General algorithm of making the solution
An
example of algorithm for solving a problem
(the linear motion)
- Read the text; make sure you understand each word
- Make a sketch; mark on the sketch all the important moments of time,
intervals of motion; locations of a body (bodies); displacements of a body
(bodies); velocities; accelerations
- Write for each interval of the motion the equation by using the
notations
- Substitute into the formulas all known magnitudes
- Solve the obtained set of equations
- Read the text
- Make a sketch (the body, the direction of its
motion, the initial and final position)
- Point on the sketch all the necessary/important
instants of time and intervals of motion
- Notate the each specific instant of time, the each
specific displacement and the each specific speed
- Wright for each intervals of motion the equation of
the linear motion with a constant velocity using the given notations
- Substitute into the formulas all known magnitudes
- Solve the obtained set of equations
Technology of
making a solution
Reflection of the
process of the creating of a solution
I. Psychology of creating of a solution
1. Convince yourself that the problem has a solution
2. Convince yourself that you can find/create the solution of the problem;
it is not really important can you do it absolutely independently or with
engaging of somebody’s help (teacher’s, friend’s)
3. Formulate some simple to perform operations/actions, from which it
would be possible to begin a solution, something that is possible to proceed in
conditions of a problem
4. Make a chose what the action are you going to do right now and do it
("enter into a cold water"), convince yourself that it is possible to
reflect the problem, to think about the problem and to do some actions on the
problem
5. Keep acting and acting, make different attempts to obtain any new
information from the text of the problem, try various variants of operations,
fix their outcomes. If the problem is still not solved, proceed to the algorithm
of creating a solution
6. Fix/record the specific gap between the goal of the problem (unknown)
and the state achieved in the solution of the problem
7. "Convert your ignorance (lack of knowledge) into a key to a
solution ":
- Analyse the reasons/premises for organization of your previous
activities, think about why you have been acting like you have been acting (what
has forced you to act in that way). The reason for errors were made or for you
got stuck lies either in an inaccuracy of your premises, or in their
insufficiency (you have made a mistake at some step or you do not have all the
necessary information)
- Formulate the new question to the problem, the answer on which could allow
you to make a new step in a solution of the problem;
- Locate search areas to find the answer on the question, formulate
methods of searching of the answer
- Find the answer on the raised question, formulate additional obtained
information
- Formulate a hypothesis on a method of a solution of the problem
(determine the sequence of steps which could lead to the solution)
- Check up the hypothesis; proceed the (hypothetical) method of the
solution
- Get the result, if not yet, ask yourself the set of questions: Am I
really want to solve this problem, Am I sure in my success, Who can assist me
in my work, Am I ready to start, Do I get myself in circles doing again and
again the same attempts/steps, Why have I started to do this, not that, Because
of what premises I proceed 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 is it possible
to try to do in order to bypass or to remove an obstacle and why is this?
- Get the result, if not yet, go back to #6
II. Technique for creating a solution
1. Analysis of a situation:
Select (and formulate the reasons for your selection):
- Key objects
- Main interactions between objects
-
- Have you met the similar situation before?
2. Abstractization and schematisation:
- Determine main empirical terms used for the description of the physics
situation of the problem
- Make the visual image of the situation (draw a detailed picture)
- Link empirical terms to appropriate physics concepts (locate the
appropriate region of physics);
3. Statement of a problem in 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 closest to the situation considering to the set of variables
standing for key parameters
5. Mathematical description
- Define 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
- Determine 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
- Fix/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
- Analyze the obtained results in point of view of their reasonableness,
naturalness, consider the possible limiting cases
III. Logic of creating a solution
Corresponding to the algorithm described above, the below is 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 they are participating,
do they experience any changes
- What is having an influence on the objects, do some objects act on another,
are there some interactions
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) are used to describe/indicate
properties of both bodies and processes
- What is the way to represent each object and what is happening to it on
a sketch
- What theoretical categories/terms Physicists use to describe the similar
objects and processes
- What is a possible "translation" of the text of the problem into
a theoretical language?
3. Determination of the type of a model:
- What are main physical quantities (terms, categories) are used for the
description of a situation
- What physical phenomena can be described by using the same physical
quantities (terms, categories)
- What are main parameters of classification are used to select
appropriate model
- What are values of these parameters for our problem?
- What is the name of the physics model/models which have the same values
of the same parameters?
4. Mathematical description:
- What are main physical quantities are used for the description of the
selected models
- What are main physical quantities from above connected by some physical
relations/dependents?
- By what kind of equations are the physical quantities connected
5. Solution:
- What are physical quantities used in the equations which a relevant to
the selected model/models
- Can we stand 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?
- Ether the obtained solutions are reasonable or they contradict to our
experience?
IV. Reflection of the process of the
creating of a solution
- Analyse the process of the
creating of a solution: - about what, in what sequence, for what reason, with
what outcome it was necessary to think during a creating a solution; what happened
during the reasoning; what problems were overcame; what kind of emotions have
been experienced
- Analyse the solution found: - is the method of the creating of the
solution applicable to the given problem only or it can be generalized for the
class of problems; what indicators determine this class of problems (by using
which indicators can a problem been assigned to the given class)
- State a general method for problem solving of the problem from the
given class/set of problems
The most important part of the mental activity leading to the structuring
the developed skills needed for creating the solution of a problem is the
reflecting the own activities performed during the problem solving process.
Technically, this part of the activities is carried out by answering to a
number of the questions pointing to oneself, such as: "Am I really want to
solve this problem?", "Am I sure in my success?", "Who can
assist me in my work?" , “Am I ready to start?", "Do I get
myself in circles doing again and again the same attempts/steps?",
"Why have I started to do this, not that? ", "Because of what
premises I proceed 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 is it possible to try to do in order to bypass or to
remove an obstacle and why is this? " And etc…
All the textbooks start the solutions from
writing down the necessary equations, which then get applied to solve the
problem. Reading this way to solve a problem, students keep being curios, how
did the author know what kind of equations to choose? I teach my students that
writing down the necessary equations is the final step of analysis! Physics is
done after that! Math is starting. The main cause for misunderstanding Physics
and for disability to solve Physics problems is the lack of experience of making
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 one of the most interesting problems
of creating new efficient educational tools, which belong to psychology,
neurology and educational science. Any algorithm, as also any written or spoken
text, has a sequence of words, which a connected to each other in series. But a
brain works out a huge amount of signals simultaneously, a brain do not used to
work in series, it used to work in parallel, making “parallel calculations” as
such as computer with a lot of parallel working processors. The structure of
the information translated to students do not correspond to the structure of
information a brain used to deal with. Hence there must be a
certain/specific process brain is using for translating one kind of the
structure into another. The effectiveness of this kind of translation has to
have a direct influence on the effectiveness of the mastering of a subject.
Usually this process happens in a natural way without a purposed influence of a
teacher. I believe, if researchers could find the ways of working with such
translating process it would lead to new approaches for constructing educational
tools and organising lessons.