Method of determination of difficulty of physics problems

In the given work the way/mechanism of determination of complexity/difficulty of physics problems/task is offered. The main approach of the mechanism seems to be useful for creating a learning software which is to be able to adjust itself on the weaknesses and strongnesses of students.

We will represent visually each a unit of information contained by the subject such as Physics or Math as a circle. The logical/operational connections/dependences between these units of information will be represented by lines drawing from one circle to another one. Altogether the lines and the circles are picturing a graph, which is mapping the subject. Every circle is the vertex of the graph, every line is an edge. This kind of graphs is like a map for the ocean of subject knowledge, where the circles are the islands of information, and the lines are the ways to get form an island to an island.

In Physics each the vertex can be assigned to a certain Physics term/category, and the lines between them is representing the certain operational connection, i.e. an equation/formula, which involves the terms. We name this kind of a graph as a System Operational Connected Categories (SOCC).

The simplest structural unit of any SOCC you can see on Fig. 1

At any instant of time the knowledge of a subject for any person can be represented by some SOCC. During the time and because of learning SOCC of any person being developing.

The SOCC is a visual reflection of existing categories of a subject and logical correlations of them. The statements connecting categories should allow to make the certain logical operations on the categories. To construct a SOCC of a subject it is convenient first to determine all main statements/equations, which are most important for study. All the categories included in selected statement will by represented by the circles, and all the statements will be represented by the lines. The construction of SOCC requires from the teacher/scientists a deep methodological analysis of bases of the subject. It is obviously, there can be different SOCC for the same subject or the part of a subject.

Let's name as an objective system of categories (OSOCC) the SOCC reflecting the modern scientific submissions in investigated discipline, i.e. OSOCC represent point of view of a teacher on the subject, or to be more exact, OSOOC represent point of view of a contemporary methodologists of the subject.

Let's name a subjective system of categories (SSOCC) the SOCC reflecting knowledge of a student. The learning process can be represented as a process of developing and adjusting of a subjective system of categories to objective.

SSOCC + learning = OSOCC

Below we consider as an example of an objective system of connected categories the SOCC of the unit "a kinematics of a point mass", stated within the Physics curriculum for high school with a major in engineering.

The Fig. 2 below represents the OSOCC for that unit of the subject (we use squares instead of circles).

Certainly, the selected statements do not have an equivalent importance. Every teacher can make the own selection, proceeding from own submissions on a level of importance of the statements and categories.

Each statement used in the SOCC is to be described separately.

Below we have the set of main statements used for the SOCC represented on the Fig. 2.

Such concepts as "vector", "derivative", "integral" are assumed known by students from a calculus (these kind of inter-subjectal connections could be represented as lines directed to outside the SOCC).

The table of statements corresponded to the SOCC shows the line and the statement are assigned to this line.

For example, 1 – 4 means the line leading from the category # 1 to the category # 4, and in this article we omit the definitions of the categories (just to save a paper).

1 – 4 DS= DS_X i + DS_Y j + DS_Z k	4 - 11 - 16 V_C = DS/Dt
1 - 5 V = V_Х i + V_Y j + V_Z k	5 - 6 - 15 t = V/V
1 – 7 position vector r – vector, which is leading from an origin of a reference system to a body	6 - 13 - 15, 3 - 13 A_n = Vt¢
	6 - 15 - 17 At = V¢t
	6 - 13 - 18 R = V²/A_n
1 - 14 A = A_X i + A_Y j + A_Z k	8 - 10 - 11 V_av = S/Dt
2 - 4 - 5 DS= ò V dt	1 - 7 - 19 coordinates – components of a position vector
2 - 6 - 8 S = ò V dt	1 - 7 - 19 coordinates – components of a position vector
3 - 5 - 7 V = r¢	9 - 11 - 12 A_C = DV/Dt
3 - 8 V = S¢	2 - 9 - 14 DV = ò A dt
3 - 5 - 14 A = V¢	7 - 11 - 20 the law of kinematics, i.e. the law of changing of a position vector during time
4 - 7 Dr = DS
13 - 14 - 17 A = A_t + A_n

The construction of both the OSOCC and the SSOCC allows to present quantitative criteria of complexity/difficulty of the physics tasks/problems.

