Students’ difficulties with mathematical proof and transition from secondary to university mathematics are key topics within mathematics education research. In this talk, we report on research with the Southampton Mathematics department and A-level mathematics teachers. In the transition from A-level maths and further maths to undergraduate mathematics, the topic of proof always is a big challenge for students. In our study, we analysed answers to a ‘proof by induction’ task from first-year undergraduate mathematics students. Findings show that many students find the proof by induction process challenging. Results illustrate the difficulties students face when they are asked to engage with a proof by induction task within the Calculus context and provide insight into the transition from A-level maths to undergraduate maths. We highlight how a multidisciplinary team of mathematics specialists (mathematics education researchers, secondary maths teachers, mathematicians) created a resource to support A-level teachers, trialled in this academic year. The booklet ‘Thinking about Proof’ supports A-level teachers in teaching proof and facilitating a smoother transition to university mathematics. 

1. The challenge of proof in the
transition from A-level
mathematics to university
Christian Bokhove, Athina Thoma

2. Introduction
- Results from first-year undergraduate
students’ difficulties on proof by
induction.
- The Mathematics in Transit Project.

3. How this work came about
 Conversations with colleague David Gammack
from the mathematics department a few years
ago.
 What difficulties do students demonstrate in
their first calculus exam?
 Athina Thoma joined our analysis.
 Focused our current analysis on proof by
induction.

4. Gauss and Proof by induction
• “Add the numbers from 1 to 100”
1 2 3 … 98 99 100
100 99 98 … 3 2 1
101 101 101 … 101 101 101
1 2 3 … n-2 n-1 n
n n-1 n-2 … 3 2 1
n+1 n+1 n+1 … n+1 n+1 n+1
• The formula for adding the numbers from 1 to n is
1
2
× 𝑛 × (𝑛 + 1)
Portrait of Gauss by Christian Albrecht Jensen
Image from: https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss
30 April 1777 – 23 February 1855
The answer is
1
2
× 100 × 101 =
5050

5. Gauss and Proof by induction
• The formula for adding the
numbers from 1 to n is
𝑛×(𝑛+1)
2
• How do we know that this is
true for all natural numbers?
• Show that this is true for the
base case: n=1
P(1)
• Assume that it is true for n=k
and prove that it is true for
n=k+1
P(k) ⇒ P(k + 1)
• Conclude that this is true for all
natural numbers.
• The first domino falls
P(1)
• The kth domino falls then the
k+1th domino falls
P(k) ⇒ P(k + 1)

6. Investigating undergraduate students’ scripts
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 with proof by induction in a Calculus exam

7. 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)

8. Context and Data
First-year Calculus course (pre-Covid)
Duration: 2 hours
Our data include:
- 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.

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
• With differentiation

11. Students’ 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

12. Results

13. Results

14. Conclusion – Our next steps
Students’ scripts illustrate difficulties with:
• The induction process;
• The other subject domain: differentiation;
• A combination of induction and differentiation;
We want to explore proof by induction both in secondary school but also in
other modules at university level.

15. Mathematics in Transit project
• A team of teachers and researchers from post-16
schools, the Southampton Education School and the
School of Mathematics.
• We explored curriculum materials to find the topic that
is considered most challenging in the transition from
A-Levels to University
• We decided that the most useful topic was ‘proof’.
School of Mathematics: David Gammack, Lu Heng Sunny Yu
Teachers from post 16 schools
• Sarah Roberts – Barton Peveril College, Eastleigh
• Frances Downey – St Edward’s School, Poole
Southampton Education School: Christian Bokhove, Athina Thoma

16. School University
• London Mathematical Society materials
• A-level curriculum specs
• First-year undergraduate materials
• School Mathematics Project materials

18. School Mathematics
Project (SMP)

19. Proof by induction
Direct proof
Proof by exhaustion Proof by contradiction

20. Proof by induction
Direct proof
Proof by exhaustion Proof by contradiction

21. Proof by induction
Direct proof
Proof by exhaustion Proof by contradiction
Proof chapters:
• Introduction
• Examples – solved in
detail
• Historical facts where
appropriate
• Exercises to try

22. Proof by induction
Direct proof
Proof by exhaustion Proof by contradiction

23. Future plans
• Further research in UG assessment
• Investigating proof by induction in other modules
• Looking at other proof methods
• Booklet
• Consider how the booklet is used
• Create new booklets on the next topics
• Considering expanding to other countries
curricula
• Research and Knowledge Exchange hand-in-
hand

24. Thank you & Questions
Thank you to our colleagues:
David Gammack, Lu Heng Sunny Yu – School of Mathematics
Sarah Roberts – Barton Peveril College, Eastleigh
Frances Downey – St Edward’s School, Poole