A Joint Assessment of Reasoning about General Statements in Mathematics and Biology


  • Libuše Samková Faculty of Education University of South Bohemia Jeronýmova 10 České Budějovice
  • Lukáš Rokos Department of Biology, Faculty of Education, University of South Bohemia in České Budějovice, Czech Republic
  • Lukáš Vízek Department of Mathematics, Faculty of Science, University of Hradec Králové, Czech Republic




argumentation, biology education, Concept Cartoons, formative assessment, future primary school teachers, mathematics education


This contribution belongs to a larger empirical study that focuses on issues related to the implementation of inquiry-based learning and formative assessment in science and mathematics education, while it also refers to the issue of STEM education. Here, we discuss the two topics from the perspective of professional preparation of primary school teachers. We employ an educational tool called Concept Cartoons and perceive it as a common diagnostic tool for investigating modes of reasoning about general statements in arithmetic, geometry and biology. The presented qualitative exploratory empirical study maps and codes various kinds of reasoning that can be identified with the tool and investigates possibilities of a joint coding procedure. As a result, it provides a conversion table between various modes of reasoning in the three subject domains. The arisen code categories cover the field of generic examples, including the initial stages so that they can be used for scaffolding the process of learning the foundations of deductive reasoning. The joint approach to reasoning in mathematics and biology shows how argumentation and formative assessment can be understood equally and developed simultaneously in both school subjects. It helps us to see how the two school subjects can be integrated didactically.


Alibali, M. W. and Sidney, P. G. (2015) ‘Variability in the natural number bias: who, when, how, and why’, Learning and Instruction, Vol. 37, pp. 56–61. http://doi.org/10.1016/j.learninstruc.2015.01.003

Artigue, M. and Blomhøj, M. (2013) ‘Conceptualizing inquiry-based education in mathematics’, ZDM Mathematics Education, Vol. 45, No. 6, pp. 797–810. http://doi.org/10.1007/s11858-013-0506-6

Balacheff, N. (1988) ‘Aspects of proof in pupils’ practice of school mathematics’, in Pimm, D. (ed.) Mathematics, Teachers and Children, London: Hodder & Stoughton, pp. 216–238.

Berland, L. K. and Reiser, B. J. (2009) ‘Making sense of argumentation and explanation’, Science Education, Vol. 93, No. 1, pp. 26–55. http://doi.org/10.1002/sce.20286

Berthelsen, B. (1999) ‘Students’ naive conceptions in life sciences’, MSTA Journal, Vol. 44, No. 1, pp. 13–19.

Braund, M. (1997) ‘Primary children’s ideas about animals with and without backbones’, Education 3-13, Vol. 25, No. 2, pp. 19–24. http://doi.org/10.1080/03004279785200191

Buchbinder, O. and Cook, A. (2018) ‘Examining the mathematical knowledge for teaching of proving in scenarios written by pre-service teachers’, in Buchbinder, O. and Kuntze, S. (ed.) Mathematics Teachers Engaging with Representations of Practice, Cham: Springer, pp. 133–154. http://doi.org/10.1007/978-3-319-70594-1_8

Bulková, K., Medová, J. and Čeretková, S. (2020) ’Identification of crucial steps and skills in high-achievers’ solving complex mathematical problem within mathematical contest’, Journal on Efficiency and Responsibility in Education and Science, Vol. 13, No. 2, pp. 67–78. http://doi.org/10.7160/eriesj.2020.130202

Clipart Library (2019) Ocean floor black and white, [Online], Available: http://clipart-library.com/coloring/kTMegybTj.gif [14 Nov 2020].

Confrey, J. and Kazak, S. (2006) ‘A thirty-year reflection on constructivism in mathematics education in PME’, in Gutiérrez A. and Boero P. (ed.) Handbook of Research on the Psychology of Mathematics Education: Past, Present and Future, Rotterdam: Sense, pp. 305–345. http://doi.org/10.1163/9789087901127_012

Corp, A., Fields, M. and Naizer, G. (2020) ‘Elementary STEM teacher education: Recent practices to prepare general elementary teachers for STEM’, in Johnson C. C., Mohr-Schroeder, M. J., Moore, T. J. and English, L. D. (ed.) Handbook of Research on STEM Education, New York: Routledge, pp. 337–348. http://doi.org/10.4324/9780429021381-32

Cullen, S., Fan, J., van der Brugge, E. and Elga, A. (2018) ‘Improving analytical reasoning and argument understanding: a quasi-experimental field study of argument visualization’, npj Science of Learning, Vol. 3, No. 21. http://doi.org/10.1038/s41539-018-0038-5

Dabell, J., Keogh, B. and Naylor, S. (2008) Concept Cartoons in Mathematics Education, Sandbach: Millgate House Education.

