International Electronic Journal of Mathematics Education

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• Blomqvist, A., & Sumpter, L. (2024). What is functional thinking? An ontological analysis of different definitions. In Psychology of Mathematics Education Conference (PME 47), Auckland, New Zealand, 17-21 July, 2024 (Vol. 1, pp. 109-109). IGPME.
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• Chimoni, M., Pitta-Pantazi, D., & Christou, C. (2018). Examining early algebraic thinking: insights from empirical data. Educational Studies in Mathematics, 98(1), 57-76. https://doi.org/10.1007/s10649-018-9803-x
• Chimoni, M., Pitta-Pantazi, D., & Christou, C. (2023). Unfolding algebraic thinking from a cognitive perspective. Educational Studies in Mathematics, 114(1), 89-108. https://doi.org/10.1007/s10649-023-10218-z
• Ding, R., Huang, R., & Deng, X. (2023). Multiple pathways for developing functional thinking in elementary mathematics textbooks: A case study in China. Educational Studies in Mathematics, 114(2), 223-248. https://doi.org/10.1007/s10649-023-10237-w
• Georges, J. S. (1946). Teaching functional thinking in mathematics. School Science and Mathematics, 46(8), 733-748. https://doi.org/10.1111/j.1949-8594.1946.tb05946.x
• González-Polo, R. I., & Castaneda, A. (2024). Progressive mathematics of functions in secondary school students using a free-fall activity. International Electronic Journal of Mathematics Education, 19(1), Article em0765. https://doi.org/10.29333/iejme/14108
• Häggström, J. (2005). Begreppet funktion i historisk belysning [The concept of function in historical lighting]. Normat, 53(2), 82-92.
• Inhelder, B., & Piaget, J. (1958). The growth of logical thinking from childhood to adolescence: An essay on the construction of formal operational structures (translated by Anne Parsons and Stanley Milgram). Routledge & Kegan Paul Ltd. https://doi.org/10.1037/10034-000
• Juter, K. (2012). The validity of students’ conceptions of differentiability and continuity. In C. Bergsten, E. Jablonka, & M. Raman (Eds.), Evaluation and comparison of mathematical achievement: Dimensions and perspectives (pp. 121-130). Linköping University.
• Kaput, J. J. (2008). What is algebra? What is algebraic reasoning? In J. Kaput, D. W. Carraher, & M. L. Blanton (Eds), Algebra in the early grades (pp. 5-17). Routledge. https://doi.org/10.4324/9781315097435-2
• Kieran, C. (2022). The multi-dimensionality of early algebraic thinking: Background, overarching dimensions, and new directions. ZDM–Mathematics Education, 54(6), 1131-1150. https://doi.org/10.1007/s11858-022-01435-6
• Kjeldsen, T. H., & Lützen, J. (2015). Interactions between mathematics and physics: The history of the concept of function–Teaching with and about nature of mathematics. Science & Education, 24(5), 543-559. https://doi.org/10.1007/s11191-015-9746-x
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• Moss, D. L., Boyce, S., & Lamberg, T. (2020). Representations and conceptions of variables in students’ early understandings of functions. International Electronic Journal of Mathematics Education, 15(2), Article em0564. https://doi.org/10.29333/iejme/6257
• Obitko, M., Snasel, V., Smid, J., & Snasel, V. (2004). Ontology design with formal concept analysis. CLA, 128(3), 1377-1390.
• Pang, J., & Sunwoo, J. (2022). Design of a pattern and correspondence unit to foster functional thinking in an elementary mathematics textbook. ZDM–Mathematics Education, 54(6), 1315-1331. https://doi.org/10.1007/s11858-022-01411-0
• Pinto, E., Cañadas, M. C., & Moreno, A. (2022). Functional relationships evidenced and representations used by third graders within a functional approach to early algebra. International Journal of Science and Mathematics Education, 20(6), 1183-1202. https://doi.org/10.1007/s10763-021-10183-0
• Pittalis, M., Pitta-Pantazi, D., & Christou, C. (2020). Young students’ functional thinking modes: The relation between recursive patterning, covariational thinking, and correspondence relations. Journal for Research in Mathematics Education, 51(5), 631-674. https://doi.org/10.5951/jresematheduc-2020-0164
• Pitta-Pantazi, D., Chimoni, M., & Christou, C. (2020). Different types of algebraic thinking: An empirical study focusing on middle school students. International Journal of Science and Mathematics Education, 18(5), 965-984. https://doi.org/10.1007/s10763-019-10003-6
• Radford, L. (2006). Algebraic thinking and the generalization of patterns: A semiotic perspective. In S. Alatorre, J. Cortina, M. Sáiz, & A. Mendez (Eds.), Proceedings of the 28^th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (vol. 1, pp. 2-21). PME-NA.
• Ramírez, R., Cañadas, M. C., & Damián, A. (2022). Structures and representations used by 6^th graders when working with quadratic functions. ZDM–Mathematics Education, 54(6), 1393-1406. https://doi.org/10.1007/s11858-022-01423-w
• Sidorov, G., Gelbukh, A., Gómez-Adorno, H., & Pinto, D. (2014). Soft similarity and soft cosine measure: Similarity of features in vector space model. Computación y Sistemas, 18(3), 491-504. https://doi.org/10.13053/CyS-18-3-2043
• Smith, E. (2008). Representational thinking as a framework for introducing functions in the elementary curriculum. In J. Kaput, D. Carraher, & M. Blanton (Eds.), Algebra in the early grades. Lawrence Erlbaum Associates/Taylor & Francis Group and National Council of Teachers of Mathematics.
• Somasundram, P., Akmar, S. N., & Eu, L. K. (2019). Pattern generalisation by year five pupils. International Electronic Journal of Mathematics Education, 14(2), 353-362. https://doi.org/10.29333/iejme/5719
• Sproesser, U., Vogel, M., Dörfler, T., & Eichler, A. (2022). Changing between representations of elementary functions: Students’ competencies and differences with a specific perspective on school track and gender. International Journal of STEM Education, 9, Article 33. https://doi.org/10.1186/s40594-022-00350-2
• Stephens, A. C., Fonger, N., Strachota, S., Isler, I., Blanton, M., Knuth, E., & Murphy Gardiner, A. (2017). A learning progression for elementary students’ functional thinking. Mathematical Thinking and Learning, 19(3), 143-166. https://doi.org/10.1080/10986065.2017.1328636
• Sumpter, L. (2016). Two frameworks for mathematical reasoning at preschool level. In T. Meaney, O. Helenius, M. L. Johansson, T. Lange, & A. Wernberg (Eds.), Mathematics Education in the Early Years: Results from the POEM2 Conference, 2014 (pp. 157-169). Springer.
• Tanışlı, D. (2011). Functional thinking ways in relation to linear function tables of elementary school students. The Journal of Mathematical Behavior, 30(3), 206-223. https://doi.org/10.1016/j.jmathb.2011.08.001
• Veith, J. M., Beste, M. L., Kindervater, M., Krause, M., Straulino, M., Greinert, F., & Bitzenbauer, P. (2023). Mathematics education research on algebra over the last two decades: Quo vadis? Frontiers in Education, 8. https://doi.org/10.3389/feduc.2023.1211920
• Vollrath, H-J. (1986). Search strategies as indicators of functional thinking. Educational Studies in Mathematics, 17(4), 387-400. https://doi.org/10.1007/BF00311326
• Warren, E. A., & Cooper, T. J. (2005). Introducing functional thinking in year 2: A case study of early algebra teaching. Contemporary Issues in Early Childhood, 6(2), 150-162. https://doi.org/10.2304/ciec.2005.6.2.5
• Warren, E. A., Cooper, T. J., & Lamb, J. T. (2006). Investigating functional thinking in the elementary classroom: Foundations of early algebraic reasoning. The Journal of Mathematical Behavior, 25(3), 208-223. https://doi.org/10.1016/j.jmathb.2006.09.006
• Warren, E. A., Miller, J., & Cooper, T. J. (2013). Exploring young students’ functional thinking. PNA, 7(2), 75-84. https://doi.org/10.30827/pna.v7i2.6131
• Wilkie, K. J., & Ayalon, M. (2018). Investigating years 7 to 12 students’ knowledge of linear relationships through different contexts and representations. Mathematics Education Research Journal, 30, 499-523. https://doi.org/10.1007/s13394-018-0236-8
• Wilkie, K. J., & Clarke, D. M. (2016). Developing students’ functional thinking in algebra through different visualisations of a growing pattern’s structure. Mathematics Education Research Journal, 28(2), 223-243. https://doi.org/10.1007/s13394-015-0146-y
• Xolocotzin, U., Medrano-Moya, A. M., & Rojano, T. (2022). Starting points: Understanding children’s pre-instructional intuitions about function tables. ZDM–Mathematics Education, 54(6), 1363-1376. https://doi.org/10.1007/s11858-022-01424-9
• Zhu, X., Liu, H., & Peng, X. (2023). A study on similarity and difficulty evaluation of elementary school mathematics application problems based on cosine similarity and AHP. Curriculum and Teaching Methodology, 6(20), 108-115. https://doi.org/10.23977/curtm.2023.062017

