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There are many theories on how to teach students to solve physics problems. A theory – any theory – is simply an intellectual tool, which we 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 hope, might work for you, too.
First we need to see the difference between a problem and a task.
If a person has “a problem” to solve and knows the solution and just have to apply it (retrieve from the memory and re-act), it is not a problem - it is a task; if a person does not know the solution and have to create it – this is a problem. Using this language, there are only two situations: (a) one has to perform a task; or (b) one has to solve a problem by creating a new (for the person) procedure, i.e. one has to create a brand new (for the one) solution; of course, one must utilize some of the knowledge previously stored in the memory and we could call this action “creative utilization of previous knowledge”.
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. 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. Of course, 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 task, we have two options: (a) the task is the same but due to some features it is camouflaged as a different one - transfer of knowledge is not required; (b) the task is different and is really new for the brain and the brain does not have the solution (at least in full) in its storage - 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”).
Based on this theory, 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 before all 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.c.a. critically). Although due to the definition of “creativity” the act of creating something new should be seen as the result of an insight, a teacher can help a student to get to that insight as close as possible using the “algorithm” described below.
I. Psychology of creating of a solution
1. Convince yourself that the problem has a solution (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 you do it absolutely independently or with engaging of somebody’s help (teacher’s, friend’s); convince yourself that it is possible to think about the problem, to do some actions related to the solution of the problem, to reflect on your actions.
3. Formulate 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.
7. "Convert your “defeat” into a key to a solution " (quoting Viktor Zarezki):
- Analyse the reasons for your previous activities, think about why have you been acting like you have been acting (what has forced you to act in that way – your assumptions, your beliefs). 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 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 do you 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 on a method for creating a solution of the 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 for solving a problem, What else can be tried out in order to bypass or to remove an obstacle and why is this?
- 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.
II. 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 situation of 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.
- 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.
III. Logic of creating a solution
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?
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?
3. Determination of the type of a model:
- 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?
- 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?
IV. Reflection on the process of creating a solution
- 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 most important part of the mental activity leading to the development of skills needed for creating a solution of a problem is 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 questions pointing 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?", and etc..
All textbooks start solutions from writing down the necessary equations, which then get applied 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? In reality, 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 inability 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 belongs to psychology, neurology and educational science. Any algorithm, like any written or spoken text, has a sequence of words (or symbols), which connected to each other in a specific way, mostly linear (like the text you are reading right now). But a brain analyses a huge amount of signals simultaneously, a brain does not work linearly, it works making at the same time a lot of “parallel calculations” (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 a brain used to deal with. Hence there must be a certain/specific process a brain is using for transforming one kind of the structure into another. 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.
©Valentin Voroshilov (1999)
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