Learning and Instruction 16 (2006) 433e449 www.elsevier.com/locate/learninstruc

Advantages of studying processes in educational research Bernhard Schmitz* Institute for Psychology, Darmstadt University of Technology, Alexanderstr. 10, 64283 Darmstadt, Germany

Abstract It is argued that learning and instruction could be conceptualized from a process-analytic perspective. Important questions from the field of learning and instruction are presented which can be answered using our approach of process analyses. A classification system of process concepts and methods is given. One main advantage of this kind of process research is the possibility to study trajectories of learning over time. It also allows the performing of intraindividual analyses. A number of empirical examples are presented which demonstrate the advantages of our approach to performing process analyses. The examples regarding individual trajectories deal with evaluation data of a training program to enhance self-regulated learning. In the discussion, limitations of the approach are described in reference to alternative ways of analyzing processes. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Processes; Time-series; Intraindividual analyses; Analysis of change

This journal deals with the scientific analyses of learning and instruction. From an intuitive perspective learning can be regarded as a process. However, it is generally difficult to observe the learning process directly. Instead, one can only look at the results of learning, the products. Sometimes, we are interested in the sequence of states starting from the tabula rasa or some previous knowlegde at the beginning of learning, passing certain stages of practicing, forgetting, rehearsing, to the final result of the learning procedure. From our point of view, this ordered sequence of learning states defines the overall learning process. Our conception of learning as a process emphasizes its cumulative nature. A similar line of argumentation is valid for instruction. It will be argued that instruction can be regarded fruitfully as a process, but very often instructional procedures are analyzed by comparing information in the pre- and postinstruction phase, at best combined with a control group approach. Another way to look at instruction is therefore to view it as an intervention into the learning process. Hence, it too can be conceived of as a process.

1. Introduction to the analysis of processes Among the basic complementary themes that persist from the Greek philosophers to the present are the concepts of stability and change (cf. Schmitz, 2001a). It is often referred to the ideas of Heraclitus that change seems to be an ubiquitous phenomenon ‘‘all is flowing’’ (panta rhei) or ‘‘you cannot get twice into the same river’’. The opposite position was held by Parmenides, who claimed ‘‘change is an illusion’’. The themes stability and change are basic * Tel.: þ49 6151 162015; fax: þ49 6151 166638. E-mail address: [email protected] 0959-4752/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.learninstruc.2006.09.004

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concepts within psychology and education, especially with respect to learning and instruction. The basic questions are whether learning processes are stable or changing, and whether and how instruction has an influence on learning. The aim of this article is to provide an overview regarding precise research questions and adequate methods for studying stability and change in learning and instruction from a process-analytical perspective. It seems to be rather uncommon in educational and psychological research to study processes in the way we prefer. Therefore, it could be helpful first to show e in a more intuitive way e which type of questions can be answered using the kind of process analyses presented here. The article is organized in the following way: after giving some examples of questions that could be studied in process research, we propose a classification of research questions which can be answered using this approach and expose the related methods. Finally, we introduce ways to study processes for some of these questions by referring to artificial and empirical examples. Since process studies seem to be unfamiliar to many researchers, we attempt to clearly define the difference between the way learning is usually studied (interindividual or cross-sectional approach) and within the process (intraindividual approach).

1.1. Example 1: Analyses of individual learning trajectories First, consider a simple learning process: a student wants to learn his English vocabulary. The main question is, How can we describe the learning process from the beginning to the end (when some kind of competence is acquired)? As a measure, we use the number of words correctly reproduced from a list of 30 items that have to be learned. In the common one-group pretesteposttest design, researchers study the learning process using two measurement occasions: before the learning starts and after the end of the learning period. During the pretest, the student is asked to solve a number of tasks which he or she has to solve again in the posttest. The results are shown in Fig. 1 (broken line): in the pretest, none of the tasks is solved, while 24 of the 30 tasks of the posttest are correctly solved. This kind of measurement is typical for many studies of learning (despite the fact that only one student is in the focus of the example here). In describing this learning process one could say that the student has learned to solve some tasks. But it does not really provide insight into the process of learning. Some of the following questions cannot be answered on the basis of the information in Fig. 1 (broken line): - Is the learning continuous or discontinuous, with one or more jumps? - Are there plateaus or drawbacks? - Does the amount of knowledge follow some simple trend, e.g. linear or quadratic? 30 Beginning Effective

Plateau

Forgetting

Beginning

learning

Effective learning

Plateau

Forgetting Beginning

Effective learning

Plateau

Forgetting

Learned words

25

20

15

10

Process Pre-post Measurement

5

0 1

4

7

10

13

16

19

22

25

28

31

34

Time / Phases Fig. 1. The learning of vocabulary: measures of reproduction for one individual before and after learning (broken line) and for a series of measurements (continuous line).

