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.)

• I realize that I haven't defined mathematical thinking. Keith Devlin addresses this in his blog entry What is mathematical thinking? 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.

Solution 1:

The assumption of this question, that engineers do not have the "basic skills in mathematical thinking" is wrong. Engineers have the same three steps (that Terence Tao has described in his blog) in their own fields. A simple example for the special case of an electronics engineer might be the following: Pre-rigorous stage: Have a simple light-bulb circuit, perhaps with a switch, imagine a current flowing, you open the switch the light-bulb goes off, etc, simple experiments even with transistors perhaps. Rigorous stage: Kirschoff's law, transistor input-output relationships, operational amplifiers, etc. Post-rigorous stage: I once witnessed the following. A huge circuit does not work for some reason that we could not figure out. A seasoned professor came, had a look for a few seconds, and just suggested, "throw a capacitor in there," and surprise surprise, it worked. He did not bother himself writing down the equations and solving them.

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).

On the other hand, I also know engineers who know a lot of "high-level" mathematical tools, simply because they need those tools.