# Mathematical thinking skills for engineering students

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.

• I can only recall once taking a course in which the textbook was oriented around engineering. The textbook was "Fundamentals of Complex Analysis with Applications to Engineering and Science" (amazon.com/…) but, if memory serves, the class spent more time using the professor's own typed notes than thinking about applications; thus, I leave a comment rather than an answer. Nevertheless, you might take a look at this quite readable book and see if it accomplishes what you wish. May 24 '14 at 8:25
• At least in computing/programming, all reasoning about a program is "mathematical thinking" May 24 '14 at 22:29
• @vonbrand: In particular, there are formal methods approaches to software and hardware engineering.
– J W
May 25 '14 at 6:32
• Nothing extremely constructive, but I should note that your experience may be limited to your locale. Traditionally in francophone locales engineering mathematics courses do not really lose much in rigour compared to their "basic sciences" counterparts. In particular, what is "proven", "heuristically justified", or "merely asserted" are often clearly demarcated. Though whether the students actually absorb the difference is up for discussion. Unfortunately, the French tradition also involves professors writing their own lecture notes instead of relying on textbooks, so ... May 27 '14 at 10:54
• ... it is a bit difficult to give concrete recommendations of textbooks as you asked for. (And FWIW, the courses I am most familiar with are differential geometry courses for civil/mechanical engineers and material scientists: geometry may lend itself more naturally to developing mathematical thinking than other subjects.) May 27 '14 at 10:56

I come at this question both having been a computer science student in undergraduate but also having studied mathematics education in graduate school. In my computer science program, they spent very little time actually teaching us about the process of solving problems - what little we learned about this came from trial and error. We had a lot of really mathematically intense classes, but almost no training in mathematical thinking.

We did have a class on techniques of proof, which was called "discrete math" and included combinatorics and things like series, sequences, and recursion, and recurrence relations as well. So in looking at your question, I looked around and found an open textbook for a discrete math class for engineers: http://faculty.uml.edu/klevasseur/ADS2_zips/ADS_Fundamentals_V2-0.pdf - you can probably find similar things in other textbooks on discrete math - that tends to be where formal mathematical thinking is taught to engineers.

By contrast, in my math education program, we spent most of the program focusing on mathematical problem solving and looking at the process and heuristics of how people solve mathematical problems. So the other thing I searched for was about teaching "problem solving" and "mathematical problem solving" to engineers. Here's the two things that seemed relevant:

A chapter on teaching problem solving to engineering students from a book on teaching engineering: https://engineering.purdue.edu/ChE/AboutUs/Publications/TeachingEng/chapter5.pdf

A study of how some engineering students used mathematical problem solving in their capstone courses along with a lot of theoretical reflection: http://monicacardella.com/mc/wp-content/uploads/2012/06/cardella2005_ASEE_engineeringstudentsmath.pdf

This is not explicitly an Engineering Mathematics textbook, but take a look at The Art of Problem Solving's Calculus text.

The AoPS textbooks are intended for students with a strong interest in mathematical competitions. As such, they include a lot of "non-routine" problems taken from past problem solving competitions, or based on such problems.

For a "Calculus for Engineers" course, I'd probably stick with a more traditional textbook and supplement with materials from the AoPS text.