The Social Dynamics of
Mathematics Course-Taking for
Boys and Girls
Kenneth A. Frank
Michigan State University
Chandra Muller
University of Texas
Kathryn Schiller
SUNY Albany
with
Robert Crosnoe
Sam Field
Amy Langenkamp
Jennifer Pearson
Kelly Raley
Catherine Riegle-Crumb
Anna Strassmann Mueller
University of Texas
Support
• *This work was supported by grants from the National
Institute of Child Health and Human Development
(1R01HD040428-01) from the National Science
Foundation (REC-0126167) to the Population Research
Center, University of Texas at Austin. Additionally, this
research uses data from Add Health, a program project
designed by J. Richard Udry, Peter S. Bearman, and
Kathleen Mullan Harris, and funded by a grant P01HD31921 from the National Institute of Child Health and
Human Development, with cooperative funding from 17
other agencies.
• Opinions reflect those of the authors and do not
necessarily reflect those of the granting agencies. The
authors are solely responsible for the contents.
Motivation
“The adolescent is choosing how to invest
time, and … the choices depend greatly
on the social system surrounding them.”
page 346 of Coleman’s “Vision for Sociology”.
Friendship Influence
B
C
A
D
Critique of Friendship Influence
• Friends accept adolescents for who they
are (Harter and Fischer 1999; Lightfoot 1997; Bearman and Bruckner 2001)
• Adolescents more likely to select friends
with similar interests than to be influenced
by existing friends (Dornbusch 1989; Matsueda and Anderson 1998)
But Peers can be Influential
• Friends provide
– Emotional support
– Information
– Instrumental resource (e.g., homework)
• As rational actors, adolescents may seek
friends or draw on them for resources
→Adolescents can be influenced by those
with whom they would like to be friends
(potential friends) (Giordano 2003)
Who are Potential Friends?
• Potential friends are theoretically
important, but how to find them
– Similar interests and characteristics (Blau)
– Opportunities to observe others’ behaviors
and interactions (Leiffer)
– Opportunities to initiate interaction (Leiffer)
School as Social Organization
Defines Structure for Forming
Friendships
• School is dominant social institution for
adolescents (Coleman 1961)
• School structures opportunities for
interaction (Moody 2001)
• School cultivates certain interactions
(Quiroz et al 1997)
Conceptualizations of Social
Organization of Schools
• Academic Tracks
– Defined by course levels
• Honors/college prep, general, vocational
– Implications for exposure to academic
material and opportunities for advancement
• Gamoran; Hallinan; Oakes
• Salience of academic tracks?
– Age of detracking (Hallinan)
– May never have been tracks, especially
outside of core areas (Lucas)
Return of the Social Category
• Jocks, nerds, burnouts, leading crowd
• Observed by Coleman in Adolescent
Society (1961)
• Still relevant
– Ethnographies (Eckert 1989; Leslie, McClure, and Oaxaca 1998;
Foley 1990; Milner 2004)
– Psychologists (Barber, Eccles, and Stone 2001)
– Economists (Akerlof and Kranton 2002)
Influence of Members of a Social
Category
• Adolescents conform to those in the social
category as part of identity (Akerlof and
Kranton [AK] 2002, Coleman 1961)
• Identity: perceived relationship with group
• Identity: “both a student’s assigned
category and the payoffs associated with
self-image” (AK: 2002: 1172).
How is Identity Influential?
Effects of Identity in AK Utility
function
1 2
1


