136.pdf, 2003 June 20-2003 June 24

Table of Contents

Peer Influence in Educational Reform
A System Dynamics Approach

Don R. Morris
Miami-Dade County Public Schools
1500 Biscayne Blvd., Ste. 225, Miami, Florida 33132
Tel: 1 305 995 7531 / Fax: 1 305 995 7521
donr.morris@worldnet.att.net

The concept of peer influence in public education is examined in the context of its effect on
student achievement. A system dynamics model based on a positive feedback interpretation
of peer influence has been developed, and applied to gain insight into claims that the concept
can be employed in efforts to raise the academic performance of disadvantaged students.
Aggregated model results are placed in context of achievement data for a large school
district and used to investigate certain of the assumptions of an educational reform that is
currently gaining popularity—the Economic Integration of Schools.

Keywords: Peer influence; Positive feedback; Economic integration of schools;
Educational reform

The past few years have seen a substantial escalation of the debate over the quality and
usefulness of educational research. The ever-increasing pressure on the public schools to
show results has increased the demand for research that will produce policies and
practices that produce those results. The Bush administration and the Congress have
joined in that demand. Last year, Education Week reported that “The phrase . . .
‘scientifically based research’ . . . . appear[s] more than 100 times in the reauthorization
of the Elementary and Secondary Education Act, which requires practices based on
research for everything from the provision of technical assistance to schools to the
selection of anti-drug-abuse programs” (Olson & Viadero, 2002).

In all this clamor for “scientifically based research” it should surprise no one that there is
no mention of feedback or any kind of dynamic analysis. The idea of feedback is not
unknown in educational research, but researchers have by and large given it wide berth.
Some years ago, the authors of a popular text on alternative educational research
methodologies, in advising their readers on the use of causal diagrams, had this to say:
There is usually a temptation to add reciprocal or back-effect arrows [to the diagram]... . We do not
advise such causal flows. . . . They can be modeled by computer (see Gaynor, 1980 and Forrester,

1973, for good examples), but they rapidly bewilder the human brain (“after all, everything affects
everything else”). (Miles & Huberman, 1984, p. 150 [note 7])

This passage tells us that educational researchers are not unaware of the concept of
feedback—some have even heard of system dynamics—but most are far from
comfortable with the idea. Nevertheless, feedback is given lip service (one not
infrequently sees references to “snowballing” and “multipliers”) even while being
sidestepped methodologically.

In this paper I examine an element of student interaction that is often considered
instrumental in the quest for educational improvement, and that is arguably best
understood as a function of feedback. I apply a simple positive feedback model to a
concept both old and (coming around again) “new”—that of peer influence as a means of
raising the academic performance of disadvantaged students—and examine the results in
the context of the achievement data of a large school district. Some of the assumptions of
an up and coming educational reform—the movement for the Educational Integration of
Schools (EIS)—are critiqued in light of the model results.

Peer Influence as a Vehicle for Improving Student Achievement
Achievement, poverty, and the ubiquity of reform

The display in Figure | reveals the relationship between academic achievement and
poverty in a large urban Florida school district over a 10-year period.'! The measures of
achievement are the standardized (z scored) seventh-grade math results by school by
year. The measure of poverty is the percent of students eligible for Free and Reduced-
price Lunch (FRL). FRL is a measure available to every school system, and in common
use by educational researchers both as a measure of the degree of poverty and as an
indicator for SES. The clearly linear shape of the descent of the scores as FRL increases
is striking. This pattern is representative of all grades in the district, and nationally. The
prevalence and persistence of this condition is what the demand for “scientifically based
research” is about.

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School Achievement Averages

Percent FRL

Figure 1 The distribution of seventh grade math scores by school by year for the Miami-Dade school district over the
period 1990-2000. Each symbol indicates an annual aggregate test score for a middle school seventh grade. The
scores have been standardized to combine the results of approximately 48 schools across 10 years and two versions of
the test. Source: Compiled from the District and School Profiles, Office of Educational Planning, Miami-Dade
County Public Schools, Miami, Florida.

There has been an unbroken stream of reform movements over the past 40 years, all
proposing to remedy the situation reflected in Figure 1. Among the most recent of these
reforms is the drive for the Economic Integration of Schools. I single it out here because
it names peer influence as one of the major forces of potential change. A direct
descendant of the movement for racial integration, EIS advocates that all of the nation’s
public school students attend middle class schools. The number of students in districts
that have adopted the approach has grown from approximately 20,000 students in 1999 to
more than 400,000 in 2002 (Kahlenberg, 2002). In a succinct description of the concept,
Weicker & Kahlenberg (2002) write:

Studies find that a child growing up in a poor family has reduced life chances, but attending a school
with large numbers of low-income classmates poses a second, independent strike against him or her.
All students—middle class and poor—perform worse in high-poverty schools. According to
Department of Education statistics, low-income children attending middle-class schools perform
better, on average, than middle. children attending high-poverty schools. . . . virtually all of the
essential features that educators identify as markers of good schools are much more likely to be
found in middle-class than in high-poverty schools. (p. 9)

The key assumption of EIS is that low-income students will improve in academic
achievement when placed in schools where a majority of students are middle class. One
of the proposition’s most ardent proponents, Richard Kahlenberg (2000), states it as an
obligation:
To better promote genuinely equal educational opportunity, every schoolchild in America should
have the right to attend a middle-class school. Using a system of public school choice, school

officials should ensure that in all public schools, a majority of students come from middle-class
households. (p. 1)

A second major assumption, a supplement to the first, is that there is an asymmetry of
effect that will protect middle-class students from adverse reactions so long as they
constitute the majority of the school’s students. That is, adding low-income students to a
middle-class student body will not affect the academic performance of middle-class
students, so long as the school’s FRL percentage does not rise above 50 percent.
Kahlenberg states this flatly: “At the same time [that low-income children are benefiting
academically], middle-class kids are not hurt academically, so long as schools remain
majority middle class” (2000, p. 4).

It is well to note that if and to the extent that EIS realizes its goals, the result must be that
all schools in a district will gravitate toward the district’s mean FRL percentage. The
best that a school district can hope to do in meeting the EIS goals is to have each school
have the same mix of low-income and middle-class students as the district mean. For
many districts, this will mean that all schools in the district will place somewhere near the
middle of the FRL range.

Assuming that they are moved to middle-class schools, how will this improvement in the
performance of low-income students come about? Kahlenberg identifies three variables
as the keys to the success of middle-class schools. Those schools have “more motivated
and well behaved peers, more active and influential parents, and . . . the very best
qualified teachers” (2002). It is the emphasis on peers that is the focus here. Elsewhere
he has elaborated on the role of peers:

Classmates provide students with what has been called a “hidden curriculum.” Children teach each
other things all day long. In high-poverty schools, students have lower aspirations and academic
achievement may be looked down on. Low-income kids are three times as likely to be disruptive
and twice as likely to cut class as middle class kids. . . . By contrast, students in middle-class
schools are much more likely to be exposed to peers with high aspirations (2000, p. 4).

Two of the named major variables, parent support and teacher quality, have received
ample attention. The association of parent activism with SES is well documented. A
recent study has reemphasized the difficulties of poor schools in finding and keeping
experienced and capable teachers (Olson, 2002). These variables, and the more general
topic of reform, have also been addressed within the field of system dynamics (see
Roberts 1974, 1975; Clauset & Gaynor 1984, 1985). The research would appear to be
consistent with the key EIS assumptions stated above. Questions arise, however, with
respect to the third—peer influence. At the elementary level, where peer influence is
subordinate to parent and teacher influences, this may be a negligible concern, but there
is a general consensus that peer influence plays a significant role in the educational
experience of middle and high school students. I now turn to an examination of the
research concerning the effect of peer influence on achievement.

Early optimism

The EIS assumptions reflect early ideas about the positive effects of peer influence. It
has long been known that socioeconomic status (SES) is very strongly associated with
student aspirations and achievement. Studies from the 1950s and 1960s showed also that
students of lower SES have higher grades and are more likely to aspire to college, if they
attend schools where there are large proportions of high-SES students (e.g., Boyle 1966;
Haller & Butterworth 1960; Krauss 1964). In 1965 Campbell and Alexander proposed a
statistical model in which they identified the mechanism of peer influence with the
probability of acquaintance with students who were highly motivated to achieve. They
did not attempt to describe in detail how friendships among students of different
backgrounds came about.

