I have two engineering degrees, each with a mathematics minor. Wish I could say that I chose that minor because “there’s a great deal of mathematics in engineering.” Not really… I took mathematics because generally it was the easy “gut” course any semester; mathematics learning came very easy – at that time, much easier than history or philosophy or similar.
Move forward in time fifty years or so: I’m now an emeritus engineering professor after almost thirty years at the University of Connecticut, having eleven years earlier in industry, and sprinkled with a variety of consulting opportunities. How many differential equations did I solve in this time? ONE that I can recall. How many integrals did I do? Very few unless you consider the easy ones (such as the integral of Sin(x)dx or xdx). BUT how many times did I use the “basics” of mathematics (calculus)? I.e., “integral of a function equals area under the curve.” Very often, routinely… FIRST PERSPECTIVE: Unless one is teaching mathematics, only the fundamentals of calculus will be important. Too bad the math departments, textbook authors, and faculty don’t think about that…
Now, thinking back on my faculty period. For students in my classes or for my advisees, I was always eager to facilitate their learning. With my mathematics minors, I was frequently able to assist with clearing up confusion – in their math courses or for the math considerations in their engineering courses. The most common difficulty that I identified: When their were multiple paths to use, they had no idea how to decide which one to use! They knew how to use one if selected for them; but no clue how to choose. Learning this, my facilitating was quite straightforward: help them figure out what to look for in order to optimize the likelihood of choosing correctly. SECOND PERSPECTIVE: It’s not enough to facilitate the learning of procedures; optimizing their use must include selecting the best procedure to use. Too bad math departments, textbook authors, and faculty don’t think about that.
My final thoughts for this blog relates to the context of mathematics use. Consider the components WITHIN mathematics: There are the math facts (e.g., 1+1=2 or the previously noted “integral equals area under the curve”), the math procedures, choosing the best procedures, using those procedures correctly, and assessing the outcome for correctness. I’ve already noted the issue with choosing the best procedures. And assessing seems to be expected routinely but not really common among students. The other three components are staples of most math courses and textbooks.
BUT, too bad most mathematics courses, even so-called APPLIED mathematics courses, don’t contextualize the mathematics. Before “doing the math,” the situation encountered has to be understood and MODELED – developing the appropriate mathematics equations, etc. to enable analysis. After “doing the math” then, one has to use the math outcomes to answer the questions identified in understanding the situation. THIRD PERSPECTIVE: There is very little real-world situations included in mathematics courses to assist with gaining experience as well as motivation by the students.Too bad math departments, textbook authors, and faculty don’t think about that.
So the subsequent courses (such as engineering or economics) then should build on robust mathematics learning. In dealing with situations, the student, and later the employee, seeks an overall outcome that is SUCCESSFUL – the efforts are USEFUL in dealing with the situation. Note that I personally never use “correct” as because of uncertainty, assumptions, approximations, etc., we don’t ever know the right answer; the best we can hope for is useful input to dealing with the situation. FOURTH PERSPECTIVE: Too often, the source of “non-usefulness” of efforts are the non-math components of those efforts: the modeling and/or the interpreting. Too bad mathematics departments, textbook authors, and faculty (AS WELL AS THE OTHER “USER” DEPARTMENTS) don’t think of that!!!