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- Is the learning behavior characterized by high or low variability over time? - Are there phases of learning exhibiting different qualitative learning behavior? - Does the learning behavior show some regularity or rhythm (e.g., a daily rhythm)? One way to obtain more information is to introduce a design with more measurement occasions: in our fictitious example, we measure a sequence of 12 periods within each of three learning days. Fig. 1 depicts the shape of the learning curve (continuous line). The figure represents the learning of words from a foreign language. For each 5-min interval, the number of words learned correctly is shown. During the first period, the learning material is unknown and one has to become familiar with the topic and the task, which is a gradual process. After becoming accustomed to the situation and the task, the amount of learning increases. At this point, the motivation is high and one feels the increasing competence. Afterwards, the person becomes tired, the concentration diminishes and the person tries to compensate for this by enhancing his effort. Even so the result is stagnation. The individual draws consequences and decides to take a break. During the break the individual forgets parts of the learned material. Learning is continued the next day. Again, habituation to the situation is necessary but then the rate of learning speeds up. On days 2 and 3, the shape of the learning trajectory is similar to that on day 1 but it starts at a higher level. At the end of day 3, we measure the amount of learned words which turns out to be 24 (cf. Fig. 1). Note that to understand the learning process, we did not only use the descriptive information about the learned material but we added pieces of information whereby an interpretation based on motivational mechanisms is possible. The discrepancies between the two lines of Fig. 1 illustrate that the simple pre-post measurement (broken line) does not help to give insight into the characteristics of the learning process, whereas the data which establish the continuous line in Fig. 1 give a precise description of the process. This example raises two questions: How can the univariate learning trajectory be described and how are changes to be explained? Remember that it was shown that a two-occasion measurement can be rather misleading regarding the true trajectory. 1.2. Example 2: Analyses of synchronous relationships between components of learning processes In the aforementioned part of example 1, we studied the trajectory of a single variable. Now, we turn to the fictitious relationship between two variables. We are interested in the study of the development and change of daily learning behavior of university students performing complex tasks at home. Therefore, we study important aspects of selfregulation following the model of Schmitz (2001b). The main variables of the self-regulation model include positive affect and the results of learning. One aspect of the learning result is the degree of understanding of the learned material. The central question of this study was, Do understanding and positive affect show similar developments over an observation period of 8 days? There are two strategies to answer this question. The first strategy entails measuring understanding and positive affect on two occasions, e.g., on day 1 and day 3, cf. Fig. 2a. On day 1, the understanding is higher and the positive affect lower. On day 3, the understanding is lower and the positive affect higher. This observation would lead us to the assumption that for this individual understanding and positive affect develop differently.

a)

b)

c)

Understanding Positive Affect

Understanding Positive Affect

Understanding Positive Affect

Fig. 2. The relationship between positive affect and understanding of learned material. (a) Two occasions as base for interindividual correlations. (b) Intraindividual synchronous correlations for one individual. (c) Asynchronous intraindividual correlations for one individual.

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If we study a group of individuals showing similar behavior as the one presented, the computation of a correlation would result in a negative coefficient. As shown in example 1, evidence is given that these two measurement occasions are neither sufficient to study the development of understanding and positive affect during the observed or any other period nor sufficient to study the similarity between the development of understanding and positive affect during that period. The second strategy is therefore, to collect data on multiple occasions for both understanding and positive affect. Fig. 2b shows the variation for understanding and positive affect measured each day over a period of 8 days. Now it becomes clear that understanding and positive affect actually show a very similar development during that period. Up to now, we tried to answer the descriptive question of the similarity of both developments. If one is interested in the explanation of the process of understanding, additional or different information is required. If we know, for instance, that positive affect is measured before understanding, the following explanation is valid with high probability (changes in positive affect influence changes in understanding, cf. Fig. 2c). In both examples, it should be shown that two-occasion measurements compress the results of learning processes, whereas a deeper insight can be gained if more data points are collected. Explanations for the special kind of process type can be based on the inclusion of additional information. Note that until now we have restricted the analyses to a single individual.

2. Overview of process analyses 2.1. Definition of process Following these more intuitive examples, we give a definition of processes. The common use of the term process in context of learning research refers to at least two concepts: (a) process as a sequence of states and (b) process as the mechanism or transformation of one state into another. Our definition of process is related to the first concept. In contrast to most empirical research, which is based on more trait-oriented concepts, process research primarily deals with the study of change. The basic concept is not a stable trait but a state, (cf. Cattell, 1952; Nesselroade, 1988), which is a measure of a phenomenon at one point in time without making the assumption that it has to be stable over time but instead can change rapidly. Because states can vary dramatically over time, they have to be measured quite often. This leads to the following definition: an observed or manifest process y(t) is defined as the repeated measurement of the same variable(s) for one entity over time (t ¼ 1, 2, 3, .). Most often, the entity will be an individual but it can also be an aggregated unit, e.g., a group of individuals, a class or a school. The process can be univariate or multidimensional. We assume that we cannot observe the true process directly because it may be disturbed by measurement error. A latent process is defined as a sequence of random variables x(t) over time (t ¼ 1, 2, 3, .). The relationship between latent and manifest processes is y(t) ¼ x(t) þ e(t), where e(t) represents some error in measurement which can be compared to that in classical test theory. (As in linear structural equation models, a more general approach is y(t) ¼ Hx(t) þ e(t), with H being some matrix specifying the weights by which the latent variables can be transformed.) Therefore, there are two corresponding definitions of processes on the level of the observed variable and for the underlying phenomenon corresponding to the conceptualization of a latent variable. Note that the definition includes univariate and multivariate processes. Note also that, in addition to the aforementioned definition of quantitative processes, one can also define processes for qualitative variables as referred to in the discussion, cf. Gottmann and Roy (1990). For this kind of process, the relationship between latent and observed processes described above does not hold. So far, we have referred to the first concept of process as a sequence of states. But interestingly, both conceptions (the sequence of states and the transformation rule) are highly related. In the area of linear and nonlinear dynamics (cf. Molenaar & Raymakers, 1998), the transformation between the series of states of the latent process will be described by the equation x(t þ 1) ¼ Fx(t) þ w(t þ 1), with F as a transformation matrix and w(t þ 1) as an error component. That means the actual state can be seen as the transformation of the earlier state (and possibly disturbed by some error process). Under general conditions, it can be shown that the latent process x(t þ 1) can be estimated recursively based on y(t þ 1) and x(t). Therefore, under some general conditions both concepts of processes mentioned above are mathematically equivalent, e.g., if one has the sequence of observed states y(t þ 1), one can compute an estimation of the transformation rule F describing the mechanism by which state x(t) is transformed into x(t þ 1) (conversely, if one has the transformation rule F and a starting state, one can estimate the whole series of states). It is important to know that this transformation corresponds to the common understanding of a process as a mechanism which translates a state of

