# Writing a Calculus textbook for a course I am creating

Using LaTeX, I am attempting to write a single-variable calculus textbook that gives the reader an understanding of calculus and its applications without a lot of the fluff I have seen in many other textbooks (I should mention that I am a rising high school senior going into multivariable calculus this coming year). In my opinion, if a student is impressed by the high quality, ease of understanding, and straightforwardness of the textbook, then with good teaching, the student would successfully learn the material as opposed to memorizing facts.

I'm not just writing a textbook. I also plan to hold a class online through Google Hangouts or a similar medium that goes through the textbook as well as practice exercises.

I have already assembled a group of friends to go through the textbook as I write it. These people range from people who have not yet started AP Calculus AB but were decent in Pre-Calculus (who will read it for understandability), people between Calculus AB and BC (who will read it to make sure I haven't overcomplicated it), and people who have completed Calculus BC and above (who will read it for correctness and suggestions on additions/deletions).

Does the idea for this class sound viable? Also, how would you recommend I maintain the organization of the documents for each chapter/lesson?

• Your second question is technical, I think, and probably doesn't belong here. First question - yes. I hope your class is great fun. I would be interested in seeing what you do. Aug 8 '15 at 3:05
• this "fluff" you mention, is that the applications? Aug 8 '15 at 18:35
• Tangential to your question: One thing I've noticed is that I'm extremely bad at teaching things which I've learned in the last year or two because I miss all sorts of little things and I don't have enough context to "do no harm" to the students' future selves. Your team may benefit from a math grad/postdoc/professor, not to do daily stuff but to be something like a consultant dealing with the little things: making sure reasonable-sounding things are actually true, you're making notation choices that won't hurt them in the future, etc. etc. Aug 8 '15 at 19:51
• I'd like to eco @JamesS.Cook's question: Could you elaborate on what you consider 'fluff'? It may be that you lack the context and experience to recognize the value of something that really should not be left out. (Of course, if that's true, it is questionable whether your text's future users would learn much from that material either, so maybe it is fluff after all.) Aug 9 '15 at 14:05
• Just out of curiosity (and since this topic has gotten some activity recently): how goes the writing? Did you finish in a few months, or were the commenters correct about the expectation that this might take a few years? Aug 13 '19 at 3:41

I am fully aware that much of this answer does not belong in this forum. On the other hand, I believe that many mathematics educators are interested in producing materials and would benefit from this advice, which I have gained over a few years of authoring. All in all, I think mathematics educators generally would benefit from these ideas so I will post them.

I have been writing resources for my students for a number of years and am currently experimenting with collating things online into a textbook/forum environment. For this I am using my own domain name and a WordPress website. I think that the idea of writing your textbook in $\LaTeX$ is a great idea because it will ensure a high quality product. May I suggest a couple of ideas for you:

