Consider tennis and mitosis: clear explanations are not enough.
Mathematicians apparently regard as self-evident the proposition that
the focus of their writing should be to give clear explanations.
No-one would dispute that in mathematics-related writing, or indeed almost any
non-fiction writing, clear explanations are necessary.
But are they sufficient?
Well, sometimes. When I read a Wikipedia article or a research paper,
I don't expect anything more than a clear explanation, and this may extend
to a graduate level monograph.
But let's come down a level, and
consider an undergraduate textbook or a popular science book
(I regard a textbook as intended to teach you to do something,
and a popular science book as intended to plant
some knowledge into your head).
At this level, I assert clear explanations are not enough.
As folklore says:
Learning mathematics is like learning to play tennis:
you learn almost nothing by being
told how to play, you learn a little by watching someone else play,
but you learn most from the experience of actually doing it yourself.
Of course we all know this -- that's why textbooks have exercises and why
teachers assign homework.
But my impression is that authors put most of their effort into their explanations,
and not enough into planning what the reader can do to
use the material in a non-textbook setting.
- teaching Mathematics is easy; it's getting people to learn the stuff
You, a representative Educated Reader, can probably say a couple of roughly accurate
sentences about mitosis and meiosis, especially if I don't require you
to remember which is which!
If your knowledge of this comes from reading a popular science book rather than
taking a formal course, then I judge the book to be a success.
Of course there are prerequisites for becoming a success -- the topic and writing style
must succeed in getting people to read it, and the material should be correct! -- but granted that,
what makes a book have intellectual value rather than being mere transitory entertainment
is that there is some "take home message" that the reader remembers permanently.
So: authors quite properly should put a lot of effort into careful page-by-page exposition,
but also need to remember that to many readers the material will be as unfamiliar to them as mitosis and meiosis
are to you.
Think about the couple of sentences you want the reader to remember for ever,
and mold some exposition around them.
What's my point?
Alas I have no magic formula for actually doing what I recommend above -- I just want authors to
keep these desiderata in mind.
Apropos "take home message" you may have realized that my
choice of the rarely juxtaposed words tennis, mitosis is designed to make this page
memorable to you!