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learning to another state. Therefore, processes can describe the generation mechanism of learning as well as the series of learning states. 2.2. Analyses of samples and populations The examples above deal with questions regarding processes for one individual. But often it is more important to study populations or samples, because we are not as much interested in the process of learning for an individual as we are for an aggregate unit (a group), e.g., a class of students. Note that the line of argumentation for examples 1 and 2 holds also for samples. Here as well, two-occasion measurements cannot tell very much about the kind of trajectory or the similarity in developments. However, it is important to recognize that we often try to answer questions about processes based on only two-occasion measurements. If we want to answer questions about the kind of development, whether it is continuous or discontinuous, linear or nonlinear, single or multi-staged, whether two or more processes are similar across time, we need multiple occasion measurements, regardless of the level of aggregation. 2.3. Classification of questions for process studies: main tasks for research on learning and instruction In order to present a classification of questions which can be answered best through process studies, we elaborate the dimensions which play a main role in this classification schema: The main tasks for research on learning and instruction e which may be similar for most empirical research in psychology and education e have to deal with at least four topics: description, explanation, prediction and modification/optimization (Zimbardo & Gerrig, 1996). Therefore, we describe how these topics can be dealt with in process research. 2.3.1. Description An important question that concerns the description of processes is the stability of learning. Are the results of learning stable? What happens when the learning process ceases? Will the knowledge gained at the end of instruction be stable or follow the shape of the forgetting curve (studies of forgetting look mostly at short time periods and short times of instruction)? Are there sleeper effects? Is there any regularity or rhythm in learning, any maxima, minima, or turning points during the learning process? 2.3.2. Explanation Explanations are indirectly contained in the introductory examples. We elaborated on the reasons why the learning trajectory changes its shape. Hypotheses based on explanations (causal factors) can be tested in two different ways. First, we could test whether there are time-lagged correlations. In our example, changes in positive affect were followed by changes in understanding of the learned material. Second, we can design an (quasi-) experiment, e.g., we may conceptualize and implement a training for enhancing learning and we observe knowledge development some time before (baseline phase) and after this training intervention (post-intervention period). Such a design is very powerful to detect intervention-related changes. 2.3.3. Prediction Another broad class of questions is related to the prediction of behavior. As done in ordinary (not time-related) statistics, the methods of description can be applied for prediction concerning process analyses. For instance, if there is a correlation between variables x and y, it can be mathematically transformed into a regression from y to x. Similarly, in process analyses, a synchronous relationship between variables can be transformed into a regression equation. In addition to this prediction based on the relationship between two (or more) variables, for process studies it is also possible to make predictions based on univariate time trends. For example, if we know motivation is growing linearly during the observed time span, we can predict the value of an individual or sample for later occasions by extrapolating the observed trend. 2.3.4. Modification and optimization Modification and optimization can be examined using the schema already used for explanations. If we know that motivation increases as knowledge increases, we could use that relationship to further increase knowledge: thus we try to enhance motivation. If we know that a certain variable (e.g. training yes/no) has an effect on another variable

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(e.g. motivation), we can use this knowledge to design an intervention (here: a motivation-focused training). The effect of this intervention can be studied by collecting process data regarding the pre- and post-intervention period, e.g. by collecting multiple occasion measurements before and after motivation training. This method of studying intervention effects is equivalent to the method of testing explanations. Therefore, questions of modification and optimization can be studied using an intervention analysis. It should be noticed that the above classification corresponds directly to the schema used by Krapp (2001). Following Cronbach (1957), he distinguishes between the correlational and the experimental approach. The correlational approach can be regarded with respect to descriptive and predictive purposes, whereas the experimental approach is mainly related to explanation. 2.4. Classification schema for process analyses For a systematic overview of some important kinds of questions that can be dealt with in process analyses, we give a classification schema which builds on the basic dimensions. This classification is shown in Table 1. It also contains the related methods. The classification dimensions are level of aggregation, type of general task (description, explanation, prediction, modification), and number of variables. Note that, in order to keep the table short, the table is not exhaustive and we have not elaborated all possible combinations, e.g., the dimension level of aggregation is not combined with all questions. Column 1 of Table 1 contains the level of aggregation. It is useful to keep in mind that the level of aggregation is very important for our process approach. Each question in process research can be examined on an intraindividual level or on the level of a population or sample (aggregate level). The second column lists the kinds of general questions or main tasks relevant to process analyses. The third important classification dimension of a process approach (column 3) is the number of variables. One can study univariate, bivariate or multivariate processes (cf. example 1 and 2 in the introduction). Specific questions of process analyses are listed in column 4. Note that the list of specific questions is not complete but contains only selected important examples. The column ‘‘method’’ contains tips for the technical manifestation of the specific form of process analyses.

