A General “Algorithm” for Creating a Solution
to a Physics Problem
There are many theories on how to
teach students to solve physics problems. A theory – any theory (a.k.a. a set of our views about specific
issues) – is simply an intellectual tool/instrument,
which we (humans) use to organize our actions. While it works, we use it,
when it stops working, it is a time for a new one – just like a hammer. Below
is my theory, which works for me, and I am sure would work for every physics
teacher.
First, we need to establish the
difference between a problem and a task (an important part of the definition of
Intelligence).
If a person has “a problem” to solve,
i.e. has to achieve a specific goal, and knows the solution (what actions in
what order to use) and just have to apply that solution (retrieve from the
memory and re-enact), it is not a “problem” - it is a task.
When a person does not know the
solution and have to create it –
that is a problem.
If a student learned how to perform
a task, one can repeat it in the future as many times as one has to repeat the
same task (of course, assuming no memory limitations).
The key word is “the same”.
Our brain is a powerful pattern
recognition machine. As soon as it recognizes the task, it retrieves from the
memory the sequence of the actions, which has to be performed to succeed, to achieve
the goal. Here, we assume that that particular brain is capable of storing and
retrieving the information and governing the actions required for fulfilling
the task (otherwise we have to discuss a case of learning disabilities).
If a brain does not recognize the assignment
which is supposed to be a task, we have two options: (a) the assignment is the
same (a task) but due to some its features it is camouflaged as a different one
- transfer of knowledge is not required, instead previously obtained knowledge
has to be regained and reused; (b) the assignment is actually different from
any previously done (a problem) and is really new for the brain and the brain
does not have the solution (at least in full) in its storage – in this case the
transfer of knowledge is not possible; there is nothing to transfer (naturally,
we could talk about how much the new task is different from ones done in the
past, but this conversation is not directly related to “knowledge transfer”).
These two examples show that term “transfer
of knowledge” is not very helpful for describing a problem-solving process.
Based on this analysis, every
teacher has to teach students to two different practices: (a) how to perform
specific tasks (the set of those tasks should be specified by a curriculum);
(b) how to create a solution to a problem which has not been solved in the past
(by that person); the latter practice, in turn, requires a practice in making a
conclusion regarding the familiarity of the given assignment - is it the same
as one from the past (is it a task?) or different (is it a problem?)?
Development of such a skill also requires specific practice.
Teaching how to find a
solution means mostly teaching how to recognize the old task in the new one and
to apply the appropriate method (which worked in the past).
Teaching
how to create/invent solutions (actions, procedures) which have not been
presented/trained before (at least in full) means teaching thinking creatively
(a.k.a. critically).
Although due to the definition of
“creativity”, the act of creating
something new should be seen as the result of an insight (hence,
unpredictable), a teacher can help a
student to get to that insight as close as possible using the “algorithm”
described below.
1. Convince
yourself that the problem has a
solution, i.e. solvable (no one starts acting unless he or she believes “this
can be done!”, or, being forced into acting).
2. Convince
yourself that you can create the
solution of the problem (no one starts acting unless he or she believes “I can
do this!”, or, being forced into acting); it is not really important can one do
it absolutely independently or with engaging somebody for help (a teacher, a
friend, the Internet); convince yourself that it is possible to think about the problem, that you can think about the problem, and can do some actions related to
the solution of the problem, and can reflect on your actions.
3. Formulate
(say out loud) some simple operations/actions you can perform, some steps you
can do to begin a solution, something that is possible to do right now (under
the conditions described in a problem).
4. Make a
choice of what action are you going to do right now and do it ("enter into
a cold water").
5. Keep
acting and acting, make different attempts to obtain any new information from
the text of the problem, try various versions of your actions in different
order, reflect on their outcomes.
6. Reflect
on what is the difference between the goal of the problem (an unknown) and what
you have achieved.
- Analyse the
reasons for your previous activities, think about why you have been acting like you have been acting (what has “forced”
you to act in that way – your assumptions, your beliefs). If you made a mistake
and something did not work out, you got stuck, the reason that to happen is either
inaccuracy or insufficiency of your premises/assumptions (at a certain step of
you work you have made a mathematical or logical mistake, or you do not have
all the necessary information).