[I]t is necessary to assume only that friendship choices are randomly distributed in the system. As
the average socioeconomic status in a school rises, the more often will individuals at each status
level choose friends of high status—simply because there are proportionately more of them
available to be chosen. We can then explain the observed association between the average status of a
school and the educational aspirations of its students in terms of the intervening variable of
interpersonal influence by an individual's friends. (1977, p. 20)

Campbell and Alexander found that friendships with high-SES students accounted for
virtually all the independent statistical influence of school SES in their study.

In these early aggregate models of peer influence—which focused mainly on high school
students and their college aspirations—the dominance (in a school) of one set of values,
consistent with high achievement and the behavior that supported it, is assumed. There is
a subset of students in the school who are exceptionally high achievers, who personify
these values and set an example with their behavior. The prestige of these high-achieving
students (call them the core) inspires others to emulate them and adopt those values.
While not a great many students are members of this core, the goal of achievement is a
value of all, and low achievement (i.e., failure) is rare. Other students and new students
who come to the school are likely to be drawn into high achievement by contact with this
core of leaders. As long as students come to the school voluntarily as individuals, there
will remain only one dominant set of values, unchallenged among the student body.
Private schools and boarding schools fit the description well.

In the wake of the Coleman Report (1966), peer relationships came to be considered
particularly important as a way of improving achievement, and served as one of the
cornerstones of the rationalization for the racial integration of schools. As Harold Howe,
Commissioner of Education from 1966 to 1968, explained it, one of the conclusions that
belatedly emerged from the Coleman Report’s confusion of findings was that “who one
went to school with was important” (interview from the video Against All Odds, 1989).

In the earlier research, the concept of peer influence had been applied in a context of
young adults and near-adults. After Coleman, with the introduction of racial integration,
the concept was pushed down the grades to apply to children of earlier ages. Whether the
behavior of younger children is so greatly influenced by peer opinions is open to
question. In 1977 Erickson warned that much of what is studied as peer influence may
actually be parent manipulation.

What often appear to be consequences of social relationships in schools could conceivably result
from the tendency of more concerned parents within a given SES stratum to find ways of placing
their children in schools that look superior, schools that are most commonly found in high-SES
neighborhoods. Because of the support and advantages provided by such parents, these same
children will perform at and aspire to higher levels than will their peers (from the same SES strata)
in “inferior” schools. But it will seem that the superior achievement and aspiration result from the
influence of the high-SES youngsters who predominate in the “superior” schools or the “better”
programs found in these schools. To extend the argument, children of exceptionally concerned,
supportive parents may not only be placed in “better” schools, but may seek the friendship of
students in these schools who seem likely to help them do well in their studies. Since the latter
youngsters will be drawn for the most part from high-SES backgrounds, it will appear that the high
attainment of the children of uncommonly supportive parents is a result of friendship links with
high-SES children, whereas its real source is the home. (Erickson, 1977, pp. 6-7)

Behavior and achievement

As the racial integration issue faded, and the difficulties of raising achievement remained,
there were many attempts to reexamine the function of peer influence. There was a
renewed emphasis on behavior, and a new perception of the connection between the
changes in behavior and the interaction with peers. One of these reexaminations was a
reassessment of the middle grades and their characteristics. Middle schools differ sharply
from elementary schools in a number of ways. Departmentalized instruction replaces the
single classroom teacher with a number of subject specialists less likely to be familiar
with individual students. It is a different and unfamiliar environment also for parents,
who are less likely to be acquainted with all their children's teachers and the expectations
of secondary education. The transition to the middle level of education is also marked by
a number of abrupt changes in the students: in achievement and motivation to achieve, in
behavior, and in self-esteem. Anderman and Maehr (1994) have noted that "motivation,
self-concept of ability, and positive attitudes toward school decrease, particularly during
grades six and seven" (p. 288). As parent and teacher influences weaken, peer
relationships become dominant and behavioral problems accelerate (Urdan & Maehr,
1995). Thus middle schools are marked by abrupt changes in the roles and relationships,
a lack of familiarity among teachers, students and parents, and—partly as a consequence
of these things—misbehavior well in excess of elementary school averages, whatever the
FRL percentage.”

Especially since the 1980s, there has been a reemphasis on the relationship between
behavior and achievement in the middle grades, and in the 1990s the characteristics of
middle-school behavior formed the basis of an alternative interpretation of the role of
peer influence. Some research has emphasized the feedback loop of reciprocal causation
among a selected group of variables in producing and sustaining student failure. Straits
(1987), for example, cites several studies which show that "age-grade retardation is a
cumulative or snowballing process" (p. 40). Weishaw and Peng (1993) list a dozen
references of research between 1960 and 1990 that "suggest a reciprocal causal
relationship between achievement and behavior" (p. 5). Kohn (1994) has noted that
"Some [researchers] say that self-esteem and achievement are causally related. . . . [And]
some writers insist that the relationship is reciprocal, with self-esteem and academic
achievement each affecting the other" (p. 275). Kaplan, Peck, and Kaplan (1994)
constructed a structural model and reported that "The causal chain whereby early school
failure leads to feelings of self-rejection in the school environment . . . which in turn
influence disposition to deviance . . . which itself influences academic failure . . . found
strong support in this analysis" (p. 169).

Tying these student-level studies inferring reciprocal causation to group-level peer
interaction patterns is a logical next step. Peer influence became a critical variable in
explaining chronic student under-performance, particularly in the middle grades. Urdan
and Maehr (1995) described the reciprocal interaction of many of the variables related to
academic failure in a dynamic scenario. They wrote:

[A] student that begins to experience failure in school . . . may begin to develop negative attitudes
about schoolwork and exert less effort in school. On the basis of these attitudes, the student may
select a friend with similarly negative feelings and attitudes toward school, and these two students
can reinforce and strengthen each other's negative orientations toward academic achievement. . . .
Over time, these attitudes may lead to sustained underachieving behavior, which in turn might
cause these students to be placed in a low-ability track with other peers who have negative
orientations toward school and school work. In this case, academic failure (an antecedent) leads to
the social goal of seeking approval from a negatively oriented peer, which leads to increased
negativity toward school and even lower achievement (a consequence). This consequence, in turn,
leads to the additional antecedent of being surrounded by negatively oriented peers, and a cyclical
pattern of causes and effects is created. (p. 231)

Here we have the classic peer influence model from the 1950s “in reverse,” so to speak.
Rather than leading to greater achievement, it de-emphasizes achievement while
encouraging other interests. The model identifies one group, one dominant core of
leaders, and one set of values. Membership is voluntary, and by deduction the power of
the attraction to membership is proportional to the size of the core membership with
respect to the entire student body.

Multiple peer groups and more complex interactions

At this point; the similarity to complementary scenarios such as that of Campbell-
Alexander are all too obvious. | Whereas the optimistic Campbell-Alexander
interpretation seeks to utilize peer influence to explain achievement, the pessimistic
Urdan-Maehr version seeks to explain declines in achievement. Peer influence is now
broadened to apply not only to friends of high status, but to friends of low status, friends
who frequently misbehave, and so on. Urdan and Maehr (1995) acknowledged this fact:

Most researchers now assume that peers can have either a negative or a positive influence on
adolescents' attitudes and behavior. In particular, peers can either encourage adolescents to view
their school experiences positively, or encourage them to see school as an uninteresting or hostile
place. The outcomes for any specific adolescent depend on the characteristics of the peers with
whom the adolescent spends most of his or her time. (Berndt & Keefe, quoted in Urdan & Maehr,
1995, p. 220)

Thus there are two core groups—high achievers and counter-achievers—with very
different attitudes and behaviors, occurring together in varying proportions in every
school where peer influence is a dominant force. Peer influence as a factor to take into
account is not expected to appear until the middle grades. The influence of low-income
core groups is expected to be more pronounced in middle school than in high school. Not
only does peer influence emerge there, but the core of the low-income group of students
is most likely to be intact and most vocal during those years. In high school, as Bidwell
and Friedkin (1989) point out, “if a student has strong ties to school friends who
themselves do not value educational attainment, the student may stay in school for a time
to enjoy the friendship, but student and friends alike will probably leave school as soon as
it is practicable to do so” (p. 463).

Like attracts like. Applying simultaneous dual models of discrete groups, as discussed
earlier, we find that intra-actions among members of these groups generate positive
feedback loops that produce changes in the group achievement level, separately within
each group. There is not much useful research on the process of the cross-interactions of
two groups, although there is a literature from the heyday of racial integration on the
difficulties of cross-interaction.

