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.