Pattern Emergence and Pattern Transition in ...ddl/articlesCopy/YanFischerMicrodevVariatn.JDP2007.pdfing strategy, over the nine sessions (see Schlagmüller & Schneider, 2002, Figure

Pattern Emergence and Pattern Transition in

Microdevelopmental Variation: Evidence of

Complex Dynamics of Developmental Processes

Zheng YanEducational and Counseling PsychologySchool of EducationUniversity at Albany, [email protected]

Kurt FischerMind, Brain, & Education ProgramHarvard Graduate School of EducationHarvard [email protected]

Abstract: An important but challenging task is to understand microdevelopmental varia-tions in learning and developmental processes—how patterns of performance change overshort time periods with growing skill and knowledge or with failure to learn. This studyanalyzed microdevelopmental variations in learning a basic computing skill. Based on 119microdevelopmental episodes of 30 students learning a computer program in four sessionsover one semester, we used both microdevelopmental trajectories for describing each in-dividual’s variations in performance and a dynamic systems approach for interpreting pat-tern emergence and pattern transition of these variations. The results of the study suggestthat each individual’s variations of microdevelopment are pervasive and complex, show-ing three basic patterns of microdevelopmental variation—unstable, fluctuating, and sta-ble—and four basic trends in change of microdevelopment variation—disorganization, re-gression, improvement, and stabilization. These basic patterns and trends provide adynamic picture of developmental processes as pattern emergence and pattern transitionin variation rather than as a ladder-like linear progression.

Introduction

Microdevelopmental variation can be considered the common, spontaneous fluctu-ation of people’s performance within a short period of time. It has been extensively re-ported in the microdevelopmental literature for the past ten years (e.g., Hsu & Fogel,2003; Granott & Parziale, 2002; Kuhn, Gracia-Mila, Zohar, & Anderson, 1995; Siegler,

39

1994, 2006; Yan & Fischer, 2002). Among existing microdevelopmental studies, strongconsistency exists in approaches to collecting microdevelopmental data by intensivelyassessing psychological changes over a short period of time (Kuhn, 1995; Siegler,1995, 2006). In contrast, significant divergence exists in approaches to analyzing mi-crodevelopmental data by using different units of analysis (Fogel, 2006; Granott, 1998;Molenaar & Valsiner, 2005; Siegler, 1987) and different frameworks of analysis (Fis-cher & Bidell, 2006; Lavelli, Pantoja, Hsu, Messinger, & Fogel, 2004; van Geert,1998). As a result, this divergence in data analysis leads toward diverse and even con-tradictory descriptions and interpretations of microdevelopmental variations. The pre-sent study was intended to use individual microdevelopmental trajectory (Yan & Fis-cher, 2002) as the unit of analysis and dynamic skill approach (Fischer & Bidell, 2006)as the framework of analysis to examine microdevelopmental variations. We hope toprovide empirical evidence of the complex dynamics of developmental processes andto advance the current understanding of developmental variability in the organizationand growth of human activities in context, a central task of dynamic skill theory.

Unit of Analysis for Describing Microdevelopmental Variations

The issue of the unit of analysis in the history of developmental science can betraced back at least to William Stern’s concept of the person as a complex unit (for areview, see Kreppner, 1992). For many years, both learning curves and growth curveshave been used to represent and analyze group-based linear trends of psychologicalchange over time (e.g., Shock, 1951; Thorndike, 1913, Thurstone, 1919). In the 1950s,several methodologists questioned the way of averaging learning curves as the unit ofanalysis to study development (e.g., Bahrick, Fitts, & Briggs, 1957; Estes, 1956; Sid-man, 1952). Starting from the 1970s, longitudinal methodologists further challengedthe conventional treatment of averaging longitudinal data and proposed three units ofanalysis, intra-individual variability (relatively rapid and reversible changes), intra-individual changes (relatively stable developmental changes), and inter-individual dif-ferences (highly stable changes even over a long time period) (Baltes, Reese, & Nes-selroade, 1977; Nesselroade, 1991, 2001; Nesselroade & Baltes, 1979; Nesselroade &Molenaar, 2003). Despite theoretical advances in the unit of analysis, however, aver-aging has been and still is one of the most common statistical methods used in empir-ical behavioral research (Brown & Heathcote, 2003).

In existing microdevelopmental research, the unit of analysis generally falls intoone of three types on the basis of how microdevelopmental data are aggregated:double-aggregated microdevelopmental trends that aggregate data across both indi-viduals and trials, single-aggregated microdevelopmental trends that aggregate dataacross either trials or individuals, and non-aggregated microdevelopmental trajecto-ries in which no aggregation is used across individuals or trials. Each unit of analysisserves different research purposes in studying microdevelopmental variations, accom-panied by different data analysis approaches.

The most frequently used basic unit of analysis in microdevelopmental research isthe double-aggregated microdevelopmental trend (e.g., Goldin-Meadow & Alibali,

40 ZHENG YAN AND KURT FISCHER

2002; Kuhn, Gracia-Mila, Zohar, & Anderson, 1995; Miller & Aloise-Young, 1995;Robinwitz, Grant, Howe, & Walsh, 1994; Schlagmüller & Schneider, 2002). This typeof microdevelopmental study primarily focuses on group-based trends and normallyuses parameters (e.g., group mean score or group average percent) as the basic unit ofanalysis that aggregates both performance sequences (the first aggregation) and par-ticipating individuals (the second aggregation) to demonstrate important trends amongdifferent groups.

