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 Title:
 A study of the use of Markov chain statistical procedures with category type observation data
 Creator:
 Dudasik, Margaret Anne, 1943
 Publication Date:
 1977
 Language:
 English
 Physical Description:
 ix, 125 leaves : ; 28 cm.
Subjects
 Subjects / Keywords:
 Classroom observations ( jstor )
Classrooms ( jstor ) Markov chains ( jstor ) Matrices ( jstor ) Modeling ( jstor ) Observational research ( jstor ) Statistical models ( jstor ) Statistics ( jstor ) Teachers ( jstor ) Transition probabilities ( jstor ) Dissertations, Academic  Subject Specialization Teacher Education  UF Interaction analysis in education ( lcsh ) Markov processes ( lcsh ) Subject Specialization Teacher Education thesis Ph. D
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 bibliography ( marcgt )
nonfiction ( marcgt )
Notes
 Thesis:
 ThesisUniversity of Florida.
 Bibliography:
 Bibliography: leaves 120124.
 General Note:
 Typescript.
 General Note:
 Vita.
 Statement of Responsibility:
 by Margaret Anne Dudasik.
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 University of Florida
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 University of Florida
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 This item is presumed in the public domain according to the terms of the Retrospective Dissertation Scanning (RDS) policy, which may be viewed at http://ufdc.ufl.edu/AA00007596/00001. The University of Florida George A. Smathers Libraries respect the intellectual property rights of others and do not claim any copyright interest in this item. Users of this work have responsibility for determining copyright status prior to reusing, publishing or reproducing this item for purposes other than what is allowed by fair use or other copyright exemptions. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder. The Smathers Libraries would like to learn more about this item and invite individuals or organizations to contact the RDS coordinator(ufdissertations@uflib.ufl.edu) with any additional information they can provide.
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A STUDY OF THE USE OF MARKOV CHAIN STATISTICAL PROCEDURES WITH CATEGORY TYPE OBSERVATION DATA
By
MARGARET ANNE DUDASIK
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1977
ACKNOWLEDGEMENTS
The author wishes to express deep gratitude to each member of her committee. The committee chairman, Dr. Elroy Bolduc, provided the initial guidance which led to the choice of the topic. His assistance in the planning and writing phases was invaluable. Dr. Alan Agresti, through his excellent teaching, interested the author in the field of statistics in general and stochastic processes in particular. His expert advice on procedures was essential to the study. The author wishes to thank Dr. John Grant for his excellent instruction in programing which made it possible for the author to write the programs for this study. His support and cooperation during the pursuit of the degree was appreciated. Although he was very busy, Dean John Newell made valuable suggestions and criticisms in the preparation of the final drafts of the dissertation. Over two years before preparation of the final work, Dean Newell patiently read the author's first attempt at the definition of the problem to be studied, and he gave much needed encouragement. To Dr. Robert Soar and Ms. Ruth Soar go thanks for the use of data from their Project FollowThrough evaluation study. They both also provided suggestions in the preparation of the review of the literature.
A number of dear friends helped the author in the critical last stages of preparation of the manuscript. The excellent typing of the
draft as well as the final copy was done by Ms. Nancy Waters. Her dependability and skill relieved the author of many concerns. For her editorial suggestions and her moral support, a special thanks goes to Ms. Marian Tillotson. The author expects to return the favor some day. The list of friends and colleagues who were supportive during this time in the author's life is too long to enumerate, but to each go sincere thanks.
Finally, the author wishes to express her thanks to Mr. Stephen Dudasik. He encouraged a spirit of selfactualization as did the author's parents, Mr. and Mrs. Roy Pierce. During every phase of the graduate program and the preparation of this dissertation, Stephen provided support and understanding in the face of the author's selfdoubt, and his editorial assistance in the preparation of the final copy was extremely helpful. Without his encouragement the author probably would not have pursued the degree; therefore, to him is extended her deepest gratitude.
TABLE OF CONTENTS
ACKNOWLEDGEMENTS ... ........
LIST OF TABLES
LIST OF FIGURES ....... ..................
ABSTRACT ......... .....................
CHAPTER
I THE RESEARCH PROBLEM ..... ............
Statement of the Problem ... ...........
Rationale for Need for the Study .........
Organization ...... .... ............
II REVIEW OF THE LITERATURE ... ...........
Observation Instruments ... ..........
Direct Observation to Study Teaching ....
Statistical Procedures Used with Observation
Markov Chain Theory in Research ........
Summary and Conclusions . . . . .........
III PROCEDURES ....... ..................
Sample ........ ....................
Transformation of Data ... .............
Markov Chain Model ... .. ...........
Application of Markov Chain Properties . .
Summary .......... ..............
IV RESULTS ....... ...................
Tests of Fit of the Markov Chain Model. . .
Application of Markov Chain Properties. . .
vii viii
2
2
4
6
7
9
Instruments. . 13
. . . . . . . 33 . . . . . . . 36
ï¿½ . . . . . . . 39
ï¿½ . . . . . . 40 . . . . . . . . 42 ï¿½ . . . . . . . 43 ï¿½ . . . . . . . 51 . . . . . . . . 62
64
ï¿½ . . . . . . . 65 . . . . . .7. 70
. . . . . . . . . . . . . . . . . . . . 88
Page
. . . . . . . . . . . . . ii
S . . . . . I . . . . . . . . . . . . . . . . . . . vi
Summary . . .
TABLE OF CONTENTScontinued
CHAPTER Page V SUMMARY AND IMPLICATIONS ........ .............. 92
Objectives .. ..... ........................ 92
Activities, Results and Implications ... ........... . 93
Problems for Future Research .... ............... . 97
APPENDIX
A Observation Systems ...... ................... . 99
AI. Summary of Categories for the Reciprocal Category
System ........ ........................ . 100
A2. Categories for Flanders Interaction Analysis System. 102
A3. Conversions of Categories: Reciprocal Category
System to Flanders Interaction Analysis System
Modified ........ ...................... . 104
B Raw Data Used in the Study ...... ................ 105
C Output From the Complete Analysis of the Second Fall Observation of Teacher A ........ ..... .......... ..113
BIBLIOGRAPHY .......... .............................. 120
BIOGRAPHICAL SKETCH ......... ....................... 125
LIST OF TABLES
Table Page
1 Summary of chisquare and Tau values .... ............ . 67
2 Equilibrium matrix versus proportion in each category ... 75 3 Mean recurrence times ........ .................... . 77
4 Mean and standard deviation of the passage time into
category 6 ......... ......................... . 81
5 Median of passage time into category 6 ... ........... . 84
6 Mean and standard deviation of passage time into category
6 or 7 ........... ........................... .. 86
7 Probability of absorption into categories 6 and 7 ...... ... 89
LIST OF FIGURES
Figure Page
1 Matrix for Bales' display of data .... .............. . 18
2 Summary of statistical procedures in research literature 32 3 Summary of data sources ....... ................... . 41
Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy
A STUDY OF THE USE OF MARKOV CHAIN STATISTICAL PROCEDURES WITH CATEGORY TYPE OBSERVATION DATA By
Margaret Anne Dudasik
June, 1977
Chairman: Dr. Elroy J. Bolduc, Jr.
Major Department: Subject Specialization Teacher Education
This study explored the use of the Markov chain statistical model
with category type observation data. The relative goodnessoffit of the data to the assumptions of independence, onestep dependence, and
twostep dependence was investigated using Goodman and Kruskal's Tau
(T b) and a chisquare (X 2) test statistic. Using a X2 test it was found
that an assumption of onestep dependence or longer was preferable to
an assumption of independence of data items in the chain. A comparison
of Tb values for onestep and twostep tally matrices gave no clear
indication of which was a better model. Some categories were more twostep dependent, while others were more onestep dependent.
Indications that a onestep assumption was not completely invalid,
and the productive past use of the onestep assumption by researchers,
viii
led the author to assume the data to be a Markov chain. Various properties of Markov chains were applied to the observation data and the results interpreted in terms of classroom behavior. A transition probability matrix, an equilibrium matrix, and the mean recurrence time were constructed for each data set. Absorbing categories were created and the distribution of the passage times into the designated categories was investigated.
CHAPTER I
THE RESEARCH PROBLEM
In the past twenty years there has been increasing interest in the study of classroom interaction, and many researchers have sought to develop effective instruments to record classroom behavior. In fact, more than 100 such observation instruments have been published. The statistical treatment of information collected in the classroom, however, has been much less diverse than the recording instruments being used.
Observation data have been traditionally reported in matrix form as frequencies of the occurrence of pairs of events. One of the most influential scholars in the field, Ned Flanders, conceptualized classroom interaction as pairs of events with the first of the pair influencing the event which follows. According to Flanders,
Teaching behavior, by its very nature, exists in a context
of social interaction. The acts of teaching lead to reciprocal contacts between the teacher and the pupils, and the
interchange itself is called teaching. Techniques for
analyzing classroom interaction are based on the notion that
these reciprocal contacts can be perceived as a series of
events which occur one after another. Each event occupies
a small segment of time, and the chain of events can be
spaced along a time dimension. It is clear that the event
was influenced by what preceded.
(Flanders, 1970, p. 1)
Flanders' hypothesis regarding dependence of sequential events
is the fundamental premise of the mathematical theory of finite Markov chains. Application of the Markov chain model to the analysis of observation data, therefore, is a logical extension of the traditional treatment of such data. Probabilities rather than frequencies are displayed in the matrix, and the probability of the occurrence of ensueing events becomes the basis of statistical comparison. The probabilistic relationship between an event and the event immediately following is called onestep dependence.
Statement of the Problem
The assumption of onestep dependence suggests the application
of Markov chain statistical procedures, and the present study explores the heuristic value of the Markov model. The specific questions addressed by this dissertation are:
1. Does category type observation data form a Markov chain?
That is, are the observed behaviors onestep dependent,
or is an assumption of independence or twostep dependence
more appropriate?
2. How can the Markov chain model be used to analyze observation data?
Rationale for Need for the Study
Traditionally, researchers such as Flanders using category type observation systems have assumed onestep dependence and have used a matrix to display the frequencies of pairs of behavior categories.
The present study will extend the traditional approach in two ways. First, the assumption of onestep dependence, as opposed to independence or twostep dependence of data items, was tested in a unique way. Second, the full Markov chain model was applied to the data.
A few studies (Pena, 1969; Hartnett and Rumery, 1973) have tested observation data for the onestep dependence assumption with conflicting findings. In the present study a proportional reduction in error (PRE) measure was used in the analysis of onestep and twostep dependence rather than the chisquare type statistic used by earlier researchers. The use of the PRE allows a comparison of the fit of the onestep and twostep models, and the statistic is interpreted as the proportional reduction in error.
The second unique feature of the present study was the use of a number of properties of finite Markov chains to generate statistics from observation data. Some researchers (Bellack, Kliebard, Hyman, and Smith, 1966; Rausch, 1965) have used the basic elements of the Markov chain model, but no research reviewed carried the analysis beyond the use of the transition probability matrix. The complete analysis of data presented in this research will include such statistics as the mean recurrence time of categories, long run probabilities of each behavior, the creation of absorbing states, the mean and variance of movement into the absorbing states, and absorption probabilities. Although the model is quite complex, the statistics which it generates are quite easily understood because they are in the form of probabilities, means and variances.
The most important feature of the Markov chain model is the
emphasis placed upon probabilities. By analyzing the probability of the events which follow a given event rather than the frequency of particular pairs of behaviors, classroom observation may be abstracted in terms of trends or patterns of pupilteacher interaction. During a particular recording session, for example, the teacher might ask a question only three times, and two of the three questions might have to be clarified before they can be answered. The frequencies in the appropriate cells of the matrix would be limited to a total of three for the entire row representing "teacher question." If probabilities are used, however, analysis.would show that, with probability .667, clarification was needed after a question. With probability .333, clarification was not needed. There are other easily understood statistics that can be generated from the Markov chain model which provide meaningful abstractions for the researcher or the teacher.
Rosenshine and Furst lament that, "observation systems appear to have been used more to document desired behavior than to test hypotheses" (1973, p. 161). In the next chapter, observation research is examined in order to assess the.heuristic value of the Markov chain model, and the discussion will demonstrate that many testable hypotheses can be generated by a Markov analysis of observation data.
Organization
The dissertation is organized around a central theme of exploration, namely, the validity and possible application of a particular
statistical model, the Markov chain. The topics of the dissertation include (1) traditional statistical procedures applied to observation data, (2) the procedures employed in the present study, and (3) the comparative value of the two approaches. Data analyzed using the Markov model are presented, and several suggestions for further research are offered.
More specifically, Chapter II contains a review of three types of literature: observation literature, statistical procedures used in the past with category type observation data, and research programs in education and other fields which have made use of the Markov chain model. A detailed discussion of the procedures applied to research information is presented in the third chapter, and the order of the presentation was logically derived from the computer program used by this author for the processing of observation data. The fourth chapter, paralleling the organization of the third, contains an analysis of the results, while the fifth and final chapter of the dissertation summarizes the findings and suggests possible topics for future research programs.
Although the present study is necessarily limited in scope, the implications for further research are significant. The Markov model is employed today by analysts in a number of fields, but a key question which remains to be answered in education concerns the validity and applicability of the model as applied to classroom observation studies.
CHAPTER II
REVIEW OF THE LITERATURE
The published research on teaching which used direct observation
is extensive and diverse. The purpose of this chapter is not to review this broad area exhaustively but rather to put into perspective the feasibility and value of the application of the Markov chain model to category type observation data.
The first section of this chapter will be a discussion of observation instruments in general and the category type instruments in particular. The next broad topic for consideration is the research areas in which observation instruments have been used. An overview of the types of studies which have been conducted will help to determine whether the model proposed in this dissertation can be used in the types of research generally carried out using observation instruments. The third section of this chapter is a discussion of the traditional analytic procedure used with category type observation data and the final section of the chapter is a review of some research studies which have used a Markov chain todel for the analysis of data. The types of statistics which were derived from the research data are focused upon in order to evaluate the usefulness of using the Markov chain model with category type observation data in the study of teaching.
Observation Instruments
The interaction which occurs in the classroom is extremely
complex. In the typical situation there may be one teacher and 30 students, and each of these individuals is constantly giving and receiving verbal and nonverbal cues. Because all behavior cannot be observed, the researcher must decide in advance what aspects of classroom behavior to study. Once this decision is made, an observation instrument is selected or developed to capture the desired dimensions of classroom interaction.
Classroom observation instruments exist in large numbers. Simon and Boyer (1970) reported 92 different systems for observing classroom behavior, while Rosenshine and Furst (1973) in their review of six references estimated that more than 120 such systems existed. Dunkin and Biddle (1974) and Rosenshine and Furst (1973) reviewed observation research in an attempt to classify the broad range of instruments.
One classification of observation systems was by recording procedure, and Rosenshine and Furst (1973) defined three types. They called a system in which an event was recorded each time it occurred a category type system. If an event which occurred during a specified time period was recorded only once, the system was designated a sign system. The third type of instrument, the rating type, required the observer to estimate the relative frequency of specified events.
The present study was restricted to category type observation data. This type of instrument is a comprehensive list of mutually
exclusive behaviors. The classroom observer classifies each occurrence during the observation period into one of the categories. A record can be made from a live observation, an audio tape or video tape.
A very simple example of a category type recording system might be the following:
Category number Behavior
1 Teacher talk 2 Student talk
3 Confusion or silence
The recording of these categories during classroom observation varies according to the unit of analysis selected. For example, the unit of analysis used by Flanders (1965), Lohman et al. (1967), and Ober et al. (1968), and the one employed in this dissertation, is an arbitrary time unit. If a unit of three seconds is used, the recording of the categories in the example above over a period of 15 seconds might yield the following sequence:
3 3 1 1 2 3
The period begins with silence or confusion followed by about six seconds of teacher talk, three seconds of student talk and another period of silence or confusion. In the application of such a system, the order of the observed behavior categories is generally preserved.
A second unit of analysis which is used with category type observation instruments involves specified chains of behavior. Bellack et al. (1966) and Smith et al. (1967), defined segments of verbal interaction as units of analysis. A category name for a sequence might be "responding," and an entire segment of the interaction would be coded "responding." Whatever followed this behavior would be the next code in the record. Just as in studies which used a time unit of analysis, research procedures with behaviors as the analytical units preserved the order of the occurrences.
Direct Observation to Study Teaching
The purpose of reviewing research which used direct observation to study teaching is to survey the types of problems that were studied using observation instruments. This survey is to provide background information for the discussion of the relative contribution of the application of Markov chain statistical procedure to observational data.
Reviews of observation research by Medley and Mitzel (1963), Flanders and Simon (1970), Soar (1972), Rosenshine and Furst (1973), and Dunkin and Biddle (1974) are notable. Flanders and Simon, Dunkin and Biddle, and Soar emphasized results of research studies, whereas Rosenshine and Furst and Medley and Mitzel surveyed all aspects of the field of direct observation with emphasis upon methods, instrumentation, and general types of research.
As an introduction to the chapter on research using direct observation of teaching in the Second Handbook of Research on Teaching, Rosenshine and Furst observed that:
The research on teaching in natural settings to date
has tended to be chaotic, unorganized and selfserving.
The purpose of this chapter is to ease the reader into
the maze of instrumentation and research which has
focused on teaching in natural and seminatural settings.
There seems to be no simple route through the chaos
which has developed.
(1973, p. 122)
The Rosenshine and Furst classification scheme will serve as a guide for the maze in the present discussion. They classified research using observation of classroom behavior into the following categories: classroomfocused research, teachingskills research, and curriculummaterials research.
In classroomfocused research the authors found that the primary emphasis had been placed upon description of classroom interaction and investigation of relationships between classroom occurrences and pupil growth on achievement measures. Descriptive research of the type done by Bellack et al..(1967), and by others was intended to provide baseline data and concepts which would verify theories of instruction and lead to correlational and experimental studies. Correlational research such as that conducted by Soar and Soar (1973) and by Flanders (1965), aimed at establishing relationships between classroom behavior and educational outcomes, yielded some consistent results.
Rosenshine and Furst (1973, pp. 156158) found that the 50 studies they reviewed indicated positive correlations between student growth and the following teacher characteristics: clarity, variability, enthusiasm, task orientation, and indirectness, Correlations between teacher criticism and student growth were negative in 12 of 17 studies of this nature reviewed. Some experimental classroomfocused research programs were also reported. In six such studies (Rosenshine and Furst, 1973) teachers were trained in the use of different types of discourse. Behavior in the experimental group classrooms was found to be significantly different than behavior in the classrooms of nontrained teachers.
The productivity of classroomfocused research was questioned by various authors (for example, Dunkin and Biddle, 1974; Gage, 1968); however, Gage expressed hope for the productivity of future research when he stated that:
The field of research on teaching is today engaged
in continuous and extensive analysis of its approaches
and theoretical formulations . . .. More complex research designs capable of taking more categories of
significant variables into account are being propounded . . .. The faith persists that educationally significant differences can be consistently produced
in the future as new intellectual and material resources are brought to bear on educational problems.
(Gage, 1968, p. 403)
The second major grouping of studies reviewed by Rosenshine and Furst was teachingskills research. An example of how observation data were used in a teacher training program was a study by Limbacher
(1971) in which effects of microteaching were investigated. Each session was videotaped and coded using Flanders Interaction Analysis System (FIAS). One of the hypotheses tested was that the group with microteaching experience would have a higher ratio of Indirect to Direct behavior (See page 23).
Flanders' categories for coding were used differently in a study by Furst (1965), which recorded differences between student teachers who had been taught FIAS and those who had not. The experimental group was found to be more accepting of student ideas and less rejecting of student initiating behavior. The Furst study used an observation system as a training tool whereas the Limbacher study used direct observation to monitor behavior of the teachers. Rosenshine and Furst summarized the possible uses of observation instruments in teacher training:
Fortified with acceptable criterion measure, investigators could use existing observational systems to
study the behaviors of those teachers and relate the skillsrelevent behaviors to the measures of growth, and they could compare the behavior and the outcomes
for trained and untrained groups of teachers.
(1973, p. 126)
The third major category of research suggested by Rosenshine and Furst, was curriculummaterials research. The primary way in which Taba et al. (1964) and Bissell (1971) used descriptive systems was to document the implementation of curriculum materials. In contrast, Siegel and Rosenshine (1972), using an observation system, validated the
importance of variables emphasized within a program. In a third approach, Soar (1972) used the monitoring capability of observation systems to compare several Project FollowThrough programs. After a factoranalysis of data from several observation systems, the factors were examined to determine if they correlated with pupil gain and if they discriminated among programs.
Through the "maze of research," the primary feature of direct observation studies is description of classroom behavior. The description of behavior using observation instruments has been used in research studies to explore interaction patterns, to discriminate among situations or programs, to monitor the implementation of programs or behavior, and to correlate behaviors with pupil variables such as attitudes or gain on achievement measures. As statistical procedures are reviewed in the next section, it must be kept in mind that one criterion for evaluation of a procedure is in terms of the problem to be studied.
Statistical Procedures Used with Observation Instruments
Two major areas of emphasis, classroom climate and sequential patterns of behaving, emerged in studies using category type observation instruments. Each of these branches of research developed distinct statistical procedures which were routinely applied.
Classroom Climate Research
Major studies which influenced the procedures used in the classroom climate studies were completed by Anderson (1939), Bales (1950), Withall (1949), Flanders (1965), and Soar (1966). Withall explained that the concept of classroom climate in his study represented "the emotional tone which is a concomitant of interpersonal interaction. It is a general emotional factor which appears to be present in interactions occurring between individuals in facetoface groups" (1967, p. 49).
One of the earliest efforts to analyze classroom behavior in
terms of climate was by H. H. Anderson (1939). The study focused upon two types of behavior, labeled "Domination" and "Integration." Domination was characterized by use of force, commands, shame, threats, and blame. "Integration" designates behavior that was more flexible, such as support, acceptance, etc. The purpose of the study was to develop a reliable technique for recording behavior in terms of domination and integration. An observer was expected to record the frequency of teacher behaviors in terms of twenty carefully defined categories. Eight of the categories described dominative behavior and nine categories were for integrative behavior. The other three codes were for neutral behavior. When a student was directly addressed, a record was kept of which student it was, but not of his behavior. In the collection of data the order of occurrences was not preserved.
In the Anderson study, classes were observed for a period of three hours, and every observation summary showed that total number of dominative statements, found by summing the frequencies of the eight categories so defined, was greater than the total number of integrative statements. The ratio of dominativetointegrative ranged from 2:1 to 5:1. The mean number of dominative and integrative contacts per hour was calculated for each teacher's morning and afternoon classes, and bar graphs were used to display the results. Similarities were observed for the same teacher and different patterns were observed between teachers.
Data regarding contact with individual students were shown on a broken line graph in the form of mean number of contacts of each type per child. Children were assigned numbers along the xaxis, and frequency of contact was shown on the yaxis. The Anderson research led to development of a reliable observation instrument with high reliability between observers of the same class sections, and the analysis suggested many areas for further research.
The work by Robert Bales in the 1940's had a major influence
on the development of theories of socialemotional climate. The purpose of his research was to develop a technique for studying and describing small groups. For a study published in 1950, data were drawn from a number of small group settings. Bales' research technique, called Interaction Process Analysis, was primarily designed to record the behavior of small groups, but it had direct classroom
application as well. Bales, unlike Anderson, believed that "sequential analysis is of particular interest in the development in the method, since it is on the assumption of a kind of idealized sequence, of perhaps several similar kinds, that the categories are arranged in their present order" (1950, p. 8). Bales defined twelve categories of verbal behavior with the first six paired with the last six. For example, category 5 was "gives opinion," and category 8 was "asks question." To facilitate coding, an Interaction Recorder (Bales and Gerbrands, 1948) was used by the observer. The instrument consisted of moving tape on which a sequential record was kept of observed categories of behavior of both the speaker and the receiver.
In his analysis of interaction data, Bales used a wide variety of summary techniques. Because it was not yet known what statistical techniques would produce insight from the data, Bales tried a number of approaches. Rates of occurrence calculated in percentages gave an indication of the frequency of use of certain categories. The observation data were partitioned into tenminute periods, and frequencies of certain behavior were studied over successive time periods in order to isolate changes in the group over time.
The sequence of observed behaviors was broken into adjacent
pairs, and Bales observed that, "ignoring repetitions, we see a number of expected tendencies according to pairs of categories . " (1950, p. 128). For example, the sequence 67786 produces the pairs (6,7), (7,7), (7,8), and (8,6). Bales chose to ignore repetitions like (7,7),
and the other pairs were tabulated in a matrixlike array (Figure 1). Cell "I", for example, would contain the frequency count for (6,7) pairs.
After initial tabulation, the array was collapsed by letting area A include categories 1 through 3, B categories 4 through 6, C categories
7 through 9, and D categories 10 through 12 (Figure 1). Within each cell of the new table, Bales recorded an observed frequency, an expected frequency, the difference between the two, and a ratio of the difference to the expected frequency. It is unclear how the expected value was obtained, but it was not the same as that obtained when the expected value for the cell was calculated using the chisquare test.
When he plotted the differencetoexpectedfrequency ratio for successive time periods on a graph, Bales found that certain patterns emerged.
To give the theoretical interpretation, it tells us whether a given functional problem received more or
less than its usual amount of attention during the
subperiod . ... The peak rate of each pair of categories appears within the meeting in the same order
in which the pairs of categories are arranged on the
observation list, which in turn is an order suggested by a priori assumptions about the hierarchial nesting relations of the various functional problems involved
in interactions systems.
(1950, p. 136)
Using frequencies derived from temporally ordered data, Bales
calculated a number of indices. The "Index of Difficulty of Communication" was computed in the following manner: the frequency of use of
A
1 2 3
1 A 2
3 4 B 5
6 7 C 8
9 10 D 11
12
B
4 5 6
7 8 9
D
10 11 12
A B C D
Figure l.Matrix for Bale's display of data
category 7 divided by the frequency for category 7 plus the frequency for category 6. The "Index of Directness of Control" was given by the following, where r. represents the frequency of use of the category i.
1
r4 + r r4
r4 + r6 r5 + r6
Six such indices were defined and used in the analysis of data.
Information gathered on who initiated and who received in the
interaction was displayed in a matrix in a manner similar to the technique used above with pairs of categories. Also, indices were computed for individuals by summing their use of certain categories.
On the basis of his findings, Bales postulates "that there is a
series of concomitant changes in ideological emphasis among the members of the group" (1950, p. 172). Many specific directions for further study were suggested by this research, and the statistical methods employed had great influence on later work in the field of observation analysis.
In the late 1940's, John Withall, a student of Herbert Thelen and Carl Rogers at the University of Chicago, conducted research on the construction of a valid and reliable categorization system for socialemotional climate (Withall, 1949). Teacher verbal responses were found to take seven forms in his Climate Index. Three of thesesupport, acceptance and questionswere classified as "learner centered," and three other types of statementsdirective, reproof and teacher self
supportwere "teacher centered." One category was reserved for administrative or neutral statements.
The frequency of occurrence was recorded for each of the seven categories. Because there were several observations of each teacher, it was useful to convert raw frequencies into percentages and to compute a mean for each teacher. As a part of his validation procedures, Withall categorized his observations and computed the Integrative to Dominative ratio, using Anderson's System. Using the Climate Index, a ratio of learner centered statements to teacher centered statements was calculated. The two ratios were not equal for the same observation, but Withall found that they were comparable. Conclusions of the study were that classroom climate could be measured empirically, with high reliability and with validity, using the Climate Index.
Ned Flanders used Withall's observation instrument in his dissertation research at the University of Chicago in 1949 and again in an article published in 1951. Using the seven categories of the Climate Index and a Qsort of verbal statements, Flanders measured student attitudes during and after the learning periods. The pulse and skin resistance of students were monitored during learning periods and evaluation periods. "From the association of particular kinds of teacher behavior with all recorded student behaviors, inferences were made concerning cause and effect cycles in the teacherstudent interaction" (1951, p. 101). The seven student groups were exposed to a learning situation in which there was more teacher centered (TC) behavior and then to another class in which there was more learner centered (LC) behavior. Flanders found
that teacher centered behavior elicited hostility/withdrawal from students while learner centered behavior elicited problem oriented behavior and a decrease in student anxiety.
Flanders came to his 1955 study in New Zealand with the tools discussed thus far (Flanders, 1965). The purpose of his series of studies in the late 1950's was to validate the Flanders Interaction Analysis System (FIAS) for verbal behavior categorization, to measure student academic achievement and student attitudes, and to correlate the achievement and attitude scores with verbal patterns in the classroom.
The first departure from tradition in the study was the observation analysis system. Flanders defined ten categories (Appendix A2 ). Seven were for teacher talk, two for student talk and one for silence or confusion. The previous system for classroom use had not included student talk or the nonverbal category of silenceconfusion. Teacher talk was broken down into "indirect influence" statements and "direct influence" statements (p. 20). Flanders followed the lead of Anderson and Withall in the definition of the climate concepts. The coding was made every three seconds, and the order of the sequence was preserved as in the Bales System. Great care was taken in the definition and explanation of each category so that the set was able to define any occurrence in the classroom and so that each event would fall into only
one category.
In Flanders' research, he needed to know how the data from the observation system should be summarized. John Darwin, a statistician at the Applied Mathematics Laboratory in Wellington, New Zealand, provided some direction in the analysis of the observation data. Darwin (1959; Flanders, 1965) found that the data were onestep dependent, a Markov chain, using Hoel's test (Hoel, 1954), and he suggested that a matrix was appropriate for data tabulation since it preserved the sequence pairs. Darwin also suggested a chisquare statistical test
2
(Darwin's X ) which could be used to determine if two matrices of observation data were significantly different.