Let Nc be a number of categories, i.e. the vertexes, the “informational islands", which are essentially/necessary for creating the solution of the problem given to a student, i.e. without using of these categories it is impossible to construct the solution of the problem. Let Ns be a number of the statements/equations by which the essential categories are connected on the SOCC. These statements are represented by the lines connecting the selected categories on the SOCC. Let N be a total number of the categories and the statements which are necessary to solve the given task (N = Nc + Ns). The number N is the measure/value of complexity of the task; the grater N, the more difficult the given problem, at least because of grater amount of information involved in the solution.

Among the all possible statements connecting the essential categories some of them can be unnecessary for the solution of the problem. Let Nis be a number of all statements from the set of Ns statements which are really necessary to use for constructing of the solution of the problem (in general case Nis £ Ns).

Let’s go further.

All categories essential the solution can be divided into two sets: the first is the set of categories which there are directly in the text of the problem, i.e. a student can just read them in the text (let Ncc be a number of this categories); the second set contents categories which are absent in the text of the problem (let Nch be a number of this categories: obviously, Nc = Ncc + Nch). Let Nisc be a number of the statements necessary to solve the problem and which are directly presented in the text of the problem (for example, as the title), and Nish is a number of the statements necessary for the solution, but the information on which is absent in the text (we shall refer to such a kind of statements, the statement for which there is at least one category that is included in the statement but is absent in the text of the problem).It is obviously, Nis = Nisc + Nish.

If all the necessary for the solution categories and statements would by known by a student (it means the student was taught before to this specific material and he/she keep the information in the memory and can extract the information from the memory), hence in this case complexity/difficulty of the problem would be determined mostly by a number of categories and statements, on which necessity the student should make a guess, i.e. the difficulty of the problem would be determined by the number Nh = Nch + Nish). Therefore, number Nh can be accepted as a quantitative criterion of the difficulty. It is noticeable, that this number is determined only by the problem. Let us name the number Nh as an absolute objective difficulty of the problem (AODP). The general meaning of the AODP is how many specific information is necessary to use and specific operations are necessary to perform to construct the solution (if the solution would be constructed by a person who is really good at problem solving). To make a comparison of difficulties of different problems the relative objective difficulty of the problem (RODP) can be applied, which is defined as a ratio of “hidden” categories and statements to all categories and statements necessary for solving of the problem: RODP = Nh/N.

Far more possibilities for researches and more important situation from point of practice arises, when a student does not know all the categories and/or statements necessary for the solution. It means, that a subjective system of categories of the student has a difference from the objective system necessary to solve the problem. We can say that the students SSOCC graph is not equal to the OSOCC. There are two main differences between the graphs; either the students graph dose not have some essential vertexes and/or lines, or the students graph has wrong vertexes and/or lines.

In this case at least two more parameters can be defined, which describe the mutual relation between the subjective and objective systems. Let Nca be a number of categories necessary to solve the problem, but are not known by the student (those vertexes/categories are absent on the student’s SOCC). Let Nsa be a number of the statements necessary for the solution, but also are not known by the student (those lines/categories are absent on the student’s SOCC). Obviously, the more numbers Nca and Nsa, the more complicated for that student is the problem offered to the one, even if the objective complexity is insignificant.

Intuitively it is clear, that the process of solving of a problem with Nca = Nsa = 0 has an essential difference from the process of solving of a problem with Nca ¹ 0 and/or Nsa ¹ 0. In the latter case a student first should determine what categories and/or the statements are absent and find a way to obtain a missing information (or, for a more complicated situation, to make by he/herself).

We have to distinguish two strictly different situation. The first one is a situation when all the categories and statements necessary for the solution have been introduced to a students before, but at the moment of the solving of the problem some of them could be forgotten by the student. And there is principally different situation, when some of the categories and statements are new for the student, that is he/she have to create them during the constructing the solution. It is obviously, the process for the solving of problems of the latter type includes an essential creative action of a student, i.e. during the solving this kind of a problem a student have to perform research–like activities. This kind of a problems, when a student have to detect insufficiency of his/her knowledge and/or skills and find the way to overcome it, is a crucial element of the theory of developed education by Elkonin D. and Davidov V. For this kind of problems we can define on more measure of the difficulty which is the number of all the categories and statements necessary for the solving of the problem, but which are unknown for the student. Let us name this number as an absolute subjective difficulty of the problem. (ASDP), by a definition ASCP = Nu = Ncu + Nsu.