Table 1 Categories of questions that can be answered using process studiesa Level of analyses

General question

Number of variables

Specific question

Method

Kind of trajectory, trends Phases Rhythm/regularities Chaos

Trend analyses Detection of mean changes Autocorrelation spectral analysis Lyapunov exponent

Synchronous relationship Lagged relationship

Synchronous correlation Lagged correlation MARIMA

Structure

Dynamic factor analysis

(Quasi) experiment Cross-lagged relationship

Intervention analysis Lagged correlation MARIMA

Individual/aggregate Description Univariate

Bivariate

Multivariate Individual/aggregate Explanation

Prediction

Compare description

Modification/optimization

(Compare intervention) (Quasi) experiment Cross-lagged relationship

Intervention analysis Lagged correlation MARIMA

Each question can be dealt with on the intraindividual and on an aggregate level (cf. column 1). For details, see text. a To read the table: determine the respective general question, the number of variables and the specific question, then you get hints for the appropriate method, e.g., if you are interested in the description of multivariate structures, you can use dynamic factor analyses.

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2.5. Overview of selected questions which are important for the analysis of processes In correspondence with Table 1, we give examples of important questions of process analyses. We only comment on a part of the table, which is not complete in many respects. The sequence of presentation is derived from Table 1, and the topics elaborated on are: -

Univariate description: types of trajectories Aggregation for univariate trajectories Bivariate synchronous relationships on the intra- and interindividual levels Asynchronous relationships Evaluation of intervention effects.

2.5.1. Description of univariate processes: types of trajectories The description of univariate trajectories on the individual level is covered in example 1 in Section 1. Therefore, we switch to the processes for univariate trajectories on the aggregate (group) level. Examples of trajectories on the group level are given by Bretz, Weber, Gmel, and Schmitz (1999), by Schmitz (1987) and Schmitz (1989). The method of studying processes on the group or aggregate level can lead to serious problems if one is interested in determining the form of a learning curve which should be valid for individuals, too. For example, does the form of the group curve correspond to the form of the individual curve? The upper part of Fig. 3 depicts four individual learning trajectories: Individuals 1 and 2 show an erratic and abrupt learning trajectory, whereas Individuals 3 and 4 show erratic forgetting. The aggregate (sample) trajectory shows a discontinuous rise and fall with four changes. If one tries to deduce the form of the individual learning curves from this sample, the results would be misleading (see also Bakan, 1954). Although there is not a single individual with a combined ascending and descending trajectory and no individual who shows more than one change, the sample curve leads to such conclusions. In sum, it can be useful to first study the form of the curve at the individual level. Similar suggestions stem from Bakan (1954), Estes (1956), and Rogosa (1995).

Aggregated trajectory

4 students

Individual trajectories

Student 1

Student 2

Student 3

Student 4

Fig. 3. The effect of aggregation: relation between group level and individual level trajectories.

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Univariate trajectories could also be analyzed to detect possible rhythms (using autocorrelations or spectral analysis) or whether the process shows characteristics of chaos (appropriate tools are, e.g. computation of Lyapunov exponent, cf. Abarbanel, 1996). 2.5.2. Bivariate processes on the individual level So far, the focus has been on univariate processes. But learning can be more effectively studied by taking more than one variable into account. We will refer to example 2 from Section 1 to demonstrate the dynamic interplay between positive affect before starting to solve a learning task and the degree of understanding of the learning material for one individual. Note that these relationships are studied over time. Therefore, it can be categorized as a process-analytic approach, whereas usually the relationship between variables is studied cross-sectionally and hence across individuals. Note also, that the longitudinal approach using two-occasion measurements cannot deal with relationships over time, and therefore, cannot be used to analyze processes in the sense defined above. The main question in process research regarding relationships is whether the variables covary over time (cf. Schmitz & Wiese, 1999). One result could be that the time courses of the variables run parallel (cf. Fig. 2b). Even more meaningful than synchronous relationships are asynchronous relationships (cf. Fig. 2c). Both variables in Fig. 2c show the same pattern but variable 2 (understanding of the learning material) lags behind variable 1 (positive affect). Such time-lagged relationships are used to study causal relationships between variables. Note that such cross-lagged relationships are also analyzed in longitudinal studies with only two occasions of measurement but the information gained in such studies may differ considerably from that in process studies. 2.5.3. Idiographic vs. nomothetic analysis: problems of aggregation for a bivariate process Table 2 (cf. Schmitz, 2000) gives an example of the difference between cross-sectional (interindividual) and process (intraindividual) relationships between two variables. In column one, the data of two variables (e.g., positive affect and understanding) provided by four individuals on one occasion is presented. Computing the correlation between the two variables yields a coefficient of .0. This is the cross-sectional correlation for occasion 1. The cross-sectional correlation for occasion 2 is computed similarly to occasion 1 and also yields a coefficient of .0, as do the cross-sectional correlations of occasions 3 and 4. Obviously, the cross-sectional correlation for the four occasions is identical (.0). From a process (intraindividual) perspective, in row 1 the data of the two variables (positive affect and understanding) are displayed for subject 1 over the four occasions. Computing the correlation for this individual over time yields a coefficient of 1.0. The same procedure for subjects 2, 3, and 4 leads to a process

Table 2 Hypothetical example regarding the relation between intra- and interindividual correlation for positive affect and understandinga Subj.

Variables

Occasions

Mean across time

Intraindividual correlation

1

2

3

4

1

Pos. affect Understanding

120 101

120 99

101 120

99 120

110 110

1.0

2

Pos. affect Understanding

120 99

120 101

99 120

101 120

110 110

1.0

3

Pos. affect Understanding

80 99

80 101

99 80

101 80

90 90

1.0

4

Pos. affect Understanding

80 101

80 99

101 80

99 80

90 90

1.0

Correlations of means (aggregated across time) þ1.0 Aggregated across individuals Pos. affect Understanding One-occasion cross-sectional correlation a b

100 100 .00

Table after Schmitz (2000, p. 85). Mean of one-occasion cross-sectional correlation.