More recently, Pena (1969) investigated the Darwin chisquare because the statistic tended to be too large (the test too powerful) to distinguish between matrices of observation data. In the first phase of the study, Pena tested the length of dependence in sequence of classroom observation data using Hoel's test and found that the betterfitting model assumed twostep dependence rather than onestep dependence of the Markov chain. Harnett and Rumery observed that "the results reported by Pena concerning the Markovian properties of interaction data are misleading" (1973, p. 2). The .criticism was based upon the fact that Pena obtained long chains, 2,398 to 11,756 tallies, of FIAS data by combining data for a single teacher across subjects and by combining data across five teachers to form sets of observation data for each subject area. Harnett and Rumery used individual classroom observations of 167 to 555 codes in length in their study of dependence assumptions.
The results of the application of Hoel's test for length of dependence indicated that the observation data sets were onestep dependent, or Markov chains.
Tabulation of pairs of observed categories into a matrix as in
Bales' studies provided the basis for analysis by Flanders. The matrix was used for the representation of data primarily because it preserved one step of the sequence. Flanders stated that "our own estimates indicate that sequence pairs probably account for about 60% of the interdependence between events when our ten categories are involved" (1967, p. 372). Cells and groups of cells became the primary objects of analyses.
Another very important technique used by Flanders was the calculation of a ratio of indirect to direct statements. A simple percentage frequency or frequency per 1000 tallies was calculated for each category, and two ratios were computed. The "big I/D ratio" consisted of column totals for categories 1, 2, 3, and 4, divided by tallies in 5, 6, and 7. The "small i/d ratio" was the same except that categories 4 and 5 were not used (Flanders, 1965, p. 35). Histograms were used to display these various statistics and compare teachers.
The observation techniques introduced and the statistical methods used had a profound influence in the field of interaction analysis. In a more recent book, Flanders (1970) suggested most of the same statistical methods that he had used in his 1965 publication.
In the years that followed the publication of these first Flanders studies, many researchers used the FIAS, or modifications of it, as well
as the statistical methods used by Flanders (e.g., Limbacher, 1971; Lohmon, 1967). Modifications of FIAS generally took the same form as the original instrument but expanded certain categories into several. The Reciprocal Category System (RCS, See Appendix Al) by Ober, Wood, and Roberts (1968) was one system which was derived from the Flanders work. The seven categories of teacher behavior in FIAS were expanded to nine in the Reciprocal Category System. The same nine categoreis were applied to student behavior and were coded with numbers 11 through 19. This increased the number of student behavior categories from two to seven. The recording of parallel categories for students and teacher had the advantage of allowing comparison of student and teacher behavior. Silence or confusion was recorded as one category in FIAS but was recorded separately in RCS.
With the availability of high speed computers, additional techniques came into use in classroom climate research. In the midsixties, Soar (1965) used FIAS in combination with the Observation Schedule and Record, a sign system developed by Medley and Mitzel (Simon and Boyer, 1970). The use of several instruments produced a record of nonverbal as well as verbal behaviors. FIAS data were tabulated in a matrix, and cell frequencies were used in conjunction with Observation Schedule and Record data, vocabulary gain scores, and reading improvement data. A factor analysis was done on the relationships between observation data and vocabulary and reading gains.
Two dimensions of classroom behavior were taken from the results of the observation schedules on the basis
of factor analysis. The dimension of control behavior
was measured by the revised i/d ratio for rows 8 and 9 of the Interaction Analysis. This is a ratio made
up of teacher behaviors which occur immediately after a pupil stops talking . . .. The clearest dimension of
emotional climate which emerged from the analysis was made up of expressions of hostility, criticism or negative feelings.
(Soar, 1965, p. 246)
A further analysis of these data by Soar (1972) revealed a nonlinear relationship among the variables under consideration. Insight was gained from plotting pupil growth on the yaxis and teacher indirectness along the xaxis. Graphs drawn for reading, vocabulary, and creativity gains showed that the fit of the data to the curve was better than the fit to a straight line (Soar, 1972, pp. 8991).
In a research program concerning the differences in pupil classroom behavior and achievement before and after summer vacation, Soar and Soar (1973) used a modification of FIAS by Ober called the Reciprocal Category System. Also used were three sign systems, the Florida Taxonomy of Cognitive Behavior, Teacher Practices Observation Record, and the Florida Climate and*Control System. The statistical analysis of data was essentially the same as that found in the 1965 study. A large number of significant linear relationships was found between pupil pretest performance and classroom behavior. Student gains on various tests were found to have a curvilinear relationship to teacher structure and control, with the optimum level of structure depending upon the particular outcome desired. Throughout the study, the use
of nonlinear as well as linear analysis proved to be a useful technique.
Sequential Pattern Research
The second major area of research employing category type instruments involved sequential patterns of classroom behavior. Studies of this type were conducted by Bellack et al. (1966), Smith et al. (1967), and Taba et al. (1964). Classroom behavior was recorded, and then ventures (Smith et al., 1967), teaching cycles (Bellack et al., 1966), or discussion patterns (Taba et al., 1964) were coded and analyzed. Research in this area has been descriptive as well as correlational. Researchers studying the sequential patterns of behavior in the classroom were more interested in longer chains of behavior than in the pairs analyzed in classroom climate studies. The studies characterized as sequential also included multiple descriptors applied to any one utterance; therefore, data for this type of analysis were recorded, transcribed and then coded. For example, the teacher phrase, "where did George Washington live?" might be coded:
teacher/question/information sought
Smith et al. (1967) analyzed classroom discourse in terms of strategies used to produce certain outcomes derived from the stated objectives of the teacher. Seventeen high school teachers of various subjects were involved in the study, and five class sessions were recorded
for each. The transcripts of classroom verbal behavior were broken into "ventures," or topic segments. Eight types of ventures were defined, and plays, shorter units of a few sentences within ventures, were coded. Ventures were displayed on a type of "flow chart." The venture diagrams were not necessarily linear; that is, various branches were found to exist in ventures with the same lable. Sequences of consecutive plays were put into a matrix to show the frequencies of sequential pairs. One of the main contributions of this descriptive study was the coding of verbal behavior into ventures and plays and the use of diagrams to display the results.
Bellack et al. (1963, 1965, 1966) did research using sequential patterns in the description and analysis of the linguistic behavior of teachers and students. "The language game" which teachers and students played by certain well defined rules was explored. Discourse was recorded and later transcribed, and the unit of analysis used in coding the transcripts was the "pedagogical move." The four major categories of moves were structuring, soliciting, responding, and reacting. Each pedagogical move was coded as follows (1966, p. 16):
1) Speaker (teacher, pupil or audio visual device)
2) Type of pedagogical move
3) Substantive meaning
4) Susstantivelogical meaning
5) Number of lines in (3) and (4)
6) Instructural meaning
7) Instructurallogical meaning
8) Number of lines in (6) and (7)
Moves were found to occur in the classroom discourse in certain patterns, and Bellack et al. designated twentyone such patterns as
particular teaching cycles. The cycles continued from one to five moves, and the frequency of various codings was expressed in terms of percentages. Total lines in the moves, for example, were calculated and converted into percentages for teachers and pupils. Another such summary statistic, a cycle activity index, was found by computing the number of cycles per minute.
A matrix was used by Bellack for tabulation of various categories and cycles of verbal behavior. Using temporarilyordered teaching cycles, Bellack
sought to determine whether certain cyclical
patterns and dimensions of cycles tend to influence
the subsequent patterning of pedagogical moves.
Statistically, this was described through a Markov
chain to determine the transition probabilities
of moving from one state to another. Taking types
of teaching cycles as states, the probabilities
of moving from one type of cycle to another were investigated as a way of determining whether one
pattern of pedagogical move tends to influence
immediately subsequent patterning.
(Bellack et al., 1965, p. 158)
A transition probability matrix was also used to summarize data. In the following example from Bellack's work, data on the initiators of cycles were summarized (1966, p. 208).
Teacher Pupil Audio Visual T 89.5% 10.4% 0.1%
P 59.4% 40.6% 0%
AV 15.4% 0% 84.6%
The three categories of initiators used were teacher, pupil and audio visual (movies, overhead projector, or other similar equipment). Each row of the display represented the initiator of a cycle, and the columns indicated the initiators of the cycle immediately following. The interpretation of the example above is as follows: Given that the pupil initiated the last cycle, there is a 59.4% chance (a .594 probability) that the teacher will initiate the next cycle. The sum of the elements of a row is 100%. This display of data supported the theory that once he had initiated a cycle a pupil was more likely to initiate the next cycle. Regarding the use of a transition probability matrix, Bellack commented that
The temporal analysis, then, by establishing
statistical relationships between prior and
following states defined here as teaching
cycles, permits a more refined description of classroom verbal behavior by greatly improving one's ability to see relationships
among classroom variables.
(Bellack et al., 1959, p. 169)
Taba and Elzey (1967) studied sequential patterns in social studies classrooms which were using Taba's curriculum materials. Verbal behavior was recorded, transcribed, and coded to reflect the speaker, the function, and the level of thought for each "thought unit." In a later publication, Taba and Elzey noted that
The multiple coding scheme makes it impossible to
deplict the flow of the classroom discussion by
charting the sequence of transactions between
the teacher and the children, the charges in
the level of thought during the discussion,
and the effects of these strategies upon the
level and the direction of thought.
(Taba and Elzey, 1964, p. 493)
Four patterns of discourse emerged as a result of the "flow
chart" presentation of data. The chart consisted of level of thought plotted on the vertical axis and time or thought units (numbered) along the horizontal axis. At each unit, a notation was made to indicate the speaker and the function of the unit. It was found that, in order to sustain higher levels of thought processes, prevention of premature entry into the level was important.
Bellack, Smith, and others used predefined chains of behavior as their unit of analysis. Several researchers recently have used computers to scan the observation data for all possible chains of three or more categories in length. Collet and Semmel (1970), for example, perfected a computer program which facilitated tabulation of chains of various lengths. Dolley (1974) used this program to do a descriptive study of motherchild interaction. Along the same lines, Campbell (1975) presented a strong case for using actual counts of chains of more than two steps. He called his approach macroanalysis, as compared to microanalysis which involves only the tabulation of adjacent pairs of categories. Campbell stated that "macroanalysis could serve as a stimulus for further descriptive research. Perhaps a new round of such research is needed to determine those patterns which are currently used by teachers at various grade levels" (1975, p. 26s).
Summary of Statistical Techniques
Figure 2 on page 32 summarizes the statistical analysis methods used by the researchers discussed in the preceding section. In his work on classroom climate, Flanders used all of the methods used by Anderson, Bales and Withall. Flanders considered the matrix appropriate because the observation data were earlier found by Darwin to fit the assumption of onestep dependence of the Markov chain model. He did not use the transition matrix which is generally used for the representation of Markov chain data. A few years later, Soar built upon the methods of Flanders and introduced the use of multiple systems and factor analysis. The study of sequences of behavior began in the early nineteen sixties, and Smith led the way with observation instrument techniques. Bellack expanded on this coding system and introduced a range of statistical methods for analysis of data. Taba, in answer to her own needs in curriculum evaluation, presented still another model for the presentation of data.
It should be noted that Smith as well as Bellack drew heavily upon the interaction analysis work done by Flanders. Bellack's use of the transition matrix, for example, was an extension of Flanders' use of the tally matrix. Bellack, however, made fuller use of the Markov chain properties of the data in his research. Rather than a descriptive display, Bellack used the transition probability matrix as a predictive model. Bellack, Smith, and others used predefined sequences of categories called ventures or cycles, and the occurrence
Anderson (1939) Bales (1950) Withall (1949) Flanders (1965) Limbacher (1971) Lohmon et al. (1967) Soar (1966) Smith et al. (1967) Taba & Elzey (1964) Bellack et al. (1966) Dolley (1974) Collet & Semmel (1970) Campbell (1975)
a) x
(I) C) .1; ro 04 ) U) .,.j M)r (a U) )
0 H F 41J > U)
4I C 0) Q4 ~ H >4 4IJ 0 (n Ci) (13 En U) 01 ) H 0 0) r,
0)a) 4' 0 H4x a) ) x x H k 04
0 U 0 05  tf c0
a ) si ( 54 (0 k 54 (a :I U (n~ r,4~ 0 :E Hq U L x x xII x x x x x x x x x x x.x x x x x x x x x x x x x x x x x x x
x x x x
x x x x x x X X X X X x
x x x
Figure 2.Summary of statistical procedures in research literature
and order of these patterns were studied. More recently, longer chains, which were not predefined but rather emerged from the data, were used by Campbell as well as by Collet and Semmel.
Markov Chain Theory in Research
A Russian mathematician, A. A. Markov, first developed the basic concepts of Markov chains in the early 1900's. Markov chain theory was refined in the years that followed and was widely used in such fields as biology, chemistry, physics, astronomy, and engineering. It has only been since the early 1950's that social and behavioral scientists have begun to use this mathematical model.
Future uses of the elegant model developed by Markov can best be identified by considering applications which have proven productive in the past. The theme in all prior studies has been the usefulness of the Markov chain model to describe and predict. The few areas of research which will be briefly reviewed here are personnel administration and organizational growth, social mobility, clinical applications, and interaction analysis.
The areas of personnel* administration and organizational growth have used Markov chains to model research problems. Rowland and Sovereign, in their study of manpower replacement problems (1969), used a matrix of probabilities of transition from one job classification to another. Similarly, Vroom and MacCrimmon (1968) described the career movement of certain personnel. States of the Markov chain were defined in these studies by the level or function of the job held
by an individual. Movement from one state to another over time was displayed on a transition probability matrix, and powers of the matrix were used to predict turnovers, promotions, and personnel needs for future years. Using the same techniques, Adleman (1958) studied organizational growth in terms of the size distribution of firms in the steel industry. He used a Markov model to describe change in the size of firms and to predict eventual equilibrium size distribution. Enrollment studies, such as that conducted by Mohrenweiser (1969), used techniques similar to those used in the study of industrial growth and personnel needs. The advantage of the application of a Markov chain model in research on personnel administration and organizational growth was not only its power to describe the observed movement from one state to another, but also the power of the model to predict future movement and eventual equilibrium.
These same qualities of description and prediction make Markov
chain procedures useful in the study of social mobility and labor supply shifts. Glass and Hall (1954) and other sociologists used a transition probability matrix to describe probabilities of membership in a particular social class over time. They also compared movement at different periods of history by comparing rows of the respective transition matrices. The predictive capability of the model was particularly useful in studies attempting to forecast manpower needs and supply for various career fields.
Aside from the large scale descriptivepredictive research programs, the Markov chain model has been meaningfully used in clinical
settings. Meredith (1974) reported a particularly interesting use of the full statistical model in his comparison of various programs in a mental hospital. In addition to the commonly used transition probability matrix to describe probabilistically movement from one state to another, Meredith used death and exit from the hospital as absorbing states. Using Markov procedures, he studied the movement of patients into these states. Conclusions of the research were that the Markov model provided a tool for comparisons of programs, for longrange evaluation of performance, and for description of existing conditions.
The final type of research reviewed which used Markov chain procedures was interaction analysis. The fields of sociology and speech have produced studies of this type. The contributions of Bales (1950), such as the use of a matrix for the display of data, was discussed in the previous section. In a more recent study, Hawes and Foley (1973) used transition probability matrices to compare the use of thirteen behavioral categories or states in interaction involving interviewers classified as directive, moderately directive and nondirective. Predicted longterm proportions of each type of behavior were calculated and compared with the actual proportion of each category in the data sets. As Hawes and Foley state, "the Markov statistics and graph present a picture of ways the communication system behaves over time" (1973, p. 219).
In a slightly different study, Raush (1965) coded the social behavior of normal and disturbed children in order to study the interaction sequence. Transition probability matrices were constructed and used to compare groups.
The specificity of the methods allows us ... to
achieve something more than the clinically obvious.
Given the situation, given the nature of the group,
we can, knowing the actions of one child, predict
rather well what another child will do . . .. Specifically, we can detail which situations are likely to induce which behaviors and how particular situations are likely to modify the contingencies
between stimulus and response acts.
(Raush, 1965, p. 497)
The research by Hawes and Foley and Raush was similar in many
ways to the work of Bellack et al. (1966) who studied sequential patterns of behavior in the classroom. The researchers assumed their data to be onestep dependent, and they used a Markov chain model in the construction of transition probability matrices from observation data.
In the observation literature reviewed, no classroom observation studies were found in which category type data were analyzed using the Markov chain model. Two research studies (Pena, 1969; Hartnett and Rumery, 1973) applied statistical tests to category type observation data to determine if the Markov chain model assumptions were met. The results were conflicting and there were, therefore, no clear indications of the viability of the model for use with category type observation data. No observation studies reviewed used any Markov chain procedures other than the transition probability matrix.
Summary and Conclusions
Aside from absolute statistical tests of assumptions of a model, guidance in the choice of procedures in a particular type of research
must come from an examination of the purpose of the work, the traditional methods in the field, and similar studies which have used the proposed procedures. In the case of the proposed use of the Markov chain model with category type observation data, indications of the possible productive applications of the model were found in the survey of literature related to observation studies, procedures generally used with category type data, and research in other fields in which the model had been used.
Direct observation data have been used to describe classroom behavior, to validate curricula and materials, to train teachers in specific skills, and to correlate with pupil gains. The primary function of the observation system is to record and describe behavior in the form of specific codes, and, in some cases, to provide feedback of information to the teacher. In some of the Markov studies, such as those conducted by Meredith (1974) and Raush (1965), the Markov chain model was demonstrated to provide useful summary statistics, descriptions, and meaningful feedback to clients.
The traditional procedures used by observation analysis researchers also indicate the possible productiveness of Markov chain modeling of observation data. The primary assumption of the Markov model, onestep dependence, was implicit in the work of most researchers since Bales, and the assumption was also manifest in the use of a matrix of tabulation of data pairs. Bellack et al. and Flanders stated that their data were approximately Markov chains. Bellack et al. even used the transition
probability matrix, a basic concept in Markov chain theory. Indications are that a more complete application of the Markov chain model would prove productive.
The final consideration in the review of the literature was
whether or not research studies had effectively used the statistical model under consideration. The works of Raush (1965) and Hawes and Foley (1973) are such studies. Both focused upon social interactions, and both used a category type system to code behavior. The application of elements of the Markov chain model, though not the entire model, was both meaningful and productive in the analysis of their data.
A review of the literature suggests that at least an exploration of the complete application of the Markov chain model is warranted. The transition probability matrix which has been used with category type data is but a first step. Other properties such as absorbing states, mean passage times, and absorption probabilities could have meaning when applied to observation data. These statistics and how they are computed are the topic discussed in Chapter III.
CHAPTER III
PROCEDURES
The research literature in observation analysis strongly suggests that the Markov chain model is useful in the analysis of observation data. In fact, several researchers (e.g., Hoel, 1954; Darwin, 1959; Pena, 1969; Hartnett & Rummery, 1973) have specifically tested observation data for the length of the. dependence of chains of interaction data. In the present study, a similar problem is studied; however, instead of an absolute test of dependence, a comparative approach is taken.
In Chapter III the data source for the present investigation is discussed. The data in their original form was not entirely suited to analysis as a Markov chain. The transformations which were made and the reasons for these changes are explained in detail.
The remainder of the present chapter is a discussion of the Markov chain mathematical model and how data are tested for the model. The discussion begins with a general description of a Markov process, and attention is given to whether observation data fit this description. Hypotheses for testing the length of dependence in a chain are presented, and the appropriate statistical tests are indicated. In the final sections, various properties of a Markov chain are discussed and their computing formulas are given. The specific results gained
from application of the procedures discussed in the present chapter are presented in Chapter IV.
Sample
This study involves the application of the Markov chain model to what might be considered typical observation data, and characteristics of the particular sample were not of direct importance to the research. The data used were Reciprocal Category System data transformed to Flanders Interaction Analysis System. The data were gathered in an evaluative research study of Project FollowThrough (Soar, 1973). The national program known as Project FollowThrough (Maccoby and Zellner, 1970) was designed to improve education in kindergarten through third grade for children from impoverished environments. The Soar study contained information gathered from eight different programs within FollowThrough. In addition to Reciprocal Category System data, the following classroom observation measures were used in the evaluation study: Florida Climate and Control System, Teacher Practices Observation Record, Florida Taxonomy of Cognitive Behavior, and global rating of activities. All subjects were observed at least once during each year of the threeyear study. A factor analysis of data from each instrument was used.
Seventeen FollowThrough program classrooms were observed three
times during the last year of study. Data from four of these seventeen classrooms were used in the present study. Subjects were selected based upon their scores on a factor generated from Florida Classroom
Climate System (FLACCS) data. This factor, called "strong control" (Soar, 1973, p. 52), includes such teacher behavior as warning, criticizing, spanking, frowning, and finger shaking. Two teachers who were high on this factor, and two who were low on this factor were selected for inclusion in the present study. Extreme levels of this particular factor provided contrasting data for analysis.
The four classrooms used in the research were all first grade
level. The two "high control" groups, A and B, were northeastern inner city schools, and the two "low control" groups, C and D, were midwestern small and mediumsized cities. A breakdown of the sample is shown in Figure 3.
Classroom FLACCSI
Teacher Factor Score
A 61.56 1 Northeast inner city B 62.74 1 Northeast inner city
C 44.22 1 Midwest small town D 45.78 1 Midwest middle town
Figure 3. Summary of data sources
Observation data were collected by Soar using the Reciprocal Category System. Three fiveminute time periods in midmorning were audiotaped and later coded. Three such observations were made of each classroom. The first observation for each group was made in early September, the second in midOctober and the last in late January or February.
Transformation of Data
The data used in the present study were originally collected using the twenty categories of the Reciprocal Category System (RCS) by Ober (See Appendix Ai). The number of empty cells in the matrix display due to the nonoccurrence of certain behaviors suggested a transformation of the data (See Appendix A3). A comparison of RCS and Flanders Interaction Analysis System (FIAS; see Appendix A2) revealed that groups of RCS categories could recode into one FIAS category without loss of meaning. The only exceptions to the parallel natures of RCS and FIAS were found in categories 1, 2, and 3 of the two systems; these three categories had overlapping meanings,
The first three categories in RCS were placed in a single category which included teacher accepting behavior, warming the climate, and amplifying student ideas. These are similar behaviors, and for the purpose of this study no distortion of data results from the collapsing of the behaviors into one category. Categories 4, 7, and 15 in RCS were left intact except for a change in the number used to identify the state. The RCS codes 5 and 6 were combined into one category corresponding to the single code in FIAS representing "lecturing." Categories 8 and 9, "corrects" and "cools the climate," became "criticism," the seventh category in FIAS. The RCS has several categories, 11 through 14 and 16 through 19, for "pupil interaction"; these were grouped into one category. Finally, category 10, "silence,"
and category 20, "confusion," were grouped together as they are in the FIAS. The transformation of the data using FIAS as a guide reduced the number of categories of behavior from twenty to eight.
While the codes after transformation were not completely analogous to the original categories, the differences were believed to be unimportant for the purpose of the present study. The general breakdown of behavior types in the transfomed data is very similar to the widely used FIAS. The advantage of RCS over FIAS most often noted is the additional student behaviors which can be coded. Many of the student behaviors do not routinely occur during an observation session. Because the statistical procedures used in the present study would not be appropriate should a number of pairs of categories not occur, a transformation of data, i.e., collapsing some categories, was indicated. Since categories were not arbitrarily combined, the extent to which the sequence is onestep dependent should not be affected by the recoding.
Markov Chain Model
A mathematical model for category type observation data whose
value is suggested by the nature of the data as well as by recent research literature in the field is the Markov chain model. Considering the recorded categories as states of the chain and the specified recording time intervals as time periods, the data intuitively form a chain of observed states.
A Markov chain process is characterized by the following information:
There are a given set of states (Sl, s2, s3, ..., sr }" The process is in one and only one state at a given time. It moves from one state to another in successive time periods, and each move is called a step. The probability that the process moves from state i, si, to state j, sj, in one step depends only on s.. In other words, if X t + 1 = s represents the state at time t + 1, then
j
P (Xt + 1 = sj Xt = si' Xt 1 = k' ... Xo = s)
= P(Xt = sj xt_1 = So)
The probability of s. given the entire sequence preceding it is the
J
same as the probability of s. knowing only that s. immediately preceded. An observation Xt + , is independent of the observation two steps before, Xt _ 1' only if Xt is known. The property is called onestep dependence. In addition, in a Markov chain process the initial state of the process must be specified.
Some of the necessary conditions of the Markov chain model are satisfied by the nature of the rules for collection of category type observation data, and the central question involves the extent to which the sequence satisfies the length of dependence assumption. In any observation system, there must be a given set of categories or states, and one and only one of these is recorded in a given time interval. The process moves from one category to another in successive time periods. By convention, the code for "silence" is used to begin the
recording of each observation set; therefore, the initial state or category is automatically specified. One problem area is the dependence of the occurrence of a state on one or more preceding states. This aspect of the Markov model needs further study before it can be determined to what extent the model fits observation data.
Tests of Fit of the Markov Chain Model
The pattern of dependence of one state or category upon preceding ones is not obvious from the nature of the category type observation system being used to record events, nor is the pattern of dependence indicated by the nature of interaction in the classroom. Statistical tests are therefore called for to indicate if the states are independent of previous events, primarily dependent on one preceding step, or dependent upon two or more steps.
The statistical tests of interest are:
Null Hypothesis 1: Successive steps are independent.
Alternative Hypothesis 1: Dependence is onestep or longer.
Null Hypothesis 2: Dependence is onestep.
Alternative Hypothesis 2: Dependence is twosteps or longer.
The general procedures'indicated by these statistical tests of hypotheses are as follows:
1) The construction of onestep and twostep tally matrices
from the raw observation data.
2) The computation of a chisquare statistic for the matrices.
3) The computation of Goodman and Kruskal's Tau for the matrices
as an indication of the proportion reduction in error using
the one and twostep dependence assumptions.
Tally Matrices
Raw data (See Appendix B ) consists of a sequence of numbers representing observed categories of behavior. The first step after the tranformation to new codes in processing of the data is to tally the observations into two types of matrices, a onestep matrix and a set of twostep matrices. To illustrate the construction of these matrices, the set of states {i, 2, 3, 4, 5, 6, 7, 8} forms an observed sequence 8, 3, 1, 2, 4, 3, 8. The sequence might be an example of observation codes recorded using eight predefined categories where codings are made at specified time intervals.
Onestep of the sequence is preserved by the construction of a matrix. Using the sequence,
8, 3, 1, 2, 4, 3, 8
adjacent codes are paired. The pattern 8 3, 3  1, 1  2, 2  4,
4  3, 3  8 can be preserved in a matrix. The rows and columns of the matrix are labeled to correspond with the set of states. The onestep tally matrix, T, is constructed by counting the number of times that a particular state i is followed by category j to form a total n.. in each cell of the matrix.
47
time = t + 1
1 2 3 j 8
T =1 nl n12 n13 nlj n18
2 n21 n22 n23 n 2j n 28 3 n 31 n 32 n33 ï¿½ 3j n38
time t
i nil ni2 ni3 . ii8
8 n8l n82 n 83 n8j n 88
Using the sample sequence, the first tally indicated by the pair
8  3, is in row eight column three. Next a mark for 3  1 is made in row three column one. The process continues in this manner until the last pair 3  8, in the sequence is recorded. The total in each cell, nij, represents the number of times the pair i  j occurred in
the sequence being analyzed.
Using the same example, the construction of eight twostep tally
matrices is illustrated. These matrices preserve twosteps or chains
of three elements of the sequence.
time = t + 1
1 2 3
T(j) = 1
2
time = t+l
nljl n ji n 2jl
n 3jl
nlj2 nlj3 n2j2 n2j3 n3j2 n3j3
nijl nij2 n8j1 n8j2
S nljk S n2jk
* n3jk
S nlj8
n2j8 n 3j8
nijk
n 8jk
n j8
One matrix is constructed for each state. The matrix number, j, represents the state at time t; whereas, the row, i, represents the preceding state and the column, k, represents the state which follows. The entry in matrix T(j) row i, column j is nij and this is the number of times ijk
the sequence i  j  k is found in the raw data set. In the illustration +
I 4+ +
8, 3, 1, 2, 3, 3, 8
the first tally is made in matrix T(3) in row eight, column one. The pattern continues as indicated until all triplets have been tallied.
Chisquare Test
The chisquare test is designed to determine whether or not the frequencies which have been empirically derived differ significantly from those which are expected under certain theoretical circumstances. Applying the chisquare statistic to the test of null hypothesis 1, the assumption is that there is no difference in the state at time = t + 1 knowing the state at time = t. The chisquare statistic is computed as follows for the onestep matrix:
Let n.. = tally count in cell ij
i3
8
r. total of row i = Z n., j=l 13
8
n= r.
i=l
8
c. = total of column j = n.
3 i113
r.. c.
eij = expected number in cell ij = n 3
n
2 8 8 (n.  e. )
x =: j 1e I]
i=l j=l ei
The obtained statistic has (8  1)(8  1) or 49 degrees of freedom
(df). For tests in which the degrees of freedom are this large the chisquare (X ) is converted to a z statistic in the following way:
2
z = 2X  2df  1
The z statistic is a normal deviate with unit variance; the chisquare test corresponds to a single tail ztest. If the z statistic is significantly nonzero at the predetermined level, null hypothesis 1 of independence can be rejected in favor of the alternative hypothesis of onestep or more dependence.
In the twostep tally matrices there are likely to be many empty
cells. For this reason, the chisquare test statistic is not appropriate for use in the test of null hypothesis 2.
