How lower-secondary pupils approach and perceive understanding mathematics
PDF (English)

Klíčová slova

Nižší sekundární vzdělávání
vnímání vlastního porozumění
hloubkové porozumění
algoritmické porozumění
strategický přístup lower-secondary pupils
perception of one’s own understanding
deep understanding
algorithmic understanding
strategic approach

Jak citovat

Novotná, G. (2023). How lower-secondary pupils approach and perceive understanding mathematics. Scientia in Educatione, 13(2), 36-50.


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.
PDF (English)


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.)

Bartley, S.R., & Ingram, N. (2018). Parental modelling of mathematical affect: Self-efficacy and emotional arousal. Mathematics Education Research Journal, 30, 277–297.

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.

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.

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.

Cohen, L., Manion, L., & Morrison, K. (2007). Research methods in education. Routledge.

Desoete, A., & De Craene, B. (2019). Metacognition and mathematics education: An overview. ZDM Mathematics Education, 51, 565–575.

Eccles, J. S., & Wigfield, A. (2002). Motivational beliefs, values and goals. Annual Review of Psychology, 53, 109–132.

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.

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

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).

Hannula, M. S. (2014). Affect in mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 23–27). Springer.

Hejný, M. (2012). Exploring the cognitive dimension of teaching mathematics through scheme-oriented approach to education. Orbis scholae, 6(2), 41–55.

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

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.

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.

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.

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

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.

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. 7

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.

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

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.

Middleton, J.A. (2014). Motivation in mathematics learning. In S. Lerman (Ed.), Encyclopedia of mathematics education. Springer.

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.

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].

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.

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

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

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.

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.

Sierpinska, A. (1994). Understanding in mathematics. The Falmer press.

Skemp, R.R. (1991). Mathematics in the primary school. Billing & Sons Ltd.

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.

Star, J.R. (2005). Reconceptualizing procedural knowledge. Journal for Research in Mathematics Education, 36(5), 404–411.

Tocci, C., & Engelhard, G. (1991). Achievement, parental support and gender differences in attitudes toward mathematics. The Journal of Educational Research, 84(5), 280–286.

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.

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.

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.

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.

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. 3

Yurt, E. (2014). The predictive power of self-efficacy sources for mathematics achievement. Education and Science, 39(176), 159–169.

Zimmerman, B. J. (2000). Self-efficacy: An essential motive to learn. Contemporary Educational Psychology, 25(1), 82–91.