A writer friend has said that her book-length manuscript has arrived on the page with scarcely any chapters at all: should she put them in? Terry Pratchett doesn't, says another writer. A fellow workshopper was really bothered by how my novel (The Mathematics of Love, since you ask) had several parts to shape a bigger architecture, but not an equal number of chapters in each. One highly successful writer of light women's fiction doesn't put the chapters in till she's written the whole thing, because only then does she know where they should be. Whereas I plan in chapters right from the beginning, like a skyscraper lift-shaft, built round the crane, and round which all else is built, and Scrivener makes that easy. But it also makes the write-it-all-out-and-decide-later method easy.
So what's going on? The question of when (and if) you put a chapter break is really one about what chapters are for, in a novel. There is a rhythm and shape to our experience of a novel - as I was exploring here - and chapters are central to it. And to some extent most of these also apply to the bigger breaks into parts, and the smaller breaks - asterisked, or just double-line-spaced - within a chapter. So these are some of the ways that a break might help to shape that experience: some reasons for whether and where to put one:
1) the reader needs a break - a reason to turn the light off or stop and get up. If Poe's definition of a short story is one that the reader can read "as a sitting", then a long story - a novel - must be one which is too long to be read at a sitting. Though you could always take the book with you, to the bathroom ...
2) the reader needs a break, a pause, to absorb what's gone before, before they get embroiled in what's about to happen. This could either be what I've taken to calling a Quiet Pause - a moment of reflection and understanding of what the stuff you've just read might mean for what you'll read next. Or it could be a Loud Pause, where the scary implications of the last scene have a moment to flower into full Triffid-hood in your imagination.
3) the writer wants the reader to have sense that time passes or we change setting. It uses the reader's experience of reading-time, in a tiny way, to evoke a sense of time passing in the events in the story.
4) the writer wants to change point-of-view. Since I'm a great believer in and advocate of the moving point of view, you won't get me to agree that this is a good reason - if it's the only reason - to break a chapter. But I do recognise that not everyone is confident in handling a moving point of view, and if you want to play safe by your more narrow-minded editors and teachers, and do it this way, I wouldn't dream of stopping you.
5) the reader doesn't need a break - the last thing they want is a break - but the narrator is whisking them away nonetheless, to show them something else and so prolong agony of waiting to find out What Happened Next ... This is harder to make use of if you have an internal, character-narrator, since the reader's more likely to feel cheated by the character deliberately witholding what happened next, rather than the tension coming about quite naturally from the need to catch up with the action in what stage directions call "another part of the field". It also, on the whole, doesn't work to break a chapter and then have the action pick up again at exactly the same point and place: readers feel that as an artificial cranking-up of the tension - again, a bit of a cheat.
6) it's the end of the scene. Some novels (specially novels by those for whom film is the primary narrative form) essentially have one scene per chapter, like a train made of carriages. The chapter break is the draughty moment between the really big, real-time, full-show big moments of change and, that's also how the train bends round corners and curves up hills.
7) this moment has big significance: it's where the EastEnders drum-roll might come in. Like enjambment in poetry, the last few lines of a chapter gain extra importance as they linger in the air while we turn the page, and because of that, experienced readers tend to read a Loud Pause in even if the actual action isn't all that dramatic. The risk for the writer is that you get addicted to the drama of the drum-roll, and habitually jerk us away to the next scene, and the novel loses its sense of continuous narrative and becomes a collection of abrupt chunks. The quieter, more fluent narrative move out of one stage of the story to the next are harder to write, but sometimes much more effective because they take the reader where you want them to go. And anyway, we're not scriptwriters.
8) the writer needs to show the bigger architecture of the story: not so much "This is where the story pauses" as "This is where the story enters a new phase." This is just about the only time when having a chapter-break in the middle of a scene might make sense, and even be rather effective in exploiting the more experienced reader's awareness: if you want to mark, very clearly that this was the big moment of change.
And that's it. I can't think of any more reasons to put a break into a narrative, but maybe you can - and if so, do please put them in the comments.