Mathematical Reasoning Requirements in Swedish National Physics Tests

  • Helena JohanssonEmail author


This paper focuses on one aspect of mathematical competence, namely mathematical reasoning, and how this competency influences students’ knowing of physics. This influence was studied by analysing the mathematical reasoning requirements upper secondary students meet when solving tasks in national physics tests. National tests are constructed to mirror the goals stated in the curricula, and these goals are similar across national borders. The framework used for characterising the mathematical reasoning required to solve the tasks in the national physics tests distinguishes between imitative and creative mathematical reasoning. The analysis process consisted of structured comparisons between representative student solutions and the students’ educational history. Of the 209 analysed tasks, 3/4 required mathematical reasoning in order to be solved. Creative mathematical reasoning, which, in particular, involves reasoning based on intrinsic properties, was required for 1/3 of the tasks. The results in this paper give strong evidence that creative mathematical reasoning is required to achieve higher grades on the tests. It is also confirmed that mathematical reasoning is an important and integral part of the physics curricula; and, it is suggested that the ability to use creative mathematical reasoning is necessary to fully master the curricula.


Creative mathematical reasoning Mathematical reasoning Physics tests Swedish national assessment Upper secondary school 


  1. Basson, I. (2002). Physics and mathematics as interrelated fields of thought development using acceleration as an example. International Journal of Mathematical Education in Science and Technology, 33(5), 679–690. doi: 10.1080/00207390210146023.CrossRefGoogle Scholar
  2. Bing, T. (2008). An epistemic framing analysis of upper level physics students’ use of mathematics (Doctoral dissertation). Retrieved from
  3. Björk, L.-E. & Brolin, H. (2001). Matematik 3000, kurs A och B [Mathematics 3000, course A and B]. Stockholm, Sweden: Natur och Kultur.Google Scholar
  4. Björk, L.-E. & Brolin, H. (2006). Matematik 3000, kurs C och D  [Mathematics 3000, course C and D]. Stockholm, Sweden: Natur och Kultur.Google Scholar
  5. Björkqvist, O. (2001). Matematisk problemlösning [Mathematical problem solving]. In B. Grevholm (Ed.), Matematikdidaktik: Ett nordiskt perspektiv (pp. 115–132). Lund, Sweden: Studentlitteratur.Google Scholar
  6. Blum, W. & Niss, M. (1991). Applied mathematical problem solving, modelling, applications, and links to other subjects—state, trends and issues in mathematics instruction. Educational Studies in Mathematics, 22(1), 37–68. doi: 10.1007/BF00302716.CrossRefGoogle Scholar
  7. Boesen, J. (2006). Assessing mathematical creativity (Doctoral dissertation). Umeå, Sweden: Umeå University.Google Scholar
  8. Boesen, J., Lithner, J. & Palm, T. (2010). The relation between types of assessment tasks and the mathematical reasoning student use. Educational Studies in Mathematics, 75(1), 89–105. doi: 10.1007/s10649-010-9242-9.CrossRefGoogle Scholar
  9. 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. Journal of Mathematical Behavior, 33, 72–87. doi: 10.1016/j.jmathb.2013.10.001.
  10. Clement, J. (1985). Misconceptions in graphing. In L. Streefland (Ed.), Proceedings of the ninth international conference for the psychology of mathematics education (pp. 369–375). Utrecht, The Netherlands: Utrecht University.Google Scholar
  11. Dall’Alba, G., Walsh, E., Bowden, J., Martin, E., Masters, G., Ramsden, P. & Stephanou, A. (1993). Textbook treatments and students’ understanding of acceleration. Journal of Research in Science Teaching, 30(7), 621–635. doi: 10.1002/tea.3660300703.CrossRefGoogle Scholar
  12. Department of Applied Educational Science (2011-10-06). Some earlier given test in physics . Retrieved from
  13. diSessa, A. A. (2004). Contextuality and coordination in conceptual change. In E. Redish & M. Vicentini (Eds.), Proceedings of the international school of physics “Enrico Fermi:” Research on physics education (pp. 137–156). Amsterdam, Netherlands: ISO Press/Italian Physics Society.Google Scholar
  14. Doorman, L. M. & Gravemeijer, K. P. E. (2009). Emergent modeling: Discrete graphs to support the understanding of change and velocity. ZDM-The International Journal on Mathematics Education, 41, 199–211. doi: 10.1007/s11858-008-0130-z.CrossRefGoogle Scholar
