Suppose one were writing a book aimed at high-school students (and their teachers), where "high-school" in the US means grades 9,10,11,12 (where college/university starts at 13). The book is not a textbook, but rather "math enrichment." The book attempts to expose such students to proofs, and to research questions in mathematics. I have questions about the best way to cite the research questions. Here are four alternatives:

(1) With introductory explanation, treat it just like a research paper, e.g., "this was proved in [ABC99]." One point would be train students to understand such references and how to access them (increasingly easy via the Internet).

(2) Instead, use Notes, gathered at the end of each chapter, something like this,1 and then the referenced note says, 1This was proved in [ABC99]. Again, advanced explanation would be needed to interpret "[ABC99]."

(3) Have no interruptions in the text—no citations, no footnotes—but at the end of the chapter in a Notes section, explain the references and history etc, citing backwards to previous pages.

(4) Similar to (3), but have all Notes gathered at the end of the book. This seems to be the current standard in academic treatises.

I'd appreciate your opinions!

  • 2
    $\begingroup$ I like (2) and especially appreciate digital versions in which you can click the number and be shown the reference. $\endgroup$ Commented Oct 18, 2020 at 19:02
  • $\begingroup$ How many students do you know that are good enough in math, have the capacity and the time to read a book and are willing to spend their free time reading a math book? P(good in math)\times P(have time & capacity | good in math) \times P(want to spend his free time reading math|...) \approx 0.08 * 0.1 * 0.4=0.0032\approx 0.3 \%. I was generous with the percentages and I assumed that every student will know about this book, that they are willing to pay, that there's no difference between genders and between races, .... $\endgroup$
    – user5402
    Commented Oct 20, 2020 at 12:22
  • $\begingroup$ @user5402: Certainly the number is small, but the aim is not for mass-market appeal. I remember reading One Two Three... Infinity by George Gamow in high school, and studying Martin Gardner's Scientific American articles. $\endgroup$ Commented Oct 20, 2020 at 13:49
  • 1
    $\begingroup$ @user5402 I think the number is not enormous, but there definitely is a market for well explained mathematics topics that are interesting but don’t require alot of background in mathematics. For instance, 3 Blue 1 Brown on YouTube has over 3 million followers. His comment sections are full of people who clearly aren’t your standard mathematics crowd. You can tell from all the “Man I hated math before but I love this stuff” comments. $\endgroup$
    – wgrenard
    Commented Oct 20, 2020 at 17:00
  • $\begingroup$ @JosephO'Rourke Those books are the exceptions to the rules; most such books don't sell. Also these books (just like "what is mathematics?") were made in a completely different era; now in a few seconds you can find the answer to any question. $\endgroup$
    – user5402
    Commented Oct 22, 2020 at 18:11

2 Answers 2


Research papers require references to verify where your data comes from. Schoolkids don't care where you get your data, extra references or footnotes just irritate them. If the presented information is incorrect, you are the one who should bear the grilling.

Regarding footnotes vs inline references, as a reader - and a former schoolkid as we all were - I prefer short inline notes where it makes sense, like "the theorem bla-bla says bla-bla. Because the proof is too long or complex or needs knowledge of this and that, which supposedly you don't possess yet, etc. we are not going to prove it here. If interested, see [ABC99] for proof". I can read this quickly as I read the theorem and decide whether I want to see the proof. No context switching. [ABC99] will be fully defined at the end of the book in the list of the references.

Footnotes tell me nothing aside that there is some additional information pertaining the prior sentence. What kind of information is it? Do I care? I would not know until I suspend my reading to look at the footnote. And by the way, I am used to Arabic numbers to mean footnotes, that is, notes on the same page. Notes at the end of the chapter or the book are usually denoted with Roman numbers, but I think there is no hard rule on that.

Anyway, unless you want to plop a whole proof into the footnote so it would take half a page in tiny typeface, I'd rather have in on the same page than at the end of the chapter or the book, because switching between the main text and the footnote does not require to leave the page. This requires careful formatting to ensure that footnote is always on the same page where it is referred to. Choosing between notes at the end of the chapter and the end of the book I prefer the latter, it is easier to flip to the back of the book than to search where the damn chapter ends (I don't use bookmarks).

And please do not mention something that is defined two or twenty pages later or not defined at all, Core-Plus Math is notorious of this, very irritating. I expect to see definitions of all important things before these things are used.


My advice is to use the least obtrusive method, probably 4, from your list. And to also minimize the number of citations. What you are considering a feature (oh goodie, we get footnotes) is really more of a bug or at least a pill for someone not already used to wading through research papers. You've got enough headwinds already by combining two difficult topics (proofs and research areas) rather than picking one. Don't make the citation density the camel-back straw.


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