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:
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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.
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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.
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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.
Best Answer
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.