Types of Generalization Made by Pupils Aged 12–13 and by Their Future Mathematics Teachers

Mária Slavíčková

doi:10.14712/18047106.1722

Vol. 11 No. 2 (2020): sciED Research Reports

Types of Generalization Made by Pupils Aged 12–13 and by Their Future Mathematics Teachers

Mária Slavíčková, ¹

Comenius University Bratislava

DOI: https://doi.org/10.14712/18047106.1722

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Slavíčková, M. (2021). Types of Generalization Made by Pupils Aged 12–13 and by Their Future Mathematics Teachers. Scientia in Educatione, 11(2), 40-52. https://doi.org/10.14712/18047106.1722

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Abstract

This paper seeks to establish what kind of arguments pupils (aged 12–13) use and how they make their assumptions and generalizations. Our research also explored the same phenomenon in the case of graduate mathematics teachers studying for their masters’ degrees in our faculty at that time. The main focus was on algebraic reasoning, in particular pattern exploring and expressing regularities in numbers. In this paper, we introduce the necessary concepts and notations used in the study, briefly characterize the theoretical levels of cognitive development and terms from the Theory of Didactical Situations. We set out to answer three research questions. To collect the research data, we worked with a group of 32 pupils aged 12–13 and 19 university students (all prospective mathematics teachers in the first year of their master’s). We assigned them two flexible tasks to and asked them to explain their findings/formulas. Besides that, we collected additional (supportive) data using a short questionnaire. The supporting data concerned their opinions on the tasks and the explanations. The results and limited scope of the research indicated what should be changed in preparing future mathematics teachers. These changes could positively influence the pupils’ strategies of solving not only flexible tasks but also their ability to generalize.