Goodman and Kruskal's Tau
Hypothesis 2 can be tested by looking at the proportional reduction of errors (PRE) using a model. The Goodman and Kruskal Tau (Tb, Blalock, 1972, pp.300302) is such a measure for contingency tables. The Tb statistic takes on values between zero and one, with one representing an explanation of all error by the proposed model. Tau is equal to
number of errors with row unknown  number of errors with row known
number of errors with row known
The computing formula which is appropriate for use with the one and twostep tally matrices in the eight category illustration is
2 2
8 8 n.. 8 c
x ï¿½ (I ) E
T b= i=l j=l r . j=1 n
8 C
n E
j=l n
The Tau can be computed directly for the onestep tally matrix.
For the twostep case, a Tb is computed for each of the eight twostep matrices in the example. A pooled estimate for an overall Tb is found by calculating a weighted average of the eight Tau values.
Rather than a direct test of null hypothesis 2, a comparison of the proportional reduction in error seems indicated by the comparative nature of the study. The Tau resulting from the onestep tally matrix can be compared to the pooled Tau from the twostep matrices.
Application of Markov Chain Properties
In this section several of the applications of Markov chain properties are discussed in some detail. These procedures were applied to the transformed observation data used in this study and discussed in Chapter IV.
If the onestep dependent Markov chain model fits category type observation data, there are several Markov chain properties that could be applied. The following quantities are of interest in the analysis of a Markov chain:
1) The transition probability matrix.
2) Powers of the transition matrix and the equilibrium matrix.
3) The mean recurrence time.
4) The means and variances of passage time into absorbing
categories.
5) The probabilities of absorption into two or more absorbing
states.
These quantities are discussed in detail below.
Transition Probability Matrix
The transition probability matrix (P) is a matrix of cells, pij, such that each entry represents the probability that state i will be followed immediately by state j in the chain.
time = t + 1
1 2 3
time = t
Pll P21
P31
Plij
p2j
p3j
P13
P23 P33
Pil Pi2 Pi3
P18
P28 P38
pij
P81 P82 P83 P .
An estimate for pij is obtained from the onestep tally matrix by dividing each entry of the tally matrix by the row total. The computing formula is the following:
n,,
Pij) r.3
1
The sum of the entries in each row of the transition matrix is one; that is, given that the process is in state i, the probability that some state will follow is one. While the relative incidence of the various pairs of categories is not indicated by the transition matrix, the pattern of movement is indicated by the probabilities found in the cells of the matrix.
Powers of the Transition Probability Matrix and the Equilibrium Matrix
The transition matrix, P, gives the probabilities of transition from a given category to a second category in one time period. The
2
square of this matrix, P x P or P , is a transition probability matrix for which the probability of transition is calculated over two time periods. Each entry in P2 p (2), represents the probability of state j following state i two time periods later. The matrices P , P and
5
P represent the transition probabilities over three, four and five time periods, respectively. In general, PN is the matrix of transition probabilities over N time periods, and PN is computed by taking the Nth power of matrix P.
As higher powers of P are computed the column values of the matrix become closer in value to other entries in the same column. Any prearranged limit for the difference can be attained by taking sufficiently high powers of P. As a result of this process, the rows of matrix P N become nearly identical.
1 2 3 ï¿½ j 18
P 1 a11 12 a 3* a1. i a 18
2 a11 a12 a13 a1 a 18
8 a11 a12 a13 alj . a18
The equilibrium matrix is defined as a row matrix consisting of one row of PN
11(all a12, a13, alj, .18)
( 7( I' T2 ' 73 ' " '" 7j ' " ' 78 )
The matrix H is a display of the long run probabilities of the occurrence of each state. The probability of state j is given by Ir., regardless of the initial state. If all transitions are carried out during the process, the proportion of time in each category should be approximately equal to the corresponding element of the 11 matrix.
Mean Recurrence Time
The mean recurrence time, sii, for category i represents the number of time periods expected to pass from the moment category i is observed until it is observed again. Mean recurrence times are computed using the elements from matrix 11,
1 1
Mean and Variance of Passage into Absorbing Categories
Any state in a Markov chain which is always followed by itself is called an absorbing state. For example, suppose the states being observed are the conditions of hospital patients. Obviously, the state death would only be followed by death so it is an absorbing state. In the transition matrix, the probability of death following state death is one; all other transition probabilities in the row would be zero.
The presence of an absorbing state allows for study of movement into that category. If no such state is present in the Markov chain, it might be useful to create an absorbing state. Suppose, for example, that state or category 3 is of particular interest; that is, the researcher may need to know which states precede it in the chain of events. In order to study movement into category 3, the transition probability matrix, P, can be altered making 3 an absorbing state. Row three of P is changed to all zero except transition probability p33 which is equal to one.
time = t + 1
1 2 3
Pil
P21
0
P12
P22
0
Pil Pi2 Pi3 P81 P82 P83
Plp
P2j
0
Pij
p8j
To facilitate computation of the mean and variance of passage times into the absorbing state, the transition probability matrix is put in canonical form, P'. The absorbing category is shifted to the top row and the columns are interchanged so that the cell containing probability one is in the upper left corner.
J
8
time = t
P18
P28 ï¿½0
time = t + 1
3 1 2 4 8
P =3 1 0 0 0 0 0 0 0
1 P13 pl P12 P14 P18 2 P23 P21 P22 P24 P28 4 P43 P41 p42 p44 P48
8 P83 P18 P28 P84 P88
Within matrix P' four submatrices are defined as follows:
1 0
Matrix I is an identity matrix, the dimensions of which depend upon the number of absorbing categories. Matrix 0 is a matrix of all zero elements. The transition probabilities from nonabsorbing states into absorbing state(s) are contained in matrix R. Matrix Q is the matrix of transition probabilities from nonabsorbing categories into nonabsorbing categories.
The expected or mean number of time intervals spent in each nonabsorbing category before eventual absorption can be calculated. Matrix M, called the fundamental matrix, consists of the mean passage times for each state. The fundamental matrix, M, is given by the following, where I is an identity matrix of the same dimensions as matrix
M = (I
The inverse of I  Q is a matrix M such that M . (I  Q) = 1.
Let the matrix M be symbolized
1 2 4
m 12 m 22 m 42
m 18 m 28 m 48
m 14
m 24
m81 m82 m84
m 88
Each entry, mij, in the matrix represents the mean number of time intervals that i is expected to be followed by some specified j in the chain before the absorbing state is reached. It is as if all action
has been artifically stopped when state 3, the absorbing category, is entered. In this way a close study of events preceding this move can be made. Furthermore, given that the sequence is in state i, the total number of time intervals expected to pass before the sequence reaches the absorbing state is found by:
in = m + m + M + +.
I il i2 i4 i8
The mean total passage time for state i before absorption or movement, in this case into category 3, is given by mi..
21
The variance of the passage time provides additional information on the movement into an absorbing state(s). The variance matrix, VAR, is computed using the fundamental matrix, M, and a matrix Md consisting of zero elements except for the principal diagonal which contains elements from the principal diagonal of M. The identity matrix I is of
2
the same dimensions as M. Matrix M is a matrix with elements mi from the fundamental matrix M. The matrix VAR is calculated:
VAR = M (2 . Md  I)  M
Each element, vij, of VAR is the variance of the corresponding mean, mij, in the fundamental matrix M.
The variance for the total time before absorption, is the result of the following calculations:
60
TOVAR = (2 . M  I) M  M p p2
Vector M
p
2
Mp has m. as
p2 1
is a column matrix with m. as elements. The matrix
1
elements. The matrix takes the form:
1 2 4 5
TOVAR = 1
2 4
b15
b 25 b 45
b12
b 22 b 42
8
b 18
b28 b48
b81 b82 b84 b85
Additional information on the distribution of the passage time from one state to another is provided by the median. Let h(k
(k)
represent the probability of first passage from state i to state j
in k time periods. A specifiedh is computed
13
h(k) (k) (kl) (1) h M2)
ij=  h.ij P.. h k
h() (kl)
 ij Pjj
bll1 b 21 b 41
(k) k
where p ( is an element of Pk. If the sum of the first passage time is computed and
ni n(k)
E h. < .50 , E h. . > .50
kl 13 k=l 13 then n equals the median of the passage time from category i to category j.
Absorption Probabilities
If more than one absorbing category is present, the probability of entry into either of the states can be found. A matrix F of probabilitities is computed
F=M . R
where matrix M is the fundamental matrix and R is a section of P', the matrix of transition probabilities in canonical form.
Matrix F has the same number of rows as M; that is, there is one row for each nonabsorbing category. The number of columns in F is equal to the number ofabsorbing states in P'.
If states 5 and 7 are absorbing, the matrix of absorption probabilities is
5 7
F =1 f15 f17
2 f25 f27 3 f35 f37 4 f45 f47 6 f65 f67
f85 f87
An element, f ij, of matrix F is the probability of absorption into category j rather than the other absorbing category(s) given that state i has occurred. If state 3 occurs, f35 equals the probability that absorption will be into state 5,and f37 is the probability of absorption into state 7. The chain will eventually be in one of the absorbing categories and the sum of the elements of a row of F is therefore one. In certain problems, the probability of movement into one of several selected categories may be of interest. If the chain does not have absorbing categories and they are created, the element f.. gives the probability that if state i occurs, category j will occur before the other absorbing category(s).
Summary
The transition probability matrix, the equilibrium matrix, mean recurrence time, and the study of movement into absorbing states
63
abstract from a Markov chain powerful patterns of movement. In Chapter IV, discussion focuses upon the results obtained when the above procedures were applied to the transformed observation data described earlier in Chapter III.
CHAPTER IV
RESULTS
In the statement of the problem in Chapter I, the research questions were
1. Do category type observation data form a Markov chain?
2. How can the Markov chain model be used to analyze observation data?
These two questions emerged from a review of interaction analysis literature. The literature suggested that an assumption of onestep dependence was made by most researchers. A discussion of Markov chain model attributes in Chapter III indicated that the onestep dependence assumption of the model is the only criterion for a Markov chain which is not clearly satisfied by the nature of category type observation data. An attempt was made in the present study to approach the length of dependence in a comparative way. Toward this end, tests of hypotheses were constructed to compare the degree to which each of the following assumptions are satisfied by these data: independence assumption, onestep dependence assumption,.and twostep dependence assumption. The results of these tests are discussed in the first section of this chapter.
The second research question may seem premature since it has not been determined if the data form a Markov chain. A review of the literature revealed that most researchers implicitly made the assumption of onestep dependence in the display and analysis of data in matrix form. The matrix display was first used by Flanders (1965) based upon advice from Darwin, a statistician. Since these techniques assuming onestep dependence have become traditional and have proved to be productive and useful, it is highly probable that researchers will continue to use the approach. If the onestep assumption is made, however, it is logical to explore the use of all aspects of the implicit model, the Markov chain model. It was in the spirit of exploration that the properties of the Markov chain model were applied to observation data.
Tests of Fit of the Markov Chain Model
The data sets used in the present study were from observation of four Project FollowThrough classrooms. Each classroom was observed three times, once in early fall (Fl), again later in the fall (F2) and in the Winter (W). Two classrooms were classified as high on a factor called "strong control"; these data sets were identified as teacher A and teacher B in the analysis. Teacher C and teacher D were low on the control factor. The raw data from each observation period (Appendix B) were transformed from the Reciprocal Category System Codes into modified Flanders Interaction Analysis System Categories (See Appendix A3). A complete analysis was done of each individual
observation for each teacher, and for each teacher the data were pooled for an analysis. The pooled sets were called "all" in the resulting tables of statistics. An example of a complete analysis is found in Appendix C). The data source for the example was the second fall observation of teacher A. Test of Independence Versus OneStep or Longer Dependence
The first hypothesis was an exploration of whether there is any dependence a mong the categories in the sequence obtained from an observation. If the frequencies in the onestep tally matrix are what can be expected by random assignment of tallies, then the completed chisquare (X 2) will be small. The transfomed data consisted of eight codes. Data were summarized into an eight by eight onestep tally ma2 2
trix, and a X statistic was computed. The X had 49 degrees of freedom.
2.
For large degrees of freedom X2 is transformed into a normal deviate z statistic with unit variance, and a one tailed ztest is used.
A formal statement of the statistical test is:
Null Hypothesis 1: Successive steps are independent.
Alternative Hypothesis 1: Dependence is onestep or longer.
In Table 1 the statistics for each observation data set are summarized. In each case, X2 was sufficiently large that when corrected to a z statistic z was nonzero at the .001 level of significance. Thus the null hypothesis of independence was rejected in favor of an alternative hypothesis of onestep or longer dependence.
Table l.Summary of chisquare and Tau values
N Chisquare za Onestep Twostep Tau pooled Tau
A F  1 300 336 16.1 .190 .197
F  2 301 224 11.3 .152 .182 W 377 572 24.0 .257 .110 All 980 884 32.2 .170 .117 B F  1 388 315 15.3 .164 .126
F  2 236 260 13.0 .217 .188 W 357 268 13.2 .237 .141 All 983 849 31.4 .175 .099 C F  1 339 269 13.3 .131 .178
F  2 374 303 14.8 .188 .111 W 307 342 16.3 .200 .236 All 1022 645 26.1 .112 .135 D F  1 295 336 16.1 .245 .185
F  2 440 253 12.6 .126 .098 W 586 128 6.1 .049 .044 All 1323 511 22.1 .068 .080
Table I.Continued
T(1) T(2) T(3) T(4) T(5) T(6) T(7) T(8)
.286(25 ) .233(46) .097(56) .115(65) .786(6) .261(55) .351(19) .099(27) .069(27) .133(39) .175(55) .213(51) .250(3) .142(84) .415(11) .332(30) .048(21) .138(23) .038(158).343(13) .611(7) .142(31) .057(87) .366(36) .064(73) .147(108).055(272) .094(129) .327(16) .142(170)1.095(117) .280(94) .341(7) .166(50) .143(70) .164(43) .318(9) .089(58) .088(22) .088(128) .000(22) .104(55) .441(25) .250(29) .306(10) .132(69) .282(10) .359(15) .000(2) .248(36) .136(157).139(32) .000(1) .123(32) .167(5) .121(91) .116(31) .038(141).179(252).098(104).281(21) .058(160).100(37) .061(236) .092(23) .071(44) .191(72) .124(30) .727(6) .156(86) .144(33) .340(44)
 (0) .211(22) .050(13) .299(16)  (0) .137(36) .077(85) .152(98) .268(22) .148(50) .211(27) .207(41) .754(7) .288(66) .133(43) .302(50) .104(47) .052(117) .082(213) .122(87) .488(14) .189(189) .056(162) .247(192) .357(5) .078(22) .138(91) .581(10)  (0) .245(28) .073(62) .280(76) .093(27) .086(83) .144(40) .394(13) .000(4) .053(134).123(31) .110(107) .082(26) .134(32) .032(116).144(33)  (0) .025(183)  (0) .Q38(194) .046(58) .030(137) .160(247 .116(56) .000(4) .027(345) .078(96) .096(379',
az= 2X 2 /2df 1
bNumber in the matrix is enclosed in the parentheses.
Test of OneStep Dependence Versus TwoStep or Longer Dependence
After rejecting the hypothesis of independence in favor of a onestep or longer dependence hypothesis, the question became, how many steps does the dependence span? The answer to this question was not expected to be a definitive one; therefore a comparative approach was designed.
The statistics were to test the following: Null Hypothesis 2: Dependence is onestep.
Alternative Hypothesis 2: Dependence is twostep or longer.
The Goodman and Kruskal's Tau (T ) was computed for each onestep
b
tally matrix, and a pooled value for Tau was computed from each set of eight twostep matrices. The techniques were designed to test whether or not the proportional reduction of errors or Tb for the tostep tally matrices was significantly greater than the Tb value for the corresponding onestep tally matrix. In this manner the twostep or longer dependence assumption was compared with the onestep dependence assumption.
The results of the computation described above were summarized
into Table 1. In the analysis of 10 of the 16 data sets, the onestep Tb was larger than the pooled twostep Tb; in the remaining six cases, the twostep Tb values were greater. A closer look at the individual Tau values for the twostep matrices revealed great variation in these values. For example, the F1 observation of teacher A produced Tau values for the twostep tally matrices ranging from .0971 for T(l) to
.7857 for T(5). Under the circumstances a pooled estimate of Tb has questionable value. An examination of the individual Goodman and Kruskal Tau values for the twostep matrices revealed that certain matrices had higher Tb values. Of the ten cases in which Tb was the nonzero for T(5), nine represented the largest Tb recorded for that observation. The twostep matrix T(5) contained a record of the chain at t 1 and t + 1 where the state at time t was category 5 (teacher corrects or cools). Other patterns might have been present had there been more observation codes in certain categories.
The results summarized in Table 1 indicate that a formal test of the null hypothesis was not appropriate; rather a comparison of the relative fit of the two models was warranted. The evidence suggests that certain categories are more onestep dependent while others are more twostep dependent. Category 5 (teacher cools or corrects), for example, had high Tb values for most twostep matrices. Due to the nature of classroom interaction, it might be expected that the categories of behavior which precede and follow a correction by the teacher are linked or form a twostep dependence pattern. The conclusion based upon the results is that category type observation data does not clearly satisfy a onestep dependence assumption nor does it satisfy completely a twostep dependence assumption. The onestep dependence assumption appears to be at least as good as the twostep dependence assumption except for particular states. The Markov chain is therefore clearly suggested as a possible model for the analysis of observation data. Perhaps a change in the categories or time unit would produce data which would fit the assumption of the model more closely.
Application of Markov Chain Properties
The tests of hypotheses suggested that, while the data chains were not completely onestep dependent, at least an exploration of the application of the Markov chain model is warranted. The review of literature also suggested that properties of the model could be meaningfully applied to classroom observation data chains. The results of the application of selected properties will be discussed in the following sections, with emphasis placed upon the meanings of the results in terms of classroom behavior.
Transition Probability Matrix
The onestep tally matrix of each data set was used to estimate
the transition probabilities of all pairs of categories. Each element of the tally matrix wad divided by the row total so that
n..
Pij =n. "
1
The resulting matrix, P, consisted of probabilities pi. =i Given 1J1)
that category i occurred, pij is the probability that category j would follow. The rows of the transition probability matrix represent the category at time t, while the columns of the matrix represent the category at time t + 1.
An example of a transition probability matrix is found in the complete analysis of the second fall observation of teacher A in Appendix C. Row 7 of the matrix is as follows:
1 2 3 4 5 6 7 8
0.0 0.273 0.182 0.091 0.0 0.182 0.0 0.273
Given that category 7 behavior (pupil initiates) was observed, the probability of observing category 1 type behavior (teacher warms, accepts, amplifies) was 0.0 for this observation. The most likely behavior after category 7 was category 2 (teacher elicits) and category 8 (silence or confusion) each with a probability of occurrence of 0.273.
The transition probability matrix is a powerful abstraction from the raw data. Rather than a display of the actual frequency of occurrence of pairs of categories of behavior, the transition matrix is a predictive model. In the example above, the analysis suggests that student initiating behavior was followed approximately 27% of the time by teacher eliciting behavior, 27% of the time by silence or confusion, 18% of the time by teacher responses or pupil responses, and 9% of the time by teacher directive behavior. It seems that in this observation, pupil initiating behaviors were very short in duration since no pairs of 7  7 were observed. One might conclude that the student was being cut off by the teacher. A closer look reveals that this was not necessarily the case. Category six (student response) and silence or confusion are expected to follow category seven with probability .46 while teacher categories follow with probability .54. The focus of the analysis of observation data with the aid of a transition probability
matrix is upon the relative frequency of transitions from a given category to each of the categories rather than upon the frequency of such movement in a particular observation. Powers of the Transition Probability Matrix and the Equilibrium Matrix
Powers of the transition probability matrix were computed for each set of observations. The elements in each column became less variable as successive powers were computed. It was found that P5 was sufficient in all cases to insure that the range of the elements in each particular column was less than or equal to .03 (any specified maximum range could have been used). The equilibrium matrix was formed from
5
the first row of P
The transition probability matrix, P, for a Markov chain gives
the probabilities of transition from a given category to a second category in one time period. The square of this matrix, P x P or P , has elements which represent the probability of movement from a given category to another category over two time periods. A similar interpreta3 4
tion may be applied to P , P , etc. In the analysis of category type observation data in the preSent study, P5 was a matrix of probabilities which indicated that, no matter which category occurred at time t, the state at t + 5 depended only on the state to which the transition was
5
made. For an example of P , see Appendix C. The information in the higher powers of the transition matrix was redundant; therefore, a row
N
vector was defined as one row of P
The equilibrium matrix, H, is the row vector containing the probabilities of movement into each category in the longrun. Furthermore, 7. represents the proportion of the total number of observed categories that can be expected to fall into every category. If, for example, i = .21, category 1 is expected to comprise 21% of the total number of behaviors in the observation record.
In many observation data sets, the predicted proportion agreed with the actual proportion of occurrence of each behavior category. A summary of the proportion rounded to two decimal places is found in Table 2. The longrun prediction and the actual proportions are very nearly equal in thirteen of the sixteen cases. Subject A's first fall observation, subject B's winter observation, and subject C's second fall observation were the only data sets which produced differences of .04 or more in the predicted and actual proportion of observations in each category. One explanation for these differences might be that the transition probability matrices were not taken to a high enoughpower before the equilibrium matrix was defined. Mean Recurrence Time
The mean recurrence time was computed from the equilibrium matrix for each category in each data set, and the reciprocal of each element of the equilibrium matrix was rounded off to the nearest whole number (Table 3). The largest and smallest means for each observation were enclosed in brackets and underlined, respectively.
Table 2.Equilibrium matrix versus proportion in each category A B
F F  2 W All F 1 F 2 W All
a b a b a b a b a b a b a b a b 1 .08 .08 .09 .09 .06 .06 .07 .07 .02 .02 .09 .09 .01 .01 .03 .03
2 .16 .15 .13 .13 .06 .06 .11 .11 .13 .13 .23 .23 .12 .10 .14 .14
3 .20 .19 .19 .18 .43 .42 .28 .28 .19 .18 .10 .11 .37 .44 .26 .27
4 .21 .22 .17 .17 .03 .03 .13 .13 .11 .11 .12 .12 .10 .09 .11 .11
5 .02 .02 .01 .01 .02 .02 .02 .02 .03 .02 .04 .04 .00 .00 .02 .02
6 .19 .18 .28 .28 .10 .08 .18 .17 .15 .15 .32 .29 .11 .09 .17 .16
7 .06 .06 .04 .04 .22 .23 .12 .12 .06 .06 .05 .04 .01 .01 .04 .04
8 .08 .09 .09 .10 .09 .10 .09 .10 .33 .33 .06 .07 .28 .26 .23 .24
a/ Longrun prediction. b/ Actual proportion.
Table 2.Continued
SC D F 2 Fi  2 W All F 1 F 2 W All
a b a b a b a b a b a b a b a b 1 .07 .07 .01 .01 .08 .07 .05 .05 .02 .02 .06 .06 .05 .04 .04 .04
2 .13 .13 .06 .06 .17 .16 .12 .11 .08 .07 .19 .19 .05 .05 .10 .10
3 .21 .21 .26 .30 .09 .09 .21 .21 .37 .31 .09 .09 .20 .20 .19 .19
4 .09 .09 .05 .04 .14 .13 .09 .09 .03 .03 .03 .03 .06 .06 .04 .04
5 .02 .02 .00 ï¿½ .00 .03 .02 .01 .01  .00 .01 .01  .00 .00 .00
6 .26 .25 .10 .10 .19 .22 .19 .18 .10 .09 .31 .30 .31 .31 .26 .26
7 .10 .10 .25 .23 .12 .14 .16 .16 .17 .21 .07 .07 .00 .00 .07 .07
8 .12 .13 .28 .26 .19 .16 .19 .19 .22 .26 .25 .24 .33 .33 .29 .29
a/ Longrun prediction.
b/ Actual proportion.
Table 3.Mean recurrence times
A
F 1 F 2
12
6
5
5
[52]
5 16 12
11
8
5
6 [99]
4
27 11
W All
18 17
2
30
[54] 10
5
11
13
9
4
8 [611
6
9 11
B
F 1
[58]
8
5
9 38
7 17
3
F  2 W All1 F  1
12
4
10
8
[26]
3
22 16
142
9 3
10 [305]
9 71
4
31
7
4
10
[46]
6 27
4
14
8
5
[56]
4 10
8
C
F  2 W All
177 17
4
22 [336] 10
4
4
12
6 11
7 [39]
5
9
5
22
9
5
12
[74]
5
6
5
D
F  1 F  2 W All
[43] 12
3
30
10
6
4
16
5
11
34 [108]
3
14
4
_________ 4 1 ~ U
22 18
5
18
3
[586]
3
23 10
5
24 [324]
4
14
3
NOTE: highest values are enclosed in brackets; lowest values are underlined. NOTE: All numbers are rounded off to the nearest whole number.
The mean recurrence time represents the number of time periods expected to pass before the designated category is expected to occur again in the Markov chain. In the analysis of observation data, these means give an indication of the relative frequency with which categories of behavior are expected to occur. According to Table 3, the most frequently recurring category for teacher A in three of the four data sets was category 3 (teacher responds, initiates). In the winter observation, for example, when category 3 is observed, it is expected that category 3 will occur again four time periods later. Four of the six smallest recurrence times for teacher A were teacherbehavior categories, while two were pupil responsebehavior categories. The largest means for teacher A were all in category 5 (teacher corrects or cools).
Based upon the chart of means, some general conclusions about the classrooms can be made. Subject A seems to be using category 3 and category 4 behavior frequently. Both of these are directive, or teacherdominative, behaviors. Pupils responded, but only in one observation was pupil initiating behavior frequent. Basing opinion only upon this table of means, one might classify classroom A as a rather structured classroom. Indeed, teacher A was one of the two teachers who were chosen because of a high score on a control factor (See Chapter III, data section).
The teachers with low scores on the control factor (who were expected to have a less structured classroom) were teacher D and teacher C. In both cases the frequency of category 8 (silence or confusion)
might indicate that time was allowed for students to think before responding, and therefore more frequent responses by pupils were obtained. If no factor score were available, the same kind of insight into behavior patterns based upon the mean recurrence times could be gained.
Absorbing Categories
The sequence of data produced by coded classroom behavior does not contain any absorbing category. If there were such categories, the techniques described in Chapter III could be used to study movement into the absorbing category(s). In the present study, category
6 (student response) and category 7 (student initiates) were the created absorbing states.
Creation of one Absorbing State
The transition probability matrix, P, for each set of observation data was altered as described in Chapter III to create category *6 as an absorbing state. The matrix, P, in canonical form (see an example in Appendix C) was used to define the matrices R and Q. The fundamental matrix, M, was computed from matrix Q and an identity matrix. Each element, mij, of matrix M, is the mean number time periods in state j before absorption given that the chain is in state i. The first row of matrix M from the second fall observation of teacher A is
1 2 3 4 5 7 8
1 1.254 0.546 0.937 0.972 0.046 0.164 0.329
Given that category 1 (warms, accepts, amplifies) behavior occurred the number of time periods expected in each of the nonabsorbing categories is given by the number in that column. The total number of time periods expected before category 6, the absorbing state, is entered is given by the sum of the row elements. In the example above, the mean total passage time after category 1 occurs is 4.252. The vectors of mean passage times for each observation are given in Table 4, with each entry rounded to the nearest whole number.
Additional information on the movement into category 6 is provided by the variance of the passage times. The variance for each state in the fundamental matrix, M, was computed and the variance matrix, VAR, with elements v.. is the variance corresponding to the mean, mi (See Appendix C for an example of these matrices). The variance of total passage times was computed. The elements of vector TOVAR are the variances of the total passage times. The use of the standard deviation, the square root of the variance provides a more complete description of expected patterns of movement into the absorbing category.
A summary of the mean, m, and standard deviation, s, rounded off to the nearest whole number is found in Table 4. The analysis shows that in each case the shortest time before passage into the absorbing category, 6 (student response), is from category 2 (teacher elicits). This interaction is expected; however, the various data sets differ in other respects. In the observation data such as from teacher A
Table 4.Mean and standard deviation of the passage time into category 6
A B
P
F 1 F 2 W All F 1 F 2 W All
m S m s m s m s m s m s m s m s
1 7 6.5 4 2.8 11 12.1 7 6.2 9 7.0 3 1.7 12 13.6 7 6.1 2 3 5.0 1 1.4 3 6.8 2 3.9 3 5.0 1 1.4 8 12.1 3 4.6 3 8 6.6 4 2.8 14 12.5 8 6.4 8 6.9 3 1.7 17 14.5 8 6.3 4 8 6.6 4 3.0 5 9.5 6 6.2 7 6.9 2 1.4 13 13.9 6 6.0 5 6 6.1 4 2.8 11 12.3 6 6.2 8 7.0 3 1.7 14 13.9 7 6.2 7 8 6.6 4 2.8 15 12.6 8 6.5 8 7.1 4 1.7 14 13.9 7 6.3
8 9 6.7 4 3.2 15 12.6 8 6.5 9 7.1 3 2.0 14 14.1 8 6.3
NOTE: All means (m) were rounded to the nearest whole number and all standard deviations (s)
were rounded to the nearest tenth.
Table 4.Continued
C D
En F F  2 W All F 1 F  2 W All
M s m s m s m s m s m s m s m s
1 5 5.4 15 13.7 5 3.2 6 5.7 14 12.2 3 2.2 3 2.0 2 2.8 2 4 5.0 8 12.2 3 2.8 4 5.0 3 7.1 2 1.7 2 2.0 2 2.2 3 6 5.6 15 13.7 5 3.3 7 5.8 12 11.9 4 2.4 3 2.0 4 3.2 4 7 5.7 15 13.7 4 3.3 7 5.9 12 12.3 3. 2.4 3 2.0 4 3.0 5 6 5.7 16 13.7 5 3.3 7 5.8   2 2.0   2 2.2
7 7 5.7 14 13.7 5 3.3 7 5.8 15 12.6 4 2.4 4 2.0 5 3.2 8 1 5.7 14 13.7 3 3.2 7 5.9 14 7.7 3 2.4 3 2.0 4 3.2
NOTE: All means (m) are rounded to the nearest whole number and all standard deviations (s)
were rounded to the nearest tenth.
