# Forged in failure: engagement patterns for successful students repeating calculus

## Abstract

Although there is extensive research on attrition in gatekeeper courses and students’ cognition about calculus concepts, there is one population in introductory calculus that remains understudied: those who failed their initial course and chose to repeat it rather than change majors. These students can provide insight into overcoming poor mathematics affect and major persistence. This case study follows eight students repeating calculus from their second try at undergraduate calculus until they graduated or left the university; six graduated with either a mathematics major or mathematics minor. While participants identified several reasons for their success in the repeated course (processing their initial failure, having a better instructor in the repeated course, and participating regularly in the formative assessments), only participation in formative assessment led to the long-term cognitive and behavioral engagement required for long-term success.

## Keywords

Calculus Leading event Patterns of engagement## Notes

## References

- Andersson, A., Valero, P., & Meaney, T. (2015). “I am [not always] a maths hater”: Shifting students’ identity narratives in context.
*Educational Studies in Mathematics, 90*(2), 143–161.CrossRefGoogle Scholar - Bandura, A. (1977). Self-efficacy: Toward a unifying theory of behavioral change.
*Psychological Review, 84*(2), 191–215.CrossRefGoogle Scholar - Bandura, A. (1993). Perceived self-efficacy development and functioning.
*Educational Psychologist, 28*, 117–148.CrossRefGoogle Scholar - Bellas, M. L. (1999). Emotional labor in academia: The case of professors.
*The Annals of the American Academy of Political and Social Science, 561*(1), 96–110.CrossRefGoogle Scholar - Black, L., Williams, J., Hernandez-Martinez, P., Davis, P., Pampaka, M., & Wake, G. (2010). Developing a ‘leading identity’: The relationship between students’ mathematical identities and their career and higher education aspirations.
*Educational Studies in Mathematics, 73*(1), 55–72.CrossRefGoogle Scholar - Black, P., & McCormick, R. (2010). Reflections and new directions.
*Assessment & Evaluation in Higher Education, 35*(5), 493–499.CrossRefGoogle Scholar - Black, P., & Wiliam, D. (1998). Inside the black box: Raising standards through classroom assessment.
*Phi Delta Kappan*, 139–148.Google Scholar - Black, P., & Wiliam, D. (2009). Developing the theory of formative assessment.
*Educational Assessment, Evaluation, and Accountability, 21*(1), 5–31.CrossRefGoogle Scholar - Braathe, H. J., & Solomon, Y. (2015). Choosing mathematics: The narrative of the self as a site of agency.
*Educational Studies in Mathematics, 89*(2), 151–166.CrossRefGoogle Scholar - Bressoud, D. M., Carlson, M. P., Mesa, V., & Rasmussen, C. (2013). The calculus student: Insights from the mathematical Association of America national study.
*International Journal of Mathematical Education in Science and Technology*, 1–15.Google Scholar - Brown, M., Brown, P., & Bibby, T. (2008). “I would rather die”: Reasons given by 16-year-olds for not continuing their study of mathematics.
*Research in Mathematics Education, 10*(1), 3–18.CrossRefGoogle Scholar - Burks, S. V., Carpenter, J., Götte, L., & Rustichini, A. (2001).
*Overconfidence is a Social Signalling Bias. Discussion Paper # 4840*(pp. 1–43). Bonn: Institute for the Study of Labor.Google Scholar - Chen, P., & Zimmerman, B. (2007). A cross-national comparison study on the accuracy of self-efficacy beliefs in middle school mathematics students.
*The Journal of Experimental Education, 75*(3), 221–244.CrossRefGoogle Scholar - Chiesa, M., & Robertson, A. (2000). Precision teaching and fluency training: Making maths easier for pupils and teachers.
*Educational Psychology in Practice, 16*(3), 297–310.CrossRefGoogle Scholar - Chiu, M., & Klassen, R. (2010). Relations of mathematics self-concept and its calibration with mathematics achievement: Cultural differences among fifteen-year-olds in 34 countries.
*Learning and Instruction, 20*(1), 2–17.CrossRefGoogle Scholar - Clark, I. (2011). Formative assessment: Policy, perspectives and practice.