References

Almossawi, A. (2016). An illustrated book of bad arguments. Scribe Publications. Search in Google Scholar
Becker, J.R., & Rivera, F. (2006). Establishing and justifying algebraic generalization at the sixth grade level. 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. 465–472). PME, Charles University. Search in Google Scholar
Blanton, M. L. (2008). Algebra and the elementary classroom: Transforming thinking, transforming practice. Heinemann. Search in Google Scholar
Blanton, M. L., & Kaput, J. J. (2005). Characterizing a classroom practice that promotes algebraic reasoning. Journal for Research in Mathematics Education, 36(5), 412–446. https://doi.org/10.2307/30034944 Open DOI Search in Google Scholar
Breiteig, T., & Grevholm, B. (2006). The transition from arithmetic to algebra: to reason, explain, argue, generalize and justify. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings 30th Conference of the International Group for the Psychology of Mathematics Education (vol. 2, pp. 225–232). PME, Charles University. Search in Google Scholar
Brousseau, G. (1997). Theory of didactical situations in mathematics. Kluwer. Search in Google Scholar
Eisenmann, P., Přibyl, J., Novotná, J., Břehovský, J., & Cihlář, J. (2017). Volba řešitelských strategií v závislosti na věku [Choice of heuristic strategies with respect to age]. Scientia in educatione, 8(2), 21–38. https://doi.org/10.14712/18047106.432 Open DOI Search in Google Scholar
Friel, S.N., & Markworth, K.A. (2009). A framework for analyzing geometric pattern tasks. Mathematics Teaching in the Middle School, 15(1), 24–33. https://doi.org/10.5951/mtms.15.1.0024 Open DOI Search in Google Scholar
Hallagan, J. E., Rule, A.C., & Carlson, L. F. (2009). Elementary school pre-service teachers’ understanding of algebraic generalizations. The Mathematics Enthusiast, 6(1), 201–206. https://scholarworks.umt.edu/tme/vol6/iss1/16 Search in Google Scholar
Kaput, J. (1995). A research based supporting long term algebra reform? Paper presented at the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (17th, Columbus, OH, October 21–24, 1995). Retrieved May 25, 2020 from https://files.eric.ed.gov/fulltext/ED389539.pdf Search in Google Scholar
Kaput, J. (1999). Teaching and learning a new algebra. In E. Fennema, & T.A. Romberg (Eds.), Mathematics classrooms that promote understanding (pp. 134–156). Routlege. Search in Google Scholar
Lannin, J. (2004). Developing mathematical power by using explicit and recursive reasoning. The Mathematics Teacher, 98(4), 216–223. https://www.jstor.org/stable/27971686 Search in Google Scholar
Lee, L., & Wheeler, D. (1987). Algebraic thinking in high school students: Their conceptions of generalization and justification [Research Report]. Concordia Universtiy, Departmenf ot Mathematics. Search in Google Scholar
Makowski, M.B. (2020). The written and oral justifications of mathematical claims of middle school pre-service teachers. Research in Mathematics Education, advance online publication https://doi.org/10.1080/14794802.2020.1777190 Open DOI Search in Google Scholar
Mason, J. (1996). Expressing generality and roots of algebra. In N. Bernarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra. Perspectives for research and teaching (vol. 18, pp. 65–86). Springer. https://doi.org/10.1007/978-94-009-1732-3 5 Open DOI Search in Google Scholar
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 Open DOI Search in Google Scholar
Novotná, J., Eisenmann, P., & Přibyl, J. (2015). Impact of heuristic strategies on pupils’ attitudes to problem solving. Journal on Efficiency and Responsibility in Education and Science, 8(1), 15–23. https://doi.org/10.7160/eriesj.2015.080103 Open DOI Search in Google Scholar
Piaget, J. (1980). Adaptation and Intelligence: Organic Selection and Phenocopy. University of Chicago Press. Richardson, K., Berenson, S., & Staley, K. (2009). Prospective elementary teachers use of representation to reason algebraically. Journal of Mathematical Behavior, 28(2), 188–199. https://doi.org/10.1016/j.jmathb.2009.09.002 Open DOI Search in Google Scholar
Rivera, F.D., & Becker, J.R. (2005). Figural and numerical models of generalization in algebra. Mathematics Teaching in the Middle School, 11(4), 198–203. Search in Google Scholar
Rivera, F.D., & Becker, J.R. (2007). Abduction–induction (generalization) processes of elementary majors on figural patterns in algebra. Journal of Mathematical Behavior, 26(2), 140–155. https://doi.org/10.1016/J.JMATHB.2007.05.001 Open DOI Search in Google Scholar
Rivera, F.D., & Becker, J.R. (2008). Middle school children’s cognitive perceptions of constructive and deconstructive generalizations involving linear figural patterns. ZDM Mathematics Education, 40(1), 65–82. https://doi.org/10.1007/s11858-007-0062-z Open DOI Search in Google Scholar
Russel, S. J., Schifter, D., & Bastable, V. (2011). Developing algebraic thinking in the context of arithmetic. In J. Cai, & E. Knuth (Eds.), Early algebraization, advances in mathematics education (pp. 43–69). Springer. https://doi.org/10.1007/978-3-642-17735-4 4 Open DOI Search in Google Scholar
Slavíčková, M., & Vargová, M. (2019). Pupils stage of a cognitive development in correlation with solving algebraic tasks. In ICERI 2019 proceeding (pp. 7260–7268). IATED. https://doi.org/10.21125/iceri.2019.1726 Open DOI Search in Google Scholar
Strand, K., & Mills, B. (2014). Mathematical content knowledge for teaching elementary mathematics: A focus on algebra. The Mathematics Enthusiast, 11, 385–432. https://scholarworks.umt.edu/cgi/viewcontent.cgi?article=1307&context=tme Search in Google Scholar
Van de Walle, J.A., Karp, K. S., & Bay-Williams, J.M. (2012). Elementary and middle school mathematics: Teaching developmentally (9th ed.). Pearson. Search in Google Scholar
Van Dooren, W., Verschaffel, L., & Onghena, P. (2002). The impact of preservice teachers’ content knowledge on their evaluation of students’ strategies for solving arithmetic and algebra word problems. Journal for Research in Mathematics Education, 33(5), 319–351. https://doi.org/10.2307/4149957 Open DOI Search in Google Scholar
Van Dooren, W., Verschaffel, L., & Onghena, P. (2003). Pre-service teachers’ preferred strategies for solving arithmetic and algebra word problems. Journal of Mathematics Teacher Education, 6, 27–52. https://doi.org/10.1023/a:1022109006658 Open DOI Search in Google Scholar
Watson, A., & Geest, E.D. (2005). Principled teaching for deep progress: Improving mathematical learning beyond methods and materials. Educational Studies in Mathematics, 58(2), 209–234. Search in Google Scholar