  15. Ekbom, L., Lillieborg, S., Larsson, S., Ölme, A. & Jönsson, U. (2004). Tabeller och formler för NV- och TE- programmen [Tables and formulas for the NV- and TE- programs] (5th ed.). Stockholm, Sweden: Liber.Google Scholar
  16. English, L. & Sriraman, B. (2010). Problem solving for the 21st century. In L. English & B. Sriraman (Eds.), Theories of mathematics education (pp. 263–290). Berlin, Heidelberg: Springer.CrossRefGoogle Scholar
  17. Engström, S. (2011). Att vördsamt värdesätta eller tryggt trotsa. Gymnasiefysiken, undervisningstraditioner och fysiklärares olika strategier för energiundervisning [To respectfully value or confidently defy. Upper secondary physics, teaching traditions and physics teachers’ different strategies for energy education] (Doctoral dissertation). Retrieved from
  18. Garden, R. A., Lie, S., Robitaille, D. F., Angell, C., Martin, M. O., Mullis, I. V. S. et al. (2006). TIMSS advanced 2008 assessment frameworks. Boston, MA: TIMSS & PIRLS International Study Center, Lynch School of Education, Boston College.Google Scholar
  19. Hanna, G. & Jahnke, H. N. (2002). Another approach to proof: Arguments from physics. ZDM-The International Journal on Mathematics Education, 34(1), 1–8. doi: 10.1007/BF02655687.Google Scholar
  20. Haylock, D. (1997). Recognising mathematical creativity in schoolchildren. ZDM-The International Journal on Mathematics Education, 29(3), 68–74. doi: 10.1007/s11858-997-0002-y.CrossRefGoogle Scholar
  21. Johansson, H. (2013). Mathematical Reasoning in Physics Tests Requirements. Relations, Dependence (Licentiate Thesis). Göteborg, Sweden: University of Gothenburg.Google Scholar
  22. Lesh, R. & Zawojewski, J. (2007). Problem solving and modeling. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 763–804). Charlotte, NC: Information Age.Google Scholar
  23. Lithner, J. (2008). A research framework for creative and imitative reasoning. Educational Studies in Mathematics, 67(3), 255–276. doi: 10.1007/s10649-007-9104-2.CrossRefGoogle Scholar
  24. Marongelle, K. A. (2004). How students use physics to reason about calculus tasks. School Science and Mathematics, 104(6), 258–272. doi: 10.1111/j.1949-8594.2004.tb17997.x.CrossRefGoogle Scholar
  25. Michelsen, C. (2005). Expanding the domain—variables and functions in an interdisciplinary context between mathematics and physics. In A. Beckmann, C. Michelsen & B. Sriraman (Eds.), Proceedings of the 1st international symposium of mathematics and its connections to the arts and sciences (pp. 201–214). Schwäbisch Gmünd, Germany: The University of Education.Google Scholar
  26. Ministry of Education and Research (2001). Samverkande styrning: om läroplanerna som styrinstrument [Interacting governance. About the curricula as means for policy change] Ministry publications series Ds 2001:48. Retrieved from
  27. Mulhall, P. & Gunstone, R. (2012). Views about learning physics held by physics teachers with differing approaches to teaching physics. Journal of Science Teacher Education, 23(5), 429–449. doi: 10.1007/s10972-012-9291-2.CrossRefGoogle Scholar
  28. National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  29. Next Generation Science Standards Lead States (2013). Next generation science standards: For states, by states (HS-PS). USA: Achieve, Inc. on behalf of the twenty-six states and partners that collaborated on the NGSS.Google Scholar
  30. Nguyen, N.-L. & Meltzer, D. (2003). Initial understanding of vector concepts among students in introductory physics courses. American Journal of Physics, 71(6), 630–638. doi: 10.1119/1.1571831.CrossRefGoogle Scholar
  31. Organisation for Economic Co-operation and Development (2009). PISA 2009 Assessment framework—key competences in reading, mathematics and science. Retrieved from
  32. Palm, T., Boesen, J. & Lithner, J. (2011). Mathematical reasoning requirements in upper secondary level assessments. Mathematical Thinking and Learning, 13(3), 221–246. doi: 10.1080/10986065.2011.564994.CrossRefGoogle Scholar
  33. Pólya, G. (1954). Mathematics and plausible reasoning (vols. I and II). Princteton, NJ: Princeton University Press.Google Scholar
  34. Planinic, M., Milin-Sipus, Z., Katic, H., Susac, A. & Ivanjek, L. (2012). Comparison of student understanding of line graph slope in physics and mathematics. International Journal of Science and Mathematics Education, 10(6), 1393–1414. doi: 10.1007/s10763-012-9344-1.CrossRefGoogle Scholar