Schlagmüller and Schneider (2002), for example, examined how third and fourthgraders acquired an organizational memory strategy in an experimental study. Themicrodevelopmental data were collected based on 22 children’s performance of sort-ing and recalling 20 picture cards over nine weekly sessions. In one of a series of sub-sequent group-based data analyses, group mean recall was estimated by aggregatingthe performance on the 20 picture cards (the first aggregation) across 22 individuals(the second aggregation). On the basis of the group mean recall as the basic unit ofanalysis, Mann-Whitney tests were conducted and trend plots were constructed to il-lustrate trends of mean recall between two groups, those with or without using a sort-ing strategy, over the nine sessions (see Schlagmüller & Schneider, 2002, Figure 2,p.306). The findings of this analysis showed differences in microdevelopmental trendsbetween two groups of children who sorted items according to categories and childrenwho did not sort, but they failed to reveal both intra-individual performances andinter-individual differences

An increasing number of microdevelopmental studies use the single-aggregatedmicrodevelopmental trends as the basic unit of analysis (e.g., Corbetta & Thelen, 1996;de Weerth, van Geert, & Hoijtink, 1999; de Weerth, van Geert, 2002; Hsu & Fogel,2003; Spencer, Vereijken, Diedrich, & Thelen, 2000; Thelen, Corbetta, & Spencer,1996; Vereijken & Thelen, 1997). For example, Thelen and her associates (Thelen,Corbetta, & Spencer, 1996) used single-aggregated microdevelopmental trends as thebasic unit of analysis to study how infants’ reaching behaviors developed in their firstyear. The microdevelopmental data were collected based on four infants’ weekly per-formance on a sequence of trials for approximately 25 weeks. In the data analysis, in-dividual average speed was estimated by aggregating the weekly reaching perfor-mance on the trial sequences for each infant. Using the weekly individual averagespeed as the basic unit of analysis, individual developmental trends of four infantswere constructed to examine each infant’s development of reaching skills (see Thelen,Corbetta, & Spencer, 1996, Figures 1, 2, 3, & 4, pp. 1065, 1068, 1070, & 1073, respec-tively). They found that infants’ reaching movements initially were jerky and tortuousand then became smother and straighter; however, this improvement was actually non-linear with plateaus and regressions for each infant as well as large variations amongthe four infants, revealing complex patterns in microdevelopmental variations.

Different from the Thelen study focusing on individual-based microdevelopmen-tal trends with cross-trial aggregation, a small number of microdevelopmental studieshave focused on non-aggregated microdevelopmental trajectories as the basic unit ofanalysis (e.g., Cooney & Troyer, 1994; Granott, 2002; Parziale, 1997; Yan & Fischer,2002). That is, these studies have directly examined each individual’s performance se-quences without across-individual or across-trial aggregation. For example, Granott

Pattern Emergence and Pattern Transition in Microdevelopmental Variation 41

and her collaborators (Fischer & Granott, 1995; Granott, 2002) studied how two adultlearners, as a pair, came to understand a computer robot. The microdevelopmental datawere collected based on this pair’s performance sequence over 30 minutes. In the dataanalysis, individual knowledge level was assessed for these two adults’ understand-ing of the robot on the performance sequences. On the basis of changes of individ-ual knowledge level over time, individual microdevelopmental trajectories were con-structed as the basic unit of analysis to examine each pair’s changes in understandingof the robot over 200 interchanges (see Granott, 2002, Figure 8.2, p. 225). Granott ob-served complex recurring progression-regression sequences: The two collaboratingstudents started with a very low level of performance and then gradually built up to arelatively high level of performance. However, when they encountered a new featureof the robot or the situation, their performance collapsed back abruptly to a low level,and then they began to rebuild their knowledge again. In the study, many complex pat-terns of microdevelopmental variations were observed.

Both the Thelen study and the Granott study demonstrate the strength of focusingon individual-based changes rather than group-based changes. These methods can an-alyze individual-based trends (the Thelen study) or individual-based trajectories (theGranott study) to reveal unusually complex patterns in microdevelopmental variation.However, due to the difficulty of collecting this type of highly dense data, these andsimilar studies have used only a small number of subjects, thus missing the chance ofexamining a wide spectrum of microdevelopmental variations of different learners andlimiting the generalizability of the findings (e.g., four infants in Thelen, Corbetta, &Spencer, 1996; four infants in de Weerth, van Geert, & Hoijtink, 1999; eight infants inVereijken & Thelen, 1997; eight children in Gelman, Romo, & Francis, 2002). In ad-dition to the small sample size, these studies normally take place over a short timespan within one single session (e.g., 40 minutes in Granott, 2002; 60 minutes in Yan& Fischer, 2002) and consequently might miss opportunities for observing and captur-ing meaningful pattern changes if these changes were to occur across multiple ses-sions over a longer period of time (Granott, 2002; Karmiloff-Smith, 1979; Lee &Karmiloff-Smith, 2002; Lewis, 2002; Siegler & Svetina, 2002).

Analytic Framework for Interpreting Microdevelopment Variations

There are two analytic frameworks for conceptualizing microdevelopmental vari-ations. A traditional analytic framework considers microdevelopment a process of lin-ear progress moving from lower levels to higher ones, and attempts to reduce com-plexity observed in microdevelopmental variations into certain statistical parametersso that linear statistics can handle the data analysis. In contrast, a dynamic systems ap-proach conceptualizes microdevelopmental variations as true manifestations of an un-derlying dynamic system and strives to undercover dynamic patterns from complexmicrodevelopmental variations (e.g., Fischer & Bidell, 1998; Lavelli, Pantoja, Hsu,Messinger, & Fogel, 2004; Fogel, 2006; Siegler, 2006; Thelen & Corbetta, 2002; The-len & Smith, 2006; van Geert, 1991, 1994, 1998).


Specifically, there are two strengths of the dynamic systems approach for inter-preting microdevelopmental variations. First, in addition to the simple linear trajec-tory, the dynamic systems approach has the capacity to entertain a large variety ofcomplex nonlinear trajectories, including logistic growth trajectories, stage-wise dis-continuous trajectories, S-shaped curves, U-shaped curves, and various combinationsof these forms (e.g., Case & Okamoto, 1996; Fischer & Bidell, 2006; van Geert,1991, 1994, 1998). It can further be used to identify dynamic patterns behind variouscomplex trajectories.

