Skip to main content
SpringerLink
Log in

Extrema points: concept images, mis-in and mis-out examples

  • Published:
Educational Studies in Mathematics Aims and scope Submit manuscript

Abstract

This paper reports on students’ conceptions of minima points. Written assignments and individual interviews uncovered salient, concept images, as well as erroneous mis-out examples that mistakenly regard examples as non-examples and mis-in examples that mistakenly grant non-examples the status of examples. We used Tall and Vinner’s theoretical framework to analyze the students’ errors that were rooted in mathematical and in real-life contexts.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Data availability

The datasets are available from the corresponding author upon reasonable request.

References

  • Ayres, F., & Mendelson, E. (2012). Schaum’s outline of calculus (5th ed.). McGraw-Hill.

    Google Scholar 

  • Bezuidenhout, J. (2001). Limits and continuity: Some conceptions of first-year students. International Journal of Mathematical Education in Science and Technology, 32, 487–500.

    Article  Google Scholar 

  • Bills, L., Dreyfus, T., Mason, J., Tsamir, P., Watson, A., & Zaslavsky, O. (2006). Exemplification in mathematics education. In J. Novotna, H. Moraova, M. Kratka, & N. Stehlikova (Eds.), Proceedings of the 30th PME International Conference (vol. 1, pp. 126–154). PME.

  • Biza, I. (2021). The discursive footprint of learning across mathematical domains: The case of the tangent line. The Journal of Mathematical Behavior, 62, 1–12. https://doi.org/10.1016/j.jmathb.2021.100870

    Article  Google Scholar 

  • Borasi, R. (1996). Reconceiving mathematics instruction: A focus on errors. Ablex.

  • Chick, H. (2009). Choice and use of examples as a window on mathematical knowledge for teaching. For the Learning of Mathematics, 29(3), 26–30 https://www.jstor.org/stable/25594563

    Google Scholar 

  • Duru, A., Koklu, O., & Jakubowski, E. (2010). Pre-service mathematics teachers’ conceptions about the relationship between continuity and differentiability of a function. Scientific Research and Essays, 5, 1519–1529.

    Google Scholar 

  • Fennema, E., Carpenter, T. P., Franke, M. L., Levi, L., Jacobs, V. R., & Empson, S. B. (1996). A longitudinal study of learning to use children’s thinking in mathematics instruction. Journal for Research in Mathematics Education, 27, 403–434.

    Article  Google Scholar 

  • Fischbein, E. (1987). Intuition in science and mathematics: An educational approach. Reidel.

    Google Scholar 

  • Fujita, T. (2012). Learners’ level of understanding of the inclusion relations of quadrilaterals and prototype phenomenon. The Journal of Mathematical Behavior, 31, 60–72. https://doi.org/10.1016/j.jmathb.2011.08.003

    Article  Google Scholar 

  • Hill, H. C., Ball, D. L., & Schilling, S. G. (2008). Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers’ topic-specific knowledge of students. Journal for Research in Mathematics Education, 39(4), 372–400. https://doi.org/10.5951/jresematheduc.39.4.0372

    Article  Google Scholar 

  • Juter, K. (2017). University students’ understandings of concept relations and preferred representations of continuity and differentiability. CERME10.

    Google Scholar 

  • Leikin, R., Leikin, M., Waisman, I., & Shelley, S. (2013). Effect of the presence of external representations on accuracy and reaction time in solving mathematical double-choice problems by students of different levels of instruction. International Journal of Science and Mathematics Education, 11, 1049–1066. https://doi.org/10.1007/s10763-012-9389-1

  • Mason, J. (2006). Distinctive ontogeny: Ontological musings inspired by two discerning gentlemen of distinction(s), David Tall & Eddie Gray. In A. Simpson (Ed.), Retirement as process and concept: A Festschrift for Eddie Gray and David Tall (pp. 153–164). Charles University.

