SpringerLink
Log in

Learning from erroneous examples in the mathematics classroom: do students with different naïve ideas benefit equally?

  • Original Research
  • Published:
Instructional Science Aims and scope Submit manuscript

Abstract

Research suggests that troubleshooting activities that require students to reflect on teacher-crafted erroneous examples; i.e., erroneous solutions to problems that correspond to widespread naïve ideas, are beneficial to learning. One possible explanation to these beneficial effects is that troubleshooting activities encourage students to test the quality of their own naïve ideas, not only the ones driving the erroneous examples, thereby improving learning. Few studies have addressed this claim, and the results are inconsistent. These studies, however, were not designed to examine the extent to which students with different naïve ideas benefit from troubleshooting activities. Here, ten 9th grade classes took part in a field experimental study that applied a pre-post-test design after finishing a unit on exponents. Students in each class were randomly assigned to a troubleshooting (114 students) or a self-diagnosis activity (112 students). Self-diagnosis activities are considered to directly nudge students to examine the quality of their own naïve ideas by requiring them to reflect on their solutions. The troubleshooting and self-diagnosis activities both capitalized on the pre-test problems. Both groups increased their proficiency in exponents to a comparable extent from the pre-test to the immediate and the delayed post-test. Troubleshooting students with different naïve ideas detected the errors in the erroneous examples equally well, and their error detection significantly and positively correlated with their self-repair of their own naïve ideas. These findings suggest that all the students benefitted from troubleshooting activities, regardless of whether their own naïve ideas resembled the ones driving the erroneous examples or not.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (France)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

Notes

  1. In only one school, due to time constraints, the delayed post-test was administered two weeks after the immediate post-test. A t-test for independent samples revealed no significant difference in the delayed post-test scores between this school and the other schools (t(224) = 0.738, p = 0.461).

  2. d = Cohen’s d effect size. Based on Hattie (1999), an effect size of d ≥ 0.400 is needed for an educational intervention to have practical significance.

  3. Of the 114 troubleshooting students, 6 students did not respond at all to the questions on the specific troubleshooting worksheet that went with the pre-problem. Accordingly, these students were removed from the analysis. Thus, all the percentages corresponding to this case are for the remaining 108 troubleshooting students.

References

  • Anderson, J. R. (2010). Cognitive psychology and its implications (7th ed.). Worth Publishers.

    Google Scholar 

  • Andrade, H. L. (2010). Students as the definitive source of formative assessment: Academic self-assessment and the self-regulation of learning. In H. Andrade & G. Cizek (Eds.), Handbook of formative assessment (pp. 90–105). Routledge.

    Chapter  Google Scholar 

  • Andrade, H. L. (2019). A critical review of research on student self-assessment. Frontiers in Education, 4(87), 1–13.

    Google Scholar 

  • Atkinson, R. K., Renkl, A., & Merrill, M. M. (2003). Transitioning from studying examples to solving problems: Effects of self-explanation prompts and fading worked-out steps. Journal of Educational Psychology, 95, 774–783.

    Article  Google Scholar 

  • Baars, M., Vink, S., van Gog, T., de Bruin, A., & Paas, F. (2014). Effects of training self-assessment and using assessment standards on retrospective and prospective monitoring of problem solving. Learning and Instruction, 33, 92–107.

    Article  Google Scholar 

  • Black, P., Harrison, C., & Lee. C. (2003). Assessment for Learning: Putting It into Practice. Open University Press.

  • Booth, J. L., Lange, K. E., Koedinger, K. R., & Newton, K. J. (2013). Using example problems to improve student learning in algebra: Differentiating between correct and incorrect examples. Learning and Instruction, 25, 24–34.

    Article  Google Scholar 

  • Borasi, R. (1994). Capitalizing on errors as “springboards for inquiry”: A teaching experiment. Journal for Research in Mathematics Education, 25(2), 166–208.

    Google Scholar 

  • Bransford, J. D., & Schwartz, D. L. (1999). Rethinking transfer: A simple proposal with multiple implications. Review of Educational Research, 24, 61–100.

    Google Scholar 

  • Brown, G. T. L., & Harris, L. R. (2013). Student self-assessment. In J. McMillan (Ed.), The SAGE handbook of research on classroom assessment (pp. 367–393). SAGE.

    Chapter  Google Scholar 

  • Chi, M. T. H. (1997). Quantifying qualitative analyses of verbal data: A practical guide. The Journal of the Learning Sciences, 6(3), 271–315.

