# Mathematical Discourses on Propositional Equivalence: An Exploration through the Commognitive Lens

### Abstract

This paper describes the mathematical learning of purposively sampled second-year Bachelor of Secondary Education mathematics majors from a state university in Bulacan province on propositional equivalence concept within the Logic and Set Theory course via a commognitive lens. This small-scale study employed exploratory qualitative research with one class recording, one focus group, and select activity outputs. Four participants in the focus group were sampled based on commognitive conflict occurrences. The teacher-researcher operated as a co-participant in the mathematical discourse. The dean's approval and participants' informed consent were observed, explaining the research objectives and confidentiality scope. The findings present accounts and descriptions of participants' mathematical discourses through the commognitive lens: word use, visual mediators, endorsed narratives, and routine practices that describe Logic and Set Theory discourses on the propositional equivalence concept from a participationist's learning standpoint.

### References

Biggs, J., & Tang, C. (2007). Designing intended learning outcomes. In Teaching for Quality Learning at University (3rd ed., pp. 64–90). USA: The McGraw-Hill Companies.

Bullock, J.O. (1994). Literacy in the language of mathematics. The American Mathematical Monthly, 101(8), 735–743.

Commission on Higher Education (2017). CMO no. 75, s. 2017: Policies, standards, and guidelines for bachelor of secondary education. Retrieved September 14, 2021. https://ched.gov.ph/wp-content/uploads/2017/11/CMO-No.-75-s.-2017.pdf

Daher, W.M. (2020). Grade 10 students' technology-based exploration processes of narratives associated with the sine function. Eurasia Journal of Mathematics, Science and Technology Education, 16(6), em1852.

Emre-Akdoğan, E., Güçler, B., & Argün, Z. (2018). One student's discursive development on rotation in relation to instruction from a commognitive perspective. In E. Bergqvist, M. Österholm, C. Granberg, & L. Sumpter (Eds.), Proceedings of the 42nd Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 403–410). Umeå, Sweden: PME.

Epp, S. (2011). Discrete mathematics with applications (4th ed.). Boston: Cengage Learning.

Fraenkel, J.R., & Wallen, N.E. (2009). How to design and evaluate research in education (7th ed.). New York: McGraw-Hill Higher Education.

Gerstein, L.J. (1996). Introduction to mathematical structures and proofs. New York: Springer.

Ho, W.K., Lim, S.H., Tay, E.G., Leong, Y.H., & Teo, K.M. (2019). Passing a proof message: student-teacher communication through a commognitive lens. In G. Hine, S. Blackley & A. Cooke (Eds.), Proceedings of the 42nd Annual Conference of the Mathematics Education Research Group of Australasia - Mathematics Education Research: Impacting Practice (pp. 700–706). Adelaide, Australia: MERGA.

Ioannou, M. (2018). Investigating the discursive shift in the learning of group theory: analysis of some interdiscursive commognitive conflicts. Proceedings of CERME 10 (pp. 2097–2104), Feb 2017, Dublin, Ireland.

Koriat, A. (2019). Confidence judgments: The monitoring of object-level and same-level performance. Metacognition and Learning, 14(3), 463–478.

Koshy, T. (2004). Discrete mathematics with applications. USA: Elsevier Academic Press. Lavie, I., Steiner, A., & Sfard, A. (2019). Routines we live: from ritual to exploration. Educational Studies in Mathematics, 101, 153–176.

Lestari, A.S.B., Nusantara, T., Susiswo, S., Chandra, T.D., & Indrawatiningsih, N. (2021). Exploring the argumentation skills of prospective teachers based on commognitive approach using Moodle LMS. TEM Journal, 10(3), 1370–1376.

Martin-Molina, V., González-Regeña, A.J., Toscano, R., & Gavilán-Izquierdo, J.M. (2020). Differences between how undergraduate students define geometric solids and what their lecturers expect from them through the lens of the theory of commognition. Eurasia Journal of Mathematics, Science and Technology, 16(12), em917.

Mpofu, S., & Mudaly, V. (2020). Grade 11 rural learners understanding of functions: a commognition perspective. African Journal of Research in Mathematics, Science and Technology, 24(2), 156–168.

Nardi, E., Ryve, A., Stadler, E., & Viirman, O. (2014). Commognitive analyses of the learning and teaching of mathematics at university level: the case of discursive shifts in the study of Calculus. Research in Mathematics Education, 16(2), 182–198.

