When I ran classes for Oxford maths undergraduates, I thought I’d reached the pinnacle of teaching. I was tasked with educating some of brightest minds in society, and delivered my lessons with a vigour that is rarely seen in the hallowed halls of the Oxford Mathematical Institute. My students were engaged and showed clear signs of intellectual growth. It was hardly Dead Poets Society, but I retired from undergraduate teaching with a touch of satisfaction in the impact I had on my students.

So when I was approached by my friends to teach their children -  four girls, aged five to eight - I figured I was up to the challenge. How hard could it be to impart the fundamental building blocks of mathematics, when I had guided undergraduates through reams of rigorous analytical proofs?

Very hard, it turns out. It wasn’t long before I was yearning for a return to the chalkboards of Oxford.

An intellectual battle

Maths lectures are crammed thick with theorems and proofs (especially in the pure modules I taught). Homework assignments demand that undergraduates apply this knowledge to solve novel problems. It’s tough work  -  that’s where tutorials come in.

The Oxford maths tutorial is a 60-minute tour of the week’s most salient ideas, and a run-through of solutions to the most challenging problems. My main charge as a tutor was to bring clarity to the haze of ideas students had been exposed to. Given that I had laboured through the very same syllabus myself only a few years earlier, this was no lean task. Planning my lessons was an intellectual battle against deep subject matter. But, with some mental effort, the occasional Google search and a sneaky look at my brightest students’ model solutions, I usually came well-prepared.

My lessons included a heavy component of direct instruction  -  when you’re navigating epsilon-driven analysis proofs, it’s often the only way to go. My students left the tutorials secure in their understanding of the main concepts. Job done.

Bloody difficult

My primary students presented an altogether different challenge. The subject matter was second nature. But direct instruction was no longer a foolproof approach. I’ll never forget the savage glance one of my students directed at me in my first lesson with her, as I attempted to explain the concept of monetary change. The concept seemed so obvious, and yet she wasn’t having any of it. I was simmering inwardly. What’s wrong with her? How can she still not get it? I’ve long believed a student’s failure to grasp a concept only reflects limited teaching methods. So instead I directed my ire at myself. What was I doing wrong?

That’s when it hit me. Primary teaching is bloody difficult.

With Oxford undergraduates, I knew that, provided my explanations were well thought-out and succinctly delivered, students would be able to overcome their mental blockages. But even my best explanations were proving futile with primary pupils. So I hit Google again.

After perusing some teacher-community sites, I purchased a mock cash register. The next lesson with my student was more interactive and geared towards supporting her to feel mathematical objects and discover concepts for herself. This did the trick - she was running her own mock mini-store in no time. On occasions since, direct instruction has proven more effective.

An act of intellectual empathy

My experiences suggested that the easier something is to understand, the harder it is to teach. Subject matter expertise  -  or content knowledge - is only one piece of the teaching puzzle. So much for George Bernard Shaw’s dictum of “He who can, does. He who cannot, teaches.”

The education researcher Lee Shulman knew this back in the 1980s when he conceived the idea of Pedagogicial Content Knowledge (PCK). The premise of PCK is that to teach effectively, educators must interpret subject-matter knowledge within the context of each child’s learning. In other words, it’s not just the understanding of concepts alone that matters, but the ability to present that idea in the most appropriate way, given the child’s needs. Teaching concepts is an act of intellectual empathy; it requires a sensitive understanding of how your students think and see the world. Primary students think in diverse, often messy ways  - they require more empathy than the typical Oxford undergraduate.

While teachers’ content knowledge must necessarily deepen as students progress through the curriculum, the demands on their Pedagogical Content Knowledge peak in the primary years.

Teaching primary school students requires high PCK, low CK. Quite the opposite for Oxford undergrads.

Blank stares and fuzzy sentiments

Perception is a funny thing. When I tell people that I tutored Oxford maths undergraduates, they marvel at me with an automatic acceptance of my teaching prowess. But when I tell them I now work with primary students, they instead smile adoringly and offer a fuzzy sentiment that barely departs from “Aww, that’s cute”.

There’s a widespread belief that teaching becomes harder as the subject matter deepens. My experience says otherwise. Working with kids has left me with an enduring respect for primary teachers. If you are overcome with warm thoughts at the idea of teaching kids, just consider how you might react when a child stares at you blankly despite your best efforts to impart understanding to them. It’s a daunting prospect, I assure you.

Junaid Mubeen is a research mathematician-turned-educator, working at the nexus of education, innovation and technology. He blogs here, and tweets as @fjmubeen

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