2
U i ( N )  p  w  ni ei  ei   (1  p )  I N  t (1  ni )  (ei  e( N )) 
2 
2



Pursuit of
Human Capital
Status of
Category
Fit in
Category
Effort to
Conform to
Category
Human capital (p):
wage rate per unit of skill (w)
the ability of person i (ni )
and the effort of person i (ei).
Identity (1-p):
social status associated with the nerd category (IN),
identity loss from an adolescent’s distance from the ideal
ascriptive characteristics in the nerd category (t),
effort (ei) adolescent exerts to fit into her social category,
where e(N) represents the ideal effort level (on academics)
for an adolescent who is a nerd.
Interpreting the Utility Function
• Niche picking: balance between status of social
category against fit and effort needed to conform
to norms
• psychological link between micro (through
individual perception) and macro level (social
category) phenomenon within social
organizations
• schools can maximize academic effort given
student composition:
– different standards for different categories
– Maintains engagement for those with lower ability
Limitations of Conceptualization of Social
Categories
• Categories (and tracks) are a priori and fixed
– Common across schools
– Decided by researchers or administrators
– Unchanged by adolescent behavior and over time
• Difficult to measure
– “How can we know that being in one social category
or another reflects anything more than individual
tastes and endowments? For example, there may be
no such thing as a group of nerds: those who are
called nerds may just be those who are smarter and
more academically inclined. The empirical task is to
establish that membership in a social category,
independent of tastes, affects behavior” (AK: page
1176).
Limitations of Conceptualization of Influence
in AK Function
• Friends valuable because
– Emotional support
– Information
– Instrumental resource (e.g., homework)
• But AK focus on psychological costs of misfit
• Recall: Friends likely have:
– Similar interests
– Observable interactions
– Opportunities for interaction
• But social categories not rooted in social
experience (based on perceptions)
Alternative Conceptualization:
Local Positions
• Clusters of students who take courses
together
• Preserves duality of data (students and
courses)
• See Field, Frank et al. (Social Networks,
2006)
Conceptualization of Clustering in 2-Mode data
From Skvoretz and Faust’s adaptation of p* models
Actor 2-stars
Event 2-stars
Identifying Local Positions
• Apply cluster analysis to the set of
transcripts in each school
– Transcripts represent people and experiences
• Define model relating local position
membership to course-taking
• Maximize criterion from model
Model of Effect of Position
Membership on Coursetaking
Define θ1 from the following model for
xij , an indicator of whether student i took course j :
 p[ xij  1] 
log 
  0  1 same positionij

 1  p[ xij  1] 


where same lpositionij takes a value of 1 if
student i and course j are in the same local position, 0 otherwise.
When θ1 is large, participation is concentrated within positions.
θ1 =log of odds ratio in following table
Identify Emergent Positions:
Criterion to Maximize
Position
Membership
Social Structure of Position: Concentration of participation in events
Maximize (AD/BC) =
Participation within position x Non participation in other positions
Participation in other position x Non-participation in position
Table 1
Courses in Local Positions in Cyprus High
Local
Position
Other (non-focal) courses taken
by at least 20% of students a
Average course
overlap between
pair of students
(weighted course
overlap)
Focal Courses
1
English 1, General Biology, Algebra 1,
World History, Work Experience, Dropout
Prevention, Keyboarding
Sports
6.01 (1.77)
2
Agricultural (introduction), Developmental
English, Carpentry, Urban Problems
Sports, health, Biology, Algebra,
World History
3.12 (.79)
3
Introduction to Technology, Agricultural
(intermediate), English 1 (honors) , Biology
(honors)
Chemistry, US government
3.22 (1.47)
Chemistry, sports, US history,
Algebra 2
2.60 (.63)
4
French 1, Agricultural(Applied)
5
English 3, Chemistry (introductory) ,
Spanish 1, Mathematics Review, Test
Taking
Sports, Band, Us History,
Algebra 2
4.69 (.92)
6
Algebra 3, Business Computer
Programming, Music History (10)
Chemistry
2.77 (2.38)
7
Auto Mechanics (1) , Physical Education
(10)
Chemistry, Geometry
3.38 (1.29)
Art I, Biology 2, Spanish 2,
2.54 (.89)
8
Table 2
Estimated Student Composition of Local Positions at Cyprus High for the 94-95 School Yeara
Local
Number
Estimated Average %
Position of
Populatio Grade
Female
b
Students n (94-95) Level
in (94-95)
Mean
Parental
Education
(1=less than
grade 8 to
7=college)
Mean
Hours
Worked
(Nonsummer
Week)
Days
per
week
Played
Sports
Average
AHPVT
Score
1
3
32
9
1
5.15
0
4.57
84.84
2
7
71
9
.16
3.94
.11
5.75
83.13
3
5
91
9/10
1
4.81
0
4.35
86.83
4
10
78
11/12
.58
4.64
1.16
5.63
87.83
5
5
51
11/12
.05
2.55
3.17
4.50
95.89
6
3
25
11/12
.46
4.59
1.58
6
98.19
7
4
45
11/12
0
3.99
1.63
6
109.22
8
8
66
11/12
.80
4.97
2.18
2.82
91.20
Two-Mode Sociogram
of Students and Courses at
Cyprus High
(Hypothetical Data)
A
B
G
C
H
D
F
E
Advantages of Local Positions
• Unique in each school
– Algorithm applied uniformly to transcript data in each
Add Health school, clusters differ by school
• Focused around experiences – courses
• Emergent from student choices and school
constraints
• Original data come from transcripts measuring
social experience independent of behavior or
attitude in survey
Research Question
• How do members of local positions influence
engagement in academics?
– AK: adolescents conform to mean behavior
– Frank et al.: adolescents adopt popular
behaviors to gain friendships
Influence within Local Positions:
Norms Via Rewards and Sanctions
?
Norms Via Rewards and Sanctions
?
Norms Via Rewards and Sanctions
?
Reminder: Original Utility function
1 2
1