Second, the dynamic systems approach considers development as a process oftransforming from one dynamic pattern to another in diverse directions rather than aprocess of simple linear progress moving from lower static levels to higher ones in asimple forward direction. It can uncover essential mechanisms that underlie variouscomplex patterns and explain how one dynamic pattern can be transformed into an-other when important elements in a dynamic system interact with each other in anevolving manner. For example, van Geert (1998) built a dynamic growth model basedon classic Piagetian and Vygotskian theories and examined a fundamental develop-mental mechanism that contributes to dynamic transition among various developmen-tal patterns, such as continuous and discontinuous changes, abrupt changes, and mi-crodevelopmental transitions.

To further understand the true complexity and dynamics of microdevelopmentalvariation, the present study (a) used the non-aggregated microdevelopmental trajec-tory as the basic unit of analysis to represent microdevelopmental variations, (b) ap-plied dynamic systems theory as the primary conceptual framework to analyze thecomplex patterns of microdevelopmental variations, (c) recruited a relatively largenumber of participants and examined more than 100 trajectories to understand howpatterns of microdevelopmental variation emerged and changed over a short period oftime, (d) covered a longer time span to examine each student’s performance on fourprojects over one semester, and (e) applied typological analysis to analyze patternemergence and pattern transition in microdevelopmental variations. The study focusedon two research questions: (1) What patterns of variations in cognitive microdevelop-ment emerged over a short period of time? (2) How did these microdevelopmentalvariation patterns change over a relatively longer period time? According to the dy-namic skill theory, complex microdevelopmental variations over both a short term anda relatively longer term should be observed and certain dynamic patterns shouldemerge and evolve.

Method

Participants

Thirty students who enrolled in an introductory statistics course at a graduate schoolin the northeastern US participated in the study voluntarily. Among them were 9 malesand 21 females, 63% of whom were master’s students and 37% doctoral students.


These students were very diverse in terms of prior educational training, professionalbackground, statistical knowledge, and computer experience. Their ages ranged fromapproximately 20 years to 40 years.

Task

The task used in the study was based on actual homework assignments using SAS,a widely used statistical software program, to analyze data. Students were required tocomplete four SAS projects as homework assignments with an interval of approxi-mately one month between the projects over one semester. Normally, it takes sevenbasic steps to finish a SAS project: creating a DAT file, creating a SAS file, creating aCOM file, executing the COM file, examining the LOG file, checking the LIS file, andfinally, printing the LOG file. The seven-step sequence had to be followed in all fourSAS projects in order to obtain the data analysis results on the computer network sys-tem. The present study focused on how each student proceeded with this seven-stepbasic sequence using SAS rather than on various statistical analysis procedures (e.g.,conducting a t-test or a χ2-test). Consequently, focusing on the same SAS basic se-quence across different projects made it possible to analyze students’ microdevelopmentin learning the same basic SAS procedure across four different projects while minimiz-ing the potential for a practice effect on students due to repeated measures over time(Diggle, Liang, & Zegger, 1994; Nesselroade & Baltes, 1979; Singer & Willet, 2003).In the study, students must follow the basic procedure to create a data file (the DAT file)for each of the four projects, but each time they had different data files (e.g., college ad-missions data or final exam data) rather than repeatedly using the same data file.

In the study, participants first received a brief introduction to the use of the SASprogram with one step-by-step demonstration by the course instructor at the beginningof the course. They then came to a research laboratory four times during the semesterand worked on the four SAS projects with a teaching assistant in a one-to-one inter-active context. All participants used the same computer that was set up to function asa terminal of the computer network system. Each SAS session lasted approximatelyone hour. During each session, when students encountered difficulties in executing theseven-step sequence, the teaching assistant offered appropriate help so that they wereable to continue and finish their own SAS projects. The help that the teaching assis-tant provided was carefully controlled according to two basic rules: (a) the helper onlyanswered the questions each student asked during his or her work with SAS, and (b) the helper did not provide extra intervention or initiate lengthy instruction beyondanswering the student’s question. These rules were intended to maximize the opportu-nity for observing authentic performance while helping students learn SAS.

The Coding System

This study used the Microdevelopmental Scale for Assessing Cognitive Complex-ity of Performance in Learning SAS in a Network System (Yan, 1998, 2000) to code


students’ performance according to the cognitive complexity of their SAS perform-ances in context. This microdevelopmental scale was developed based on the hierar-chical complexity scale in dynamic skill theory (Fischer, 1980; Fischer & Bidell,2006; Fischer, Yan, & Stewart, 2003), which is supported by extensive empirical evi-dence of scaling (e.g., Dawson, 2003; Dawson & Wilson, 2004; Fischer & Bidell,2006). Similar scales have been developed and used successfully in several micro-developmental studies (e.g., Bidell, 1990; Bidell & Fischer, 1994; Granott, 1993,2002; Parziale, 1997, 2002; Schwartz, 2000; Schwartz & Sadler, in press).

As shown in Table 1, the scale has six different levels of cognitive complexity withseveral transitional sub-levels within each level. Whenever a breakdown of a student’sexecution of the seven-step sequence occurs, the cognitive complexity of the student’sperformance is analyzed and coded. Generally speaking, levels of performance gofrom typing skills as the lowest performance level, to the production of one meaning-ful unit of a computer command, one complete command statement, one logic com-mand sequence, a coherent multi-sequence flow, and finally to the highest level of flu-ent navigation through innovative pathways within computer systems.

For instance, when a student types “EDIT” as a regular English word after the $prompt but does not know the particular meaning of EDIT in the VMS computer oper-ating system (creating and opening a new file rather than reopening and editing anexisting file), the cognitive complexity of this performance is coded at the level ofsensory-motor systems (capable of typing EDIT, but does not know the particularmeaning of EDIT) and assigned a score of 3. If a student types “EDIT” and shows thathe or she understands EDIT as a command for creating and opening a new file, the cog-nitive complexity of this performance is considered higher than the previous perfor-mance, and is coded at the level of single representational sets (capable of understand-ing only one component of a computer command) and assigned a score of 4. If a studenttypes “EDIT PROJECT1.SAS” and knows the logical relationship between EDIT as asystem command and PROJECT1.SAS as a file name, then the cognitive complexityof this performance is considered even higher than the previous two, and is coded at thelevel of representational mappings (capable of understanding the relation between twomajor components of a computer command) and assigned a score of 5.

