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.
Thank you for visiting,
Dr. Valentin Voroshilov
Education Advancement Professionals
To learn more about my professional experience:
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
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.
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.
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