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I am writing an ODEs textbook for second year students and I would like to get inspirations on general good designs on undergraduate textbooks taught in the first two years (i.e. calculus, linear algebra, real analysis and ODEs ) that enhance student understanding .

Q: Can you recommend some design principles that you like seeing or would like to see in a math undergraduate textbook and books that exemplify it?

I was debating whether to put this post here or in the math-educators stackexchange, but I am curious to hear of design strategies seen in research/graduate textbooks that haven't trickled down to undergraduate textbooks. But if it doesn't fit here, tell me and I will promptly remove it.

One reference I found is "Designing Science Textbooks to Enhance Student Understanding " but I would like to hear more of them. Here are some design principles:

  • a key difference with undergraduate students as opposed to graduate students, is that one should spent more time motivating the material. One idea is introducing methods and theorems through examples and especially applied ones (eg. from physics and economics). The design principle here is going from the concrete to the abstract. Di Prima's ODEs textbook achieves this beautifully.

  • I personally enjoyed graduate textbooks that also provided me with short code programs and guided exploratory exercises. Like "Computational Methods for Fluid Dynamics" by Peric etal. It is also done in many ODE textbook such as Boyce's. This is doable with ODEs if you are working with MATLAB, which provides ODE solver packages.

  • In terms of designing exercises, I liked it when the first few questions were broken down into multiple baby questions, which also taught me how to ask questions so as to make a large question more manageable. I saw this in Pugh's real analysis textbook.

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    $\begingroup$ Here's an MO post related to your endeavor, Undergraduate ODE textbook following Rota. $\endgroup$ – Mike Pierce Jul 13 '18 at 5:49
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    $\begingroup$ I think an excellent index and excellent bibliography are essential. I tend to dismiss books that don't have both. You may find, when you come to compile your index, that you end up at least partially re-writing your book, because it will force you to clarify concepts, etc. I think that is as it should be. My favourite bibliography is from Spivak's Calculus, a great undergraduate text. $\endgroup$ – James Smith Jul 13 '18 at 13:20
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    $\begingroup$ This seems to me to be clearly off-topic here, more appropriate for matheducators.SE. $\endgroup$ – Ben Crowell Jul 13 '18 at 23:33
  • $\begingroup$ @Crowell I want to get feedback from other researchers also on techniques found in graduate textbooks that enhance student understanding but have yet to trickle down in undergraduate literature. I personally find that undergraduate textbooks tend to follow a very similar exposition whereas upper undergraduate and graduate textbooks have more diverse styles. For example, I found all the references in Stork's answer very helpful. But if many of you think that this is not a good enough reason, then I will close my own post. $\endgroup$ – Thomas Kojar Jul 14 '18 at 17:29
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I've published a number of undergraduate and graduate science books, some heavy in mathematics, but no true mathematics textbooks. I've thought long and hard about how to design and craft them, and have several professional calligraphers, type designers, book designers in my immediate family, and they (and of course my students) have given lots of great feedback.

I think the first thing any author must address is "why another book?" Because writing a book is such an ordeal, you should really have the sense that you have something important and unique to say, or new viewpoints, that will energize you through the inevitable burden of writing. Much of your design decisions will stem from your (preferably) unique pedagogical views.

My personal writing style (and indeed academic/professional style) is to be as visual as possible. (Try to get your publisher to agree to full-color graphics.) Many students will remember topics better with careful graphics, find topics by flipping through the book faster, and so on. Two of my favorite math book presentations are Visual group theory by Nathan Carter and An illustrated theory of numbers by Martin Weissman, both of which should give you ideas. This latter uses a great $\LaTeX$ style sheet, which I am sure is freely available. Also, be sure to read the master--Ed Tufte--and his great books, such as Envisioning information.

Take time making great figures!! I spent a week programming a three-dimensional Voronoi tesselation (with data points), and as far as I can tell my book Pattern classification (2nd ed., 2000) was the first to have it. Likewise, I wrote the first scientific paper on auto-random-dot stereograms (remember Magic Eye?) and my book Seeing the light was the first to include one. Students remember these!