100 100 .00

100 100 .00

100 100 .00

.00b

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(intraindividual) correlation of 1.0. Again, all correlations are equal (1.0). In sum, this example shows that all interindividual correlations are the same as are all intraindividual correlations, but inter- and intraindividual correlations differ strongly! Hence, these two kinds of correlations are conceptually different, refer to different ways of collecting data, and can have completely different numerical values. The bivariate relationship, mentioned in the example above, shows the difference between the analysis of an individual (intraindividual analysis) and of a sample (cross-sectional). One important advantage of this kind of process approach is the possibility to study individual relationships. But, as the example in Table 2 shows, one need not stay on the level of individual relationships, but one can also study them on the group level: two simple procedures demonstrate this idea. Procedure 1 is the categorization of the computed intraindividual correlations, e.g., positive, zero or negative. Applying procedure 1 to the example mentioned above results in the following conclusion: there are four negative intraindividual correlation coefficients. Procedure 2 is the computation of a mean value across these individual parameters: referring to the example, a value of 1 results, leading to the interpretation that the average intraindividual correlation is 1. Note that Krapp (2001) and Renkl (2001) also point to the problem of inferences from interindividual to intraindividual relationships. In psychology, there has been a controversy between advocates of a more idiographic way of research (cf. Allport, 1960) and a more nomothetic one (cf. Holt, 1962). The above method of aggregating individual relationships can be regarded as a methodological approach that can contribute to the solution of the idiographicenomothetic controversy. A sophisticated method to perform this aggregation is the HLM-approach (Bryk & Raudenbush, 1987), if the database is longitudinal. Another research strategy which emphasizes the necessity to study individual behavior instead of focusing on groups or variables is the person-oriented approach, cf. Von Eye and Bergman (2003), s. also Molenaar (2004). 2.5.4. Process analyses for studying asynchronous multivariate relationships In the discussion of explanation, it was argued that if time-lagged relationships exist, they can with high probability be interpreted in terms of causality. Although this interpretation is for many cases correct, one has to be cautious as time lag does not necessarily prove causality. For example, night always follows day but day does not cause night. The best way to perform dynamic analyses for multivariate processes is by using multivariate time-series. We cannot go into details here, but introductions are given in Schmitz (1987), Schmitz (1989), Schmitz (1990) and Schmitz and Skinner (1993). Consider the process y(t) ¼ fy(t  1) þ e(t). This equation states that a process y(t) can be predicted from its own past y(t  1) with the regression coefficient f. An error component e(t) has to be added. This is called an autoregressive equation. If y(t) is bi- or multivariate, each term has to be conceptualized as a vector and the scalar ‘‘f’’ turns out to be a matrix, which can be written (in the bivariate case) as y1 ðtÞ ¼ f11 y1 ðt  1Þ þ f12 y2 ðt  1Þ þ e1 ðtÞ y2 ðtÞ ¼ f21 y1 ðt  1Þ þ f22 y2 ðt  1Þ þ e2 ðtÞ In these equations, the transformation from the state of y at time t  1 to the state of time t is described. Information about the transition is contained in the f matrix. For more information see Schmitz (1990) and Schmitz and Skinner (1993). The f-coefficients represent the type of the causal mechanism of the bivariate process. In the sample case of a bivariate relationship between control beliefs (y1) and performance (y2), the coefficient f21 describes the time-lagged effect of control belief on performance, whereas f12 describes the time-lagged effect of performance on control belief. By estimating the matrix of the f-parameters, one can describe how this dynamic system works. One should keep in mind that the problems of inference from interindividual to intraindividual analyses (cf. Krapp, 2001; Renkl, 2001) are also valid for asynchronous relationships, cf. Schmitz (2000) for an example. The equations describe the particular case of a bivariate autoregressive model. The special importance of the more general class, the so called multivariate autoregressive moving average models (MARMAmodels), is that it can be shown that every process (which is stationary, meaning that it does not change its basic mechanisms) can be modeled as a sum of a deterministic component and a MARMA-process. This fact is known as the WOLD-decomposition theorem (cf. Lu¨tkepohl, 1993).

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2.5.5. Process analyses for studying modification/optimization (explanation) effects using intervention analyses In the discussion of the tasks of psychology it was explained that most kinds of modification/optimization and also explanation within instruction, could be analyzed from an intervention framework (cf. Schmitz, 1987; Schmitz, 1989). Any treatment, which is supposed to have an impact on an ongoing process, can be studied from an intervention perspective. In the usual evaluation of treatments performed by a pre-post experimental control-group design, results give no information about the stability and the time trajectory. A combination of first computing individual intervention analysis and then aggregating the results across individuals is performed in Schmitz, Stanat, Sang and Tasche (1996). 3. Examples from empirical analyses The presentation of empirical examples has to be selective. Therefore, we can only refer to a rather small number of the procedures introduced in the theoretical part. The sequence of the examples follows the line in the theoretical part, starting with univariate analyses and then introducing bivariate analyses. 3.1. Univariate variables For univariate variables we provide examples regarding trajectories. First, we will demonstrate individual analyses for univariate trends. Data are taken from a study in self-regulation, cf. Perels, Gu¨rtler, and Schmitz (2005). Students answered questions in diaries regarding their self-regulation competence for a period of 49 days. One aspect of selfregulation is the time taken for homework. Fig. 4 shows the trajectory for one subject (Subject a) over the 49 day observation period. For this subject, the daily learning time decreased linearly (b0 ¼ 39.4; b1 ¼ .65*, R** 2 ¼ .29). For Subject b a quadratic trend (b0 ¼ 77.6; b1 ¼ 5.2; b2 ¼ .19*; R** 2 ¼ .30) best described the development of the learning time, whereas Subject c showed a linearly ascending trend (b0 ¼ 26.5; b1 ¼ 5.0*; R** 2 ¼ .28). Subjects d and e differed in the variability of learning time (SD ¼ 14.5; SD ¼ 65.4, respectively) and the trajectory for the whole group did not show any kind of trend, but interestingly, a rhythm could be identified (r(lag 7) ¼ .35*). As described in Section subject a

subject c

subject b

60

80

Time (min)

