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14ИННОВАЦИИ И ПРОБЛЕМЫ В МИРОВОМ ОБРАЗОВАТЕЛЬНОМ
ПРОСТРАНСТВЕ
УДК 371.26 DOI 10.7442/2071-9620-2015-1-8-16
ББК 4.028
S.K. Kaldybaev
(International Ataturk- Alatoo University, Bishkek, Kyrgyz Republic)
BASIC CONCEPTS OF PEDAGOGICAL MEASUREMENTS THEORY
In modern conditions of society dimension plays an important role. There is no scope of practice of human activities that does not use the measurement results. The development of science and technology has indissoluble links with to the increasing role of the measurements. Measurement is one of the methods of scientific knowledge. Czech philosopher K. Berka notes that modern science has grown up from a measurement, and has established itself only by measurement. By studying the questions, included in the definition of a range of concepts, defining their functions, pedagogical measurement of the ordinary, empirical knowledge, enriched the ranks of unfizical measurements and which turned it into the applied theory. Today pedagogical measurement has its own conceptual apparatus and operates its facilities and specific objects, reflects significant internal communication of the testing ofpedagogical object and the laws of their development.
Keywords: measurement, quantity, theory of measurement, measuring instruments, nominal scale, ordinal scale, interval scale, the scale relations.
С.К. Калдыбаев
(Международный университет Ататурк-Алатоо, г. Бишкек, Кыргызская Республика)
ОСНОВНЫЕ ПОНЯТИЯ ПЕДАГОГИЧЕСКОЙ ТЕОРИИ ИЗМЕРЕНИЙ
В современных условиях развития общества измерение играет большую роль. Нет ни одной сферы практической человеческой деятельности, где не используются результаты измерений. Развитие науки и техники имеет неразрывные связи с возрастанием роли измерений. Измерение является одним из методов научного познания. Изучая вопросы определения входящих в свой круг понятий, определяя свои функции, педагогические измерения из обыденных, эмпирических знаний, пополнили ряд внефизических измерений и превратились в прикладную теорию. Педагогические измерения имеют свой понятийный аппарат и оперируют своими объектами и специфическими предметами, отражают существенные внутренние связи исследуемого педагогического объекта и
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Ключевые слова: измерение, величина, теории измерений, средства измерений, номинальная шкала, порядковая шкала, интервальная шкала, шкала отношений.
1. Physical and non-physical measurement. The Big Soviet Encyclopedia defines measurement as the "operation by means of which the relation between one value (being measured) and another homogeneous value (taken as the measurement unit) is defined" [4, p. 77]. The Philosophical Encyclopedic Dictionary defines measurement as the cognitive process [13, p. 202].
According to the definition, measurement assumes doing the following:
1) choosing the measurement object;
2) determination of the measured value;
3) choosing the measurement tool (instrument);
4) influencing the object with the tool;
5) getting the numeric value.
One can notice that measurement is intrinsically connected with the required value. If the value is observable, i.e. its magnitude (number) can be derived from the value itself, such a procedure is called a direct (fundamental) measurement. Length and mass measurement are examples of direct measurements.
This principle can be illustrated by the following scheme:
-►
operation
value
magnitude
However, many values are not observable. Measurement of such values depend on measurement of other values. Thus, the key task of measurement is to obtain such values. For example, the volume of the object and the density of a substance
are determined with lengths, sides and heights of the object, volume and density of the substance. The process of obtaining magnitudes for such values is called an indirect measurement. Such classification exists for physical measurements.
non- observable value
observable -w values operation
values operatio
Mathematical methods have penetrated into social sciences thus allowing to refer to use of measurement methods and procedures in social sciences and led to the emergence of notion defined as a "non-physical measurement". Such kinds of measurement are used in economics (e.g., profit measurement), sociology (for determining of preferences given by a human to one or another social event), in pedagogical science (revealing of knowledge acquisition level), in psychology (intellectual capabilities measurement).
