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Steps in Physics Problem-Solving Method (Teaching Assistant) Here is a modified version of the problem solving method as described by the U. of Minnesota Physics Education group. 1. Focus the problem: Develop your own, qualitative (i.e. conceptual) description of the problem. What is the physical situation and what do you need to figure out? First visualize the events taking place in the problem, using sketches where appropriate. Write down in your own words a simple statement of what it is you need to find out. Write down a list of physics ideas that might be relevant and describe an approach you think you might be able to use. The point of all this is to get the problem straight in your own mind, and further to communicate the problem to the grader or anyone else who may need to know what the problem actually is. When you finish this step, you should never have to look back at the problem statement again because it will be internalized. 2. Describe the physics: In this step you should use your qualitative understanding of the problem to plan a quantitative solution. First, simplify the problem by describing it with (another) diagram involving simple physical objects and essential physical quantities. The objects could be the parts of a physical system, such as blocks, wheels, pulleys, etc. The quantities could be forces or velocities or energies, etc. Then, using the physical ideas assembled in step 1, write down the basic equations you know that specify how these quantities are related to one another mathematically. These basic relations may be in the form of Newton's laws, conservation of energy or momentum, etc. The results of this step contain all the relevant information so you should not need to refer to step 1 again. 3. Plan a solution: In this step you translate the physics description from step 1 and the set of general relations in step 2 into a set of specific equations that represent the problem mathematically. Write an outline of how you will solve these equations to see if they will yield a solution before you go through the effort of actually doing any mathematics. 4. Execute the plan: In this step you actually solve the equations you have come up with in the planning phase. Combine the equations as planned to get first an algebraic solution. Then plug in all the unknown quantities, including units, to determine actual values for the desired unknown or "target" quantities. 5. Evaluate the answer: Finally, check the answer to see that it is properly stated, reasonable, and that you have actually answered the question asked. Tools you can use for checking include, for instance, checking for dimensional consistency and taking special limiting cases where you know the solution. Check to be sure the algebraic solutions don't blow up or become imaginary for reasonable physical values of the given information. Steps in Physics Problem-Solving Method- 1 Comments to the teaching assistant: 1. Focusing the problem: Often we find that a student cannot describe the problem without reading it. Does this mean the student is inarticulate, even though in his or her mind there is really a workable understanding of the problem, or does it mean the problem is not really comprehended? Experience suggests that the student has not understood the problem well enough to begin working on it. This difficulty involves reading comprehension. For the novice, it is very, very difficult to get enough of the problem in view at once to see the essential features. Get the student to restate the problem in his or her own words. If you listen to this restatement carefully, it can suggest some things. At first, the description should be qualitative. Does the student use the vocabulary of the problem appropriately? If not, this is a place to start. Sometimes the student will represent quantities using words that don’t usually represent quantities. One useful diagnostic is the use of prepositions. For instance, forces are caused by things (sometimes fields) and act on other things. An inappropriate preposition would be between, when speaking of a single force, and a suspicious one would be of. As another example, electric current flows through a resistor as a result of a voltage or potential difference between one end and the other. But one cannot speak of the number of volts through some circuit element. These language points may not always indicate internal confusion, but more often than not they do. If the problem assignment includes a figure or picture, can the student come up with an approximate figure without looking back at the given one? If there is no figure, can the student invent some sort of figure or diagram? The usual approach for the instructor here is to get the student to go over certain parts of the problem statement and elaborate. You might say something like, “I’m not sure I see what you really mean by…?” If there is a contradiction it is best to lead the student into discovering this by himself or herself. It is unfortunately very easy for the student to get a negative feeling from the process. So, on the one hand it is quite important to pick up on and reinforce anything the student might be trying to say that is correct. But on the other hand students often present a sort of Rorschach blot and expect the instructor to project a correct interpretation onto it. So they need to be able to explain these ambiguities. Another consideration is that, from the student’s point of view, it may not seem wise to expose too much ignorance to the instructor who, after all, has to grade the students as well as coach them. The fact that students will be working in groups makes it possible for some of the group members to help correct misconceptions. There may be less of a stigma associated with this peer-to-peer interaction. After getting some reasonable idea of the physical situation, a list of things to find out, or the target quantities, is the next thing to have. Comparing this list to a list of basic concepts that might be involved will sometimes give a hint as to Steps in Physics Problem-Solving Method- 2 a possible strategy. Problem statements will often contain catch phrases such as “moving very slowly,” or, “rolls without slipping,” that in turn contain hidden information. What does such a condition imply one should assume? These are things to be learned. 