300 60

40

200 40 20

100 20 0

0 1

7

49

1

7

subject d

Time (min)

300

49

1

7

Day

Day

aggregated

subject e

300

49 Day

100

200

200

100

100

80 60 40 20

0

0 1

49

7 Day

1

7

49 Day

Fig. 4. Learning trajectories for six individuals and the sample.

1

49

7 Day

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1, one of the advantages of the process approach is the possibility to analyze group as well as individual processes. The present example clearly demonstrates that individual trajectories can differ significantly, and can be quite different from the sample trajectory. For similar examples using a very large database s. Sang, Schmitz, and Tasche (1993). Now we turn to bi- or multivariate synchronous and asynchronous relationships. 3.2. Bivariate synchronous relations Consider the relationship between positive affect and understanding during a learning period. Theoretically, different relationships can be expected. If positive affect leads to more motivation and more intensive learning and hence, understanding, the correlation might be positive. On the other hand, positive affect could also lead to an overly enthusiastic learning which leads one to not concentrate on details, to neglect analytic thinking and therefore, to limit understanding. Thus, no relationship or even a negative correlation could be expected. We now present the intraindividual relationship for two subjects from empirical data, stemming from diary information. Fig. 5 shows the bivariate trajectories for these two individuals. For Subject f, the relationship between positive affect and understanding of the learned material results is r ¼ .72** (T ¼ 35 days), whereas for Subject g this relationship results as r ¼ .00. A different mechanism seems to operate for both individuals. This could not have been detected without process analyses. Again, it turns out that the intraindividual correlations can also be quite different for bivariate relationships. 3.3. Multivariate causal relations One important application of process analyses is the analysis of causal relationships. In the data presented here, homework behavior is studied from the perspective of self-regulated learning, cf. Perels et al. (2005) and Schmitz and Wiese (2006). Regarding the model of self-regulated learning, we consider three important variables: selfefficacy, effort, and goal attainment. The results of multivariate time-series analyses for two individuals are presented subject f

6

Understanding Positive Affect

5,5 5

Amount

4,5 4 3,5 3 2,5 2 1,5 1 1

3

5

7

9 11 13 15 17 19 21 23 25 27 29 31 33 35

Day subject g

6 5,5

Understanding Positive Affect

5

Amount

4,5 4 3,5 3 2,5 2 1,5 1 1

3

5

7

9 11 13 15 17 19 21 23 25 27 29 31 33 35

Day Fig. 5. Bivariate trajectories of positive affect and understanding of learned material for two subjects.

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Table 3 Results of intraindividual time series analyses: phi-matrices, and residual correlations for two individuals Student

Variables

Phi-matrix

Synchronous correlations

Self-efficacy

Effort

Student A

Self-efficacy Effort Goal-attainment

.34* .01 .30*

.16 .19 .04

Student B

Self-efficacy Effort Goal-attainment

.01 .13 .24

.11 .14 .02

Goal-attainment .32* .24 .04 .01 .01 .09

Self-efficacy

Effort

Goal-attainment

1. .40* .24

.40* 1. .41*

.24 .41* 1.

1. .68* .53*

.68* 1. .76*

.53* .76* 1.

*p < .05.

in Table 3, which on the one hand contains the causal (time-lagged) relationships given by the phi-matrix, and on the other hand, the synchronous relationships of the variables. As mentioned above, the phi-matrix (cf. the f matrix explained above for the bivariate case) contains the lagged relationships between the variables and has to be interpreted as follows: the column variables have a time-lagged effect on the row variables. Student A shows the following time-lagged relationships: In addition to the univariate autoregressive effect of self-efficacy, self-efficacy is raised one time unit after goal attainment is enhanced, and goal attainment changes one time unit after self-efficacy is enhanced. In contrast, Student B does not show any time-lagged relationship because the phi-matrix is not significant for any element. Table 3 also contains the pattern of synchronous correlations of these variables, which are different for these students, too. 3.4. Intervention analyses

Using Learning Aids (tables, graphical representations)

Awell-known procedure for testing intervention effects is the interrupted time-series design following Campbell and Stanley (1963). As an example, we present the evaluation results of a self-regulation training combined with mathematical problem solving. In the baseline period and the post-intervention period, the amount of learning aids used was observed in a group of N ¼ 28 students by standardized diaries. On day 22, a training session was conducted to instruct students when and how to use graphical representations as learning aids while doing math problems. In Fig. 6, the usage 3,5

3

2,5

2

1,5

1

7

22

49

Day Fig. 6. Effect of a training intervention on the usage of learning aids.