Non-physical measurements include mostly latent (subtle, not evident) values. They are measured indirectly through their transformation into observable values which are further subject to measurement and
appropriate interpretation together with the measured object. It is explained by the fact that the studied object can be characterized by several interconnected properties with specific structures. In this case, it is necessary to obtain corresponding numbers with their structures and relations to get the values.
To carry out non-physical measurements one shall do the following:
1) set an obj ective and choose the obj ect for study;
2) analyze the set of characteristics (properties) of the object and identify the measured characteristics in the object;
3) identify values corresponding to characteristics;
4) establish a measurement unit and acceptable measurement standard;
5) apply the measurement standard to the measured value;
6) obtain a numeric characteristic of object properties of a quantitative model characterizing the object.
Therefore, measurement is a complicated and multifaceted concept. Object properties (characteristics) are always interrelated in some way. They are subtle but shall be detected. The values corresponding to these characteristics shall also be interconnected
and interrelated in the same way. To make these numbers isomorphic towards the values, their properties shall reflect these links and relations. Such properties of objects and values are called empirical relational systems (ERS), and such numbers - numerical relational systems (NRS). In this case, measurement will be characterized as an isomorphic (or homomorphic) reflection of ERS in NRS which is presented as a scheme below:
ERS isomorphic NRS
reflection
This provision gave grounds for development of different measurement theories. The representation measurement theory is considered to be a classical one according to which measurement is a process of assigning numbers to measured objects. The core of representation is that the numerical system presents, i.e. represents the measured object for a conscious study of object properties. This theory is associated with N.R, Campbell, one of the founders of the measurement theory.
Study of measurement operations nature led to the development of an operational measurement theory. This theory was developed by S. Stevens. According to his concept, the measurement theory is a theory of scales based upon "properties of scale type transformation and acceptable statistical operations on empirical data depending upon these types" [11, p. 142]. The representation theory framework included a formal theory as well, defining measurement as a "homomorphic reflection of some empirical relational system (empirical structure) onto some numeric relational system (numeric structure)". This definition was suggested by A. Tarski. The empirical relational system shall be understood as a variety of object properties with relations and operations. ro There are other classifications of
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measurement theories. In terms of scientific * development one can identify fundamental ^ and applied measurement theories which,
in their turn, are divided into: physical, mathematical, non-physical (sociological, psychological, pedagogical, etc.), algorithmic, etc.
2. Nature of pedagogical measurements. Pedagogical measurement refers to the area of non-physical measures being a part of a fundamental, representation measurement theory. It should be taken into account that this kind of measurement, as distinct from an indirect measurement used in natural sciences, requires performance of additional procedures. For example, when determining the prism capacity, a formula V=Sh is explicitly set, then, simple transformations help to determine the sides of prism base, and its height is set or obtained in some way. Pedagogical science does not stipulate obtaining of a specific "formula" for setting a measurement pattern. In most cases it is impossible to accurately establish properties of pedagogical phenomena being measured.
The measurement shall be related to the technical categories by its nature. The term "measurement" itself refers more to technical procedures. In that sense many researchers define measurement as a "process of assigning numbers for representation of properties" (N.R. Campbell), "assigning of numbers to the objects or events in accordance with the rules" (S. Stevens), "homomorphic reflection of an empirical relational system onto a numerical relational system" (A. Tarski).
Social science gives the following definition: "Measurement is nothing but one of the ways to get information from the phenomenon being observed consisting of correspondence between the object of social reality and a specific numerical system" [2, p. 89]. The nature of measurement in psychology is defined as an "identification of quantitative characteristics of psychic phenomena being studied" [9, p. 132-133].
N.M. Rosenberg stated the following definition of measurement in the 70-s: "Pedagogical measurement is a cognitive process where an isomorphic empirical relational system is assigned numeric values characterizing some properties of pedagogical objects or phenomena, or indicate the class to which they relate on the basis of the previously obtained numerical system (or the system of classes)" [10, p. 15]. V.S. Avanesov defined a pedagogical measurement as a "process of numeric reflection of levels revealing the personal quality of interest" [1, p. 15]. MB. Che-lyshkova characterizes measurement as a "process of establishing correspondence between the characteristics of learners being evaluated and points of empirical scale in which the relations between different characteristic evaluations are expressed by numeric sequence properties" [14, p. 63].
