International Electronic Journal of Mathematics Education

International Electronic Journal of Mathematics Education Indexed in ESCI
Encountering Proportional Reasoning During a Single Algebra Lesson: A Microgenetic Analysis

A correction notice for this article is published at:
Lundberg, A. L. V. (2022). Correction to Encountering Proportional Reasoning During a Single Algebra Lesson: A Microgenetic Analysis. International Electronic Journal of Mathematics Education, 17(3), em0693. https://doi.org/10.29333/iejme/12082

APA
In-text citation: (Lundberg, 2022)
Reference: Lundberg, A. L. V. (2022). Encountering Proportional Reasoning During a Single Algebra Lesson: A Microgenetic Analysis. International Electronic Journal of Mathematics Education, 17(1), em0673. https://doi.org/10.29333/iejme/11571
AMA
In-text citation: (1), (2), (3), etc.
Reference: Lundberg ALV. Encountering Proportional Reasoning During a Single Algebra Lesson: A Microgenetic Analysis. INT ELECT J MATH ED. 2022;17(1), em0673. https://doi.org/10.29333/iejme/11571
Chicago
In-text citation: (Lundberg, 2022)
Reference: Lundberg, Anna L. V.. "Encountering Proportional Reasoning During a Single Algebra Lesson: A Microgenetic Analysis". International Electronic Journal of Mathematics Education 2022 17 no. 1 (2022): em0673. https://doi.org/10.29333/iejme/11571
Harvard
In-text citation: (Lundberg, 2022)
Reference: Lundberg, A. L. V. (2022). Encountering Proportional Reasoning During a Single Algebra Lesson: A Microgenetic Analysis. International Electronic Journal of Mathematics Education, 17(1), em0673. https://doi.org/10.29333/iejme/11571
MLA
In-text citation: (Lundberg, 2022)
Reference: Lundberg, Anna L. V. "Encountering Proportional Reasoning During a Single Algebra Lesson: A Microgenetic Analysis". International Electronic Journal of Mathematics Education, vol. 17, no. 1, 2022, em0673. https://doi.org/10.29333/iejme/11571
Vancouver
In-text citation: (1), (2), (3), etc.
Reference: Lundberg ALV. Encountering Proportional Reasoning During a Single Algebra Lesson: A Microgenetic Analysis. INT ELECT J MATH ED. 2022;17(1):em0673. https://doi.org/10.29333/iejme/11571

Abstract

This case study explores how 12-13-year-old students encounter proportional reasoning while working with geometric patterning tasks using concrete materials. The focus is on the students’ use of spontaneous concepts when first dealing with such patterns in the context of collaborative work. Based on video recordings of a single lesson, a microgenetic analysis was performed to identify students’ learning trajectories, starting with students familiarizing themselves with pattern structure, followed by engagement in proportional reasoning, and ending with students perceiving a new technique to handle a situation where proportional reasoning did not suffice. While some student groups were able to move along the whole trajectory, most groups, when facing challenges, regressed to simpler techniques. The results provide new insights into students’ learning trajectories, which can be used to support students’ progress in the context of student-teacher interaction.

Disclosures

Declaration of Conflict of Interest: No conflict of interest is declared by author(s).

Data sharing statement: Data supporting the findings and conclusions are available upon request from the corresponding author(s).