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[1] Almossawi, A. (2016). An illustrated book of bad arguments. Scribe Publications. Search in Google Scholar

[2] Becker, J.R., & Rivera, F. (2006). Establishing and justifying algebraic generalization at the sixth grade level. 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. 465–472). PME, Charles University. Search in Google Scholar

[3] Blanton, M. L. (2008). Algebra and the elementary classroom: Transforming thinking, transforming practice. Heinemann. Search in Google Scholar

[4] Blanton, M. L., & Kaput, J. J. (2005). Characterizing a classroom practice that promotes algebraic reasoning. Journal for Research in Mathematics Education, 36(5), 412–446. https://doi.org/10.2307/30034944 Open DOI Search in Google Scholar

[5] Breiteig, T., & Grevholm, B. (2006). The transition from arithmetic to algebra: to reason, explain, argue, generalize and justify. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings 30th Conference of the International Group for the Psychology of Mathematics Education (vol. 2, pp. 225–232). PME, Charles University. Search in Google Scholar

[6] Brousseau, G. (1997). Theory of didactical situations in mathematics. Kluwer. Search in Google Scholar

[7] Eisenmann, P., Přibyl, J., Novotná, J., Břehovský, J., & Cihlář, J. (2017). Volba řešitelských strategií v závislosti na věku [Choice of heuristic strategies with respect to age]. Scientia in educatione, 8(2), 21–38. https://doi.org/10.14712/18047106.432 Open DOI Search in Google Scholar

[8] Friel, S.N., & Markworth, K.A. (2009). A framework for analyzing geometric pattern tasks. Mathematics Teaching in the Middle School, 15(1), 24–33. https://doi.org/10.5951/mtms.15.1.0024 Open DOI Search in Google Scholar

[9] Hallagan, J. E., Rule, A.C., & Carlson, L. F. (2009). Elementary school pre-service teachers’ understanding of algebraic generalizations. The Mathematics Enthusiast, 6(1), 201–206. https://scholarworks.umt.edu/tme/vol6/iss1/16 Search in Google Scholar

[10] Kaput, J. (1995). A research based supporting long term algebra reform? Paper presented at the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (17th, Columbus, OH, October 21–24, 1995). Retrieved May 25, 2020 from https://files.eric.ed.gov/fulltext/ED389539.pdf Search in Google Scholar

[11] Kaput, J. (1999). Teaching and learning a new algebra. In E. Fennema, & T.A. Romberg (Eds.), Mathematics classrooms that promote understanding (pp. 134–156). Routlege. Search in Google Scholar

[12] Lannin, J. (2004). Developing mathematical power by using explicit and recursive reasoning. The Mathematics Teacher, 98(4), 216–223. https://www.jstor.org/stable/27971686 Search in Google Scholar

[13] Lee, L., & Wheeler, D. (1987). Algebraic thinking in high school students: Their conceptions of generalization and justification [Research Report]. Concordia Universtiy, Departmenf ot Mathematics. Search in Google Scholar

[14] Makowski, M.B. (2020). The written and oral justifications of mathematical claims of middle school pre-service teachers. Research in Mathematics Education, advance online publication https://doi.org/10.1080/14794802.2020.1777190 Open DOI Search in Google Scholar