- Formulate the new question to the problem,
the answer to which could allow you to make a new step in a solution of the
problem.
-
Locate/state the areas/topics for the search to find the answer to the new
question, formulate methods for searching the answer to the new question.
- Search and
find the answer to the new question, state what additional information you do
have now.
- State what
new steps could have been used now (with this new information), establish the
new sequence of steps which could lead to the solution (basically, this is your
hypothesis of a method for creating the solution of a problem).
- Check your
hypothesis, proceed, try your new approach.
- Get the
result, if not there yet, ask yourself the following of questions: Am I really
want to solve this problem, Am I sure of my success, Who can assist me in my
work, Am I ready to start, Do I get myself thinking in circles repeating again
and again the same steps, Why have I started to do this but not that,
Because of what ideas I proceeded my reasoning in this way, How can it be done
in a different way, What can I try to do instead
of doing this, What can be the obstacle preventing me from solving a problem,
What else can be tried out in order to bypass or to remove an obstacle and why
could it work?
- Get the
result, if not there yet, go back to part 6. To help yourself in creating a
solution, reread the technique for creating a solution in physics.
1. Analysis
of a situation:
Select (and
formulate the reasons for your selection):
- Key
objects.
- Main
interactions between objects.
- Main
processes happening to objects.
- Have you
met the similar situation before?
2.
Abstractization and schematisation:
- Determine
main empirical terms (everyday words) used for the description of the physics in
the problem.
- Make the
visual image of the situation (draw a detailed picture).
-
Link/connect the empirical terms to appropriate physics concepts (state/locate
the appropriate region/area/topic of physics).
3.
Translation of a problem into theoretical language:
- Find the
correspondence between empirical terms and theoretical terms (“a car” = “an
object”, etc.).
- Translate
the text of the problem from empirical language into theoretical
4.
Determination of a model:
- Select
main parameters describing the objects and processes (formulate the reasons for
the selection).
- Select key
parameters describing a situation as a whole.
- Determine
variables for chosen parameters.
- Correlate/compare
the chosen variables with the variables for similar physical models.
- Determine
classes of the phenomena most relevant to the situation described in the
problem.
- Select
models the closest to the situation considering the set of variables used to
the algebraically description of the key physical parameters/quantities.
5.
Mathematical description:
- Establish
the correspondence between specific objects, processes, quantities essential to
the considered situation and the general (abstract, theoretical) objects,
processes, quantities describing the chosen classes of the phenomena and
models.
- Establish
the set of main categories essential to the description of selected classes of
the phenomena and corresponding models.
- State main
laws and definitions relevant to classes of the selected phenomena and models.
- Write main
algebraic statements/expressions corresponding to the laws and definitions.
6. Solution:
- Substitute
the given numbers in the stated equations.
- Perform
the mathematical transformations necessary for determination of the values of
the quantities.
- Analyse
the obtained results from the point of view of their reasonableness,
naturalness, consider the possible limiting cases.
Corresponding
to the technique described above, the below is a description of mental
operations which have to be realised at each stage of the solution; this part
of mental work consists of the answers to the following questions:
1. Analysis
of a situation:
- What can
we say about objects (bodies, things) in the condition of a problem?
- What is
happening to the objects, in what processes are they participating, do they
experience any changes, what changes?
- What is
influencing the objects, do some objects act on another, are there some
interactions, what interactions?
- What main
properties should be listed for each object and each process?
2.
Abstractization and schematisation:
- What words
(usually they are nouns) are used to name the objects/bodies?
- What words
(usually they are verbs) are used to describe the processes (what is happening to
the objects)?
- What words
(usually they are adjectives or adverbs) are used to describe/indicate
properties of both bodies and processes?
- How can
you visually represent each object and what is happening to it using a picture?
- What
theoretical categories/terms a physicist would use to describe the similar
objects and processes?
- What is a
possible "translation" of the text of the problem into a theoretical
language?
- What are
the main physical quantities (terms, categories) used for the description of a
situation?