When students’ flow of executing the seven-step sequence was interrupted, theteaching assistant offered timely help so that students could continue their work withSAS. The teaching assistant’s help to students was coded on the same scale, fromlower levels of help (e.g., explaining one command, TYPE) to the higher ones (e.g.,explaining two sequentially ordered statements, from creating a COM file to execut-ing the COM file). Note that the teaching assistance’s help could be at a level lowerthan that of students’ breakdowns (the scaffolding of “pushing,” as was the most fre-quent case in the study) or higher than that of students’ breakdowns (the scaffoldingof “pulling,” in some cases).

A second rater independently coded 20% of the data and the Cohen’s Kappa sta-tistic was 0.94. According to Bakeman and Gottman’s (1997) criteria, the interrateragreement for the scale is considered excellent. This is generally consistent with inter-rater agreement reported in other microdevelopmental studies (e.g., Bidell, 1990; Gra-nott, 1993; Parziale, 1997; Schwartz, 2000).



Table 1. Microdevelopmental Scale for Assessing Cognitive Complexity of Performance inLearning SAS in a Network System

Code Description Score

3.0

3.0

4.0

4.0

4.4

5.0

5.0

5.8

SM3

RP1

RP2

LEVEL OF SENSORY-MOTOR SYSTEMS

DefinitionA learner can coordinate various pairs of keyboard movements and screen moves simultane-ously and shows competent typing skills. This is the lowest performance observed in the study.

Example: Typing competently without understandingA student read the course handout and quickly typed INFILE PART2. This example shows thatthis learner is competent at typing the words, but he simply copied the exact words from anexample given by the instructor. He did not know that the exact meaning of PART2 was ademonstrating data file the teacher used in the class rather than the data file the student cre-ated a few minutes ago for himself. The correct file name is MOM2 instead of PART2.

LEVEL OF SYSTEMS OF SENSORY-MOTOR SYSTEMS OR LEVEL OF SINGLE REPRESENTATIONAL SETS

DefinitionA learner can independently coordinate two systems, the type-look action system and therequest-execution semantic system simultaneously. This formulates a basic unit of executableinstruction, such as a VMS command or a SAS file.

Example 1: A typical VMS commandA student said, “OK, I want to change the SAS file,” and then started to type EDIT after the $prompt. In this example, the student intercoordinated the type-look action system (typingEDIT) and the request-execution semantic system (changing the SAS file) and produced aVMS command EDIT. This student was not just typing a string of letters E-D-I-T, but was ac-tually producing one of the basic commands in the VMS operating system.

Example 2: Compounding SAS file componentsA student said, “Oh, I have to change the file,” and then typed ASSIGN2.SAS. In this exam-ple, the student understood well that the SAS file includes a file name, ASSIGN2, and a filetype, SAS. The student integrated these two components into a more complex but integratedunit, a complete SAS file.

LEVEL OF REPRESENTATIONAL MAPPINGS

DefinitionA learner can coordinate a specific VMS command with a specific SAS file to create a com-plete instruction for execution.

Example 1: A typical SAS statementAfter receiving the notification from the system, a student typed EDIT ASSIGN.LOG. In thisexample, the student coordinated a system command, EDIT, with a SAS file, ASSIGN2.LOG,and produced a proper instruction in order to view the LOG file. A complete instruction is thebasic unit in a SAS programming sequence.

Example 2: Not knowing about the next stepAfter creating and checking a SAS file, a student read the class handout and said, “The nextstep is to create a . . . COM file.” In this example, the student figured out the logic sequencebetween having a SAS file and having a COM file, but did not yet show how to make this se-quence executable, a skill requiring intercoordinating the two steps simultaneously. This is atypical “next step problem.”


Table 1. Continued

Code Description Score

6.0

6.0

6.4

7.0

7.0

7.4

8.0

8.0

RP3

AB1

AB2

LEVEL OF REPRESENTATIONAL SYSTEMS

DefinitionA learner can coordinate a variety of system commands with a variety of SAS files accordingto both the context and the syntax, e.g., the DIR command requires no SAS file, the EDITcommand needs one SAS file, and the COPY command demands two SAS files. This resultsin an intended executable sequence with clear logic and behavioral evidence.

Example 1: A typical sequenceWithin a few seconds, as a student first typed EDIT ASSIGN2.DAT and then quickly typedEDIT ASSIGN2.SAS. In this example, the student coordinated two executable instructionsand built an adjacent sequence.

Example 2: Compounding with more componentsAfter creating three files, SURVEY.DAT, SURVEY.SAS, and SURVEY.COM, a student typedDIR SURVEY.* In this example, this student not only showed clear knowledge about the logicsequence, but also more knowledge about using the DIR command to search for specific files.

LEVEL OF SYSTEMS OF REPRESENTATIONAL SYSTEMS, WHICH IS ALSOLEVEL OF SINGLE ABSTRACT SETS

DefinitionA learner can coordinate two systems, the VMS-SAS system and the computer conceptual sys-tem, to reach a good understanding of the hierarchical relationship between an operating sys-tem and an application. This leads to both fluent sequential flows in constructing various VMScommands and different SAS files and clear conceptual understanding of the VMS-SAS sys-tem architecture.

Example 1: A sound understanding of the VMS-SAS systemTo answer a question, “What does the dollar sign mean to you?” a student replied, “It tells mewhat mode I am in, for example, I can type MAIL. It is a command shell.” Here, this studentcorrectly used the computer concept of command shell to specify the function of the dollarsign as the VMS system prompt. This understanding was associated with the student’s fluentsequential flow in using VMS commands and SAS files.

Example 2: Compounding with SAS/VMS features When talking about processing the COM file, a student said, “We have to have a COM file.Give it to the sever to run it.” Here, this student correctly used the concept of server to explainthe particular function of a COM file in the VMS-SAS system. This understanding was asso-ciated with this student’s rather fluent sequential flow in using VMS commands and SAS files.

