Skip to main content
SpringerLink
Log in

Competencies and behaviors observed when students solve geometry proof problems: an interview study with smartpen technology

  • Original Paper
  • Published:
ZDM – Mathematics Education Aims and scope Submit manuscript

Abstract

Decades of research have established that solving geometry proof problems is a challenging endeavor for many students. Consequently, researchers have called for investigations that explore which aspects of proving in geometry are difficult and why this is the case. Here, results from a set of 20 interviews with students who were taught proof in school geometry are reported. Students who earned A or B course-grades in the proof unit(s) were asked to share their thinking aloud while solving two proof tasks using smartpens. Student thinking was analyzed for two subgroups—students who were successful with both proofs (n = 7) and who were unsuccessful with both proofs (n = 13). Large differences were observed in how often students in the two groups exhibited certain competencies and behaviors. The largest gaps occurred in the ways in which students attended to the proof assumptions, attended to warrants in their proofs, and demonstrated logical reasoning.

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
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

Notes

  1. Our interpretation of this claim is not that there are some countries teaching proof well, but, rather, that not enough documentary evidence to support a claim that teaching proof is a failure in all countries exists.

  2. A small number of data glitches occurred with the smartpen technology.

  3. This is useful information for viewing figures in the Findings.

  4. Rubrics were included in Senk’s (1983) dissertation.

  5. This hypothesis was confirmed when we coded these data later on. We chose not to include these details because they over-complicate the reporting of the findings and were not very enlightening.

  6. P14 refers to the 14th participant. P14–P20 were the SPs, and P1–P13 were the NPs.

  7. We “cleaned” up the transcripts slightly by removing “ums,” “uhs,” and so forth, for ease of reading.

  8. Due to a technology glitch, P10’s markings in this diagram are off-center but were sensibly placed on the paper.

References

  • Balacheff, N. (1988). Aspects of proof in pupils’ practice of school mathematics. In D. Pimm (Ed.), Mathematics, teachers and children (pp. 216–235). London: Hodder & Stoughton.

    Google Scholar 

  • Battista, M. T. (2007). The development of geometric and spatial thinking. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 843–908). Charlotte: Information Age Publishing.

    Google Scholar 

  • Boeije, H. (2002). A purposeful approach to the constant comparative method in the analysis of qualitative interviews. Quality and Quantity, 36(4), 391–409.

    Article  Google Scholar 

  • Burton, L., & Morgan, C. (2000). Mathematicians writing. Journal for Research in Mathematics Education, 31(4), 429–453.

    Article  Google Scholar 

  • Cai, J. (1994). A protocol-analytic study of metacognition in mathematical problem solving. Mathematics Education Research Journal, 6(2), 166–183.

    Article  Google Scholar 

  • Carlson, M. P., & Bloom, I. (2005). The cyclic nature of problem solving: An emergent multidimensional problem-solving framework. Educational Studies in Mathematics, 58(1), 45–75.

    Article  Google Scholar 

  • Cirillo, M. (2018). Engaging students with non-routine geometry proof tasks. In P. Herbst, U. H. Cheah, P. R. Richard, & K. Jones (Eds.), International perspectives on the teaching and learning of geometry in secondary schools (pp. 283–300). Berlin: Springer.

    Chapter  Google Scholar 

  • Cirillo, M., & Herbst, P. (2011). Moving toward more authentic proof practices in geometry. The Mathematics Educator, 21(2), 11–33.

    Google Scholar 

  • Cirillo, M., & Hummer, J. (2019). Addressing misconceptions in secondary geometry proof. Mathematics Teacher, 112(6), 410–417.

    Article  Google Scholar 

  • Cirillo, M., Murtha, Z., McCall, N., & Walters, S. (2017). Decomposing mathematical proof with secondary teachers. In L. West (Ed.), Reflective and collaborative processes to improve mathematics teaching (pp. 21–32). Reston: NCTM.

    Google Scholar 

  • Creswell, J. W., & Poth, C. N. (2016). Qualitative inquiry and research design: Choosing among five approaches. Thousand Oaks: Sage publications.

    Google Scholar 

  • Döhrmann, M., Kaiser, G., & Blömeke, S. (2012). The conceptualisation of mathematics competencies in the international teacher education study TEDS-M. ZDM Mathematics Education, 44(3), 325–340.

    Article  Google Scholar 

  • Donaldson, M. L. (1986). Children’s explanations. Cambridge: Cambridge University Press.

    Book  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: Information Age Publishing.

    Google Scholar 

  • Healy, L., & Hoyles, C. (1998). Technical report on the nationwide survey: Justifying and proving in school mathematics. London: University of London.

    Google Scholar 

  • Herbst, P. G., & Brach, C. (2006). Proving and doing proofs in high school geometry classes: What is it that is going on for students? Cognition and Instruction, 24(1), 73–122.

    Article  Google Scholar 

  • Iannone, P. (2009). Concept usage in proof production: Mathematicians’ perspectives. In F.-L. Lin, J.-H. Feng, G. Hanna, & M. de Villiers (Eds.), Proof and proving in mathematics education: ICMI Study 19 conference proceedings, vol 1 (pp. 220–225). Taipei: National Taiwan Normal University.