With the idea of two separate groups together, vying for dominance on the basis of core-
prestige attraction, we begin to think of two different cultures or backgrounds that
coexist. Each student has a family and neighborhood where he/she has grown up, learned
to behave, formed basic habits and outlook, and—even during the school year—spends
most of his/her time. When some students are poor (or otherwise set apart) and others are
affluent, those formative habits and behaviors are apt to be mutually antagonistic. There
is an inclination to remain apart, even when thrown together in the same school
environment. Efforts to bring them together may sometimes result in hostility rather than
friendships, as the racial integration experience revealed (e.g., Amir, 1969; Eisenman,
1969). In other cases, special efforts may be required to foster opportunities for
interaction. McPartland (1969) argued that to achieve (racial) interactions, the groups
had to be integrated at the classroom level. Even special efforts may not be enough to
achieve the desired result. One recent report from Chicago (Banchero & Little, 2002)
notes that despite great integrative efforts on the part of school administrators in affluent
Chicago-area schools, the test scores of poor and minority students remained unchanged
even as those of their more affluent peers increased in response to the additional efforts.

Impediments to peer influence can also occur in the form of organizational rules that
make it difficult for individuals to interact. While all these impeding causes—social or
organizational—are different, they all have the same effect. As they increase, the peer
influence process becomes more restricted. That is, restricting individual interaction,
whether by social precedent or by organizational rules, reduces the opportunities for peer
influence to occur, for better or worse.

Given that cross-group interactions may face obstacles, such interactions do occur. I have
found little research in this area of describing the process of cross-group interaction, and
more is needed, but some conclusions follow from common sense and reason. Let us
assume that there are two groups, and that there are no salient obstacles blocking
interaction across the groups. What will govern that interaction—what will it look like?

Recall the process within each group. Each core consists of that sub-group of individuals
that embodies and excels at those values that the group holds in common, and each core
possesses those leadership qualities that encourage others to follow and emulate its
examples. As a consequence, each core attracts an active following from among its own
group over the course of the year. As a core group gains momentum, it not only
increases its own group of active followers, it also detracts from the prestige (the
“attractive force”) of the other group’s core, such that its followers do not find it as
attractive, and may fall away from it, becoming less interested in its values and practices.
This is an indirect cross-interaction effect, reducing the other core’s influence over its
own potential followers.

A direct effect of one group on the values and behavior of the other would seem to
require some considerable dominance. If one group becomes dominant enough (beyond
some threshold, say), an attraction of sorts to the non-core members of the other group
can develop. Even while retaining their own values and preferences, these other-group
members may find themselves persuaded to go along with the majority. These
disenchanted members of the less successful group will not be drawn to embrace the
values of the other group’s core, but they may be persuaded to emulate the behavior of
the other group’s members. The main idea here is that the power of the attraction of a
core leadership is much stronger for the members of their own group, who share their
background and values, than it is for members of another group. For example, students
who are at risk of failure may not be persuaded to excel academically when they find
themselves in an overwhelmingly high achieving school, but they may be impelled to
improve their work sufficiently to avoid what in that environment will be seen as the
stigma of failure. The converse should hold for not-at-risk students who find themselves
in a very anti-achievement environment, and their achievement level should decline. It is
more a case of captivation, perhaps, than attraction?

To summarize, peer influence is most likely to play a major role in determining
achievement, in the secondary grades. In middle school, an adequate level of maturity
(more freedom from parental supervision) coincides with less familiarity with teachers
(more teachers and less time with each), and greater opportunity for broad interaction
with other students (changing classes hourly, for example), to bring peer influences to a
maximum. Peer influence can work either for or against academic achievement. There is
the possibility of competition and uncertainty of outcomes when groups of opposing
values are strongly represented in the same school. Finally, while the within-group peer
influence process appears to be easy and “automatic,” there are many obstacles that can
slow cross-group influence and even bring it to a halt. Among these are not only social
and cultural forces, but organizational and policy factors as well.
Modeling the Effects of Peer Influence

Based on the foregoing discussion, a model of the peer influence concept has been
constructed. A diagram and equation list for the model are given in the appendix. The
narrative will concentrate on elaborating the major aspects of the model.

In the model, an enrollment of 300 is drawn from a neighborhood or community. The
interaction of a pre-determined “social climate” and the random variable Risk
Determination determine whether a student entering the grade will be one who is a
member of the Low Risk or the High Risk group. Each student make his/her way
through the year, ending up among the High Achievers, Average Achievers, Under
Achievers, or Counter Achievers.

The mark of this progression through the year is the Encounter, simulated by a second
random generator. One should think of the Encounter as the determining event in a
cumulative series of experiences that the student has come upon—the turning point or
decision point that results in determining his/her performance level. Every student
accumulates experiences as he/she progresses through the year. The Encounter is
conceptualized as a culmination of these experiences that results in a choice to continue
on with the performance characteristic of his/her group, or to embrace the values of its
most dedicated members. Although the model is sequenced through regular iterations, it
is the Encounter that should be seen—rather than “time” per se—as the major unit. As a
result of this process, the student body will be redistributed by the end of the school year.
This section explains the model process.

Description of the model

The basic structure There are two identical submodels. Each represents a group of
students and their values or orientation. One group consists of students who are oriented
to academic achievement—the Low Risk group. The Low Risk submodel is intended to
correspond to the Campbell-Alexander concept discussed in the literature review. The
other consists of those not oriented to academic achievement—the High Risk group,
which corresponds to the Urdan-Maehr description of low achieving students reinforcing
each other.

Within each group there is a subculture—called the “Core’”—that represents the “values”
predominant for each group, and serves as the source of behaviors that members of the
group presumably desire to imitate. These Cores are initially weighted to be about a
fourth of the size of each group, and are called: for the Low Risk group, High
Achievement, and for the High Risk group, Counter Achievement.

Figure 2 displays the submodel for the Low Risk group.’ When a student is identified as
Not At Risk, then depending on the value of Encounter, he/she may become a High
Achiever, or simply an Average Achiever. If the value of Encounter—a random
variable—is greater than the HiAch Fraction, then that student will join the high
achievers. This will increase the HiAch Fraction, making it slightly more likely that the
next Not At Risk Student will become a High Achiever. The feedback loop is positive.

The flow equation in which this is achieved—for the submodel as diagrammed in Figure
2—is shown in equation (1).
At Risk HiAch Fraction

Exceptional Performar High Achievement

Risk Determinant

OY

Encounter

Community

Not at Risk Student

Satisfactory Performance inseace RBoentabk

Figure 2. The Low Risk sub-group. Students who are not at risk become either high achievers or average achievers in
this simplified diagram, depending on the value of Encounter.

Exceptional Performance = IF (Not_At_Risk_Student=1) AND (0)
(Encounter< HiAch Fraction) THEN 1 ELSE 0

In this equation, the feedback loop depends only on the size of High Achievement
relative to Average Achievement. As such, it is most effective when the sizes of both
stocks are small (say, less than 10). As the structure is applied here, however, the stocks’
contents become relatively large, and in fact are deliberately initialized with large
numbers (a weighting by another factor, Climate, to be discussed later). When the core
fraction (in this case the HiAch Fraction) is small relative to the sum of both stocks, the
outcomes over a model run quickly stabilize. The same is true, of course, for the High
Risk group, where the feedback is through the CntrAch Fraction. The results of a model
run in which the feedback is entirely dependent upon the core fractions alone, as
described for the Low Risk group here, is shown in Figure 4A, subsequent to the
discussion of the attraction functions considered next.

Peer attraction The feedback loops in the model, however, are not dependent on the
Core size alone. The cores (i.e., High Achievment and Counter Achievement), which
exercise influence over their respective groups, are set to be in competition with each
other for strength of influence. This is accomplished by adding a function to the flows of
the core stocks of both submodels that has the effect of increasing or decreasing the core
fraction. For the Low Risk group that function is called Ach Attraction, and the flow
equation is reproduced as equation (2).

Exceptional_Performance = IF (Not_At_Risk_Student=1) AND (2)
(Encounter<Ach_Attraction* HiAch Fraction) THEN 1 ELSE 0

The reader should compare this to equation (1). Unlike the first equation, this one
(identical in every other respect) adds Ach Attraction as the multiplier of HiAch Fraction.
The Ach Attraction can double the size of HiAch Fraction, or reduce it to zero, depending
on the relationship obtaining between the two cores.
Ach Attraction, the function added, is displayed as equation (3), and takes on a range of
values from 0 to +2. If High Achievement is greater than Counter Achievement, the
value of Ach Attraction is greater than 1, and the product of Ach Attraction and HiAch
Fraction is more likely to be greater than Encounter, sending the student to the High
Achievement stock. Conversely, if Counter Achievement is the larger, the value of Ach
Attraction is less than 1, reducing the probability that the student will become a High
Achiever.