*Florida Journal of Education Administration and Policy, 4*(2), 158–180.Google Scholar - Di Martino, P., Coppola, C., Mollo, M., Pacelli, T., & Sabena, C. (2013). Preservice primary teachers’ emotions: The math-redemption phenomenon. In A. M. Lindmeier & A. Heinze (Eds.),
*Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education*(Vol. 2, pp. 225–232). Kiel, Germany: PME.Google Scholar - Elliot, D. L., & Gillen, A. (2013). Images and stories: Through the eyes of at-risk college learners.
*International Journal of Qualitative Studies in Education, 26*(7), 912–931.CrossRefGoogle Scholar - Ellis, J., Kelton, M. L., & Rasmussen, C. (2014). Student perceptions of pedagogy and associated persistence in calculus.
*ZDM*, 1–13.Google Scholar - Esmonde, I. (2009). Ideas and identities: Supporting equity in cooperative mathematics learning.
*Review of Educational Research, 79*(2), 1008–1043.CrossRefGoogle Scholar - Fernández-Plaza, J. A., & Simpson, A. (2016). Three concepts or one? Students’ understanding of basic limit concepts.
*Educational Studies in Mathematics, 93*(3), 315–332.CrossRefGoogle Scholar - Fredricks, J. A., Blumenfeld, P. C., & Paris, A. H. (2004). School engagement: Potential of the concept, state of the evidence.
*Review of Educational Research, 74*(1), 59–109. http://rer.sagepub.com/content/74/1/59. Accessed 15 Nov 2017.CrossRefGoogle Scholar - Frenzel, A. C., Pekrun, R., & Goetz, T. (2007). Perceived learning environment and students' emotional experiences: A multilevel analysis of mathematics classrooms.
*Learning and Instruction, 17*(5), 478–493.CrossRefGoogle Scholar - Gallagher, E., Bones, R., & Lambe, J. (2006). Precision teaching and education: Is fluency the missing link between success and failure?
*Irish Educational Studies, 25*(1), 93–113.CrossRefGoogle Scholar - Goldin, G. A., Epstein, Y. M., Schorr, R. Y., & Warner, L. B. (2011). Beliefs and engagement structures: Behind the affective dimension of mathematical learning.
*ZDM, 43*(4), 547–560.CrossRefGoogle Scholar - Gueudet, G. (2008). Investigating the secondary-tertiary transition.
*Educational Studies in Mathematics, 67*, 237–254.CrossRefGoogle Scholar - Halverson, L. R., Woodfield-Porter, W., Graham, C. R., Hernrie, C., & Borup, J. (2013). Measuring cognitive and affective engagement in blended and online settings. In
*Paper presented at the 10th Annual Sloan Consortium, Milwaukee, WI*.Google Scholar - Hannula, M. S. (2006). Motivation in mathematics: Goals reflected in emotions.
*Educational Studies in Mathematics, 63*(2), 165–178.CrossRefGoogle Scholar - Hardy, N. (2009). Students’ perceptions of institutional practices: The case of limits of functions in college level calculus courses.
*Educational Studies in Mathematics, 72*(3), 341–358.CrossRefGoogle Scholar - Hembree, R. (1990). The nature, effects, and relief of mathematics anxiety.
*Journal for Research in Mathematics Education, 21*(1), 33–46.CrossRefGoogle Scholar - Hoffman, B. (2010). I think I can but I’m afraid to try: The role of self-efficacy beliefs and mathematics anxiety in problem solving efficiency.
*Learning and Individual Differences, 20*(3), 276–283.CrossRefGoogle Scholar - Imada, T., & Ellsworth, P. C. (2011). Proud Americans and lucky Japanese: Cultural differences in appraisal and corresponding emotion.
*Emotion, 11*(2), 329–345.CrossRefGoogle Scholar - Ingram, J. (2012).
*Whole class interaction in the mathematics classroom: A conversation analytic approach*(Unpublished doctoral dissertation, University of Warwick).Google Scholar - Kajander, A., & Louric, M. (2005). Transition from secondary to tertiary mathematics: McMaster University experience.
*International Journal of Mathematical Education in Science and Technology, 36*(2–3), 149–160.CrossRefGoogle Scholar - Karpinski, A. C., & D’Agostino, J. V. (2012). The role of formative assessment in student achievement. In
*International Guide to Student Achievement*, (p 202).Google Scholar - Klassen, R. (2007). Using predictions to learn about the self-efficacy of early adolescents with and without learning disabilities.