The number Nu/N, which is equal to a ratio of the number of unknowns for the student to the number of the all the categories and statements necessary for the solution, we shall name as relative subjective difficulty of the problem (RSDP).

Teachers have different interpretations for such terms as "a creative task/problem", "a non-standard task/problem", "a complicated task" etc. In the offered approach as a complicated problem can be considered a problem the solution of that involves a lot of categories and statements, all of them have been introduced to a student before, hence for that kind of problems Nu = 0, but Nh >> 0. In this situation we can assume that a student has been studied all the necessary algorithms, but the number of logical/mental steps necessary to construct the solution is rather big.

If all the categories and statements have been taught to a student but he/she forgot some of them, we can say that Nu > 0, and such a problem could be difficult for the student, but not creative.

The problem/tasks with Nu > 0 can be considered as creative if the teacher on purpose have designed an absence of some knowledge, which a student has to eliminate during the solving of the problem.

As the non-standard such a problem can be considered, for which Nu = 0, i.e. all the knowledge necessary for the solution have been presented to a student before, but the part of them was never used earlier in the consideration of similar problem.

Let's consider now some examples of determination of difficulty of physics problems.

The problem #1 has the following conditions: the kinematics law of motion is given as …. Find coordinates of a body in three seconds after the beginning of the motion.

To solve this problem it is necessary to know categories 7, 11, 19 and 20 and two statements such as 7 - 19 and 7 -20. The category 7 is not presented directly at the text. Both statements also should be considered as not presented directly. Hence we have: N = 6, AODP = Nh = 3, RODP = Nh/N = 0.5.

Problem #2: the kinematics law of motion is given as …. Find the value of the velocity of the body in three seconds after the beginning of the motion.

The solution of this problem requires knowledge of categories 6, 11, 20, which are presented in the text directly; and also categories 5 and 7, which are not presented in the text directly. Also we have to know statements 7 - 20, 5 - 7 and 5 - 6, which have to be considered as not presented directly. Hence, for this problem we shall receive the following: N = 8, AODP = Nh = 5, RODP = Nh/N = 0.625.

We can see that the problem # 2 is more difficult as the problem # 1in point of view knowledge we have to use to solve the problems. But for the student who is equally aware with the necessary theory and practice (Nu = 0) both of the problem would be equally easy to solve.

Problem #3: the kinematics law of motion is given as …. Find the value of the acceleration of the body in three seconds after the beginning of the motion.

Categories of this problem: directly presented in the text are 11, 14, 20; not presented directly in the text are 5, 7. The statement of the problem 7 - 20, 5 - 7, 5 - 14 are all not presented directly in the text.

We have for this problem N = 8, AODP = 5, RODP = 0.625. We can see that the objective difficulties of the problems #2 and # 2 are the same.

Let's introduce an example of determination of subjective difficulty of the problem. Let say the student has solved successfully the problem #1, but has not solved the problem #2. It means, that he/she is not familiar with category "acceleration" and do know the connection between velocity and acceleration. Hence, for the problem #2 and for thit student we have Ncu = Nsu = 1, that leads to ASDP = Nu = 2, RSDP = Nu/N = 0.25.

This example can be considered as an example of "X-ray picturing" of the subjective system of operationally connected categories of the student. As a result of such kind of "scanning" we can see the specific defect of the system (absence of the category 14 and the link 5 - 14).

Let's repeat one more time that the objective difficulty of a problem is determined by a structure of objective SOCC only and does not depend on the knowledge of a person, but the subjective difficulty can be different for different students even for the same problem being solving.

The further developing of the described method has to involve quantification of the difficulties leaded from such steps of the constructing of the solution as a translating the text of the problem from everyday language into theoretical language and recognising the appropriate theoretical model.