I have heard many times (as numbers of other
physics teachers have) a student saying that he or she understands physics but
“just cannot solve problems”. I always say in return, that physics is one of
the clearest (and I would add easiest) subjects, since physics has a very straightforward logic which underlines the thought
process.
However, in order to teach how to solve physics
problems a teacher should not be focusing on demonstrating how to
solve specific physics problems (!), but instead should demonstrate the
thinking process happening in the mind of an
expert problem solver when that expert is constructing a solution for a given
problem (this approach is based on a certain thinking algorithm; however, it requires much
more time than a regular course can usually provide).
When a student does not know how to solve a
problem, he or she should not ask a question “how to solve this problem?”, or “how did you solve this problem?”, but instead
should ask a question “how can I come up
with the solution?”, “how did you come up with this solution?”
And a
teacher should be able to demonstrate the thinking process behind the creating
of that solution.
In this paper I offer a short description of a
general thinking process of an expert physics problem-solver (FYI: this approach
should be mandatory for professional preparation or development of school science
teachers: every STEM teacher must learn some physics (in
addition to the own subject), and then must reflect on how he or she learned it
– to be able to apprehend the mental processes needed to be happened in order
to acquire that specific physics knowledge or skill). This is specifically true
for teachers teaching
computer science.
One of the common complaints coming to a teacher
form a student is “I don’t understand this”.
However, NOT
everything in physics requires
understanding – this is a very common misunderstanding.
Students often say: “I do not understand”, but
in actuality it usually means: “I do not know the basics”.
There are many important physical concepts,
which do not require understanding, but require rote memorizing; in a standard
physics textbook all those concepts are laid out in paragraphs like “facts to
learn” (or similar). Well, of course, some understanding is required at this
level, for example, understanding of the meaning of the words and sentences
(that is why I do not write this paper in Russian). The set of the facts to be
memorized comprises that knowledge, which most people mean when saying: “I know physics”.
The next level of understanding comes when
students are making connections between the just learned (usually abstract)
concepts and their knowledge of everyday life, which came from their everyday
experience. The existence of these connections makes students to say: “I
understand you, I understand this”. This level of understanding might help
students to explain some phenomena they observe around them, but usually is not
enough for making them being able to solve physics problems.
The true (actual, complete) understanding underlines the ability to apply previously
accumulated knowledge for analyzing new specific physical situations
and comes with the experience of solving specific physical problems, and
then reflecting on the process of the creating the solutions.
This level of understanding is tested by the
ability to solve physics problems not congruent to previously
solved problems (“congruent” simple means “identical”; please, visit http://gomars.xyz/mocc.htm for the difference between
congruent, analogous, similar, and like problems).
This level of understanding demonstrates the
existence of strong connections between knowledge
(a) acquired using rote memorization, (b) developed in everyday life, (c)
developed during previous problems solving activities, and actions, which have to be performed in order to solve a problem.
The best (and only!) way for achieving this kind
of understanding is solving physics problems. Learning solving physics problems
is like learning how to drive a car, or how to swim; no one can do it just by
watching how other people do that; it requires a lot of personal practice,
preferably guided by an experienced instructor.
If you read a text of a problem and you know
what to do, it is not really a problem; it is rather just a training exercise,
a task. A real problem happens when you do not know what to do and have to
construct the solution on a spot.
As I often like to say, study physics without
solving problems is the same as learning how to swim with never entering water.
To learn how to swim is necessary to swim, i.e. to lie down on water, start
moving hands and legs, and see what happens. At the very first time, certainly,
you will fail; you will drink some water and will not make any progress in
moving ahead, but gradually, a try after a try you will be doing better and
better. And the time will come when the first actual swim is accomplished! You
can swim now!
Precisely the same situation happens when you
need to find a solution to a physics problem (and, actually, to any other
problem in life). To learn how to solve problems it is necessary to solve
problems, i.e.: to read a text of a problem, to imagine as clearer as possible
a described situation, to draw a picture, to write down formulas, and to try to
make sense of them. And of course, at some point in constructing your solution
you might get confused or make a mistake. Getting confused and making mistakes
is a natural thing when solving problems (any problems). Mistakes are inevitable
and unavoidable. I make an even stronger statement: making mistakes is the necessary
element of learning how to solve problems.
True learning actually gets triggered only when
a mistake had been made and we start thinking about how to correct it (people
say “learn from your mistakes”, but in reality, there is no other way to
learn!). Of course, when we make a mistake, we usually feel some discomfort
(one of the legacies of a poor schooling, I believe). We need to be able to
overcome that discomfort. In fact, we need to embrace it, because this is the
sign of a true learning happening inside us. The feeling will pass and be
replaced with the joy of having a problem solved (the # 1 goal of any teacher is to lead a
student through these emotional stages; “make physics/math/anything fun” is
a wrong and misleading
approach because it will not leave students with knowledge and skills, the
right approach is “make physics joy-able”).
