Aurora Fernández-León, José María Gavilán-Izquierdo, Rocío Toscano


This paper studies how four primary-school in-service teachers develop the mathematical practices of conjecturing and proving. From the consideration of professional development as the legitimate peripheral participation in communities of practice, these teachers’ mathematical practices have been characterised by using a theoretical framework (consisting of categories of activities) that describes and explains how a research mathematician develops these two mathematical practices. This research has adopted a qualitative methodology and, in particular, a case study methodological approach. Data was collected in a working session on professional development while the four participants discussed two questions that invoked the development of the mathematical practices of conjecturing and proving. The results of this study show the significant presence of informal activities when the four participants conjecture, while few informal activities have been observed when they strive to prove a result. In addition, the use of examples (an informal activity) differs in the two practices, since examples support the conjecturing process but constitute obstacles for the proving process. Finally, the findings are contrasted with other related studies and several suggestions are presented that may be derived from this work to enhance professional development.


conjecturing; proving; primary-school in-service teachers; professional development; research mathematicians

Full Text:



Alibert, D., & Thomas, M. (2002). Research on mathematical proof. In D. Tall (Ed.), Advanced mathematical thinking (pp. 215-230). Dordrecht: Kluwer. https://doi.org/10.1007/0-306-47203-1_13

Astawa, I. W. P., Budayasa, I. K., & Juniati, D. (2018). The process of student cognition in constructing mathematical conjecture. Journal on Mathematics Education, 9(1), 15-26. https://doi.org/10.22342/jme.9.1.4278.15-26

Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389-407. https://doi.org/10.1177/0022487108324554

Blömeke, S., Kaiser, G., König, J., & Jentsch, A. (2020). Profiles of mathematics teachers’ competence and their relation to instructional quality. ZDM Mathematics Education, 52(2), 329-342. https://doi.org/10.1007/s11858-020-01128-y

Boero, P. (Ed.). (2007). Theorems in school: From history, epistemology and cognition to classroom practice. Rotterdam: Sense Publishers.

Boero, P., Garuti, R., & Lemut, E. (2007). Approaching theorems in grade VIII: Some mental processes underlying producing and proving conjectures, and conditions suitable to enhance them. In P. Boero (Ed.), Theorems in school: From history, epistemology and cognition to classroom practice (pp. 249-264). Rotterdam: Sense Publishers.

Burton, L. (1998). The practices of mathematicians: What do they tell us about coming to know mathematics? Educational Studies in Mathematics, 37(2), 121-143. https://doi.org/10.1023/A:1003697329618

Burton, L. (2004). Mathematicians as enquirers: Learning about learning mathematics. Dordrecht: Kluwer. https://doi.org/10.1007/978-1-4020-7908-5

Carrillo-Yañez, J., Climent, N., Montes, M., Contreras, L. C., Flores-Medrano, E., Escudero-Ávila, D., Vasco, D., Rojas, N., Flores, P., Aguilar-González, A., Ribeiro, M., & Muñoz-Catalán, M. C. (2018). The mathematics teacher’s specialised knowledge (MTSK) model. Research in Mathematics Education, 20(3), 236-253. https://doi.org/10.1080/14794802.2018.1479981

Chong, M. S. F., Shahrill, M., & Li, H-C. (2019). The integration of a problem-solving framework for Brunei high school mathematics curriculum in increasing student’s affective competency. Journal on Mathematics Education, 10(2), 215-228. https://doi.org/10.22342/jme.10.2.7265.215-228

Fernández-León, A., Gavilán-Izquierdo, J. M., & Toscano, R. (2020). A case study of the practices of conjecturing and proving of research mathematicians. International Journal of Mathematical Education in Science and Technology, https://doi.org/10.1080/0020739X.2020.1717658

Hidayah, I. N., Sa’dijah, C., Subanji, R., & Sudirman, S. (2020). Characteristics of students’ abductive reasoning in solving algebra problems. Journal on Mathematics Education, 11(3), 347-362. http://doi.org/10.22342/jme.11.3.11869.347-362

Huang, C-H. (2016). Examples as mediating artifacts in conjecturing. American Journal of Educational Research, 4(19), 1295-1299. https://doi.org/10.12691/education-4-19-3

Knuth, E. J. (2002). Secondary school mathematics teachers’ conceptions of proof. Journal for Research in Mathematics Education, 33(5), 379-405. https://doi.org/10.2307/4149959

