Solving a Typical High School Physics Problem

The task to be analyzed is essentially, “Solving a Typical High School Physics Problem”. Although this may seem to be an overly general one, it actually is not, for the requisite knowledge and necessary mental operations, as well as the logical solution procedures outlined are basically the same for all variations of this level of task (i.e. all specific problems), as this paper is intended to demonstrate. Thus, although the specific mental information represented will vary with the problem encountered, the gist of the discussion of this analysis is applicable to them all. Before breaking down this task, let us first briefly justify this choice. Firstly, for those considering taking up high school physics teaching, while I am hardly an “expert” in solving physics problems, I do feel competent enough to be of assistance to others in performing this task. Secondly, I feel that this task is important, not necessarily as an end in itself, but for its applicability to real world problems. The student attacking a problem is typically in the formal operational stage of development and soon will be turned “loose” into the world. I feel, perhaps over-zealously, that proficiency in the problem solving can be of valuable assistance in developing deductive reasoning abilities that the student can use on problems to be faced the rest of their lives, even if one doesn’t plan on being a mathematician or a scientist. The development of this ability should be stressed in public schooling and the physics course is a logical place to accomplish this. Having justified the validity and attached the appropriate importance to the task, we now analyze it in detail. Basically, the task involves the following components. First, the problem at hand, written or orally administered is translated into a mental “picture” of what is going on physically. This initially involves a scanning and holding operation, that is, key words, terms and numbers are noted and kept in mind throughout the rest of the task. In order to do this, of course, the student must possess a reasonable high level of reading and verbal ability. If foundations in these areas are not firmly established in the problem solver, the task must be deferred until such are built up. More will be said on this later. Next, one must classify the aforementioned key words and generate the appropriate mental pictures. When new words are encountered, these too must be associated with our stored images (with either the student or the teacher bridging the gap). Even with adequate ability in these areas, however, this step might not be easy, as it also relies on the individual’s physical intuition, which widely varies with his/her experiences. For example, one may have difficulty visualizing in three dimensions. For this reason, various class demonstrations can be very useful in building up this ability (more on this later). These “mental pictures” can take various forms. For example, one may translate the physics term “simple harmonic motion” into a mass (or masses) on the end of a spring or a ball moving in a frictionless valley; potential energy signifies a rock on top of a mountain; an electric dipole is two charged “balls” on a stick; current consists of tiny spheres (electrons) “bumping” into each other and pushing onward, to name just a few. This list can be extended indefinitely and grow steadily with time one’s experience level. The student should have such analogies stored pictorially in their memory, even if not the associated terms. As one’s knowledge grows, the term-visual picture combination becomes a building block for more complex concepts. After the “pictures” are established, the symbolic and/or mathematical representation must be generated. This is essentially an ordering and relating operation. In this portion of the task, the entire (sometimes lengthy) word problem is neatly organized into a few equations, which are then related to algorithms of mathematics. This step is of equal importance and probably is more difficult that the preceding steps. To initiate this process, the problem solver must possess a working knowledge of the algorithms of algebraic manipulation, geometry, and trigonometry and, as the problem complexity increases (beyond typical high school curricula), calculus and differential equations become necessary. As above, if the student is not proficient in these areas, they must be brought up to speed before the task can be undertaken (i.e. there is a defined “minimum” level of proficiency is called for). This point will be elaborated on below. Here, the representation is more or less standardized and should not vary with the individual. Taking the above examples again, simple harmonic motion implies an equation; F = -kx (force is proportional to distance), with solutions of the form x = sin t or cost. Here, a basic physical law is drawn upon (Newton’s F= ma), and equated to the above, then algebraic and calculus relations are utilized to solve the resulting differential equation to obtain a solution. However, one need not go through this procedure each time this concept is encountered, for the equations and the physical situation are stored together as a “concept”. A similar example would be storing the equation P.E. = mgh, with the potential energy mental picture described above (rock on top of a mountain). It must be noted that the above steps are not always performed in the same sequence, as often problems require one to interpret the (given) mathematical relationships physically. As stated above, however, the pictorial-symbolic representation is, in most cases, stored together under the concept so that the seemingly reversed problem sequence actually involves the same mental operations. Once both the physical picture and symbolic representations have been formed, one can then begin the actual rationalization process, leading to the solution of the problem (i.e. Problem Solving Process). In doing this, it is often helpful to draw some type of picture so that one’s mind can be freed of constantly redoing the first step (pictorial representation). The same applies to the symbolic representation (i.e. write down equations). Now, one must devise a plan of attack for the problem at hand. We do this by drawing upon our storehouse of knowledge (i.e. we ask ourselves numerous questions). Typical of these are; “Have I seen this problem or a similar problem before?”, If so, “Can I remember the plan of attack?” If not, “Where might I find it (textbooks, notes, etc.)