How lower-secondary pupils approach and perceive understanding mathematics

Gabriela Novotná

doi:10.14712/18047106.2202

Vol. 13 No. 2 (2022): Scientia in educatione Research Reports

How lower-secondary pupils approach and perceive understanding mathematics

Gabriela Novotná, ¹

Katedra matematiky a didaktiky matematiky,PedF UK

DOI: https://doi.org/10.14712/18047106.2202

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Novotná, G. (2023). How lower-secondary pupils approach and perceive understanding mathematics. Scientia in Educatione, 13(2), 36-50. https://doi.org/10.14712/18047106.2202

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Abstract

Mathematics requires deep thinking. It may be educationally beneficial if pupils are aware of their own shortcomings in understanding. We investigated how pupils of lower-secondary schools in Prague perceive their understanding mathematics, with a particular focus on whether they have distinctive attitudes towards the quality of their understanding. In the first, quantitative stage, a diagnostic test of surface knowledge and a questionnaire about mathematical understanding were developed and administrated. Using a factor analysis, t-tests and methods of descriptive statistics, we created indices of understanding and ascertained that the respondents were often mixing various levels of depth of their understanding mathematics. The quality of a pupil’s understanding was also influenced by many latent factors, including strategic approach to learning, volition to remember facts, ability to solve tasks independently, perfectionism, and also, to some extent, the parental view of mathematics. In the second stage of the research, some individual semi-structured interviews were conducted to illustrate and validate the results. The findings of the study highlight the need to raise pupils’ awareness of the quality of their mathematical understanding, since it may influence their willingness to deepen their knowledge in mathematics and subsequently their school performance.

Keywords lower-secondary pupils mathematics perception of one’s own understanding deep understanding algorithmic understanding strategic approach