Ko, Y. (2010). Mathematics teachers’ conceptions of proof: Implications for educational research. International Journal of Science and Mathematics Education, 8(6), 1109-1129. https://doi.org/10.1007/s10763-010-9235-2

Lakatos, I. (1976). Proof and refutations: The logic of mathematical discovery. Cambridge, UK: Cambridge University Press. https://doi.org/10.1017/CBO9781139171472

Lave, J., & Wenger, E. (1991). Situated learning: Legitimate peripheral participation. Cambridge, UK: Cambridge University Press. https://doi.org/10.1017/CBO9780511815355

Lesseig, K. (2016). Conjecturing, generalizing and justifying: Building theory around teacher knowledge of proving. International Journal for Mathematics Teaching and Learning, 17(3), 1-31.

Lynch, A. G., & Lockwood, E. (2019). A comparison between mathematicians’ and students’ use of examples for conjecturing and proving. Journal of Mathematical Behavior, 53, 323-338. https://doi.org/10.1016/j.jmathb.2017.07.004

Martin, W. G., & Harel, G. (1989). Proof frames of preservice elementary teachers. Journal for Research in Mathematics Education, 20(1), 41-51. https://doi.org/10.2307/749097

Martín-Molina, V., González-Regaña, A. J., & Gavilán-Izquierdo, J. M. (2018). Researching how professional mathematicians construct new mathematical definitions: A case study. International Journal of Mathematical Education in Science and Technology, 49(7), 1069-1082. https://doi.org/10.1080/0020739X.2018.1426795

Mason, J., Burton, L., & Stacey, K. (1982). Thinking mathematically. London: Addison Wesley.

Melhuish, K., Thanheiser, E., & Guyot, L. (2018). Elementary school teachers’ noticing of essential mathematical reasoning forms: Justification and generalization. Journal of Mathematics Teacher Education, 23(1), 35-67. https://doi.org/10.1007/s10857-018-9408-4

Morselli, F. (2006). Use of examples in conjecturing and proving: An exploratory study. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (vol. 4, pp. 185-192). Prague: PME.

Muñoz-Catalán, M. C. (2009). El desarrollo profesional en un entorno colaborativo centrado en la enseñanza de las matemáticas: El caso de una maestra novel [Professional development in a collaborative context focused on mathematics teaching: The case of a novice teacher] (Doctoral dissertation). Retrieved from http://rabida.uhu.es/dspace/handle/10272/2949

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

National Governors Association Center for Best Practices, & Council of Chief State School Officers. (2010). Common core state standards for mathematics. Retrieved from http://www.corestandards.org/Math

Oflaz, G., Polat, K., Altayl? Özgül, D., Alcaide, M., & Carrillo, J. (2019). A Comparative research on proving: The case of prospective mathematics teachers. Higher Education Studies, 9(4), 92-111. https://doi.org/10.5539/hes.v9n4p92

Oktaviyanthi, R., Herman, T., & Dahlan, J. A. (2018). How does pre-service mathematics teacher prove the limit of a function by formal definition? Journal on Mathematics Education, 9(2), 195-212. https://doi.org/10.22342/jme.9.2.5684.195-212

Ouvrier-Buffet, C. (2015). A model of mathematicians’ approach to the defining processes. In K. Krainer & N. Vondrová (Eds.), Proceedings of the 9th Congress of the European Society for Research in Mathematics Education (pp. 2214-2220). Prague: Charles University in Prague, Faculty of Education and ERME.

Podkhodova, N., Snegurova, V., Stefanova, N., Triapitsyna, A., & Pisareva, S. (2020). Assessment of mathematics teachers’ professional competence. Journal on Mathematics Education, 11(3), 477-500. https://doi.org/10.22342/jme.11.3.11848.477-500

Polya, G. (1954). Mathematics and plausible reasoning: Patterns of plausible inference. Princeton, NJ: Princeton University Press.