?” If I haven’t seen this problem type before, “What references (other textbooks, tables, etc.) might be of assistance?” “Can I apply principles from these sources to the problem at hand?” and so forth. An example of the above might be a problem of two masses connected by a spring. First draw a picture. Then, write down equations: m1a1 = -kx1 and m2a2 =-kx2 (x1 and x2 are displacements of masses m1 and m2 from their equilibrium position). Next, ask questions: 1). Have I seen a similar problem before?-yes (mass on a spring) 2). Do I remember the plan of attack-no (say) 3). Where to look-Physics textbook, lecture notes or online? One may well remember certain problem “tricks” as pictorial representations of the problem steps themselves (i.e. successive sets of equations containing certain key “steps”). In other cases, certain words or colors may “key” one in to the book to look into to obtain the solution. The exact steps to follow in devising the plan are really not cut and dry, as they must be developed through practice, experience and expanded representation systems. For example, knowing the plan of attack for solving the mass-spring problem, one can then devise a new plan for two masses connected by a spring, etc. Next, one carries out the plan, making sure that each step is justified and “done correctly”. Sooner or later, depending on time constraints and the quality of the plan, a solution will be obtained. This solution, in many cases, ultimately arises from the student’s own trial and error and originality (at least in the student’s mind!). We must now check its meaning. Does it make sense, physically? Is it consistent with our physical intuition? For example, if a “negative” mass arises out of the model, or a velocity exceeds the speed of light, in conflict with the theory of relativity (to cite ridiculous extremes), we know that something is amiss. If an inconsistency is found, we must re-check our plan to verify its contents. If it is logical and, if we find no errors in our execution, we must then re-examine the original problem to determine if its conditions can, indeed be satisfied. This re-cycling process continues until a logical “solution” is obtained (either a “correct” solution or an explained inconsistency in the problem). Finally, the solution can now be “stored” (either the plan of attack, or where to find it) for use in solving other problems in the future. This is the remembering operation. As discussed elsewhere, an appropriate retrieval clue will usually have to be attached for retrieval in the future. As mentioned above, color, words, etc. are typical of what are used. In all of the previous discussion, it was assumed that the leaner was already at a certain “high” stage of mental development. Since the task is to be performed by high school students (usually), this should be a reasonable assumption. Therefore, primitive knowledge requirements will only be sketched. Firstly, it is assumed that the learner possesses a moderately high level of reading and verbal competence. Thus, he or she should have “mastered” the following: 1) the concept of letters, 2) combination of letters into words, 3) recognition of printed or written words, 4) word-object (or concept) association and, 5) reading comprehension (i.e. following written directions and reading for specific information. In a parallel fashion, he or she should be mathematically competent, having acquired the following abilities: 1) the concept of numbers, 2) numerical ordering, 3) number-object association and, 4) simple math operations (+, -, x, /). In addition, a working knowledge of simple algebraic manipulation of numbers, geometric concepts (triangles, lines, planes) and some trigonometric concepts (sin, cos, etc.). As stated previously, without these foundations, the task cannot realistically be completed. Of course, accurate estimation of just how much of these skills the student possesses must be done very carefully. An interesting application of the above process can be cited whereupon a class demonstration, involving a balance beam apparatus, which consisted of a beam with various, equally spaced pegs on it is balanced on a fulcrum. Metal rings are then placed on various pegs and the class is asked to determine which side would fall. This is an obvious example of the torque principle; however, even people who had supposedly never heard of the concept eventually were able to solve the problem correctly. One can conclude that they went through a hierarchy similar to that shown (minus a few steps)-i.e. they: 1) formed mental pictures (from previous experience-perhaps with a “teeter-totter”), 2) assigned a symbolic representation-(1 peg = 1 unit from fulcrum, 2 pegs= 2, etc.), 3) devised a plan-of-attack- (multiply the number of rings by units of peg, side with greatest total will fall) and, 4) stored this plan in memory- (were able to solve more “complex” problems henceforth). That trial and error was involved was demonstrated by a number of incorrect responses, initially. Now that the task has been analyzed, a few pedagogically-related comments will be made. Firstly, there are a number of difficulties that could emerge while performing this task. One is simply “working for an answer”. This occurs, especially when under time constraints and entails the student manipulating symbols in any way (logical or not), simply to obtain the desired answer. Needless to say, this procedure will be of little use in future situations (particularly when “answers” are not provided). The solution to this behavior is emphasizing process rather than products in teaching and quality over quantity in assigning problems. Another phenomenon arises when an apparently well informed student cannot solve problems or cannot “take tests”. Here, one must instill the importance of approaching each problem with a “clear mind”, for even an expert can have difficulty if the mind is pre-occupied. Also, emphasis on general learning, rather than grades (inclusion of factors other than solely examination scores in student evaluations) may aid here. Among additional positive teacher inputs that should be practiced include: 1) presenting an organized framework of material (to assist students in “channeling” various important concepts), 2) providing demonstrations (like that cited above) to aid students in building up a storehouse of physical “pictures” in their minds and, 3) use of guided discovery methods in teaching (letting students draw upon knowledge from previous courses or life, in general, when introducing new material). If these guidelines are pursued, mastery of the task of solving a “typical high school physics problem” can be a valuable learning experience from which to draw upon when encountering problems in later life.