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of learning aids as well as the mean usage for the baseline and the post-intervention period is shown. A time-series intervention analysis is performed, comparing means and taking into account the serial dependency of the diary observations, cf. Schmitz (1990). It can be seen that after the respective training session, the usage of study aids increased (baseline mean ¼ 2.2, the intervention effect is .39, t ¼ 5.6, df ¼ 47, p < .001, R** 2 ¼ .40). 4. Discussion First, we want to summarize the argumentation in this article. Second, we discuss possibilities of applying this kind of process analysis for research in cognitive learning. Then, the frequency of application of process research is compared to its possible relevance. Finally, limitations of this approach are considered and an outlook given. This article points out the benefits of considering learning and instruction from a process perspective. In the beginning, some questions were presented, which could be answered best using process analyses. Because the term process is used with widely differing notions and with various respects, a definition was given. The advantages of this kind of process approach were discussed based on artificial examples as well empirical data. To get an overview regarding process concepts, a classification system, which also describes the appropriate methods, was given. We presented the results of analyses of univariate trajectories, bivariate synchronous relationships, and multivariate causal relationships. In addition, we demonstrated how to perform an intervention analysis. Principally, individual and interindividual analyses can be performed for each of the questions regarding process research described in Table 1. The empirical examples in this article are taken from our research group and are related to different fields of research in learning and instruction, excepting the field of cognitive learning. Researchers interested in this topic may ask whether and how the methods of process analyses discussed in this article could be applied to their field. To give some examples to answer this question, we refer to the following research topics: forgetting, learning, and problem solving. 4.1. Cognitive research 4.1.1. Forgetting and learning The studies of forgetting using nonsense syllables by Ebbinghaus are well known. The retention curve found by Ebbinghaus describes the percentage of time saved to relearn the previously learned material as a function of the delay. Initial forgetting is rapid but the rate of forgetting slows down over time, s. Anderson (1999, p. 227). Interestingly, with respect to our topic, the research object was a single individual: Ebbinghaus. The retention curve is a good example of an observed univariate process. In a similar way, learning is often studied via the time needed to perform certain tasks. It is shown that with increasing practice the time for performance is reduced. This relationship can be modeled via a logarithmic curve, cf. Anderson (1999). The study of the effects of distributive practice compared to massive practice has a long tradition in learning research and the results can be illustrated as two different retention functions. For distributive practice, the retention function starts at a lower level but remains high compared to massed practice, which at the beginning leads to high retention with the amount of retention decreasing more quickly than for distributive practice (cf. Anderson, 1999, p. 237). This means that process analyses of these different learning styles lead to curves with the same logarithmic function but with a different starting level and a different pace of decreasing. 4.1.2. Process approach and problem solving In the problem-solving literature, the tower of Hanoi is a well-known research topic (cf. Anderson, 1999). To provide an example, we discuss the three pegs and four disks problem. The task is to move all four disks of different sizes from peg A to peg C. The disks can be moved from any peg to any other peg. Only the top disk on a peg can be moved, and it can never be placed on a smaller disk. How can the process-analytical approach outlined in this article be applied to this kind of problem solving? The first step could be to describe the process of problem solving as a sequence of moves by the sequence of coordinates of the disks. This procedure is very well-known from chess. The result of this description is a process: each move is described by the new coordinates of the moved disk. With respect to process analyses, the result is a sequence of coordinates. Note that the measure is a qualitative one in this case. We did not elaborate on qualitative processes. This process description as a sequence of coordinates does not give insight into the mechanisms in the head of the problem solver. For this purpose, one could compare different kinds of solutions, e.g., an optimal solution is compared to empirical solutions of a group of problem solvers. Analyzing sequences of

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moves described in this way for different individuals may be a good starting point for deriving ways to understand problem-solving strategies. In addition to this description of the process, one could also study quantitative parameters of this problem-solving sequence. One way to do this could be to measure the time needed for each move. This would give us a univariate process, consisting of the sequence of the duration of the moves. Now, the hypothesis that the duration for the moves will decrease with decreasing distance to the goal could be examined. A nice way to perform a bivariate time series analysis for this problem-solving task is described by Anderson (1999, p. 318). Anderson tests the hypothesis that individuals will find a solution to the problem by deriving subgoals. One kind of subgoal in this example could be that first the largest disk on peg A has to be moved to peg C. Anderson relates the time needed for a move to the number of goals which were pursued for each move. This can be viewed as a bivariate time-series where the number of the move is related to its duration and the number of subgoals pursued for this move. By performing the analysis of empirical time-series for these variables, Anderson can show that there is a positive relationship between the number of goals and the duration of a move. The example demonstrates that there are different ways to apply process analyses for this task. For instance, one could use descriptive or hypothesis testing procedures, and one could also apply univariate or bivariate time-series. There are numerous ways to analyze this kind of problem solving, and certainly, there are many ways to study problem solving without using methods of process research. In our view, process analyses can be regarded as a research tool. The concept of this manuscript is to enlarge the toolbox of researchers in the field of learning and instruction by methods of process research. The metaphor of the toolbox may also help to understand the limits of this article. Analyses of processes may be one element of the toolbox. Hopefully, it may contain numerous other tools. Note also that our way of process analysis is definitely not the only way to study processes. 4.2. The frequency of application of process studies in empirical research The main goal of this article was to describe the advantages of process research and to categorize main questions and ways of how they can be answered especially by using a process approach. If the importance of process research would be clear to the audience, there would be no need for this article. But if one looks at the empirical realization of process studies, by using the term ‘‘process’’ in databases like ERIC or Psycinfo, the result is that there exists a high number of studies using the phrase ‘‘process’’, which, however, do not include any kind of process analysis described here. The search for process studies in the sense defined here, as analyzing the sequence of states measured frequently, leads to very few hits. One can find only a few process studies, cf. Schmitz (1996). It can be concluded that there seems to be a discrepancy between the power, importance and actual application of process studies. What are the reasons for this discrepancy? Presumably, there are theoretical as well as more practical reasons: (1) The main reason is that very often there is no need for process research because researchers have other kinds of questions. For example, if one is interested in the relationship of variables for one point in time. (2) Sometimes the assumptions of the methods of process analyses are not fulfilled and therefore, process analyses cannot be applied (see below). (3) Other kinds of modeling were applied which do not belong to our class of process approach. (4) The trait concept, which is still dominant in most kinds of empirical educational research, assumes that traits are stable over time. Traits do not change or change only slightly. Corresponding to the trait theory, many instruments (questionnaires) are still constructed based on the assumptions of classical test theory, which assumes that the true value is constant over time. Therefore, process analysis is not needed. (5) To collect data suitable for process analyses is far more expensive with respect to researchers and subjects. (6) To perform process analyses needs more expertise with respect to statistical procedures. This list of reasons why not to perform process studies does not claim to be exhaustive. Regarding the plausibility of the arguments, the first three are the most convincing. Whether trait theory or the assumptions of classical test theory are justified can best be tested using process data. If the variables turn out to be stable over time, the trait approach might be appropriate. With respect to the arguments regarding practicability, economic resources clearly play a role in decisions regarding design and procedure. But if theoretical considerations or previous empirical results lead to the conclusion that behavior should best be described as a process, economic considerations cannot lead one to apply

