Pre-service Teachers’ General and Specific Arguments in Real Number Contexts
Keywords:General and specific arguments, real numbers, mathematics subject-matter beliefs, pre-service teachers
A study of pre-service elementary school teachers’ subject-matter beliefs regarding real numbers related to infinity, i.e., division by zero and denseness of the real number line, was conducted at a Swedish university. Data were collected twice during the respondents’ teacher education using questionnaires and interviews on both occasions. The data were analysed in terms of subject-matter beliefs from traces of students’ concept images, focusing on general and specific arguments in different contexts. The results show that the students often used conflicting general and specific arguments to explain the same phenomena. Some concrete arguments used for explanation represented other mathematical structures than the ones intended. Several issues indicate students’ need for further work with the concepts studied to raise awareness of their capabilities. Implications for mathematics teaching on teacher education programmes are discussed concerning the results.
Bryman, A. (2016). Quantitative and qualitative research: Further reflections on their integration. In J. Brannen (Ed.), Mixing methods: qualitative and quantitative research (pp. 57–80). Routledge. https://doi.org/10.4324/9781315248813
Coles, A., & Sinclair, N. (2019). Re-thinking ‘concrete to abstract’ in mathematics education: Towards the use of symbolically structured environments. Canadian Journal of Science. Mathematics and Technology Education, 19, 465–480. https://doi.org/10.1007/s42330-019-00068-4
Crespo, S., & Nicol, C. (2006). Challenging preservice teachers’ mathematical understanding: The case of division by zero. School Science and Mathematics, 106(2), 84–97. https://doi.org/10.1111/j.1949-8594.2006.tb18138.x
Dimmel, J. K., & Pandiscio, E. A. (2020). When it’s on zero, the lines become parallel: Preservice elementary teachers’ diagrammatic encounters with division by zero. Journal of Mathematical Behaviour, 58, 100760. https://doi.org/10.1016/j.jmathb.2020.100760
Fischbein, E. (2001). Tacit models and infinity. Educational Studies in Mathematics, 48(2), 309–329. https://doi.org/10.1023/A:1016088708705
Flick, U. (2018). Triangulation in data collection. In U. Flick (Ed.), The SAGE handbook of qualitative data collection (pp. 527–544). SAGE Publications.
Font, V., & Contreras, A. (2008). The problem of the particular and its relation to the general in mathematics education. Educational Studies in Mathematics, 69(1), 33–52. https://doi.org/10.1007/s10649-008-9123-7
Gobo, G. (2018). Upside down: Reinventing research design. In U. Flick (Ed.), The SAGE handbook of qualitative data collection (pp. 65–83). SAGE Publications.
Juter, K. (2007). Students’ conceptions of limits, high achievers versus low achievers. The Mathematics Enthusiast, 4(1), 53–65. https://doi.org/10.54870/1551-3440.1058
Juter, K. (2019). University students’ general and specific beliefs about infinity, division by zero and denseness of the number line. Nordic Studies in Mathematics Education, 24(2), 69–88.
Karakus, F. (2018) Investigation of pre-service teachers’ pedagogical content knowledge related to division by zero. International Journal for Mathematics Teaching and Learning, 19(1), 90–111.
Katz, K. U., & Katz, M. G. (2010). When is .999... less than 1? The Mathematics Enthusiast, 7(1), 3–30. https://doi.org/10.48550/arXiv.1007.3018
Lajoie, C., & Mura. R. (1998). The danger of being overly attached to the concrete: The case of division by zero. Nordic Studies in Mathematics Education, 6(1), 7–21.
Levenson, E., Tsamir, P., & Tirosh, D. (2010). Mathematically based and practically based explanations in the elementary school: Teachers’ preferences. Journal of Mathematics Teacher Education, 13(4), 345–369. https://doi.org/10.1007/s10857-010-9142-z
Mason, J., & Pimm, D. (1984). Generic examples seeing the general in the particular. Educational Studies in Mathematics, 15(3), 277–289. https://doi.org/10.1007/BF00312078
Moreira, P. C., & David, M. M. (2008). Academic mathematics and mathematical knowledge needed in school teaching practice: some conflicting elements. Journal of Mathematics Teacher Education, 11(1), 23–40. https://doi.org/10.1007/s10857-007-9057-5
Sbaragli, S. (2006). Primary School Teachers' beliefs and change of beliefs on Mathematical Infinity. Mediterranean Journal for Research in Mathematics Education, 5(2), 49–75.
Shulman, L. S. (1986). Those who understand knowledge growth in teaching. American Educational Research Association, 15(2), 4–14. https://doi.org/10.2307/1175860
Stohlmann, M., Cramer, K., Moore, T., & Maiorca, C. (2014). Changing pre-service elementary teachers’ beliefs about mathematical knowledge. Mathematics Teacher Education and Development, 16(2), 4–24.
Swedish Research Council. (2017, June). Good research practice. https://www.vr.se/download/18.5639980c162791bbfe697882/1555334908942/Good-Research-Practice_VR_2017.pdf
Takker, S., & Subramaniam, K. (2019). Knowledge demands in teaching decimal numbers. Journal of Mathematics Teacher Education, 22(5), 1–24. https://doi.org/10.1007/s10857-017-9393-z
Tall, D. (2001). Natural and formal infinities. Educational Studies in Mathematics, 48(2), 199–238. https://doi.org/10.1023/A:1016000710038
Tall, D., & Tirosh, D. (2001). Infinity: The never-ending struggle. Educational Studies in Mathematics, 48(2), 129–136. https://doi.org/10.1023/A:1016019128773
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. https://doi.org/10.1007/BF00305619
Thanheiser, E., Whitacre, I., & Roy, G. (2014). Mathematical content knowledge for teaching elementary mathematics: A focus on whole-number concepts and operations. The Mathematics Enthusiast, 11(2), 217–266. https://doi.org/10.54870/1551-3440.1303
Tsamir, P. (1999). The transition from comparison of finite to the comparison of infinite sets: Teaching prospective teachers. Educational Studies in Mathematics, 38(1–3), 209–234. https://doi.org/10.1023/A:1003514208428
Tsamir, P. (2001). When ‘the same’ is not perceived as such: The case of infinite sets. Educational Studies in Mathematics, 48(2), 289–307. https://doi.org/10.1023/A:1016034917992
Tsamir, P., & Sheffer, R. (2000). Concrete and formal arguments: The case of division by zero. Mathematics Education Research Journal, 12(2), 92–106. https://doi.org/10.1007/BF03217078
Tsamir, P., & Tirosh, D. (2002). Intuitive beliefs, formal definitions and undefined operations: Cases of division by zero. In G. C. Leder, E. Pehkonen & G. Törner (Eds.), Beliefs: A hidden variable in mathematics education? Mathematics Education Library, Vol 31 (pp. 331–344). Springer.
Vamvakoussi, X., & Vosniadou, S. (2012). Bridging the gap between the dense and the discrete: The number line and the “rubber line” bridging analogy. Mathematical Thinking and Learning, 14(4), 265–284. https://doi.org/10.1080/10986065.2012.717378
Yopp, D. A., Burroughs, E. A., & Lindaman, B. J. (2011). Why it is important for in-service elementary mathematics teachers to understand the equality .999… = 1. The Journal of Mathematical Behavior, 30(4), 304–318. https://doi.org/10.1016/j.jmathb.2011.07.007