This is a talk Athina Thoma and Christian Bokhove gave at the BSRLM summer conference on 11 June 2022.

1. Proof by induction in Calculus
Investigating first-year students’
examination scripts
Athina Thoma, Christian Bokhove, David Gammack

2. Introduction
Assessment shows students what lecturers value about their subject
(Smith et al., 1996; van de Watering et al., 2008)
Students’ transition from secondary to university mathematics
(e.g., Gueudet, 2008)
Our study:
Students’ difficulties on proof by induction Calculus examination task

3. Literature review - Differentiation
• Differentiation symbols (e.g., Orton, 1983)
• Derivative-Rate of Change (e.g., Jones, 2017)
• Graphical representation (e.g., Orton, 1983; Jones and Watson,
2018)
• Derivative as function and derivative at a point (e.g., Park, 2015)

4. Literature review – Proof by induction
• Base step (e.g., Harel, 2002; Stylianides, Stylianides, and Philippou,
2007)
• Inductive step - Proving P(k+1) instead of proving P(k) ⇒ P(k + 1) (e.g.,
Dubinsky, 1989; Stylianides, Stylianides, and Philippou, 2007)
• Following the steps without understanding (e.g., Baker, 1996)

5. Context and Data
First-year Calculus course (pre-Covid)
Duration: 2 hours
Our data consists:
- Module materials
- Examination questions and relevant
solutions created by the lecturer
- Students’ scripts
4 questions on limits, continuous
functions, Mean value theorem,
critical points and points of
inflection, derivatives and
induction, integrals, Taylor
polynomials, and ODEs.

6. Statistical overview - marks
Mean Median SD
Total Score (out of 100) 63.72 66.00 18.89
Score to Question 2 (out of 25) 17.38 18 4.49
Score to Question 2c (out of 8) 3.37 3 2.35

7. Lecture materials

8. Lecture materials

9. Lecture materials

10. Similar inductions to part (c) have been done in the lectures and on homework
problems.
Learning outcomes: Part (c) tests the student’s understanding of logical thinking.
Task & solution – Data analysis
We analysed students’ scripts
focusing on students’ writing
regarding:
• Proof induction process
• Differentiation

16. Conclusion
Students’ scripts illustrate difficulties with:
• Induction process
• Differentiation
• Both the induction process and differentiation

17. Next steps
- Interviews with the lecturers who teach Year 1 modules:
- Introduction of Proof by Induction in their module.
- Expectations of students’ knowledge regarding Proof by Induction prior to coming to their
module.
- Their perspectives on students’ difficulties with Proof by Induction (examination and
coursework)
- Mathematics in Transit (MiT) supporting students’ transition from secondary to
university
- Secondary Mathematics teachers
- Lecturers from the School of Mathematics
- Researchers from Southampton Education School
Follow @cbokhove

18. References
Baker, J. D. (1996). Students' Difficulties with Proof by Mathematical Induction..
Dubinsky, E. (1989). Teaching mathematical induction II. Journal of Mathematical Behavior, 8(3), 285-304.
Gueudet, G. (2008). Investigating the secondary–tertiary transition. Educational studies in mathematics, 67(3), 237-254.
Harel, G. (2002). The development of mathematical induction as a proof scheme: A model for DNR-based instruction. In S. Campbell, & R.
Zaskis (Eds.), Learning and teaching number theory: Research in cognition and instruction (pp. 185–212). New Jersey: Ablex Publishing
Corporation.
Jones, S. R. (2017). An exploratory study on student understandings of derivatives in real-world, non-kinematics contexts. The Journal of
Mathematical Behavior, 45, 95-110.
Jones, S. R., & Watson, K. L. (2018). Recommendations for a “target understanding” of the derivative concept for first-semester calculus
teaching and learning. International Journal of Research in Undergraduate Mathematics Education, 4(2), 199-227.
Orton, A. (1983). Students' understanding of differentiation. Educational studies in mathematics, 14(3), 235-250.
Park, J. (2015). Is the derivative a function? If so, how do we teach it?. Educational Studies in Mathematics, 89(2), 233-250.
Smith, G., Wood, L., Coupland, M., Stephenson, B., Crawford, K., & Ball, G. (1996). Constructing mathematical examinations to assess a
range of knowledge and skills. International Journal of Mathematical Education in Science and Technology, 27(1), 65-77.
Stylianides, G. J., Stylianides, A. J., & Philippou, G. N. (2007). Preservice teachers’ knowledge of proof by mathematical induction. Journal
of Mathematics Teacher Education, 10(3), 145-166.
Van de Watering, G., Gijbels, D., Dochy, F., & Van der Rijt, J. (2008). Students’ assessment preferences, perceptions of assessment and
their relationships to study results. Higher Education, 56(6), 645-658.

19. Thank you