    Article  Google Scholar 

  • Chi, M. T. H. (2000). Self-explaining expository texts: The dual processes of generating inferences and repairing mental models. In R. Glaser (Ed.), Advances in Instructional Psychology (pp. 161–238). Lawrence Erlbaum Associates.

    Google Scholar 

  • Chi, M. T. H. (2008). Three types of conceptual change: Belief revision, mental model transformation, and categorical shift. In S. Vosniadou (Ed.), Handbook of research on conceptual change (pp. 61–82). Erlbaum Associates Inc.

    Google Scholar 

  • Chi, M. T. H., Bassok, M., Lewis, M. H., Reimann, P., & Glaser, R. (1989). Self-explanations: How students study and use examples in learning to solve problems. Cognitive Science, 13(2), 145–182.

    Google Scholar 

  • Chi, M. T. H., de Leeuw, N., Chiu, M. H., & LaVancher, C. (1994). Eliciting self-explanations improves understanding. Cognitive Science, 18(3), 439–477.

    Google Scholar 

  • Clark, R. C., Nguyen, F., & Sweller, J. (2006). Efficiency in learning: Evidence-based guidelines to manage cognitive load. Pfeiffer.

  • Cohen, J. (1992). A power primer. Psychological Bulletin, 112(1), 155–159.

    Article  Google Scholar 

  • Dkeidek, I., Mamlok-Naaman, R., & Hofstein, A. (2012). Assessment of the laboratory learning environment in an inquiry-oriented chemistry laboratory in Arab and Jewish high schools in Israel. Learning Environments Research, 15(2), 141–169.

    Article  Google Scholar 

  • Double, K. S., McGrane, J. A., & Hopfenbeck, T. N. (2020). The impact of peer assessment on academic performance: A meta-analysis of control group studies. Educational Psychology Review, 32(2), 481–509.

    Article  Google Scholar 

  • Dunlosky, J., Rawson, K. A., Marsh, E. J., Nathan, M. J., & Willingham, D. T. (2013). Improving students’ learning with effective learning techniques: Promising directions from cognitive and educational psychology. Psychological Science in the Public Interest, 14(1), 4–58.

    Article  Google Scholar 

  • Durkin, K., & Rittle-Johnson, B. (2012). The effectiveness of using incorrect examples to support learning about decimal magnitude. Learning and Instruction, 22(3), 206–214.

    Article  Google Scholar 

  • Eilam, B. (2002). Passing through a western-democratic teacher education: The case of Israeli-Arab teachers. Teacher College Record, 104(8), 1656–1701.

    Article  Google Scholar 

  • Gick, M. L., & Holyoak, K. J. (1983). Schema induction and analogical transfer. Cognitive Psychology, 15(1), 1–38.

    Article  Google Scholar 

  • Glogger-Frey, I., Fleischer, C., Grüny, J., Kappich, J., & Renkl, A. (2015). Inventing a solution and studying a worked solution prepare differently for learning from direct instruction. Learning and Instruction, 39, 72–87.

    Article  Google Scholar 

  • Glogger-Frey, I., Gaus, K., & Renkl, A. (2017). Learning from direct instruction: Best prepared by several self-regulated or guided invention activities? Learning and Instruction, 51, 26–35.

    Article  Google Scholar 

  • Glogger-Frey, I., Treier, A. K., & Renkl, A. (2022). How preparation-for-learning with a worked versus an open inventing problem affect subsequent learning processes in pre-service teachers. Instructional Science, 50(3), 451–473.

    Article  Google Scholar 

  • Große, C. S., & Renkl, A. (2007). Finding and fixing errors in worked examples: Can this foster learning outcomes? Learning and Instruction, 17(6), 612–634.

    Article  Google Scholar 

  • Hartmann, C., van Gog, T., & Rummel, N. (2020). Do examples of failure effectively prepare students for learning from subsequent instruction? Applied Cognitive Psychology, 34(4), 879–889.

    Article  Google Scholar 

  • Hartmann, C., van Gog, T., & Rummel, N. (2021). Preparatory effects of problem solving versus studying examples prior to instruction. Instructional Science, 49(1), 1–21.

    Article  Google Scholar 

  • Hartmann, C., van Gog, T., & Rummel, N. (2022). Productive versus vicarious failure: Do students need to fail themselves in order to learn? Applied Cognitive Psychology, 36(6), 1219–1233.

    Article  Google Scholar 

  • Hattie, J. (1999). Influences on student learning. Retrieved Augost 14, 2020, from http://projectlearning.org/blog/wp-content/uploads/2014/02/Influences-on-Student-Learning-John-Hattie.pdf.