The folk wisdom tells: “If you didn’t succeed
first time, try and try again”. This rule is partially correct. If you keep doing the same thing you keep
repeating the same mistake (Einstein’s “insanity”), unless you have to dig
a deep hole. Every new trial has to be in some way different from the previous
ones. That means that when you make a mistake you should figure out what went
wrong (or at least to make a guess what might have gone wrong), so you would
not repeat the same mistake again, i.e. you
should reflect on the thinking you
have used to get to this point in your solution.
The very first difficulty many students run into
when they have to solve a problem is “how to begin”? Usually I answer: “Try
something, anything”, ask a question: “What can you do”? and
do it.
There is set of specific learning aids which could be offered to a
student to help him or her to begin the problem-solving process. It is usually
very much helpful to give students some general description of the work our
mind does when a problem is presented to it.
Let's imagine that you are invited to a party.
You come, and there are so many unfamiliar people over there. What do you
usually do in this kind of a situation? You usually are trying to find somebody
familiar and approach him or her.
The exactly same thing is happening when we
start solving a problem. Our brain is a powerful pattern-recognition “computer”,
and the first thing it does in a problematic situation is starting looking for
familiar patterns. And it always finds them, even if we do not feel that way.
So, if you do not know what to do, your
brain knows, so just trust it and do the first thing which comes to mind,
but DO IT!
Of course, you can help your brain to find the
appropriate pattern faster and with more confidence by using various learning
aids (the general algorithm for developing a solution, a picture, a MOCC, a
dictionary, a classification table; please, visit www.GoMars.xyz for examples).
When you are looking for a familiar person, your
brain automatically analyzes a set of indicators, like a face expression,
voice, talking manners, posture, gestures, etc. Every physics problem has some
indicators/parameters, too, which differ one problem from another, but also
which combine similar problems into a certain cluster of problems. When you
recognize to which cluster this problem belongs, you can immediately employ the
method you used in the past for solving similar problems, or at least you can
try using similar strategy, reflect on it, and correct it, if needed.
Physics studies specific phenomena, i.e.
specific processes happening to various objects.
Phenomena are the first things we all observe
from our birth. We feel a lot of things: we can see, we can smell, we can touch
objects and hear sounds. And we have developed many words we use to describe
these phenomena to each other. But in science we have to use a specific
language, which represents a purified/simplified/specified version of an everyday
language (the main reason for using a specific language is to minimize
misunderstanding between scientists – everyone
has to be on the same page).
Hence, when we read a text of a problem we must perform
a translation from an everyday language into its scientific version. This is a
skill which every expert problem solver has, but which usually has been
developed without conscious efforts (during
professional practice), however which can be trained, too.
Another important thing to remember when solving
physics problems is that in physics we NEVER can solve any real-world problem,
because all real-world problems are simply too complicated (because in the real
world “everything is connected to everything” and the number of connections is
huge)! We always must make some simplifications, some assumptions, which make
the situation described in a problem being manageable. Instead of actual
objects we use idealizations, i.e. abstract objects, which do not exist in
nature but have the same important properties as the real objects in a problem.
For example, we do not draw the Earth to scale keeping its exact shape with all
the oceans and continents, we just draw a circle. When solving a problem, it is
important to make a clear statement and keep track of the assumptions which
have been made, because (a) our solution is limited by these assumptions, and
(b) if something goes wrong, maybe it is because one of our assumptions was
incorrect and we have to rethink them all ti find the wrong one.
Since we do not deal with the actual world, but
rather with an imaginary world which, in a way, is a reflection of the actual
world, having a good imagination is as
useful as being good at math.
Physics studies what happens to the objects
around us and why. Some objects are huge, some tiny, some very fast, some not
moving at all. We use a specific language to name physical objects (using
nouns), to describe their properties (using adjectives), to name processes
happening to the objects (using verbs) and to describe the properties of the
processes (using adverbs). Any textbook gives a sufficient description of that
language and how to apply it for describing the physical world. If you need to
solve a problem and not sure how to start, at least you can start looking for
nouns (at least some of them describe important objects), and verbs (at least
some of them describe important processes), and then draw a picture which would
show those objects and illustrate those processes – it is always a good
starting point.
Almost every word we have to use to analyze the
situation described in a problem has a very specific meaning (given by its definition) and everyone must know that meaning exactly/literally; usually we call
such special words/terms as physical quantities and use letters (a.k.a.
variables) for a short representation of those quantities. A sentence which
describe that meaning is a definition of a quantity – we have to know all
important definitions. Later we use theses quantities as equations we write.
Each equation represents a specific connection between quantities/variables (and
can be visualized by a small map).
Physicists, as all scientists, are always
looking for patterns (looking for patterns is a job description of a scientist;
the mission of a science
is making prediction based on the known patterns). A pattern is a process
which repeats itself (as long as we do not drastically change the conditions of
the happening phenomenon). When we find a pattern (might take a while to prove
that it is actually a regular pattern of nature), we call it “a law”. We use
laws to predict what might happen under certain circumstances and to build
devises, which we want to use for our purposes. Ideally, a law should be
written in a mathematical form, i.e. as an equation, so we could use math to
derive our predictions. In physics there are only two fundamental kinds of
equations: every important equation in physics is ether a definition or a law.