RAND Mathematics Study Panel. (2003). Mathematical proficiency for all students: Toward a strategic research and development program in mathematics education. Santa Monica, CA: RAND Corporation. Retrieved from https://www.rand.org/pubs/monograph_reports/MR1643.html

Rasmussen, C., Wawro, M., & Zandieh, M. (2015). Examining individual and collective level mathematical progress. Educational Studies in Mathematics, 88(2), 259-281. https://doi.org/10.1007/s10649-014-9583-x

Rasmussen, C., Zandieh, M., King, K., & Teppo, A. (2005). Advancing mathematical activity: A practice-oriented view of advanced mathematical thinking. Mathematical Thinking and Learning, 7(1), 51-73. https://doi.org/10.1207/s15327833mtl0701_4

Siswono, T. Y. E., Hartono, S., & Kohar, A. W. (2020). Deductive or inductive? Prospective teachers’ preference of proof method on an intermediate proof task. Journal on Mathematics Education, 11(3), 417-438. http://doi.org/10.22342/jme.11.3.11846.417-438

Stylianides, A. J., & Ball, D. L. (2008). Understanding and describing mathematical knowledge for teaching: Knowledge about proof for engaging students in the activity of proving. Journal of Mathematics Teacher Education, 11(4), 307-332. https://doi.org/10.1007/s10857-008-9077-9

Stylianides, A. J., & Stylianides, G. J. (2018). Addressing key and persistent problems of students’ learning: The case of proof. In A. J. Stylianides & G. Harel (Eds.), Advances in mathematics education research on proof and proving: An international perspective, ICME-13 monographs (pp. 99-113). Cham: Springer International Publishing AG. https://doi.org/10.1007/978-3-319-70996-3_7

Treffers, A. (1987). Three dimensions. A model of goal and theory description in mathematics education: The Wiskobas project. Dordrecht: D. Reidel Publishing Company. https://doi.org/10.1007/978-94-009-3707-9

Turrisi, P. A. (Ed.). (1997). Pragmatism as a principle and method of right thinking: The 1903 Harvard lectures on pragmatism. Albany, NY: State University of New York Press.

Tymoczko, T. (Ed.). (1998). New directions in the philosophy of mathematics. Princeton, NJ: Princeton University Press.

Ubuz, B., & Yayan, B. (2010). Primary teachers’ subject matter knowledge: Decimals. International Journal of Mathematical Education in Science and Technology, 41(6), 787-804. https://doi.org/10.1080/00207391003777871

U.K. Department of Education (2014). National curriculum in England: Mathematics programmes of study. Retrieved from https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study

Van Dormolen, J. (1977). Learning to understand what giving a proof really means. Educational Studies in Mathematics, 8(1), 17-34. https://doi.org/doi:10.1007/BF00302502

Watson, F. R. (1980). The role of proof and conjecture in mathematics and mathematics teaching. International Journal of Mathematical Education in Science and Technology, 11(2), 163-167. https://doi.org/10.1080/0020739800110202

Weber, K. (2008). How mathematicians determine if an argument is a valid proof. Journal for Research in Mathematics Education, 39(4), 431-459.

Weber, K., & Dawkins, P. C. (2018). Toward an evolving theory of mathematical practice informing pedagogy: What standards for this research paradigm should we adopt? In A. J. Stylianides & G. Harel (Eds.), Advances in mathematics education research on proof and proving: An international perspective, ICME-13 monographs (pp. 69-82). Cham: Springer International Publishing AG. https://doi.org/10.1007/978-3-319-70996-3¬_5

Weber, K., Inglis, M., & Mejia-Ramos, J. P. (2014). How mathematicians obtain conviction: Implications for mathematics instruction and research on epistemic cognition. Educational Psychologist, 49(1), 36-58. https://doi.org/10.1080/00461520.2013.865527

Weber, K., & Mejia-Ramos, J. P. (2011). Why and how mathematicians read proofs: An exploratory study. Educational Studies in Mathematics, 76(3), 329-344. https://doi.org/10.1007/s10649-010-9292-z

Yilmaz, R. (2020). Prospective mathematics teachers’ cognitive competencies on realistic mathematics education. Journal on Mathematics Education, 11(1), 17-44. http://doi.org/10.22342/jme.11.1.8690.17-44

DOI: https://doi.org/10.22342/jme.12.1.12800.49-72


  • There are currently no refbacks.

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

Journal on Mathematics Education
Doctoral Program on Mathematics Education
Faculty of Teacher Training and Education, Universitas Sriwijaya
Kampus FKIP Bukit Besar
Jl. Srijaya Negara, Bukit Besar
Palembang - 30139
email: jme@unsri.ac.id

p-ISSN: 2087-8885 | e-ISSN: 2407-0610

Creative Commons License
Journal on Mathematics Education (JME) is licensed under a Creative Commons Attribution 4.0 International License.

View My Stats