How I Prepared a History of Mathematics Course

Every June, after the international exams end, our school sets aside about a month for Oxbridge preparation. There is no regular timetable during this period — teachers can design their own content. I had quite a bit of freedom. The school called it a “super curriculum activity,” meaning you could run whatever you wanted, as long as it had value for students.

I chose history of mathematics.

Why I wanted to teach this course

It was not a spur-of-the-moment decision. I have a habit in my regular classes: when teaching a theorem, I often mention how it came about. For example, when covering the Fundamental Theorem of Calculus, I would explain that Newton and Leibniz discovered it independently, and that their approaches were quite different. Over time, I noticed that students responded to these “tangents” more enthusiastically than to the theorems themselves.

This led me to believe that understanding how knowledge develops may matter more for learning mathematics than doing a few extra problems.

Many students carry an unspoken assumption — that theorems and proofs were simply “always there,” that whatever the textbook says is how things have always been. But that is not the case. Every clean-looking definition emerged through iteration. The discovery of irrational numbers once triggered a philosophical crisis. It took mathematicians two hundred years to put calculus on rigorous foundations. When students know this, their attitude toward difficult concepts changes. Instead of thinking “I must be stupid,” they think “this thing took centuries to figure out.”

The exams ended, the preparation window opened, and the opportunity was there.

Finding the right book

I started with Morris Kline’s Mathematical Thought from Ancient to Modern Times. It is a classic in the field — four volumes, extremely comprehensive. But after reading through it, I felt it was not suitable for high school students. It focuses on the evolution of mathematical thought — which schools of thought existed in which periods, how they related to each other. This is valuable for academic research, but for an elective course, there was too little actual mathematics. Students would hear a parade of names and dates without seeing the math itself.

I also looked at other popular histories of mathematics. They were either too shallow, turning into collections of “mathematician anecdotes,” or too deep, reading like monographs for graduate students.

By chance I found Stillwell’s Mathematics and Its History. Each chapter centres on a specific mathematical topic — the Pythagorean theorem, Greek number theory, infinite series, projective geometry — explains the mathematics first, then places it in historical context. Mathematics is the subject; history is the setting. That was exactly what I wanted.

Designing the course structure

There were only seven students. A pure lecture format felt like a waste — seven people listening to me for two hours, when they could be thinking for themselves.

I settled on a rhythm: I would lecture in the first week, and students would investigate and present in the second. One investigation task per week, no more.

Designing those tasks was the hardest part. Stillwell’s textbook has plenty of exercises, but doing exercises is not investigation. I needed to distil directions from the exercises that required genuine thinking, and let students choose a topic to explore.

Take the first unit — the Pythagorean theorem. The first draft of the task was “use the formula to generate five Pythagorean triples and explain their geometric meaning.” After writing it, I realised it was wrong. It was a drill. Students would finish it and move on without really thinking. I revised it three times. The final version offered three choices: rational parametrisation and rational points on the unit circle, the discovery of √2 and incommensurability, or the axiomatisation of the distance formula. Each direction required a proof and a discussion of its historical significance. Students had to pick one direction and go deep, rather than skimming everything.

This “choose one of three” structure carried through all 11 units of the course.

I learned quite a bit myself

One unit covered Eastern mathematics, which interested me greatly.

Growing up in China, I had heard about the Pythagorean theorem and Zu Chongzhi since childhood. But I had never seriously studied what ancient Chinese mathematics actually accomplished. While preparing this course, I read the proof of the Pythagorean theorem in the Zhoubi Suanjing — Zhao Shuang’s 弦图, which uses four right triangles to form a square. Quite elegant. I also learned about the relationship between ancient Chinese astronomy, the calendar, and mathematics: math was not just an abstract pastime for scholars, but was tightly connected to calendar-making, land surveying, and engineering.

Teaching this course was also a form of learning for the teacher. You think you understand the Pythagorean theorem, but when you go back to two-thousand-year-old source texts and see how people from that era understood and proved it in their own way, your understanding of the theorem changes.

A student who left an impression

One student in the class was particularly drawn to the philosophical aspects of the course.

During the lecture on Zeno’s paradoxes, most students found “Achilles and the Tortoise” an amusing brain teaser. He wanted to know: what assumption is this paradox actually challenging? Later, when we reached Gödel’s incompleteness theorems, he gave a presentation on the content of the theorems, their impact on the foundations of mathematics, and their connection to computability theory. The quality of that presentation exceeded my expectations.

He was later admitted to the University of Oxford to read Computer Science and Philosophy. In a school interview, he mentioned that this history of mathematics course helped him see the connections between mathematics, philosophy, and computer science, and broadened his perspective.

A student from an elective course voluntarily bringing it up in a formal interview — that was not something I had expected.

Seven students at the end of the course

After the course ended, the school asked me to do an assessment. Seven students gave presentations, and I scored them on four dimensions: content quality, presentation skills, visual aids, and Q&A.

Overall, the students’ grasp of mathematical history was adequate, and their presentation skills were not a major issue. But one common problem stood out: most students relied heavily on the textbook and lacked personal expression. When preparing their presentations, they essentially restated the textbook content, rarely adding their own understanding or judgements.

“Grasping motivation” was also generally weak. Students could say “Euclid proved this theorem,” but could not explain “why he wanted to prove it” or “where this theorem sat in the mathematical system of the time.” They knew the what, but not the why.

This actually reflects a core difficulty in teaching history of mathematics: historical stories are easy to listen to, but truly internalising them into one’s own understanding requires students to actively think about “why.” And “why” is precisely the hardest thing to teach.

Next time I run this course, I will strengthen the “motivation analysis” requirement in the investigation tasks. It is not enough for students to study what a theorem says. They also need to answer: what problem was this theorem created to solve? What did it change?