How do prospective primary teachers approach area conservation? An analysis based on intra-mathematical connections

Sofía Caviedes Barrera; Genaro de Gamboa; Èdgar Ribot-Llobet; Edelmira Badillo

doi:10.29333/ejmste/19476

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Sofía Caviedes Barrera ¹ ^* , Genaro de Gamboa ¹ , Èdgar Ribot-Llobet ¹ , Edelmira Badillo ¹

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¹ Departament de Didàctica de la Matemàtica i de les Ciències Experimentals, Universitat Autònoma de Barcelona, Barcelona, SPAIN^* Corresponding Author

Abstract

This study examines the intra-mathematical connections evidenced in groups of prospective primary school teachers’ written productions when solving and justifying an area-conservation task comprising an initial task and follow-up questions. A mixed-methods design combined qualitative content analysis with exploratory co-occurrence and clustering analyses. Procedure and justification were among the most frequent connections identified, whereas equivalent procedure and alternate representation appeared less frequently, particularly in the follow-up questions. A cluster map and cross-specification sensitivity analysis showed that some similarities among groups recurred across retained alternative specifications, whereas broader grouping boundaries were more sensitive to methodological choices. Overall, the findings indicate that area conservation was expressed through different combinations of mathematical objects and connection types, highlighting the complexity of justification and the potential of such tasks for teacher education.

Keywords

intra-mathematical connections
area conservation
prospective primary school teachers
mixed methods
initial teacher education

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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Research Article

EURASIA J Math Sci Tech Ed, Volume 22, Issue 10, October 2026, Article No: em2938