3. The scale and its meaning for pedagogical measurements. Measurement is closely connected with modelling of an objective reality and determination of numeric characteristics of this model and, finally, with the statement of conclusion on the object being studied. Naturally, this is a methodological part of measurement. Moreover, the key difference of measurement from any other procedure of object study lies in the study of the object itself via its quantitative indicator which stipulate the "accession" of a scientific aspect of object properties study.
On the other hand, identification of object properties is not yet a measurement as these properties have not become measurable yet. That's why it is necessary to obtain such values (variables) which will be both
measurable and directly connected with object properties. Only after determining of such values, measurement becomes possible. In this connection, one of the methodological issues of the term "measurement" includes the notion of a "measurement scale".
Until the 50-s years of the XXth century, when the notion of a measurement scale obtained its scientific status, measurement had been performed without strict consideration of its level. While analyzing the process of development of mass measurement in his work, V.I. Mikheev gives examples of similarities with pedagogical measurements [6, p. 25-26] and points out that the only difference between the measurements consists in the fact that mass measurement is finished by its nature. In the XlXth century an international standard for mass measurement was established as well as the measurement unit. These issues are still open-ended for pedagogical science as "the measurement technique and the corresponding measurement theory have not been developed yet" [6, p. 28]. Nevertheless, in the 50-s of the XXth century a certain progress in the measurement theory development was observed as the scale types were developed that become the landmark for a systematic study of pedagogical phenomena properties.
Well, what is a measurement scale in general and in a pedagogical sense?
1. A scale is generally understood as a measure, i.e. a scaling ratio. These are so-called marks equally spaced on the tools to determine the number of constituent measures (mass, length, volume, etc.). In mathematics, e.g., a scale denotes a scaling ratio used to compare the properties of different subjects. For physical measurement this term is defined as a measurement tool, and more precisely, as an element of such tool, evenly divided marks of the measurement tool. Complying with these terms, the literature gives a definition characterizing these actions.
2. Sometimes the value (an indicator, a variable) itself denotes a scale. A so-called hardness scale, for example, is a wide-known
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scale named after the German scientist F. Mohs (the Mohs scale). One can find lots of cases in pedagogical measurements where a specifically designated value characterizes a scale.
For example, indicators "high level", "mean level" and "low level" are identified as a scale of properties measurement that can be entitled as a "learner competence level". The value "n years" may be characterized as the scale for determination of the pedagogical phenomenon property entitled as a "pedagogical experience". Numeric values such as "2", "3", "4", "5" can make a measurement scale of a property "academic progress".
From this perspective, the concept of scale is interpreted as a measurement standard. A scale is formed by means of measured values (variables, indicators). In case of identification of the obtained scale and the value, the values become the scales. In other words, if the value of a measured property is a measurement model, or standard, it certainly becomes a measurement scale.
3. The formal measurement theory interprets a scale as a measurement itself. All the procedure in this case acquires the values of a measurement scale. American researchers P. Suppes and J. Zines give the following definition to the term "scale": "Let's assume that U is an empirical relational system, f is a function which reflects U in a subsystem N in a homomorphic way ... Let's call an ordered triad <U,N,f> a scale" [11, p. 19].
But it would be a mistake to think that the concept of scale replaces the concept "measurement" completely. In this case, it becomes a kind of a pre-stated rule, or instruction, for measurement, i.e. a measurement procedure. The scale can be understood both as a process and as a result. Being a process, it includes the procedure of > reflecting the properties and relations of the .2 measured object, properties and relations of ^ measured values (indicators). Being a result, * a scale represents a final measurement w "prototype".
4. The concept of scale is sometimes interpreted as a measurement tool for some value characterizing the object. The history of measurement has numerous examples when any tests, questionnaires or surveys serving as measurement tools were entitled as measurement scales. Such examples include the "Binet-Simon Scale", "Likert Scale", "Thurstone Scale", "Guttman Scale", etc. It shall be noted that not all measurement tools become measurement scales.