Ach_Attraction = 1+(High_Achievement-Counter_Achievement) / (3)
(Counter _Achievement+High Achievement)

The counterpart of HiAch Attraction is called CntrAch Attraction, and has the same
function with respect to the High Risk group. Together they link the two submodels into
one peer influence system. Figure 3 shows this linking of the two submodels. The flows
(Exceptional Performance and Poor Performance) are affected by changes in both cores,
and feed information from both back into the flow equation. When FRL is at or near
50%, the possibility of sudden shifts in the dominance of the cores is highly probable. In
the center of the FRL range, both groups start equal; neither has an advantage. At all
other points, one core has at least a slight initial edge over the other. If High
Achievement is larger, then each addition to it multiplies the effect by some increment

Satisfactory Performance (Al oraee Aahley anant

At Risk

Encounter HiAch Fraction|

Not at Risk Student

Risk Determinant

Exceptional perf ormanc®

High Achiel ema

‘Ach Attraction

ChtrAch Attraction

Community

Auras Counter Achievément

Risk Determinant CntrAch Fraction

At Risk Student

Encounter

At Risk q

Substandard Perfomance Under Achievement

Figure 3 The peer attraction functions. The functions Ach Attraction and CntrAch
Attraction enhance the feedback of their respective “fraction” functions, HiAch
Fraction and CntrAch Fraction.
above unity, and there is a non-linear positive feedback. This advantage is “double-
edged.” Every gain of one core diminishes the position of the other. For the smaller
core, the multiplier is negative, causing the core to decrease more rapidly. Away from
the center (FRL = 50), there is always a smaller core that has a disadvantage to overcome,
and that initial disadvantage increases with distance from the FRL center.

The effect on the behavior of the model is substantial. Panel B of Figure 4 shows the
results of a run with Ach Attraction and CntrAch Attraction added to the model. This
result should be compared with panel 4A, which shows an identical run but without the
Attraction functions.

A B

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oO
oO 50 100 150 200 250 300 0 50 100 150 200 250 300
Encounters Encounters

Figure 4 Model runs without and with the Attraction functions added. Two copies of the model attached to the same
random generators were run simultaneously at an FRL fraction of 0.4 and Climate weighting of 50. The results permit
the comparison of model performance with and without the Ach Attraction and CntrAch Attraction functions. Panel A
shows the result when feedback to the Core functions High Achievement (1) and Counter Achievement (4) is only from
the HiAch Fraction and CntrAch Fraction respectively. There is not a great deal of interaction with the other sub-
groups—Average Achievement (2) and Under Achievement (3), Panel B shows the result when the Attraction
functions are added.

“Crossovers” Finally, there is a way for a student to “cross over” and perform at the
same level as an average member of the other group—though he/she cannot progress to
the other group’s core level. By cross over, | mean that a Not-At-Risk student, for
example, will perform at the level of an At-Risk student, and vice versa. A very
dominant core can attract members of the other group’s non-core to its own group. Ifa
core grows large relative to the whole student body, presumably its influence will
overwhelm even the members of the other group, who will respond by imitating the
behavior of the dominant group’s members. Thus as the core of the High Risk group, for
example, grows very large relative to the whole, some Not-At-Risk students will begin to
perform in the same manner as their At-Risk cousins. That is, their achievement scores
will deteriorate substantially.

In the model, this is effected by rerouting the Not-At-Risk student away from the Low
Risk stocks and to the Under Achievement stock. The equation by which the Not-At-
Risk student “crosses over” from the Low Risk to the High Risk group is equation (4).

Declining_Performance = IF (Not_At_Risk_Student=1) (4)

AND (Exceptional_Performance=0)
AND (Encounter<Convert_to_Hi Risk) THEN 1 ELSE 0
This says that if a Not-At-Risk Student is not directed to High Achievement, then the
student is tested become a member of the High Risk group. If the Counter Achievement
influence is strong enough to persuade him/her to cross over, then the Under
Achievement stock is incremented by one. Otherwise, the student goes to Average
Achievement by default. The process is diagrammed in Figure 5.

High Achievement

>

Exceptional Performance

Not at Risk Styaént

Average Achievernent

>

satfigtify Performance

Total
PetCntrAch

Encoufter

GpM@ert to Hi Risk

‘Community =

Declining Performance

CntrAch Fraction

fro |

der Achiev enfeft

Poor Performance Counter Achievement

Figure 5 Possible destinations for a Not At Risk Student. This simplified diagram shows the paths for a Not At Risk
Student to High Achievement, Under Achievement (as a crossover), or Average Achievement. The inset displays the
graph for the connector Convert to Hi Risk.

How strong is strong enough? The Not-At-Risk student has a probability of becoming an
At-Risk student based on the Counter Achievement sub-group’s size relative to the sum
of the contents of all four stocks. Convert-to-Hi-Risk is a graphic function that increases
at a non-linear rate as the Counter Achievement core’s percent of the entire student
enrollment increases, equaling 0.05 when the core is 20 percent of the enrollment, 0.165
at 50 percent, and 0.595 when the core is 90 percent of the enrollment. The graph of this
function is displayed as an inset of Figure 5. I have no empirical data by which to
estimate the shape of the Convert-to-Hi-Risk function. The choice rests on logic and
plausibility.

Within the crossover process there is a negative feedback effect that works as follows.
When a Not-At-Risk student goes to the Under Achievement stock, the denominator of
the CntrAch Fraction is increased slightly, in turn slightly decreasing the probability that
At-Risk students will become Counter Achievers. This then inhibits the probability that
the Convert-to-At-Risk value will exceed any given Encounter value, and crossovers
become less likely. However, the conditions which promote such Counter Achievement
dominance are most likely to occur when there is a smaller percentage of Not At Risk
students in the system. As a result, the effect of the negative feedback on the behavior of
the model is negligible.

The constants The social characteristics of the school for which peer influence is to be
simulated are determined by three constant values chosen prior to running the model.
Those constants are: the fraction of students who are At Risk (determined by selecting the
FRL of the school); the Core (that initial weighting that will represent the group’s
values—achievement or non-achievement); and Climate (the degree of force from
whatever source that inhibits student interaction and the communication of peer
influence).

First, the proportion of students At Risk is determined as follows. The SES composition
of a school’s student body is a basic determinant of its overall achievement pattern. The
concept of FRL is used in the model to represent SES. As explained earlier, FRL
represents the percent of students in a school who qualify for Free or Reduced-price
Lunch, and it is read as the reverse of SES. That is, where SES is inversely proportional
to low academic performance and poverty, FRL is directly proportional to those
variables.

Since the percent At Risk is assumed to be linearly related to FRL (and convincingly so,
the relationship graphed in Figure 1| is representative of such relationships), I use a linear
equation to derive the proportion of the student body that will be At Risk directly from
FRL. It is:

AT_RISK = a + b*FRL = 0.03 + 0.94*FRL (5)

where FRL and At Risk are fractions rather than percents, and where a and b are
constants that can be changed to fit the information relevant to the system to be modeled.
Here I have chosen to keep At Risk very close to the value of FRL, assigning a value of
0.03 to a and 0.94 to b. This means that when FRL is zero about 3 percent of the student
population will be at risk.

Second, the Core size is estimated to be initialized at one-fourth the group size (where the
groups—named High Risk and Low Risk—are the proportions At Risk and Not At Risk).
I have chosen 0.25 as the core size, constant for this paper. The decision is not arbitrary;
it is based on observations of the example district’s performance over time in the 1980s
and 1990s on a criterion-referenced test for graduation given to go" (later 10") graders,
wherein about 75 percent consistently pass. The reasoning is that the persistent 25
percent failure rate represents an unchanging core of students who are irreconcilably
uninterested in academic achievement. Having no comparable measure with which to
estimate the achievers, I assume that the cores are symmetric and fixed, for the purposes
of this paper.

Climate is the third constant. Among sociologists, school climate usually refers to the
SES makeup alone. I use the term here to name the constant indicating the strength and
degree of rigidity of the structure, both social (ethnic preference and the like) and
organizational (rules and policies). In the model it is a number to be chosen by the
modeler. Choose a smaller number and the model behavior is more volatile (simulating
greater ability of students to interact with each other). It may be assumed to represent
environmental conditions, a program or policy, or a combination of the two. Choose a
larger number and there is less deviation from the initial pre-determined conditions as the
run progresses (students have less ability/interest in interacting, and are consequently less
affected by peer influences).