- What
physical phenomena can be described by using the same physical quantities
(terms, categories)?
- What are
the main parameters of classification used to select an appropriate model?
- What are
the values of these parameters for your problem?
- What is
the name of the physics model(s) which has/have the same values of the same
parameters?
4.
Mathematical description:
- What are
the main physical quantities used for the description of the selected models?
- What of
the main physical quantities from above are connected by some physical
relations/dependents?
- By what
kind of equations are the physical quantities connected?
5. Solution:
- What
physical quantities used in the equations which are relevant to the selected
model/models?
- Can we
select appropriate variables (letters) for the physical quantities using in the
model and can we write the equation corresponded to connections between them?
- What
numerical values can be substituted in the equations for the labels/letters of
the quantities (corresponded variables)?
- How many
unknowns and algebraic equations are obtained as the result of the
substitution?
- How can we
solve the obtained set of equations?
- Do the
obtained solutions look reasonable or they contradict to your experience?
- Analyse
the process of creating a solution: - about what, in what sequence, for what
reason, with what outcome it was necessary to think during creating a solution;
what happened during the reasoning; what problems had been overcome; what kind
of emotions have been experienced?
- Analyse
the solution found: - is the method of creating the solution applicable to only
the given problem or it can be generalized for the class of problems; what
indicators/parameters determine this class of problems (by using which
indicators can a problem be assigned to the given class)?
- State a
general method for solving a problem from the given class/set of problems.
The “algorithm”
looks large, but its essence can be described in one simple rule:
after every
action ask yourself the same question “what could
be done now”, and do it!
The most
important part of the mental activities leading to the development of skills
needed for creating a solution of a problem is reflecting on the own activities performed during the problem-solving
process. Technically, this part of the activities is carried out by
answering a number of questions directed to oneself, such as: "Am I really
want to solve this problem?", "Am I sure of my success?",
"Who can assist me in my work?" , “Am I ready to start?",
"Do I get myself thinking in circles doing again and again the same
steps?", “What can I do now?”, and etc..
All
textbooks and handbooks on problem-solving recommend to start a solution from drawing
a picture, then writing down the necessary equations, and apply those equations
to solve the problem. When reading this approach for solving a problem, students do not know how did the author
know what kind of equations to choose? Choosing the right equations is the
crucial part of the reasoning, which remains in the mind of in expert and inaccessible
to students. That is why the whole problem-solving
process looks for students like a miracle, and that is why students are
convinced that they cannot do the same.
In reality, writing down the necessary equations is the final step of analysis!
In reality, writing down the necessary equations is the final step of analysis!
Physics is done after that!
Mathematics
starts.
The main
cause for misunderstanding Physics and for inability to solve Physics problems
is the lack of experience in practicing the analysis which leads to the
necessary equations! This is the focus, the main goal and the most valuable
result of Physics
education.
The
described “algorithm” introduces the approach to one of the most difficult
problems of creating new efficient educational tools, which ae based on the advances
in psychology, neurology and educational science.
Any
algorithm, like any written or spoken text, has a sequence of words (or
symbols), which are connected to each other in a specific way, mostly linear
(like the text you are reading right now). But a brain analyses simultaneously a
huge amount of signals; a brain does not
work linearly, it works making a lot of “parallel calculations” at the same
time (like a computer with a lot of processors working in parallel). The
structure of the information translated to students does not correspond to the structure of the information which is being processed
by a brain. Hence there must be a certain/specific process a brain is using
for transforming one kind of the structure (linear) into another (topological).
The effectiveness of this kind of transformation has to have a direct influence
on the effectiveness of mastering a subject. Usually this process happens in a
natural way without a purposed influence from a teacher. I believe, if a
teacher could effectively help a brain to make a transformation between
different types of structuring information a brain is processing, it would lead
to better learning outcomes of students (and it would require new approaches
for constructing educational tools and organising lessons). The presented
“algorithm” is one of the instruments a teacher can use when helping a brain of
a student with organising its (brain’s) work.
Thank you for visiting,
Dr. Valentin Voroshilov
Education Advancement
Professionals
To learn more about my
professional experience:
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