LEVEL OF ABSTRACT MAPPINGS

DefinitionA learner can coordinate the VMS-SAS system with the EVE-SAS system to achieve a strongconceptual understanding of the architecture of these two systems. This leads to flexible and in-novative pathways in navigating among the systems and fluent navigation skills in the systems.

Example: Navigating within EVE editorAfter reading one COM file, unlike most students’ efforts to close the file, a student with aprofessional programming background pressed the CTRL key and the Z key to bring up aprompt of COMMAND and then typed OPEN ASSIGN4.COM to open another COM file.This creative way of moving from one system to another system without even closing previ-ous files reveals the student’s solid understanding of the architecture among VMS, SAS, andEVE systems and her/his strong skill of navigating among different systems.

Data Analysis

The study used the non-aggregated microdevelopmental trajectory as the basic unitof analysis for analyzing microdevelopmental variations. Specifically, the microdevel-opmental trajectories without aggregation across trials or individuals were plotted byusing microdevelopmental levels based on the microdevelopmental scale as the Y-axisand microdevelopmental steps based on students’ performance, such as typing one filename or one SAS command, as the X-axis. In addition, the frequency and the level ofhelp offered by the teaching assistant were included in the trajectory as an inherentpart of the interactions between the student and the teaching assistant. Thus, a mi-crodevelopmental trajectory can be considered a series of cognitive footprints of howan individual student’s level varies over multiple steps in completing a SAS projectwith one-to-one assistance.

To answer the first research question of what patterns of microdevelopmental vari-ations emerges over a SAS project, 120 microdevelopmental trajectories based on allthe performance sequences of 30 students on four projects were plotted and a typolog-ical analysis was used for describing patterns of the microdevelopmental variations ineach individual’s microdevelopmental trajectories. This analysis was based on com-parison and classification among individual microdevelopmental trajectories ratherthan aggregation of means across trials and individuals. In a sense, this analysis isanalogous to a conventional heart examination where a doctor analyzes a group of pa-tients’ heart problems with electrocardiography (ECG) that records a series of electri-cal waves during each beat of the heart. The first task is to run an ECG on each pa-tient and print out all ECG diagrams that display the heart dynamics of each patient.The doctor then examines complex patterns of ECG diagrams for each patient to di-agnose what typical heart problems each patient might have.

To answer the second research question of how the microdevelopmental variationpatterns change across four SAS learning sessions, 30 microdevelopmental trendswere first plotted. Since each student completed a total of four SAS projects that couldbe viewed as four waves in a multi-wave longitudinal study (Willett, 1997), students’trajectories over four projects represented four-wave microdevelopmental trends. Atypological analysis was then conducted to examine whether the patterns of micro-developmental variation changed over time. Building on the above example, this isanalogous to a doctor’s analysis of a patient’s four ECG diagrams taken at the fourpoints in time to investigate whether the patient recovers over time under a particularmedical treatment.

Results

Patterns of Microdevelopmental Variations Within One Project

On the basis of 30 students’ performance on four SAS projects over one semester,a total of 119 microdevelopmental trajectories were plotted, with one student’s perfor-mance in one project missing due to a technical failure in recording. These trajecto-


ries show enormous complexity of microdevelopmental variation. Four microdevelop-mental trajectories are presented in Figure 1 to illustrate the complex variations ob-served among these 119 trajectories.

As shown in Trajectory 103, for example, Student 3 started her Project 1 with threeperformances at skill levels 5 and 6 (Steps 1, 2, and 3) and then was unable to sustainher high-level performance sequence. She asked two questions of the teaching assis-tant, who answered with the requested information (Steps 4 and 5). Next she madefour consecutive performances moving from level 5 to level 6 again (Steps 6, 7, 8, and9). To overcome another breakdown she received assistance again (Step 10), followedby five performances at levels 4 and 5 (Steps 11 to 15). Throughout the hour-long SASsession, her performance levels constantly varied between lower levels and higher lev-els. She finished the first project with 46 steps and received 10 instances of help. The


Trajectory 103

Step

Leve

l

00 10 20 30 40

2

4

6

8

Trajectory 230

Step

Leve

l

00 10 20 30 40

2

4

6

8

Trajectory 326

Step

Leve

l

00 2010 4030 6050 80 9070

2

4

6

8

Trajectory 422

Step

Leve

l0

0 10 3020 40 60 7050

2

4

6

8

FIGURE 1. Four microdevelopmental trajectories (Trajectory 103, Trajectory 230,Trajectory 326, and Trajectory 422) by four students (Student 3, 30, 26, and 22) onfour different projects (Project 1, 2, 3, and 4) illustrating pervasive and complex vari-ations in microdevelopment in learning a computer program in the tutor-tutee context.The student performance is marked with small circles and the help provided by theteaching assistant is marked with solid small squares. Level on the Y-axis = Micro-developmental Level. Step on the X-axis = Task Step. Note that in this study three dig-ital numbers were used to label one of 119 microdevelopmental trajectories: the firstdigital number indicates the number of the project, from 1 to 4, and the second andthird number represent the number of students, from 01 to 30. With this labeling sys-tem, one can easily see what specific project and which specific student a microdevel-opmental figure represents.

help provided by the teaching assistant, often higher or lower than the student’s cur-rent performance levels, enabled her to continue and finish the project. Similar to thistrajectory, other microdevelopmental trajectories in Figure 1 varied substantially notonly in performance level at each step, but also in number of helps received, overallshape of trajectories, time used for completing the project, and other aspects. In fact,all of the 119 individual microdevelopmental trajectories in the study look idiosyn-cratic and complicated, exhibiting a high degree of complexity of microdevelopmen-tal variation.

