A few months ago, I asked a question on teaching engineers mathematical thinking skills over at MSE. I also asked it a little later at The Mathematics Teaching Community, but traffic on that site is very low. (Update: the site is currently unavailable, but I will leave the link up in case it comes back online.) Now that we have MESE, I would like to ask it here, after first having checked on Meta whether doing so would be considered acceptable.
Preamble: In my experience, many introductory engineering mathematics textbooks these days tend to skip proofs and discuss logic only in the context of digital electronics. On the other hand, I can imagine that engineering students (and others) could benefit from developing basic skills in mathematical thinking* beyond the cookbook approach. See, for instance, Keith Devlin's Introduction to Mathematical Thinking course. I hasten to add that many engineering mathematics textbooks do have strong points, such as numerous examples of mathematics applied to solving engineering problems. For the record, I currently use Engineering Mathematics: A Foundation for Electronic, Electrical, Communications and System Engineers by Croft et al. The material covered includes derivatives, integrals, complex numbers, matrices, differential equations, Laplace transforms and Fourier series.
Question: Can you suggest engineering mathematics textbooks that do contain material on mathematical thinking and/or share your experience(s) in teaching introductory mathematics to engineering students where you went beyond the cookbook approach?
$\text{*}$ I realize that I haven't defined mathematical thinking. Keith Devlin addresses this in his blog entry What is mathematical thinking? In particular, he writes:
Mathematical thinking is a whole way of looking at things, of stripping them down to their numerical, structural, or logical essentials, and of analyzing the underlying patterns. Moreover, it involves adopting the identity of a mathematical thinker.
See also Terry Tao's There's more to mathematics than rigour and proofs and the anonymous answer to the question What is it like to understand advanced mathematics? However, please note that I am focusing on introductory courses and basic skills in this question.
Related question: Is there any difference between teaching calculus for math and engineering students?