https://doi.org/10.29333/ejmste/19476

Publication date: 02 Oct 2026

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Article Downloads: 9

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References

Battista, M. T. (1999). The importance of spatial structuring in geometric reasoning. Teaching Children Mathematics, 6(3), 170-177. https://doi.org/10.5951/TCM.6.3.0170
Battista, M. T., Clements, D. H., Arnoff, J., Battista, K., & Borrow, C. V. A. (1998). Students’ spatial structuring of 2D arrays of squares. Journal for Research in Mathematics Education, 29(5), 503-532. https://doi.org/10.2307/749731
British Educational Research Association. (2018). Ethical guidelines for educational research (4th ed.). British Educational Research Association.
Businskas, A. (2008). Conversations about connections: How secondary mathematics teachers conceptualize and contend with mathematical connections [PhD thesis, Simon Fraser University].
Caviedes, S., de Gamboa, G., & Badillo E. (2023a). Mathematical objects that configure the partial area meanings mobilized in task solving. International Journal of Mathematical Education in Science and Technology, 54(6), 1092-1111. https://doi.org/10.1080/0020739X.2021.1991019
Caviedes, S., de Gamboa, G., & Badillo, E. (2023b). Preservice teachers’ knowledge mobilized in solving area tasks. Journal on Mathematics Education, 14(1), 35-54. https://doi.org/10.22342/jme.v14i1.pp35-54
Caviedes, S., de Gamboa, G., & Badillo, E. (2024). Mathematical connections involved in area measurement processes. Research in Mathematics Education, 26(2), 237-257. https://doi.org/10.1080/14794802.2024.2370333
Caviedes, S., de Gamboa, G., Ribot-Llobet, E., & Badillo, E. (2026). Conexiones intramatemáticas en tareas que involucran la conservación de área [Intramathematical connections in tasks requiring area conservation]. In B. Barquero, C. de Castro, A. Gómez-Escobar, A. M. Oller-Marcén, & M. D. Torres (Eds.), Investigación en Educación Matemática XXIX (pp. 163-170). SEIEM.
Caviedes, S., Gamboa, G. D., & Badillo, E. (2022). Knowledge used by preservice teaching students when comparing areas of 2D figures. Uniciencia, 36(1), 639-658. https://doi.org/10.15359/ru.36-1.41
Cobb, P., Confrey, J., DiSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational Researcher, 32(1), 9-13. https://doi.org/10.3102/0013189X032001009
Cohen, L., Manion, L., & Morrison, K. (2017). Research methods in education. Routledge. https://doi.org/10.4324/9781315456539
Creswell, J. W., & Plano Clark, V. L. (2023). Revisiting mixed methods research designs twenty years later. In C. N. Poth (Ed.), Handbook of mixed methods research designs (pp. 21-36). SAGE. https://doi.org/10.4135/9781529614572.n6
D’Amore, B., & Fandiño Pinilla, M. I. (2007). Relaciones entre área y perímetro: Convicciones de maestros y de estudiantes [Relationships between area and perimeter: Teachers’ and students’ beliefs]. Revista Latinoamericana de Investigación en Matemática Educativa, 10(1), 39-68.
de Gamboa, G., Badillo, E., & Font, V. (2023). Meaning and structure of mathematical connections in the classroom. Canadian Journal of Science, Mathematics and Technology Education, 23(2), 241-261. https://doi.org/10.1007/s42330-023-00281-2
de Gamboa, G., Caviedes, S., & Badillo, E. (2024). The role of intra-mathematical connections in learning decimal numbers. Advances of Research in Mathematics Education, 25, 131-149. https://doi.org/10.35763/aiem25.6399
Dolores-Flores, C., & García-García, J. (2017). Intra-mathematics and extra-mathematics connections that occur when solving calculus problems in a context: A case study in higher level education. Bolema: Boletim de Educação Matemática, 31(57), 158-180. https://doi.org/10.1590/1980-4415v31n57a08
Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1-2), 103-131. https://doi.org/10.1007/s10649-006-0400-z
Ekawati, R., Kohar, A. W., Imah, E. M., Amin, S. M., & Fiangga, S. (2019). Students’ cognitive processes in solving problem related to the concept of area conservation. Journal on Mathematics Education, 10(1), 21-36. https://doi.org/10.22342/jme.10.1.6339.21-36
Eli, J., Mohr-Schroeder, M., & Lee, C. (2011). Exploring mathematical connections of prospective middle grades teachers through card-sorting tasks. Mathematics Education Research Journal, 23(3), 297-319. https://doi.org/10.1007/s13394-011-0017-0
Evitts, T. (2004). Investigating the mathematical connections that preservice teachers use and develop while solving problems from reform curricula [PhD thesis, Pennsylvania State University].
Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Reidel.
García-García, J., & Dolores-Flores, C. (2018). Intra-mathematical connections made by high school students in performing calculus tasks. International Journal of Mathematical Education in Science and Technology, 49(2), 227-252. https://doi.org/10.1080/0020739X.2017.1355994
García-García, J., & Dolores-Flores, C. (2021). Exploring pre-university students’ mathematical connections when solving calculus application problems. International Journal of Mathematical Education in Science and Technology, 52(6), 912-936. https://doi.org/10.1080/0020739X.2020.1729429
Gómez, E. E. (2009). Una nota metodológica sobre los análisis cualitativos. El análisis de las relaciones entre los elementos: El análisis de las frecuencias y co-ocurrencias [A methodological note on qualitative analyses. Analysis of the relationships between elements: Analysis of frequencies and co-occurrences]. Theoria, 18(1), 57-67.
Hatisaru, V. (2023). Mathematical connections established in the teaching of functions. Teaching Mathematics and its Applications: An International Journal of the IMA, 42(3), 207-227. https://doi.org/10.1093/teamat/hrac013
Hill, C. E., Knox, S., Thompson, B. J., Williams, E. N., Hess, S. A., & Ladany, N. (2005). Consensual qualitative research: An update. Journal of Counseling Psychology, 52(2), 196-205. https://doi.org/10.1037/0022-0167.52.2.196
Hong, D. S., & Runnalls, C. (2020). Examining preservice teachers’ responses to area conservation tasks. School Science and Mathematics, 120(5), 262-272. https://doi.org/10.1111/ssm.12409
Huang, H.-M. E. (2017). Curriculum interventions for area measurement instruction to enhance children’s conceptual understanding. International Journal of Science and Mathematics Education, 15, 1323-1341. https://doi.org/10.1007/s10763-016-9745-7
Kamii, C., & Kysh, J. (2006). The difficulty of “length× width”: Is a square the unit of measurement? The Journal of Mathematical Behavior, 25(2), 105-115. https://doi.org/10.1016/j.jmathb.2006.02.001
Kordaki, M. (2003). The effect of tools of a computer microworld on students’ strategies regarding the concept of conservation of area. Educational Studies in Mathematics, 52(2), 177-209. https://doi.org/10.1023/a:1024065107302
Kospentaris, G., Spyrou, P., & Lappas, D. (2011). Exploring students’ strategies in area conservation geometrical tasks. Educational Studies in Mathematics, 77(1), 105-127. https://doi.org/10.1007/s10649-011-9303-8
Lehmann, T. H. (2023). Learning to measure the area of composite shapes. Educational Studies in Mathematics, 112(3), 531-565. https://doi.org/10.1080/14794802.2024.2304329
Mamona-Downs, J., & Papadopoulos, I. (2006). The problem-solving element work relate to the concept of area. In Proceedings of the 30 Conference of the International Group for the psychology of Mathematics Education. PME.
McDonald, N., Schoenebeck, S., & Forte, A. (2019). Reliability and inter-rater reliability in qualitative research: Norms and guidelines for CSCW and HCI practice. Proceedings of the ACM on Human-Computer Interaction, 3(72), 1-23. https://doi.org/10.1145/3359174
Murphy, C. (2012). The role of subject knowledge in primary prospective teachers’ approaches to teaching the topic of area. Journal of Mathematics Teacher Education, 15(3), 187-206. https://doi.org/10.1007/s10857-011-9194-8
Piaget, J., Inhelder, B., & Szeminska, A. (1981). The child’s conception of geometry. Norton and Company.
Rodríguez-Nieto, C. A., Font Moll, V., Borji, V., & Rodríguez-Vásquez, F. M. (2022). Mathematical connections from a networking of theories between extended theory of mathematical connections and onto-semiotic approach. International Journal of Mathematical Education in Science and Technology, 53(9), 2364-2390. https://doi.org/10.1080/0020739X.2021.1875071
Runnalls, C., & Hong, D. S. (2020). Half the base or half the height?: Exploring a student’s justification of ½ × base × height. International Journal of Mathematical Education in Science and Technology, 51(4), 604-613. https://doi.org/10.1080/0020739X.2019.1576928
Sari, P., & Ng, S. F. (2022). Exploring quantitative relationship through area conservation activity. Journal on Mathematics Education, 13(1), 31-50. https://doi.org/10.22342/jme.v13i1.pp31-50
Sari, Y. M., Fiangga, S., El Milla, Y. I., & Puspaningtyas, N. D. (2023). Exploring students’ proportional reasoning in solving guided-unguided area conservation problem: A case of Indonesian students. Journal on Mathematics Education, 14(2), 375-394. https://doi.org/10.22342/jme.v13i1.pp31-50
Tashakkori, A., & Creswell, J. W. (2007). The new era of mixed methods. Journal of Mixed Methods Research, 1(1), 3-7. https://doi.org/10.1177/2345678906293042
Zacharos, K. (2006). Prevailing educational practices for area measurement and students’ failure in measuring areas. The Journal of Mathematical Behavior, 25(3), 224-239. https://doi.org/10.1016/j.jmathb.2006.09.003