  • Hausmann, R. G. M., & VanLehn, K. (2007). Explaining self-explaining: A contrast between content and generation. In R. Luckin, K. R. Koedinger, & J. Greer (Eds.), Artificial intelligence in education: Building technology rich learning contexts that work (Vol. 158, pp. 417–424). IOS Press.

  • Heemsoth, T., & Heinze, A. (2014). The impact of incorrect examples on learning fractions: A field experiment with 6th grade students. Instructional Science, 42(4), 639–657.

    Article  Google Scholar 

  • Heemsoth, T., & Heinze, A. (2016). Secondary school students learning from reflections on the rationale behind self-made errors: A field experiment. The Journal of Experimental Education, 84(1), 98–118.

    Article  Google Scholar 

  • Hieggelke, C. J., Maloney, D. P., O’Kuma, T. L., & Kanim, S. (2006). E&M TIPERs: Electricity & magnetism tasks. Addison Wesley.

    Google Scholar 

  • Holmes, N. G., Day, J., Park, A. H. K., Bonn, D. A., & Roll, I. (2014). Making the failure more productive: Scaffolding the invention process to improve inquiry behaviors and outcomes in invention activities. Instructional Science, 42(4), 523–538.

    Article  Google Scholar 

  • Huisman, B., Saab, N., van den Broek, P., & van Driel, J. (2019). The impact of formative peer feedback on higher education students’ academic writing: A Meta-Analysis. Assessment & Evaluation in Higher Education, 44(6), 863–880.

    Article  Google Scholar 

  • Kapur, M., & Bielaczyc, K. (2012). Designing for productive failure. Journal of the Learning Sciences, 21(1), 45–83.

    Article  Google Scholar 

  • Kapur, M. (2014a). Productive failure in learning math. Cognitive Science, 38(5), 1008–1022.

    Article  Google Scholar 

  • Kapur, M. (2014b). Comparing learning from productive failure and vicarious failure. The Journal of the Learning Science, 23(4), 651–677.

    Article  Google Scholar 

  • Kapur, M. (2016). Examining productive failure, productive success, unproductive failure, and unproductive success in learning. Educational Psychologist, 51(2), 289–299.

    Article  Google Scholar 

  • Kawasaki, M. (2010). Learning to solve mathematics problems: The impact of incorrect solutions in fifth grade peers’ presentations. Japanese Journal of Developmental Psychology, 21(1), 12–22.

    Google Scholar 

  • Li, H., **ong, Y., Hunter, C. V., Guo, X., & Tywoniw, R. (2020). Does peer assessment promote student learning? A meta-analysis. Assessment & Evaluation in Higher Education, 45(2), 193–211.

    Article  Google Scholar 

  • Loibl, K., & Leuders, T. (2018). Errors during exploration and consolidation – the effectiveness of productive failure as sequentially guided discovery learning. Journal Fur Mathematik-Didaktik, 39(1), 69–96.

    Article  Google Scholar 

  • Loibl, K., & Leuders, T. (2019). How to make failure productive: Fostering learning from errors through elaboration prompts. Learning and Instruction, 62, 1–10.

    Article  Google Scholar 

  • Loibl, K., Roll, I., & Rummel, N. (2017). Towards a theory of when and how problem solving followed by instruction supports learning. Educational Psychology Review, 29(4), 693–715.

    Article  Google Scholar 

  • Loibl, K., & Rummel, N. (2014). Knowing what you don’t know makes failure productive. Learning and Instruction, 34, 74–85.

    Article  Google Scholar 

  • Loibl, K., Tillema, M., Rummel, N., & van Gog, T. (2020). The effect of contrasting cases during problem solving prior to and after instruction. Instructional Science, 48(2), 115–136.

    Article  Google Scholar 

  • Maloney, D. P. (2011). An overview of physics education research on problem solving. In C. Henderson & K. A. Harper (Eds.), Getting Started in PER, Reviews in PER Vol. 2. American Association of Physics Teachers.

  • Markic, S., Eilks, I., Mamlok-Naaman, R., Hugerat, M., Kortam, N., Dkeidek, I., & Hofstein, A. (2016). One country, two cultures - a multi-perspective view on Israeli chemistry teachers’ beliefs about teaching and learning. Teachers and Teaching, 22(2), 131–147.

    Article  Google Scholar 

  • McLaren, B. M., Adams, D. M., & Mayer, R. E. (2015). Delayed learning effects with erroneous examples: A study of learning decimals with a web-based tutor. International Journal of Artificial Intelligence in Education, 25(4), 520–542.

