A published statement that the physics student needs to memorize values of physical constants is questioned. It is pointed out that understanding, not memorization, should be aimed at. Requirement for the latter should be minimized, though it is reasonable to demand order-of-magnitude knowledge of important constants which have direct experimental significance. Values of constants are categorized as a crude kind of "craftsman's knowledge" which may be obtained genuinely only from experience, not by learning.
He goes on to give the doctrine of a physics teacher who said:
I have always felt it a necessary part of physics that students know the numerical values of a few important constants such as the free-fall acceleration, the sea-level atmospheric pressure, the density, specific heat, and latent heats of water, the charge and mass of the electron, epsilon_0, mu_0, etc.....For examinations students should be expected to have these numbers memorized.
Whitaker's rebuttal to this is simple: although he can from memory specify a few of these values, he cannot remember them all, and furthermore, that he can find no no reason to believe that his inability to remember these values has in any way "done me the slightest harm," he says.
Whitaker begins the next section of his article with the title "Memory or Understanding?," which I will soon make clear, I hope, is the wrong question, and belies a severe misconception of the actual interplay between memorization and understanding. Anyway, he continues:
Let us return to the often-heard student complaint the physics is just "a lot of facts to be learned." The lecturer will certainly reply that what should be aimed at rather is a general understanding of the principles and concepts of a particular branch of physics.
Later he says:
Necessarily we must support the teacher's view; we must encourage in students (and aim at for ourselves) a perhaps painfully built up understanding of issues, rather than a reliance on superficial memorizing. But it is important that we also attempt to see things from the student's point of view. The understanding we aim at must be, in large part, a creation of experience, which we build up as we study different problems, or the same problems from different viewpoints.
The last point of this article that I want to quote is:
I believe the general rule should be that while the student must be expected to come to grips with those formulas, equations, and so on that form an essential part of the structure of the subject, we should make great efforts to reduce them to an absolute minimum.
OK, that's it for presenting quotes from Whitaker. Now I want to basically agree with his thesis, yet simultaneously argue against it! Whitaker has presented us a bunch of platitudes to deal with. His most egregious platitude is this: "Necessarily we must support the teacher's view; we must encourage in students (and aim at for ourselves) a perhaps painfully built up understanding of issues, rather than a reliance on superficial memorizing." Well, who would argue for "superficial memorizing" or for "superficial" anything for that matter?! The content and thus the importance of Whitaker's thesis lies entirely in how he defines the terms "learn," "memorize," "understand," "concepts," and "experience" -- term which he hasn't bothered to define for us.
It's clear that Whitaker sees the definitions of these terms as generally uncontroversial and agreed upon. I don't believe this to be true, especially within the physics community, or any scientific community, except perhaps cognitive psychology. What would we find if we asked a dozen physicists to define each of these terms? In any case, I will define them now to get directly to the source of my objections to Whitaker's presentation.
Memorize -- any process by which information is stored in the brain for instant, accurate, on-demand retrieval. Thus, any piece of information that has been memorized is said to be "in memory."
I'm going to leave the term "concept" as an undefined term!
Primitive Concept -- a concept that is used within a particular theory but is left undefined in that theory.
Derived Concept -- a concept that is logically, mathematically, graphically, or by fiat built up out of other derived concepts or out of primitive concepts or both.
Complex Concept -- a concept that has been modeled as having "parts." For example, the concept of Newtonian mechanics has the parts: point particles, aggregates of point particles, action-at-a-distance forces, etc.
Understand -- a process of finding meaningful and contextual relationships among all the "parts" of a complex concept.
Learn -- Memorize! (It may not agree with your definition of "memorize" but it agrees with mine.) Rote learning -- the use of ANY form of repetition to accomplish memorization.
Activity -- any action of human participation that is more than mere reading or listening to a lecture.
Experience -- the active mental participation in an activity designed to facilitate memorization without rote and the memorization of the relationships among conceptual parts by the use of a so-called "hands-on" approach to learning. This definition is not exactly to my liking because of its misconception of the true role of rote learning. What learning by activity really gives us is a memory of a series of events that we can, if we're good at visualization, play back in our minds for the purpose of "mining" it for key points that we can map into conceptual parts and their relationships, in such a way as to achieve understanding. However, the act of replaying this memory is the use of repetition and so is by my definition the use of rote learning.
On to a critique of Whitaker's points. One of his points is that instruction should be geared to facilitating the student's understanding of physical concepts rather than memorizing mere trivial physical constants that can be looked up as needed. I agree with this. In fact, I submit that tests should not grade heavily on precise numerical answers because 1) tests are given to determine conceptual understanding, not mastery of arithmetic which should be presumed anyway, 2) tests are too time limited to worry about numerical details that can easily be a source of arithmetic errors which say nothing about the student's ability to get right numerical answers in the real world, and 3) homework is the right place to ask students to reduce the answer to precise numerical form. An added benefit of reducing answers to algebraic form is to reinforce to students that the premature substitution of numbers into algebraic expressions will eliminate the chance of them finding an algebraic or trigonometric simplification to make along the way by use of an identity.
Another of his points is that memorization is seen only as an opposing "learning" strategy to that of conceptual understanding, not as a preparation to it. In every complex concept there are parts to it that must be memorized before the relationships among these parts can be discovered. And after they are discovered, they too must be set to memory! Thus I am not arguing for less memorization in physics, but for even more. But not memorization of silly physical constants that can be looked up at one's need, but for the conceptual underpinnings that one needs in memory before deeper physical concepts can possibly be integrated into one's understanding.
Even if you don't agree with my definition of "understand," you should still have a definition of it, so try to define it yourself. It makes no sense to me at all to place enormous weight to the importance of understanding, yet never provide a definition of it, nor to even have a mental model of what understanding really is! This is perhaps the greatest oversight of physics and mathematics education. It's probably a good idea to provide a model of understanding to students as well.
Patrick, Nov 2001