A combination of these constants determines the distribution of initial values or
weightings among the four stocks (High Achievement, Average Achievement, Under
Achievement, Counter Achievement), according to the sub-groups they represent. If, for
example, a school starts the school-year (run) with an FRL fraction of 0.4, it exists in an
environment that is made up of more than half Not At Risk Students. This in turn will
constitute a slight bias in favor of an exponential growth in the Low Risk Core, which
with a value of 0.25 has a potential size of one-fourth the size of the Low Risk group.
However, a Climate weighting of 50 will introduce a resistance to the feedback and
restrain growth of the core that gains dominance .

In this example, the initialization of the High Achievement stock is given by equation (6):

INIT High_Achievement = (1-AT_RISK) *Core*Climate (6)
= 0.59*0.25450 = 7.4

The other stock in the Low Risk group, Average Achievement, is initialized by equation
(7):

INIT Average_Achievement = (1-AT_RISK) *(1-Core) *Climate (7)
= 0.59*0.75*50 = 22.3

For the High Risk group, the initial values of Counter Achievement and Under
Achievement are 5.1 and 15.2, respectively. Even at a modest 10 percent below the
halfway point in the FRL range, the initial ratio of the cores High Achievement to
Counter Achievement is about 7 to 5.

Example runs Figure 6 shows examples of model performance, with each sub-group
shown as a percent of the whole, changing with respect to the others as the run
progresses. The columns of panels on the left (A, C, E) display the results of runs with a
Climate weighting of 50, and the column on the right (Panels B, D, and F) shows results
with a Climate weighting of 100. The runs on the left show a great deal more variation
from beginning to end than do those on the right.

The rows represent runs at different FRL percentages. In the top and bottom rows, the
FRL percent is at an extreme end of the range and one or the other of the Cores quickly
dominates, forcing the other Core to near zero. At the top, with an FRL percent of 0.1,
the High Achievement Core dominates, climbing from around 20% to 70% over the
course of the run. In the bottom row, it is the Counter Achievement Core that dominates.
Note that in all cases, any gain in a Core is at the expense of its own non-core sub-group,
and that the losing Core’s non-core sub-group gains modestly in percentage as its Core
falls.

In the center row of the Figure, the Cores begin equally matched. In the example shown
the Counter Achievement Core gains dominance, but increases to less than 40% in Panel
C, and in Panel D—where the Climate weighting is higher, it finishes at less than 30%.
Two points should be noted about the model’s performance at the FRL midpoint. First,
both Cores have the same opportunity to gain dominance. Second, the maximum that

either is likely to attain will be less than 50%.

A B
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0 50 100 150 200 250 300 0 50 100 150 200 250 300
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Encounters Encounters

Figure 6 Example runs of the peer influence model. Each left-right pair of panels represent simultaneous runs by two
models joined to the same random variables (Risk Determination and Encounter), and representing two different Climate
weightings, 50 on the left and 100 on the right. The runs at the top row are at FRL=0.1, at the middle row, FRL=0.5, and at
the bottom row, FRL=0.9. The lines represent the percent of the total sum across the four Achievement stocks constituted
by (1) High Achievement, (2) Average Achievement, (3) Under Achievement, and (4) Counter Achievement.
Placing the model in a district context

To place the Peer Influence model in the context of the EIS reform, it will be necessary to
examine it from the perspective of an analysis aggregated to the district level. The
procedure by which this is done is described next.

Expressing model results as a school’s “achievement score” If the model is a
reasonable representation of the actual functioning of peer influence in the middle grades,
then there should be some reflection of the model’s outcomes in the behavior of actual
systems. With some manipulation it is possible to compare the model behavior with
empirical data relevant to questions raised concerning the EIS reform. Consider first that
the peer influence model produces results for one grade in one school, for one year. A
district is made up of many schools, which produce outcomes annually over a specified
time period.

Looking back to Figure 1, the data displayed in that graph consists of 480 data points
from approximately 48 middle school a grades (the number varied over the years) over
a 10 year period. The poverty levels of the schools varied over most of the FRL range.
The graph presented there is reproduced in Figure 7 as panel A. The data points plotted
there are school-level achievement test scores.

Zscores

0 10 20 30 40 50 60 70 80 90 1000 10 20 30 40 50 60 70 80 90 100
PercentFRL PercentFRL

Figure 7 Seventh grade test scores by school by year, paired with peer influence model results. Panel A is a
reproduction of the MDCPS seventh grade scores from Figure 1. Panel B displays the results of model runs produced
by assigning the model the identical FRL percentages as each of the school scores in the data set displayed at left.

For comparison, runs from the model have been assembled in a similar manner. We can
conceive of any student body as made up of the groups and sub-groups described in the
Model section. We can further assume that the students in each sub-group will
collectively average a score within a given percentile range on some hypothesized
achievement test. First, I assigned each model sub-group an average percentile score.
Students in the High Achievement sub-group were assumed to deliver an average
between 85 and 95 percentile points. Next comes the Average Achievement sub-group,
scoring between 55 and 65. The Under Achievement sub-group was presumed to score
persistently below the 50" percentile, and was assigned the range 35-45. Finally, the
Counter Achievement sub-group was assumed to average between 5 and 15 percentile
points.
I then ran the model 480 times, each time assigning to the run an FRL value from the
original data set. The value of Climate was held constant at 50. The result was a set of
values for each model sub-group for each run, based on the poverty level (FRL
percentage) given to the school.

Next, each of the sub-groups, in each of the 480 runs of the model, was given a randomly
selected score within its assigned percentile range, and those scores summed across all
sub-groups to yield a “school-level test score” for each of the 480 runs. These “test
scores” were then divided by “year” in the same manner as the original data set, and z-
scores computed.

The results of these efforts are displayed in Figure 7B. A comparison with panel A
demonstrates the strong similarity of outcomes. The plot from the model run is much
tighter, of course, reflecting the fact that there is only one variable acting on the model
plots, peer influence. There are an unknown number of variables affecting the data plot,
one of which is assumed to be peer influence.

Model outcomes and data patterns The model results are now directly comparable to the
data in Figure 7A, and similar data sets, provided it is clear what is being compared. The
logic of the model and experience with it indicate that outcomes—in particular which
group will emerge dominant at the end of the year—become increasingly unpredictable in
the middle of the range, as FRL approaches 50%. This of course follows from the fact
that the groups are more evenly matched in that area.

This suggests that the variation in model outcomes should change in a predictable way
across the range of FRL. In the model-generated data of Figure 7B, the “test scores”
should show more variability from one run to the next in the middle of the FRL range
than at the extremes, and—since we know that our simulated peer influence is the
cause—there is no ambiguity in the interpretation of the resulting pattern.

One way to examine that variability is to take measurements of the variability of the
school score—standard deviations, say—across small increments of the FRL range, and
then observe the pattern that they form across that range. The expectation is that there
will be less variation near the ends of the range, and more in the center.

To uncover the pattern, I divided each data set into 5 percentage point intervals of the
FRL range and found the standard deviation for the group of data points within each
interval.” These results are shown in panel A of Figure 4. The line emphasizing the
patterns is a scatterplot smooth produced by a local regression program called lowess (see
Cleveland, 1979).

If the model is a reasonable interpretation of peer influence, then we expect that pattern to
be reproduced in similar empirical data sets where peer influence is a major variable.
From the survey of the research and principles developed earlier, we expect peer
influence to be at a maximum in the middle grades, and to be overshadowed by parent
and teacher influences at earlier grades. Let us now examine the test data from the
MDCPS seventh grade more closely, and add to that data a comparable data set from an
elementary grade, in the present case, ten years of math test data from elementary 6th
grade.° Standard deviations across the FRL range for the 7th grade data set and from the
1989-1999 annual test scores of elementary 6" grades (N = 485) were extracted using the
methods described above. The results are shown in panels B and C of Figure 8,
respectively.

Standard Deviation

0 1 20 3 40 50 60 70 0 9 19000 10 20 30 40 50 60 70 80 90 1000 10 20 30 40 50 60 70 80 90 100
Percent FRL Porcent FRL Percent FRL

Figure 8 Patterns of variation of scores across the FRL range in three data sets. Panel A shows the variation for the
model runs shown in Figure 7B. Panel B shows the variation for the seventh grade data plotted in Figure 7A. Panel C
shows the variation in a data set of elementary sixth grade test scores. The scatterplot smooths tracing out the patterns
are lowess lines (Cleveland, 1979).

