Students’ use of definitions of independence across different visualizations
DOI:
https://doi.org/10.52041/serj.986Keywords:
Independence, Conditional probabilities, Visualizations, Natural frequenciesAbstract
The independence of two events can be defined using either joint probabilities or conditional probabilities. Although mathematically equivalent, these definitions highlight different probabilistic structures that may not be equally transparent and accessible across different visualizations. In this study, 319 secondary school students in Germany solved three probabilistic problems involving two binary events, each presented with a fully completed visualization (2×2 table, double tree, or net diagram) in two different information formats (probabilities or natural frequencies). Students’ performance in correctly analyzing the (in)dependence of events across these versions and their use of the definitions of independence were analyzed. The participants showed a particularly low performance in problems presented with probability double trees, even though the relevant probabilities are explicitly displayed. Moreover, 49% of students consistently applied the joint probability definition of independence across all visualizations. These findings suggest that many students rely on stable rather than flexible (or even adaptive) patterns of definition use, highlighting the need to explicitly connect multiple properties of independence in instruction.
References
Ahlgren, A., & Garfield, J. (1991). Analysis of the Probability Curriculum. In R. Kapadia & M. Borovcnik (Eds.), Chance Encounters: Probability in Education (pp. 107–134). Springer. https://doi.org/10.1007/978-94-011-3532-0_4
Andam, E. A., Awuah, F. K., & Obeng-Denteh, W. (2025). Analysis of Grade 11 Students’ Conceptual Understanding in Probability Concepts: A Perspective of Skemp Understanding Theory. East African Journal of Education Studies, 8(2), 358–374. https://doi.org/10.37284/eajes.8.2.2987
Australian Curriculum, Assessment and Reporting Authority. (2015). Mathematics: Sequence of content. https://docs.acara.edu.au/resources/Mathematics_-_Sequence_of_content.pdf
Batanero, C., & Álvarez-Arroyo, R. (2024). Teaching and learning of probability. ZDM – Mathematics Education, 56(1), 5–17. https://doi.org/10.1007/s11858-023-01511-5
Batanero, C., & Borovcnik, M. (2016). Statistics and Probability in High School. SensePublishers. https://doi.org/10.1007/978-94-6300-624-8_3
Batanero, C., Chernoff, E. J., Engel, J., Lee, H. S., & Sánchez, E. (2016). Research on Teaching and Learning Probability. Springer. https://doi.org/10.1007/978-3-319-31625-3_1
Batanero, C., Contreras, J. M., Díaz, C., & Sánchez, E. (2015). Prospective Teachers’ Semiotic Conflicts in Computing Probabilities from a Two-Way Table. International Electronic Journal of Mathematics Education, 10(1), 3–16. https://doi.org/10.29333/iejme/288
Biehler, R. (2016). Professional Development for Teaching Probability and Inference Statistics with Digital Tools at Upper Secondary Level. 13th International Congress on Mathematical Education, Hamburg, Germany.
Binder, K., Krauss, S., & Bruckmaier, G. (2015). Effects of visualizing statistical information – an empirical study on tree diagrams and 2 × 2 tables. Frontiers in Psychology, 6:1186. https://doi.org/10.3389/fpsyg.2015.01186
Binder, K., Krauss, S., & Wiesner, P. (2020). A New Visualization for Probabilistic Situations Containing Two Binary Events: The Frequency Net. Frontiers in Psychology, 11:750. https://doi.org/10.3389/fpsyg.2020.00750
Blitzstein, J. K., & Hwang, J. (2019). Introduction to Probability (2nd ed.). Chapman and Hall/CRC. https://doi.org/10.1201/9780429428357
Böcherer-Linder, K., & Eichler, A. (2019). How to Improve Performance in Bayesian Inference Tasks: A Comparison of Five Visualizations. Frontiers in Psychology, 10:267. https://doi.org/10.3389/fpsyg.2019.00267
Borovcnik, M. (2012). Perspectivas múltiples sobre el concepto de probabilidad condicional. Avances de Investigación En Educación Matemática, 2, 5–27. https://doi.org/10.35763/aiem.v1i2.32
Cordani, L. K., & Wechsler, S. (2006). Teaching Independence and Exchangeability. 7th International Conference on Teaching Statistics, Salvador, Brazil.