  • Mason, J., & Pimm, D. (1984). Generic examples: Seeing the general in the particular. Educational Studies in Mathematics, 15, 277–289. https://doi.org/10.1007/BF00312078

    Article  Google Scholar 

  • McGowen, M.A., & Tall, D. (2010). Metaphor or Met-Before? The effects of previous experience on practice and theory of learning mathematics. The Journal of Mathematical Behavior, 29, 169–179. https://doi.org/10.1016/j.jmathb.2010.08.002

  • Nardi, E. (2007). Amongst mathematicians: Teaching and learning mathematics at university level. Springer.

  • Ng, O. L. (2021). How ‘tall’ is the triangle? Constructionist learning of shape and space with 3D pens. International Journal of Mathematical Education in Science and Technology, 52, 1426–1432. https://doi.org/10.1080/0020739X.2020.1844910

    Article  ADS  Google Scholar 

  • Orhun, N. (2012). Graphical understanding in mathematics education: Derivative functions and students’ difficulties. Procedia - Social and Behavioral Sciences, 55, 679 – 684. https://doi.org/10.1016/j.sbspro.2012.09.551

  • Pape, S. J., & Tchoshanov, M. A. (2001). The role of representation in developing mathematical understanding. Theory Into Practice, 40, 117–127.

    Article  Google Scholar 

  • Petty, O., & Jansson, L. (1987). Sequencing examples and non-examples to facilitate concept attainment. Journal for Research in Mathematics Education, 18(2), 112–125.

    Article  Google Scholar 

  • Pimm, D. (1987). Speaking mathematically: Communication in mathematics classrooms. Routledge and Kegan Paul.

    Google Scholar 

  • Ryan, J., & Williams, J. (2007). Children’s mathematics 4-15: Learning from errors and misconceptions. Open University Press, McGraw-Hill Education.

    Google Scholar 

  • Santagata, R. (2005). Practices and beliefs in mistake-handling activities: A video study of Italian and US mathematics lessons. Teaching and Teacher Education, 21, 491–508.

    Article  Google Scholar 

  • Sierpińska, A. (1987). Humanities students and epistemological obstacles related to limits. Educational Studies in Mathematics, 18, 371–397. https://doi.org/10.1007/BF00240986

    Article  Google Scholar 

  • Shepherd, M. D., Selden, A., & Selden, J. (2012). University students’ reading of their first-year mathematics textbooks. Mathematical Thinking and Learning, 14(3), 226–256. https://doi.org/10.1080/10986065.2012.682959

    Article  Google Scholar 

  • Selden, A., & Selden, J. (2001). Tertiary mathematics education research and its future. In D. Holton (Ed.), The teaching and learning of mathematics at university level: An ICMI Study (pp. 237–254). Springer.

    Google Scholar 

  • Sevimli, E. (2018). Understanding students’ hierarchical thinking: A view from continuity, differentiability and integrability. Teaching mathematics and its applications: An international journal of the IMA, 37, 1–16. https://doi.org/10.1093/teamat/hrw028

    Article  Google Scholar 

  • Stylianou, D. A. (2002). On the interaction of visualization and analysis: The negotiation of a visual representation in expert problem solving. The Journal of Mathematical Behavior, 21(3), 303–317. https://doi.org/10.1016/S0732-3123(02)00131-1

    Article  Google Scholar 

  • Tall, D. O. (1981). The transition to advanced mathematical thinking: Functions, limits, infinity and proof. In D. A. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 495–511). Macmillan.

    Google Scholar 

  • Tall, D. O., & Vinner, S. (1981). Concept image and concept definition in mathematics, with particular reference to limits and continuity. Educational Studies in Mathematics, 12, 151–169. https://doi.org/10.1007/BF00305619

    Article  Google Scholar 

  • Thomas, G. B., Weir, M. D., Hass, J. R., & Giordano, F. R. (2004). Thomas’ calculus (11th ed.). Addison-Wesley.