1. Consider using ShareLaTeX as your editor, because it becomes easy to collaborate, to work on your document in various locations with no tex installation, and is quite nice and easy to keep organised. The two problems are: no SageTeX; and only the tex files are autosynced (the pdf can be previewed in browser but must be downloaded manually to keep, not a huge deal though)
2. Use the subfiles package for $\LaTeX$ and write each chapter as a separate .tex file - this improves preview compilation time and allows you to focus on the structure as well as the content more easily
3. Write your preamble in a .sty file separately and call it in the preamble of the main document. Your subfiles will automatically pick up the style from the main document. This means you only need to make changes to the preamble in one place.
4. Download geogebra, use it to draw your pictures, then export as TiKz and copy/paste into your document instead of drawing things manually. This will save a lot of time (the code often requires tweaking but it is still faster).
5. Have you thought about how your textbook might display online? Will you use a pdf embedder or some kind of TeX2html solution? How will students refer to specific parts of your textbook? Can they copy/paste quotes? Can they right-click to see the TeX commands you have used to produce equations? This is an interesting post on the topic.
6. In your forum, can they use TeX to write mathematics online? If not, how else can they communicate mathematically using their keyboards? Check out this solution - it is impressive what they have done.
7. From a safeguarding point of view, how will you make sure that the online communication remains friendly and is not used for trolling/bullying? Does your school have guidelines for people creating online forums? Could you be held accountable for anything bad that may happen online?
8. Writing problems for your textbook will be one of the largest tasks. Get help with this. Generally, your main concerns should be sequencing (later problems must depend upon earlier ones, not the other way round) and comprehensiveness. Be sure to account for special cases or else have a rationale for omitting them. If you can do a problem with variables from $\mathbb{N}$, what about positive fractions? then what about negative integers? Then what about negative fractions? Every changeable aspect of a problem adds a new 'dimension of variability' and it is important to account for them all. You would be amazed at how many of my students can factorise $x^{2}+x-2$ but not $-x^{2}-x+2$. How will you train readers to prepare for variation, to be adaptable?
9. Write a manifesto for your textbook - what is it you don't like about others? Be specific: why is the existing textbook deficient for the purpose of learning mathematics? How will yours be different? Imagine that paper is a scarce resource and you are having to justify printing your book versus an existing textbook - why should they print yours and no-one else's? This will help your book remain distinct and not fall into the trap of lazy writing (it is very, very tempting to gloss over certain points so you can get to the stuff you already have good ideas for)
10. Try and read a few academic papers about (a) the use of mathematics textbooks and (b) the problems students commonly face with calculus. There are a lot of people doing a lot of research which largely gets ignored by textbook authors (at least here in the UK, where the standard books are merely reference guides). If more textbooks were research based, the they would be far more relevant to their purpose. For example here or here are two important UK based mathematics education research journals (I think you can get a free account and get a few articles per month for free)

Yours is a sprawling question which pretty much draws on everything the mathematics educator has to worry about. Don't be daunted by this, but do make sure that you start your work in the best way so that you minimise redrafting. It is better to have a clear idea about your methods and purpose before you begin or else you duplicate your workload with every revision (I have learned this the hard way). I really hope your book sees the light of day and that the online community around it becomes fruitful for all those involved.

• Wow, thanks for all of this information and advice. 1. I am using something very similar to ShareLaTeX: Overleaf. 4. I have Geogebra, but I didn't know it could be used to export to TikZ. This will be useful, as 80% of the time I have spent has been spent trying to learn TikZ. 5. One of the things I think is advantageous about having separate documents to be concatenated later is that on the site where the students can access materials, I will put each separate chapter on a separate page for each lesson we have, in addition to the entire book. Aug 12 '15 at 13:53
• 6. My original plan was to use a LaTeX extension in [Slack]() but I was not originally satisfied with this idea, as it requires the student to know LaTeX, which is not something I feel I can expect. 7. A person with moderator powers on Slack can see and delete comments of that type. Abusive students would be dealt with accordingly. 8. My first class will have the lower number of problems as I described in my discussion with James, but I will continue to add problems to the book, but still not have an excessive number of them. I want to have my first class by the time I graduate. Aug 12 '15 at 13:59
• I hadn't heard of slack. Maybe using it in combination with some WYSIWYG type app such as codecogs.com/latex/eqneditor.php could be a solution - your site could have a permalink to the editor in the menu bar. I find that typesetting equations is one of the biggest barriers to online learning at this level. [As it happens, with a live class I once gave a lesson on LaTeX as an enrichment at the end of a pre-university course and they loved it, especially the fact that they could put it on their university applications and CVs (resumés).]
– Shai
Aug 12 '15 at 14:37
• It looks like I forgot to add the link for Slack. Here it is. I agree that typesetting equations presents a challenge, and that link seems like it could work well. Something else I could do is explain the LaTeX that would be needed in a certain week in a 10-minute prologue to the class period. At the moment, I am working on the parts of the site from my end, which includes setting up the website and writing the textbook. Aug 12 '15 at 15:40
• I use overleaf for our senior capstone course. In principle they are supposed to work collaboratively. In practice, it's just nice because the templates are there and it is so easy to get started. I love that site. Aug 12 '15 at 22:17

Although I admire the enthusiasm of the OP, I'd like to make a point similar to @guest's post: There are quite a few calculus textbooks out there. Each thumbnail below is the cover of a calc textbook. I would be wary of investing too much time in such a crowded environment without investigation of how it is already populated.

Also, there are free, high-quality online calculus courses available, e.g., this Coursera course. Not quite as crowded an environment, but still not easy to compete with the professionals.