F  1, teacher D, F  1, and teacher C, F  2, the mean total passage time from category 2 into the absorbing category was less than the means from the other categories into the absorbing category indicating that state 2 might be expected to precede state 6 by fewer time periods than other categories. In most of the cases presented, the standard deviation is large relative to the mean, indicating a positively skewed distribution. In four cases , teacher A winter observation, teacher D first fall observation, teacher B winter visit and teacher C second fall observation, the means and standard deviations are large. A few of the medians of the passage time into state 6 are found in Table 5. Additional verification of the skewness is provided by a comparison of the means and the medians. The distribution of the passage times in these cases does not appear to differ in shape (skewness) from the others.
The creation of an absorbing category facilitated a study of
movement into that state. Certain types of behaviors are particularly desirable or undesirable in classroom interaction, and these can be focused upon by making them absorbing categories in the chain of observation data. It may not otherwise be observed that a particular behavior rather than others precedes the behavior of interest. The absorbing category property of Markov chain theory in effect stops all progress of the chain when the absorbing category is reached and thereby facilitates analysis of movement into the designated category.
Table 5.Median of passage time
State Teacher A (F  2) C (W)
Median Mean Median Mean
1 3 4 3 5
2 1 1 1 3 3 3 4 3 5 4 3 4 3 4 5 3 4* 3 5 7 2 4 4 5 8 3 4 1 3
into category 6
Creation of two Absorbing States
The creation of two absorbing states allows study of the categories which precede either of them. In the present study, both student behavior categories 6 and 7 were changed to absorbing states in the manner described in Chapter III. An example of the canonical form which resulted from matrix P' is found in Appendix C. The mean passage time, the variance of the passage time, the mean total passage time, and the variance of the total passage time were each computed as for the one absorbing category. Each of these statistics apply to movement into either absorbing category. Since both student behavior categories were changed to absorbing states, the study focuses upon those behavior categories which precede student verbal behavior.
A summary of the mean, m, and standard deviation, s, of passage
into category 6 or 7 is given in Table 6. Although category 2 (teacher elicits) still generally has the smallest mean passage time, a general trend is not as clear as the trend that was observed when only category
6 was absorbing. An inspection of individual observation data sets suggests that the distributions are not skewed as badly as they were when one absorbing category was used. The means are generally greater than the standard deviation.
Since more than one absorbing category was created, the relative probability of entry into each absorbing state can be studied. A matrix F of probabilities of entry into category 6 or 7 was computed for each observation data set. A row i of matrix F contains two elements
Table 6.Mean and standard deviation of passage time into category 6 or 7
A B
P
Fi F  2 W All F  1 F  2 W All
m s m s m s m s m s m s m s m s
1 5 3.9 4 2.4 4 2.6 4 2.8 6 4.2 3 1.0 10 11.0 5 4.2 2 1 1.4 1 4.1 1 1.0 1 1.4 2 3.2 1 1.0 7 9.9 2 3.3 3 6 4.0 3 2.4 4 2.8 4 2.8 4 4.0 2 1.0 14 11.9 5 4.5 4 5 4.1 3 2.4. 2 1.4 3 2.6 4 4.0 2 1.0 11 11.4 4 4.1 5 4 3.6 4 2.2 4 2.6 4 2.8 4 4.0 3 1.0 12 11.4 5 4.2 8 6 4.3 4 2.6 2 1.4 3 2.4 6 4.5 2 1.4 11 11.5 6 4.5
NOTE: All means (m) were rounded to the nearest
were rounded to the nearest tenth.
whole number and all standard deviations (s)
Table 6.Continued
IVcD
4J
F 1 F  2 W All F 1 F 2 W All
m s m s m s m s m s m s m s m s 1 3 2.4 1 0.0 3 1.4 3 2.0 6 4.1 2 1.4 3 2.0 3 2.0 2 2 2.0 2 2.2 2 1.4 2 1.7 2 2.6 1 1.0 2 2.0 2 14 3 3 2.4 4 3.5 3 1.7 3 2.2 5 4.1 2 1.4 2 2.0 3 2.0 4 4 2.4 3 2.8' 2 1.4 3 2.0 4 3.7 2 1.4 3 2.0 3 2.0 5 4 2.4 4 2.8 2 1.7 3 2.2   2 1.4   2 1.7
8 3 2.4 2 2.4 1 0.1 2 1.7 3 3.3 2 1.4 3 2.0 3 2.0
NOTE: All means (m) were rounded to the nearest
were rounded to the nearest tenth.
whole number and all standard
deviations (s)
fi6 and f i7 Given that state i occurs, fi6 is the probability that absorption will be into category 6; that is, category 6 will occur in the chain before category 7. Absorption into one of the categories will eventually occur; therefore, the sum of fi6 and f is one.
The matrix F for each observation data set is displayed in Table 7. Category 2 (teacher elicits) seems to be absorbed into category 6 with high probability in most of the data sets. In only 11 of 96 pairs of probabilities was the probability of absportion into category 7 greater than the probability of absorption into category 6.
The creation of two or more absorbing categories enables a powerful comparison of the likelihood of two or more behaviors. This technique could be especially useful if one state is a desirable behavior type and the other an undesirable type. If category 7 were desirable, then subject C might use silence to increase the probability of category 7 behavior.
Summary
The second research question guiding the present study was, how can the Markov chain model be used to analyze observation data? The preceding pages contain a discussion of the application of various aspects of the model and an interpretation of the results. In each example, the results of the statistical analysis using the Markov chain model were meaningful in the context of classroom interaction. These proportions, means, standard deviations, and probabilities are summary
Table 7.Probability of absorption into categories 6 and 7
4) 4J
2
3
4
5
8
A B Fi F 2 W All Fi F  2 W All
6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 .68 .32 .85 .15 .49 .51 .63 .37 .62 .38 .88 .12 .85 .15 .80 .20
.78 .22 .98 .02 .88 .12 .88 .12 .91 .09 .96 .04 .90 .10 .93 .07
.73 .27 .87 .13 .34 .66 .56 .44 .60 .40 .76 .24 .79 .21 .70 .30
.68 .32 .82 .18 .75 .25 .67 .33 .63 .37 .92 .08 .84 .16 .79 .21
.79 .21 .91 .09 .48 .52 .70 .30 .49 .51 .86 .14 .84 .16 .71 .29
.67 .33 .85 .15 .11 .89 .39 .61 .65 .35 .72 .28 .80 .20 .73 .27
NOTE: All values were rounded to the nearest hundredth.
Table 7.Continued
4)
1
2
3
4
5
8
B
F 2 W All
A
F1 F  2 W All Fi
6 7 6 7 6 7 6 7 6 7
.79 .21 .00 1.00 .67 .33 .60 .40 .48 .52 .72 .28 .56 .44 .84 .16 .72 .28 .91 .09 .61 .39 .28 .72 .55 .45 .50 .50 .58 .42 .52 .48 .18 .82 .52 .48 .44 .56 .49 .51 .62 .38 .18 .82 .51 .49 .49 .51 .45 .55 .15 .85 .55 .45 .37 .63 .29 .71
NOTE: All values were rounded to the nearest hundredth.
6 7
1.00 .00 1.00 .00 1.00 .00 1.00 .00
.99 .01
6 7 .88 .12 .94 .06 .82 .18 .84 .16 .96 .04 .77 .23
6 7 .85 .15 .95 .05 .67 .33 .76 .24 .92 .08 .71 .29
91
statistics which allow the researcher to focus upon trends of behavior. The easily understood statistics could provide meaningful feedback to the teacher and thereby aid in desired behavior changes in the classroom.
A discussion of the conclusions generated from the results of the analysis is found in the next chapter. Chapter V also outlines possible future research regarding the applications of the Markov chain model to observation data.

Full Text 
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A STUDY OF THE USE OF MARKOV CPIAIN STATISTICAL PROCEDURES WITH CATEGORY TYPE OBSERVATION DATA By MARGARET ANNE DUDASIK A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1977
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ACKNOWLEDGEMENTS The author wishes to express deep gratitude to each member of her committee. The committee chairman, Dr. Elroy Bolduc, provided the initial guidance which led to the choice of the topic. His assistance in the planning and writing phases was invaluable. Dr. Alan Agresti, through his excellent teaching, interested the author in the field of statistics in general and stochastic processes in particular. His expert advice on procedures was essential to the study. The author wishes to thank Dr. John Grant for his excellent instruction in programing which made it possible for the author to write the programs for this study. His support and cooperation during the pursuit of the degree was appreciated. Although he was very busy. Dean John Newell made valuable suggestions and criticisms in the preparation of the final drafts of the dissertation. Over two years before preparation of the final work. Dean Newell patiently read the author's first attempt at the definition of the problem to be studied, and he gave much needed encouragement. To Dr. Robert Soar and Ms. Ruth Soar go thanks for the use of data from their Project FollowThrough evaluation study. They both also provided suggestions in the preparation of the review of the literature. A number of dear friends helped the author in the critical last stages of preparation of the manuscript. The excellent typing of the ii
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draft as well as the final copy was done by Ms. Nancy Waters. Her dependability and skill relieved the author of many concerns. For her editorial suggestions and her moral support, a special thanks goes to Ms. Marian Tillotson. The author expects to return the favor some day. The list of friends and colleagues who were supportive during this time in the author's life is too long to enumerate, but to each go sincere thanks . Finally, the author wishes to express her thanks to Mr. Stephen Dudasik. He encouraged a spirit of selfactualization as did the author's parents, Mr. and Mrs. Roy Pierce. During every phase of the graduate program and the preparation of this dissertation, Stephen provided support and understanding in the face of the author's selfdoubt, and his editorial assistance in the preparation of the final copy was extremely helpful. Without his encouragement the author probably would not have pursued the degree; therefore, to him is extended her deepest gratitude. iii
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TABLE OF CONTENTS Page ACKNOWLEDGEMENTS ii LIST OF TABLES vi LIST OF FIGURES. vii ABSTRACT viii CHAPTER I THE RESEARCH PROBLEM 1 Statement of the Problem 2 Rationale for Need for the Study 2 Organization. ... 4 II REVIEW OF THE LITERATURE 6 Observation Instriaments 7 Direct Observation to Study Teaching. 9 Statistical Procedures Used with Observation Instruments. . 13 Markov Chain Theory in Research 33 Summary and Conclusions 35 III PROCEDURES. . 39 Sample 40 Transformation of Data 42 Markov Chain Model 43 Application of Markov Chain Properties 51 Summary Â• 62 IV RESULTS 64 Tests of Fit of the Markov Chain Model 65 Application of Markov Chain Properties 70 Summary 88 iv
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TABLE OF CONTENTS Â— continued CHAPTER Page V SUMMARY AND IMPLICATIONS . 92 Objectives 92 Activities, Results and Implications 93 Problems for Future Research ...... 97 APPENDIX A Observation Systems 99 A1. Summary of Categories for the Reciprocal Category System 100 A2. Categories for Flanders Interaction Analysis System. 102 A3. Conversions of Categories: Reciprocal Category System to Flanders Interaction Analysis System Modified 104 B Raw Data Used in the Study 105 C Output From the Complete Analysis of the Second Fall Observation of Teacher A 113 BIBLIOGRAPHY 120 BIOGRAPHICAL SKETCH 125 v
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LIST OF TABLES Table Page 1 Summary of chisquare and Tau values 67 2 Equilibrium matrix versus proportion in each category. ... 75 3 Mean recurrence times 77 4 Mean and standard deviation of the passage time into category 6 81 5 Median of passage time into category 6 84 6 Mean and standard deviation of passage time into category 6 or 7 86 7 Probability of absorption into categories 6 and 7 89 vi
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LIST OF FIGURES Figure Page 1 Matrix for Bales' display of data 18 2 Sxammary of statistical procedures in research literature . . 32 3 Summary of data sources 41 vii
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Abstract of Dissertation Presented to the Graduate Coxincil of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy A STUDY OF THE USE OF MARKOV CHAIN STATISTICAL PROCEDURES WITH CATEGORY TYPE OBSERVATION DATA By Margaret Anne Dudasik June, 1977 Chairman: Dr. Elroy J. Bolduc, Jr. Major Department: Subject Specialization Teacher Education This study explored the use of the Markov chain statistical model with category type observation data. The relative goodnessoff it of the data to the assumptions of independence, onestep dependence, and twostep dependence was investigated using Goodman and Kruskal's Tau 2 2 (T^) and a chisquare (x ) test statistic. Using a X test it was found that an assiamption of onestep dependence or longer was preferable to an assumption of independence of data items in the chain. A comparison of values for onestep and twostep tally matrices gave no clear indication of which was a better model. Some categories were more twostep dependent, while others were more onestep dependent. Indications that a onestep assumption was not completely invalid, and the productive past use of the onestep assumption by researchers. viii
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led the author to assume the data to be a Markov chain. Various properties of Markov chains were applied to the observation data and the results interpreted in terms of classroom behavior. A transition probability matrix, an equilibrium matrix, and the mean recurrence time were constructed for each data set. Absorbing categories were created and the distribution of the passage times into the designated categories was investigated. ix
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CHAPTER I THE RESEARCH PROBLEM In the past twenty years there has been increasing interest in the study of classroom interaction, and many researchers have sought to develop effective instruments to record classroom behavior. In fact, more than 100 such observation instruments have been published. The statistical treatment of information collected in the classroom, however, has been much less diverse than the recording instruments being used. Observation data have been traditionally reported in matrix form as frequencies of the occurrence of pairs of events. One of the most influential scholars in the field, Ned Flanders, conceptualized classroom interaction as pairs of events with the first of the pair influencing the event which follows. According to Flanders, Teaching behavior, by its very nature, exists in a context of social interaction. The acts of teaching lead to reciprocal contacts between the teacher and the pupils, and the interchange itself is called teaching. Techniques for analyzing classroom interaction are based on the notion that these reciprocal contacts can be perceived as a series of events which occur one after another. Each event occupies a small segment of time, and the chain of events can be spaced along a time dimension. It is clear that the event was influenced by what preceded. (Flanders, 1970, p. 1) 1
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2 Flanders' hypothesis regarding dependence of sequential events is the fundamental premise of the mathematical theory of finite Markov chains. Application of the Markov chain model to the analysis of observation data, therefore, is a logical extension of the traditional treatment of such data. Probabilities rather than frequencies are displayed in the matrix, and the probability of the occurrence of ensueing events becomes the basis of statistical comparison. The probabilistic relationship between an event and the event immediately following is called onestep dependence. Statement of the Problem The assumption of onestep dependence suggests the application of Markov chain statistical procedures, and the present study explores the heuristic value of the Markov model. The specific questions addressed by this dissertation are: 1. Does category type observation data form a Markov chain? That is, are the observed behaviors onestep dependent, or is an assumption of independence or twostep dependence more appropriate? 2 . How can the Markov chain model be used to analyze observation data? Rationale for Need for the Study Traditionally, researchers such as Flanders using category type observation systems have assumed onestep dependence and have used a matrix to display the frequencies of pairs of behavior categories.
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3 The present study will extend the traditional approach in two ways . First, the assumption of onestep dependence, as opposed to independence or twostep dependence of data items, was tested in a unique way. Second, the full Markov chain model was applied to the data. A few studies (Pena, 1969; Hartnett and Rumery, 1973) have tested observation data for the onestep dependence assumption with conflicting findings . In the present study a proportional reduction in error (PRE) measiare was used in the analysis of onestep and twostep dependence rather than the chisquare type statistic used by earlier researchers. The use of the PRE allows a comparison of the fit of the onestep and twostep models, and the statistic is interpreted as the proportional reduction in error. The second unique feature of the present study was the use of a number of properties of finite Markov chains to generate statistics from observation data. Some researchers (Bellack, Kliebard, Hyman, and Smith, 1966; Rausch, 1965) have used the basic elements of the Markov chain model, but no research reviewed carried the analysis beyond the use of the transition probability matrix. The complete analysis of data presented in this research will include such statistics as the mean recurrence time of categories, long run probabilities of each behavior, the creation of absorbing states, the mean and variance of movement into the absorbing states, and absorption probabilities. Although the model is quite complex, the statistics which it generates are quite easily understood because they are in the form of probabilities, means and variances.
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. 4 The most important feature of the Markov chain model is the emphasis placed upon probabilities. By analyzing the probability of the events which follow a given event rather than the frequency of particular pairs of behaviors, classroom observation may be abstracted in terms of trends or patterns of pupilteacher interaction. During a particular recording session, for example, the teacher might ask a question only three times, and two of the three questions might have to be clarified before they can be answered. The frequencies in the appropriate cells of the matrix would be limited to a total of three for the entire row representing "teacher question." If probabilities are used, however, analysis would show that, with probability .667, clarification was needed after a question. With probability .333, clarification was not needed. There are other easily understood statistics that can be generated from the Markov chain model which provide meaningful abstractions for the researcher or the teacher. Rosenshine and Furst lament that, "observation systems appear to have been used more to document desired behavior than to test hypotheses" (1973, p. 161) . In the next chapter, observation research is examined in order to assess 'the .heuristic value of the Markov chain model, and the discussion will demonstrate that many testable hypotheses can be generated by a Markov analysis of observation data. Organization The dissertation is organized around a central theme of exploration, namely, the validity and possible application of a particular
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statistical model, the Markov chain. The topics of the dissertation include (1) traditional statistical procedures applied to observation data, (2) the procedures employed in the present study, and (3) the comparative value of the two approaches . Data analyzed using the Markov model are presented, and several suggestions for further research are offered. More specifically. Chapter II contains a review of three types of literature: observation literature, statistical procedures used in the past with category type observation data, and research programs in education and other fields which have made use of the Markov chain model. A detailed discussion of the procedures applied to research information is presented in the third chapter, and the order of the presentation was logically derived from the computer program used by this author for the processing of observation data. The fourth chapter, paralleling the organization of the third, contains an analysis of the results, while the fifth and final chapter of the dissertation summarizes the findings and suggests possible topics for future research programs. Although the present study is necessarily limited in scope, the implications for further research are significant. The Markov model is employed today by analysts in a number of fields, but a key question which remains to be answered in education concerns the validity and applicability of the model as applied to classroom observation studies .
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CHAPTER II REVIEW OF THE LITERATURE The published research on teaching which used direct observation is extensive and diverse. The purpose of this chapter is not to review this broad area exhaustively but rather to put into perspective the feasibility and value of the application of the Markov chain model to category type observation data. The first section of this chapter will be a discussion of observation instruments in general and the category type instruments in particular. The next broad topic for consideration is the research areas in which observation instrxnaents have been used. An overview of the types of studies which have been conducted will help to determine whether the model proposed in this dissertation can be used in the types of research generally carried out using observation instruments. The third section of this chapter is a discussion of the traditional analytic procedure used with category type observation data and the final section of the chapter is a review of some research studies which have used a Markov chain rtiodel for the analysis of data. The types of statistics which were derived from the research data are focused upon in order to evaluate the usefulness of using the Markov chain model with category type observation data in the study of teaching. 6
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7 Observation Instruments The interaction which occurs in the classroom is extremely complex. In the typical situation there may be one teacher and 30 students, and each of these individuals is constantly giving and receiving verbal and npnverbal cues. Because all behavior cannot be observed, the researcher must decide in advance what aspects of classroom behavior to study. Once this decision is made, an observation instrument is selected or developed to capture the desired dimensions of classroom interaction. Classroom observation instruments exist in large numbers. Simon and Boyer (1970) reported 92 different systems for observing classroom behavior, while Rosenshine and Furst (1973) in their review of six references estimated that more than 120 such systems existed. Dunkin and Biddle (1974) and Rosenshine and Furst (1973) reviewed observation research in an attempt to classify the broad range of instruments. One classification of observation systems was by recording procedure, and Rosenshine and Furst (1973) defined three types. They called a system in which an event was recorded each time it occurred a category type system. If an event which occurred during a specified time period was recorded only once, the system was designated a sign system. The third type of instrument, the rating type, required the observer to estimate the relative frequency of specified events. The present study was restricted to category type observation data. This type of instrument is a comprehensive list of mutually
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8 exclusive behaviors. The classroom observer classifies each occurrence during the observation period into one of the categories. A record can be made from a live observation, an audio tape or video tape. A very simple example of a category type recording system might be the following: Category number Behavior 1 Teacher talk 2 Student talk 3 Confusion or silence The recording of these categories during classroom observation varies according to the unit of analysis selected. For example, the unit of analysis used by Flanders (1965) , Lohman et al . (1967) , and Ober et al . (1968) , and the one employed in this dissertation, is an arbitrary time unit. If a unit of three seconds is used, the recording of the categories in the example above over a period of 15 seconds might yield the following sequence : 3 3 1 1 2 3 The period begins with silence or confusion followed by about six seconds of teacher talk, three seconds of student talk and another period of silence or confusion. In the application of such a system, the order of the observed behavior categories is generally preserved.
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9 A second unit of analysis which is used with category type observation instruments involves specified chains of behavior. Bellack et al . (1966) and Smith et al. (1967) , defined segments of verbal interaction as units of analysis . A category name for a sequence might be "responding," and an entire segment of the interaction would be coded "responding." Whatever followed this behavior would be the next code in the record. Just as in studies which used a time unit of analysis, research procedures with behaviors as the analytical units preserved the order of the occurrences. Direct Observation to Study Teaching The purpose of reviewing research which used direct observation to study teaching is to survey the types of problems that were studied using observation instruments. This survey is to provide background information for the discussion of the relative contribution of the application of Markov chain statistical procedure to observational data. Reviews of observation research by Medley and Mitzel (1963) , Flanders and Simon (1970) , Soar (1972) , Rosenshine and Furst (1973) , and Dunkin and Biddle (1974) are notable. .Flanders and Simon, Dunkin and Biddle, and Soar emphasized results of research studies, whereas Rosenshine and Furst and Medley and Mitzel surveyed all aspects of the field of direct observation with emphasis upon methods, instrumentation, and general types of research.
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10 As an introduction to the chapter on research using direct observation of teaching in the Second Handbook of Research on Teaching , Rosenshine and Furst observed that: The research on teaching in natural settings to date has tended to be chaotic, unorganized and selfserving. The purpose of this chapter is to ease the reader into the maze of instrumentation and research which has focused on teaching in natural and seminatural settings. There seems to be no simple route through the chaos which has developed. (1973, p. 122) The Rosenshine and Furst classification scheme will serve as a guide for the maze in the present discussion. They classified research using observation of classroom behavior into the following categories: classroomfocused research, teachingskills research, and curriculiammaterials research. In classroomfocused research the authors found that the primary emphasis had been placed upon description of classroom interaction and investigation of relationships between classroom occurrences and pupil growth on achievement measures. Descriptive research of the type done by Bellack et al . .(1967) , and by others was intended to provide baseline data and concepts which would verify theories of instruction and lead to correlational and experimental studies. Correlational research such as that conducted by Soar and Soar (1973) and by Flanders (1965) , aimed at establishing relationships between classroom behavior and educational outcomes, yielded some consistent results.
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11 Rosenshine and Furst (1973, pp. 156158) found that the 50 studies they reviewed indicated positive correlations between student growth and the following teacher characteristics: clarity, variability, enthusiasm, task orientation, and indirectness. Correlations between teacher criticism and student growth were negative in 12 of 17 studies of this nature reviewed. Some experimental classroomfocused research programs were also reported. In six such studies (Rosenshine and Furst, 1973) teachers were trained in the use of different types of discourse. Behavior in the experimental group classrooms was found to be significantly different than behavior in the classrooms of nontrained teachers. The productivity of classroomfocused research was questioned by various authors (for example, Dunkin and Biddle, 1974; Gage, 1968); however. Gage expressed hope for the productivity of future research when he stated that: The field of research on teaching is today engaged in continuous and extensive analysis of its approaches and theoretical formulations .... More complex research designs capable of taking more categories of significant variables into account are being propounded .... The faithpersists that educationally significant differences can be consistently produced in the future as new intellectual and material resources are brought to bear on educational problems. (Gage, 1968, p. 403) The second major grouping of studies reviewed by Rosenshine and Furst was teachingskills research. An example of how observation data were used in a teacher training program was a study by Limbacher
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12 (1971) in which effects of microteaching were investigated. Each session was videotaped and coded using Flanders Interaction Analysis System (FIAS) . One of the hypotheses tested was that the group with microteaching experience would have a higher ratio of Indirect to Direct behavior (See page 23) . Flanders' categories for coding were used differently in a study by Furst (1965) , which recorded differences between student teachers who had been taught FIAS and those who had not. The experimental group was found to be more accepting of student ideas and less rejecting of student initiating behavior. The Furst study used an observation system as a training tool whereas the Limbacher study used direct observation to monitor behavior of the teachers. Rosenshine and Furst summarized the possible uses of observation instruments in teacher training: Fortified with acceptable criterion measure, investigators could use existing observational systems to study the behaviors of those teachers and relate the skillsrelevent behaviors to the measures of growth, and they could compare the behavior and the outcomes for trained and untrained groups of teachers. Â• (1973, p. 126) The third major category of research suggested by Rosenshine and Furst, was curriculummaterials research. The primary way in which Taba et al . (1964) and Bissell (1971) used descriptive systems was to document the implementation of curriculum materials. In contrast, Siegel and Rosenshine (1972) , using an observation system, validated the
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13 importance of variables emphasized within a program. In a third approach. Soar (1972) used the monitoring capability of observation systems to compare several Project FollowThrough programs. After a factoranalysis of data from several observation systems, the factors were examined to determine if they correlated with pupil gain and if they discriminated among programs. Through the "maze of research," the primary feature of direct observation studies is description of classroom behavior. The description of behavior using observation instrximents has been used in research studies to explore interaction patterns, to discriminate among situations or programs, to monitor the implementation of programs or behavior, and to correlate behaviors with pupil variables such as attitudes or gain on achievement measures. As statistical procedures are reviewed in the next section, it must be kept in mind that one criterion for evaluation of a procedure is in terms of the problem to be studied. Statistical Procedures Used with Observation Instruments Two major areas of emphasis, classroom climate and sequential patterns of behaving, emerged in studies using category type observation instruments. Each of these branches of research developed distinct statistical procedures which were routinely applied.
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14 Classroom Climate Research Major studies which influenced the procedures used in the classroom climate studies were completed by Anderson (1939) , Bales (1950) , Withall (1949) , Flanders (1965) , and Soar (1966) , Withall explained that the concept of classroom climate in his study represented "the emotional tone which is a concomitant of interpersonal interaction. It is a general emotional factor which appears to be present in interactions occurring between individuals in facetoface groups" (1967, p. 49) . One of the earliest efforts to analyze classroom behavior in terms of climate was by H. H. Anderson (1939) . The study focused upon two types of behavior, labeled "Domination" and "Integration." Domination was characterized by use of force, commands, shame, threats, and blame. "Integration" designates behavior that was more flexible, such as support, acceptance, etc. The purpose of the study was to develop a reliable technique for recording behavior in terms of domination and integration. An observer was expected to record the frequency of teacher behaviors in terms of twenty carefully defined categories. Eight of the categories described dominative behavior and nine categories were for integrative behavior. The other three codes were for neutral behavior. When a student was directly addressed, a record was kept of which student it was, but not of his behavior. In the collection of data the order of occurrences was not preserved.
