Copyright © 2015 jsd
Let’s start with a few basic, well-established ideas:
This comes as a shock to many students. All-too-often, they’ve been taught, year after year, that if they just rote-memorize some glib definition and regurgitate it on the test, they’ll get a good grade. However, in the real world, dictionary-style definitions leave a lot to be desired. Students need to learn this, the sooner the better.
In particular, many of the most important concepts (in physics and elsewhere) are so fundamental that they simply cannot be defined in terms of anything more fundamental. One place where you see this with particular clarity is in Euclidean geometry, where the fundamental objects – points, lines, and planes – are emphatically and explicitly not defined. They acquire meaning from how they are used, and not otherwise. In physics, energy is very much in this category: energy “is” as energy “does”. It is vastly better to learn how energy behaves than to rote-memorize some verbiage about what energy allegedly “is”.
Knowledge can be thought of as a high-dimensional lattice of nodes connected by arcs. Concepts are the nodes. A great deal of knowledge is stored in the arcs connecting the nodes.
This idea (3) is related to the previous idea (2) as follows: On the first turn around the spiral, you lay out a bunch of ideas. They are not yet connected, because there is nothing to connect them to. On the second turn around the spiral, you start building up connections.
Principle (4a) speaks of primacy as in emphasis, whereas (4b) speaks of priority as in chronology. I strongly prefer (4a). I say the ideas should be first and last. The concepts should be the foundation below and the finial on top.
Even more importantly, even though principle (4b) sounds nice in theory, it is mostly impossible in practice. It is inconsistent with other, more-important principles. At best, it is open to multiple interpretations, some of which are approximately correct and some not.
Consider for example the term “atom”. There are fat books on the subject of atomic physics. It could take years to develop a professional-grade understanding of how atoms behave. For most of that time, the term “atom” is attached to an inchoate, highly imperfect concept.
The usual dictionary definition of “atom” is “the smallest component of an element having the chemical properties of the element” – but that’s deplorable, because “element” is defined in terms of atoms. You could describe an atom in terms of protons, neutrons, and electrons – but then you would have to define those terms. You could describe things in terms of their mass, charge, and spin – but if you want to be logical and hierarchical, in the style of Euclid’s Elements, you would have have to define those terms before using them. And so on, ad infinitum.
When I want to introduce the idea of atoms, it takes a couple of pages to give the most basic descriptions. See reference 1.
There are innumerable other examples of words for which it is difficult or impossible to give a usable dictionary-style definition. In all of these cases, the best we can do is to present “Some tiny part of the idea before the name”. In contrast, if we insist on “the whole idea before the name” we are demanding the impossible.
Also note that principle (4b) is at best misleading, insofar as we tend to give names to the nodes in the lattice of knowledge, but not to the arcs, even though a great deal of the knowledge is carried by the arcs, in accordance with principle (3). The nodes are named early (before very much conceptual knowledge is laid down) whereas the arcs usually aren’t named at all.
There is an important idea hiding behind principle (4a), even if the statement doesn’t clearly capture it. The idea is that you start using a name without properly explaining it, the students will cook up their own mental models for what you are talking about ... and most of the models will be spectacularly wrong. Anybody with teaching experience has seen this phenomenon.
Nevertheless, it remains absolutely necessary to use terms that are not yet fully understood. Familiar examples include Euclidean terms such as points, lines, and planes, as well as physics terms such as atoms and energy. Other examples abound. These are sometimes called “undefined terms” but I prefer to say that they are implicitly defined, or retroactively defined.
This leads us to the mirror image of principle (4b): Because we are necessarily throwing around terms that are not yet fully explained, students will cobble up various misconceptions. (For the next level of detail about misconceptions and how to deal with them, see reference 2.) We can try to minimize the misconceptions, but we cannot reduce them to zero. Overly-strict adherence to principle (4b) would come at a terrible price, at the expense of the vastly more important principles, including (3) powerful connection-based learning, (2) the spiral approach, and (1) a deep understanding of how things behave (rather than a glib definition of what they «are»). Principle (4a) is not a problem, because is stated in terms of emphasis (not chronology).
To say the same thing another way, deduction is overrated. Euclid’s Elements is often touted as a paragon of systematic deduction, where each theorem is proved using previous theorems and axioms. (As a minor point, Euclid is not nearly as systematic is he’s cracked up to be; on many occasions he makes assumptions that are not covered by the stated axioms. However, let’s not worry about that; it’s a fixable problem.) More importantly, it must be emphasized that that points, lines, and planes are not deduced. They are not defined in terms of previously-defined terms. They acquire meaning gradually, by induction not deduction.
Physics is even more conspicuously not deductive. Results are not obtained “in order”. Feynman compared physics in particular and knowledge in general to a grand tapestry. A forgotten fact was like a hole in the tapestry, and could be repaired by weaving up from the bottom, down from the top, in from the sides, or all of the above. There are no axioms that take precedence over theorems; fact A can be used to explain fact B and vice versa.
Folks who think all ideas can be deduced in terms of previously-known ideas are seriously fooling themselves. Folks who think words can be defined in terms of previously-known words are seriously fooling themselves.
Furthermore: Good deduction is wildly different from good pedagogy. The order in which things are proved in a mathematical treatise is not the same as the order in which things are learned by human beings. Even professional mathematicians often come up with new ideas by informal guessing, then spiral back to make a rough sketch of a proof, and then spiral back a few more times to construct a complete formal proof, and then spiral back yet again to make it more concise and polished.
Math students are taught some ideas multiple times, year after year, with ever-increasing levels of formality and rigor applied to the same basic idea.
Copyright © 2015 jsd