References

Bandura, A. (1994). Self-efficacy. In V. S. Ramachaudran (Ed.), Encyclopedia of human behavior (vol. 4, pp. 71–81). Academic Press. (Reprinted in H. Friedman [Ed.], Encyclopedia of mental health. Academic Press, 1998.) Search in Google Scholar
Bartley, S.R., & Ingram, N. (2018). Parental modelling of mathematical affect: Self-efficacy and emotional arousal. Mathematics Education Research Journal, 30, 277–297. https://doi.org/10.1007/s13394-017-0233-3 Open DOI Search in Google Scholar
Behr, M., Lesh, R., Post, T., & Silver, E. (1983). Rational number concepts. In R. Lesh, & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 91–125). Academic Press. Search in Google Scholar
Chvál, M. (2013). Změna postojů českých žáků k matematice během školní docházky [Change of attitudes of Czech pupils towards mathematics during school attendance]. Orbis scholae, 7(3), 49–71. https://doi.org/10.14712/23363177.2015.13 Open DOI Search in Google Scholar
Code, W., Merchant, S., Maciejewski, W., Thomas, M., & Lo, J. (2016). The mathematics attitudes and perceptions survey: an instrument to assess expert-like views and dispositions among undergraduate mathematics students. International Journal of Mathematical Education in Science and Technology, 47(6), 917–937. https://doi.org/10.1080/0020739X.2015.1133854 Open DOI Search in Google Scholar
Cohen, L., Manion, L., & Morrison, K. (2007). Research methods in education. Routledge. https://doi.org/10.4324/9780203029053 Open DOI Search in Google Scholar
Desoete, A., & De Craene, B. (2019). Metacognition and mathematics education: An overview. ZDM Mathematics Education, 51, 565–575. https://doi.org/10.1007/s11858-019-01060-w Open DOI Search in Google Scholar
Eccles, J. S., & Wigfield, A. (2002). Motivational beliefs, values and goals. Annual Review of Psychology, 53, 109–132. https://doi.org/10.1146/annurev.psych.53.100901.135153 Open DOI Search in Google Scholar
Entwistle, N., & Entwistle, A. (1992). Developing, revising, and examining conceptual understanding: the student experience and its implications. Centre for Research on Learning and Instruction, University of Edinburgh. Search in Google Scholar
Garofalo, J., & Lester, F. (1985). Metacognition, cognitive monitoring and mathematical performance. Journal for Research in Mathematics Education, 16(3), 163–176. Search in Google Scholar
Hammann, L.A., & Stevens, R. J. (1998). Metacognitive awareness assessment in self regulated learning and performance measures in an introductory educational psychology course. Paper presented at the Annual Meeting of the American Educational Research Association (San Diego, CA, April 13–17, 1998). Search in Google Scholar
Hannula, M. S. (2014). Affect in mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 23–27). Springer. Search in Google Scholar
Hejný, M. (2012). Exploring the cognitive dimension of teaching mathematics through scheme-oriented approach to education. Orbis scholae, 6(2), 41–55. https://doi.org/10.14712/23363177.2015.39 Open DOI Search in Google Scholar
Hejný, M., & Kuřina, F. (2009). Dítě, škola a matematika: konstruktivistické přístupy k vyučování [Child, school and mathematics: Constructivist approaches to teaching]. Portál. Search in Google Scholar
Hemmings, B., Grootenboer, P., & Key, R. (2011). Predicting mathematics achievement: The influence of prior achievement and attitudes. International Journal of Science and Mathematics Education, 9, 691–705. https://doi.org/10.1007/s10763-010-9224-5 Open DOI Search in Google Scholar
Hiebert, J., & Lefevre, P. (2009). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1–27). Routledge. Search in Google Scholar
Hong, S., Yoo, S., You, S., & Wu, C. (2010). The reciprocal relationship between parental involvement and mathematics achievement: Autoregressive cross-lagged modeling. Journal of Experimental Education, 78(4), 419–439. https://doi.org/10.1080/00220970903292926 Open DOI Search in Google Scholar
Hrabal, V., & Pavelková, I. (2010). Jaký jsem učitel [What kind of a teacher I am]. Portál. Search in Google Scholar
Ivankova, N.V., Creswell, J. S., & Stick, S. L. (2006). Using mixed-methods sequential explanatory design: From theory to practice. Field Methods, 18(1), 3–20. https://doi.org/10.1177/1525822X05282260 Open DOI Search in Google Scholar
Kieran, C. (2013). The false dichotomy in mathematics education between conceptual understanding and procedural skills: An example from algebra. In Leatham, K.R. (Ed.), Vital directions for mathematics education research (pp. 153–171). Springer. https://doi.org/10.1007/978-1-4614-6977-3 7 Open DOI Search in Google Scholar
Kieren, T. (1976). On the mathematical, cognitive and instructional foundations of rational numbers. In R. A. Lesh (Ed.), Number and measurement. Papers from a research workshop (pp. 101–150). Columbus, Ohio. Search in Google Scholar
Mareš, J. (1998). Styly učení žáků a studentů [Learning styles of pupils and students]. Portál. Search in Google Scholar