To become a scale, these tools must meet certain requirements and be able to perform the functions of a scale. First of all, they should be closely connected with the measured object. Secondly, each item on the test or survey question should act as an indicator, i.e. the value or the indicator characterizing the object. In the third place, these tools should be able to reflect the empirical relations system in the numerical relational system (a system of symbols). Hence, this tool helps to create the whole procedure, the scale pattern. Only in this situation we can speak about measurement tools as a measurement scale.
4. Characteristics for scale typology. For a long time physical measurements have used the scaling ratio and the tool calibrated scale. The research continued, and development of measurement tools enabled to use a standard scaling measure. Research on this issue was finalized in 60-s of the 20th century in the work of an American psychologist S. Stevens [11]. Two interdependent characteristics formed the basis for the scale typology:
1) demonstration of object properties:
a) identity forming the basis for object classification, i.e. the objects can be referred to one or another class;
b) transitive property enabling to organize objects in a certain order, e.g., according to one's rank;
c) metricity implying a measurement unit which enables to accurately determine the differences between the objects;
d) presence of a precise reference point.
2) acceptable transformations of numbers corresponding to demonstration of object properties:
a) one-to-one transformation, identity relation (operations: equal - not equal);
b) strongly monotone transformation, order relation (operations: equal, not equal, greater/less than);
c) linear transformations, i.e., difference quotient (operations of multiplication and addition);
d) similarities, i.e., relation of relation (operations of multiplication and division).
Basing upon these characteristics, S. Stevens determined 4 types of scales: a nominal scale, an ordinal scale, an interval scale and relation scale. Let us characterize each of them and give examples of their application in pedagogical science.
5. Nominal scale. It is characterized by the object under study being assigned the numbers (or symbols) for the purpose of their classification and identification. Studied objects in this scale are grouped by their identical measured property. .This scale can only distinguish objects by their names, but does not allow to range, or to put them in a certain order. Numbers or symbols assigned to the objects are just marks (identifiers) of the corresponding classes. These numbers and symbols do not suit for arithmetic operations. Examples may include names of groups at the faculty or names of forms in school, a sex or a nationality, codifiers of students and academic staff data bases, etc.
During the study of the object properties their objects categories are identified, and, depending on them, numbers or symbols are chosen. To identify these categories, one can use numbers: for example, "man" is marked by the number "1", while the category "woman" is marked by the number "2", or: the category "man" is marked by the symbol "M", and "woman" is marked by the symbol "w". When a student data base is developed, the item "faculty" has a certain number of categories (for example, there are five faculties; the codes may be 1, 2, 3, 4, 5). The property "year of study" will be determined with the help of numbers 1, 2, 3, 4, 5. The property "group" can also have numeric values depending on the number of groups
(for example, 35 groups). In this case, the code (number) 453 signifies the code address of the fifth-year student from the 4th faculty who studies in the 3d group.
The nominal scale uses the following mathematical and statistical operations:
1) mode calculation, i.e. determination of frequently occurred values;
2) calculation of percentage ratio of the studied properties;
3) determination of an interaction factor for the property and the object.
As one can see, this scale provides a minimum information, but stipulates wide range of objects and a choice of symbols for identification.
6. The ordinal scale puts objects in order according to the measured properties. The literature also uses the terms "rank scale" and "order scale" to denote this scale type.
This scale ranges the object (or object properties) according to the subordination hierarchy. The ordinal scale is widely used in pedagogical science. One of its advantages is assignment of scores for demonstration of phenomena attributes according to which these properties can be ranked and put in order. The scale ranks objects and assigns scores basing on how evident is the object measured property for demonstrated attributes.