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15 In the Anderson study, classes were observed for a period of three hours, and every observation summary showed that total number of dominative statements, found by summing the frequencies of the eight categories so defined, was greater than the total number of integrative statements. The ratio of dominativetointegrative ranged from 2:1 to 5:1. The mean number of dominative and integrative contacts per hour was calculated for each teacher ' s morning and afternoon classes, and bar graphs were used to display the results. Similarities were observed for the same teacher and different patterns were observed between teachers . Data regarding contact with individual students were shown on a broken line graph in the form of mean number of contacts of each type per child. Children were assigned numbers along the xaxis, and frequency of contact was shown on the yaxis. The Anderson research led to development of a reliable observation instrument with high reliability between observers of the same class sections, and the analysis suggested many areas for further research. The work by Robert Bales in the 1940 's had a major influence on the development of theories of socialemotional climate. The purpose of his research was to develop a technique for studying and describing small groups. For a study pxiblished in 1950, data were drawn from a number of small group settings . Bales ' research technique, called Interaction Process Analysis, was primarily designed to record the behavior of small groups, but it had direct classroom
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16 application as well. Bales, unlike Anderson, believed that "sequential analysis is of particular interest in the development in the method, since it is on the assumption of a kind of idealized sequence, of perhaps several similar kinds, that the categories are arranged in their present order" (1950, p. 8) . Bales defined twelve categories of verbal behavior with the first six paired with the last six. For example, category 5 was "gives opinion/" and category 8 was "asks question." To facilitate coding, an Interaction Recorder (Bales and Gerbrands, 1948) was used by the observer. The instrument consisted of moving tape on which a sequential record was kept of observed categories of behavior of both the speaker and the receiver. In his analysis of interaction data. Bales used a wide variety of summary techniques. Because it was not yet known what statistical techniques would produce insight from the data. Bales tried a number of approaches. Rates of occurrence calculated in percentages gave an indication of the frequency of use of certain categories. The observation data were partitioned into tenminute periods, and frequencies of certain behavior were studied over successive time periods in order to isolate changes in the group over time. The sequence of observed behaviors was broken into adjacent pairs, and Bales observed that, "ignoring repetitions, we see a number of expected tendencies according to pairs of categories ..." (195.0, p. 128). For example, the sequence 67786 produces the pairs (6,7), (7,7), (7,8), and (8,6). Bales chose to ignore repetitions like (7,7),
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17 and the other pairs were tabulated in a matrixlike array (Figure 1) . Cell "I", for example, would contain the frequency count for (6,7) pairs . After initial tabulation, the array was collapsed by letting area A include categories 1 through 3, B categories 4 through 6, C categories 7 through 9, and D categories 10 through 12 (Figure 1) . Within each cell of the new table. Bales recorded an observed frequency, an expected frequency, the difference between the two, and a ratio of the difference to the expected frequency. It is unclear how the expected value was obtained, but it was not the same as that obtained when the expected value for the cell was calculated using the chisquare test. When he plotted the dif ferencetoexpectedfrequency ratio for successive time periods on a graph. Bales found that certain patterns emerged. To give the theoretical interpretation, it tells us whether a given functional problem received more or less than its usual amount of attention during the subperiod .... The peak rate of each pair of categories appears within the meeting in the same order in which the pairs of categories are arranged on the observation list, which in turn is an order suggested by a priori assumptions about the hierarchial nesting relations of the various functional problems involved in interactions systems. (1950, p. 136) Using frequencies derived from temporally ordered data. Bales calculated a number of indices. The "Index of Difficulty of Communication" was computed in the following manner: the frequency of use of
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B 2 3 4 5 6 7 8 9 10 11 12 18 B 10 11 12 B A B C D Figure 1.Â— Matrix for Bale's display of data
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19 category 7 divided by the frequency for category 7 plus the frequency for category 6. The "Index of Directness of Control" was given by the following, where r^ represents the frequency of use of the category i. r r 4 + 4 4 6 5 6 Six such indices were defined and used in the analysis of data. Information gathered on who initiated and who received in the interaction was displayed in a matrix in a manner similar to the technique used above with pairs of categories. Also, indices were computed for individuals by siunming their use of certain categories. On the basis of his findings. Bales postulates "that there is a series of concomitant changes in ideological emphasis among the members of the group" (1950, p. 172), Many specific directions for further study were suggested by this research, and the statistical methods employed had great influence on later work in the field of observation analysis. In the late 1940' s, John Withall, a student of Herbert Thelen and Carl Rogers at the University of Chicago, conducted research on the construction of a valid and reliable categorization system for socialemotional climate (Withall, 1949). Teacher verbal responses were found to take seven forms in his Climate Index. Three of these Â— support, acceptance and questions Â— were classified as "learner centered," and three other types of statements Â— directive, reproof and teacher self
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20 support Â— were "teacher centered." One category was reserved for administrative or neutral statements. The frequency of occurrence was recorded for each of the seven categories. Because there were several observations of each teacher, it was useful to convert raw frequencies into percentages and to compute a mean for each teacher. As a part of his validation procedures, Withall categorized his observations and computed the Integrative to Dominative ratio, using Anderson's System. Using the Climate Index, a ratio of learner centered statements to teacher centered statements was calculated. The two ratios were not equal for the same observation, but Withall found that they were comparable. Conclusions of the study were that classroom climate could be measured empirically, with high reliability and with validity, using the Climate Index. Ned Flanders used Withall 's observation instrument in his dissertation research at the University of Chicago in 1949 and again in an article published in 1951. Using the seven categories of the Climate Index and a Qsort of verbal statements, Flanders measured student attitudes during and after the learning periods . The pulse and skin resistance of students were monitored during learning periods and evaluation periods. "From the association of particular kinds of teacher behavior with all recorded student behaviors, inferences were made concerning cause and effect cycles in the teacherstudent interaction" (1951, p. 101) The seven student groups were exposed to a learning situation in which there was more teacher centered (TC) behavior and then to another class in which there was more learner centered (LC) behavior. Flanders found
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21 that teacher centered behavior elicited hostility/withdrawal from students while learner centered behavior elicited problem oriented behavior and a decrease in student anxiety. Flanders came to his 1955 study in New Zealand with the tools discussed thus far (Flanders, 1965) . The purpose of his series of studies in the late 1950 's was to validate the Flanders Interaction Analysis System (FIAS) for verbal behavior categorization, to measure student academic achievement and student attitudes , and to correlate the achievement and attitude scores with verbal patterns in the classroom . The first departure from tradition in the study was the observation analysis system. Flanders defined ten categories (Appendix A2 ) . Seven were for teacher talk, two for student talk and one for silence or confusion. The previous system for classroom use had not included student talk or the nonverbal category of silenceconfusion. Teacher talk was broken down into "indirect influence" statements and "direct influence" statements (p. 20) . Flanders followed the lead of Anderson and Withall in the definition of the climate concepts. The coding was made every three seconds, and the order of the sequence was preserved as in the Bales System. Great care was taken in the definition and explanation of each category so that the set was able to define any occurrence in the classroom and so that each event would fall into only one category.
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22 In Flanders' research, he needed to know how the data from the observation system should be summarized. John Darwin, a statistician at the Applied Mathematics Laboratory in Wellington, New Zealand, provided some direction in the analysis of the observation data. Darwin (1959; Flanders, 1965) found that the data were onestep dependent, a Markov chain, using Hoel's test (Hoel, 1954), and he suggested that a matrix was appropriate for data tabulation since it preserved the sequence pairs. Darwin also suggested a chisquare statistical test 2 (Darwin's X ) which could be used to determine if two matrices of observation data were significantly different. More recently, Pena (1969) investigated the Darwin chisquare because the statistic tended to be too large (the test too powerful) to distinguish between matrices of observation data. In the first phase of the study, Pena tested the length of dependence in sequence of classroom observation data using Hoel's test and found that the betterfitting model assumed twostep dependence rather than onestep dependence of the Markov chain. Harnett and Rumery observed that "the results reported by Pena concerning the Markovian properties of interaction data are misleading" (1973, p. 2). The criticism was based upon the fact that Pena obtained long chains, 2,398 to 11,756 tallies, of FIAS data by combining data for a single teacher across subjects and by combining data across five teachers to form sets of observation data for each subject area. Harnett and Rumery used individual classroom observations of 167 to 555 codes in length in their study of dependence assumptions.
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23 The results of the application of Hoel's test for length of dependence indicated that the observation data sets were onestep dependent, or Markov chains. Tabulation of pairs of observed categories into a matrix as in Bales" studies provided the basis for analysis by Flanders. The matrix was used for the representation of data primarily because it preserved one step of the sequence. Flanders stated that "our own estimates indicate that sequence pairs probably account for about 60% of the interdependence between events when our ten categories are involved" (1967, p. 372) . Cells and groups of cells became the primary objects of analyses . Another very important technique used by Flanders was the calculation of a ratio of indirect to direct statements. A simple percentage frequency or frequency per 1000 tallies was calculated for each category, and two ratios were computed. The "big I/D ratio" consisted of column totals for categories 1, 2, 3, and 4, divided by tallies in 5, 6, and 7. The "small i/d ratio" was the same except that categories 4 and 5 were not used (Flanders, 1965, p. 35). Histograms were used to display these various statistics and compare teachers. The observation techniques introduced and the statistical methods used had a profound influence in the field of interaction analysis. In a more recent book, Flanders (1970) suggested most of the same statistical methods that he had used in his 1965 publication. In the years that followed the publication of these first Flanders studies, many researchers used the FIAS, or modifications of it, as well
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24 as the statistical methods used by Flanders (e.g., Limbacher, 1971; Lohmon, 1967) . Modifications of FIAS generally took the same form as the original instrxoment but expanded certain categories into several. The Reciprocal Category System (RCS, See Appendix A1) by Ober, Wood, and Roberts (1968) was one system which was derived from the Flanders work. The seven categories of teacher behavior in FIAS were expanded to nine in the Reciprocal Category System. The same nine categoreis were applied to student behavior and were coded with numbers 11 through 19. This increased the number of student behavior categories from two to seven. The recording of parallel categories for students and teacher had the advantage of allowing comparison of student and teacher behavior. Silence or confusion was recorded as one category in FIAS but was recorded separately in RCS. With the availability of high speed computers, additional techniques came into use in classroom climate research. In the midsixties, Soar (1965) used FIAS in combination with the Observation Schedule and Record, a sign system developed by Medley and Mitzel (Simon and Boyer, 1970) . The .use of several instruments produced a record of nonverbal as well as verbal behaviors. .FIAS data were tabulated in a matrix, and cell frequencies were used in conjunction with Observation Schedule and Record data, vocabulary gain scores, and reading improvement data. A factor analysis was done on the relationships between observation data and vocabulary and reading gains.
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25 Two dimensions of classroom behavior were taken from the results of the observation schedules on the basis of factor analysis. The dimension of control behavior was measured by the revised i/d ratio for rows 8 and 9 of the Interaction Analysis . This is a ratio made up of teacher behaviors which occur immediately after a pupil stops talking .... The clearest dimension of emotional climate which emerged from the analysis was made up of expressions of hostility, criticism or negative feelings. (Soar, 1965, p. 246) A further analysis of these data by Soar (1972) revealed a nonlinear relationship among the variables under consideration. Insight was gained from plotting pupil growth on the yaxis and teacher indirectness along the xaxis. Graphs drawn for reading, vocabulary, and creativity gains showed that the fit of the data to the curve was better than the fit to a straight line (Soar, 1972, pp. 8991). In a research program concerning the differences in pupil classroom behavior and achievement before and after summer vacation. Soar and Soar (1973) used a modification of FIAS by Ober called the Reciprocal Category System. Also used were three sign systems, the Florida Taxonomy of Cognitive Behavior, Teacher Practices Observation Record, and the Florida Climate and' Control System. The statistical analysis of data was essentially the same as that found in the 1965 study. A large number of significant linear relationships was found between pupil pretest performance and classroom behavior. Student gains on various tests were found to have a curvilinear relationship to teacher structure and control, with the optimum level of structure depending upon the particular outcome desired. Throughout the study, the use
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26 of nonlinear as well as linear analysis proved to be a useful technique . Sequential Pattern Research The second major area of research employing category type instruments involved sequential patterns of classroom behavior. Studies of this type were conducted by Bellack et al . (1966), Smith et al . (1967), and Taba et al . (1964). Classroom behavior was recorded, and then ventures (Smith et al ., 1967), teaching cycles (Bellack et al . , 1966) , or discussion patterns (Taba et al . , 1964) were coded and analyzed. Research in this area has been descriptive as well as correlational. Researchers studying the sequential patterns of behavior in the classroom were more interested in longer chains of behavior than in the pairs analyzed in classroom climate studies. The studies characterized as sequential also included multiple descriptors applied to any one utterance; therefore, data for this type of analysis were recorded, transcribed and then coded. For example, the teacher phrase, "where did George Washington live?" might be coded: teacher/question/information sought Smith et al . (1967) analyzed classroom discourse in tems of strategies used to produce certain outcomes derived from the stated objectives of the teacher. Seventeen high school teachers of various subjects were involved in the study, and five class sessions were recorded
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27 for each. The transcripts of classroom verbal behavior were broken into "ventures," or topic segments. Eight types of ventures were defined, and plays, shorter units of a few sentences within ventures, were coded. Ventures were displayed on a type of "flow chart." The venture diagrams were not necessarily linear; that is, various branches were found to exist in ventures with the same lable. Sequences of consecutive plays were put into a matrix to show the frequencies of sequential pairs. One of the main contributions of this descriptive study was the coding of verbal behavior into ventures and plays and the use of diagrams to display the results. Bellack et al . (1963, 1965, 1966) did research using sequential patterns in the description and analysis of the linguistic behavior of teachers and students. "The language game" which teachers and students played by certain well defined rules was explored. Discourse was recorded and later transcribed, and the unit of analysis used in coding the transcripts was the "pedagogical move." The four major categories of moves were structuring, soliciting, responding, and reacting. Each pedagogical move was coded as follows (1966, p. 16) : 1) Speaker (teacher, pupil or audio visual device) 2) Type of pedagogical move 3) Substantive meaning 4) Susstantivelogical meaning 5) Number of lines in (3) and (4) 6) Instructural meaning 7) Instructurallogical meaning 8) Number of lines in (6) and (7) Moves were found to occur in the classroom discourse in certain patterns, and Bellack et al . designated twentyone such patterns as
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28 particular teaching cycles. The cycles continued from one to five moves, and the frequency of various codings was expressed in terms of percentages. Total lines in the moves, for example, were calculated and converted into percentages for teachers and pupils. Another such summary statistic, a cycle activity index, was found by computing the number of cycles per minute. A matrix was used by Bellack for tabulation of various categories and cycles of verbal behavior. Using temporarilyordered teaching cycles, Bellack sought to determine whether certain cyclical patterns and dimensions of cycles tend to influence the subsequent patterning of pedagogical moves. Statistically, this was described through a Markov chain to determine the transition probabilities of moving from one state to another. Taking types of teaching cycles as states, the probabilities of moving from one type of cycle to another were investigated as a way of determining whether one pattern of pedagogical move tends to influence immediately subsequent patterning. A transition probability matrix was also used to summarize data. In the following example from Bellack 's work, data on the initiators of cycles were summarized (1966, p. 208) . (Bellack et al 1965, p. 158) Teacher Pupil Audio Visual T P AV 89.5% 59.4% 15.4% 10.4% 40.6% 0% 0.1% 0% 84.6%
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29 The three categories of initiators used were teacher, pupil and audio visual (movies, overhead projector, or other similar equipment) . Each row of the display represented the initiator of a cycle, and the columns indicated the initiators of the cycle immediately following. The interpretation of the example above is as follows: Given that the pupil initiated the last, cycle, there is a 59.4% chance (a .594 probability) that the teacher will initiate the next cycle. The sum of the elements of a row is 100%. This display of data supported the theory that once he had initiated a cycle a pupil was more likely to initiate the next cycle. Regarding the use of a transition probability matrix, Bellack commented that The temporal analysis, then, by establishing statistical relationships between prior and following states defined here as teaching cycles, permits a more refined description of classroom verbal behavior by greatly improving one's ability to see relationships among classroom variables. (Bellack et_al . , 1959, p, 169) Taba and Elzey (1967) studied sequential patterns in social studies classrooms which were using Taba's curriculum materials. Verbal behavior was recorded, transcribed, and coded to reflect the speaker, the function, and the level of thought for each "thought unit." In a later publication, Taba and Elzey noted that The multiple coding scheme makes it impossible to deplict the flow of the classroom discussion by charting the sequence of transactions between
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30 the teacher and the children, the charges in the level of thought during the discussion, and the effects of these strategies upon the level and the direction of thought. (Taba and Elzey, 1964, p. 493) Four patterns of discourse emerged as a result of the "flow chart" presentation of data. The chart consisted of level of thought plotted on the vertical axis and time or thought units (numbered) along the horizontal axis. At each unit, a notation was made to indicate the speaker and the function of the unit. It was found that, in order to sustain higher levels of thought processes, prevention of premature entry into the level was important. Bellack, Smith , and others used predefined chains of behavior as their unit of analysis. Several researchers recently have used computers to scan the observation data for all possible chains of three or more categories in length. Collet and Semmel (1970), for example, perfected a computer program which facilitated tabulation of chains of various lengths. Dolley (1974) used this program to do a descriptive study of motherchild interaction. Along the same lines, Campbell (1975) presented a strong case for using actual counts of chains of more than two steps. He called his approach macroanalysis , as compared to microanalysis which involves only the tabulation of adjacent pairs of categories. Campbell stated that "macroanalysis could serve as a stimulus for further descriptive research. Perhaps a new round of such research is needed to determine those patterns which are currently used by teachers at various grade levels" (1975, p. 263) .
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31 Summary of Statistical Techniques Figure 2 on page 32 summarizes the statistical analysis methods used by the researchers discussed in the preceding section. In his work on classroom climate, Flanders used all of the methods used by Anderson, Bales and Withall. Flanders considered the matrix appropriate because the observation data were earlier found by Darwin to fit the assumption of onestep dependence of the Markov chain model. He did not use the transition matrix which is generally used for the representation of Markov chain data. A few years later. Soar built upon the methods of Flanders and introduced the use of multiple systems and factor analysis. The study of sequences of behavior began in the early nineteen sixties, and Smith led the way with observation instrument techniques. Bellack expanded on this coding system and introduced a range of statistical methods for analysis of data. Taba, in answer to her own needs in curriculum evaluation, presented still another model for the presentation of data. It should be noted that Smith as well as Bellack drew heavily upon the interaction analysis work done by Flanders. Bellack' s use of the transition matrix, for example, was an extension of Flanders' use of the tally matrix. Bellack, however, made fuller use of the Markov chain properties of the data in his research. Rather than a descriptive display, Bellack used the transition probability matrix as a predictive model. Bellack, Smith, and others used predefined sequences of categories called ventures or cycles, and the occurrence
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32 H a 3 +J tn 0) X H Â•H M o 4> M 0) Â»] , c: 0) >i P Â•H Â•H u 3 > n3 W 4) o o >^ H e P >i W 3 u 0) rH >1 41 0 tn CO c u (0 01 to >i Q) Â•H 0 Q) C (d
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33 and order of these patterns were studied. More recently, longer chains, which were not predefined but rather emerged from the data, were used by Campbell as well as by Collet and Semmel. Markov Chain Theory in Research A Russian mathematician, A. A. Markov, first developed the basic concepts of Markov chains in the early 1900 's. Markov chain theory was refined in the years that followed and was widely used in such fields as biology, chemistry, physics, astronomy, and engineering. It has only been since the early 1950 "s that social and behavioral scientists have begun to use this mathematical model. Future uses of the elegant model developed by Markov can best be identified by considering applications which have proven productive in the past. The theme in all prior studies has been the usefulness of the Markov chain model to describe and predict. The few areas of research which will be briefly reviewed here are personnel administration and organizational growth, social mobility, clinical applications, and interaction analysis. The areas of personnel' administration and organizational growth have used Markov chains to model research problems. Rowland and Sovereign, in their study of manpower replacement problems (1969) , used a matrix of probabilities of transition from one job classification to another. Similarly, Vroom and MacCrimmon (1968) described the career movement of certain personnel. States of the Markov chain were defined in these studies by the level or function of the job held
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34 by an individual. Movement from one state to another over time was displayed on a transition probability matrix, and powers of the matrix were used to predict turnovers, promotions, and personnel needs for future years. Using the same techniques, Adleman (1958) studied organizational growth in terms of the size distribution of firms in the steel industry. He used a Markov model to describe change in the size of firms and to predict eventual equilibrixim size distribution. Enrollment studies, such as that conducted by Mohrenweiser (1969) , used techniques similar to those used in the study of industrial growth and personnel needs. The advantage of the application of a Markov chain model in research on personnel administration and organizational growth was not only its power to describe the observed movement from one state to another, but also the power of the model to predict future movement and eventual equilibrium. These same qualities of description and prediction make Markov chain procedures useful in the study of social mobility and labor supply shifts. Glass and Hall (1954) and other sociologists used a transition probability matrix to describe probabilities of membership in a particular social class over time. They also compared movement at different periods of history by comparing rows of the respective transition matrices. The predictive capability of the model was particularly useful in studies attempting to forecast manpower needs and supply for various career fields. Aside from the large scale descriptivepredictive research programs, the Markov chain model has been meaningfully used in clinical
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35 settings. Meredith (1974) reported a particularly interesting use of the full statistical model in his comparison of various programs in a mental hospital. In addition to the commonly used transition probability matrix to describe probabilistically movement from one state to another, Meredith used death and exit from the hospital as absorbing states. Using Markov procedures, he studied the movement of patients into these states. Conclusions of the research were that the Markov model provided a tool for comparisons of programs, for longrange evaluation of performance, and for description of existing conditions. The final type of research reviewed which used Markov chain procedures was interaction analysis. The fields of sociology and speech have produced studies of this type. The contributions of Bales (1950) , such as the use of a matrix for the display of data, was discussed in the previous section. In a more recent study, Hawes and Foley (1973) used transition probability matrices to compare the use of thirteen behavioral categories or states in interaction involving interviewers classified as directive, moderately directive and nondirective . Predicted longterm proportions of each type of behavior were calculated and compared with the actual proportion of each category in the data sets. As Hawes and Foley state, "the Markov statistics and graph present a picture of ways the communication system behaves over time" (1973, p. 219). In a slightly different study, Raush (1965) coded the social behavior of normal and disturbed children in order to study the interaction sequence. Transition probability matrices were constructed and used to compare groups.
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36 The specificity of the methods allows us ... to achieve something more than the clinically obvious. Given the situation, given the nature of the group, we can, knowing the actions of one child, predict rather well what another child will do ... . Specifically, we can detail which situations are likely to induce which behaviors and how particular situations are likely to modify the contingencies between stimulus and response acts. (Raush, 1965, p. 497) The research by Hawes and Foley and Raush was similar in many ways to the work of Bellack et al . (1966) who studied sequential patterns of behavior in the classroom. The researchers assumed their data to be onestep dependent, and they used a Markov chain model in the construction of transition probability matrices from observation data . In the observation literature reviewed, no classroom observation studies were foxand in which category type data were analyzed using the Markov chain model. Two research studies (Pena, 1969; Hartnett and Rumery, 1973) applied statistical tests to category type observation data to determine if the Markov chain model assumptions were met. The results were conflicting and there were, therefore, no clear indications of the viability of the model for use with category type observation data. No observation studies reviewed used any Markov chain procedures other than the transition probability matrix. Summary and Conclusions Aside from absolute statistical tests of assumptions of a model, guidance in the choice of procedures in a particular type of research
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37 must come from an examination of the purpose of the work, the traditional methods in the field, and similar studies which have used the proposed procedures. In the case of the proposed use of the Markov chain model with category type observation data, indications of the possible productive applications of the model were found in the survey of literature related to observation studies, procedures generally used with category type data, and research in other fields in which the model had been used. Direct observation data have been used to describe classroom behavior, to validate curricula and materials, to train teachers in specific skills, and to correlate with pupil gains. The primary fiinction of the observation system is to record and describe behavior in the form of specific codes, and, in some cases, to provide feedback of information to the teacher. In some of the Markov studies, such as those conducted by Meredith (1974) and Raush (1965), the Markov chain model was demonstrated to provide useful summary statistics, descriptions, and meaningful feedback to clients. The traditional procedures used by observation analysis researchers also indicate the possible productiveness of Markov chain modeling of observation data. The primary assumption of the Markov model, onestep dependence, was implicit in the work of most researchers since Bales, and the assumption was also manifest in the use of a matrix of tabulation of data pairs. Bellack et al . and Flanders stated that their data were approximately Markov chains. Bellack et al. even used the transition
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38 probability matrix, a basic concept in Markov chain theory. Indications are that a more complete application of the Markov chain model would prove productive . The final consideration in the review of the literature was whether or not research studies had effectively used the statistical model under consideration. The works of Raush (1965) and Hawes and Foley (1973) are such studies. Both focused upon social interactions, and both used a category type system to code behavior. The application of elements of the Markov chain model, though not the entire model, was both meaningful and productive in the analysis of their data. A review of the literature suggests that at least an exploration of the complete application of the Markov chain model is warranted. The transition probability matrix which has been used with category type data is but a first step. Other properties such as absorbing states, mean passage times, and absorption probabilities could have meaning when applied to observation data. These statistics and how they are computed are the topic discussed in Chapter III.
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CHAPTER III PROCEDURES The research literature in observation analysis strongly suggests that the Markov chain model is useful in the analysis of observation data. In fact, several researchers (e.g., Hoel, 1954; Darwin, 1959; Pena, 1969; Hartnett & Rummery, 1973) have specifically tested observation data for the length of the. dependence of chains of interaction data. In the present study, a similar problem is studied; however, instead of an absolute test of dependence, a comparative approach is taken. In Chapter III the data source for the present investigation is discussed. The data in their original form was not entirely suited to analysis as a Markov chain. The transformations which were made and the reasons for these changes are explained in detail. The remainder of the present chapter is a discussion of the Markov chain mathematical model and how data are tested for the model. The discussion begins with a general description of a Markov process, and attention is given to whether observation data fit this description. Hypotheses for testing the length of dependence in a chain are presented, and the appropriate statistical tests are indicated. In the final sections, various properties of a Markov chain are discussed and their computing formulas are given. The specific results gained 39
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40 from application of the procedures discussed in the present chapter are presented in Chapter IV. Sample This study involves the application of the Markov chain model to what might be considered typical observation data, and characteristics of the particular sample were not of direct importance to the research. The data used were Reciprocal Category System data transformed to Flanders Interaction Analysis System. The data were gathered in an evaluative research study of Project FollowThrough (Soar, 1973) . The national program known as Project FollowThrough (Maccoby and Zellner, 1970) was designed to improve education in kindergarten through third grade for children from impoverished environments. The Soar study contained information gathered from eight different programs within FollowThrough. In addition to Reciprocal Category System data, the following classroom observation measures were used in the evaluation study: Florida Climate and Control System, Teacher Practices Observation Record, Florida Taxonomy of Cognitive Behavior, and global rating of activities. All subjects were observed at least once during each year of the threeyear study. A factor analysis of data from ' each instrument was used. Seventeen FollowThrough program classrooms were observed three times during the last year of study. Data from four of these seventeen classrooms were used in the present study. Subjects were selected based upon their scores on a factor generated from Florida Classroom
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41 Climate System (FLACCS) data. This factor, called "strong control" (Soar, 1973, p, 52), includes such teacher behavior as warning, criticizing, spanking, frowning, and finger shaking. Two teachers who were high on this factor, and two who were low on this factor were selected for inclusion in the present study. Extreme levels of this particular factor provided contrasting data for analysis. The four classrooms used in the research were all first grade level. The two "high control" groups, A and B, were northeastern inner city schools, and the two "low control" groups, C and D, were midwestern small and mediumsized cities. A breakdown of the sample is shown in Figure 3 . Classroom FLACCS 1 Teacher Factor Score ^^^'^^ ^^^""^ Northeast inner city Northeast inner city Midwest small town Midwest middle town Figure 3. Summary of data sources Observation data were collected by Soar using the Reciprocal Category System. Three fiveminute time periods in midmorning were audiotaped and later coded. Three such observations were made of each classroom. The first observation for each group was made in early September, the second in midOctober and the last in late January or February. A 61.56 1 B. 62.74 1 C 44.22 1 D 45,78 1
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42 Transformation of Data The data used in the present study were originally collected using the twenty categories of the Reciprocal Category System (RCS) by Ober (See Appendix A1) . The nvunber of empty cells in the matrix display due to the nonoccurrence of certain behaviors suggested a transformation of the data (See Appendix A3) . A comparison of RCS and Flanders Interaction Analysis System (FIAS; see Appendix A2) revealed that groups of RCS categories could recede into one FIAS category without loss of meaning. The only exceptions to the parallel natures of RCS and FIAS were found in categories 1, 2, and 3 of the two systems; these three categories had overlapping meanings, The first three categories in RCS were placed in a single category which included teacher accepting behavior, warming the climate, and amplifying student ideas. These are similar behaviors, and for the purpose of this study no distortion of data results from the collapsing of the behaviors into one category. Categories 4, 7, and 15 in RCS were left intact except for a change in the number used to identify the state. The RCS codes 5 and 6 were combined into one category corresponding to the single code in FIAS representing "lecturing." Categories 8 and 9, "corrects" and "cools the climate," became "criticism," the seventh category in FIAS. The RCS has several categories, 11 through 14 and 16 through 19, for "pupil interaction"; these were grouped into one category. Finally, category 10, "silence,'
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43 and category 20, "confusion," were grouped together as they are in the FIAS. The transformation of the data using FIAS as a guide reduced the number of categories of behavior from twenty to eight. While the codes after transformation were not completely analogous to the original categories, the differences were believed to be unimportant for the purpose of the present study. The general breakdown of behavior types in the transfomed data is very similar to the widely used FIAS. The advantage of RCS over FIAS most often noted is the additional student behaviors which can be coded. Many of the student behaviors do not routinely occur during an observation session. Because the statistical procedures used in the present study would not be appropriate should a number of pairs of categories not occur, a transformation of data, i.e., collapsing some categories, was indicated. Since categories were not arbitrarily combined, the extent to which the sequence is onestep dependent should not be affected by the receding. Markov Chain Model A mathematical model for category type observation data whose value is suggested by the nature of the data as well as by recent research literature in the field is the Markov chain model. Considering the recorded categories as states of the chain and the specified recording time intervals as time periods, the data intuitively form a chain of observed states.
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44 A Markov chain process is characterized by the following information: There are a given set of states {s^, s^, s^, s^}. The process is in one and only one state at a given time. It moves from one state to another in successive time periods, and each move is called a step. The probability that the process moves from state i, s^, to state j, Sj , in one step depends only on s^. In other words, if ^ ^ = s_. represents the state at time t + 1, then ^ ^\ H 1 = = ^' \ 1 = \' = V The probability of s_. given the entire sequence preceding it is the same as the probability of s_. knowing only that s^ immediately preceded An observation X^ ^ ^ is independent of the observation two steps before, _ J/ only if X^ is known. The property is called onestep dependence. In addition, in a Markov chain process the initial state of the process must be specified. Some of the necessary conditions of the Markov chain model are satisfied by the nature of the rules for collection of category type observation data, and the central question involves the extent to which the sequence satisfies the length of dependence assumption. In any observation system, there must be a given set of categories or states, and one and only one of these is recorded in a given time interval. The process moves from one category to another in successive time periods. By convention, the code for "silence" is used to begin the
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45 recording of each observation set; therefore, the initial state or category is automatically specified. One problem area is the dependence of the occurrence of a state on one or more preceding states. This aspect of the Markov model needs further study before it can be determined to what extent the model fits observation data. Tests of Fit of the Markov Chain Model The pattern of dependence of one state or category upon preceding ones is not obvious from the nature of the category type observation system being used to record events, nor is the pattern of dependence indicated by the nature of interaction in the classroom. Statistical tests are therefore called for to indicate if the states are independent of previous events, primarily dependent on one preceding step, or dependent upon two or more steps. The statistical tests of interest are: Null Hypothesis 1: Successive steps are independent. Alternative Hypothesis 1: Dependence is onestep or longer. Null Hypothesis 2: Dependence is onestep. Alternative Hypothesis 2: Dependence is twosteps or longer. The general procedures' indj.ca ted by these statistical tests of hypotheses are as follows: 1) The construction of onestep and twostep tally matrices from the raw observation data. 2) The computation of a chisquare statistic for the matrices. 3) The computation of Goodman and Kruskal's Tau for the matrices as an indication of the proportion reduction in error using the one and twostep dependence assumptions.