  35. Pålsgård, J., Kvist, G. & Nilson, K. (2005a). Ergo Fysik A [Ergo Physics A]. Stockholm, Sweden: Liber.Google Scholar
  36. Pålsgård, J., Kvist, G. & Nilson, K. (2005b). Ergo Fysik B [Ergo Physics B]. Stockholm, Sweden: Liber.Google Scholar
  37. Redish, E. F. (2003). Teaching physics with the physics suite. Hoboken, NJ: Wiley.Google Scholar
  38. Redish, E. F. & Gupta, A. (2009). Making meaning with math in physics: a semantic analysis. Paper presented at GIREP 2009, Leicester, United Kingdom. Retrieved from
  39. Schoenfeld, A. H. (1985). Mathematical problem solving. Orlando, FL: Academic.Google Scholar
  40. Schoenfeld, A. H. (1992). Learning to think mathematically: problem solving, metacognition, and sense-making in mathematics. In D. Grouws (Ed.), Handbook for research on mathematics teaching and learning (pp. 334–370). New York, NY: Macmillan.Google Scholar
  41. Silver, E. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing. ZDM-The International Journal on Mathematics Education, 29(3), 75–80. doi: 10.1007/s11858-997-0003-x.CrossRefGoogle Scholar
  42. Sumpter, L. (2013). Themes and interplay of beliefs in mathematical reasoning. International Journal of Science and Mathematics Education, 11(5), 1115–1135. doi: 10.1007/s10763-012-9392-6.CrossRefGoogle Scholar
  43. Swedish National Agency for Education (2000). Ämne - Fysik, gymnasieskolan [Syllabuses for upper secondary Physics]. Retrieved 2014-05-22 from
  44. Swedish National Agency for Education (2003). Lusten att lära – Med fokus på matematik: Nationella kvalitetsgranskningar 2001–2002 [Lust to learn—with focus on mathematics: National quality inspections 2001–2003]. Stockholm, Sweden: Fritzes.Google Scholar
  45. Swedish National Agency for Education (2005). Skolverkets Provbank. Hur används den och vad tycker användarna? [The National Test bank. How it is used and what do the users think?]. Stockholm, Sweden: Skolverket.Google Scholar
  46. Swedish National Agency for Education (2006). Curriculum for the non-compulsory school system Lpf 94. Stockholm, Sweden: Fritzes.Google Scholar
  47. Swedish National Agency for Education (2009a). TIMSS Advanced 2008. Svenska gymnasieelevers kunskaper i avancerad matematik och fysik i ett internationellt perspektiv [TIMSS Advanced 2008. Swedish upper secondary students’ knowledge in advanced mathematics and physics from an international perspective]. Stockholm, Sweden: Fritzes.Google Scholar
  48. Swedish National Agency for Education (2009b). Hur samstämmiga är svenska styrdokument och nationella prov med ramverk och uppgifter i TIMSS Advanced 2008? [How aligned are the Swedish policy documents and national tests with the framework and the tasks in TIMSS Advanced 2008?]. Stockholm, Sweden: Fritzes.Google Scholar
  49. Swedish National Agency for Education (2014-05-22). Tabell [Table] 3B. Retrieved from
  50. Swedish Schools Inspectorate (2010). Fysik utan dragningskraft. En kvalitetsgranskning om lusten att lära fysik i grundskolan [Physics without attraction. An evaluation about the lust to learn physics in elementary school]. Report No. 2010:8. Retrieved from
  51. Tasar, M. F. (2010). What part of the concept of acceleration is difficult to understand: The mathematics, the physics, or both? ZDM-The International Journal on Mathematics Education, 42, 469–482. doi: 10.1007/s11858-010-0262-9.CrossRefGoogle Scholar
  52. Tuminaro, J. (2002). How student use mathematics in physics: A brief survey of the literature. College Park, MD: University of Maryland Physics Education Research Group. Retrieved from
  53. Winter, H. (1978). Geometrie vom Hebelgesetz aus – ein Beitrag zur Integration von Physik- und Mathematikunterricht der Sekundarstufe I [Geometry from the lever rule—a contribution to the integration of physics—and mathematics education at upper secondary school]. Der Mathematikunterricht, 24(5), 88–125.Google Scholar
  54. Wyndham, J., Riesbeck, E. & Schoult, J. (2000). Problemlösning som metafor och praktik [Problem solving as metaphor and as practice]. Linköping, Sweden: Institutionen för tillämpad lärarkunskap, Linköpings universitet.Google Scholar

Copyright information

© Ministry of Science and Technology, Taiwan 2015

Authors and Affiliations

  1. 1.University of GothenburgGothenburgSweden

Personalised recommendations