2. Describing the physics: This involves breaking the problem down into parts, separating the physical system into components, and figuring out what physical quantities are involved. A problem statement with a properly realistic context will often supply unnecessary information. Students need to figure out what is extraneous. Sometimes masses are unnecessary, or sometimes irrelevant lengths or distances will come up. In the laboratory, it is easy to take quite a lot of irrelevant data. In a mechanics context, the physical system is often separated into components and each is simplified to a free body diagram. In a collision it is usually appropriate to consider before and after the collision for each of the objects involved, and so on. So, to begin putting the problem into mathematical language, the physical system can usually be decomposed and simplified. It is important to know, for instance, not only the conditions under which total mechanical energy is conserved, but also that the reason we look for this to be the case is that it lets one relate speed at one point to speed at another. Having these ideas in mind will be important for formulating a strategy. A good plan here is to use the list of possible concepts that might be involved, weeding out the ones where the conditions are not appropriate, and then writing down a list of appropriate formulas. It is best if this is not done as a treasure hunt. Students sometimes scan the current chapter looking for a formula, or preferably an example problem solution, that seems to contain the right letters. One way to tell when this is going on is when a student asks what a certain letter means in one of the formulas, or has clearly confused tension with period of rotation. A student question of this sort should be answered with another, carefully chosen question. Finally, this process should result in a list of potentially useful relations among symbols representing the physical quantities actually involved. Sometimes there will also be unknown quantities that one does not want to know. The student may need to write the formulas down anyway, because the extra quantities might either cancel out or be eliminated later by algebra. 3. Planning a solution: Ideally this would consist of a complete plan for starting with the information given—or in the laboratory with the given situation—and proceeding through measurements and analysis to extracting the desired result. But really there are always places where one does not know exactly what to do, or where something unexpected comes up. So in fact it is sometimes a strategy that seems as if it should work. There are usually choices to make and relations Steps in Physics Problem-Solving Method- 3 to think of. As the plan is executed, it will almost always happen that it needs to be modified, either a little bit or a lot. It is important that the students themselves should make the plan. The instructor needs to keep an eye on the groups to see that each plan is basically sound and capable of producing the desired results in the allotted time. Sometimes, time permitting, it is beneficial for the students to learn from mistakes. But recalcitrant equipment and logistical problems usually burn up enough of the laboratory period so that there is not sufficient time left for free experimentation. This is why the instructor should interject some questions when the group is making an obvious blunder or not noticing an equipment problem. The group measurement plans should contain a brief statement of the problem or objective, a statement of the series of activities to take place, what data need to be collected and why, some mention of what the group members will be doing, .i.e. when taking data and so on, how the data will be analyzed, e.g. what variable might be graphed versus what other variables, and how the target quantities will be gotten from this. Serious thought should be given to how to judge the size of measurement errors and their effect on the target quantities. Some use should be made of the particular group roles: task master, recorder and skeptic, as defined in the syllabus. However, all of this has to be kept reasonably short. At least a complete outline of this plan must exist before the team begins taking data. It will be written up nicely later as part of the report. Ideally the laboratory period should consist mostly of exploring equipment making measurements and making preliminary graphs to test that the measurement plan is working. When planning a solution, starting from fundamental principles such as conservation of energy and momentum, involves algebra, particularly eliminating between two or more equations, it can use up too much time. This is true especially for the non-calculus students. When time gets too short a small amount of help