How to cite this article

APA

Barrera, S. C., de Gamboa, G., Ribot-Llobet, È., & Badillo, E. (2026). How do prospective primary teachers approach area conservation? An analysis based on intra-mathematical connections. Eurasia Journal of Mathematics, Science and Technology Education, 22(10), em2938. https://doi.org/10.29333/ejmste/19476

Vancouver

Barrera SC, de Gamboa G, Ribot-Llobet È, Badillo E. How do prospective primary teachers approach area conservation? An analysis based on intra-mathematical connections. EURASIA J Math Sci Tech Ed. 2026;22(10):em2938. https://doi.org/10.29333/ejmste/19476

AMA

Barrera SC, de Gamboa G, Ribot-Llobet È, Badillo E. How do prospective primary teachers approach area conservation? An analysis based on intra-mathematical connections. EURASIA J Math Sci Tech Ed. 2026;22(10), em2938. https://doi.org/10.29333/ejmste/19476

Chicago

Barrera, Sofía Caviedes, Genaro de Gamboa, Èdgar Ribot-Llobet, and Edelmira Badillo. "How do prospective primary teachers approach area conservation? An analysis based on intra-mathematical connections". Eurasia Journal of Mathematics, Science and Technology Education 2026 22 no. 10 (2026): em2938. https://doi.org/10.29333/ejmste/19476

Harvard

Barrera, S. C., de Gamboa, G., Ribot-Llobet, È., and Badillo, E. (2026). How do prospective primary teachers approach area conservation? An analysis based on intra-mathematical connections. Eurasia Journal of Mathematics, Science and Technology Education, 22(10), em2938. https://doi.org/10.29333/ejmste/19476

MLA

Barrera, Sofía Caviedes et al. "How do prospective primary teachers approach area conservation? An analysis based on intra-mathematical connections". Eurasia Journal of Mathematics, Science and Technology Education, vol. 22, no. 10, 2026, em2938. https://doi.org/10.29333/ejmste/19476