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inadequate data and methods of analyses. Regarding the kind of data analyses, some researchers might be unfamiliar with these techniques. Although more complex procedures exist in the time-series literature (ARIMA modeling, cf. Glass, Willson, & Gottman, 1975; Schmitz, 1990; dynamic factor analyses, Molenaar, DeGooijer, & Schmitz, 1992; Musher-Eizenman, Nesselroade, & Schmitz, 2002), some basic methods, like computing intraindividual correlations and trends, could be used rather easily. 4.3. Limitations and outlook The limitations with respect to the style of presentation in this article stem from the aim that this introduction into process research in education should convince a larger audience, and therefore, we attempt to stick mainly to methods that are easy to apply. Clearly, within this article we could not go into much detail. The description of the concepts, the elaboration of methods and the presentation of empirical examples had to be rather short and introductory. Because of this article’s introductory style, we could not elaborate on the precise nature of the models, and used mathematical formulas only in rare cases. This may have led to some loss in precision. In addition, we could not clearly specify the assumptions which must be fulfilled to apply a certain method of analysis. In addition to the limitations stemming from and related to the style of presentation, there are limitations of content, e.g., the kind of process models which correspond to methods of data analysis and also limitations to the methods of data collection. Regarding the kind of models and analyses, we concentrated on time-series models because they were very well-suited for process modeling and for process analyses. But clearly, each model and method is built based on certain assumptions. For each kind of assumption, we refer to some other method which may be more appropriate in case of the assumption is not fulfilled. For process analyses, we assume that the behavior we want to study is time varying. But in case of stable behavior, which does not change over time (i.e., trait-related behavior), there is no need for process analyses. Time-series models assume that behavior is stationary, which means that main parameters of the process (the mean, the variability and the auto-correlative structure of the process) do not change; otherwise, methods to induce stationary state have to be applied, e.g. differencing or detrending. Most time-series models are also conceptualized as linear models. In case of nonlinearity, chaos-models could be applied, (cf. e.g., Newell & Molenaar, 1998). We did not deal with questions of dynamic dimension analyses, but for these there exist powerful methods like dynamic factor analyses (cf. Molenaar et al., 1992). If the modeling of the kind of development is of interest, latent growth-curve models may be appropriate (Stoolmiller, 1995). If complex periods and rhythms are to be studied, methods of spectral analyses can be applied. If the time dimension is not discrete, differential equation models can be used. In the case that the assumption of an intervally scaled measurement cannot be fulfilled, one should apply Markov models (Gottmann & Roy, 1990). Other methods specially suitable for the prediction of processes come from the ACT theory (Anderson, 1999) or from the theory of neuronal nets (Wassermann, 1989). Questions dealing with samples of individuals and series of observations measured over time can often be dealt with using hierarchical linear modeling HLM (Raudenbush, 1995). The list of assumptions looks quite restrictive. Note, however, that although the class of time-series models is based on these assumptions, they are not as restrictive as ordinary methods of statistical analysis, like analyses of variance or regression analyses, because they allow for serial dependency. In this paragraph, we discussed the limitations of one class of process models and methods of analysis. But another aspect of process research deserves further discussion: the method of data collection. Our examples are based mainly on diary data. As self-report measures, they are accompanied by the known problems of this method. In addition, even more than in cross-sectional research, good compliance of the subjects is needed. Otherwise, a high dropout rate might result. Another problem is that a diary can lead to reactive effects: the behavior which is observed can change as an effect of self-observation, cf. Schmitz and Wiese (2006). To avoid such difficulties of diaries, one could use observation data, archival data or physiological data. Examples of such procedures are given in Fahrenberg and Myrtek (1996). As we argued, one of the advantages of process research is the possibility to study individual processes. This kind of research fits into the category of a person-oriented approach (cf. Bergman & Magnusson, 1997; Von Eye & Bergman, 2003). There are numerous kinds of methods belonging to a person-oriented approach which cannot be dealt with here. But note that the main idea of this article is to advocate process research. This kind of research offers as a special case the study of individual analyses, which is an important subgroup of process analyses, but not the main focus of the present article.

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The suggestion to consider more often a process-analytic approach should not be misunderstood as a plea for process studies as the only way of conceptualizing learning and instruction. In addition, the argumentation to include more studies of individual processes within empirical research should not be understood as anabolishment of the nomothetic approach. On the contrary, we advocate combining intra- and interindividual analyses. We hope that this introduction to process research can reduce the discrepancy between the importance and power of the process approach and its still rare empirical application.

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