might need to be provided, but this should be avoided if possible. The following strategy for handling simultaneous equations (Gauss elimination) is usually easy for the students to appreciate: Start with your set of equations. Draw three circles: In them put (a) the variables with known values (b) the target variables, and most importantly (c) the variables whose values you do not know and do not need. Known Desired Don’t know and don’t want You should place only one quantity in the desired circle at a time. Then choose a variable from the third circle (a “pivot”) and solve for it in terms of the other variables. Then one equation is used up, so mark it used. Replace (substitute for) that variable in each of the other equations. Thus one of the don’t-know- Steps in Physics Problem-Solving Method- 4 don’t-want variables no longer appears. One less variable, one less equation. Take another equation and solve for one of the other don’t-know-don’t-wants. Then mark that equation as used up and substitute out that variable in the other equations, and so on. Pretty soon you have only equations involving what you know and what you want. Then, if you had the right number of equations to start with, you can solve for the unknown. Eventually you can get all the target variables this way. Students will sometimes try copying an example problem from their texts. The textbooks should not usually be open in the lab. Reading the text is done before the lab. In any event, this should not be part of the solution planning. 4. Executing the plan: All the group members need to be involved in taking data at some point. If during part of the measurement only one or two can be active, the others should be paying attention. It will sometimes be necessary to ask some questions to the team members who do not seem to be engaged. It is not acceptable that some of the students should be standing about and, “have no clue” as to what the group is doing. When a group is lagging behind the others, and there is no obvious equipment problem or other reason, ask which member is the task master and remind him or her to keep an eye on the time. When there is an obvious blunder, find out who the skeptic is and start asking some questions, perhaps taking the role of a supervisor. The result of the group activity should be a set of reasonable data, appropriate to the measurement plan. These should include units and notes about procedures that will be needed later in the analysis. When the plan includes making a graph, there should be at least a preliminary graph before the end of the lab period to ensure that the data behave as expected, and can lead to the desired results. Each student should get a copy of the data. Data from each group will be checked on exit from the laboratory. Executing the plan also includes carrying out the analysis. as much of this as possible should be done within the laboratory period. It should follow the measurement plan. The analysis should be accompanied by a clear explanation that could be followed by, for example, a physics student in another group, or of course the instructor who must grade the lab reports. It has t be clear which physical principles are being used and how the necessary information is extracted. One way of explaining might be that at least one example of any relevant calculation could be worked out in detail with appropriate comments. If the plan has to be changed in the midst of the analysis, this should be explained. The measurement plan should provide for error estimation. One technique that is quite general and easy to apply is to make redundant measurements. If a given measurement is taken five times, the average provides a better value and the spread gives an error estimate. This idea is developed in the first laboratory of Part I (Measurement Error). Thus the group activity should involve some type of error monitoring. Steps in Physics Problem-Solving Method- 5 5. Evaluation of the solution: This means assessing the quality of the answer, including an estimate of the numerical errors involved in finding and reporting a quantitative solution. There are quite a few ways to detect errors in the analysis, such as checking units and taking limiting cases. The students need to start thinking of these as ways to find and fix mistakes. A general theme of the current laboratory course, which was not stressed as much previously in the physics labs, is developing a realistic treatment of measurement errors. Formerly an error analysis consisted of having some “known” value for the answer (gotten from a table or something) then subtracting it from the measured value, dividing by the known value and then multiplying by 100%. This is not a good model either for how experimental science actually works or for how one estimates errors in a numerical value obtained on the job. It really should be clear from the start that one is not going to have a correct value with which to compare a result. When reporting a numerical answer obtained via measurements and calculations there is a responsibility to provide a realistic estimate of the error, or of the quality of the answer. This is almost as important as the problem solution itself. Finally, there is a meta-objective in performing the laboratory work. The student encounters some important physical principles and learns basic laboratory procedures. At the end of a laboratory period, the groups will be called upon to tell which basic learning objectives the activities have addressed. some but not all of these are listed in the lab books. Steps in Physics Problem-Solving Method- 6