    Google Scholar 

  • Tirosh, D. (2000). Enhancing prospective teachers’ knowledge of children’s conceptions: The case of division of fractions. Journal for Research in Mathematics Education, 31(1), 5–25. https://doi.org/10.2307/749817

    Article  Google Scholar 

  • Tirosh, D., & Tsamir, P. (1996). The role of representations in students’ intuitive thinking about infinity. International Journal of Mathematical Education in Science and Technology, 27(1), 33–40. https://doi.org/10.1080/0020739960270105

    Article  Google Scholar 

  • Tirosh, D., & Tsamir, P. (2022). Mis-out and mis-in concept images of parallelograms: The case of Tal. International Journal of Science and Mathematics Education, 20, 981–997. https://doi.org/10.1007/s10763-021-10175-0

    Article  ADS  Google Scholar 

  • Tsamir, P., & Ovodenko, R. (2013). University students’ grasp of inflection points. Educational Studies in Mathematics, 83, 409–427. https://doi.org/10.1007/s10649-012-9463-1

    Article  Google Scholar 

  • Tsamir, P., & Tirosh, D. (2023). Mis-in and mis-out concept images: The case of even numbers. Educational Studies in Mathematics. https://doi.org/10.1007/s10649-022-10183-z

  • Tsamir, P., Tirosh, D., & Levenson, E. (2008). Intuitive nonexamples: The case of triangles. Educational Studies in Mathematics, 69, 81–95. https://doi.org/10.1007/s10649-008-9133-5

    Article  Google Scholar 

  • Ubuz, B. (2007). Interpreting a graph and constructing its derivative graph: Stability and change in students’ conceptions. International Journal of Mathematical Education in Science and Technology, 38(5), 609–637. https://doi.org/10.1080/00207390701359313

    Article  Google Scholar 

  • Ulusoy, F. (2021). Prospective early childhood and elementary school mathematics teachers’ concept images and concept definitions of triangles. International Journal of Science and Mathematics Education, 19, 1057–1078. https://doi.org/10.1007/s10763-020-10105-6

    Article  ADS  Google Scholar 

  • Vinner, S. (1990). Inconsistencies: Their causes and function in learning mathematics. Focus on Learning Problems in Mathematics, 12(3–4), 85–98.

    Google Scholar 

  • Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In D. O. Tall (Ed.), Advanced Mathematical Thinking (pp. 65–81). Kluwer.

    Google Scholar 

  • Vinner, S., & Dreyfus, T. (1989). Images and definitions for the concept of function. Journal for Research in Mathematics Education, 20(4), 356–366.

    Article  Google Scholar 

  • Watson, A., & Mason, J. (2005). Mathematics as a constructive activity: Learners generating examples. Erlbaum.

    Google Scholar 

  • Willingham, J. C., Strayer, J. F., Barlow, A. T., & Lischka, A. E. (2018). Examining mistakes to shift student thinking. Mathematics teaching in the middle school, 23, 324–332. https://doi.org/10.5951/mathteacmiddscho.23.6.0324

    Article  Google Scholar 

  • Zaslavsky, O., & Shir, K. (2005). Students’ conceptions of a mathematical definition. Journal for Research in Mathematics Education, 36(4), 317–346 http://www.jstor.org/stable/30035043

    Google Scholar 

  • Zorich, V. A. (2004). Mathematical analysis I. Springer-Verlag.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Science and Technology Education, Tel-Aviv University, Tel-Aviv, Israel

    Pessia Tsamir, Regina Ovodenko & Dina Tirosh

Authors
  1. Pessia Tsamir
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Regina Ovodenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dina Tirosh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dina Tirosh.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tsamir, P., Ovodenko, R. & Tirosh, D. Extrema points: concept images, mis-in and mis-out examples. Educ Stud Math 115, 13–33 (2024). https://doi.org/10.1007/s10649-023-10273-6

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10649-023-10273-6

Keywords

Search

Navigation