2
U i ( N )  p  w  ni ei  ei   (1  p )  I N  t (1  ni )  (ei  e( N )) 
2 
2



Pursuit of
Human Capital
Status of
Category
Fit in
Category
Effort to
conform
Human capital (p):
wage rate per unit of skill (w)
the ability of person i (ni )
and the effort of person i (ei).
Identity (1-p):
social status associated with the nerd category (IN),
identity loss from an adolescent’s distance from the ideal
ascriptive characteristics in the nerd category (t),
effort (ei) adolescent exerts to fit into her social category,
where e(N) represents the ideal effort level (on academics)
for an adolescent who is a nerd.
Alternative Utility Function
1


U i (C )    w  ni (ei )  (ei  hi ) 2   
2


Pursuit
of
Human
Capital
Value of
Help
 1
1
2
*
2

(
e

e
(
C
))


(
e

e
(
C
))
 2 i

 2 i

Conformity to
Mean
hi
Help towards academic endeavors
e (C )
Mean level of effort in local position
e* (C )
Note π+ρ+κ=1.
Conformity to mean
(weighted by popularity)
Mean level of effort in local position
(weighted by popularity in local position)
Maximizing Utility with Respect to
Effort (first order equations)a
ei   w  ni   hi   e(C )   e (C )
*
=
Advance
in math?
=
Effort
Value of
human
capital
Educational
Expectations
aAssuming
π+ρ+κ=1.
Access to help
Number of
Friends
Mean of others
Mean
math
level of
others
in social
contexts
Mean
(weighted by
popularity) of
others
Mean math
level
(weighted by
popularity) of
others
in social
contexts
Data
•
•
•
•
Add Health data, nationally representative
Longitudinal: control for prior behavior
Dependent Variable: advanced in math (or not) from 94-95 to 95-96
Network:
– Exposure through course overlap (also have exposure through friends,
but sample size reduced)
• Three Levels:
– students (n=4,000)
– within local positions (n=770)
– within schools (n=78)
• All analysis weighted at level 1 for individual weight
• Unit-specific effects reported with robust standard errors used to
compensate for possible misspecification of distribution of error
terms (conservative).
Controls in Models
•
•
•
•
•
•
•
•
Ever been retained
Level of math in 1994-95
Math grades for 1994-95
Grade level in Fall 1994
Educational expectations
Latino/a, Black
Parental education
Indicator of whether data were available to
construct the measure of friends’ mean math
level
Interpretation
• Boys and girls more likely to advance if they
–
–
–
–
–
Have high expectations
Parents disappointed if they don’t go
Never retained
Had high grades
Were at relatively low math levels
• Effects associated with local positions
– Girls more likely to advance if they occupy local positions in
which other girls taking high levels of math
• Interpretation: girls advance to make friends – pursuit of social
capital.
• Effect size: whole extra class of 21 girls advancing out of 9th grade
containing 350 girls.
• Applies especially to girls at relatively low levels of math
Predicted increased stratification between members of low
and high local positions
on account of conformity to local position means (for girls).
Gender Interpretation
• Boys may identify more and perceive more
competence with math than girls because
of societal expectations (Eccles)
 boys use social resources to do well in math,
girls use math to gain social resources.
• Girls may be more savvy about social
contexts
Realized Social Consequences:
Pursuit of Friendships
• Do adolescents make friends if they
advance and are in local positions in which
math is associated with popularity?
• Model of change in popularity as a
function of advancing and mean level
math, weighted for popularity
• Answer: Girls: yes; Boys: no
Realized Social Consequences:
Value of Friendships
• Do adolescents who have friends perform
better in next course?
• Model of Math GPA in 95-96 (controlling
for Math GPA in 94-95)
• Answer: Girls: no; Boys: yes
Conceptual Summary of Influence
• Social consequences
– Boy may not be able to gain popularity by advancing in math
– Girl may not be able to draw on friends to do well in math
• Not just differences in adolescents’ perceptions of math
(Eccles)
• Agents of influence are
– Potential nominators (girls)
– Existing friends (boys)
• Schools can elicit effort indirectly
– Determining type and composition of local positions
– Symbolic action to elevate popularity of desired behaviors
Conceptual Summary regarding
Local Positions
• Emergent level between individual and
school
– Individual Agency versus Organizational
Constraint
• More than just dyads
• Group mechanisms operate through local
positions – adolescents pursue social
capital
Limitations
• Know courses, not classes
• Don’t know extracurricular well
• Only a sample in most schools