Since one cannot capture complex dynamics of microdevelopmental variationbased on a few single performances, Vygotsky’s (1978) classic concept of the zone ofproximal development (ZPD) was used to holistically identify emergent patterns be-hind the remarkable complexity among the 119 microdevelopmental trajectories. Ac-cording to Vygotsky, ZPD “is the distance between the actual developmental level asdetermined by independent problem solving and the level of potential development asdetermined through problem solving under adult guidance or in collaboration withmore capable peers.” (1978, p. 86). Based on this definition, especially extending Vy-gotsky’s notions of “the actual developmental level” and “the level of potential devel-opment,” three zones of development can be categorized in a hierarchical order (fromlower, intermediate, to higher) and presented in a vertical order (from bottom, mid-dle, to top): (1) the zone of potential development that is below the potential develop-ment level (where individuals cannot do a task even with help), the developmentalzone in the bottom, (2) the zone of proximal development that is between the poten-tial development level and the actual development level (where individuals can do atask with help), and (3) the zone of actual development that is above actual develop-ment level (where individuals can do a task independently without help). In otherwords, whether or how much scaffolding is needed is a critical indication of individ-uals’ current ability (van Geert, 1994, 1998).

Based on Vygotsky’s concept of ZPD, we observed three general types of micro-developmental trajectories that represent three major patterns of microdevelopmentalvariations over one SAS project: (a) unstable trajectories that show significant mi-crodevelopmental variation when students performed below the zone of proximal de-velopment (i.e., within the potential development zone) and needed an excessiveamount of help to finish a project, (b) fluctuating trajectories that show moderate vari-ation when students performed within the zone of proximal development and neededa reasonable amount of help to finish a project, and (c) stable trajectories that showlittle variation when students performed above the zone of proximal development (i.e., within the actual development zone) and finished a project with a minimumamount of help. Quantitatively, trajectories with the number of helps in the upper 25thpercentile, with 10 or more helps, were labeled unstable. Trajectories with number ofhelps between the 25th and 75th percentiles, with 4–9 helps, were labeled as fluctuat-ing. Trajectories with the number of helps in the lower 25th percentile, 0–3 helps, werelabeled as stable trajectories. These three patterns are illustrated below.

Unstable trajectories. The upper panel of Figure 2 provides two examples of thetrajectory with a very unstable pattern of microdevelopmental variations, indicatingthat students were working below the zone of proximal development. For Trajectory


135, Student 35 received a total of 38 instances of help during Project 1. The shape ofthe trajectory looks extremely complex, with few clusters of higher-level perfor-mance. For Trajectory 326, Student 26 received 33 instances of help at various levelsduring Project 3 and the trajectory also looks extremely complex, showing no clustersof higher-level performance. Overall, the two trajectories suggest that these two stu-dents were having a very difficult learning experience during the project. Even withfrequent help, they were unable to sustain their performance level. As a result, theirmicrodevelopmental trajectories featured a very unstable dynamic process. For them,the SAS task was far too difficult, and even the timely scaffoldings did not bring themto a level of competence using the program. In the study, 27 unstable trajectories wereobserved, accounting for 23% of the 119 trajectories.


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FIGURE 2. Prototypical microdevelopmental trajectories of unstable ones (Trajectory135 and 326), fluctuating ones (Trajectory 206 and 122), and stable ones (Trajectory101 and 327). The student performance is marked with small circles and the help pro-vided by the teaching assistant is marked with solid small squares.

Fluctuating trajectories. The middle panel of Figure 2 provides two examples ofthe trajectory with a fluctuating pattern of microdevelopmental variations, indicatingthat students were working within the zone of proximal development. For Trajectory206, Student 6 received eight instances of help during Project 2. The trajectory alsolooks relatively complex, with several clusters of higher-level performances at or nearlevel 6. For Trajectory 407, there were seven instances of help that Student 7 received,distributed through Project 4. This student completed the project within 40 steps, withmany of them performed at level 6. The shape of the trajectory is moderately complex,showing several clusters of higher-level performances and one period of rapid oscil-lation. These two fluctuating trajectories from two different students for two differentprojects indicate a feature of transition between stable and unstable systems. Thesestudents did not appear to possess a solid knowledge of how to use SAS. They expe-rienced frequent ups and downs, but with the help from the teaching assistant they hadapparently learned certain basic knowledge, showing some work in progress and var-ious clusters of higher-level performance. They obviously needed some help but notwith all the procedural steps all the time. As a result, their microdevelopmental trajec-tories showed the feature of fluctuation, indicating a dynamic process that is charac-terized by moving from a relatively unstable system into a relatively stable one. Therewere a total of 56 fluctuating trajectories observed in the study, accounting for 47% ofthe 119 trajectories.

Stable trajectories. The lower panel of Figure 2 provides two examples of the tra-jectory with a stable pattern of microdevelopmental variations and indicates that stu-dents were working above the zone of proximal development. For Trajectory 101, Stu-dent 1 received help only once in the beginning of Project 1. The shape of this trajectoryis close to a straight line. The student finished the whole project within 19 steps. ForTrajectory 327, Student 27 received help only twice in the middle of Project 3. The tra-jectory appears as a straight line with a small W-shaped curve in the middle. These twoequilibratory trajectories from two different students at two different projects share thefeature of stabilization: These students appeared to possess a solid knowledge of SAS

and needed little help during thewhole project. Thus, their microde-velopmental trajectories showed a sta-bilized dynamic process. There were atotal of 36 stable trajectories observedin the study, accounting for 30% ofthe 119 trajectories.

Figure 3 shows the change innumber of the three types of micro-developmental trajectories over time,showing an increase in stable trajec-tories and a decrease in fluctuatingtrajectories, but with no statisticallysignificant change in unstable trajec-tories. Further, repeated measures


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analysis of variance shows no linear trend across the four projects, F(1, 29) = 2.085, p = .159, partial Eta2 = .076, but a significant nonlinear cubic trend, F(1, 29) = 6.333,p = .018, partial Eta2 = .179, indicating that overall the students accelerated their learn-ing during the semester. This is consistent with what was observed during the study;approximately one quarter of the students improved their performance at the end of thesemester, moving from the oscillatory trajectory to the equilibratory trajectory.