[15] Mason, J. (1996). Expressing generality and roots of algebra. In N. Bernarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra. Perspectives for research and teaching (vol. 18, pp. 65–86). Springer. https://doi.org/10.1007/978-94-009-1732-3 5 Open DOI Search in Google Scholar

[16] 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 Open DOI Search in Google Scholar

[17] Novotná, J., Eisenmann, P., & Přibyl, J. (2015). Impact of heuristic strategies on pupils’ attitudes to problem solving. Journal on Efficiency and Responsibility in Education and Science, 8(1), 15–23. https://doi.org/10.7160/eriesj.2015.080103 Open DOI Search in Google Scholar

[18] Piaget, J. (1980). Adaptation and Intelligence: Organic Selection and Phenocopy. University of Chicago Press. Richardson, K., Berenson, S., & Staley, K. (2009). Prospective elementary teachers use of representation to reason algebraically. Journal of Mathematical Behavior, 28(2), 188–199. https://doi.org/10.1016/j.jmathb.2009.09.002 Open DOI Search in Google Scholar

[19] Rivera, F.D., & Becker, J.R. (2005). Figural and numerical models of generalization in algebra. Mathematics Teaching in the Middle School, 11(4), 198–203. Search in Google Scholar

[20] Rivera, F.D., & Becker, J.R. (2007). Abduction–induction (generalization) processes of elementary majors on figural patterns in algebra. Journal of Mathematical Behavior, 26(2), 140–155. https://doi.org/10.1016/J.JMATHB.2007.05.001 Open DOI Search in Google Scholar

[21] Rivera, F.D., & Becker, J.R. (2008). Middle school children’s cognitive perceptions of constructive and deconstructive generalizations involving linear figural patterns. ZDM Mathematics Education, 40(1), 65–82. https://doi.org/10.1007/s11858-007-0062-z Open DOI Search in Google Scholar

[22] Russel, S. J., Schifter, D., & Bastable, V. (2011). Developing algebraic thinking in the context of arithmetic. In J. Cai, & E. Knuth (Eds.), Early algebraization, advances in mathematics education (pp. 43–69). Springer. https://doi.org/10.1007/978-3-642-17735-4 4 Open DOI Search in Google Scholar

[23] Slavíčková, M., & Vargová, M. (2019). Pupils stage of a cognitive development in correlation with solving algebraic tasks. In ICERI 2019 proceeding (pp. 7260–7268). IATED. https://doi.org/10.21125/iceri.2019.1726 Open DOI Search in Google Scholar

[24] Strand, K., & Mills, B. (2014). Mathematical content knowledge for teaching elementary mathematics: A focus on algebra. The Mathematics Enthusiast, 11, 385–432. https://scholarworks.umt.edu/cgi/viewcontent.cgi?article=1307&context=tme Search in Google Scholar

[25] Van de Walle, J.A., Karp, K. S., & Bay-Williams, J.M. (2012). Elementary and middle school mathematics: Teaching developmentally (9th ed.). Pearson. Search in Google Scholar

[26] Van Dooren, W., Verschaffel, L., & Onghena, P. (2002). The impact of preservice teachers’ content knowledge on their evaluation of students’ strategies for solving arithmetic and algebra word problems. Journal for Research in Mathematics Education, 33(5), 319–351. https://doi.org/10.2307/4149957 Open DOI Search in Google Scholar

[27] Van Dooren, W., Verschaffel, L., & Onghena, P. (2003). Pre-service teachers’ preferred strategies for solving arithmetic and algebra word problems. Journal of Mathematics Teacher Education, 6, 27–52. https://doi.org/10.1023/a:1022109006658 Open DOI Search in Google Scholar

[28] Watson, A., & Geest, E.D. (2005). Principled teaching for deep progress: Improving mathematical learning beyond methods and materials. Educational Studies in Mathematics, 58(2), 209–234. Search in Google Scholar

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