Calum H - Tutors - Keystone Tutors

• GCSE Maths
• A Level Maths
• A Level Further Maths
• IB Maths
• STEP Maths
• Undergraduate Maths

I was a Graduate Teaching Assistant at the University of Manchester for 4 years, teaching 8 different courses, completing 248 hours of teaching mathematics. The topics I taught ranged from some final year modules in complex analysis and number theory to foundation students, which I was helping understand the content of A level maths for their courses in engineering.

I now tutor students at both school and university level, ranging from GCSE students to undergraduate students.

Here are some comments regarding my teaching that were mentioned on a survey taken after the course was finished:

• ''I like the maths tutorials, the graduate teaching assistant is really helpful with going over topics that we don’t understand.'' (3rd year mathematics student)
• ''Well explained. Tutorial very helpful.'' (1st year mathematics student)
• ''The tutorials by Calum Hughes were extremely helpful in clearing any doubts regarding the contents of the lectures. '' (Foundation year aerospace engineering student)

From 2018-2022, I was studying for a Masters in Mathematics at the University of Sheffield, in which I won the T.M. Flett prize in Pure Mathematics twice in the years 2020 and 2022 for the best performance in Mathematics. I had such a good time learning about maths and got really involved in problem solving, helping out in teaching STEP to A level students and getting involved in the UK maths challenge.

This love of maths made me want to pursue it further! From 2022-2026, I was completing a PhD in pure mathematics from the University of Manchester. This yielded research publications, and I gave over 30 research talks to a range of different levels of understanding, including to undergraduates and experts in the field. I also gave a couple of outreach talks to A level students hoping to study mathematics.

Mathematics can be challenging to teach because of its abstract nature. While some students are comfortable working symbolically from the outset, many find it difficult to engage with material that can initially feel detached from intuition or concrete experience. I therefore believe it is important to build understanding before focusing on technical proficiency alone.

When introducing a new topic, I like to begin by developing the key underlying ideas through visualisations, simple examples, or intuitive “toy” models. Establishing this conceptual foundation helps students gain confidence and appreciate why the mathematics works, rather than seeing it as a collection of disconnected techniques.

Once this foundation is in place, I gradually increase the complexity of the examples and problems we consider. Success in mathematics requires both understanding and fluency: students need repeated practice in applying methods until they become familiar and natural. I think of this process similarly to physical training — if you wanted to run a marathon, you would not attempt the full distance immediately, but would instead build endurance progressively over time.

By combining conceptual understanding with carefully structured practice, I aim to help students develop both confidence in the material and strong academic results.

When I am not thinking about maths, you will probably find me running, cycling, climbing or playing the piano!