, 76(1), 49–63.
   Web of Science ®Google Scholar