Dofner, T., Förtsch, C., Germ, M. and Neuhaus, B. J. (2018) ‘Biology instruction using generic framework of scientific reasoning and argumentation’, Teaching and Teacher Education, Vol. 75, pp. 232–243. http://doi.org/10.1016/j.tate.2018.07.003

Dolin, J. and Evans, R. (ed.) (2018) Transforming assessment: Through an interplay between practice, research and policy, Cham: Springer. http://doi.org/10.1007/978-3-319-63248-3

van Dooren, W., Lehtinen, E. and Verschaffel, L. (2015) ‘Unraveling the gap between natural and rational numbers’, Learning and Instruction, Vol. 37, pp. 1–4. http://doi.org/10.1016/j.learninstruc.2015.01.001

Dorier, J.-L. and Maass, K. (2014) ‘Inquiry based mathematics education’, in Lerman S. (ed.) Encyclopedia of Mathematics Education, Dordrecht: Springer, pp. 300–304. http://doi.org/10.1007/978-94-007-4978-8_176

English, L. D. (2016) ‘STEM education K-12: perspectives on integration’, International Journal of STEM Education, Vol. 3, No. 3. http://doi.org/10.1186/s40594-016-0036-1

Erduran, S. and Jiménez-Aleixandre, M. P. (ed.) (2007) Argumentation in science education: Perspectives from classroom-based research, Dordrecht: Springer. http://doi.org/10.1007/978-1-4020-6670-2

Fujita, T. and Jones, K. (2007) ‘Learners’ understanding of the definitions and hierarchical classification of quadrilaterals: Towards a theoretical framing’, Research in Mathematics Education, Vol. 9, No. 1, pp. 3–20. http://doi.org/10.1080/14794800008520167

Furtak, E. M., Hardy, I., Beinbrech, C., Shavelson, R. J. and Shemwell, J. T. (2010) ‘A framework for analyzing evidence-based reasoning in science classroom discourse’, Educational Assessment, Vol. 15, No. 3–4, pp. 175–196. http://doi.org/10.1080/10627197.2010.530553

Galbraith, P. L. (1981) ‘Aspects of proving: A clinical investigation of process’, Educational Studies in Mathematics, Vol. 12, No. 1, pp. 1–28. http://doi.org/10.1007/BF00386043

Hadar, N. (1977) ‘Children’s conditional reasoning’, Educational Studies in Mathematics, Vol. 8, No. 4, pp. 413–438. http://doi.org/10.1007/bf00310946

Hallström, J. and Schönborn, K. J. (2019) ‘Models and modelling for authentic STEM education: reinforcing the argument’, International Journal of STEM Education, Vol. 6, No. 22. http://doi.org/10.1186/s40594-019-0178-z

Harel, G. and Sowder, L. (1998) ‘Students’ proof schemes: Results from exploratory studies’, in Schoenfeld, A. H., Kaput, J., Dubinski, E. and Dick, T. (ed.) Issues in Mathematics Education (Research in Collegiate Mathematics Education III, Vol. 7), Providence: AMS, pp. 234–283. http://doi.org/10.1090/cbmath/007/07

Harel, G. and Sowder, L. (2007) ‘Toward comprehensive perspectives on the learning and teaching of proof’, in Lester, F. (ed.) Second Handbook of Research on Mathematics Teaching and Learning, Charlotte: NCTM, pp. 805–842.