• I wonder how many editions of Stewart are in the graphic above. That said, I do agree that there are at least dozens of quality commercial calculus texts. Of these quality texts, each one fails in numerous ways because they face the marketplace which puts negative pressure on excellence. (I'll blame the market rather than authors since it is less unappreciative of their already monumental effort). Calculus texts try to cover so much ground for such an math-illiterate audience, it is not an easy task. I still think, better books can and will be written from experienced teachers... Aug 14 '19 at 14:11
• @JamesS.Cook: Yes, maybe 11 thumbnails for Stewart. But still... Aug 14 '19 at 14:14
• Also there are hundreds, if not thousands, "lecture notes" on line, and also full books (in everything but paper publishing), and even open textbooks (search for it). Feb 14 '20 at 17:04

You mention the desire for a book with content and exercises, but less fluff. But also, seem not to have done a lot of comparison research (other commenters recommend this to you, also). Two specific texts to look at would be: Schaum's Outline and Granville, et al. Both have a short, simple presentation. And there is definitely a student market for books like this versus the doorstops that entice purchasing committees.

Even if you still feel the need for something different, better, I think what you do will be better if you have looked at other texts with the same objective as yours.

P.s. Yes, I realize the question is old, but the intention of this Q&A site is to create archived Google-able answers for others with similar questions in the future. For instance, the "related questions" list freely pushes up other questions and there is no time limit on responses.

This indirectly answers a number of implicit or explicit questions, but obviously not this entire huge question; hope it's useful, if you are still trying this project, but certainly to others wishing to try similar projects.

• A number of upcoming (free) textbooks, including at least one already-used free calculus book, are using or switching to the "write once, read anywhere" philosophy of PreTeXt. It has a definite learning curve, but you may find it a useful way to keep everything together.

• Another online LaTeX editor (and full Linux environment) is SageMath Cloud. Overkill for just TeXing but if you want to start adding computation or easily adding certain graphics then using it with SageTeX works very nicely.

• Perhaps most importantly, if you really want to crowdsource (that wasn't 100% clear in your post) things, you may wish to post your code in an online distributed repository for people to comment on or even submit change requests, such as Github or Bitbucket or Gitlab. They may seem intimidating (and I wouldn't ask anyone to learn git as a first revision control system, but apparently nowadays everyone else disagrees) but is very, very useful for open projects (and others, as long as you only let authorized people see the source). Even trivial typo fixes like this one I recently noted on a book I've been reading can be easily crowdsourced this way (though it's no guarantee they will be!).

In my opinion, if a student is impressed by the high quality, ease of understanding, and straightforwardness of the textbook, then with good teaching, the student would successfully learn the material as opposed to memorizing facts.

may work for very motivated students as yourself, but you will soon find that the "with good teaching" part is the really challenging piece. And perhaps you will someday be one such great teacher, good luck!

First of all, this is a huge task. Plan on spending a few years on this.

Second, Newton said he was able to see farther because he stood on the shoulders of giants. Check what is available already (here are plenty of good to excellent free textbooks and lecture notes around).

If the above doesn't unearth something close to what you want, by all means go ahead. But when writing a totally new/different textbook, make sure (by asking e.g. here, by reading up on pedagogy, by doing limited experiments) that your approach really is better.

• I understand the magnitude of my project, though I don't expect to take a few years; I expect to finish the textbook in several months. I say this because I do not (initially) intend to go to publication, nor have the book distributed on a large scale. I strive to present the information in a concise but useful manner. With regards to your second point, I wish to add my take on the material to the pot; if other people have done something similar, so be it. I wish to contribute something that is my own (and I also don't want to violate copyright laws). Aug 12 '15 at 2:57
• As for the methodology of the course (having a textbook and going through it in online moderated discussions), I am taking a number theory course offered by Stanford in almost the exact same form, and I have found it to be extremely successful in teaching me the material. I believe it depends on the student's learning style, but if a student thinks they would adapt well to my style of teaching and has a passion for learning math, I would be glad to accept them into the course. Aug 12 '15 at 2:59
• "I expect to finish the textbook in several months." Progress report? Oct 21 at 7:28