Patterns of Change in Microdevelopmental Variations Across Four Projects

Besides the examination of the basic patterns of microdevelopmental variationswithin one project, we further analyzed patterns of change in microdevelopmentalvariations across four projects among 30 students. Specifically, if one student’s trajec-tories across the four projects are all unstable, we classified them as the disorganiza-tion trend. If one student’s trajectories across the four projects are all stable, we clas-sified them as the stabilization trend. If one student’s trajectories move toward lessstability, showing fluctuating regression over time (e.g., from stable to fluctuating),we used a term of regression trend. If one student’s trajectories change toward morestability, showing fluctuating progression (e.g., from unstable to fluctuating), we usedimprovement to label them. Figures 4, 5, 6, and 7 provide prototypical examples ofthese four microdevelopmental trends.

The trend of disorganization. As shown in Figure 4, Student 26 received 31 in-stances of help in Project 1; 27 in Project 2; 33 in Project 3; and 29 in Project 4. Allfour trajectories showed tremendous variations, with frequent ups and downs all theway to the end, but without many clusters of higher-level performance. This patterntransition moving from Projects 1 to 4 indicates a tendency toward dynamic disorgan-ization and reveals that the student was experiencing serious difficulties in complet-ing the task and made little progress over one semester, even with much help from theteaching assistant.

The trend of regression. As shown in Figure 5, Student 22 started with seven helpsin Project 1 and three helps in Project 2; the shapes of the two trajectories appearedquite flat with evident clusters of higher-level performance. However, this trend didnot continue. In Projects 3 and 4, the student received 33 and 29 helps, and the shapeof the trajectories turned into an unstable pattern. This pattern transition moving fromfluctuating trajectories into unstable ones or from stable trajectories into fluctuatingones indicates a tendency of change toward dynamic regression and reveals that, withan increasing amount of scaffolding, this student was still getting more and more con-fused in using SAS over time, showing a clear regression rather than progression.

The trend of improvement. As shown in Figure 6, Student 36 started with 18 helpsin Project 1, a relatively poor performance among the 119 trajectories, but only needednine helps in Project 2, although the shapes of the trajectory still appear quite up anddown. In Projects 3 and 4, this student only received one and two help(s) respectively,and performance continued to improve. The shape of the trajectories turned into a



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FIGURE 4. Prototypical microdevelopmental trend of disorganization (Subject 26).The student performance is marked with small circles and the help provided by theteaching assistant is marked with solid small squares.

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FIGURE 5. Prototypical microdevelopmental trend of regression (Subject 22). Thestudent performance is marked with small circles and the help provided by the teach-ing assistant is marked with solid small squares.

strong straight line. This pattern transition moving from Projects 1 to 4 indicates a ten-dency of change toward dynamic improvement and reveals that this student was ex-periencing significant learning difficulties in the beginning, but made substantialprogress over time. As a result, the unstable trajectory shown in Project 1 evolved intoa fluctuating trajectory in Project 2 and then a stable trajectory in Projects 3 and 4.

The trend of stabilization. This trend is evident when a student’s four microdevel-opmental trajectories from Project 1 to Project 4 show a clear tendency of dynamicstabilization, with stable patterns across sessions. As shown in Figure 7, Student 1 typ-ically needed little help in order to complete the projects, and showed consistentlyhigher level performances from the first project to the last one. The shapes of thesefour trajectories are very close to a straight line, indicating that this student showed asolid knowledge of SAS and was able to finish all the projects independently.

Figure 8 presents the distribution of the four types of trends. Among the total of 30 students in the study, four students showed a trend of disorganization, nine studentsa trend of regression, 14 students a trend of improvement, and three students a trendof stabilization. These differences in the four types of microdevelopmental trends arestatistically significant, χ2 (3, N = 30) = 10.27, p < .05, suggesting that these 30 stu-dents had significantly different types of learning experiences with SAS over one se-mester: With the scaffolding from the teaching assistant, nearly half of the studentssubstantially improved their performance and nearly half of the students experiencedvarious degrees of difficulty.


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FIGURE 6. Prototypical microdevelopmental trend of improvement (Subject 36). Thestudent performance is marked with small circles and the help provided by the teach-ing assistant is marked with solid small squares.

Discussion

The present study is one of the earliest empirical efforts that systematically examinepatterns of microdevelopmental variations with a relatively large number of individualsover a relatively long period of time from the perspective of microdevelopmental dynam-ics. The findings of the study offer two important insights to the current understanding ofmicrodevelopmental variation in particular and developmental processes in general.

First, the study empirically demonstrates real complexity of microdevelopmentalvariations with the non-aggregated microdevelopmental trajectory as the basic unit ofanalysis, as complex as clinical ECG diagrams. The findings of the study indicate that,just as objects invisible to the eye become visible under a microscope, the process oflearning SAS by 30 students over four projects from a non-aggregated trajectory per-spective shows extremely complex variations, which we might never see at a regularlevel of observation using the double- or single-aggregated tends. Through intensiveobservation of students’ performance in learning SAS, one can see the whole spectrumof the enormous variations of complex change over time. The microscopic process oflearning SAS, represented by various microdevelopmental trajectories, microdevelop-mental trends, as well as microscopic details, presents empirical evidence of the on-going process of learning SAS individually or as a group.

One might consider these complex variations as the result of some artifact, such asthe scale used, instead of reflecting true patterns of microdevelopment. This possible


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FIGURE 7. Prototypical microdevelopmental trend of stabilization (Subject 01). Thestudent performance is marked with small circles and the help provided by the teach-ing assistant is marked with solid small squares.

explanation can be ruled out by empirical evidence that different patterns occur in dif-ferent tasks and situations (e.g., Bidell & Fischer, 1994; Granott, 1993, 2002; Parziale,1997, 2002; Schwartz, 2000; Schwartz & Sadler, in press). Granott (2002), for exam-ple, used a similar microdevelopmental scale based on dynamic skill theory to exam-ine pairs of novices performing an open-ended exploratory task and found a differenttrajectory—recurring upward movement in level with frequent fluctuation, withoutthe ceiling levels observed in the present study. Thus, it is unlikely that such differenttrajectories would be the artifact of an assessment instrument.