D’Amelio, A. (2009). Undergraduate Students Difficulties with Independent and Mutually Exclusive Events Concepts. The Mathematics Enthusiast, 6(1–2), 47–56. https://doi.org/10.54870/1551-3440.1133
Díaz, C., Batanero, C., & Contreras, J. M. (2010). Teaching Independence and Conditional Probability. Boletín de Estadística e Investigación Operativa, 26(2), 149–162.
Estrada, A., & Díaz, C. (2006). Computing Probabilities from Two Way Tables: An Exploratory Study with Future Teachers. 7th International Conference on Teaching Statistics, Salvador, Brazil.
Falk, R. (1986). Conditional Probabilities: Insights and Difficulties. 2nd International Conference on Teaching Statistics, Victoria, Canada.
Gigerenzer, G., & Hoffrage, U. (1995). How to improve Bayesian reasoning without instruction: Frequency formats. Psychological Review, 102(4), 684–704. https://doi.org/10.1037/0033-295X.102.4.684
Heitele, D. (1975). An Epistemological View on Fundamental Stochastic Ideas. Educational Studies in Mathematics, 6(2), 187–205. https://doi.org/10.1007/BF00302543
Johnson, E. D., & Tubau, E. (2015). Comprehension and computation in Bayesian problem solving. Frontiers in Psychology, 6:938. https://doi.org/10.3389/fpsyg.2015.00938
Kaplan, A. (1964). The Conduct of Inquiry: Methodology for Behavioral Science. Routledge. https://doi.org/10.4324/9781315131467
Khan, A., Breslav, S., Glueck, M., & Hornbæk, K. (2015). Benefits of visualization in the Mammography Problem. International Journal of Human-Computer Studies, 83, 94–113. https://doi.org/10.1016/j.ijhcs.2015.07.001
Krauss, S., Weber, P., Binder, K., & Bruckmaier, G. (2020). Natürliche Häufigkeiten als numerische Darstellungsart von Anteilen und Unsicherheit – Forschungsdesiderate und einige Antworten. Journal für Mathematik-Didaktik, 41(2), 485–521. https://doi.org/10.1007/s13138-019-00156-w
Krüger, K., Sill, H.-D., & Sikora, C. (2015). Didaktik der Stochastik in der Sekundarstufe I. Springer. https://doi.org/10.1007/978-3-662-43355-3
Kultusministerkonferenz. (2012). Bildungsstandards im Fach Mathematik für die Allgemeine Hochschulreife. https://www.kmk.org/fileadmin/Dateien/veroeffentlichungen_beschluesse/2012/2012_10_18-Bildungsstandards-Mathe-Abi.pdf
Kunzelmann, A. K., Binder, K., Fischer, M. R., Reincke, M., Braun, L. T., & Schmidmaier, R. (2022). Improving Diagnostic Efficiency with Frequency Double-Trees and Frequency Nets in Bayesian Reasoning. MDM Policy & Practice, 7(1), 1–11. https://doi.org/10.1177/23814683221086623
Linde, W. (2024). Probability Theory. A First Course in Probability Theory and Statistics (2nd ed.). De Gruyter. https://www.degruyterbrill.com/document/doi/10.1515/9783111325064/html
Luchins, A. S. (1942). Mechanization in problem solving: The effect of Einstellung. Psychological Monographs, 54(6), i–95. https://doi.org/10.1037/h0093502
Maslow, A. (1966). The Psychology of Science: A Reconnaissance. Gateway Editions.
McDowell, M., & Jacobs, P. (2017). Meta-analysis of the effect of natural frequencies on Bayesian reasoning. Psychological Bulletin, 143(12), 1273–1312. https://doi.org/10.1037/bul0000126
Molnar, A. (2018). Language and Lexical Ambiguity in the Probability Register. In C. Batanero & E. J. Chernoff (Eds.), Teaching and Learning Stochastics: Advances in Probability Education Research (pp. 23–37). Springer. https://doi.org/10.1007/978-3-319-72871-1_2
Mošna, F. (2023). Secondary and university students’ understanding of independence and conditional probability. Scientia in Educatione, 14(1), 15–26. https://doi.org/10.14712/18047106.3013
Nabbout-Cheiban, M. (2017). Intuitive Thinking and Misconceptions of Independent Events: A Case Study of US and French Pre-Service Teachers. International Journal of Research in Undergraduate Mathematics Education, 3. https://doi.org/10.1007/s40753-016-0038-x
Nasser, R. (2007). Deciding Whether There is Statistical Independence or Not? Journal of Mathematics and Statistics, 3(3), 151–162. https://doi.org/10.3844/jmssp.2007.151.162
NCTM. (2000). Principles and Standards for School Mathematics. NCTM. https://www.nctm.org/Standards-and-Positions/Principles-and-Standards/Principles,-Standards,-and-Expectations/
O’Grady, C. (2022). Unlucky numbers. Science, 379(6629), 228–233.
