One common piece of advice often given to first-year maths undergraduates is to read mathematical texts with pen in hand. Mathematical writing is dense, so you will undoubtedly need a separate piece of paper to work out the details.

As an extension of this, I recently tried overannotating a set of lecture notes. I printed the full set of notes, and armed with my pencil, I set out to destroy the pages.

Here’s why. When trying to read ordinary books ‘well’, I annotate abundantly: for normal fiction, I typically leave some kind of mark on each page; for poetry, I shred the page. So, if I wanted to keep the same annotations-per-flop rate, this meant annotating much more. Overannotating was worth a try.

The delusive thing about mathematical texts is that they’re full of italics and boxes, so it feels as if someone has already annotated the text for you¹. I tried ignoring the styling, sometimes underlining the bold-faced ‘Theorem’, ‘Definition’ and ‘Lemma’, just to make the text mine.

No human faces #

Here are some of my favourite ways for conquering a page:

• Explaining: Adding explanatory notes, whether of ’trivial’ claims or anything explicitly left to the reader, is the obvious way of conquering a page. One can also underline motivations and intuitions that particularly resonate. I found it quite useful to give a one-sentence summary of an earlier exercise or lemma invoked in a proof, so I wouldn’t have to look up the result upon rereading (unproductive friction); this was also a good way of revising the previous result².
• Renaming: Name any unnamed theorems; rename named theorems. One can also come up with more descriptive aliases for definitions. If you have a secret, internal language for every result in the script, that’s a good thing – it means you’ve appropriated the material.
• Celebrating: Sometimes it feels good writing ‘OK’ in the margin after verifying a particularly tedious result.
• Memorising: One can also annotate with a view towards memorisation. If some mnemonic springs to mind, note it down. When reading proofs, one can either underline The Observation, or, in the case of several proof steps, enumerate them.

That’s just to name a few lines of attack. There’s plenty of room to be creative – you could also just take your favourite technique for understanding a piece of maths, and do that in the margin rather than in your notebook³.

My rules were quite simple: adding anything except for human faces was allowed. As a guiding principle, I tried annotating to the point where someone else could see what I’d read. But it was largely anarchy on the page.

This way of reading proved surprisingly fun, and it feels as though it helped for retaining the material. Ironically, seeing the mess I’d made also turned out to be rather satisfying – mess became a way of tracking progress.

Obviously you can overannotate a mathematical text too, but it takes a lot to get to that point. I certainly haven’t gotten there yet.

Nodding and doodling #

While you can read an ordinary book for an audience without taking your eyes of the book, the same would never work for a mathematical manuscript. At a good maths lecture, there appears to be more back-and-forth between the lecturer and the audience. The lecturer needs input form the audience to pace the lecture.

Perhaps this provides further evidence in favour of annotating more. Scribbling in the margin is like much like nodding or passive aggressively staring into the void – it’s a way of engaging in the conversation.