The empirical literature indicates that using different basic units of analysis (i.e.,group means, individual means, or individual trajectories) produces different estimatesof the complexity of microdevelopmental variation (e.g., de Weerth, van Geert, & Hoi-jtink, 1999; Gelman, Romo, & Francis, 2002; Granott & Parziale, 2002; Hsu & Fogel,2003; Kuhn, Gracia-Mila, Zohar, & Anderson, 1995; Siegler, 1995; Thelen, Corbetta,& Spencer, 1996; Yan & Fischer, 2002). These differences can be illustrated by threegraphs based on three microdevelopmental studies published in a recent book of em-pirical papers (Granott & Parziale, 2002). Siegler (2002, p. 35, Figure 1.3) plotted thegroup-based percentage of strategy use over multiple sessions, elegantly showing ageneral trend of overlapping waves in microdevelopment but not complex variationsacross multiple trials among individuals. In contrast, Gelman, Romo, and Francis(2002, p. 280, Figure 10.3) showed more complex variation, consisting of eight indi-vidual trajectories, clearly presenting individual-based trends over time but not varia-tions across multiple sessions. Granott (2002, p. 225, Figure 8.2) found the most com-plex variation, an individual-based trajectory for the collaborative performances of apair of students working together in one session, with no aggregation of multiple in-terchanges. The results of the present study are consistent with the Granott study, butshow even more complex variations with a larger sample size over a longer time pe-riod. Given that the basic procedure to complete the computational task with SAS isrelatively fixed, the complexity of the microdevelopmental variations in acquiring


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other cognitive skills in ill-defined domains (e.g., playing chess or inventing a new de-vice) might be much larger.

The second insight that one can learn from the study is that behind the complexityof microdevelopmental variation there are certain patterns of microdevelopmental dy-namics, as dynamic as clinical ECG diagrams. ECG diagrams are often too complexfor a novice to see individual heart beat patterns immediately. But experienced doc-tors who have received specific ECG training and understand normal and abnormalheart dynamics can recognize certain patterns of heart dynamics from complex ECGsthat directly indicate typical heart problems. Likewise, researchers knowledgeableabout microdevelopmental dynamics can recognize from complex microdevelop-mental variations that the dynamic process of learning SAS takes place in the form ofcomplex pattern change rather than simple linear progress. Looking at all 119 micro-developmental trajectories in the study, it is clear that learning SAS does not alwaysshow a simple linear progress moving from lower levels to higher ones. Instead, theytypically reveal nonlinear dynamic patterns, changing from one pattern to another.Learning SAS involves constant transitions of complex patterns over time rather thanstepwise changes in magnitude of the SAS performance alone. In other words, the tra-ditional linear approach will encounter tremendous challenges in discovering and in-terpreting the enormous complexity in microdevelopmental variation; in contrast, thenonlinear dynamic approach can not only reveal a higher degree of complexity in de-velopmental processes, but also explain dynamic patterns within the complexity.

The findings of the study have important theoretical implications for understandingdevelopmental processes among individuals and groups. Understanding true micro-genesis is important for understanding true ontogenesis, which occurs distinctly in in-dividuals, even for physical growth (Lampl, Veldhuis, & Johnson, 1992) and more ob-viously for psychological development and learning (e.g., Estes, 1956; van der Maas &Molenaar, 1992). Just like doctors who use electrocardiograms to diagnose an individ-ual patient’s heart dynamics, researchers should use non-aggregated individual mi-crodevelopmental trajectories as the basic unit for both data collection and data analy-sis in order to understand microgenesis and ontogenesis. Aggregating performancescores across multiple trials or multiple individuals will substantially disturb, disrupt,and even destroy our recognition of the temporal dynamics of microgenesis, and thuswill essentially make it impossible to develop an authentic and accurate picture of on-togenesis. Furthermore, understanding true microgenesis is also important for under-standing true polygenesis, an ultimate goal for behavioral researchers to generalizeresearch findings to large populations. Like experienced doctors who have solidknowledge of the typology of cardiologic symptoms among various clinic popula-tions, researchers should first examine microgenesis and ontogenesis among individ-uals, and then move further to study typologies of developmental processes amongdifferent groups. The present study has demonstrated how the typological analysis ofa large number of individual microdevelopmental trajectories can shed light on dy-namics of microgenesis, ontogenesis, and polygenesis.

The findings of the study also have useful implications for practitioners workingin the complex real world. The present study suggests that a developmental process


primarily concerns pattern emergence and pattern transition rather than simple ladder-like linear sequences. Thus, it is important for practitioners such as educators or clini-cians to focus on pattern changes rather than the linear progress of their students ortheir patients to design effective educational or interventional programs. Specifically,it would be helpful to have “dynamics eyes” and “complex minds” to view develop-mental processes, develop sophisticated knowledge of complex patterns of behavioralchanges, acquire skills for identifying pattern changes based on both observed indi-vidual trajectories and sophisticated knowledge of pattern changes, synthesize a typol-ogy of pattern changes of individuals and groups, and effectively help individuals andgroups to meet various challenges in daily life.

In summary, the present study of pattern emergence and pattern transition in micro-developmental variation provides new evidence of complex dynamics of developmen-tal processes as transformed constantly in the real world. From the dynamic systemsviewpoint, a microdevelopmental trajectory represents a series of temporal states inwhich an individual’s multi-component skill system interacts with the multi-componenttask system in a specific context over a short time span. The three types of trajectoriesobserved in this study—unstable, fluctuating, and stable—reflect three emergent pat-terns of a dynamic system. Students change among these patterns in a continuous pro-cess of self-organization that produces the four types of trends—disorganization,regression, improvement, and stabilization—across sessions. There exits a relationbetween complex microdevelopmental variations and dynamic pattern emergence/transitions. That is, seemingly complex microdevelopmental variations reveal basicdynamic patterns of learning and development; the basic dynamic mechanisms pro-duce complex microdevelopmental variations. Put simply, complexity reveals dynam-ics, and dynamics produce complexity (van Geert, 1998).

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Pattern Emergence and Pattern Transition in ...ddl/articlesCopy/YanFischerMicrodevVariatn.JDP2007.pdfing strategy, over the nine sessions (see Schlagmüller & Schneider, 2002, Figure