References

  • Bosch, M. (1994). La dimensión ostensiva en la actividad mathemática: El caso de la proporcionalidad [The ostensive dimension in mathematical activity: The case of proportionality] [Doctoral dissertation, Universitat Autónoma de Barcelona].
  • Bosch, M., & Gascón, J. (2006). Twenty-five years of didactic transposition. International Commission on Mathematical Instruction Bulletin, 58, 51-65.
  • Burgos, M., & Godino, J. D. (2020). Semiotic conflicts in the learning of proportionality: Analysis of a teaching experience in primary education. International Electronic Journal of Mathematics Education 15(3). https://doi.org/10.29333/iejme/7943
  • Chevallard, Y. (2006). Steps towards a new epistemology in mathematics education. In M. Bosch (Ed.), Proceedings of the 4th congress of the European society for research in mathematics education (pp. 21-30). FUNDEMI-1QS and ERME.
  • Chevallard, Y., Bosch, M., & Kim, S. (2015). What is a theory according to the anthropological theory of the didactic? In K. Krainer, & N. Vondrová (Eds.), Proceedings of the ninth congress of the European society for research in mathematics education (pp. 2614-2620). Charles University in Prague, Faculty of Education and ERME.
  • Cole, M. (1996). Cultural psychology: A once and future discipline. The Belknap Press.
  • Confrey, J., Maloney, J., Nguyen, K., Mojica, G., & Myers, M. (2009). Equipartitioning/splitting as a foundation of rational number reasoning using learning trajectories. In M. Tzekaki, M. Kaldrimidou, & C. Sakonidis (Eds.), Proceedings of the 33rd conference of the international group for the psychology of mathematics education (Vol. 1, pp. 1-8). PME.
  • Cramer, K., & Post, T. (1993). Connecting research to teaching: Proportional reasoning. Mathematics Teacher, 86(5), 404-407. https://doi.org/10.5951/mt.86.5.0404
  • Del Río, P., & Álvarez, A. (2007). Inside and outside the zone of proximal development: An ecofunctional reading of Vygotsky. In H. Daniels, M. Cole, & J. Wertsch (Eds.), The Cambridge companion to Vygotsky (pp. 276-303). Cambridge University Press. https://doi.org/10.1017/CCOL0521831040.012
  • Dewey, J. (1916/1980). Logical objects. In J. A. Boydston (Ed.), The Middle Works of John Dewey, Volume 10, 1899 - 1924: Journal articles, essays, and miscellany published in the 1916-1917 period, with an introd. by Lewis E. Hahn (Vol. 10: 1916-1917, pp. 89-97). Southern Illinois University Press.
  • Euclid, & Heath, T. L. (1956). The thirteen books of Euclid’s elements. Dover.
  • Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Reidel Publishing Company. https://doi.org/10.1007/0-306-47235-X
  • García, F. (2005). La modelización como herramienta de articulación de la matemática escolar. De la proporcionalidad a las relaciones funcionales [Modeling as an articulation tool for school mathematics. From the proportionality to the functional relations] [Doctoral dissertation, Universidad de Jaén].
  • Goodwin, C. (1994). Professional vision. American Anthropologist, 96(3), 606-633. https://doi.org/10.1525/aa.1994.96.3.02a00100
  • Granott, N. (1998). Unit of analysis in transit: From the individual’s knowledge to the ensemble process. Mind, Culture, and Activity, 5(1), 42-66. https://doi.org/10.1207/s15327884mca0501_4
  • Granott, N., & Parziale, J. (2002). Microdevelopment: A process-oriented perspective for studying development and learning. In J. Parziale, & N. Granott (Eds.), Microdevelopment: Transition processes in development and learning (pp. 1-28). Cambridge University Press. https://doi.org/10.1017/CBO9780511489709.00
  • Heath, C., Hindmarsh, J., & Luff, P. (2010). Video in qualitative research, analysing social interaction in everyday life. SAGE. https://doi.org/10.4135/9781526435385
  • Inhelder, B., & Piaget, J. (1958). The growth of logical thinking from childhood to adolescence: An essay on the construction of formal operational structures. Routledge and Kegan, Paul Ltd. https://doi.org/10.4324/9781315009674
  • Jordan, B., & Henderson, A. (1995). Interaction analysis: Foundations and practice. The Journal of the Learning Sciences, 4(1), 39-103. httpS://doi.org/10.1207/s15327809jls0401_2