    Article  Google Scholar 

  • Newman, P. M., & DeCaro, M. S. (2019). Learning by exploring: How much guidance is optimal? Learning and Instruction, 62, 49–63.

    Article  Google Scholar 

  • Osborne, J. (2010). Arguing to learn in science: The role of collaborative, critical discourse. Science, 328(5977), 463–466.

    Article  Google Scholar 

  • Panadero, E., Brown, G. T. L., & Strijbos, J. W. (2016). The future of student self-assessment: A review of known unknowns and potential directions. Educational Psychology Review, 28(4), 803–830.

    Article  Google Scholar 

  • Peugh, J. L., & Enders, C. K. (2004). Missing data in educational research: A review of reporting practices and suggestions for improvement. Review of Educational Research, 74(4), 525–556.

    Article  Google Scholar 

  • Reinholz, D. (2016). The assessment cycle: A model for learning through peer assessment. Assessment & Evaluation in Higher Education, 41(2), 301–315.

    Article  Google Scholar 

  • Renkl, A. (1997). Learning from worked-out examples: A study on individual differences. Cognitive Science, 21(1), 1–29.

    Google Scholar 

  • Renkl, A. (2017). Learning from worked-examples in mathematics: Students relate procedures to principles. ZDM, 49(4), 571–584.

    Article  Google Scholar 

  • Rittle-Johnson, B., Loehr, A. M., & Durkin, K. (2017). Promoting self-explanation to improve mathematics learning: A meta-analysis and instructional design principles. ZDM, 49(4), 599–611.

    Article  Google Scholar 

  • Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Develo** conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93(2), 346–362.

    Article  Google Scholar 

  • Roll, I., Aleven, V., & Koedinger, K. (2011). Outcomes and mechanisms of transfer in invention activities. In L. Carlson, C. Hölscher, & T. Shipley (Eds.), Proceedings of the 33rd Annual Conference of the Cognitive Science Society (pp. 2824–2829). Cognitive Science Society.

  • Sadler, D. R. (1989). Formative assessment and the design of instructional systems. Instructional Science, 18(2), 119–144.

    Article  Google Scholar 

  • Safadi, R. (2017a). Designing efficient self-diagnosis activities in the physics classroom. The Physics Teacher, 55(9), 453–457.

    Article  Google Scholar 

  • Safadi, R. (2017b). Self-diagnosis as a tool for supporting students’ conceptual understanding and achievements in physics: The case of 8th-graders studying force and motion. Physics Education, 52(1), 014002.

    Article  Google Scholar 

  • Safadi, R. (2018). Knowledge-integration processes and learning outcomes associated with a self-diagnosis activity: The case of 5th-graders studying simple fractions. International Journal of Science and Mathematics Education, 16(5), 929–948.

    Article  Google Scholar 

  • Safadi, R. (2022). Supporting student learning from diagnosing erroneous examples when contrasting them with worked examples in the physics classroom. International Journal of Science Education, 44(2), 245–270.

    Article  Google Scholar 

  • Safadi, R., & Ababsy, R. (2020). Learning from Troubleshooting Activities when Contrasting Erroneous Examples with Worked Examples in the Physics Classroom. Physics Education, 55(5), 055024.

    Article  Google Scholar 

  • Safadi, R., & Saadi, S. (2021). Learning from self-diagnosis activities when contrasting students’ own solutions with worked examples: The case of 10th graders studying geometric optics. Research in Science Education, 51(2), 523–546.

    Article  Google Scholar 

  • Safadi, R., Safadi, E., & Meidav, M. (2017). Reflection on solutions in the form of refutation texts versus problem solving: The case of 8th graders studying simple electric circuits. Physics Education, 52(1), 015013.

    Article  Google Scholar 

  • Safadi, R., & Yerushalmi, E. (2014). Problem solving vs. troubleshooting tasks: The case of sixth-grade students studying simple electric circuits. International Journal of Science and Mathematics Education, 12(6), 1341–1366.

    Article  Google Scholar 

  • Sanchez, C. E., Atkinson, K. M., Koenka, A. C., Moshontz, H., & Cooper, H. (2017). Self-grading and peer-grading for formative and summative assessments in 3rd through 12th grade classrooms: A meta-analysis. Journal of Educational Psychology, 109(8), 1049–1066.

    Article  Google Scholar 

  • Schafer, J. L., & Graham, J. W. (2002). Missing data: Our view of the state of the art. Psychological Methods, 7(2), 147–177.