Mayring, P. (2015). Qualitative content analysis: Theoretical background and procedures. In A. Bikner-Ahsbahs, Ch. Knipping, & N. Presmeg (Eds.), Approaches to qualitative research in mathematics education. Examples of methodology and methods (pp. 365–380). Springer. Search in Google Scholar
Middleton, J.A. (2014). Motivation in mathematics learning. In S. Lerman (Ed.), Encyclopedia of mathematics education. Springer. Search in Google Scholar
Muis, K.R., Psaradellis, C., Lajoie, S.P., Di Leo, I., & Chevrier, M. (2015). The role of epistemic emotions in mathematics problem solving. Contemporary Educational Psychology, 42, 172–185. https://doi.org/10.1016/j.cedpsych.2015.06.003 Open DOI Search in Google Scholar
Novotná, G. (2020). Vnímání kvality vlastního poznání v matematice a jeho souvislost s individuálním doučováním [Perception of the quality of one’s own knowledge in mathematics and its connection to private supplementary tutoring] [Ph.D thesis, Charles University]. http://dspace.cuni.cz/handle/20.500.11956/121355 Search in Google Scholar
Novotná, G., & Janda, D. (2021). Jak vybírat úlohy do hodiny a jak sestavit vhodnou taxonomii výukových cílů? [How to choose mathematical tasks and how to use appropriate taxonomy of learning objectives]. In N. Vondrová (Ed.), Dva dny s didaktikou matematiky 2021. Sborník příspěvků (pp. 127–130). PedF UK. Search in Google Scholar
Oluwatelure, T.A., & Oloruntegbe, K. (2008). Effects of parental involvement on students’ attitude and performance in science. The Social Sciences, 3, 573–577. Search in Google Scholar
Pajares, F., & Graham, L. (1999). Self-efficacy, motivation constructs, and mathematics performance of entering middle school students. Contemporary Educational Psychology, 24(2), 124–139. Search in Google Scholar
Pajares, F., & Usher, E. L. (2008). Self-efficacy, motivation and achievement in school from the perspective of reciprocal determinism. In M. L. Maehr, S. A. Karabenick, & T.C. Urdan (Eds.), Social Psychological Perspectives (pp. 391–423). Emerald Group Publishing. Search in Google Scholar
Schöber, C., Schütte, K., Köller, O., McElvany, N., & Gebauer, M. M. (2018). Reciprocal effects between self-efficacy and achievement in mathematics and reading. Learning and Individual Differences, 63, 1–11. https://doi.org/10.1016/j.lindif.2018.01.008 Open DOI Search in Google Scholar
Sierpinska, A. (1994). Understanding in mathematics. The Falmer press. Search in Google Scholar
Skemp, R.R. (1991). Mathematics in the primary school. Billing & Sons Ltd. Search in Google Scholar
Smetáčková, I. (2018). Obliba školní matematiky a její souvislost s externím hodnocením a sebehodnocením [Preference of school mathematics and its link to external evaluation and self-evaluation]. Scientia in educatione, 9(2), 44–56. https://doi.org/10.14712/18047106.1049 Open DOI Search in Google Scholar
Star, J.R. (2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36(5), 404–411. https://doi.org/10.2307/30034943 Open DOI Search in Google Scholar
Tocci, C., & Engelhard, G. (1991). Achievement, parental support and gender differences in attitudes toward mathematics. The Journal of Educational Research, 84(5), 280–286. https://doi.org/10.1080/00220671.1991.10886028 Open DOI Search in Google Scholar
Ubuz, B., & Aydinyer, Y. (2017). Measuring striving for understanding and learning value of geometry: a validity study. International Journal of Mathematics Education in Science and Technology, 48(7), 1072–1086. https://doi.org/10.1080/0020739X.2017.1327087 Open DOI Search in Google Scholar
Usiskin, Z. (2015). What does it mean to understand some mathematics? In J.C. Sung (Ed.), The proceedings of the 12th International Congress on Mathematical Education (pp. 821–842). Springer. https://doi.org/10.1007/978-3-319-12688-3 Open DOI Search in Google Scholar
Vondrová, N., Rendl, M., Havlíčková, R., Hříbková, L., Páchová, A., & Žalská, J. (2015). Kritická místa matematiky základní školy v řešeních žáků. [Critical places of lower secondary mathematics in the solutions of pupils]. Nakladatelství Karolinum. Search in Google Scholar
Voňková, H., & Hullegie, P. (2011). Is the anchoring vignette method sensitive to the domain and choice of the vignette? Journal of the Royal Statistical Society: Series A (Statistics in Society), 173(3), 597–620. https://www.jstor.org/stable/23013513 Search in Google Scholar
Vorhölter, K., Krüger, A., & Wendt, L. (2019). Metacognition in mathematical modeling – an Overview. In S. Chamberlin, & B. Sriraman (Eds.), Affect in mathematical modeling (pp. 29–51). Springer. https://doi.org/10.1007/978-3-030-04432-9 3 Open DOI Search in Google Scholar
Yurt, E. (2014). The predictive power of self-efficacy sources for mathematics achievement. Education and Science, 39(176), 159–169. https://doi.org/10.15390/EB.2014.3443 Open DOI Search in Google Scholar
Zimmerman, B. J. (2000). Self-efficacy: An essential motive to learn. Contemporary Educational Psychology, 25(1), 82–91. https://doi.org/10.1006/ceps.1999.1016 Open DOI Search in Google Scholar