One would expect the strength of peer influence to be inconsequential in 6" and fully
developed in the al grade. If this is the case, then the pattern of variation found in the
model results graphed in Figure 8 should be clearly apparent in the 7 grade data and
absent in the elementary oe grade data.

The patterns observed in Figure 8 are consistent with the theory, and with what one
would expect given the known characteristics of the model. The 7 grade pattern is very
similar to the model pattern, while there is no hint of the expected shape in the 6" grade
graph.

The pattern associated with peer influence is not unique, and so a match of patterns will
not confirm the presence of peer influence. But while it is not unique, the match of the
pattern in the data to the pattern predicted by the model results—particularly as
augmented by the absence of a pattern where that was predicted—adds credibility and
support both the presence of peer influence in the data and the ability of the model to
reflect it.

Patterns of model results across FRL A quantitative summary of the results of the 480
runs of the model, which were graphed in the preceding section, is given in Table 1. The
information will be used in the ensuing discussion of the application of the model results
to the EIS reform. The table displays the patterns resulting from runs of the model
produced as described in the previous section, at Climate settings of 50, 100, and 500. In
the table, the initial values have been subtracted from the model outcomes, so that the
averages are based only on the distribution of the 300 “students.” Results are presented
in intervals across the FRL range, for each of the four model subgroups—that is, the final
“student distributions” are shown, and how they change across the FRL range. The
results are given as averages at the FRL intervals.

In addition, the crossovers have been separated out so that their average quantities and
patterns may be observed. A copy of the model was modified to send them to stocks
created for the purpose. For this reason, in the table, the average size of the crossovers is
given by the numbers to the right of the plus sign in the columns showing the averages of
Table 1

Average Model Outcomes by FRL Intervals

FRL | No.i ' High Average Achievement | Under Achievement Counter
lo.in | Climate f b ‘
Avg? | Interval | weighting Achievement plus crossovers plus crossovers. Achievement
Avg [| StD Avg [std Avg [stb | Avg [ StD
50 233.8 13.2 21.7 + 10.0 83 340+ 0.0 8.7 0.5 0.6
12% 43 100 219.1 148 365 + 63 109 370+ 0.0 9.6 12 14
500 140.4 136 1202 + 04 116 367+ 0.0 7.0 2.3 1.7
Initial® 64.3 192.8 32.2 10.7
50 208.5 170 246 + 100 115 560+ 0.0 9.1 0.9 1.0
20% 27 100 187.0 21.7 461 + 57 178 594+ 0.0 8.4 1.8 1.3
500 115.0 149 1183 + 03 122 619+ 0.0 9.2 46 3.0
Initial® 58.6 175.7 49.3 16.4
50 157.3 368 495 + 68 296 805+ 0.0 8.9 5.9 6.3
30% 30 100 134.7 335 721 + 33 27.7 829+ 0.0 79 7.0 5.6
500 91.2 132 1175 + O00 111 795+ 00 109 11.8 48
Initial® 51.6 154.7 70.3 23.4
50 119.3 465 580 + 55 41.1 1054+ 0.0 175 11.7 158
40% 46 100 93.3 319 84.0 + 1.1 28.3 107.7 + 0.0 15.1 13.9 10.9
500 59.6 93 1175 + 0.0 9.7 1006 + 0.0 88 22.3 64
Initial® 44.5 133.5 91-5) 30.5
50 49.7 403 982 + O07 36.1 1033 + 08 356 473 385
50% 61 100 41.8 228 1068 + 00 206 1118+ 01 207 396 21.8
500 37.0 10.2 1154 + 0.0 9.9 110.1 + 0.0 8.6 37.4 10.1
Initial® 37.2 AAI 113.3 37.8
50 11.1 138 104.7 + 0.00 142 638+ 50 403 1154 466
60% 86 100 13.4 9.0 106.0 + 0.0 109 873+ 14 295 91.9 33.3
500 20.5 62 984 + 0.0 84 1197+ 00 115 614 126
Initial® 30.2 90.5 134.5 44.8
50 49 85 80.7 + 0.0 124 452+ 81 33.5 161.1 39.7
70% 91 100 64 59 838 + 00 109 702+ 35 278 136.1 34.1
500 11.4 48 82.2 + 0.0 8.9 123.3 + 0.0 123 83.1 15.7
Initial® 23.4 70.1 154.9 51.6
50 ETA 21 54.1 + 0.0 98 273 + 95 15.9 207.5 22.2
80% 95 100 24 23 574 + 0.0 9.7 489 + 52 21.8 185.9 26.8
500 46 26 598 + 0.0 9.9 117.1 + 03 11.9 118.2 15.1
Initial® 16.2 48.7 176.3 58.8
50 0.5 08 36.1 + 0.0 8.0 218 + 85 18.2 233.0 21.7
88% 31 100 0.8 10 389 + 0.0 8.2 390+ 54 19.1 215.9 23.0
500 22 15 416 + 0.0 77° 116.2 + 1.5 134 138.6 16.3
Initial® 11.0 32.9 192.1 64.0

‘Model runs were generated using the FRL values of the data for the MDCPS 7” grades displayed in Figure 1. These results have
been grouped into intervals of 10 percentage points each over the FRL range from 15 to 85 percent. The first and last intervals
span 7-15% and 85-91%, respectively. The FRL percents are the averages across the intervals rounded to the nearest integer,
and the number of values in each interval is given in the column immediately to the right.

*The average in this column displays the average number of original non-core members in the group (on the left of the plus sign),
and the average number of non-core members of the other group converted to the group's non-core—the crossovers (on the right).

The sum of the two is the total number in the group's non-core at the end of the run.

“The term initial here refers not to the values used to initialize the stocks, but to the initial distributions reflected in those initial
values, adjusted to sum to 300 across the four “Achievement” stocks. As such, they represent the outcomes that would have
resulted had the model runs had no effect. The distribution is the same regardless of the climate weighting.

the non-core sub-groups to which they have “crossed over.”

To summarize the table contents briefly, the cores form non-linear increasing patterns in
opposite directions. The non-cores vary inversely with the cores. Of equal importance
are the standard deviations of the averages; they grow larger in the center of the FRL
range. These measures make the volatility of the middle of the FRL range evident in the
table. The variability is exceptionally high at the lowest climate value, at which value
student interaction is maximized.

In fact, all the results shown in the table are exceptionally sensitive to Climate—perhaps
most notably the crossovers. Never large in quantity, the crossovers overlap the FRL
midpoint only at low Climate values, and almost disappear altogether beyond the Climate
setting of 100.

This summary gives an indication of the information available for reference as the model
results are applied to questions arising from the proposals for the Economic Integration of
Schools reform. The interested reader is invited to peruse the table as desired.

Discussion
Implications of the model for the Economic Integration Reform

At the secondary educational level, influence over student behavior shifts away from
parents (the home environment) and more toward the social environment. There, peer
influence is but one of several variables affecting achievement that is thought to be
subject to manipulation, but it may be one of the more important for a middle-school
strategy to raise achievement.

The question before us here is this: Is it possible to apply a knowledge of peer influence
in the service of an ideology-driven reform—the Economic Integration of Schools—in
such a way that the resources that school districts currently possess can be effectively
used? If we assume that the model has adequately captured at least the gross behavior of
the peer influence phenomenon, then we may draw inferences from it concerning the
reform.

Results from the model appear to support the EIS assumptions. Examination of the
model results indicates that where FRL is above 60 percent, there is not a lot that can be
done to raise achievement. Even the students who are not at risk have little incentive to
improve their performance, and some may succumb to the pressures of conformity to the
dominant Counter-Achievement values. Below 40 percent FRL, high achievement tends
to increase whatever the policies or resources brought to bear. At first glance, then, the
case for EIS appears to be made, although concerns about transportation and
neighborhood integrity remain, unresolved since the days of racial integration (see for
example Lamm 2002). The model indicates the obvious—there is little chance for
improving achievement at high FRL, and it is practically guaranteed at low FRL. It
would seem to follow that moving poor students to middle class schools would produce
the desired results.

However, the role of peer influence in promoting achievement is more complex than it at
first appears. An examination of the details indicates that there are difficulties and
complications that give pause to an uncritical endorsement of an EIS policy. Three issues
are examined from the standpoint of the model’s implications: 1) the claim to help poor
students achieve; 2) the claim that middle class students are not harmed in the process;
and 3) the logical ramifications of an effectively applied EIS policy.