For example, marks "excellent", "good", "satisfactory" and "poor" exemplify the ordinal scale. Marks expressed as scores ("2", "3", "4", "5") are subject to ranking, one score is greater (or less) than the other. But the extent to which one score is greater or less than the other remains unknown. We don't know, for instance, to what extent the score "5" is better than the score "4". Probably, the difference between the mark "excellent" and the mark "good" is not that significant while this difference can be great for marks "good" and "satisfactory". The fact is that there is no equal intervals between these marks. Intervals on this scale exist, but they are not equal. For instance, numbers (scores) can be situated as follows:
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There is no a compliance with relations of a kind 1+4=2+3 or 3-2=5-4 on this scale. It might be that the distances 1+2+3 and 3+4+5 are equal according to the scheme. But arithmetic operations 1+2+3^3+4+5. That's why the procedure of scores (marks) averaging, i.e., calculation of average values (e.g., an average mark for a group) is not advisable for this scale. We can only determine that 1<2<3<4<5.
That's why some countries use non-numeric ordinal scales, for example, A -"excellent", B- "good", C - "satisfactory", F- "poor".
The scale gives an opportunity to perform the following operations:
1) mode;
2) median, i.e., the value of mark having the same number of score distribution on the right and on the left;
3) quintiles indicating the measures of dispersion;
4) rank correlation coefficients determining the measure of relation between two properties.
7. Interval scale. This scale includes the first three of the above-mentioned properties manifestations and has the ability to overcome the drawbacks of nominal and ordinal scales. This scale can determine to what extent one object is greater than other, i.e. to accurately determine the distance between objects. This scale is characterized by the measurement unit, but it doesn't have a natural reference point. Some conditional zero point is chosen as a reference one. Presence of measurement units enables to perform the arithmetic operations on the scale, such as addition, diminution, multiplication, raising to power. Absence of the zero pint does not allow to perform a division operation.
The interval scale includes monotonous and linear transformations. Mathematically, this statement can be expressed as a ratio of S = Ax + B, where S - the measured variable, > A - a measurement unit, x - numerical value
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of the variable in the adopted unit, B- the ^ reference point.
* The reference point depends on the
w variable nature. For example, the "level
of acquisition" has no natural reference point, thus, "low or zero acquisition level," or zero score for the assignment do not mean absence of any knowledge. Another example is the reference point for the day is midnight, zero hours which does not mean, however, "the absence of time". Anyway, nobody says it. The zero value of the date denotes some conditional moment, e.g. the "Nativity of Christ".
This scale provides an opportunity to determine to what extent one object (or property of the other object) is greater than the other. But it is impossible to determine how many times one object is greater than the other. For example, if one learner made 15 tasks correctly, and the other made only 5 tasks correctly, then, we can say that the first learner made three times as much tasks than the other but we cannot state that knowledge of the first learner are three times better than the knowledge of the second learner.
The scale gives an opportunity to perform the following transformations:
1) mode;
2) median;
3) ranked correlation coefficients;
4) average value;
5) dispersion;
6) mean-squared deviation;
7) correlation coefficient.
Comparing to the ordinal scale, the interval one has an equal distance between intervals on the numeric line. Due to the conditionality of the reference point choice, the initial value of the scale can become any number from a sequence.
-1 0 1 2 3 4 S
8. Relation scale. It provides a high level of measurement. It allows you to relate the object to a certain class, put the object in order according to the properties, to reveal to what extent one object is greater (or less) than the other. Unlike other scales, this one has a reference point whereby it can be determined to what extent one object is greater (or less) than the other or how many times a single object property is
greater (or less) than the similar property of another object. The reference point is not a conditional one but a fixed one, and number 0 means the absence of a measured property. For example, a number of books on the specific subject in the library has a reference point, and zero means there are no books on the subject.
This scale can present the numeric relation in the form of equation: P=Ax. For example, the number of first-year students is twice greater than the number of fifth-year students. This assumption can be expressed as X1=2X5 where X1 is a number of first-year students, X5 - a number of fifth-year students. Besides, one can establish equality of numeric relations. For example, X1=60 -a number of first-year students, X2=40 - a number of second-year students, X3=30 - a number of third-year students and X4=20 is a number of fourth-year students. In this case the equality of a numeric relation can be expressed as follows:
This looks as follows on a number line:
At the beginning of a relation scale all arithmetic operation may be applied as well all the mathematical and statistical ones. The relation scale measures physical quantities (length, height, mass, volume (capacity), etc.). Therefore, this scale is designated for properties having quantitative characteristics.