  • Kieran, C. (2018). Seeking, using, and expressing structure in numbers and numerical operations: A fundamental path to developing early algebraic thinking. In C. Kieran (Ed.), & G. Kaiser (Series Ed.), ICME-13 Monographs. Teaching and learning algebraic thinking with 5- to 12-year-olds: The global evolution of an emerging field of research and practice (pp. 79-106). Springer Nature. https://doi.org/10.1007/978-3-319-68351-5
  • Kilhamn, C., & Röj-Lindberg, A.-S. (2012). Seeking hidden dimensions of algebra teaching through video analysis. In B. Grevholm, P. S. Hundeland, K. Juter, K. Kislenko, & P. E. Persson (Eds.), Nordic research in mathematics education, past, present and future (pp. 299-326). Cappelen Damm Akademisk.
  • Kirsch, A. (1969). Eine Analyse der sogenannten Schlussrechnung. [An analysis of the so-called final invoice]. Mathematische-Physikalische Semsterberichte [Mathematical-Physical Semester Reports], 16(1), 41-55.
  • Küchemann, D. (2010). Using patterns generically to see structure. Pedagogies: An International Journal, 5(3), 233-250. https://doi.org/10.1080/1554480X.2010.486147
  • Lamon, S. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics (pp. 629-667). Information Age Publishing.
  • Lannin, J., Barker, D., & Townsend, B. (2006). Recursive and explicit rules: How can we build student algebraic understanding? The Journal of Mathematical Behavior, 25, 299-317. https://doi.org/10.1016/j.jmathb.2006.11.004
  • Lincoln, Y. S., & Guba, E. (1985). Naturalistic inquiry. SAGE. https://doi.org/10.1016/0147-1767(85)90062-8
  • Lobato, J., & Ellis, A. (2010). Developing essential understanding of ratios, proportions and proportional reasoning. National Council of Teachers of Mathematics.
  • Lobato, J., & Walters, D. (2017). A taxonomy of approaches to learning trajectories and progressions. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 74-101). National Council of Teachers of Mathematics.
  • Lüken, M. (2012). Young children’s structure sense. Journal für Mathematik-Didaktik [Journal for Mathematics Didactics], 33(2), 263-285. https://doi.org/10.1007/s13138-012-0036-8
  • Lundberg, A. L. V. (2011). Proportionalitetsbegreppet i den svenska gymnasiematematiken - en studie om läromedel och nationella prov [The concept of proportionality in the Swedish upper secondary school mathematics – a study of textbooks and national examination] [Licentiate thesis, Linköping University].
  • Lundberg, A. L. V., & Hillman, T. (2013). Student-driven proportional reasoning approaches to an early algebra task. In A. M. Lindmeier & A. Heinze (Eds.), Proceedings of the 37th international group for the psychology of mathematics education (Vol. 5, pp. 112). PME.
  • Lybeck, L. (1981). Arkimedes i klassen: En ämnespedagogisk berättelse [Archimedes in the classroom: A narrative on the didactics of subject matter] [Doctoral dissertation, Gothenburg University].
  • Maloney, A., Confrey, J., & Nguyen, K. H. (2014). Learning over time: Learning trajectories in mathematics education. Information Age Publishing. https://ebookcentral.proquest.com
  • Mason, J., Stephens, M., & Watson, A. (2009). Appreciating mathematics structure for all. Mathematics Education Research Journal, 21(2), 10-32. https://doi.org/10.1007/BF03217543
  • Miyakawa, T., & Winsløw, C. (2009). Didactical designs for students’ proportional reasoning: An “open approach” lesson and a “fundamental situation”. Educational Studies in Mathematics, 72(2), 199-218. https://doi.org/10.1007/s10649-009-9188-y
  • Mulligan, J., & Mitchelmore, M. (2009). Awareness of pattern and structure in early mathematical development. Mathematics Education Research Journal, 21(2), 33-49. https://doi.org/10.1007/BF03217544
  • Radford, L. (2011a). Embodiment, perception, and symbols in the development of early algebraic thinking. In B. Ubuz (Ed.), Proceedings of the 35th conference of the international group for the psychology of mathematics education (Vol. 4, pp. 17-24). PME.
  • Radford, L. (2011b). Grade 2 students’ non-symbolic algebraic thinking. In J. Cai, & E. Knuth (Eds.), Early algebraization - a global dialogue from multiple perspectives (Vol. 2, pp. 303-322). Springer. https://doi.org/10.1007/978-3-642-17735-4_17
  • Säljö, R. (1991). Learning and mediation: Fitting reality into a table. Learning and Instruction, 1(3), 261-272. https://doi.org/10.1016/0959-4752(91)90007-u