Nelson, T.O., & Narens, L. (1994). Why investigate metacognition? In J. Metcalfe & A.P. Shimamura (Eds.), Metacognition: Knowing about knowing (pp. 1–25). Cambridge, Massachusetts: The MIT Press.

Nonose, K., Kanno, T., & Furuta, K. (2012). A team cognition model derived from an analysis of reflection on cooperation. Cognition, Technology and Work, 14, 83–92.

Park J.Y. (2017). A commognitive perspective on pre-service secondary teachers' content knowledge in mathematical modelling. In G.A. Stillman, W. Blum., & G. Kaiser (Eds), Mathematical Modelling and Applications. International Perspectives on the Teaching and Learning of Mathematical Modelling (pp. 289– 299). Switzerland: Springer.

Pratiwi, E., Nusantara, T., Susiswo, S., & Muksar, M. (2020). Textual and contextual commognitive conflict students in solving an improper fraction. Journal for the Education of Gifted Young Scientists, 8(2), 731–742.

Research Center for Teacher Quality (RCTQ) (2020). Prototype syllabi on priority programs in pre-service teacher education compendium 6: Bachelor of secondary education mathematics specialization courses. Teacher Education Council and Research Center for Teacher Quality. Retrieved September 15, 2021. https://www.rctq.ph/?p=1310

Roberts, A., & Le Roux, K. (2019). A commognitive perspective on grade 8 and grade 9 learner thinking about linear equations. Pythagoras, 40(1), a438.

Rosen, K.H. (2012). Discrete mathematics and its applications (7th ed.). New York: McGraw Hill.

Schiersmann, C., Einarsdóttir, S., Katsarov, J., Lerkkanen, J., Mulvey, R., Pouyaud, J., … Weber, P. (2016). European competence standards. In European Competence Standards for the Academic Training of Career Practitioners: NICE Handbook Volume 2 (1st ed., pp. 49–62). Verlag Barbara Budrich.

Sfard, A. (2007). When the rules of discourse change, but nobody tells you: Making sense of mathematics learning from a commognitive standpoint. Journal of the Learning Sciences, 16(4), 565–613.

Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses, and mathematizing. UK: Cambridge University Press.

Sfard, A. (2018a). Commognition. In S. Lerman (Ed.), Encyclopedia of Mathematics Education. Switzerland: Springer.

Sfard, A. (2018b). On the need for theory of mathematics learning and the promise of 'commognition'. In P. Ernest (Ed.), The Philosophy of Mathematics Education Today. ICME-13 Monographs (pp. 219–228). Switzerland: Springer.

Sfard, A., & Lavie, I. (2005). Why cannot children see as the same what grown-ups cannot see as different? –early numerical thinking revisited. Cognition and Instruction, 23(2), 237–309.

Stebbins, R.A. (2008). Exploratory data analysis. In L.M. Given (Ed.), The SAGE Encyclopedia of Qualitative Research Methods (pp. 325–329). USA: SAGE Publications.

Sundstrom, T. (2014). Mathematical reasoning: Writing and proof. Allendale, England: ScholarWorks @ Grand Valley State University.

UNESCO (2021). SDG resources for educators – quality education. Retrieved from https://en.unesco.org/themes/education/sdgs/material/04

Velleman, D.J. (2006). How to prove it: A structured approach (2nd ed.). UK: Cambridge University Press.

Viirman, O. (2014). The functions of function discourse – university mathematics teaching from a commognitive standpoint. International Journal of Mathematical Education in Science and Technology, 45(4), 512–527.

Viirman, O. (2015). Explanation, motivation and question posing routines in university mathematics teachers' pedagogical discourse: a commognitive analysis. International Journal of Mathematical Education in Science and Technology, 46(8), 1165–1181.

Zuckerman, B.L., Azari, A.R., & Doane, W.E.J. (2013). Challenges related to competencies. In Advancing Technology-Enhanced Education: A Workshop Report (pp. 13–16). Institute for Defense Analyses.

**Asian Journal of Research in Education and Social Sciences**, [S.l.], v. 4, n. 2, p. 162-174, aug. 2022. Available at: <https://myjms.mohe.gov.my/index.php/ajress/article/view/18658>. Date accessed: 26 sep. 2022.