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46 Tally Matrices Raw data (See Appendix B ) consists of a sequence of numbers representing observed categories of behavior. The first step after the tranformation to new codes in processing of the data is to tally the observations into two types of matrices, a onestep matrix and a set of twostep matrices. To illustrate the construction of these matrices, the set of states {l, 2, 3, 4, 5, 6, 7, 8} forms an observed sequence 8, 3, 1, 2, 4, 3, 8. The sequence might be an example of observation codes recorded using eight predefined categories where codings are made at specified time intervals. Onestep of the sequence is preserved by the construction of a matrix. Using the sequence. 1 4 1 i 3, 1, 2, 4, 3, 8 t t f t t adjacent codes are paired. The pattern 83, 31, 12, 24, 43, 38 can be preserved in a matrix. The rows and columns of the matrix are labeled to correspond with the set of states. The onestep tally matrix, T, is constructed by counting the number of times that a particular state i is followed by category j to form a total n in each cell of the matrix.
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47 time = t + 1 tune = t 1 2 3 j 8 T = 1 J. J. "l2 "l3 n Â•J "l8 2 "21 "22 "23 Â• n . 2d Â• "28 "31 "32 "33 Â• n. . Â• "38 t "il "i2 "i3 Â•Â• Â• n . . ID Â• "i8 "81 "82 "83 '8d 88 Using the sample sequence, the first tally indicated by the pair 8 3, is in row eight column three. Next a mark for 3 1 is made in row three column one. The process continues in this manner until the last pair 38, in the sequence is recorded. The total in each cell, n^y represents the number of times the pair i j occurred in the sequence being analyzed. Using the same example, the construction of eight twostep tally matrices is illustrated. These matrices preserve twosteps or chains of three elements of the sequence.
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48 T(j) = 1 2 3 time = t + 1 "ijl "lj2 "lj3 "2jl ^^212 "2j3 n n n 33I 3j2 3]3 n. n. 13I 132 8:1 8j2 time = t + 1 'lj8 '2jk 2j8 3jk '3j8 n. ., 'ij8 8jk 8j8 One matrix is constructed for each state. The matrix number, j, represents the state at time t; whereas, the row, i, represents the preceding state and the colvunn, k, represfents the state which follows. The entry in matrix T(j) row i, column j is n. and this is the number of times the sequence i j k is found in the raw data set. In the illustration I T \ i ^ i >f T 48' 3, 1, 2, 3, 3, 8 t t t t t t
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49 the first tally is made in matrix T(3) in row eight, column one. The pattern continues as indicated until all triplets have been tallied. The chisquare test is designed to determine whether or not the frequencies which have been empirically derived differ significantly from those which are expected under certain theoretical circumstances. Applying the chisquare statistic to the test of null hypothesis 1, the assumption is that there is no difference in the state at time = t + 1 knowing the state at time = t. The chisquare statistic is computed as follows for the onestep matrix: Let n. . = tally count in cell ij Chisquare Test r . 1 = total of row i 8 = Z n. j=l n 8 = E r. i=l ^ 8 c. = total of column j = Z n. i=l e = expected number in cell i j = r , . 1 c . 3 n
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50 The obtained statistic has (8 1) (8 1) or 49 degrees of freedom (df ) . For tests in which the degrees of freedom are this large the 2 chxsquare (/ ) is converted to a z statistic in the following way: The z statistic is a normal deviate with unit variance; the chisquare test corresponds to a single tail ztest. If the z statistic is significantly nonzero at the predetermined level, null hypothesis 1 of independence can be rejected In favor of the alternative hypothesis of onestep or more dependence. In the twostep tally matrices there are likely to be many empty cells. For this reason, the chisquare test statistic is not appropriate for use in the test of null hypothesis 2. Goodman and Kruskal ' s Tau Hypothesis 2 can be tested by looking at the proportional reduction of errors (PRE) using a model. The Goodman and Kruskal Tau (T^, Blalock, 1972, pp. 300302) is such a measure for contingency tables. The T stab tistic takes on values between zero and one, with one representing an explanation of all error by the proposed model. Tau is equal to number of errors with row unknown Â— number of errors with row known The computing formula which is appropriate for use with the one and twostep tally matrices in the eight category illustration is number of errors with row known
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51 8 8 Z Z ( 2 8 Z 2 n c r . 1 n 8 Z j=l 2 c 3 n n The Tau can be computed directly for the onestep tally matrix. For the twostep case, a T, is computed for each of the eight twostep b matrices in the example. A pooled estimate for an overall T, is found b by calculating a weighted average of the eight Tau values . Rather than a direct test of null hypothesis 2, a comparison of the proportional reduction in error seems indicated by the comparative nature of the study. The Tau resulting from the onestep tally matrix can be compared to the pooled Tau from the twostep matrices. In this section several of the applications of Markov chain properties are discussed in some detail. These procedures were applied to the transformed observation data used in this study and discussed in Chapter IV. If the onestep dependent Markov chain model fits category type observation data, there are several Markov chain properties that could be applied. The following quantities are of interest in the analysis of a Markov chain: 1) The transition probability matrix. 2) Powers of the transition matrix and the equilibrium matrix. Application of Markov Chain Properties
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52 3) The mean recurrence time. 4) The means and variances of passage time into absorbing categories. 5) The probabilities of absorption into two or more absorbing states . These quantities are discussed in detail below. Transition Probability Matrix The transition probability matrix (P) is a matrix of cells, p ij such that each entry represents the probability that state i will be followed immediately by state j in the chain. time = t + 1 P = time = t 1 2 3 j 8 1 Pll P12 Pl3 Â• Â• Plj Â• Â• P18 2 P2I ^22 P23 . Â• P2j Â• Â• P28 3 P3I P32 P33 . Â• P3j Â• Â• P38 Pil Pi2 Pi3 Â• p. . . Â• Pi8 8 _P81 ^82 P83 Â• Â• Â• P8j Â• Â• Â• P88
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53 An estimate for p^_. is obtained from the onestep tally matrix by dividing each entry of the tally matrix by the row total . The computing formula is the following: n . . /V ID PÂ• = Â— ID Â• The sum of the entries in each row of the transition matrix is one; that is, given that the process is in state i, the probability that some state will follow is one. While the relative incidence of the various pairs of categories is not indicated by the transition matrix, the pattern of movement is indicated by the probabilities found in the cells of the matrix. Powers of the Transition Probability Matrix and the Equilibrium Matrix The transition matrix, P, gives the probabilities of transition from a given category to a second category in one time period. The 2 square of this matrix, P x P or P , is a transition probability matrix for which the probability of transition is calculated over two time 2 (2) periods. Each entry in P ,Â• p^^ , represents the probability of state j following state i two time periods later. The matrices P"^, P^ and 5 P represent the transition probabilities over three, four and five time periods, respectively. In general, P^ is the matrix of transition probabilities over N time periods, and P^ is computed by taking th the N power of matrix P.
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54 As higher powers of P are computed the column values of the matrix become closer in value to other entries in the same colvunn. Any prearranged limit for the difference can be attained by taking sufficiently high powers of P. As a result of this process, the rows N of matrix P become nearly identical. P^= 1 ^11 ^12 ^13 ^11 ^12 ^13 ^11 ^12 ^13 Ij Â• 18 18 18 The equilibrixam matrix is defined as a row matrix consisting of one row of P N " ~ ^^1' ^12' ^3' Â•Â•Â• ^j' Â•Â•Â• ^18^ = (ir^, TT^, TT^, ... TT . , .". . TT ) The matrix 11 is a display of the long run probabilities of the occurrence of each state. The probability of state j is given by it . , regardless of the initial state. If all transitions are carried out during the process, the proportion of time in each category should be approximately equal to the corresponding element of the IT matrix.
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55 Mean Recurrence Time The mean recurrence time, s^^, for category i represents the nvrniber of time periods expected to pass from the moment category i is observed until it is observed again. Mean recurrence times are computed using the elements from matrix IT, 1 s . . = 11 IT . 1 Mean and Variance of Passage into Absorbing Categories Any state in a Markov chain which is always followed by itself is called an absorbing state. For example, suppose the states being observed are the conditions of hospital patients. Obviously, the state death would only be followed by death so it is an absorbing state. In the transition matrix, the probability of death following state death is one; all other transition probabilities in the row would be zero. The presence of an absorbing state allows for study of movement into that category. If no such state is present in the Markov chain, it might be useful to create an absorbing state. Suppose, for example, that state or category 3 is of particular interest; that is, the researcher may need to know which states precede it in the chain of events. In order to study movement into category 3, the transition probability matrix, P, can be altered making 3 an absorbing state. Row three of P is changed to all zero except transition probability p^^ which is equal to one.
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56 time = t + 1 time = t 1 2 3 . j . . 1 CO 1 Pll P12 Pl3 Â• Â• Â• P18 2 P22 P23 Â• Â• Â• P2j Â• Â• Â• P28 3 0 0 1 0 0 Pil Pi2 Pi3 ^81 ^82 ^83 13 ^8j i8 88 To facilitate computation of the mean and variance of passage times into the absorbing state, the transition probability matrix is put in canonical form, p'. The absorbing category is shifted to the top row and the colimms are interchanged so that the cell containing probability one is in the upper left corner.
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57 time = t + 1 J Â± 4 0 0 = 3 1 0 0 0 0 0 0 0 1 Pl3 Pll P12 Pl4 Â• 2 ^23 P2I P22 P24 Â• P28 4 P43 P4I P42 P44 Â• ^48 ^83 ^18 ^28 ^84 Within matrix four submatrices are defined as follows: I 0 R Q Matrix I is an identity matrix, the dimensions of which depend upon the number of absorbing categories. Matrix 0 is a matrix of all zero elements. The transition probabilities from nonabsorbing states into absorbing state (s) are contained in matrix R. Matrix Q is the matrix of transition probabilities from nonabsorbing categories into nonabsorbing categories.
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58 The expected or mean number of time intervals spent in each nonabsorbing category before eventual absorption can be calculated. Matrix ^}, called the fundamental matrix, consists of the mean passage times for each state. The fundamental matrix, M, is given by the following, where I is an identity matrix of the same dimensions as matrix M = (I Q)""" The inverse of I Q is a matrix M such that M . (I Q) = 1. Let the matrix M be symbolized 1 2 4 8 1 ^2 ^4 Â• . m^Q 2 "Â•21 "Â•22 "^24 Â• Â• ^8 4 %1 ^2 ^4 Â• Â• "^48 8 "^81 ""82 "^84 Â• Â• Â• ^"88 Each entry, m^_., in the matrix represents the mean number of time intervals that i is expected to be followed by some specified j in the chain before the absorbing state is reached. It is as if all action
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59 has been artifically stopped when state 3, the absorbing category, is entered. In this way a close study of events preceding this move can be made. Furthermore, given that the sequence is in state i, the total number of time intervals expected to pass before the sequence reaches the absorbing state is found by: m, = m.+ m.Â„ + m. . + . . . + m.Â„ 1 il i2 i4 x8 The mean total passage time for state i before absorption or movement, in this case into category 3, is given by m^. The variance of the passage time provides additional information on the movement into an absorbing state (s). The variance matrix, VAR, is computed using the fundamental matrix, M, and a matrix consisting of zero elements except for the principal diagonal which contains elements from the principal diagonal of M. The identity matrix I is of 2 the same dimensions as M. Matrix Mis a matrix with elements m 2 13 from the fundamental matrix M. The matrix VAR is calculated: VAR = M (2 . I) d 2 Each element, v^_., of VAR is the variance of the corresponding mean, m^_. , in the fundamental matrix M. The variance for the total time before absorption, is the result of the following calculations :
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60 TOVAR =(2.MI)M MÂ„ P P2 Vector M is a column matrix with m. as elements. The matrix P 1 2 M has m. as elements. The matrix takes the form: p2 X 1 2 4 5 8 TOVAR = 1 ^1 ^2 ^14 ^5 Â• Â• ^18 2 ^21 Â•^22 ^24 '^25 Â• Â• ^28 4 ^1 ^42 ^44 ^45 Â• Â• ^48 8 ^81 ^82 ^84 ^85 Â• Â• ^88 Additional information on the distribution of the passage time (k) from one state to another is provided by the median. Let h represent the probability of first passage from state i to state j (k) in k time periods. A specified" h is computed il _ ^ (k) (k1) (1) (k2) (2) h ..p. . h.. p.. h.. p.. ID i: ID 33 13 33 , (1) (kl) h. . p . . 13 33
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61 where p.. is an element of P . If the sum of the first passage time is computed and E h!"^^ < .50 , Z hf''^ > .50 k^l k=l then n equals the median of the passage time from category i to category j . Absorption Probabilities If more than one absorbing category is present, the probability of entry into either of the states can be found. A matrix F of probabilitities is computed F = M . R where matrix M is the fundamental matrix and R is a section of P ' , the matrix of transition probabilities in canonical form. Matrix F has the same number of rows as M; that is, there is one row for each nonabsorbing category. The number of columns in F is equal to the number ofabsorbing states in P ' . If states 5 and 7 are absorbing, the matrix of absorption probabilities is
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62 F = 1 5 7 f f ^5 ^17 ^27 ^35 ^37 ^45 ^47 ^65 ^67 ^85 ^87 An element, ^ of matrix F is the probability of absorption into category j rather than the other absorbing category (s) given that state i has occurred. If state 3 occurs, f^^ equals the probability that absorption will be into state 5, and f^^ is the probability of absorption into state 7. The chain will eventually be in one of the absorbing categories and the sum of the elements of a row of F is therefore one. In certain problems, the probability of movement into one of several selected categories may be of interest. If the chain does not have absorbing categories and they are created, the element f . . gives ID the probability that if state i occurs, category j will occur before the other absorbing category (s). Summary The transition probability matrix, the equilibrium matrix, mean recurrence time, and the study of movement into absorbing states
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63 abstract from a Markov chain powerful patterns of movement. In Chapter IV, discussion focuses upon the results obtained when the above procedures were applied to the transformed observation data described earlier in Chapter III.
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CHAPTER IV RESULTS In the statement of the problem in Chapter I , the research questions were 1. Do category type observation data form a Markov chain? 2 , How can the Markov chain model be used to analyze observation data? These two questions emerged from a review of interaction analysis literature. The literature suggested that an assumption of onestep dependence was made by most researchers. A discussion of Markov chain model attributes in Chapter III indicated that the onestep dependence assumption of the model is the only criterion for a Markov chain which is not clearly satisfied by the nature of category type observation data. An attempt was made in the present study to approach the length of dependence in a comparative way. Toward this end, tests of hypotheses were constructed to compare the degree to which each of the following assumptions are satisfied by these data: independence assumption, onestep dependence assumption, and twostep dependence assumption. The results of these tests are discussed in the first section of this chapter. 64
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65 The second research question may seem premature since it has not been determined if the data form a Markov chain. A review of the literature revealed that most researchers implicitly made the assumption of onestep dependence in the display and analysis of data in matrix form. The matrix display was first used by Flanders (1965) based upon advice from Darwin, a statistician. Since these techniques assuming onestep dependence have become traditional and have proved to be productive and useful, it is highly probable that researchers will continue to use the approach. If the onestep assumption is made, however, it is logical to explore the use of all aspects of the implicit model, the Markov chain model. It was in the spirit of exploration that the properties of the Markov chain model were applied to observation data. Tests of Fit of the Markov Chain Model The data sets used in the present study were from observation of four Project FollowThrough classrooms. Each classroom was observed three times, once in early fall (F1) , again later in the fall (F2) and in the Winter (W) . Twoclassrooms were classified as high on a factor called "strong control"; these data sets were identified as teacher A and teacher B in the analysis. Teacher C and teacher D were low on the control factor. The raw data from each observation period (Appendix B) were transformed from the Reciprocal Category System Codes into modified Flanders Interaction Analysis System Categories (See Appendix A3) . A complete analysis was done of each individual
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66 observation for each teacher, and for each teacher the data were pooled for an analysis. The pooled sets were called "all" in the resulting tables of statistics. An example of a complete analysis is found in Appendix C ) . The data source for the example was the second fall observation of teacher A. Test of Independence Versus OneStep or Longer Dependence The first hypothesis was an exploration of whether there is any dependence a m ong the categories in the sequence obtained from an observation. If the frequencies in the onestep tally matrix are what can be expected by random assignment of tallies, then the completed chi2 square (x ) will be small. The transfomed data consisted of eight codes. Data were summarized into an eight by eight onestep tally matrix, and a x statistic was computed. The x had 49 degrees of freedom. 2 For large degrees of freedom x is transformed into a normal deviate z statistic with unit variance, and a one tailed ztest is used. A formal statement of the statistical test is : Null Hypothesis 1: Successive steps are independent. Alternative Hypothesis 1: Dependence is onestep or longer. In Table 1 the statistics for each observation data set are sum2 marized. In each case, x was sufficiently large that when corrected to a z statistic z was nonzero at the .001 level of significance. Thus the null hypothesis of independence was rejected in favor of an alternative hypothesis of onestep or longer dependence.
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67 Table 1. Â— Summary of chisquare and Tau values N Chisquare a z Onestep Tau Twostep pooled Tau A F F W All 1 2 300 301 377 980 336 224 572 884 16,1 11.3 24,0 32.2 .190 .152 .257 .170 .197 .182 .110 .117 F F W All 1 2 388 236 357 983 315 260 268 849 15.3 13.0 13.2 31.4 ,164 .217 .237 .179 .126 .188 .141 .099 Â£ F F W All 1 2 339 374 307 1022 269 303 342 645 13.3 ' 14.8 16.3 26.1 .131 ,188 .200 .112 .178 .111 .236 .135 D F F W All 1 2 295 440 586 1323 336 253 Â• 128 511 16.1 12.6 6.1 22.1 .245 .126 .049 .068 .185 .098 .044 .080
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68 Table 1. Â— Continued T(l) T(2) T(3) T(4) T(5) T(6) T(7) T(8) .286(25^) .233(46) .097(56) .115(65) .786(6) .261(55) .351(19) .099(27) .069(27) .133(39) .175(55) .213(51) .250(3) .142(84) .415(11) .332(30) .048(21) .138(23) .038(158] .343(13) .611(7) .142(31) .057(87) .366(36) .064(73) .147(108) .055(272) .094(129 .327(16) .142(170; .095(117) .280(94) .341(7) .166(50) .143(70) .164(43) .318(9) .089(58) .088(22) .088(128) .000(22) .104(55) .441(25) .250(29) .306(10) .132(69) .282(10) .359(15) .000(2) .248(36) .136(157) .139(32) .000(1) .123(32) .167(5) .121(91) .116(31) .038(141) .179(252) .098(104) .281(21) .058(160) .100(37) .061(236) .092(23) .071(44) .191(72) .124(30) .727(6) .156(86) .144(33) .340(44) Â— (0) .211(22) .050(13) .299(16) Â— (0) .137(36) .077(85) .152(98) .268(22) .148(50) .211(27) .207(41) .754(7) .288(66) .133(43) .302(50) .104(47) .052(117) .082(213) .122(87) .488(14) .189(189) .056(162) .247(192) .357(5) .078(22) .138(91) .581(10) ~ (0) .245(28) .073(62) .280(76) .093(27) .086(83) .144(40J .394(13) .000(4) .053(134) .123(31) .110(107) .082(26) .134(32) .032(116) .144(33) ~ (0) .025(183) ~ (0) .038(194) .046(58) .030(137) .160(247) .116(56) .000(4) .027(345) .078(96) .096(379} =/2x^ / 2dÂ£ 1 Number in the matrix is enclosed in the parenthes
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69 Test of OneStep Dependence Versus TwoStep or Longer Dependence After rejecting the hypothesis of independence in favor of a onestep or longer dependence hypothesis, the question became, how many steps does the dependence span? The answer to this question was not expected to be a definitive one; therefore a comparative approach was designed. The statistics were to test the following: Null Hypothesis 2: Dependence is onestep. Alternative Hypothesis 2: Dependence is twostep or longer. The Goodman and Kruskal's Tau (T^) was computed for each onestep tally matrix, and a pooled value for Tau was computed from each set of eight twostep matrices. The techniques were designed to test whether or not the proportional reduction of errors or T for the twob step tally matrices was significantly greater than the T value for b the corresponding onestep tally matrix. In this manner the twostep or longer dependence assumption was compared with the onestep dependence assumption. The results of the computation described above were summarized into Table 1. In the analysis of 10 of the 16 data sets, the onestep was larger than the pooled twostep T^; in the remaining six cases, the twostep values were greater. A closer look at the individual Tau values for the twostep matrices revealed great variation in these values. For example, the F1 observation of teacher A produced Tau values for the twostep tally matrices ranging from .0971 for T(l) to
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70 .7857 for T(5) . Under the circumstances a pooled estimate of T has b questionable value. An examination of the individual Goodman and Kruskal Tau values for the twostep matrices revealed that certain matrices had higher T values. Of the ten cases in which T was the b b nonzero for T(5) , nine represented the largest T recorded for that b observation. The twostep matrix T(5) contained a record of the chain at t 1 and t + 1 where the state at time t was category 5 (teacher corrects or cools) . Other patterns might have been present had there been more observation codes in certain categories . The results summarized in Table 1 indicate that a formal test of the null hypothesis was not appropriate; rather, a comparison of the relative fit of the two models was warranted. The evidence suggests that certain categories are more onestep dependent while others are more twostep dependent. Category 5 (teacher cools or corrects) , for example, had high values for most twostep matrices. Due to the nature of classroom interaction, it might be expected that the categories of behavior which precede and follow a correction by the teacher are linked or form a twostep dependence pattern. The conclusion based upon the results is that category type observation data does not clearly satisfy a onestep dependence assumption nor does it satisfy completely a twostep dependence assumption. The onestep dependence assumption appears to be at least as good as the twostep dependence assumption except for particular states. The Markov chain is therefore clearly suggested as a possible model for the analysis of observation data. Perhaps a change in the categories or time unit would produce data which would fit the assiamption of the model more closely.
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71 Application of Markov Chain Properties The tests of hypotheses suggested that, while the data chains were not completely onestep dependent, at least an exploration of the application of the Markov chain model is warranted. The review of literature also suggested that properties of the model could be meaningfully applied to classroom observation data chains. The results of the application of selected properties will be discussed in the following sections, with emphasis placed upon the meanings of the results in terms of classroom behavior. Transition Probability Matrix The onestep tally matrix of each data set was used to estimate the transition probabilities of all pairs of categories. Each element of the tally matrix wad divided by the row total so that n. . n. The resulting matrix, P, consisted of probabilities p = p . Given il ij that category i occurred, p^_. is the probability that category j would follow. The rows of the transition probability matrix represent the category at time t, while the columns of the matrix represent the category at time t + 1. An example of a transition probability matrix is found in the complete analysis of the second fall observation of teacher A in Appendix C. Row 7 of the matrix is as follows:
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72 12345678 0.0 0,273 0.182 0.091 0.0 0.182 0.0 0.273 Given that category 7 behavior (pupil initiates) was observed, the probability of observing category 1 type behavior (teacher warms, accepts, amplifies) was 0.0 for this observation. The most likely behavior after category 7 was category 2 (teacher elicits) and category 8 (silence or confusion) each with a probability of occurrence of 0.273. The transition probability matrix is a powerful abstraction from the raw data. Rather than a display of the actual frequency of occurrence of pairs of categories of behavior, the transition matrix is a predictive model. In the example above, the analysis suggests that student initiating behavior was followed approximately 27% of the time by teacher eliciting behavior, 27% of the time by silence or confusion, 18% of the time by teacher responses or pupil responses, and 9% of the time by teacher directive behavior. It seems that in this observation, pupil initiating behaviors were very short in duration since no pairs of 7 7 were observed.. One might conclude that the student was being cut off by the teacher. A closer look reveals that this was not necessarily the case. Category six (student response) and silence or confusion are expected to follow category seven with probability .46 while teacher categories follow with probability .54. The focus of the analysis of observation data with the aid of a transition probability
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73 1 matrix is upon the relative frequency of transitions from a given category to each of the categories rather than upon the frequency of such movement in a particular observation. Powers of the Transition Probability Matrix and the Equilibrium Matrix Powers of the transition probability matrix were computed for each set of observations. The elements in each column became less variable as successive powers were computed. It was found that P^ was sufficient in all cases to insure that the range of the elements in each particular column was less than or equal to .03 (any specified maximum range could have been used) . The equilibrium matrix was formed from 5 the first row of P . The transition probability matrix, P, for a Markov chain gives the probabilities of transition from a given category to a second cate2 gory in one time period. The square of this matrix, P x P or P , has elements which represent the probability of movement from a given category to another category over two time periods. A similar interpreta3 4 tion may be applied to P , P , etc. In the analysis of category type observation data in the present study, P was a matrix of probabilities which indicated that, no matter which category occurred at time t, the state at t + 5 depended only on the state to which the transition was made. For an example of P^, see Appendix C. The information in the higher powers of the transition matrix was redundant; therefore, a row vector was defined as one row of p'^.
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74 The equilibrium matrix, IT, is the row vector containing the probabilities of movement into each category in the longrun. Furthermore, TT^ represents the proportion of the total number of observed categories that can be expected to fall into every category. If, for example, = .21, category 1 is expected to comprise 21% of the total number of behaviors in the observation record. In many observation data sets, the predicted proportion agreed with the actual proportion of occurrence of each behavior category. A summary of the proportion rounded to two decimal places is found in Table 2. The longrun prediction and the actual proportions are very nearly equal in thirteen of the sixteen cases. Siibject A's first fall observation, subject B's winter observation, and subject C's second fall observation were the only data sets which produced differences of .04 or more in the predicted and actual proportion of observations in each category. One explanation for these differences might be that the transition probability matrices were not taken to a high enough power before the equilibriiam matrix was defined. Mean Recurrence Time The mean recurrence time was computed from the equilibrium matrix for each category in each data set, and the reciprocal of each element of the equilibrium matrix was rounded off to the nearest whole number (Table 3) . The largest and smallest means for each observation were enclosed in brackets and underlined, respectively.