What about the claim that lower FRL helps poor students? There are two distinct ways
in which a lower FRL can help to boost the achievement of poor students through peer
influence. First, economic integration alone—that is, reducing the FRL percentage—can
increase a High Risk group’s average achievement level, simply by reducing the
influence of the group’s own peer attraction. The weaker the attraction the fewer Under
Achieving students will be lured into the even lower achievement performance of the
Counter Achievement core. So even modest improvements in FRL status will show an
improvement in the school’s average score.

But what the EIS people really want and expect is crossover—the conversion of At Risk
Students into Not At Risk Students. Ifthe achievement environment is strong enough—
that is, if the core of high-achieving students is large enough—then some of the High
Risk group of students will leave the values of their own group behind and begin
behaving in the manner of their fellows who are not at risk of failure. However, if the
reasoning incorporated in the model is right, this will require much greater reductions in
FRL than simple majority middle class, even if the actual Convert-to-Low-Risk curve is
more generous than that used in the model. As presently constructed, for example, at the
lowest applied Climate weighting of 50, the average percentage of At Risk Students
diverted to Average Achievement at 12% FRL is 22.5 (from Table 1,
10.0/(10.0+34.0+0.5)*100=22.5). Not much, considering that the size of the whole High
Risk group at that FRL percentage is only about 15 percent of the total enrollment. And
the percent converted drops off quickly from there: 14.9% of the Under Achievement
group at 20% FRL, 7.3% at 30% FRL, and 4.5% at 40% FRL.

The numbers crossing over also decrease rapidly as the Climate weighting increases. The
core averages just do not get big enough when the climate value is large. These results
are certain to focus attention on the Convert-to-Low-Risk function that I have chosen.
The curve is not tied to empirical data (i.e., it is an “educated guess”), and it may be of
interest as a point of further research, since the crossover activity is sensitive to its shape.
However, if the concept is itself correct, it is unlikely that the number of crossovers will
increase to any great extent in the middle range of FRL where most of the schools are
likely to gravitate under an aggressive EIS policy, however generously drawn the
conversion curve.

What about the claim that EIS will not interfere with the educational progress of middle
class students? The model indicates that there is symmetry. The converse of what
happens to the High Risk group will happen to the Low Risk group. Raising the FRL
percentage even modestly—and this is what must follow from moving poor students to
middle class schools—has implications for all students. Moderate increases to a low FRL
percentage lessens the attraction of the High Achievement core, resulting in fewer middle
class students becoming High Achievers (though there will be little chance that any will
become Under Achievers). Lowering the FRL percentage down into the middle of the
FRL range, say below 40 percent, will create a situation where a dominant Counter
Achievement core could be a frequent result, further reducing the High Achievement core
and having a strong effect on the school’s average test scores, assuming of course that an
effective staff and/or programs do not counter these effects.

What are the likely implications of a widely adopted and successfully applied EIS
reform? To maximize an EIS policy is to equalize all schools in a district at the same
FRL percentage. For any district, that equal percentage will be the district mean. The
problem is that the at-risk portion of the population can be so large that a direct and
problem-free policy of economic integration is not feasible. The state of Florida, for
example, had in the school year 2001-2002 an average middle school FRL percentage of
45.9 % spread over 67 school districts (Florida Department of Education, 2003). Of
these 67 districts, only 18 (27%) had mean middle-school FRL percentages under 40
percent. Twelve (18%) had FRL percentages of over 60 percent. For these 12 districts,
EIS is not a practical solution.

The majority of districts had an FRL percentage falling between 40 and 60 percent.
Twenty-two districts (33%) had mean middle school FRL percentages in the 40 to 50
percent range, and 15 (22%) had FRL percentages between 50 and 60 FRL percent.
Together they constituted a majority of the state’s school districts in 2001-02. These
districts fall in the volatile middle of the FRL range, where the Low Risk and High Risk
groups are relatively evenly matched.

Within this “central zone” of 40 to 60 percent FRL, certain conditions prevail. For
crossover, one of the ironies is that near 50% FRL, neither core gets big enough to attract
students from the other group. Both peer influence cores are smaller, grow more slowly,
and finish the year smaller, than does a dominant core farther from the center.
Consequently, there will be fewer High Achiever middle class students, and fewer
crossovers, than at lower FRL percentages. For this reason, peer influence alone (and
thus moving students around to equalize the poverty level in order to manipulate this
variable) cannot begin to resolve the low achievement problem. It can only set some
conditions.

The model indicates, contrary to the more optimistic views, that to ensure success and
avoid problems, the integration must be of small numbers of at risk students into solidly
middle-class schools—preferably maintaining an FRL percentage of under 40%. In the
less affluent schools of the 40 to 60 FRL range, one can strive to increase the probability
of occurrence of High Achievers, driving down the Counter-Achievers. This will ensure
that the At Risk students—while still performing below average—will be amenable to
remedial programs and good teaching. The kinds of things that good administrators and
good teachers know how to do can offer a constructive challenge here, and with hope of
substantial improvement. The schools should also have a greater percentage of more
active parents than would have been the case in those schools that were previously
constituted of higher FRL percentages. On the other hand, there will not be as many high
achievers as there would have been in the schools that were previously constituted of
lower FRL percentages; this is the price paid by the middle class students. Success will
depend more than ever on skilled administration and teachers, and on carefully crafted
programs that work for all students.

In sum, the condition in which all schools fall in the 40-60 FRL range is a likely result of
the reform. Confined to this range, the reform will be a compromise for everyone most
of the time. However, the problem of under-achieving peer influence should be more
manageable, and it is after all only one of the key variables. Much will depend on staff
quality and the ability to mobilize parents—elements that should also improve with the
reform. The challenge just might lead to better educational skills and a more satisfying
experience for all.

Educational research and system dynamics

I began this paper by noting a renewed interest on the part of policy makers in
“scientifically based research” for the creation of better school policies and programs.
The question at this point is, Where in the new scheme of educational research does a
paper such as this one fit? I answer at three levels of generality.

At the most basic level, that of direct application to a particular problem, the answer is
straightforward. Taken altogether, the peer influence model seeks to fill in a part of the
“theory of change toward higher achievement” underlying the program of economic
integration described by Kahlenberg and others, making the reform’s theory more explicit
and reducing ambiguity. The model permits the identification of weaknesses in the
original assumptions as described, suggests new strategies, and poses new questions to be
evaluated by more conventional methods.

The next level of generality is that of categories within the field of educational research.
System dynamics is not common enough in educational research to have its own niche,
but it fits best in a general category called “Theories of Change,” which in its application
to program evaluation is known as program theory. Theories of Change uses a chain of
causal events, designated as the theory, to identify the mechanisms by which events are
predicted to occur. A congruence between the theoretical predictions and the actual data
outcomes is assumed to validate the theory, to be detected by an observed consistency or
pattern match between theory and reality over a series of events.

I suggest that this category also encompasses system dynamics. Theories of Change is
well suited to (almost “made for’) the application of system dynamics models, although
there has been almost no use made of them (McClintock, 1990, is the only exception of
which I am aware). There is, however, a nascent awareness of the need for the type of
system analysis that system dynamics has pioneered, as program theory researcher
Patricia Rogers (2000) has indicated.’ For these reasons, I place the present paper in this
category.

Finally, there is the most general level, that of the field of educational research itself. As
I noted earlier, assessments of that research have recently become more stringent. The
Department of Education, some members of Congress, and the Bush administration all
favor hard quantitative research, meaning that all “softer” methods are considered
subordinate to the randomized experiment and closely associated approaches. A well
reasoned and cogent superiority-but-not-exclusiveness argument in favor of the
randomized experiment has been advanced by veteran researcher Thomas Cook (2002),
who argues that softer methods are “valuable ” and “serious forms of research,” but only
as subordinate complements to the randomized experiment.

In the course of his argument, Cook explicitly recognizes the importance of the Theories
of Change approach. “Few advocates of experimentation,” he writes, “will argue against
the greater use of substantive theory to guide measurement and analysis in experimental
evaluations” (p. 194). He does not, however, accept the approach’s claim to the ability to
stand alone. Citing a list of perceived weaknesses,* he concludes that “theory-based
evaluations are useful complements to randomized experiments but not alternatives to
them” (p. 195). System dynamics then, insofar as it falls within the Theories of Change
category, shares this designation. Consequently, as a piece of educational research, I
submit that the present paper falls into the role of “useful complement to randomized
experiments.”
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Notes

1. Although the unit (the aggregate test percentile for the school) was the same across all years, the
standard deviations by year and the version of the test (the Stanford Achievement Math Comprehension
Test) varied through the years. The conversion to z scores was made for this reason.