A definition is basically an agreement between all the physicists in the world
on the meaning of a quantity/variable. Definitions come mostly from
observations of the phenomena happening to objects. In physics, a law is a well-established
mathematical connection between quantities/variables (previously defined); laws
come from experiments specifically designed to test those laws when they have
not been called “a law” yet, but just “a hypothesis”.
Of course, there are also many additional
relationships which are derived from laws and definitions by algebraic
manipulations, which also might be very useful when solving problems
(memorizing those relationship might save valuable seconds on an exam).
So, we read a text of a problem, translate the
text into scientific language, use a visual representation of what is happening
to the objects, list important quantities needed to describe properties of
objects and processes, recognize patterns, and based on all those indicators
conclude to which class of the problems this particular problem belongs (i.e. we
name the model). As soon as we named the
model we can start writing equations related to that model (because the same
equations helped in the past to solve a similar problem). Then and only then we
can start manipulating with the laws and definitions trying to create a
solution to a problem, and we can reflect on our way of thinking and make a
correction (if needed), and after practicing in doing all this for a
significant period of time we become experts in solving physics problem.
Simple!
Let’s provide a short example of thinking as a
physicist.
Let’s say, we need to find the speed of a
meteorological satellite which is orbiting the Earth. At first, we recognize in
this problem the following situation: there are two objects (the Earth and a
satellite), we assume that there is only one important interaction in the
system, i.e. the objects interact with each other only via gravitational
attraction, the Earth is not moving (another assumption), the satellite makes a
circular motion with the Earth at the center of the circle (another
assumption).
Key concepts for recognizing the physical
situation described in the problem are “gravitational attraction” and “circular
motion”. We know, that “attraction” is a kind of interactions and interactions
give rise to forces, and forces are related to motion of objects via the
Newton’s second law. We also know that for an object making a circular motion
there are specific relationships between its kinematical variables (for
example, speed, acceleration, radius). This information is already enough to
start constructing the solution. We can draw a picture, we can write the
equations we have mentioned, and start manipulating with the equations until we
get a relationship between the speed of the satellite and other important
parameters of the problem. If we got it, we are done, if not, we start looking
for a missing link or for a mistake in our previous reasoning.
Good luck!
Thank you for visiting,
Dr. Valentin Voroshilov
Education Advancement
Professionals
To learn more about my
professional experience:
P.S.
In short, we can list seven steps of a scientific
way of thinking (developed in physics – used in every science!)
1. Seeing (or imagining) things =
objects. Naming important objects.
2. Listing important properties
of important objects.
3. Seeing changes = processes.
Naming important processes.
4. Listing important properties
of important processes.
5. Describing various properties using various
parameters. Describing various states and changes using values various
parameters.
6. Stating important patterns (i.e. well-established
connections between important entities – a.k.a. laws).
7. Using important patterns to establish the
correct sequence of events (an algorithm, a solution).
Also, follow this link to: A
General “Algorithm” for Creating a Solution to a Physics Problem
http://www.cognisity.how/2018/02/Algorithm.html
P.P.S.
Many of students taking my courses are pre-med
students.
For those students the importance of taking a
physics course is far beyond getting prepared for a MCAT (in fact, a MCAT is a
more complicated exam than any of the elementary physics exams). For not physics major students the most
important outcome from study physics is assessing their own personal abilities
to guide their own thinking process when solving a problem (any problem,
for example, diagnosing a patient).
Taking
a physics course
|
Becoming
a physician
|
When study physics students have to memorize
some definitions and laws.
|
To become a doctor students have to memorize a
lot of stuff (way more than when taking physics course), for example names of
all mussels, bones, diseases, and treatments.
|
When solving a physics problem students have
to recognize the underlying model.
|
A doctor has to recognize a disease, i.e. make
a diagnosis.
|
For solving a physics problem students have to
formulate the sequence of steps leading to the solution.
|
A doctor has to formulate the course
of the treatment for treating a disease.
|
If the proposed solution of a problem did not
work, a student has to reflect on the own work and to make a correction, and
to try a new approach.
|
If the treatment did not work a doctor has to
reflect on possible reasons for that and to offer a correction.
|
I am absolutely sure of the existence of a direct correlation between an ability of
a person to learn elementary physics and the ability of that person to treat
people (or, for that matter, to do any highly intelligent work; I would
love to teach a one semester physics course for business majors or politicians).
I would not like very much if my physician (as well as my financial adviser, or
a representative) had less than A- for an elementary physics course.
P.P.P.S.
A while ago I sent a polished and extended
version of this article to “The Physics Teacher” magazine. The magazine
informed me (a) “that this manuscript
does not draw upon the vast literature on this subject” - which means, there
are not enough citations; which is natural, because it mainly draws upon
reflection of my professional experience, and (b) it is not “directed at teachers of introductory physics” (I still cannot
understand why teachers of introductory physics do not need to mull on “What
does “thinking as a physicist” mean?”, why - from the point of view
of the reviewer – a physics teacher would not be interested in this topic?
The good thing was I did not have any critique
of the content.
Dear Visitor, please, feel free to use the buttons below to share your feelings (ANY!) about this post to your Twitter of Facebook followers.
No comments:
Post a Comment