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75 I I j3 n ^ O rH CM (N ^ ^ O H O CN O H y3 iH Â«a< ro rl O H O CN XI H O O rH CTi O Ol H U3 'I' O O O O CN r~ O O iH iH 00 ro rH O .) O CN X! CTi ro O CN (N ^ CTi ^ r~ H O CN O O as O CN o (N ^ CM in iH rH O CO O O X! CN n O rH CO CM in ro O rH O ro (0 CN o
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76 CN I I U I I Em ^OrtrHOOrvlOCN (OOrHHOOCNOfN ^JinOt^OrHOro XIOOtNOOnoro in o rH O O o CM o i ro O m cn n ti O 'J XI O H o o o o CM nS ^0 Â£)(nn(NcN'3' O H O rH O CN rH cdCX)r^(T>'<3"rO(T\CN{Ti OrHOHOHrHH XI H O O 1XÂ» O o CO ID CN '3' o o CO ID o o H CN CN J Â• Â• m o o o in CN CO CN XI o rH CN cn o CN o in CN (0 o ro cn o CN o CN CN c^o in CN CO 0 Â•H c M 0 U Â•H H +J a M 0) o a, 0 u a u H 1 c +J 0 o < >l
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77 c nS Q) S I m 0) rH EH rH H n CN CN rH cn ro O rH 1 Â— 1 1 A u 1 o ' ' en H n (N 1 fa CN H o H CO 1 Â— 1 yo CN ro 1 CN CN ' rH rH 1 Â— 1 1 fa 00 in 1 1 00 in a\ CO r~ rH col All ro H CTl 00 rH 1 1 rH rH <: 00 rH H CN o n 1 Â— 1 in 1 1 O H in H H CM fa rH H CO in ( Â— 1 cn 1 1 ^1 CN rH rH rH 1 fa * CN rH ml ml 1 Â— 1 CN in 1Â— 1 inl H CN rH H CN in CO fa Eh O 2 o 2
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78 The mean recurrence time represents the number of time periods expected to pass before the designated category is expected to occur again in the Markov chain. In the analysis of observation data, these means give an indication of the relative frequency with which categories of behavior are expected to occur. According to Table 3, the most frequently recurring category for teacher A in three of the four data sets was category 3 (teacher responds, initiates). In the winter observation, for example, when category 3 is observed, it is expected that category 3 will' occur again four time periods later. Four of the six smallest recurrence times for teacher A were teacherbehavior categories, while two were pupil response behavior categories The largest means for teacher A were all in category 5 (teacher corrects or cools) . Based upon the chart of means , some general conclusions about the classrooms can be made. Subject A seems to be using category 3 and category 4 behavior frequently. Both of these are directive, or teacherdominative , behaviors. Pupils responded, but only in one observation was pupil initiating behavior frequent. Basing opinion only upon this table of means, one might classify classroom A as a rather structured classroom. Indeed, teacher A was one of the two teachers who were chosen because of a high score on a control factor (See Chapter III, data section) . The teachers with low scores on the control factor (who were expected to have a less structured classroom) were teacher D and teacher C. In both cases the frequency of category 8 (silence or confusion)
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79 might indicate that time was allowed for students to think before responding, and therefore more frequent responses by pupils were obtained. If no factor score were available, the same kind of insight into behavior patterns based upon the mean recurrence times could be gained. absorbing Categories The sequence of data produced by coded classroom behavior does not contain any absorbing category. If there were such categories, the techniques described in Chapter III could be used to study movement into the absorbing category(s). In the present study, category 6 (student response) and category 7 (student initiates) were the created absorbing states. Creation of one Absorbing State The transition probability matrix, P, for each set of observation data was altered as described in Chapter III to create category 6 as an absorbing state. The matrix, P, in canonical form (see an example in Appendix C ) was used to define the matrices R and Q. The fundamental matrix, M, was computed from matrix Q and an identity matrix. Each element, m^_., of matrix M, is the mean number time periods in state j before absorption given that the chain is in state i. The first row of matrix M from the second fall observation of teacher A is 1 2 3 4 5 7 8 1 1.254 0.546 0.937 0.972 0.046 0.164 0.329
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80 Given that category 1 (warms, accepts, amplifies) behavior occurred the niamber of time periods expected in each of the nonabsorbing categories is given by the number in that column. The total number of time periods expected before category 6, the absorbing state, is entered is given by the sum of the row elements. In the example above, the mean total passage time after category 1 occurs is 4.252. The vectors of mean passage times for each observation are given in Table 4, with each entry rounded to the nearest whole nximber. Additional information on the movement into category 6 is provided by the variance of the passage times. The variance for each state in the fundamental matrix, M, was computed and the variance matrix, VAR, with elements v. . is the variance corresponding to the mean, m. , (See ID 1] Appendix C for an example of these matrices) . The variance of total passage times was computed. The elements of vector TOVAR are the variances of the total passage times. The use of the standard deviation, the square root of the variance provides a more complete description of expected patterns of movement into the absorbing category. A summary of the mean, m, and standard deviation, s, rounded off to the nearest whole number is fo.und in Table 4. The analysis shows that in each case the shortest time before passage into the absorbing category, 6 (student response) , is from category 2 (teacher elicits) . This interaction is expected; however, the various data sets differ in other respects. In the observation data such as from teacher A
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81 rH CN I I IN 1 I H O ro ro in VD g n CO r~ 00 rH in (Tv G\ H in n rH CN rH H ro rH ro rH ro rH e CN rH 00 r~ H ro H rH 'arH rH e ro ^^^ cTi O CN ro (N ro 'S" ro r~in vD r~6cT\rocor~oocoai CN CN in in 1^ ro iÂ£) vO 6trvjco>l>vDoooo rH 00 in in ro ID tn CN H H CN rH CN H eg H E H H ro H in H rH in H in H w 00 CO o . 00 00 CM CN rH CN ro CN ro e '3' H w in o in rH e ro 00 CO IC 00 rH CN ro in 00 w c o Â•H P nj Â•H > 73
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82 00 ts) (N O fN (N CN in . CN CN n n CN n n gCNCN^'J'fNin':)' W o CN o CN o CN o CN 1 1 o o CN E CN ro ro 1 1 ro (N 1 in r^l 1^ H 'J' CM o CN CN CN e CN ro CN ro CN H ro H w CN H ri H CN H 1 CN 1 fa e '3' rH CN ri CN rH 1 1 in ri rH S UI o 00 cn CO 00 All in iri in in in in rÂ» in rM CM CO n ro ro ro CM CN ro ro ro ro ro e in rn in in in ro U CN 1^ t^ CN n CN iH ro ro iH ro iH ro ro H 1 fM a LO H CO in H in H WD H 'J' H '3H O r~ iH 1 IT) in in in in in in fa e in vo H CM ro "a* in rCO C o Â•rt Â•P rfl Â•H > O O (0 c (IS p ^3 C nS U
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83 F 1, teacher D, F 1, and teacher C, F 2, the mean total passage time from category 2 into the absorbing category was less than the means from the other categories into the absorbing category indicating that state 2 might be expected to precede state 6 by fewer time periods than other categories. In most of the cases presented, the standard deviation is large relative to the mean, indicating a positively skewed distribution. In four cases , teacher A winter observation, teacher D first fall observation, teacher B winter visit and teacher C second fall observation, the means and standard deviations are large. A few of the medians of the passage time into state 6 are found in Table 5. Additional verification of the skewness is provided by a comparison of the means and the medians. The distribution of the passage times in these cases does not appear to differ in shape (skewness) from the others. The creation of an absorbing category facilitated a study of movement into that state. Certain types of behaviors are particularly desirable or undesirable in classroom interaction, and these can be focused upon by making them absorbing categories in the chain of observation data. It may not otherwise be observed that a particular behavior rather than others precedes the behavior of interest. The absorbing category property of Markov chain theory in effect stops all progress of the chain when the absorbing category is reached and thereby facilitates analysis of movement into the designated category.
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84 Table 5. Â— Median of passage time into category 6 State Teacher A (F 2) C (W) Median Mean Medxan Mean 1 3 4 3 5 2 1 1 1 3 3 3 4 3 5 4 3 4 3 4 5 3 4" 3 5 7 2 4 4 5 8 3 4 1 3
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85 Creation of two Absorbing States The creation of two absorbing states allows study of the categories which precede either of them. In the present study, both student behavior categories 6 and 7 were changed to absorbing states in the manner described in Chapter III. An example of the canonical form which resulted from matrix is found in Appendix C. The mean passage time, the variance of the passage time, the mean total passage time, and the variance of the total passage time were each computed as for the one absorbing category. Each of these statistics apply to movement into either absorbing category, since both student behavior categories were changed to absorbing states, the study focuses upon those behavior categories which precede student verbal behavior. A summary of the mean, m, and standard deviation, s, of passage into category 6 or 7 is given in Table 6. Although category 2 (teacher elicits) still generally has the smallest mean passage time, a general trend is not as clear as the trend that was observed when only category 6 was absorbing. An inspection of individual observation data sets suggests that the distributions are not skewed as badly as they were when one absorbing category was" used. The means are generally greater than the standard deviation. Since more than one absorbing category was created, the relative probability of entry into each absorbing state can be studied. A matrix F of probabilities of entry into category 6 or 7 was computed for each observation data set. A row i of matrix F contains two elements
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86 0) 13 a u nS Â•o C nJ +J M H C H 0) o +J M 0) Id ^ +j p c 0) o p p p n3 to Tl U (d c 0) ^ M +J 0) Â» o 41 "g nS Â— 0) a c (d g o o Sh (U i4
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87 All o w 'J' H O CN O CN rH o CN CN ro ro CN CO o w O CN O CN O CN 1 1 O CN e CO CN CN ro 1 1 ro Q rH O (N 1 iH rH rH rH e CN H CN CN OJ CN rH 1 rH w kfl CN ^ r~ ro 1 1 ro ro CN IT) 1 1 CO H O 03 CN H CN CN o CN CN rH H g n CN ro ro ro CN (0 H H H H H o CN ro CN CN H U CN 1 o o CN CN LT) Â» CO 00 .00 CN CN B rH CN ro CN rH 1 M CN O CN CN CN CN CN E ro CN ro ro iH CN ro to CO o r( 4> (0 H > a u (0 Â•a c (0 p tn rH rH (d (0 c Q) rH O 41 cn 0) >4 rd c (U A ^ p +J c 0) O 41 4J +J
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88 f^g and f^^. Given that state i occurs, f^^ is the probability that absorption will be into category 6; that is, category 6 will occur in the chain before category 7. Absorption into one of the categories will eventually occur; therefore, the s\m of f . and f is one. i6 i7 The matrix F for each observation data set is displayed in Table 7. Category 2 (teacher elicits) seems to be absorbed into category 6 with high probability in most of the data sets. In only 11 of 96 pairs of probabilities was the probability of absportion into category 7 greater than the probability of absorption into category 6, The creation of two or more absorbing categories enables a powerful comparison of the likelihood of two or more behaviors. This technique could be especially useful if one state is a desirable behavior type and the other an undesirable type. If category 7 were desirable, then subject c might use silence to increase the probability of category 7 behavior. Summary The second research question guiding the present study was, how can the Markov chain model be uged to analyze observation data? The preceding pages contain a discussion of the application of various aspects of the model and an interpretation of the results. In each example, the results of the statistical analysis using the Markov chain model were meaningful in the context of classroom interaction. These proportions, means, standard deviations, and probabilities are summary
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89 PQ D CO (Ti CTi 00 00 cTi o rH in fo o n in n VD U3 CN ro O rH ro rt rn fr, CO r~ o oi VD iÂ£l 00 in ro H (N in CN CTl H VD CN in CO CTi CO '3' in CO H vD 00 ro ^ ,_j m rN ro 00 cr> in 1^ >H O H .rH o ( Ln CO rIN H in <^ CO CTi CO CO CO CN (M r o 2;
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90 (U a H p n 5 I I 0) Â•9 Eh (N I C4 I lO CO in in in in o '3' CN CN CN in o rCO CO CO H O CO CO CO in O in CM H rf H I r~ CN CO CN 00 'J' CN in CO CN 1^ in in in P 0) ^^ Â•v c +j CO Eh O in 00
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91 statistics which allow the researcher to focus upon trends of behavior. The easily understood statistics could provide meaningful feedback to the teacher and thereby aid in desired behavior changes in the classroom . A discussion of the conclusions generated from the results of the analysis is found in the next chapter. Chapter V also outlines possible future research regarding the applications of the Markov chain model to observation data.
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CHAPTER V SUMMARY AND IMPLICATIONS Direct observation of classroom behavior is an integral part of research projects which focus upon the description of classroom interaction, the study of teaching skills, and the evaluation of curriculum materials. The various types of classroom observation instruments record different aspects of the classroom interaction. In the present study, data from a category type system were used because this type of observation instrxament preserves order in addition to frequency of behavior categories. A set of data consists of a chain or sequence of category codes. The methodological question which arises revolves about the statistical procedures which prove productive and valid in the context of the given study. Objectives This dissertation was designed to study the relative validity of a onestep dependence assumption of the Markov chain model applied to category type observation data as well as the heuristic value of the model . 92
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93 Activities , Results and Implications In the initial stages of the study, an extensive review of observation literature was conducted. The review focused upon the general topics of study and the statistical procedures employed. Another search of literature in which the Markov chain model was used revealed that some of the same types of research questions were being studied as in the observation analysis literature. Individuals observing classroom behavior using category type observation instruments consistently followed certain procedures . These procedures suggested that the resulting data could be assumed to fit all the requirements for a Markov chain except the order of dependence assumption. Tests of Hypotheses The hypotheses were designed to test in a comparative way the following assumptions: independence of codes from one time period to another, onestep dependence of categories, and twostep dependence of behavior states. The chisquare statistic which resulted from onestep tally matrix was large enough to reject the hypothesis of independence in favor of a hypothesis of onestep or multistep dependence. The second test involved a null hypothesis of onestep dependence versus an alternative hypothesis of twostep dependence or more. The Goodman and Kruskal Tau values for the onestep and twostep tally matrices suggest that no clear conclusion could be drawn regarding these hypotheses.
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94 A comparison of Tau values revealed that certain categories seemed to be more twostep dependent than others which showed more evidence of being onestep dependent. Application of Markov Chain Properties Since there was no conclusive evidence to indicate that observation data do not form a Markov chain, and since elements of the Markov chain model have proved productive in previous studies, elements of the Markov chain model were applied to interaction analysis. Transition probability matrix The transition probability matrix was computed from the onestep tally matrices. This abstraction from the tally matrix changed frequencies of occurrence of pairs of categories to probabilities of the occurrence of categories following given categories. Thus the focus was no longer upon the number of times a pair occurred, but rather predicted transitions. The transition probability matrix could form the basis for a subjective comparison of observation data sets. In much of the observation literature, the goal of the researchers has been to compare behaviors using observation data. The transition probability matrix could serve as a basis for these comparisons. Furthermore, there are tests of stationarity (Goodman, 1962) which allow the researcher to determine if two transition probability matrices are significantly different .
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95 A second area in which the use of a transition probability matrix might be useful is feedback to the observed subject. Further research needs to be done to determine if the probabilities of transition are relatively stable from one observation to another. Until then the only cpmment which can be made to the observed teacher concerns the probability of certain pairs of behaviors in the classroom during an observation period. Feedback in the form of probabilities could be helpful in the modification of certain kinds of behavior in the classroom. Equilibrium matrix The equilibrium matrices were computed from powers of the transition probability matrix and represented the "longrun" probability of the observation of each category. The predicted proportion of time in each category was compared with the actual proportions of the total observations in each category. It is unclear what significance can be attached to the fact that the actual proportions were close to the equilibrium matrix values. Mean recurrence times The mean recurrence times for each state were computed based upon the equilibrium matrix. The means gave some indication of the number of time periods expected to pass before a category was reused. The mean recurrence times for an observation would provide an easily understood feedback statistic for the observed teacher or the researcher.
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96 Like the transition probability matrix, the mean provides a way in which observations or subjects could be compared. Insight could be gained into the amount of structure being used in the interaction as well as into the pattern of flow of interaction by carefully studying the mean recurrence times for a set of observation data. Absorbing categories Absorbing categories were created by altering the transition probability matrix in order to study movement into these specified cate gories. The statistics generated from this alteration of the original transition probability matrix allow a detailed study of movement into the specified categories. If a behavior is desirable or undesirable, those states which precede it in the chain and the distribution of move ment from these categories into the specified state or states are of prime interest. The matrix, F, of comparative probabilities or absorption is a powerful abstraction which allows the researcher to study the flow of interaction which precedes absorbing states. Summary of properties The two themes which emerge from the results of application of the Markov model to observation data are feedback technique and vehicle of comparison. In the observation literature to date, these are two of the primary uses of observation instruments. The present study contends that an analysis of the observation data using the properties of Markov chains is an appropriate extension of the traditional procedures.
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97 Problems for Future Research The spirit of exploration which gave rise to this research gives rise to several areas of future research. The principal questions for further study which emerged in the course of the present study are the following: 1. Would a different set of categories or a different unit of analysis (other than time) produce categorytype observation for which the onestep dependence assumption is a better fit? 2. Do observations of the same teacher produce transition probability matrices which are similar? 3. Is it possible to use the degree to which the equilibrium matrix values agree with the computed proportions of use of each category to determine the required length of a productive observation session? 4. Can the results of the application of Markov chain properties provide adequate feedback to classroom teachers to facilitate behavior modification? 5. Can the Markov chain model serve as a guide for the development of a meaningful classroom observation system? The present study was but a first attempt in the exploration of the extent to which Markov chain analysis of observation data is useful and appropriate. The ultimate test of the heuristic value of a model is whether it can be used productively in the research on classroom behavior, teaching skills, and curriculum materials. The development of an observation instrument guided by the Markov chain model as suggested above might allow the meaningful application of powerful procedures suggested in this dissertation.
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APPENDICES
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APPENDICES A
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Appendix A1. Â— Siaminary of Categories for the Reciprocal Category System Category Number Assigned to Party 1 Description of Verbal Behavior Category Number Assigned to Party 2 "WARMS" (INFORMALIZES) THE CLIMATE; Tends 11 2 3 4 5 6 to open up and/or eliminate the tension of the situation; praises or encourages the action, behavior, comments, ideas and/or contributions of another; jokes that release tension not at the expense of others; accepts and clarifies the feeling tone of another in a friendly manner (feelings may be positive or negative; predicting or recalling the feelings of another are included) . ACCEPTS ; Accepts the action, behavior, com12 ments, ideas and/or contributions of another; positive reinforcement of these. AMPLIFIES THE CONTRIBUTIONS OF ANOTHER ; Asks 13 for clarification of, builds on, and/or develops the action, behavior, comments, ideas and/ or contributions of another. ELICITS ; Asks a question or requests informa14 tion about the content subject, or procedure being considered with the intent that another should answer (respond) . RESPONDS ; Gives direct answer or response to 15 questions or requests for information that are initiated by another; includes answers to one's own questions. INITIATES ; Presents facts, information and/or 16 opinion, concerning the content, subject, or procedures being considered that are selfinitiated; expresses one's own ideas; lectures (includes rhetorical questions Â— not intended to be answered) . DIRECTS ; Gives directions, instructions, orders 17 and/or assignments to which another is expected to comply. 100
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101 Appendix A1. Â— Continued Category Number ^ rr l. , r. v. Category Number . J, ^ . .^ Description of Verbal Behavior . , 2 Assigned to Party 1 ^ Assigned to Party 2 8 CORRECTS ; Tells another that his answer 18 or behavior is inappropriate or incorrect. 9 "COOLS" (FORMALIZES) THE CLIMATE ; Makes 19 statements intended to modify the behavior of another from an inappropriate to an appropriate pattern; may tend to create a certain amount of tension (i.e., bawling out someone, exercising authority in order to gain or maintain control of the situation, rejecting or criticizing the opinion or judgement of another) . 10 SILENCE ; Pauses, short periods of silence. 20 CONFUSION ; Periods of confusion in which communication cannot be understood. Category numbers assigned to Teacher Talk when used in classroom situation. 2 Category numbers assigned to Student Talk when used in classroom situation.
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Appendix A2, Â— Categories for Flanders Interaction Analysis System, (1965) 1 ACCEPTS FEELING: accepts and clarifies the tone of feeling of the students in an \in threatening manner. Feelings may be positive or negative. Predicting or recalling feelings are included. IDIRECT INFLUENCE 2 PRAISES OR ENCOURAGES: praises or encourages student action or behavior. Jokes that release tension, but not at the expense of another individual, nodding head or saying "um hm?" or "go on" are included. 3 * ACCEPTS OR USES IDEAS OF STUDENT: clarifying, building or developing ideas suggested by a student. As teacher brings more of his own ideas into play, shift to category 5. TEACHER TALK H 4 * ASKS QUESTIONS: asking a question about content or procedure with the intent that a student answer. W U 5. * LECTURING: giving facts or opinions about content or procedure; expressing his own ideas, asking rhetorical questions. [NFLUE^ 6. * GIVING DIRECTIONS: directions, commands, or orders which students are expected to comply with. DIRECT ] 7. * CRITICIZING OR JUSTIFYING AUTHORITY: statements intended to change student behavior from unacceptable to acceptable pattern; bawling someone out; stating why the teacher is doing what he is doing; extreme selfreference. 8. * STUDENT TALKRESPONSE: talk by students in response to teacher. Teacher initiates the contact or solicits student statement . STUDENT 1 9. * STUDENT TALKINITIATION: talk initiated by students. If "calling on" student is only to indicate who may talk next, observer must decide whether student wanted to talk. 102
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103 Appendix A2. Â— Continued 10. SILENCE OR CONFUSION: pauses, short periods of silence and periods of confusion in which communication cannot be understood by the observer. There is No scale implied by these numbers . Each number is classif icatory , designating a particular kind of communication event. To write these numbers down during observation is merely to identify and enumerate communication events, not to judge them.
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Appendix A3. Â— Conversion of Categories: Reciprocal Category System to Flanders Interaction Analysis System modified Reciprocal Flanders InterCategory Category action Analysis Category System System State numbers in print out of data Warms the climate Accepts Amplifies contribution 2 3 Accepts feeling, praises, uses ideas 1,2,3 Elicits Ask question Responds Initiates 5 6 Lecturing Directs Gives directions 6 Corrects 8 Cools climate 9 Criticism Pupil response 15 Pupil response 8 6 Pupil initiation 1114 1619 Pupil initiation 9 7 Silence Confusion 10 Silence or con10 fusion 20 104
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r APPENDIX B
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Appendix B. Â— Raw Data Used in the Study TEACMtR A FALL 1 070 70710 100 70 7151 50 71 00 70 71 515151515 041 50^06 0 707 041 502 041 50 70 70 7060 60 7 1 5 15 15 15 1515 I 50 10 6071 0160O041 5020 60404 04150 70 70 70 710101006151515151515150 4 04 04150106 04 15041501063604150 7150 20 6 01060606041 50 6041 507071 0 1 00 71 61 61 01 0 10 10 lOlD 10041502060607161610041 50606 0415041 5 060 606D 7043 70 41 50 60 6060 60 70 6 061 01006061 0071 01 0070704 1 51 0 06060707 10 10 I 006 0 606060 60 6060 70 7 0 70 6060 60606 0604 1502 0 70 706071 51 51 515151 0 1004 150 7 07070 704150 70 707020 70701 0 70 70101 160 7 0 70/1&01 06060604041 60407060608070808 04150 70 7070*040416160 20 4150 204150 415 1502 0606 06 06 04 041 61 602 04 1 60 204 1 60 1 04 1601080416041602060 60 4150 4150 70 7070 7 070 7070 7 160 7041 502041 50 2 160 70 60 41 50 7 O6O0O6 FALL 2 101004150 7150415061 502071504150201 06 150 4 1 b041 5061 5161 01 51 5020 60 41 50 204 15 15 02 060*15 0607153 20 4 1 50 4 1 50 6 1 50 20 70 4 1 5060616 041 5041 5071 50204 I 50 60 70 7 0 71 6 16040 50* 150415151 01 01 007041 510 151010 0 7 04 1510 15071 50201 060 6C 7 151 5150 7071 5 1504150415041 508041505150415101 00707 0705151502041506150 50 204151 5161 50 806 06071 50 615101 50 60 4100 60 7150 50 21 50 105 07 07070706040 7070 7061 51 6 0706041 50405 0502 020 1 10 100 71 60 41 51 01 5050206060604 1502 0 707070 70 7070 7151 515150207041502 02070 707071 51 51 51 51 51 51 5 1 5 1 5 1 51 5 1 50 2 070 70 70616101610100 71 60 50 71 0101 01 00 7 07160605063607101 01 OtO 70 7060 710100415 0208 060606 0606041 50^0 61 0 1 006060 60606 060615041502060606070415031 415041 504 1 5060606 10*