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[1] Bandura, A. (1994). Self-efficacy. In V. S. Ramachaudran (Ed.), Encyclopedia of human behavior (vol. 4, pp. 71–81). Academic Press. (Reprinted in H. Friedman [Ed.], Encyclopedia of mental health. Academic Press, 1998.) Search in Google Scholar

[2] Bartley, S.R., & Ingram, N. (2018). Parental modelling of mathematical affect: Self-efficacy and emotional arousal. Mathematics Education Research Journal, 30, 277–297. https://doi.org/10.1007/s13394-017-0233-3 Open DOI Search in Google Scholar

[3] Behr, M., Lesh, R., Post, T., & Silver, E. (1983). Rational number concepts. In R. Lesh, & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 91–125). Academic Press. Search in Google Scholar

[4] Chvál, M. (2013). Změna postojů českých žáků k matematice během školní docházky [Change of attitudes of Czech pupils towards mathematics during school attendance]. Orbis scholae, 7(3), 49–71. https://doi.org/10.14712/23363177.2015.13 Open DOI Search in Google Scholar

[5] Code, W., Merchant, S., Maciejewski, W., Thomas, M., & Lo, J. (2016). The mathematics attitudes and perceptions survey: an instrument to assess expert-like views and dispositions among undergraduate mathematics students. International Journal of Mathematical Education in Science and Technology, 47(6), 917–937. https://doi.org/10.1080/0020739X.2015.1133854 Open DOI Search in Google Scholar

[6] Cohen, L., Manion, L., & Morrison, K. (2007). Research methods in education. Routledge. https://doi.org/10.4324/9780203029053 Open DOI Search in Google Scholar

[7] Desoete, A., & De Craene, B. (2019). Metacognition and mathematics education: An overview. ZDM Mathematics Education, 51, 565–575. https://doi.org/10.1007/s11858-019-01060-w Open DOI Search in Google Scholar

[8] Eccles, J. S., & Wigfield, A. (2002). Motivational beliefs, values and goals. Annual Review of Psychology, 53, 109–132. https://doi.org/10.1146/annurev.psych.53.100901.135153 Open DOI Search in Google Scholar

[9] Entwistle, N., & Entwistle, A. (1992). Developing, revising, and examining conceptual understanding: the student experience and its implications. Centre for Research on Learning and Instruction, University of Edinburgh. Search in Google Scholar

[10] Garofalo, J., & Lester, F. (1985). Metacognition, cognitive monitoring and mathematical performance. Journal for Research in Mathematics Education, 16(3), 163–176. Search in Google Scholar

[11] Hammann, L.A., & Stevens, R. J. (1998). Metacognitive awareness assessment in self regulated learning and performance measures in an introductory educational psychology course. Paper presented at the Annual Meeting of the American Educational Research Association (San Diego, CA, April 13–17, 1998). Search in Google Scholar

[12] Hannula, M. S. (2014). Affect in mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 23–27). Springer. Search in Google Scholar

[13] Hejný, M. (2012). Exploring the cognitive dimension of teaching mathematics through scheme-oriented approach to education. Orbis scholae, 6(2), 41–55. https://doi.org/10.14712/23363177.2015.39 Open DOI Search in Google Scholar