*Contemporary Educational Psychology, 32*(2), 173–187.CrossRefGoogle Scholar - Kruger, J., & Dunning, D. (1999). Unskilled and unaware of it: How difficulties in recognizing one’s own incompetence lead to inflated self-assessments.
*Journal of Personality and Social Psychology, 77*(6), 1121–1134.CrossRefGoogle Scholar - Leontiev, A. N. (1981). Problems of psychic development. Moscow: Moscow St. Un. Publ.Google Scholar
- Mahatmya, D., Lohman, B. J., Matjasko, J. L., & Farb, A. F. (2012). Engagement across developmental periods. In
*Handbook of research on student engagement*(pp. 45–63). Springer US.Google Scholar - Malmivuori, M. (2006). Affect and self-regulation.
*Educational Studies in Mathematics, 63*, 149–164.CrossRefGoogle Scholar - Martínez-Sierra, G., & García González, M. D. S. (2016). Undergraduate mathematics students’ emotional experiences in linear algebra courses.
*Educational Studies in Mathematics, 91*(1), 87–106.CrossRefGoogle Scholar - Martínez-Sierra, G., & García González, M. D. S. (2014). High school students' emotional experiences in mathematics classes.
*Research in Mathematics Education, 16*(3), 234–250.CrossRefGoogle Scholar - Merriam, S. (1998).
*Qualitative research and case study applications in education*. San Francisco: Jossey-Bass.Google Scholar - Quintos, B., & Civil, M. (2008). Parental engagement in a classroom Community of Practice: Boundary practices as part of a culturally responsive pedagogy.
*Adults Learning Mathematics, 3*(n2a), 59–71.Google Scholar - Pekrun, R. (2006). The control-value theory of achievement emotions: Assumptions, corollaries, and implications for educational research and practice.
*Educational Psychology Review, 18*(4), 315–341.CrossRefGoogle Scholar - Roh, K. H. (2010). An empirical study of students’ understanding of a logical structure in the definition of limit via the ε-strip activity.
*Educational Studies in Mathematics, 73*(3), 263–279.CrossRefGoogle Scholar - Selden, A. (2005). New developments and trends in tertiary mathematics education: Or, more of the same?
*International Journal of Mathematical Education in Science and Technology, 36*(2–3), 131–147.CrossRefGoogle Scholar - Selden, A., & Selden, J. (2002). Tertiary mathematics education research and its future. In D. Holton (Ed.),
*The teaching and learning of mathematics at university level*(pp. 237–254). The Netherlands: Springer.CrossRefGoogle Scholar - Sonnert, G., Sadler, P. M., Sadler, S. M., & Bressoud, D. M. (2015). The impact of instructor pedagogy on college calculus students’ attitude toward mathematics.
*International Journal of Mathematical Education in Science and Technology, 46*(3), 370–387.CrossRefGoogle Scholar - Stewart, J. (2006).
*Calculus early transcendentals, 6e*. Belmont: Thompson Brooks/Cole.Google Scholar - Stiggins, R., & Chappuis, J. (2005). Using student-involved classroom assessment to close achievement gaps.
*Theory Into Practice, 44*, 11–18.CrossRefGoogle Scholar - Tainio, L., & Laine, A. (2015). Emotion work and affective stance in the mathematics classroom: The case of IRE sequences in Finnish classroom interaction.
*Educational Studies in Mathematics, 89*(1), 67–87.CrossRefGoogle Scholar - Tall, D. (2008). The transition to formal thinking in mathematics.
*Mathematics Education Research Journal, 20*, 5–24.CrossRefGoogle Scholar - Tsay, J. J., Judd, A. B., Hauk, S., & Davis, M. K. (2011). Case study of a college mathematics instructor: Patterns of classroom discourse.
*Educational Studies in Mathematics, 78*(2), 205–229.CrossRefGoogle Scholar - Wheeler, D. L., & Montgomery, D. (2009). Community college students’ views on learning mathematics in terms of their epistemological beliefs: A Q method study.
*Educational Studies in Mathematics, 72*(3), 289–306.CrossRefGoogle Scholar - Windschitl, M., Thompson, J., Braaten, M., & Stroupe, D. (2012). Proposing a core set of instructional practices and tools for teachers of science.
*Science Education, 96*(5), 878–903.CrossRefGoogle Scholar - Wyatt, M. (2011). Overcoming low self-efficacy beliefs in teaching English to young learners.
*International Journal of Qualitative Studies in Education, 26*, 238–255. https://doi.org/10.1080/09518398.2011.605082 CrossRefGoogle Scholar