  • Säljö, R. (2009). Learning, theories of learning, and units of analysis in research. Educational Psychologist, 44(3), 202-208. https://doi.org/10.1080/00461520903029030
  • Säljö, R. (2011). Learning in a sociocultural perspective. In V. G. Aukrust, (Ed.), Learning and cognition in education (pp. 59-63). Elsevier.
  • Shield, M., & Dole, S. (2013). Assessing the potential of mathematics textbooks to promote deep learning. Educational Studies in Mathematics, 82(2), 183-199. https://doi.org/10.1007/s10649-012-9415-9
  • Siegler, R. (2006). Microgenetic analyses of learning. In D. Kuhn, & R. Siegler (Eds.), Handbook of child psychology (Vol. 2: Cognition, perception and language, pp. 464-510). Wiley. https://doi.org/10.1002/9780470147658.chpsy0211
  • Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26, 114-145. https://doi.org/10.2307/749205
  • Simon, M. A. (2018). An emerging methodology for studying mathematics concept learning and instructional design. The Journal of Mathematical Behavior, 52, 113-121. https://doi.org/10.1016/j.jmathb.2018.03.005
  • Stacey, K. (1989). Finding and using patterns in linear generalising problems. Educational Studies in Mathematics, 20(2), 147-164. https://doi.org/10.1007/bf00579460
  • Strømskag Måsøval, H. (2011). Factors constraining students’ establishment of algebraic generality in shape patterns: A case study of didactical situations in mathematics at a University College [Doctoral dissertation, University of Agder].
  • Swedish National Agency for Education. (2001). Compulsory school: Syllabuses. Fritzes.
  • Swedish National Agency for Education. (2011). Curriculum for the compulsory school, preschool class, and the leisure-time centre, Lgr2011 (revised). Swedish National Agency for Education. https://www.skolverket.se/publikationer?id=3984
  • The Swedish Research Council. (2007). CODEX: Rules and guidelines for research. https://codex.uu.se
  • Tjoe, H., & Torre, J. (2014). On recognizing proportionality: Does the ability to solve missing value proportional problems presuppose the conception of proportional reasoning? The Journal of Mathematical Behaviour, 33, 1-7. https://doi.org/10.1016/j.jmathb.2013.09.002
  • Van Dooren, W., De Bock, D., Hessels, A., Janssens, D., & Verschaffel, L. (2005). Not everything is proportional: Effects of age and problem type on propensities for overgeneralization. Cognition and Instruction, 23(1), 57-86. https://doi.org/10.1207/s1532690xci2301_3
  • Van Dooren, W., De Bock, D., Janssens, D., & Verschaffel, L. (2008). The linear imperative: An inventory and conceptual analysis of students’ overuse of linearity. Journal for Research in Mathematics Education, 39(3), 311-342. https://doi.org/10.2307/30034972
  • Vergnaud, G. (1983). Multiplicative structures. In R. Lesh, & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 128-175). Academic Press.
  • Vygotsky, L. (1930/1978). Mind in society: The development of higher psychological processes. In M. Cole, V. John-Steiner, S. Scribner, & E. Souberman (Eds.). Harvard University Press.
  • Vygotsky, L. (1934/1986). Thought and language (A. Kozulin, Trans.). MIT Press.
  • Weber, E., Walkington, C., & McGalliard, W. (2015). Expanding notions of “Learning trajectories” in mathematics education. Mathematical Thinking and Learning, 17(4), 253-272. https://doi.org.ezproxy.ub.gu.se/10.1080/10986065.2015.1083836
  • Wells, G. (1999). Dialogic inquiry: Towards a sociocultural practice and theory of education. Cambridge University Press. https://doi.org/10.1017/cbo9780511605895
  • Werner, H. (1948). Comparative psychology of mental development. International University Press.
  • Wertsch, J. (1985). Vygotsky and the social formation of mind. Cambridge University Press.
  • Wilkie, K. (2016). Students’ use of variables and multiple representations in generalizing functional relationships prior to secondary school. Educational Studies in Mathematics, 93(3), 333-361. https://doi.org/10.1007/s10649-016-9703-x
  • Wood, D., Bruner, J., & Ross, G. (1976). The role of tutoring in problem solving. Journal of Child Psychology and Psychiatry, 17(2), 89-100. https://doi.org/10.1111/j.1469-7610.1976.tb00381.x

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