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107 WINTER 06i61606 071016101 60 606 1 6 1 006061 6 1 6 1 0 16061 406061 61 6061716060606161 61 01606 0 606 1610 1614 10 161 01 60 60 6 16060 20 60 60 6 04 15 0204 1 5021 61 60606061 6161016161016 161406060606161016060 7150 61 0141 01610 060616161 016101606C60606060606071506 0606 0606060606061 61 60606 06 I 6 1 0 1 6063 6 16 14 101612 161 01606060 4100 71 5020 61606 0 60 6071 6 0606 041 50 20 6060 1 04 1 50 20 606 1 6 061 00207151 61 01606060607150 4 1510 160 8 0807 1510 16160 41 5060 60 8081 50 606060606 16 1606 16 10 161 014061 60 80 6060 61 61 9 191 7 1707 08060 6060606041 5020 71 51 0161 61007 10060616061 60 60 61 61 61 4060 71 50 61 01 61 6 1610161016100 006061 40 60 71 50 616101610 161 t>l 0160206061 6 02 060604 1 5020606 1 602 02 0 60 4150606041 5060 60 60 4150 2 041 502 06 C606060 6060 60 6060 6060606 041 0041 50 20 4 04 15 020* 1 502 060 60 60 60 60 6 0 60 6060 60 60 4 1504 150602 060606041 50 41 50206060 60604 0410 0415150206060606060 6060 606060606 0606 06 06060606060 60 6 TEACMEH B FALL I 20202020 06 061 01 004 041 506060 714090 615 060O0410 15060908091 406060606060606 1 0 1006060606 06101 60 5060 707060 604 15 1504 150 7100716051010100614051 50415041504 0415 1510 150 712071 51 51 508040410 150 415 04151504 151 00707151 034 1513 10041 50 41 5 1510 101006060606 10100 41 50 71 5041 50806 1506071507 15021 006081010 lOlOlOlOlOlO 10101C070 7101C10101010101010 10101010 101 0 070 7001 01 00607060604 150 71 0 I 00606 0604 151 50 4 15041 504 I 004 10041 5041 51 506 0 60 4 1502060 41502 06060/10 100 7 150 6 101 5. 160606041 5081602061 61 0070 70 6061 4061 4 10 10 04150604150 2010 714051604041 5041 5 150 7 04 150310101004041 50 410 150 80 70 410 10160404 0604 0 704151 OlOlOlOlOlOlOlOlO 1010060 7070 40 41 50 61 40 51 01 61 01 01 61 01 0 10100 7060604061406100710041 50 41 50 60 7 07151010 100 71 01 006060604070606040710 10 lOlOlOlOlOlOlOlOO 70 61405141 51 0100 7 0716101010101615101 00 710101010101010 0 70 715101010101 01 01 01 01 0070714100 710 10 100708
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108 FALL 2 2006 10 20 161 0 1608100*15071 502070 4151 5 15 102007 151 510151 5150 71 51 61 0041 508 36 06150 71507150 715080 71 50 71 50 80 71 50 80 4 1 50415041 508041508041534 1510 150 40 61 5 150607150604 15020616071 508041 0060704 1502060 7 04 15041 5041 5060 4 150 2060 41 50 2 0206070415080615041502070410041 50207 04150207 041 5 02 041 502 0 60 7 1 4 060 70 4 1 00 6 0 60 704 15 0204150415041 007041504150204 1504150415020607061 507040415041 50204 04150 2041502041 50 20 41 5041504150 20410 040 6 1502 0 61 40 40604 1 50 706 1 40 6041 502 06 0604 15020 71 50 706 160 4150 7061610 160 704 15 04 1506041 5020415 WINTER 10 100 606060606060606060 6060 6060 60606 0606 0606 0606 060 60 6060 606 10 1006060 60 6 06060606060606061 01 00606060606060606 10 10 0606060606060606060 60 606060 60606 060 6060 6 060 6060 60 6060606 060 6060 6060 6 06 06 0606 061 5 1 5060 o06 0 63 60 60 60 60 60 60 6 O6O0O6O6O6O6O6O6I 0060606061 506060606 0606 1006 101004041 5040410041 50604 150 2 02040415 15041 50606101 5041010041 51515 041010101010 061 01 0101010101010071010 100 6 15150710 100610101004060606060606 04 1 0 041 0 1 0 IOO0O6U6 101013101002101010 1010101006101 006061 0 1 00 4 1 50 604 1 00 71 0 1015041010041 50 41 00 60 71510161 00 71010 IOO0O7I 0 1 61 00 60 70 71 01 6070 70 7 10 10041 5 041 0080704040710 1 0 1 00 6 1 60 4 1 0 04 1 51 01 0 10041 50 6100 60 60 604150 70 70 710100 60 70 6 06 10 10 06 04 15071 50 41 0073 60 61 6100 70 710 100 70 7070 704150604 151015060 7101 01 00 7 100?0 60506070704151 01 50 4151 5100 70 40 7 07 Of 0606 0606 060 6
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109 TEACHER C FALL I 0 20 61 0060 7041 5081 51 50207160 2 070 6060 7 15151515151515150 60 41004100 60 4150 20 6 0604 16041 0061 6041 6080 6041 61 6061 61604 1504 1502 06041 004100 41 5041 0060604 1502 0 204 150215021502041 50604150706081008 100606060 6060 6061 607 0606 060 6041 51502 020415020704150415020615020410060415 02 0 6041 5 0204061502150204 16 10 160 4151 5 15 lt> 1515061 51 615150 615150606151 51 51 5 15 06060606061 0060 60 60 616 10 160 60 61 61 6 160o 1610 1607100606041 007 10 16 10 1 60 40 6 16101 604 061 61 61 01 6071 0160806060 7160 4 041607061007100604100416101610160415 150 615151515060 215151515151515151516 16 10 1515151515061 51 51 506071 007061 506 150 615060 7061 00 70 61 00 706100 7060 7041 0 1004 1502041 50 6150 20 71 00 7 1 00 7 100 7 1 004 15060206 150215060 7100 7 100 71007100415 15020415160615020 7100 4100 41 51 5041 506 07 10 lOlO 160604 FALL 2 16160 41515062 01 405050516161616101610 16101610 160404150607080 7160 2140 50716 16101 41 406 14071 610161 00 6141014101 00 7 07071610201406202 0041 60 41 51 40 704100 7 0 70 61 00 7071710151 00 70 410100415161016 1016101604101 60 41 51 61 60 4161616101 60 4 041515101 51 5 0604 10101010100610101004 15151 015060606041 51 51 01 604041 5060606 0 60 40 60 606151 50 60 6C610 16 1610 161 01 60 6 1610161016101 61 016101 61 016101 61 01 60 7 U70o061 506060606161 0 1 6 1 0 1 6 1 0 1 60 40 71 0 161016101010141014151610111010161014 14101610 16161 0161 01606061006 160 21610 10 16 1623 20162020200 7200 6061 6041 01 010 1 00 60606 0606 0 6060 60 60 60 6060 6060 60 60 6 0606060604160 71 0101 01 51 5060 6060 61 01 5 1 5060 60 616151010161610101010100 60 60 6 0616 16161506060 60 6151 51 5060 6061 51515 151 0 060606 06061 01 01 01 0 06 0606060 60 61 6 0610 10 1606 160 60 61 60 60 6060 60 616101010 1 0 06 06060 o 060 1)0 60 4 1 506360 60 60 6151515 06060606060606
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110 WINTER 15020 40606 1610161 0 1 6 1 0 1 6 1 0 1 604 1 504 1 5 06 041 602 06 161 0161 0 160916041 50 70 71 60 7 070 7 1 6070o0 70 71 01 51 01 bl 01 51 01 51 0161 0 161015131510161016101510151015101510 1015101510 151C161 016101510 16101 51015 1016 1016 10 15101 51 0 061 5041 50 7071 7071 5 15040 802 0 70 7150 91502041 50 20 4 1 50 7 1 50 2 070606160415041 50 50415060 70 6060 40 40 7 07 06 1 60716071 61 01 61 01 61 0160716081 607 0716 101607160 60 6160 71510 1510 15101510 10150207071 51 01510150415050604150204 150 20 7151004 041 502 060 1 04 1 50 7 1 6 1 0 1 40 7 0 716 10 16041 7071 602C606040404 I 50 8071 5 0207041 5040204041 504 150 204 1 5040 40410 10150 204 1 502041502041 504 1 502 0 70 4 1 504 07060413 15080708040604150206041 50207 07070 407150 70404061610 160 415070 71 504 06041 505 06041 5020 706 TEACHER O FALL 1 161616161616101610101010101616101610 I61ol01415101 61 616101610161016101610 161610161616101010161 61 6161616161610 1416 1410 151610101410151010101010101 7 101417161007101010101410151010141016 161 01616 161 0161 0161 610 16100 7 100 71010 100 7 ICO 7062 0041 504 1 5041 5041 5100 60 70 6 06060d04 15061515071 52004060415062020 20100415 060 40 7151 00 41 51 51 506061 5041 5 04150606060 6060 60 71 40 60 41 50 606061 506 0 60 oO 60 41 50 40 41 50 3061 40 5040 50606060 4 0415 06101010100101C110010 60 60 4150 60 6 06 1 40506141 01 60606070604 151 010101010 1010 1019101016140 5161 00 6 1 0 1 0 060 40 60 6 0606060 6 06 060 60 60 60 60 60 60 60 6 060 60 60 6 0606060606 06 06060606 0606 06 0 6060 6060 6 0614 1 606 060606060606060 606 10 10 1 0
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Ill FALL 2 1604 1 510 151 5041 5041 5041 5100 41 51 01 51 0 1610160 41504150106161004150 204100415 1 50 104101614101 60 6101010 06 1004151004 1510 160 6041 50 2041 50 61 00 1 1 0 1 0 1 00 71 51 5 1510 1610 160 61610161 0 1 60 41 50 60 61 0 1 40 5 0710160104150 71405160 4150415150 4150 4 041 5061506041 50 71 51 00 410041510101010 04 10041010 1010060415041510 15100 71006 0604 1 504 151015041510160 4150 415101516 04060 615060 60 710070 71 510041 50 2041 502 02150415020 710 10040 415150 20 7041 50204 15 10 1010 10 10041 502041502041 506041502 020606041506041504151015020410041 504 151G0415101510100 410150 4100 4150 2100 4 04 1502 04 10151 50 815101010101016100415 150204151015101504 101515151 0161 004 15 15081506 160415100 6100415 150410151015 060415151515010415151515101610151515 150 6161 40 5101510161015151 51 015101515 08 15021510 151 0101014051015020610 0415 1510 150710 10041 SC41 50415 0614070 41504 150 2 1502 1 51 015061 0 1 5 1 0 1 5060 6061 60604 04151 50 4151504101 50 41 50 20 4151 50 20 60 6 1 50o04l504 151 01 004 1 0 1 0 1 50 7 1 0040 4 1 0 1 5 15060604 151510 1015101602161 0160 21 51 5 0610 16 M INTER 1006151006151 01 51 00 61 0061 01 5100 60 610 1610061 50204 10061 51 5060410 1 5021 51 004 04 10 150 60 6100 7061 51 50 61 5 1006 150 61 506 101006101515101510100634100610061006 06 10061 502101 50 41515 06 101 50 61 50 20 61 5 0215151006101510061015100 60 6150 61504 041 00615 1 50 1 150 20 70 706100415021 0061 0 1503 1510061 00 710071 50 71 0041 0151 00204 041510151006041015101510101015101510 10 1007151510061 5061 5021 01 51 5071 5061 5 151015101510151 502101010101010101010 10 150210 151 51 50 4101 0100 70 610061 50 410 10 1506041 51 506061 50604 10041510101510 100604100610151004150 61 01 510 061 5061 0 1010151015101510150 60 7150 61510151015 07 1510151 51 01 5061 01 51 01 51 00 7061 5021 5
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112 15 151510 0 61 51 5061 5061504100615061515 06061 506 151 0 1504 i 51 51 0061 006060 7061 0 10 10 06 13 1 50 61 0 1 50 21 0061 0061 01 00 61006 15061510 151 51 002041 5041 50304 I 01 01506 060 7 151510101506061510071506150 71 006 100610U60 706101 00 70 71010151015151015 1510 1510 15061 0 060710 100410041 50 2061 00 6101006150 610 06060715151 0150 20 6100 71510 150 61 51015 151006100 40 71 50 7 04 151 00 70710071 5021 5 0215 101510 10 150204151015061510150610 100 61 510 15061 01 0 061 5041 5 1 00 6 1 5 1 0 1 00 7 10010710 0610061510 1015101 51 5061 51 015 150610101510151015 1510151510 150 61015 061510150210070 71 51 00607100610041010 10 15 1506150615101 51 01004151 51 0061 015 041504150206151502 101 01 0061 5061 01510 1006151010101 51 0061 00 706101506151006 1010
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APPENDIX C
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Appendix C. Â— Output from the Complete Analysis of the Second Fall Observation of Teacher A NUMBER OF CODtS =301. ONE STEP MATRIX FOR RAW DATA STATE 1 . 2. 3. 4. 5. 6. 7. 6 . 1 ^ . 5 . 8. 6 . 1 . 1 . 1 . 1 . 2 0. 0. 2. 1 . 0. 35 . 0 . 1 . 3 4. 8. 20. 10. 0. 10. 2. 1 . 4 0. 5. 6. 22. 0 . 1 1 . 4. 3. 5 0. 1 . 2 . 0 . 0. 0. 0 . 0. 6 19. 15. 14 . 4 Â• 2. 20. 3. 7. 7 0. 3 . 2 Â• 1 Â• 0 . 2. 0 . 3. 8 0Â« 2. 2. 7. 0. 5. 1 . 14. ROM TOTALS 27. 39. 55. 51 . 3. 84. 1 1 . 31 . TWO STEP MATRIX : STATE AT T I HE J IS 1 A STATE 1 . 2 . . S. ft . Â• . o . 1 1 Â• 0 Â• 2. 0 Â• n Â• \j . 1 . 2 0. 0. 0 . 0 . 0 . 0 . 0 . 0 . 3 1. 1 Â• 0. 0. 1 . 0. 0 . 4 0. 0 . 0 . 0 Â• 0 Â• 0 . 0. 0. 5 0. 0. 0 . 0. 0. 0 . 0 . 0 . 6 2. 4. 5. 6. 1 . 0. 1 . 0. 7 0. 0. 0 . 3 . 0 . 0 . 0 . 0 Â• 6 0. 0. 0. 0. 0. 0. 0. 0 Â• TWO STE? MATRIX : STATE AT TI ME T I S 2 STATE 1 . 2. 3. 4. 5. 6. 7. 8. I 0. 0. 0 . 0. 0. 5 . 0 . 0 . 2 0. 0. 0. 0. 0. 0. 0. 0. 3 0. 0. 0 . 1 . 0 . 6 . 0 Â• 1 Â• 4 0. 0. 0. 0 . 0. 5. 0. 0 . 5 0. 0. 0 . 0 . 0. 1 . 0. 0. 6 0. 0. 1 . 0 . 0 . 14. 0 . 0. 7 0. 0. 1 . 0. 0. 2. 0. 0 . 8 0. 0. 0 . 0 . 0 . 2. 0. 0. Two STEP MATRIX : STATE AT T IME J IS 3 STATE 1 . 2. 3. 4 . 5. 6. 7. 6. 1 0. 2. 3. 1 . 0. 1. 0. 1 . 2 1 Â• 1. 0 . 0 . 0 . 0 . 0 . 0 . 3 0. 2. 12. 3. 0. 1 Â• 1 Â• 0. 4 0. Â£Â• 0. 1 . 0. 2. 1 . 0. 5 c. 0. 1 . 1 . 0 . 0 . 0 . 0 , 6 !Â• 2. 2. 0. 6. 0. 0 . 7 0. 0. 1 Â• 1 . 0 . 0 . 0. 0. 8 0. 0. 1 . 1 . 0. 0. 0. 0 . 114
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115 TWO STEP matrix: state STATE 1 Â• 2. 3. 4 1 0* 2 Â• 0 a 3 2 0Â« 0. 0 a I 3 0* 1 . 0. 2 4 Oa 0 Â• 5 . 13 5 0Â« 0. 0 . 0 6 0* 0. 0. 1 7 0* 0. 1 a 0 8 0. 2. Oa 2 TWiJ STEP matrix: STATa STATE I , ^ a J a 1 0 \j a A a \j 2 0 Â• v a V a V 3 0* w a w a V 4 0* Oa Oa 0 Â«> 0* w # 6 0* ft a w 7 . 0Â« Oa Oa 0 8 0. 0 . 0 . 0 T MO STEP M AT n IX 1 'iT AT F STA TE I , a J a * I I , w a V/ Â• \J 2 9a ft . o a 7 a ^ 3 2Â« I ^ a 1 ft & a 4 3m 3a Â• w a V 5 0* 0 a 1/ a h# 6 4* Oa 0 Â» I w Â• & 7 bÂ» 1 , 0 Â• 0 8 0. Oa 2. 1 TliQ STEP MATkl X : state STATE 1 Â• 2. 3. 4 1 0* Oa 0 a 3 2 0* Oa Oa 0 3 0* la 0 a 0 4 0* 2. 2 a 0 b 0* Oa Oa0 6 Oa Oa 0 . 1 7 Ca Oa 0 a 0 a Oa Oa 0. 0( TMO STEP MATR IX ; STATE > STATE 1 . 2 . 3. 4. I 0. 0. Oa 0 2 0. Oa I a 0 3 Oa Oa 0 . 0 4 Oa Oa Oa 0. 5 0. Oa 0 a 0 , 6 Oa 0. 0 Â• 0 . 7 Oa Oa Oa 0. 8 Oa 2. 1 . 7 , AT T IME T IS 4 > 9 5. 6a 7a 8 a (a Oa I a 0. Oa a 0 a 0 a 0 a 0 . la Oa 4. Oa 3. I. 0. 3a 1 a 0. ) a 0 a 0 a 0 a 0 a , a Oa 3a Oa 0 a 1 a 0 a 3 a 0 a 0 a :a Oa Oa 3 a 0 a AT TI ME T I S 5 5a 6a 7a 8a ) a 0 a 0 a 0 a 0 a 1 Â• Oa 0. Oa Oa Â• Â• 0 a 0 a 0 a 0 a ) Â• 0 a 0 a 0 a 0 a 1 . 0 a Oa Oa Ca 0 a 0 a 0 a 0 a 1 a Oa Oa Oa 0 a Â• 0 . Oa Oa 0. AT T IME T I S 6 # 5a 6a 7 Â• S a 0. Oa Oa Oa 1 Â• 2 a 0 a 4 a 0 a i Â• 2 a 1 , Oa 3a 0. C a Â• 0 a 0 a 0 a 0 a Â• 0. 13. 1 a 1 a Â• 1 a 0 a Oa Oa Â« Oa 1 a Oa 1 a AT T I ME T I S 7 Â• 5. 6. 7a 8 a Â• 0 a 1 Â• 0 a 0 Â• Â• 0 a n a 0 a 0 a Â• 0. Oa Oa 1 . Â• 0 . 0 a 0 a 0 a Â• 0. 0. Oa 0 a Â• Ca 1 Â• Oa 1 , # 0 a 0 a 0 a 0 a Â• 0. 0. Oa i . AT TIME T I S a Â• 5. 6a 7a 8. Â• 0. 0. 0 a 1 . Â• Oa Oa Oa 0. Â• 0 a 0 a 0. 1 a Â• Oa 0. 0. 3. Â• 0 a 0. 0. 0. Â• 0 . 4 a 0 a 3 . Â• Oa 1 . I a 1 Â« Â• 0. Oa 0. 4 a
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116 GK TAU FOH ONESTEP = .152 1 GT TAU FOR TWO STEP 1= .0693 27, GT TAU FOR T*0 STEP 2= .1326 39 . GT TAU FOR Ti^O STEP 3= .1748 55 GT TAU FOR TWO STEP 4= .2132 51 GT TAU FOR TWO STEP 5= .2500 GT TAU FOR TmO STEP 6= .1419 84. GT TAU FOR TWO STEP 7= .4149 1 1. GT TAU FOR T*0 STEP 8= .3320 30, DEGREES OF FREEDOM = 49 CHI SQUARE ONE STEP = 224.448 TRAMSITION PRQBA8ILITY MATRIX STATE 1. 2 . 3. 4. 5. 6. 7. 8. 1 . C. 1 48 0.0 0.073 0.0 0.0 0.226 0.0 0.0 2 . 3 . 4. 5. 6. 7, 8. 0. 1 85 0. 296 0 .222 0. 03 7 0 .037 0. 037 0 .03 7 0.0 0. 051 0 .026 0. 0 0. 897 0. 0 0.026 0.145 0. 364 0 .182 0 . 0 0 . 182 0. 0 36 0.018 0.098 0. 1 18 0 .431 0. 0 0.2 16 0. 078 0 .059 0.333 0. 667 0 .0 0. 0 0.0 0. 0 0.0 0.179 0. 167 .048 0. 024 0.238 0. 0 36 C ,083 0.2 73 0. 182 0 . 091 0. 0 0. 1 82 0. 0 0 .273 0 .065 0 . 065 0 .226 0. 0 0. 161 0. 032 0.452 TRANS IT I ON STATE 1 , 2. 3< 4 . 5 . 6 . 7. 8. 1. 0.091 0.0 92 0.091 0.090 0.0 9<: 0 .091 0.089 0. 0 86 MATRI X 2. 0. 130 0.131 0. 131 0 .130 0.131 0. 130 0.129 0. 127 FIFTH 3 . 0. 189 0 .1 90 0. 1 69 0. 186 0 . 192 0. 189 0 . 186 0.179 4. 0 . 1 68 0 . 167 0. 168 0. 170 0 . 166 0 . 168 0 . 170 0 . 1 75 5. 0. 010 0.0 10 0.010 0.0 10 0.010 0. 01 0 0.0 10 0 .009 6 . 0. 260 0 .280 0.280 0.2 79 0.281 0.260 0.2 78 0.276 7. 0. 036 0.036 0.036 0. 037 0.C36 0. 036 0.037 0. 037 6. 0.095 0. 094 0 .094 0 .099 0 .091 0.0 95 0.101 0.109
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117 EQUILIBRIUM MATRIX 1 2 3 4 5 6 7 8 0. 091 0. 130 0. 169 0. 16B 0.010 0. 260 0. 036 0.095 MEAN RECURRENCE 1 1 . 006 7.664 5. 299 5. 942 99.207 3.569 2 7. 404 10 .540 SHIFTED SO THAT STATE 6 ABSORBING CANONICAL FORM 1 Â• 000 0. 0 0.0 0 Â• 0 0.0 0 .0 0 .0 0 . 0 0.037 0.146 0. 185 0 .296 0.222 0 .037 0 . 037 0. 037 0. 697 0.0 0 .0 0 .05 1 0 .026 0 .0 0 .0 0. 026 0. 162 0. 073 0. 1 45 0 . 364 0. 1 82 0 .0 0 .0 36 0. 01 8 0.216 0 .0 0 .096 0 Â• lie 0.431 0 Â• 0 0 .0 78 0. 059 0. 0 0.0 0 .333 0 .66 7 0.0 0 .0 0 .0 0 . 0 0 . 182 0.0 0.2 73 0 . 182 0. 091 0 .0 0 . 0 0. 273 0 .161 0 .0 0 .065 0 .065 0 .226 0 .0 0 Â• 0 32 0. 452 MATRIX M MEAN T I Mt IN STATE BEFORE ABSORBED 1.254 0.011 0.1b3 0. 047 0.112 0.048 0. 042 0. 546 1. 042 0.447 0. J78 0.646 0.498 0.360 0.937 0. 125 1 .910 0.546 1.315 0 .567 0. 498 0 .972 0. 124 0 .820 2.148 0. 588 0 .660 1. 034 0 .046 0. 000 0.006 0. 002 1. 004 0.002 0. 002 0 . 167 0.017 0. 148 0 .202 0. 104 1 .095 0.167 0 .329 0.075 0.257 0,370 0.1 96 0 .66 1 2 .054 TOTAL MEAN PASSAGE TIME 4,252 1 ,394 3. 751 3.&92 3.905 3.531 4. 157
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118 VARIANCE OF CELLS OF MATRIX M 0.31 8 0.01 6 0. 21 9 0.0&8 0.157 0.071 0.0&2 0.294 0.044 0.285 0. 267 0.283 0. 292 0.261 1 . 764 0 .336 1 .737 1 .242 1 .978 1 .276 1 Â• 155 2.258 0 .392 2.030 2. 465 1 .592 1 . 74 0 2.339 0.045 0.000 0 .006 0.002 0 .004 0.002 0.002 0.171 0.020 0.154 0. 200 0.113 0.105 0. 171 0 .914 0.226 0.732 1.012 0.572 1 .6 1 7 2. 1 64 VARIANCE OF TOTAL PASSAGE TIME 8.348 2.257 8 .423 6.598 7.60 3 8. 169 9.566 SHIFTED SO THAT STATE 7 ABSORBING CANONICAL FORM 1. 000 0. 0 0.0 0. 0 0.0 0,0 0.0 0.0 0 . 0 1 .000 0.0 0 . 0 0.0 0.0 0.0 0. 0 0. 037 0 .037 0 .1 48 0 . 185 0 .296 0 .222 0 .037 0. 037 0. 697 0.0 0.0 0. 0 0. 051 0.026 0.0 0.026 0. 182 0 .036 0 .073 0. 145 0 .364 0. 182 0. 0 0. 01 8 0. 216 0.07S 0.0 0. 098 0.118 0.431 0.0 0.059 0. 0 0.0 0.0 0. 333 0 . 667 0.0 0.0 0.0 0 . 161 0.032 0.0 0 . 065 0 .065 0.226 0 .0 0. 452 MATRIX M MEAN TIME IN STATE BEFORE ABSORBED 1 .247 0 .0 10 0. 167 0. 038 0. 1 08 0.035 0.470 1.034 0.380 0. 286 0. 596 0 .284 0 .851 0.116 1 .833 0.442 1 .261 0 .4 1 1 0.871 0.113 0.731 2 . 02 6 0.525 0.934 0 .046 0. 000 0 .006 0 .001 1 . 00
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119 VARIANCE OF CELLS OF MATRIX H 0.307 0.281 1.544 1.899 0.044 0.610 0.015 O.OJb 0.296 0.333 0.000 0.185 0.209 0.2t>2 1 .527 1.697 0.006 0.459 0.055 0.224 0.983 2.079 0.001 0.659 0.149 0.281 1.772 1.327 0.004 0.370 0.051 0.223 0.927 1.978 0.001 1.861 TOTAL MEAN PASSAGf: TIME 3.71 2 1.33 9 3.274 3.04 1 3.629 3. 61 8 VARIANCE OF TOTAL PASSAGE TIME 5.872 1 .633 5.89 0 5. 776 5.303 6.824 PRObABlLlTY OF ABSORPTION FROM EACH STATE 0.847 0.153 0.964 0 .016 0.865 0.135 0.bl6 0.1 84 0.905 0.095 0.848 0 .152
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BIBLIOGRAPHY Adelman, I. G. A stochastic analysis of the size of distribution of firms. Journal of the American Statistical Association, 1958, 53, 893904. Anderson, H. H. The measurement of domination and of social integrative behavior in teacher's contacts with children. Child Development , 1939, _10, 7389. Reprinted in E. J. Amidon and J. B. Houghs (Eds.), Interaction analysis; Theory, research and application . Reading, Mass.: Addison Wesley, 1967, 423. Anderson, T. W. & Goodman, L. A. Statistical inference about Markov chains. Annals of Mathematical Statistics , 1957, 28, 89110. Bales, R. F. Interaction process analysis . Cambridge, Mass.: Addison Wesley, 1950. Bales, R. F. s Gerbrands, H. The interaction recorder; An apparatus and check list for sequential content analysis of social interaction. H\jman Relations , 1949, 1^, 456463. Bales, R. F. & Strodtbeck, F. L. Phases in group problemsolving. Journal of Abnormal and Social Psychology , 1951, 46 , 485495. Reprinted in E. J. Amidon and J. B. Hough (Eds.), Interaction analysis: Theory, research and application . Reading, Mass.: Addison Wesley, 1967, 89102. Beckenboch, E. F., Drooyan, I. S Wooton, W. College algebra (3rd ed.), Belmont, Calif.: Wadsworth Publishing Company, Inc., 1973."' Bellack, A. A., Kliebard, H. M., Hyman, R. T., & Smith, F. L. , Jr. The language of the classroom . New York: Columbia University Press, 1966. Bhat, U. N. Elements of applied stochastic processes . New York: Wiley, 1972. / Bissell, J. S. Implementation of planned variation in head start . Review and summary of Stanford Research Institute interim report: First year evaluation . Washington, D. C. : Office of Child Development, U. S. Department of Health, Education and Welfare, 1971. 120
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121 Blalock, H. M. , Jr. Social Statistics . New York: McGraw Hill, 1972. Campbell, J. R. Macroanalysis : A new development for interaction analysis. Journal of Educational Research , 1975, 68 , 261269. Collet, L. S. & Semmel, M. I. The analysis of sequential behavior in classrooms and social environments : Problems and proposed solutions . Unpublished manuscript. Office of Research Services, School of Education, University of Michigan, 1970. Darwin, J. H. Note on comparison of several realizations of Markoff chains. Biometrika , 1959, 46, 412419. Dolley, D. G. Mothers as teachers: Instruction and control patterns observed in instructions of middleclass mothers with trainable mentally retarded and nonretarded children . Unp\iblished manuscript. Center for Innovation in Teaching the Handicapped, Indiana University, March 1974. Dunkin, M. J. & Diddle, B. J. The study of teaching . New York: Holt, Rinehart & Winston, 1974. Flanders, N. A. Analyzing teacher behavior . Reading, Mass.: Addison Wesley, 1970. Flanders, N. A. Interaction model of critical teaching behaviors. In E. J. Amidon and J. B. Hough (Eds.) , Interaction analysis : Theory , research and application . Reading, Mass.: Addison Wesley, 1967, 1967, 360374. Flanders, N. A. Personalsocial anxiety as a factor in experimental learning situations. Journal of Educational Research , 1951, 45 , 100110. Flanders, N. A. Teacher influence, pupil attitudes and achievement (Cooperative Research Monograph No. 12, OE25040) . Washington, D. C. : U. S. Department, of Health, Education and Welfare, 1965. Flanders, N. A. S Simon, A. Teacher effectiveness: A review of research 196066. In R. L. Ebel (Ed.), Encyclopedia of Educational Research . Chicago: Rand McNally, 1970, 14231437. Furst, N. The effects of training in interaction analysis on the behavior of student teachers in secondary schools . Paper presented at the meeting of the American Educational Research Association, Chicago, February 1965. Reprinted in E. J. Amidon and J. B. Hough (Eds.), Interaction analysis: Theory, research and application . Reading, Mass.: Addison Wesley, 1967, 315328.
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122 Gage,, N. L. Can science contribute to the art of teaching? Phi Delta Kappan , 1968, 49, 399403. Glass, D, V. & Hall, J. R. Social mobility in Great Britain: A study of intergenerational changes in status. In D. V. Glass (Ed.), Social mobility in Great Britain . London : Routledge & Kegan Paul, 1954. Goodman, L. A. Statistical methods for analyzing processes of change. American Journal of Sociology , 1962, 68, 5758. Goodman, L. A. & Kruskal, W. H. Measures of association for crossclassifications III: Approximate sampling theory. Journal of the American Statistical Association , 1963, 58_, 310364. Hartnett, B. M. & Rummery, R. E. Markov chain analysis of classroom interaction data . Paper presented at the meeting of the American Educational Research Association, New Orleans, February 1973. (ERIC Document Reproduction Service No. ED 083133). Hawes, I. C. s Foley, J. M. A Markov analysis of interview communication. Speed Monographs , 1973, 40^, 208219. Hoel, P. G. A test for Markoff chains. Biometrika , 1954, 4Â£, 168177. Kemeny, J. G. s Snell, J. L. Finite Markov chains . Princeton, N. J.: D. Van Nostrand, 1960. Limbacher, P. C. A study of the effects of microteaching experiences upon classroom behavior of social studies student teachers . Paper presented at the meeting of the American Educational Research Association, New York, February 1971. (ERIC Document Reproduction Service No. ED 046855) . Lohman, E. E. , Ober, R. & Hough, J. B. A study of the effects of preservice training in interaction analysis on the verbal behavior of student teachers. In E. J. Amidon and J. B. Hough (Eds.), Interaction analysis: Theory, research and application . Reading , Mass.: Addiston Wesley, 1967, 346359. Maccoby, E. E. & Zellner, M. Experiments in primary education, aspects of project followthrough . New York: Harcourt, Brace, Jonanovich, 1970. McNamara, J. P. Markov chain theory and technological forecasting. In S. P. Hencley and J. R. Yates, Futurism in education . Berkeley, Calif.: McCutchan Publishing Company, 1974, 301345.
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123 Medley, D. M. & Mitzel, H. E. Measuring classroom behavior by systematic observation. In N. L. Gage (Ed.), Handbook of research on teaching . Chicago: Rand McNally, 1963, 247328. Meredith, J. Program evaluation in a hospital for mentally retarded persons. American Journal of Mental Deficiency , 1974, 1Q_, 471481. Mohrenweiser, G. A. A Markov chain model for projecting school enrollments . Unpublished doctoral dissertation. University of Minnesota, 1969. Nuthall, G. A. An experimental comparison of alternative strategies for teaching concepts. American Educational Research Journal , 1968, 5^, 561584. Ober, R. L., Wood, S. E. & Roberts, A. The development of a reciprocal category system for assessing teacherstudent classroom verbal interaction . Paper presented at the meeting of the American Educational Research Association, Chicago, February 1968. Pena, D. M. Some statistical methods for interaction analysis . Doctoral dissertation. University of Michigan, 1969. Raush, H. L. Interaction sequences. Journal of Personality and Social Psychology , 1965, 2_, 487499, Rosenshine, B. Evaluation of classroom instruction. Review of Educational Research , 1970, 40, 279300. Rosenshine, B. & Furst, N. The use of direct observation to study teaching. In R. M. Travers (Ed.), Second handbook of research on teaching . Chicago: Rand McNally, 1973, 122183. Rowland, K. M. & Sovereign, M. G. Markov chain analysis of internal manpower supply. Industrial Relations , 1969, % 8899. Siegel, M. A. & Rosenshine, B. Teacher behavior and student achievement in the DISTAR program. Chicago Principals Reporter , 1972, 62, 2428. Simon, A. & Boyer, E. G. (Eds.). Mirrors for behavior: An anthology of observation instruments (14 vols.). Philadelphia: Research for Better Schools, Inc., 1970. Smith, B. O., Meux, M. O. , Combs, J., Nuthall, G. A., & Precians, R. A study of the strategies of teaching , urbana. 111. : Bureau of Educational Research, University of Illinois, 1967.
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124 Soar, R. S. An integrative approach to classroom learning . Final Report, Public Health Service Grant No. 5Rll MH 01096 and National Institute of Mental Health Grant No. 7Rll MH 02045. Philadelphia: Temple University, 1966. (ERIC Document Reproduction Service No. ED 033 749). Soar, R. S. Final report; Followthrough classroom measurement and pupil growth (19701971 ) . (Research Rep. Contract No. OEG085223943991 (286)). Gainesville, Fla. : Institute for Development of Human Resources, June 1973. Soar, R. S. Teacherpupil interaction. In J. Squire (Ed.), A new look at progressive education . Washington, D. C. : Association for Supervision and Curriculum Development, 1972, 166204. Soar, R. S. & Soar, R. M. Classroom behavior pupil characteristics and pupil growth for the school year and the summer . Gainesville, Fla.: University of Florida College of Education Foundations Department and the Institute for Development of Human Resources, 1973. Suppes, R. & Atkinson, R. C. Markov learning models for multiperson interactions . Stanford, Calif.: Stanford University Press, 1960. Taba, H. & Elzey, F. F. Teaching strategies and thought processes. Teachers College Record , 1967, 65, 524534. Taba, H. , Levine, S., & Elzey, F. F. Thinking in elementary school children . U. S. Office of Education Cooperative Research Project No. 1574. San Francisco: San Francisco State College, 1964. (ERIC Document Reproduction Service No. ED 003 285) . Vroom, V. H. s MacCrimmon, K. R. Toward a stochastic model of managerial careers. Administrative Science Quarterly , 1968, 13 , 2646. Withall, J. The development of a technique for the measurement of socialemotional climate in classrooms. Journal of Experimental Education , 1949, 17, 347361.
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BIOGRAPHICAL SKETCH The author was born in Roanoke Rapids, North Carolina, on May 13, 1943, the first child of Mr. and Mrs, Roy Pierce. After graduation from Littleton High School in 1961, she attended the University of North Carolina in Greensboro. As a member of the Phi Beta Kappa at the University of North Carolina in Chapel Hill, she graduated in June of 1965 with a B. A. degree in Mathematics Education. She taught mathematics at the elementary school, middle school, high school, teachers college, and community college levels. Three years of teaching was with the United States Peace Corps in Peru, South America. She received her Master of Education in Junior College Mathematics Education from the University of Florida in August of 1974. In the Fall of 1974, she began graduate work toward a Doctor' of Philosophy degree in Mathematics Education with minors in Mathematics and Statistics . The author is presently on the Mathematics Department Faculty in Valdosta State College in Georgia. 125
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Elroy J, hoi^c , Jr., Chairjnan Associate Hrofessor of Subjiect Specialization Teacher Education I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality Â» as a dissertation for the degree of Doctor of Philosophy. Alan G. Agreal^i Assistant Professor of Statistics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. J(Sjl(n H. Grant 0 Assistant Professor of Computer and Informational Sciences I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. John M. Newell Pr<4fessor of Foundations of Education
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Robert S. Soar Professor of Foundations of Ecjuqation This dissertation was submitted to the Graduate Faculty of the College of Education and to the Graduate Covmcil, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy, Dean, Graduate School