[14] Hejný, M., & Kuřina, F. (2009). Dítě, škola a matematika: konstruktivistické přístupy k vyučování [Child, school and mathematics: Constructivist approaches to teaching]. Portál. Search in Google Scholar

[15] Hemmings, B., Grootenboer, P., & Key, R. (2011). Predicting mathematics achievement: The influence of prior achievement and attitudes. International Journal of Science and Mathematics Education, 9, 691–705. https://doi.org/10.1007/s10763-010-9224-5 Open DOI Search in Google Scholar

[16] Hiebert, J., & Lefevre, P. (2009). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1–27). Routledge. Search in Google Scholar

[17] Hong, S., Yoo, S., You, S., & Wu, C. (2010). The reciprocal relationship between parental involvement and mathematics achievement: Autoregressive cross-lagged modeling. Journal of Experimental Education, 78(4), 419–439. https://doi.org/10.1080/00220970903292926 Open DOI Search in Google Scholar

[18] Hrabal, V., & Pavelková, I. (2010). Jaký jsem učitel [What kind of a teacher I am]. Portál. Search in Google Scholar

[19] Ivankova, N.V., Creswell, J. S., & Stick, S. L. (2006). Using mixed-methods sequential explanatory design: From theory to practice. Field Methods, 18(1), 3–20. https://doi.org/10.1177/1525822X05282260 Open DOI Search in Google Scholar

[20] Kieran, C. (2013). The false dichotomy in mathematics education between conceptual understanding and procedural skills: An example from algebra. In Leatham, K.R. (Ed.), Vital directions for mathematics education research (pp. 153–171). Springer. https://doi.org/10.1007/978-1-4614-6977-3 7 Open DOI Search in Google Scholar

[21] Kieren, T. (1976). On the mathematical, cognitive and instructional foundations of rational numbers. In R. A. Lesh (Ed.), Number and measurement. Papers from a research workshop (pp. 101–150). Columbus, Ohio. Search in Google Scholar

[22] Mareš, J. (1998). Styly učení žáků a studentů [Learning styles of pupils and students]. Portál. Search in Google Scholar

[23] Mayring, P. (2015). Qualitative content analysis: Theoretical background and procedures. In A. Bikner-Ahsbahs, Ch. Knipping, & N. Presmeg (Eds.), Approaches to qualitative research in mathematics education. Examples of methodology and methods (pp. 365–380). Springer. Search in Google Scholar

[24] Middleton, J.A. (2014). Motivation in mathematics learning. In S. Lerman (Ed.), Encyclopedia of mathematics education. Springer. Search in Google Scholar

[25] Muis, K.R., Psaradellis, C., Lajoie, S.P., Di Leo, I., & Chevrier, M. (2015). The role of epistemic emotions in mathematics problem solving. Contemporary Educational Psychology, 42, 172–185. https://doi.org/10.1016/j.cedpsych.2015.06.003 Open DOI Search in Google Scholar

[26] Novotná, G. (2020). Vnímání kvality vlastního poznání v matematice a jeho souvislost s individuálním doučováním [Perception of the quality of one’s own knowledge in mathematics and its connection to private supplementary tutoring] [Ph.D thesis, Charles University]. http://dspace.cuni.cz/handle/20.500.11956/121355 Search in Google Scholar

[27] Novotná, G., & Janda, D. (2021). Jak vybírat úlohy do hodiny a jak sestavit vhodnou taxonomii výukových cílů? [How to choose mathematical tasks and how to use appropriate taxonomy of learning objectives]. In N. Vondrová (Ed.), Dva dny s didaktikou matematiky 2021. Sborník příspěvků (pp. 127–130). PedF UK. Search in Google Scholar

[28] Oluwatelure, T.A., & Oloruntegbe, K. (2008). Effects of parental involvement on students’ attitude and performance in science. The Social Sciences, 3, 573–577. Search in Google Scholar

