# Teaching engineers mathematical thinking skills

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 engineers (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.

I would be very interested in learning about either engineering mathematics textbooks that do contain material on mathematical thinking or your experience(s) in teaching introductory mathematics to engineers where you went beyond the cookbook approach. Input from engineering students and practising engineers is also welcome, as are contributions from those involved in other fields in which mathematics is applied.

(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.)

• Subjects? For vector calc. I'd recommend Vector Calculus, Marsden, Tromba for one with examples and such, but another great one (my personal favorite), sliiightly less for engineers is Calculus in Vector Spaces, Corwin, Szczarba. – GPerez Jan 11 '14 at 14:00
• While I am very interested in the answers to this question, I think this caveat is important: most can't see beyond the one-semester horizon. It is likely to be attacked with barrage of questions like "Will this be on the test?" or "What are real life applications of this? Will I ever use this in my life?" Sometimes I feel these questions are rhetorical, they don't expect an answer, they just expect to fool themselves that it isn't important. – Mark Fantini Jan 11 '14 at 17:42
• @GPerez: Thanks for the suggestions. I've edited the question to include the textbook I use at the moment, along with some of the main subjects. (Please note that the list isn't exhaustive). – J W Jan 11 '14 at 17:47
• @Fantini: Good points. Of course, it might be possible to include such material on the test, with suitable care. For applications to real life, perhaps this blog post by Keith Devlin might shed some light. – J W Jan 11 '14 at 18:09
• You may want to look at Gilbert Strang's Introduction to Applied Mathematics. I have no idea how well it would fit with what you're teaching, but it could be a useful text to have on your shelf for reference. – Dave L. Renfro Jan 13 '14 at 16:33

In this context, the common misconception of most ($\textbf{Edit:}$ "most" is not the right word here, let's say "some") mathematicians is that they think engineers need to know where Kirschoff's law is coming from a deep mathematical perspective (whatever that is), so that they will e.g. be able to analyze complicated resistor networks constructed via infinite superpositions of $K_5$ graphs. Believe me they don't. They are also usually not interested in the existence of non-measurable functions etc., because (usually) they do not need it. A similar analogy is that a number theorist, despite being a mathematician, perhaps does not bother herself with the existence of natural numbers (I am not a number theorist, so this may be wrong).