Olofsson, P., & Andersson, M. (2012). Probability, Statistics, and Stochastic Processes. John Wiley & Sons. https://onlinelibrary.wiley.com/doi/epub/10.1002/9781118231296
Rößner, M., Binder, K., Geier, C., & Krauss, S. (2025). Students’ performance and typical errors in filling empty probabilistic visualizations with probabilities or frequencies. Educational Studies in Mathematics, 120(1), 137–167. https://doi.org/10.1007/s10649-024-10372-y
Schneps, L., & Colmez, C. (2013). Math on Trial: How Numbers Get Used and Abused in the Courtroom. Basic Books.
Schultz, P. W., & Searleman, A. (2002). Rigidity of thought and behavior: 100 years of research. Genetic, Social, and General Psychology Monographs, 128(2), 165–207.
Sloman, S. A., Over, D., Slovak, L., & Stibel, J. M. (2003). Frequency illusions and other fallacies. Organizational Behavior and Human Decision Processes, 91(2), 296–309. https://doi.org/10.1016/S0749-5978(03)00021-9
Staatsinstitut für Schulqualität und Bildungsforschung. (2025). LehrplanPLUS Bayern Gymnasium Jahrgangsstufe 11 Mathematik Fachlehrpläne. https://www.lehrplanplus.bayern.de/fachlehrplan/gymnasium/11/mathematik
Star, J. R., & Rittle-Johnson, B. (2008). Flexibility in problem solving: The case of equation solving. Learning and Instruction, 18(6), 565–579. https://doi.org/10.1016/j.learninstruc.2007.09.018
Stegmüller, N., Binder, K., & Krauss, S. (2024). How general is the natural frequency effect? The case of joint probabilities. Frontiers in Psychology, 15:1296359. https://doi.org/10.3389/fpsyg.2024.1296359
Steib, N., Büchter, T., Eichler, A., Binder, K., Krauss, S., Böcherer-Linder, K., Vogel, M., & Hilbert, S. (2025). How to teach Bayesian reasoning: An empirical study comparing four different probability training courses. Learning and Instruction, 95(102032). https://doi.org/10.1016/j.learninstruc.2024.102032
Tarr, J. E., & Lannin, J. K. (2005). How Can Teachers Build Notions of Conditional Probability and Independence? In G. A. Jones (Ed.), Exploring Probability in School: Challenges for Teaching and Learning (pp. 215–238). Springer. https://doi.org/10.1007/0-387-24530-8_10
Truran, J. M., & Truran, K. M. (1996). Statistical Independence—One Concept or Two? Implications for Research and for Classroom Practice. 8th International Congress on Mathematical Education, Seville, Spain.
Verschaffel, L. (2024). Strategy flexibility in mathematics. ZDM – Mathematics Education, 56(1), 115–126. https://doi.org/10.1007/s11858-023-01491-6
Weber, P., Binder, K., & Krauss, S. (2018). Why Can Only 24% Solve Bayesian Reasoning Problems in Natural Frequencies: Frequency Phobia in Spite of Probability Blindness. Frontiers in Psychology, 9:1833. https://doi.org/10.3389/fpsyg.2018.01833
Wells, J., Malone, U., Parkes-Hupton, H., Stonehouse, G., Wakatama, G., & Coote, G. (2023, June 5). Kathleen Folbigg pardoned after 20 years in jail, as ex-husband maintains her guilt. ABC News. https://www.abc.net.au/news/2023-06-05/kathleen-folbigg-attorney-general-provides-update/102440136
Wiesner, P., & Binder, K. (2024). Stochastische Unabhängigkeit: Die Rolle der gewählten Definition und verschiedener Visualisierungen. Stochastik in der Schule, 44(3), 21–29.