Apparently, each subsequent scale includes the properties of previous scales. A transition from a low-level measurement to a high-level one is accompanied by the expansion of acceptable transformations. At the same time, some low-level scale properties are incompatible for high-level scales. For example, an interval scale does not admit an interval scatter often used in an ordinal scale. Consequently, one can conclude that the transition from a low-level scale to a highlevel scale is impossible [7, p. 16].
Most researchers characterize the above mentioned scale types. But this is not the only scale type classification. For example, L.B. Itelson speaks about 3 types of scales (nominal, ordinal and interval) paying attention to the following main operations: registration, ordering and comparison [5, p. 55-64]. V.I. Mikheev identifies 5 scale types: a nominal scale, an ordinal scale, an interval scale, a relation scale and a difference scale [6, p. 30-31]. A.I.Orlov adds a difference scale and an absolute scale to the above mentioned four types [8, p. 40-44].
Thus, it is worthwhile to say that modern science has developed out of measurement and validated its status only due to measurement [4]. References:
1. Avanesov V.S. Fundamentals of educational measurement theory. Pedago-gicheskie izmerenia. 2004. № 1. P. 1521. [in Russian]
2. Agabekian R.L., Kirichenko M.M., Usatikov S.V. Mathematical sociology. Rostov: Fenix, 2005. [in Russian]
3. Berka K. Measurement: Concepts, theories, problems. M.: Progress, 1987. [in Russian]
4. Grand Soviet Encyclopedia. - M.: Soviet Encyclopedia, 1976. Vol. 2. [in Russian]
5. Itelson B. Mathematical and cybernetic methods in pedagogy. M., 1964. [in Russian]
6. Mikheev V.I. Modeling and theory of educational measurement. - M.: Logos, 2005. [in Russian]
7. Novikov D.A. Statistical methods in educational research (typically). - M.: MH-Press, 2004. [in Russian]
8. Orlov A.I. Measurement theory and pedagogical diagnostics. Pedagogiches-kie izmerenia. 2004. №1. P. 22-51. [in Russian]
9. Psychology: Dictionary. M.: Politizdat, 1990. [in Russian]
10. Rosenberg N.M. Measurement problems in the. Kiev: Vyshcha Shkola, 1979. [in Russian]
11. Stevens S. Experimental Psychology. V.1. - M.: Izd-vo inostrannoi literatury, 1960. [in Russian]
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12. Suppes P., Zines J. Fundamentals of the theory of measurement / Pedagogicheskie izmerenia. Collection. - M.: Mir, 1967. -P. 9-110. [in Russian]
13. Philosophical Encyclopedic Dictionary. M.: Sov. Encyclopedia, 1983. [in Russian]
14. Chelyshkova M.B. Theory and practice of designing pedagogical tests: Textbook. M.: Logos, 2002. [in Russian]
Поступила 19.02.15
About the author:
Kaldybaev Salidin Kadyrkulovich, Director, Institute for Development and Innovation of International Ataturk- Alatoo University (1/8, Str. Ankara, Bishkek, Kyrgyz Republic), Doctor of Sciences (Education), Professor, kaldibaev@rambler.ru
For citation: Kaldybaev S.K. Basic concepts of pedagogical measurements theory // Contemporary Higher Education: Innovative Aspects. 2015. № 1. Р. 8-16.
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Об авторе:
Калдыбаев Салидин Кадыркулович, директор института развития и инноваций Международного университета Ататурк-Алатоо (Кыргызская Республика, г. Бишкек, ул. Анкара, 1/8), доктор педагогических наук, профессор, kaldibaev@rambler.ru
Для цитирования: Калдыбаев С.К. Основные понятия педагогической теории измерений // Современная высшая школа: инновационный аспект. 2015. № 1. С. 8-16.
