To students, threshold concepts (TCs) (Cousin, 2006) are abnormally troublesome to learn. To practitioners their importance lies in the transformation in the "ways of thinking and practising" of the learner as Davies (2006) put it, what we engineers might call acquiring the competencies of our profession. In TC parlance the concepts are said to "be transformative", to possess the power to change the learner's very way of thinking. Meyer (2010) asserted that concepts do not present in a continuum, but fall all-or-nothing into being "threshold" or not. Being both the hardest to learn and the most influential to "identity and ways of thinking" among the ideas within a discipline, it would seem important to both learners and teachers to identify TCs. Once identified, effort can be concentrated on them, and the inordinate learning difficulty anticipated and addressed. The identification proves to be difficult for various reasons (Davies, 2006).
In this manuscript we put forward the small catalogue of concepts we believe to be central to undergraduate analogue electronics, and to be threshold. They were identified in the first place by working with students and observing where they reported difficulty, or had learning troubles exposed through assessment. Later each was evaluated against the five attributes of Meyer & Land (2003). Debate between electrical engineering (EE) educators and instances from the literature have added weight to our selection. Rountree & Rountree (2009) favoured the search for "the ways in which practitioners in related disciplines solve similar problems" offered by Davies (2006) to identify TCs, and we will go even further than this. This manuscript puts our catalogue of postulated TCs up for debate, presents some novel approaches that we are using to identify TCs, and presents some provocative suggestions regarding irreversibility and integration of TCs.
2 POSTULATED THRESHOLD CONCEPTS
We commence by describing the concepts we would offer as TCs, in no particular order, and without attempting to justify the selection. This is to ensure clarity in identification and terminology.
Thevenin's theorem/modelling. Thevenin's theorem is the first example of circuit modelling that students encounter in electronics and circuit theory. It causes learners an inordinate amount of trouble. It is not Thevenin: any model would present the difficulty.
Dynamic resistance/linear approximation. The idea that quantities can have an "AC value" as well as a "DC value" appears first and foremost in electronics in the replacement of something like a diode junction with a resistor subject to the limitation of small disturbances; its so-called "dynamic" or small-signal resistance. The same phenomenon--a tangent as a linear approximation of a curve in a local region--appears as marginal cost in economics and the [beta] of a bipolar junction transistor, and ought to be familiar from differential calculus, but the connection appears seldom to be made.
Reactive power/phasors. Perhaps the most widely-acknowledged TC in EE revolves around the complex nature of impedance and its consequences. The idea of complex electrical quantities is often encountered first by students in the use of phasors. "Reactive power" is a named example that appears commonly in the literature concerning the intellectual effort of applications involving complex quantities. Indeed, it is an idea that continues to cause conceptual difficulty even for experts (Willems, 2011). Like dynamic resistance and the calculus of tangents, this concept ought to come easily on top of complex numbers in mathematics, but the leap is apparently not easy.
Feedback/operational amplification. The operational amplifier (op-amp) is the "simplest" example of feedback. Many practitioners realise that op-amps cause learners a great deal of trouble, but the threshold was there before the op-amp was common. …