Hattie, J. and Timperley, H. (2007) ‘The power of feedback’, Review of Educational Research, Vol. 77, No. 1, pp. 81–112. http://doi.org/10.3102/003465430298487

Kattmann, U. (2001) ‘Aquatics, flyers, creepers and terrestrials – Students’ conceptions of animal classification’, Journal of Biological Education, Vol. 35, No. 3, pp. 141–147. http://doi.org/10.1080/00219266.2001.9655763

Keogh, B. and Naylor, S. (1999) ‘Concept Cartoons, teaching and learning in science: an evaluation’, International Journal of Science Education, Vol. 21, No. 4, pp. 431–446. http://doi.org/10.1080/095006999290642

Komatsu, K. (2010) ‘Counter-examples for refinement of conjectures and proofs in primary school mathematics’, Journal of Mathematical Behavior, Vol. 29, No. 1, pp. 1–10. http://doi.org/10.1016/j.jmathb.2010.01.003

Kubiatko, M. (2021) ‘Subject didactics: relevant issues’, Problems of Education in the 21st Century, Vol. 79, No. 3, pp. 340–342. http://doi.org/10.33225/pec/21.79.340

Kubiatko, M. and Prokop, P. (2007) ‘Pupils’ misconceptions about mammals’, Journal of Baltic Science Education, Vol. 6, No. 1, pp. 5–14.

Lazarou, D., Sutherland, R. and Erduran, S. (2016) ‘Argumentation in science education as a systemic activity: An activity-theoretical perspective’, International Journal of Educational Research, Vol. 79, pp. 150–166. http://doi.org/10.1016/j.ijer.2016.07.008

Lazarowitz, R. and Lieb, C. (2006) ‘Formative assessment pre-test to identify college students’ prior knowledge, misconceptions and learning difficulties in biology’, International Journal of Science and Mathematical Education, Vol. 4, No. 4, pp. 741–762. http://doi.org/10.1007/s10763-005-9024-5

Martin, W. G. and Harel, G. (1989) ‘Proof frames of preservice elementary teachers’, Journal for Research in Mathematics Education, Vol. 20, No. 1, pp. 41–51. http://doi.org/10.2307/749097

McComas, W. F. (ed.) (2002) The nature of science in science education, Boston: Kluwer. http://doi.org/10.1007/0-306-47215-5

Melhuish, K., Thanheiser, E. and Guyot, L. (2020) ‘Elementary school teachers’ noticing of essential mathematical reasoning forms: justification and generalization’, Journal of Mathematics Teacher Education, Vol. 23, pp. 35–67. http://doi.org/10.1007/s10857-018-9408-4

Miles, M. B., Huberman, A. M. and Saldaña, J. (2014) Qualitative data analysis: A methods sourcebook, 3rd edition, Thousand Oaks, CA: SAGE.

Minner, D., Levy, A. and Century, J. (2010) ‘Inquiry-based science instruction – what is it and does it matter? Results from a research synthesis years 1984 to 2002’, Journal of Research in Science Teaching, Vol. 47, No. 4, pp. 474–496. http://doi.org/10.1002/tea.20347

Moore, T. J., Johnson, A. C. and Glancy, A. W. (2020) STEM integration: A synthesis of conceptual frameworks and definitions’, in Johnson C. C., Mohr-Schroeder, M. J., Moore, T. J. and English, L. D. (ed.) Handbook of Research on STEM Education, New York: Routledge, pp. 3–16. http://doi.org/10.4324/9780429021381-2

National Research Council (1996) National science education standards, Washington, DC: National Academy Press.

Naylor, S. and Keogh, B. (2010) Concept Cartoons in Science Education, 2nd Edition [CD-ROM], Sandbach: Millgate House Education.

Naylor, S., Keogh, B. and Downing, B. (2007) ‘Argumentation and primary science’, Research in Science Education, Vol. 37, No. 1, pp. 17–39. http://doi.org/10.1007/s11165-005-9002-5

Ping, I. L. L., Halim, L. and Osman, K. (2020) ‘Explicit teaching of scientific argumentation as an approach in developing argumentation skills, science process skills and biology understanding’, Journal of Baltic Science Education, Vol. 19, No. 2, pp. 276–288. http://doi.org/10.33225/jbse/20.19.276

Prokop, P., Prokop, M. and Tunnicliffe, S. D. (2007) ‘Effects of keeping animals as pets on children’s concepts of vertebrates and invertebrates’, International Journal of Science Education, Vol. 30, No. 4, pp. 431–449. http://doi.org/10.1080/09500690701206686

Rocard, M., Csermely, P., Jorde, D., Lenzen, D., Walberg-Henriksson, H. and Hemmo, V. (2007) Science education now: a renewed pedagogy for the future of Europe, Luxembourg: Office for Official Publications of the European Communities, [Online], Available: http://www.eesc.europa.eu/resources/docs/rapportrocardfinal.pdf [4 Jun 2021].