APA

Sumpter, L., & Blomqvist, A. (2025). What is functional thinking? Using cosine similarity matrix in a semantic ontological analysis. International Electronic Journal of Mathematics Education, 20(1), em0804. https://doi.org/10.29333/iejme/15679

Vancouver

Sumpter L, Blomqvist A. What is functional thinking? Using cosine similarity matrix in a semantic ontological analysis. INT ELECT J MATH ED. 2025;20(1):em0804. https://doi.org/10.29333/iejme/15679

AMA

Sumpter L, Blomqvist A. What is functional thinking? Using cosine similarity matrix in a semantic ontological analysis. INT ELECT J MATH ED. 2025;20(1), em0804. https://doi.org/10.29333/iejme/15679

Chicago

Sumpter, Lovisa, and Anneli Blomqvist. "What is functional thinking? Using cosine similarity matrix in a semantic ontological analysis". International Electronic Journal of Mathematics Education 2025 20 no. 1 (2025): em0804. https://doi.org/10.29333/iejme/15679

Harvard

Sumpter, L., and Blomqvist, A. (2025). What is functional thinking? Using cosine similarity matrix in a semantic ontological analysis. International Electronic Journal of Mathematics Education, 20(1), em0804. https://doi.org/10.29333/iejme/15679

MLA

Sumpter, Lovisa et al. "What is functional thinking? Using cosine similarity matrix in a semantic ontological analysis". International Electronic Journal of Mathematics Education, vol. 20, no. 1, 2025, em0804. https://doi.org/10.29333/iejme/15679