One of the marks of a successful problem-solver is a habit of applying general principles to every new problem encountered. One of the marks of an ineffective problem-solver is a fixation on procedural steps. Sadly, most of the students I have encountered as a technical college teacher fall into this latter category, as well as a number of working instrument technicians.
Teachers share a large portion of the blame for this sad state of affairs. In an effort to get our students to a place where they are able to solve problems on their own, there is the temptation to provide them with step-by-step procedures for each type of problem they encounter. This is a fundamentally flawed approach to teaching, because a set of rigid procedures only works on a very specific set of problems. To be sure, your students might learn how to solve problems falling within this narrow field by following your algorithmic procedures, but they will be helpless when faced with problems not precisely fitting that same mold. In other words, they might be able to pass your exams but they will flounder when faced with real-world challenges, and you are utterly wasting their time if you are not preparing them for real-world challenges.
I am as guilty of this as any other teacher. When I first began teaching (the subject of electronics), I was dismayed at how difficult it was for students to grasp certain fundamental concepts, such as the analysis of series-parallel resistor circuits. Knowing that I had a very limited amount of time to get my students ready to pass the upcoming exam on series-parallel circuits, I decided to make things simpler for my students by repeatedly demonstrating a set of simple steps by which one could analyze and solve any series-parallel resistor circuit. Fellow instructors did the same thing, and gladly shared their procedures with me, including tips such as the use of different pen colors (black for drawing wires and components, red for writing current values and directional arrows, and blue for writing voltage values and braces) to help organize all the work. The procedure could be long-winded depending on how many nested levels of series-parallel resistors were in the circuit, but precisely followed it would never fail to yield the correct answers. Students greatly appreciated me giving them a set of step-by-step instructions they could follow.
The fallacy of this approach became increasingly evident to me as students would request repeated demonstrations on more and more example problems. I remember one particular classroom session, after having applied this procedure to at least a half-dozen example problems, that one of the students asked me to do one more example. “Are you kidding?” was the unspoken thought rushing through my mind, “How many times must you see this demonstrated before you can do it on your own?” It suddenly occurred to me that my students were not learning how to solve problems – instead, they were merely memorizing a sequence of steps including keystrokes on their calculators. Despite all my effort, the only thing I was preparing them to do successfully was pass the upcoming exam, and that was only because the exam contained exactly the same types of problems I was beating to death on the whiteboard in front of class.
What I should have been doing instead was presenting to my students only the general principles of resistor circuits, which may be neatly summarized as such:
Then, with constant reference to these principles, I should have challenged students to identify where they could be applied to circuits, beginning with the simplest of circuits and progressing to ever-increasing levels of difficulty.
It was not as though I had failed to present these principles often enough, nor that I had failed to demonstrate where these principles applied in the procedure. My fault was in giving students a comprehensive procedure in the first place, which had the unintended consequence of drawing their attention away from the fundamental principles. The simple reason why a step-by-step procedure makes any problem easier to solve is because it eliminates the need for the student to apply general principles to that problem, which is the very thing I my students actually needed to learn. To put it bluntly, a comprehensive procedure “does the thinking” for the student, because the application of general principles is already pre-determined and encoded into the steps of the procedure itself. What we get by robotically following the procedure is only an illusion of problem-solving competence. The real test of whether or not students have mastered the principles (rather than the procedure) is to check their performance on solving similar problems of different form, where the rote procedure is not applicable.
In order to teach students to approach problem-solving from a conceptual rather than procedural perspective, you must insist students show you how they make the links between general principles and the specifics of given problems. A useful tool for doing this is to have students maintain a notebook identifying and explaining general principles in their own words. You may choose to allow students the use of their own notepage or notecard on exams, as an incentive to tersely summarize all the major principles they will need to solve problems on exams.
An inverted classroom structure is well-suited for the encouragement of principle-based problem solving, in that it affords you the opportunity to see how students approach problems and to continually emphasize principles over procedures.