2. Coincident with the emergence of a stronger role for peer influence, test scores also drop substantially in
middle school, across the whole of the FRL range. For example, in the Miami-Dade district in 1999, 48
middle schools received into their 6" grades the students who had graduated from 5" grade the year before.
Of the 48, 21 had 6" grade median percentile math scores in 1999 that were lower than the 1998 median
percentiles of the 5" grades of any of the elementary schools contributing students to them. Another 12
scored lower than all save one of their contributing 5" grades. In one feeder pattern, the middle school’s
test average was lower than that of its poorest contributing elementary. (Compiled from data published by
the Office of Educational Planning, 2000.)

3. The Webster’s New World Dictionary distinguishes between attraction, which according to that source
“implies the exertion of a force such as magnetism to draw a person or thing and connotes susceptibility in
the thing drawn;” and captivation, which “implies a capturing of the attention or affection, but suggests a
light, passing influence.” GB" Edition, 1994).

4. Lam indebted to Hannon and Ruth (1994, Chapter 6), for the initial inspiration for this model structure.

5. There are more elegant ways of estimating local variance across a range. One is suggested by the work
of Efron and Tibshirani (1991). My experience in the present instance, however, particularly since the Ns
are large, indicates that the results do not differ greatly from simpler methods.

6. Although the majority of middle schools in the Miami-Dade district are of the 6-8 configuration, the
district has moved fairly recently from the 7-9 junior high school configuration, and many elementary
schools retain a sixth grade. To confuse matters more, there is a recent move toward adopting a K-8
configuration.

7. In an overview of causal models in program theory, Rogers has written that “causal models are at the
heart of program theory evaluation, yet there has been surprisingly little discussion of the different types of
causal relationships that might be useful for program evaluation” (p. 47). She acknowledges that causality
is complex, and that the simple causal chains in PTE theories are usually gross oversimplifications. She is
further aware of the possibility that the relationship between cause and effect is not linear, and notes that
feedback loops are rarely if ever included in the program logic. However, although she writes that a few
“causal models from systems theory . . . appear to be potentially useful for program theory” (p. 52), her
sole source of reference outside her own field is Senge’s Fifth Discipline. The conclusion must be that (1)
system dynamics would make a much needed contribution to this field, but (2) it is as yet little known
there.

8. Cook’s major objection is that there is no counterfactual. He points out that there may be multiple
theories that fit the patterns to be matched, and so causality cannot be convincingly established. He does
recognize Scriven’s 1976 concept of signed causes that create a pattern so unique that cause cannot be
mistakenly attributed, but he dismisses the method on the grounds that the requirements are too difficult to
be met in practice. However, anyone familiar with system dynamics knows that unique patterns are not
uncommon in even the simpler models. I have argued elsewhere (Morris, 2001) that it is possible to
establish cause based partly on Scriven’s reasoning, using a system dynamics approach, and I included an
example in which unique patterns are matched.
Appendix: The Peer Influence Model

poor Patrmance ) "ten Mace ©

Petcare ‘~ _@)

= . sch Fact pet cal
Ds pita eat LR
they Petdmance Paihia Ach Aan

‘Average Achievement

Rak High Achievement Courter Aehievement

Drqeage Aah Count

iach Fraction High Ah Count

Average Achievement (t) = Average Achievement (t - dt) + (Satisfactory Performance +
Improving Performance) * dt

INIT Average Achievement = (1-Core)*(1-AT_RISK)*Climate

Satisfactory Performance = IF (Not_At_Risk_Student=1) AND (Exceptional _Performance=0) AND
(Declining Performance=0) THEN 1 ELSE 0

Improving Performance = IF(At_Risk_Student=1) AND (Poor _Performance=0) AND
(Encounter<Convert_to_Lo_Risk) THEN 1 ELSE 0

Community (t) = Community(t - dt) + (- Poor_Performance - Substandard Performance -
Satisfactory Performance - Exceptional_Performance - Declining Performance -
Improving Performance) * dt

INIT Community = 300

Poor Performance = IF (At_Risk_Student=1) AND
(Encounter<CntrAch_Attraction*CntrAch Fraction) THEN 1 ELSE 0

Substandard Performance = IF (At_Risk Student=1) AND (Poor _Performance=0) AND
(Improving _Performance=0) THEN 1 ELSE 0

Satisfactory Performance = IF (Not_At_Risk_Student=1) AND (Exceptional _Performance=0) AND
(Declining _Performance=0) THEN 1 ELSE 0

Exceptional Performance = IF (Not_At_Risk_Student=1) AND
(Encounter<Ach_Attraction*HiAch Fraction) THEN 1 ELSE 0

Declining Performance = IF (Not_At_Risk Student=1) AND (Exceptional_Performance=0) AND
(Encounter<Convert_to_Hi Risk) THEN 1 ELSE 0

Improving Performance = IF(At_Risk Student=1) AND (Poor_Performance=0) AND
(Encounter<Convert_to_Lo_Risk) THEN 1 ELSE 0

Counter Achievement (t) = Counter_Achievement(t - dt) + (Poor_Performance) * dt
INIT Counter Achievement = Core*At_Risk*Climate

Poor Performance = IF (At_Risk_Student=1) AND
(Encounter<CntrAch_Attraction*CntrAch Fraction) THEN 1 ELSE 0

High Achievement (t) = High Achievement (t - dt) + (Exceptional_Performance) * dt

INIT High Achievement = Core*(1-AT_RISK) *Climate
Exceptional Performance = IF (Not_At_Risk_Student=1) AND
(Encounter<Ach_Attraction*HiAch Fraction) THEN 1 ELSE 0

Under_Achievement (t) = Under_Achievement (t - dt) + (Substandard Performance +
Declining Performance) * dt

INIT Under Achievement = (1-Core)*At_Risk*Climate

Substandard Performance = IF (At_Risk_Student=1) AND (Poor_Performance=0) AND
(Improving _Performance=0) THEN 1 ELSE 0

Declining Performance = IF (Not_At_Risk Student=1) AND (Exceptional_Performance=0) AND
(Encounter<Convert_to_Hi Risk) THEN 1 ELSE 0

a = 0.03

Ach Attraction = 1+(High_Achievement-Counter Achievement) /
(Counter _Achievement+High Achievement

At_Risk = a+b*FRL

At_Risk Student = IF (Risk_Determinant<=At_Risk) THEN 1 ELSE 0
Average_Ach Count = Average Achievement - INIT (Average Achievement
b = 0.94

Climate = 50

CntrAch_Attraction = 1+(Counter_Achievement-High Achievement) /
(Counter _Achievement+High Achievement

CntrAch_Fraction = Counter_Achievement/(Counter_Achievement+Under_Achievement
Core = 0.25

Counter_Ach_Count = Under_Achievement- INIT (Under_Achievement.

Encounter = RANDOM(0,1)

FRL = 0

HiAch Fraction = High Achievement/(High_Achievement+Average Achievement
High Ach Count = High Achievement - INIT(High_ Achievement

Not_At_Risk_ Student = IF (Risk Determinant>At_Risk) THEN 1 ELSE 0

PctAvAch = Average _Achievement/(Total+1)*100

PctCntrAch = Counter _Achievement/(Total+1)*100

PctHiAch = High Achievement/(Total+1)*100

PctSubAch = Under_Achievement/(Total+1)*100

Pct_HiRisk = PctSubAch+PctCntrAch

Pct_LoRisk = PctAvAch+PctHiAch

Risk Determinant = RANDOM(0,1)

Total = Under_Achievement+Counter_Achievement+Average_Achievement+High Achievement
Under_Ach Count = Counter_Achievement- INIT (Counter Achievement

Convert_to_Hi Risk = GRAPH (PctCntrAch)
(0.00, 0.015), (10.0, 0.03), (20.0, 0.05), (30.0, 0.08), (40.0, 0.125), (50.0, 0.165)
(60.0, 0.225), (70.0, 0.305), (80.0, 0.41), (90.0, 0.595), (100, 0.935

Convert_to_Lo_Risk = GRAPH (PctHiAch
(0.00, 0.015), (10.0, 0.03), (20.0, 0.05), (30.0, 0.08), (40.0, 0.125), (50.0, 0.165)
(60.0, 0.225), (70.0, 0.305), (80.0, 0.41), (90.0, 0.595), (100, 0.935

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