[29] Pajares, F., & Graham, L. (1999). Self-efficacy, motivation constructs, and mathematics performance of entering middle school students. Contemporary Educational Psychology, 24(2), 124–139. Search in Google Scholar

[30] Pajares, F., & Usher, E. L. (2008). Self-efficacy, motivation and achievement in school from the perspective of reciprocal determinism. In M. L. Maehr, S. A. Karabenick, & T.C. Urdan (Eds.), Social Psychological Perspectives (pp. 391–423). Emerald Group Publishing. Search in Google Scholar

[31] Schöber, C., Schütte, K., Köller, O., McElvany, N., & Gebauer, M. M. (2018). Reciprocal effects between self-efficacy and achievement in mathematics and reading. Learning and Individual Differences, 63, 1–11. https://doi.org/10.1016/j.lindif.2018.01.008 Open DOI Search in Google Scholar

[32] Sierpinska, A. (1994). Understanding in mathematics. The Falmer press. Search in Google Scholar

[33] Skemp, R.R. (1991). Mathematics in the primary school. Billing & Sons Ltd. Search in Google Scholar

[34] Smetáčková, I. (2018). Obliba školní matematiky a její souvislost s externím hodnocením a sebehodnocením [Preference of school mathematics and its link to external evaluation and self-evaluation]. Scientia in educatione, 9(2), 44–56. https://doi.org/10.14712/18047106.1049 Open DOI Search in Google Scholar

[35] Star, J.R. (2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36(5), 404–411. https://doi.org/10.2307/30034943 Open DOI Search in Google Scholar

[36] Tocci, C., & Engelhard, G. (1991). Achievement, parental support and gender differences in attitudes toward mathematics. The Journal of Educational Research, 84(5), 280–286. https://doi.org/10.1080/00220671.1991.10886028 Open DOI Search in Google Scholar

[37] Ubuz, B., & Aydinyer, Y. (2017). Measuring striving for understanding and learning value of geometry: a validity study. International Journal of Mathematics Education in Science and Technology, 48(7), 1072–1086. https://doi.org/10.1080/0020739X.2017.1327087 Open DOI Search in Google Scholar

[38] Usiskin, Z. (2015). What does it mean to understand some mathematics? In J.C. Sung (Ed.), The proceedings of the 12th International Congress on Mathematical Education (pp. 821–842). Springer. https://doi.org/10.1007/978-3-319-12688-3 Open DOI Search in Google Scholar

[39] Vondrová, N., Rendl, M., Havlíčková, R., Hříbková, L., Páchová, A., & Žalská, J. (2015). Kritická místa matematiky základní školy v řešeních žáků. [Critical places of lower secondary mathematics in the solutions of pupils]. Nakladatelství Karolinum. Search in Google Scholar

[40] Voňková, H., & Hullegie, P. (2011). Is the anchoring vignette method sensitive to the domain and choice of the vignette? Journal of the Royal Statistical Society: Series A (Statistics in Society), 173(3), 597–620. https://www.jstor.org/stable/23013513 Search in Google Scholar

[41] Vorhölter, K., Krüger, A., & Wendt, L. (2019). Metacognition in mathematical modeling – an Overview. In S. Chamberlin, & B. Sriraman (Eds.), Affect in mathematical modeling (pp. 29–51). Springer. https://doi.org/10.1007/978-3-030-04432-9 3 Open DOI Search in Google Scholar

[42] Yurt, E. (2014). The predictive power of self-efficacy sources for mathematics achievement. Education and Science, 39(176), 159–169. https://doi.org/10.15390/EB.2014.3443 Open DOI Search in Google Scholar

[43] Zimmerman, B. J. (2000). Self-efficacy: An essential motive to learn. Contemporary Educational Psychology, 25(1), 82–91. https://doi.org/10.1006/ceps.1999.1016 Open DOI Search in Google Scholar

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