I very much like separate little sections that work a problem and integrate the material in the rest of the body of the text. You might like to put in little questions within the text—in a different color, separated—to keep the student alert and thinking.

My preference is to have bibliographical and historical material separate at the back of each chapter, not within the body of a chapter. Students don't want to learn from sentences such as: "As Jones and Smith (1988) and later Candace and Tao (2007) showed, the signal..." Don't burden the student with history and citations: The primary material is surely difficult enough.

Another issue is whether you'll teach coding, or use programming, as part of the material. If you want the broadest adoption, write pseudo-code, so students coming with different programming backgrounds can all learn. However, if you are linked to a particular language (Mathematica, Matlab, R, etc.), then post the exact code... with helpful comments.

You will likely test market your book on your students as you write. Ask for honest feedback... including feedback about the design.

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I think the question is too broad, but I'll take a shot at making helpful remarks that are too long to fit in comments.

Ostensibly, the question is about "design principles," but the follow-up examples are largely about writing/pedagogy decisions. So this seems to be a question about how to write a textbook that's good for student learning (broadly speaking) rather than the visual/physical design.

So this question is kind of like the writing analogue of the question "Design principles for good undergraduate teaching". It's really too broad, but maybe people can share some short helpful tips. Maybe this is a question whose answer should be a "big-list".


My list of tips for writing an undergraduate textbook

(off the top of the head version)

List #1: Caveat Auctor!

  1. Teach the class many times, writing notes along the way, before writing a textbook.

  2. Look at other textbooks in the subject closely. What are you trying to offer that those books don't already offer? If you go to a publisher, they will make you go through this process, for good reasons.

  3. Have tenure, or an equivalent combination of secure employment and professional reputation.

  4. Write and publish shorter things, both research and pedagogical.

  5. Think about good writing more broadly. There's Strunk and White's "The Elements of Style". You might read Stephen King's "On Writing" and Orwell's "Why I write" for more thoughts about writing. Respect the craft.

List #2: So you've decided to go for it.

  1. Be clear. Write and edit, again and again, until you can't think of a clearer presentation, proof, definition, etc.

  2. Assign problems to students when you teach, and keep track of which ones worked particularly well as homework exercises, and which ones might need to be clarified, might need hints, and might need to be chucked.

  3. When you have to make a decision, make it intentionally, and keep an explicit record of it (for later consistency).

  4. Sometimes you have to "kill your darlings"

  5. Be receptive to suggestions, and acknowledge the assistance of countless students and colleagues who will find your typos and larger problems along the way. Hire a proofreader if possible to avoid later embarassments.

List #3: Visual design, because I care about it.

  1. Trust the experts, like Knuth. Don't attempt to hack around with typography, spacing, etc., unless you know what you're doing. I used Tufte-LaTeX since I wanted to imitate Tufte's book style and typography, but this isn't for everyone (and see #3 below).

  2. Avoid what Tufte calls chartjunk. This goes for textbookification junk like multicolored boxes around every little thing.

  3. If you're thinking about publishing, beware that very few publishers will give you free reign over how the book is designed and printed in the end.

  4. Take care to smoothly incorporate your graphics with the text. This can include inline graphics and separate "floating" figures. For consistent typography, control over line weights, shades of gray, etc., I really like TikZ/PGF.

  5. Consider illustrations with different purposes. Some might be "visual explanations". Others might be traditional data displays, e.g. graphs of functions. Others might be schematic -- a diagram to say what something is, illustration relationships between things, etc.. Before inserting a picture, it's good to complete the sentence "The goal(s) of this picture is/are..."

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  • $\begingroup$ Is the reason for having tenure so that committees might think you are writing expositions when you could be doing research? $\endgroup$ – Grad student Aug 21 '18 at 20:10

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