Rokos, L. and Samková, L. (2020) ‘Coding classroom talk from the perspective of formative assessment and inquiry-based education: a communication model for mathematics and science lessons’, in Proceedings of ICERI2020 Conference, Seville, pp. 1170–1179. http://doi.org/10.21125/iceri.2020.0318

Roubíček, F. (2014) The set of four geometric Concept Cartoons for assessing future primary school teacher's knowledge, [Internal material, unpublished].

Samková, L. (2019) ‘Investigating subject matter knowledge and pedagogical content knowledge in mathematics with the Concept Cartoons method’, Scientia in educatione, Vol. 10, No. 2, pp. 62–79. http://doi.org/10.14712/18047106.1548

Samková, L. (2020) ‘Observing how future primary school teachers reason about general statements’, in Proceedings of the 17th International Conference on Efficiency and Responsibility in Education (ERIE 2020), Prague, pp. 263–271.

Samková, L. and Tichá, M. (2017) ‘On the way to observe how future primary school teachers reason about fractions’, Journal of Efficiency and Responsibility in Education and Science, Vol. 10, No. 4, pp. 93–100. http://doi.org/10.7160/eriesj.2017.100401

Selden, A. (2012) ‘Transitions and proof and proving at tertiary level’, in Hanna, G. and de Villiers, M. (ed.) Proof and Proving in Mathematics Education, Dordrecht: Springer, pp. 391–420. http://doi.org/10.1007/978-94-007-2129-6_17

Simon, M. A. and Blume, G. W. (1996) ‘Justification in the mathematics classroom: A study of prospective elementary teachers’, Journal of Mathematical Behavior, Vol. 15, No. 1, pp. 3–31. http://doi.org/10.1016/s0732-3123(96)90036-x

Stevens, A. L., Wilkins, J. L. M, Lovin, L. H., Siegfried, J., Norton, A. and Busi, R. (2020) ‘Promoting sophisticated fraction constructs through instructional changes in a mathematics course for PreK-8 prospective teachers’, Journal of Mathematics Teacher Education, Vol. 23, No. 2, pp. 153–181. http://doi.org/10.1007/s10857-018-9415-5

Trowbridge, J. E., and Mintzes, J. J. (1988) ‘Alternative conceptions in animal classification: A cross-age study’, Journal of Research in Science Teaching, Vol. 25, No. 7, pp. 547–571. http://doi.org/10.1002/tea.3660250704

Tuset, G. A. (2019) ‘Preservice teachers’ geometrical discourses when leading classroom discussions about defining and classifying quadrilaterals’, in Proceedings of CERME11, Utrecht, pp. 3523–3530.

Valovičová, L’., Ondruška, J., Zelenický, L’., Chytrý, V. and Medová, J. (2020) ‘Enhancing computational thinking through interdisciplinary STEAM activities using tablets’, Mathematics, Vol. 8, No. 12, 2128. http://doi.org/10.3390/math8122128

de Villiers, M. (1994) ‘The role and function of a hierarchical classification of quadrilaterals’, For the Learning of Mathematics, Vol. 14, No. 1, pp. 11–18.

Vízek, L. and Samková, L. (2021) ‘Observing how future primary school teachers reason about quadrilaterals’, in Proceedings of the 18th International Conference on Efficiency and Responsibility in Education (ERIE 2021), Prague, pp. 159–167.

Yen, C.-F., Yao, T.-W. and Chiu, Y.-C. (2004) ‘Alternative conceptions in animal classification focusing on amphibians and reptiles: A cross-age study’, International Journal of Science and Mathematics Education, Vol. 2, No. 2, pp. 159–174. http://doi.org/10.1007/s10763-004-1951-z

Zazkis, R. and Chernoff, E. J. (2008) ‘What makes a counterexample exemplary?’, Educational Studies in Mathematics, Vol. 68, pp. 195–208. http://doi.org/10.1007/s10649-007-9110-4

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Samková, L., Rokos, L. and Vízek, L. (2021) ’A Joint Assessment of Reasoning about General Statements in Mathematics and Biology’, Journal on Efficiency and Responsibility in Education and Science, vol. 14, no. 4, pp. 270–287 https://doi.org/10.7160/eriesj.2021.140406.



Research Paper