    Article  Google Scholar 

  • Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 334–370). Simon and Schuster.

    Google Scholar 

  • Schwartz, D. L., & Bransford, J. D. (1998). A time for telling. Cognition and Instruction, 16(4), 475–522.

    Article  Google Scholar 

  • Siegler, R. S. (1996). Emerging minds: The process of change in children’s thinking. Oxford University Press.

    Book  Google Scholar 

  • Siegler, R. S. (2002). Microgenetic studies of self-explanation. In N. Garnott & J. Parziale (Eds.), Microdevelopment: A process-oriented perspective for studying development and learning (pp. 31–58). Cambridge University Press.

    Chapter  Google Scholar 

  • Sinha, T., & Kapur, M. (2021a). When problem-solving followed by instruction works: Evidence for productive failure. Review of Educational Research, 91(5), 761–798.

    Article  Google Scholar 

  • Sinha, T., & Kapur, M. (2021). From problem-solving to sensemaking: A comparative meta-analysis of preparatory approaches for future learning. EdAr**v, 91(5), 761–798.

    Google Scholar 

  • Top**, K. J. (1998). Peer assessment between students in colleges and Universities. Review of Educational Research, 68(3), 249–276.

    Article  Google Scholar 

  • Tulis, M., Steuer, G., & Dresel, M. (2015). Learning from errors: Process and contextual conditions. Towards a model of individual processes within contexts. In M. Gartmeir, H. Graber, T. Hascher & H. Heid (Eds.), Fehler: Ihre Funktinonen im Kontext Individueller und gesellschaftlicher Entmckliung [Mistakes: their functions in the context of individual and social development], (pp. 53–70). Münster: Waxmann.

  • Ulusoy, F. (2019). Serious obstacles hindering middle school students’ understanding of integer exponents. International Journal of Research in Education and Science (IJRES), 5(1), 52–69.

    Google Scholar 

  • van den Haak, M. J., de Jong, M. D. T., & Schellens, P. J. (2003). Retrospective vs concurrent think-aloud protocols: Testing the usability of an online library catalogue. Behaviour & information technology, 22(5), 339–351.

    Article  Google Scholar 

  • VanLehn, K. (1999). Rule-learning events in the acquisition of a complex skill: An evaluation of CASCADE. The Journal of the Learning Sciences, 8(1), 71–125.

    Article  Google Scholar 

  • Yan, Z., Lao, H., Panadero, E., Fernández-Castilla, B., Yang, L., & Yang, M. (2022). Effects of self-assessment and peer-assessment interventions on academic performance: A pairwise and network meta-analysis. Educational Research Review, 37, 100484.

    Article  Google Scholar 

  • Yerushalmi, E., Cohen, E., Mason, A., & Singh, C. (2012). What do students do when asked to diagnose their mistakes? Does it help them? II. A more typical quiz context. Physical Review Physics Education Research, 8, 020110.

    Article  Google Scholar 

  • Zamora, A., Suárez, J., & Ardura, D. (2018). Error detection and self-assessment as mechanisms to promote self-regulation of learning among secondary education students. The Journal of Educational Research, 111(2), 175–185.

    Article  Google Scholar 

  • Zheng, L., Zhang, X., & Cui, P. (2020). The role of technology-facilitated peer assessment and supporting strategies: A meta-analysis. Assessment & Evaluation in Higher Education, 45(3), 372–386.

    Article  Google Scholar 

  • Ziegler, E., & Stern, E. (2014). Delayed benefits of learning elementary algebraic trans-formations through contrasted comparisons. Learning and Instruction, 33, 131–146.

    Article  Google Scholar 

Download references

Acknowledgements

The authors would especially like to thank Rina Hershkowitz for her valuable support and comments. The authors wish to express their gratitude to the teachers who participated in this study. The authors appreciate the support of The Academic Arab College for Education in Israel-Haifa, Haifa, Israel.

Author information

Authors and Affiliations

  1. The Academic Arab College for Education in Israel-Haifa, 22 Hahashmal St., P.O. Box 8349, 33145, Haifa, Israel

    Rafi Safadi & Nadera Hawa

Authors
  1. Rafi Safadi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nadera Hawa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rafi Safadi.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Safadi, R., Hawa, N. Learning from erroneous examples in the mathematics classroom: do students with different naïve ideas benefit equally?. Instr Sci 52, 277–308 (2024). https://doi.org/10.1007/s11251-023-09648-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11251-023-09648-2

Keywords

Search

Navigation