    Google Scholar 

  • Kantowski, M. G. (1977). Processes involved in mathematical problem solving. Journal for Research in Mathematics Education, 8(3), 163–180.

    Article  Google Scholar 

  • Koedinger, K. R., & Anderson, J. R. (1990). Abstract planning and perceptual chunks: Elements of expertise in geometry. Cognitive Science, 14(4), 511–550.

    Article  Google Scholar 

  • Mamona-Downs, J., & Downs, M. (2005). The identity of problem solving. Journal of Mathematical Behavior, 24(3–4), 385–401.

    Article  Google Scholar 

  • Mayer, R. E. (1985). Implications of cognitive psychology for instruction in mathematical problem solving. In E. A. Silver (Ed.), Teaching and learning mathematical problem solving: Multiple research perspectives (pp. 123–138). London: Lawrence Erlbaum Associates.

    Google Scholar 

  • Movshovitz-Hadar, N. (2001). Proof. In L. Grinstein & S. I. Lipsey (Eds.), Encyclopedia of mathematics education (pp. 585–591). Abingdon: Routledge Falmer.

    Google Scholar 

  • Mullis, I.V.S. & Martin, M.O. (Eds.) (2013). TIMSS 2015 Assessment Frameworks. Retrieved from Boston College, TIMSS & PIRLS International Study Center. http://timssandpirls.bc.edu/timss2015/frameworks.html. Accessed 18 Nov 2020.

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

    Google Scholar 

  • Reiss, K., Klieme, E., & Heinze, A. (2001). Prerequisites for the understanding of proofs in the geometry classroom. In A. D. Cockburn & E. Nardi (Eds.), Proceedings of the 25th conference of the international group for the pscyhology of mathematics education (Vol. 4, pp. 97–104). University of East Anglia.

  • Reiss, K., Hellmich, F., & Reiss, M. (2002). Reasoning and proof in geometry: Prerequisites of knowledge acquisition in secondary school students. In A. D. Cockburn & E. Nardi (Eds.), Proceedings of the 26th Conference of the International Group for the pscyhology of mathematics education (Vol. 4, pp. 113–120). University of East Anglia.

  • Rowland, T. (1992). Pointing with pronouns. For the Learning of Mathematics, 12(2), 44–48.

    Google Scholar 

  • Rowland, T. (1999). Pronouns in mathematical talk: Power, vagueness, and generalisation. For the Learning of Mathematics, 19(2), 19–26.

    Google Scholar 

  • Schoenfeld, A. H. (1985). Mathematical problem solving. New York: Academic Press.

    Google Scholar 

  • Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning. New York: Macmillan.

    Google Scholar 

  • Senk, S. L. (1983). Proof-writing achievement and van Hiele levels among secondary school geometry students. (Doctoral Dissertation), University of Chicago, ProQuest Dissertations and Theses.

  • Senk, S. L. (1985). How well do students write geometry proofs? The Mathematics Teacher, 78(6), 448–456.

    Article  Google Scholar 

  • Sinclair, N., Pimm, D., & Skelin, M. (2012). Developing an essential understanding of geometry for teaching mathematics in grades 9–12. NCTM.

  • Sinclair, N., Cirillo, M., & de Villiers, M. (2017). The learning and teaching of geometry. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 457–488). Reston: NCTM.

    Google Scholar 

  • Livescribe Desktop Version 2.8.3.58503. (2012). [Computer Software]. https://www.livescribe.com/site/. Accessed 4 Feb 2020.

  • Weber, K. (2001). Student difficulty in constructing proofs: The need for strategic knowledge. Educational Studies in Mathematics, 48(1), 101–119.

    Article  Google Scholar 

  • Weber, K., & Leikin, R. (2016). Recent advances in research on problem solving and problem posing. In A. Gutiérrez, G. C. Leder, & P. Boero (Eds.), The second handbook of research on the psychology of mathematics education (pp. 353–382). Rotterdam: Sense Publishers.

    Chapter  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the teachers and their students for allowing them to conduct this study. They also thank Amanda Seiwell and Kevin Felice for supporting data analysis, as well as Keith Weber for useful suggestions on methods and Ron Gallimore for helpful suggestions on the manuscript. The research reported in this paper was supported with funding from the National Science Foundation (NSF; Award #1453493, PI: Michelle Cirillo). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the NSF.

Author information

Authors and Affiliations

  1. University of Delaware, Newark, USA

    Michelle Cirillo

  2. West Chester University of Pennsylvania, West Chester, USA

    Jenifer Hummer

Authors
  1. Michelle Cirillo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jenifer Hummer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michelle Cirillo.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cirillo, M., Hummer, J. Competencies and behaviors observed when students solve geometry proof problems: an interview study with smartpen technology. ZDM Mathematics Education 53, 861–875 (2021). https://doi.org/10.1007/s11858-021-01221-w

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11858-021-01221-w

Keywords

Search

Navigation