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